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Strongly correlated ultracold plasma Sadeghi Esfahani, Hossein 2016

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Strongly correlated ultracold plasmabyHossein Sadeghi EsfahaniPhysical chemistry, Sharif University of Technology, 2009Physics, Sharif University of Technology, 2008Chemistry, Sharif University of Technology, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Chemistry)The University of British Columbia(Vancouver)October 2016c© Hossein Sadeghi Esfahani, 2016AbstractUltracold molecular plasmas represent a new frontier of plasma physics. Theyoffer an easy and accessible laboratory for the study of strongly coupled Coulombsystems. Strong Coulomb coupling can give rise to exotic materials such as Wignercrystals and liquid like plasmas.In this thesis, I present a set of experiments and theoretical models that explorethe properties of ultracold plasma in great detail.A supersonic beam of nitric oxide in helium creates a cold ensemble of groundstate molecules. Upon two-color excitation, a Rydberg gas of nitric oxide evolveson a time scale of nanoseconds to form an ultracold plasma. The excited volume isimaged using a multichannel plate detector mounted on a movable grid. By movingthe detector back and forth, we can observe the expansion dynamics of the plasmaand its decay.Selective field ionization captures the relaxation of the Rydberg states to aplasma. We use a very reliable coupled-rate-equation model to understand thedecay dynamics and evolution of Rydberg gas to a plasma by accounting for all themajor processes that happen during the avalanche process.We find that a plasma evolved from an ultracold Rydberg gas expands veryslowly, exhibits long relaxation time, and shows evidence suggesting the develop-ment of spatial order.iiPrefaceThis dissertation is ultimately based on the experimental apparatus and data ofthe ultracold molecular plasma experiment on the movable grid machine in Prof.Grant’s research laboratory at University of British Columbia. Parts of experi-ments done for this thesis was carried out by a team of undergraduate and graduatestudents, however the majority of experiments, data analysis, interpretation andwritings were done by the author of this thesis. Chapter 2-4 were published previ-ously, and they are attached as the aforementioned chapters with little revision andexpansion.• H. Sadeghi et. al., Dissociation and the Development of Spatial Correlationin a Molecular Ultracold Plasma, Phys. Rev. Lett. 112, 075001 (2014)Conducted experiments and collected data published within this article.• H. Sadeghi and E. R. Grant, Dissociative recombination slows the expansionof a molecular ultracold plasma, Phys. Rev. A 86, 052701 (2012)Conducted experiments and collected data published within this article.• H. Sadeghi et. al., Molecular ion-electron recombination in an expandingultracold neutral plasma of NO+, Physical Chemistry Chemical Physics,18872 (2011)Conducted experiments and collected data published within this article.The hardware design was done primarily by Chris Rennick, except for the ad-dition of pulsed field ionization system. LabView programs that automates dataiiiacquiring process was originally developed by Chris Rennick, but was heavily cus-tomized to afford new demands.The theoretical work on coupled-rate-equation model was previously carriedout by J. Morrison and N. Saquet, but was also independently developed by my-self. The rate constants for the model were taken from the work of T. Pohl, D.Vrinceanu, and H. R. Sadeghpour.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Molecular Ion-Electron Recombination in an Expanding UltracoldNeutral Plasma of NO . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Evolution to plasma and the development of space charge 142.3.2 Expansion and electron temperature . . . . . . . . . . . . 152.3.3 Plasma decay . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22v3 Dissociative Recombination Slows the Expansion of a Molecular Ul-tracold Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 Hydrodynamic model for a plasma undergoing dissociative recom-bination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 313.2.1 Time evolution of charged-particle density . . . . . . . . 313.2.2 Time evolution of ion velocity . . . . . . . . . . . . . . . 323.2.3 Time evolution of electron temperature . . . . . . . . . . 343.2.4 Collisional processes and the evolution of quasi equilibrium 363.2.5 Relation to experimental measurements . . . . . . . . . . 373.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 Penning Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Experiment and method . . . . . . . . . . . . . . . . . . . . . . . 424.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Selective Field Ionization Study of Rydberg Gas and Plasma . . . . 525.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1.1 Selective field ionization (SFI) spectroscopy . . . . . . . . 535.1.2 Evolution of Rydberg gas to plasma . . . . . . . . . . . . 545.1.3 Factors contributing to the density of NO Rydberg gas . . 555.2 Field ionization of Rydberg states . . . . . . . . . . . . . . . . . 565.3 Experiment and method . . . . . . . . . . . . . . . . . . . . . . . 575.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.4.1 Rydberg spectra with selective field ionization . . . . . . 595.4.2 Density variation . . . . . . . . . . . . . . . . . . . . . . 635.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.5.1 Dependence of Rydberg gas evolution on principal quan-tum number . . . . . . . . . . . . . . . . . . . . . . . . . 685.5.2 Density dependence of electron avalanche . . . . . . . . . 705.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75viBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79A Coupled-Rate Equation Model . . . . . . . . . . . . . . . . . . . . . 85A.1 Shell model for coupled-rate equation . . . . . . . . . . . . . . . 88A.2 Non-uniform model results . . . . . . . . . . . . . . . . . . . . . 90A.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 97A.3 Shell model codes for rate equation model with expansion for non-uniform density . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A.3.1 Non-uniform coupled rate equation: setting up the param-eters and initial value problem . . . . . . . . . . . . . . . 99A.3.2 Non-uniform coupled rate equation: numerical integration 101A.3.3 Non-uniform coupled rate equation: testing energy andparticle conservation . . . . . . . . . . . . . . . . . . . . 105A.4 Coupled rate equations with GPU . . . . . . . . . . . . . . . . . 108A.4.1 Rate constants . . . . . . . . . . . . . . . . . . . . . . . 109A.4.2 Coupled rate equation: setting up the parameters and initialvalue problem . . . . . . . . . . . . . . . . . . . . . . . . 112A.4.3 Coupled rate equation: numerical integration . . . . . . . 117A.4.4 OpenCL kernels for parallel computing . . . . . . . . . . 127B Spectroscopy of Nitric Oxide . . . . . . . . . . . . . . . . . . . . . . 131B.0.1 Rotational levels and Rydberg series of nitric oxide . . . . 135B.0.2 Observation of a new series . . . . . . . . . . . . . . . . 137C Detailed Analysis of Field Ionization Data . . . . . . . . . . . . . . . 141D Molecular Dynamic Simulation: Parallel Implementation . . . . . . 147D.1 OpenCL kernels for molecular dynamics simulation . . . . . . . . 147D.2 Python routines for setting up the initial value problem and numer-ical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 151E Matlab Routines for Selective Field Ionization Calculations . . . . . 163E.1 List of constants and conversion functions . . . . . . . . . . . . . 163E.2 Binary data and text file handling . . . . . . . . . . . . . . . . . . 168viiList of TablesTable 2.1 Second order rate coefficients obtained by fitting the plasmasignal decay for a set of initial charged particle densities sys-tematically varied by selecting the ω1−ω2 delay. kev describesthe slow first order decrease in plasma signal at long times. . . 20viiiList of FiguresFigure 2.1 Experimental apparatus. A skimmed molecular beam passesthrough grid, G1, where it intersects counter-propagating laserbeams indicated as ω1 and ω2. As the excitation volume tran-sits the plane defined by G2, an Multi-Channel Plate (MCP) de-tector situated behind G3 collects the signal of extracted plasmaelectrons. The components surrounded by the light-grey bor-der translate together on a moving carriage. . . . . . . . . . . 10Figure 2.2 Waveforms captured by the MCP detector for different posi-tions of the moving carriage. The upper collection of signalwaveforms is obtained for a fixed ω1-ω2 delay of 50 ns. Thelower waveforms show the decay and expansion of the plasmasignal at an initial density reduced by increasing the ω1-ω2 de-lay to 100 ns. Traces have been offset vertically for clarity. . . 13Figure 2.3 Fitted plasma widths as a function of centre arrival time at G2for ω1-ω2 delay times of 0, 50 and 100 ns (top to bottom).Solid lines represent fits of Eq. 2.2 to the experimental data,yielding a fitted initial temperatures Te(0)+Ti(0) = 12.1,14.3and 15.8 K, respectively. . . . . . . . . . . . . . . . . . . . . 23ixFigure 2.4 Integrated area of the plasma signal as a function of arrival timeat the imaging grid G2 for ω1-ω2 delay times of 0, 50, 100 and200 ns (top to bottom).. The lines are integrated rate equationsfitted to the decay data. The calculations consider second-orderdecay processes in an expanding plasma with first-order elec-tron evaporation (c.f. Eqs. 2.8 and 2.9) . . . . . . . . . . . . . 24Figure 3.1 Density of electrons in the presence (black curves) and absence(red curves) of DR for expanding spherical shells. Densitiesare plotted at times (from top to bottom), t = 0,0.4 and 0.8 µs(left) and with expanded scale at later times t = 10 and 13 µs(right). Results obtained numerically for the non-dissociatingcase agree precisely with the analytical solution of the Vlasovequations for conditions given in the text. . . . . . . . . . . . 32Figure 3.2 Radial velocities of ions as a function of time in shells with ini-tial radii at: 1σ (top-left), 2σ (top-right), 3σ (bottom-left), and4σ (bottom-right) positions of the t = 0 Gaussian. Conditionsas in Fig. 3.1. Red curves describe the self-similar expansionof a non-dissociating plasma. Black curves show the effect ofdissociative recombination. . . . . . . . . . . . . . . . . . . . 33Figure 3.3 Plot points (black) marking γi as a function of time for shellswith initial radii, reading from bottom to top, of 1σ , 2σ , 3σ ),and 4σ positions of the t = 0 Gaussian. Solid red line givesγ(t) for all radii of the non-dissociating self-similar plasma . . 34Figure 3.4 Acceleration of ions scaled by the linear acceleration in a col-lisionless plasma as a function of scaled distance at differenttimes. Black curves show the scaled acceleration in a dissoci-ating system and red dotted line represents the constant scaledacceleration in the collisionless system. . . . . . . . . . . . . 35xFigure 3.5 Schematic cross section of plasma shells illustrating the vol-umes sampled by projection onto an x,y grid by plasma propa-gation in the z direction.(top waveform) Distribution of densityas a function of shell radius along z. (bottom waveform) Dis-tribution of number per shell as a function of shell radius alongz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.6 Widths of electron waveforms sampled by projection of themodel plasmas of Fig. 3.1, expanding hydrodynamically with(black) and without (red) dissociative recombination. The solidred dot-curve plots the measured width of the density distribu-tion σ(t) calculated analytically for the Vlasov expansion ofa non-dissociating Gaussian plasma. Red dots give the samequantity obtained numerically allowing a shell model plasmawithout dissociation to expand from an initial Gaussian densitydistribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 4.1 Terminal radial velocities of ions measured in a long-time limitof ion expansion for ultracold plasmas created from molecularRydberg gases of NO with initial principal quantum number,n0. Error bars determined from the residuals of Gaussian fitsto experimental waveforms produced by plasma volumes tran-siting G2. Grey points represent radial expansion velocitiespredicted for Te = 9 K by a hydrodynamical model accountingfor spatial correlation calculated stochastically (see text). . . . 44Figure 4.2 Distributions of ion-ion nearest neighbours following Penningionization and electron-impact avalanche in a predissociatingmolecular Rydberg gas of initial principal quantum number,n0, from 30 to 80. Dashed lines mark corresponding values ofaws. Calculated by counting ion distances after relaxation toplasma in 106-particle stochastic simulations. Integrated areasproportional to populations surviving neutral dissociation. . . 48xiFigure 4.3 Reduction of average ion-ion repulsion energy in Penning lat-tices formed by Rydberg gases of initial density, 1012 cm−3and principal quantum number from 30 to 80, plotted as a frac-tion of the pairwise repulsion energy difference between a gasof random ions and a perfect lattice. Data points at each valueof n0 taken from the results of four simulations. . . . . . . . . 50Figure 5.1 Experimental apparatus. A skimmed molecular beam passesthrough grid, G1, where it intersects counter-propagating laserbeams indicated as ω1 and ω2. Before the excitation volumetransits the plane defined by G2, a ramped electric field is ap-plied to G1 with a forward bias. An MCP detector situatedbehind G3 collects the signal of extracted plasma and Rydbergelectrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 5.2 Spectrum of NO Rydberg states obtained by applying a time-varying electric field right after two-photon excitation. Eachhorizontal line is a field ionization trace and contains two ormore peaks appearing at different fields. The features appearat higher field when the wavelength of the second photon isincreased. The series of peaks in the upper left corner belongsto an excited rotational level, K = 4, of 2Σ+1/2. . . . . . . . . . 60Figure 5.3 Spectrum of NO Rydberg states obtained by applying a time-varying electric field 200 ns after two-photon excitation. Eachhorizontal line is a field ionization trace and contains two ormore peaks appearing at different fields. The features appearat higher field when the wavelength of the second photon isincrease. The series of peaks in the upper left corner are theresult of overlap between two transition lines of ground stateNO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 5.4 Field ionization traces at a fixed density at time delays t = 0and 200 ns for n=59. The blue solid trace is taken at t=0 andthe dashed orange traces is taken at 200 ns. . . . . . . . . . . 62xiiFigure 5.5 Field ionization traces at a fixed density at time delays t = 0and 200 ns for n=43. The blue solid trace is taken at t=0 andthe dashed orange traces is taken at 200 ns. . . . . . . . . . . 63Figure 5.6 Field ionization traces at various densities determined by thetime delay between the two exciting laser beams. The y axisis the strength of the signal. The base line of each signal isoffset by the logarithm of the density relative to the maximum(zero delay between the two lasers). From top to bottom, thedensity becomes smaller. The traces have low field signal athigher densities. At lower densities, each curve has two peaksat high fields. . . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 5.7 Field ionization traces at various densities determined by thetime delay between the two exciting laser beams for a Rydberggas with the principal quantum number n = 49. The y axisis the amplitude of the signal in an arbitrary unit. The solidline is taken at the maximum density (zero delay between thetwo lasers). The signal is strongest and it extends to higherfields. The dashed line has a density 30% of the maximum.This trace has features of the strong signal and two peaks at90 and 140 V/cm. The dot-dashed curve has a density 5%of the maximum, and only shows small low field and slightlystronger signal at high field. . . . . . . . . . . . . . . . . . . 65Figure 5.8 Number of electrons (top graph) and surviving Rydberg states(bottom graph) after 20 ns as a function of density determinedby the delay of the two laser pulses. Black dots are experimen-tal points obtained by integrating the area under the curves infigure 5.6. In the top plot, the black dots are the integral ofthe curves from zero field to 75 V/cm, and in the bottom plot,from 75 V/cm to 250 V/cm. The green dashed line representthe coupled rate equation model for uniform distribution Ryd-berg gases. Solid red line is calculated by a similar model witha Gaussian density distribution. . . . . . . . . . . . . . . . . 67xiiiFigure A.1 Density of Rydberg gas versus Penning electron. The blackcurve is a Gaussian profile with a standard width of 800 µm.The profile is scaled by its values at the centre. The dashed redline, is the electron density after Penning ionization which isequal to density of the Rydberg gas times Penning fraction. . . 91Figure A.2 Number of electrons in each disk (total of 15 disks) as a func-tion time. Initially each disk is populated with a Rydberg gaswith a density determined by the between distance between itscentre and cylinder’s. The shell with the highest number ofelectron (top curve) initiates a fast avalanche. Shell number 7and above have a very small number of electrons. . . . . . . . 93Figure A.3 The distance of each disk from the middle of the cylinder. Thecylinder is divided into 15 disk each place at uniform interval.The rate of the expansion is faster as the disk is place at longerdistances, however, the rate of avalanche act in the oppositeway. Since the outermost disk has not evolved to a plasma, itdissipates the force generated by electron pressure to a pointthat it acts as a solid wall in front of other expanding shells. . 94Figure A.4 Similar to Fig. A.2, but instead of the distance of each disk,the thickness of it is plotted. The thickness is defined as thedistance between the centre of two adjacent disks. Some disksexpand as predicted by ambipolar expansion. However, somedisk are squeezed due to fast expansion of the inner ones. . . . 95Figure A.5 The velocity of each disk as a function time. Disks with thehigher Rydberg gas density, avalanche to a plasma quickly, andaccelerate proportional to their distance after that. Some diskswith very low electron density, are stationary at first, but aftera late avalanche (e.g. 3 µs), they are pushed forward. . . . . . 96Figure A.6 The velocity of each disk as a function of their index as a func-tion of time. The velocity grow in time and with index number,but drops very fast in regions where no plasma is formed. . . . 96xivFigure A.7 Plot of nearest neighbour distribution for two densities 1010 cm−3and 1012 cm−3. At the lower density the distances are larger.Two lines indicate the critical distance for Rydberg states withn= 50 and n= 70. The critical distance (rc) scales with squareof quantum number. For a higher n, a larger fraction of pairsare within rc, the same applies to higher density. . . . . . . . 98Figure A.8 Plot of electron density over initial Rydberg density as a func-tion of scaled time. Red traces are computed for n=30, bluefor n=50, and black for n=70. For each n, computation for fourdifferent densities is performed. . . . . . . . . . . . . . . . . 99Figure B.1 Simulated spectrum of one color REMPI on top and exper-imental observation at the bottom. The simulated spectrumwith a temperature of 2.0 K matches the experiment. . . . . . 133Figure B.2 Simulated and real spectrum of nitric oxide Rydberg excitationvia 2Σ+1/2 in presence of an external electric field. Details of theexperiment are discussed in chapter ??. . . . . . . . . . . . . 134Figure B.3 Intensity map of laser scan with field ionization, first UV laseris tunned to lower rotational level of the excited electronic Astate, however, there is an overlap with a transition J−4.5→K = 4. Red lines indicated Rydberg series converging to N+ =0,1,2 resulted from the original rotational level k = 0. . . . . 136Figure B.4 Intensity map of laser scan with field ionization, first UV laseris tunned to lower rotational level of the excited electronic Astate, however, there is an overlap with a transition J−4.5→K = 4. Green lines indicated Rydberg series converging toN+ = 1,2,3,4,5 resulted from k = 4 level. . . . . . . . . . . 137Figure B.5 Signal integrated over two regions indicated by red lines inB.3. The bottom trace is the integral of peaks between the twobottom lines and the top trace is the integral of peaks above thetop red line. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138xvFigure B.6 Signal integrated over peaks at the bottom triangle of figureB.3. The peaks correspond to a series of Rydberg states con-verging to N+ = 2 via K=4 A state. . . . . . . . . . . . . . . 138Figure B.7 Intensity map of laser scan with field ionization, first UV laseris tuned to an isolated peak that selects a K = 4 of the excitedelectronic A state. . . . . . . . . . . . . . . . . . . . . . . . . 140Figure C.1 DC electric field ionization traces at various time delay at fixeddensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Figure C.2 Pulsed field ionization early peak in time domain. Rydbergstate n= 49, ω1 power = 2µJ, ω2 power = 6mJ. Only the earlypart of the Pulsed-Field Ionization (PFI) is plotted for variousdelays. The y axis for each traces is offset by the pulse delay. 143Figure C.3 A typical scope trace shows an early prompt peak and a latepeak. The first one corresponds to arrival of hot electronsboiled off of the plasma volume and the late peak representsthe bulk volume of the plasma that travels at the speed of molec-ular beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Figure C.4 Time corrected. Pulsed field ionization early peak in time do-main. Rydberg state n = 49, ω1 power = 2µJ, ω2 power =6mJ. Only the early part of the PFI is plotted for various de-lays. The y axis for each trace is offset by the pulse delay. . . 145Figure C.5 The position of the detector as a function of arrival time of theprompt electron signal. . . . . . . . . . . . . . . . . . . . . . 146xviGlossaryUCP Ultracold plasmaMOT Magneto-Optical TrapMCP Multi-Channel PlateSFI Selective Field IonizationPFI Pulsed-Field IonizationPD Pre-DissociationDR Dissociative RecombinationxviiAcknowledgmentsI would like to express my gratitude to my supervisor, Professor Edward Grant,whose expertise, understanding, and patience, added considerably to my graduateexperience. I appreciate his vast knowledge and skill in many areas, and his as-sistance in writing reports (i.e., grant proposals, scholarship applications and thisthesis). I would like to thank the other members of my committee, Roman Krems,Valery Milner, Fei Zhou, and Andrew MacFarlane for the assistance they providedand their helpful discussions. I would like to especially thank Professor ThomasKillian for his critical review of this thesis.I would also like to thank my family for the support they provided me throughmy entire life and in particular, I must acknowledge my wife and best friend,Akram, without whose love, encouragement, I would not have finished this the-sis.Lastly, I recognize that this research would not have been possible without thefinancial assistance of NSERC, AFOSR, the University of British Columbia Grad-uate Studies, the Department of Chemistry at the University of British Columbia(Teaching Assistantships, Graduate Research Scholarships), and I express my grat-itude to those agencies.xviiiChapter 1IntroductionPlasma is commonly considered a fourth state of matter. When a gas is heated,the electrons move to higher energy levels and eventually ionize. The new prop-erties of cold plasmas absolutely contradict the common understanding of plasmaphysics. Plasmas can be formed from any known state of matter over a great rangeof temperatures under modern laboratory conditions. We focus here specifically onthe cold plasmas in which the heavier charge carriers, ions, are cooled to 1 K orlower and electrons are created with a controlled excess energy to keep the temper-ature as low as few degrees Kelvin. These plasmas can serve as simulators to helpus better understand the interior of planets and white dwarf stars. New states ofmatter including crytalline and liquidlike plasmas can be created and studied withaccess to ultracold plasmas.In an ideal plasma, the ensemble of charge particles moves freely. Most of thematter in the universe exists in the form of a plasma. Plasmas span a wide rangeof properties, from a density of few thousand atoms per cubic centimetre with atemperature of a few tens of kelvin in the earth’s ionosphere, to a density milliontimes higher than everyday material with a temperature as hot as the core of thesun. Plasmas have been studied for many years. Nowadays, scientists have themeans to create not only plasmas that exist in nature, but also ones that extend thelimits of achievable temperature and densities.Ultracold neutral plasmas, introduced a decade ago [29], exceed the limits ofconventional plasma parameters by pushing temperatures below one degree Kelvin.1These systems provide an excellent testing environment for plasma theories. Ul-tracold plasmas were first created by photo ionization of xenon atoms, laser cooledin a Magneto-Optical Trap (MOT) down to few micro Kelvin. Since it was possibleto give electrons a precise amount of initial energy, it was possible to control theexcess energy of the plasma and to create very cold plasmas. This enabled the cre-ation of a system in which the Coulomb force between charged particles exceededthe average kinetic energy. Such a condition defines the state of a strongly coupledplasma.The two fundamental properties, density and temperature, give rise to the de-gree of coupling in a plasma. Density determines the average spacing of particlesand therefore the attractive and repulsive coulomb forces; and temperature deter-mines the average kinetic energy. When the strength of interaction energy betweenthe particles exceeds that of the thermal motion, plasma becomes strongly coupledand exhibits complex and liquid like behaviour.Theoretical understanding of such systems is virtually impossible, requiringcomputationally expensive simulations. However, even the limited knowledge ofstrongly coupled systems has revealed very important findings such as modifiedmass transport– Mott effect in electron-hole plasmas [50], and non-Markovian be-havior in electron-ion energy transfer [5].In a strongly correlated systems of electrons, such as certain metals and insu-lators, interaction between conductions electrons becomes very important in deter-mining the properties of the material. Two-dimensional electrons systems exhibitinteraction effects such as fractional quantum Hall effect, metal insulator transition(MIT) [49], and enhanced effective mass [13].Strongly correlated plasmas are of great interest for scientific reasons such asastrophysical modelling and technical applications such as confined fusion. Thelaboratory scale system of an ultracold plasma provides a simulator for naturalexamples of strongly coupled plasmas in the universe. Strong coupling exists indense astrophysical systems, matter irradiated with intense laser fields, colloidal ordusty plasmas of highly charged macroscopic particles, and non-neutral trapped ionplasmas that are laser cooled until they freeze into Wigner crystals. Strong couplingcauses spatial correlation between particles which in turn affects equilibrium andcollective behaviour [31].2However, experiments showed that the initially low kinetic energy of chargedparticles in a strongly coupled ultra-cold plasma grows significantly to the pointwhere it exceeds Coulomb interaction energy, putting the plasma back in the non-coupled regime [31, 36].Our work has introduced the ultracold molecular plasmas. A supersonic beamof nitric oxide molecule seeded in Helium with the temperature of 0.5 K is inter-cepted with two lasers to create highly excited states (Rydberg states). These ex-cited state molecules interact on a short time scale and ionize, seeding an avalancheof electron-impact processes that ultimately ionize 80% of the Rydberg moleculesinitially present. The result is an ultra-cold plasma.This system, in particular, connects two of the frontiers in atomic-molecular-optical physics: the physics of Rydberg states, and the physics of ultracold plasmas.A great deal of evidence supports the idea that this plasma attains a state ofstrong coupling for reasons that are explained in chapter 4. These plasmas ex-hibit very slow expansion dynamics [39], evidence of spatial correlation as a resultof dissociative-recombination [60], and arrested relaxation to neutral atoms [66].As a consequence, ultracold molecular plasmas constitute interesting systems forstudying strong coupling and observing non-ideal behaviour.This thesis details some of the effects that hint at existence of strong coupling,such as slow expansion, arrested relaxation, and spatial ion correlation. Chap-ters to follow explore the mechanisms that enables plasma to sustain the stateof strong coupling, asking three simple questions: 1) Is the molecular ultracoldplasma formed by a Rydberg gas of NO plasma truly correlated? What facts con-firm or deny this? 2) If the plasma is strongly coupled, what mechanism(s) helpsit attain and preserve this property? Does strong coupling affect the evolution dy-namics of a dense, ultracold Rydberg gas?Few evidence, such as slow expansion and Penning lattice effect (chapter 4),suggest strong correlation. However, the former, is only a hint at strong coupling ifthe density of the plasma is as high as 1012 cm−3, otherwise simpler interpretationscan explain a slow expansion. As a result, determination of the plasma or Rydberggas density becomes very crucial in deciding the properties of ultracold molecularplasmas.This research aims to find the physical bases for experimental observation pub-3lished before I joined the group and the many more observations by myself thatfurther confirm that this plasma system behaves in a way that is difficult to under-stand with current models.In the first phase of my project, I used a number of experimental methodsto characterize the expansion, equilibration, and the decay of molecular plasmasunder various conditions. The results showed a second order decay of electronsand ions after evolution of the Rydberg gas to a plasma. This plasma expandsvery slowly, suggesting a very cold electron temperature. I developed a compu-tational model to study the behaviour of the non-uniform plasma such that it cansimultaneously describe the expansion and the decay dynamics. Using hydrody-namical models I found that the electron temperature barely exceed 30 K in themost extreme case. This electron temperature at the density of 1012 cm−3 suggestscoupling strength of Γ= 1 which has not been reported for any other plasma [61].Later I revisited my computational model to re-formulate the Vlasov expansionmodel to include the effect of non-linear processes in a dense Ultracold plasma(UCP). The Vlasov theory ignores processes such as three body recombination anddissociative recombination. By including the non-linear processes and their effecton the shape of the plasma, which is a key factor in determining the dynamics ofthe expansion, I found that plasma no longer goes through a uniform expansion butrather a slower expansion happens for central regions and faster for the periphery.While this model was successful in explaining the slow expansion with relativelyhigher conceivable electron temperatures, it was not able to explain how the plasmacould have sustained such low electron temperatures [59].These results sparked the creation of a new series of experiments, computa-tions, and models in an attempt to explain the cold electron temperature and ledto the observation of many interesting new phenomena. In an article in PhysicalReview Letters, we reported evidence of spatial ordering that results from a se-lective dissociation mechanism that removes neighbouring ions. As a result, anultracold plasma forms in a state of partially ordered structure where no two pos-itively charged particles sit closely. The ordered structure has a lower potentialenergy due to the absence of close ion-ion pairs. In a system with high repulsivepotential energy, particles mobilize quickly and heat up while forming to a moreordered structure. Since the dissociative channel causes the plasma to initially form4with an ordered structure, it can avoid disorder induced heating [60].Recently we have developed an imaging apparatus that enables us to probe themolecular plasma in Cartesian dimensions orthogonal to the propagation of themolecular beam. Surprisingly, we find that under the conditions of high densityand Rydberg gas high principal quantum number, the plasma breaks symmetry bysplitting into two lobes in a direction perpendicular to plasma propagation throughour detection system. This fission can account for all the missing pieces of thepuzzle that we have been trying to put together to obtain a concrete picture of whatgoes on in an ultra-cold plasma. Coulomb explosion creates a hole in the centreof plasma. While the ions fly away from the centre they encounter the remainingRydberg molecules in the area that have not converted to a plasma. Through elec-tron exchange they drag the Rydberg molecules with them, therefore, causing afast expansion in a direction perpendicular to plasma propagation [66].While the finding reveals an important mechanism of electron cooling, detailsin the behaviour of the system raise more questions than they answer, simulatingconjectures that could open exciting new avenues of research. Of particular interestin this system is the resulting state of two lobes, which are exhausted of energy andvery cold. Such a system is not only strongly coupled but also shows signs of manybody effects. So far most plasma theories are focused on statistical models createdfor dilute and hot systems. At high density and cold temperatures sophisticatedmany body theories are required.Chapter 2 gives a detailed account of the experimental setup and how data onplasma expansion and it decay is collected and analysed. Then this chapter willfocus on what this information tells us about the plasma formation mechanism andhow it develops and decays. This set of experiments is crucial and is the foundationof our understanding of this system. Every other experiments mentioned in the nextchapters, start more or less the same way.Chapter 3 will focus on theoretical aspects of plasma expansion in presence ofdissociative recombination. This process, unique to molecular plasmas, plays animportant role in shaping the plasma and its dynamics. Here we explore the effectof plasma density profile and dissociative recombination on the expansion and wecompare the results to a self-similar expansion theorized by Vlasov model.Chapter 4, will explore another effect of molecular dissociation channel. The5process of dissociation at early plasma formation stage, arranges ions in a con-figuration that is closer to a thermal equilibrium, thus eliminates disorder inducedheating. This effect is observed by varying the principal quantum number of a Ry-dberg gas, under the same conditions of density and molecular beam temperature.We then observe a quantum number dependence of the plasma expansion rate. Wepropose a mechanism and a simple formalism that explains the observed trend andwe relate it to the pre-correlation of ions position as a result of Penning latticeformation.Chapter 5, reviews some of the most recent experiment that we have performedto study the evolution of a Rydberg gas to a plasma. We use a ramped electric fieldto interrogate the state of the Rydberg gas as it evolves to a plasma. We directlyobserve the evolution and we draw conclusion on the properties of the system,particularly, its density. We then propose detailed mechanism of plasma formationusing coupled-rate equation model. We have confirmed some of the observationsby M. W. Shulz [66] that the plasma is in a state of arrested relaxation, that someof the decay channels are stopped.6Chapter 2Molecular Ion-ElectronRecombination in an ExpandingUltracold Neutral Plasma of NO 1Ultracold plasmas offer laboratory access to an important regime of ionized gasesin which moderate densities combine with very low ion and electron temperaturesto increase the influence of short-range interactions. The introduction of charged-particle correlations causes a deviation from ideal plasma properties, giving rise,for example, to structured radial distribution functions, suppressed electron pres-sure and vanishing binding energies.[25, 32, 44, 47] Employing well-defined ini-tial conditions and precise means of laser excitation, research on ultracold plasmaspromises to deepen our understanding of correlated systems in general and providenew methods to accurately control their properties.[38]Experiments using magneto-optical traps have formed ultracold plasmas froma range of laser cooled alkali, alkaline earth and rare-gas metastable atoms.[15,18, 27, 29, 67] Here, laser light of a selected frequency prepares either a sampleof nearly stationary ions and monoenergetic electrons, or a Rydberg gas in a sin-gle selected energy level. In either case the system evolves to form a plasma ofelectrons, ions and Rydberg atoms in quasi-equilibrium.1H. Sadeghi et. al., Molecular ion-electron recombination in an expanding ultracold neutralplasma of NO+, Physical Chemistry Chemical Physics, 18872 (2011)7Charged particle densities approach 1010 cm−3 in a typical plasma volume ofabout 1 mm3. Initially, MOT ions have the micro-Kelvin energies of the cooledatoms, and threshold photoionization can prepare electrons with temperatures aslow as a few degrees Kelvin. As the plasma forms, however, disorder-inducedheating increases the ion temperature to the order of 1 K, and super-elastic colli-sions with relaxing Rydberg states excites the distribution of electrons to 30 K ormore.Recently, we have introduced a new method to prepare and study ultracoldplasmas.[39, 40, 52, 63] This approach achieves MOT-like final conditions by pho-toionizing target atoms or molecules cooled in a seeded supersonic expansion. Ourexperiments thus far have focused on nitric oxide. Supersonic expansion coolsNO, seeded at 10 percent in He, to a moving frame temperature of about 0.7 K, asconfirmed indirectly by a measurement of the rotational state population.Double-resonant excitation operates on a small volume of these molecules toproduce a Rydberg gas with a density as high as 4× 1012 molecules cm−3 in asingle selected state. This population evolves to emit a prompt signal arising fromthe release of free electrons, proceeding thereafter to form a cold, quasineutralplasma of electrons and NO+ ions.After plasma formation, we find that a pulsed electrostatic field as low as 3Vcm−1 – far less than the field-ionization threshold of the precursor state – yieldsa small signal of electrons, but pulses with amplitudes as high as 200 Vcm−1 failto diminish the plasma waveform. These observations confirm two properties thatpoint to the formation of a cold plasma. The response to pulses of low-voltagesignifies the presence of a surface charge of electrons that is trapped by the plasmaspace charge, but extractable by a weak field. The resilience of the plasma to pulsesof high voltage indicates a Debye screening length much smaller than the diameterof the plasma, shielding the charged particles at its core.Downstream, the plasma passes through a moveable grid. A signal of elec-trons extracted from the volume of the plasma, as it transits this plane, profiles thespatial distribution of charge. We find that the plasma expands at an acceleratingrate, in accord with the behaviour expected for an electron-driven ambipolar ex-pansion. Analysis of this expansion in the simple limit of the Vlasov equationsreturns apparent electron temperatures as low as 5 K. Thus, the plasma created in a8molecular beam seems both colder and denser than that formed in a MOT, puttingthis system further into the interesting regime in which the potential and kineticenergies of intermolecular interactions balance, and electrostatic correlations gainin importance.This molecular beam ultracold plasma differs from a MOT plasma in at leastone other important way. It is composed of molecular ions, and diatomic NO+ candissociate upon recombination with an electron. We observe this. On a tens-of-microsecond timescale, the charged particle density declines. This decay occursrapidly at first. It then gives way to a residual plasma signal that persists. Onthe same timescale, we observe the plasma volume to expand. This expansiondecreases the charged particle density, and, by converting random electron motionto the radial motion of ions, reduces the electron temperature.The dissociative recombination of NO+ is well known.[1, 42, 64, 68] Con-ventional bimolecular rate laws describe a decay of charged particle density thatvaries with ion and electron density, in proportion to a rate constant that increaseswith decreasing temperature. In addition, Rydberg molecules are maintained inquasi-equilibrium with ions and electrons by three-body recombination, and thesepredissociate with rates that increase characteristically with decreasing principalquantum number.The NO+ plasma thus presents a complex, expanding system of ionizing, re-combining, and reacting particles. The question arises, of whether we can assemblean understanding of the various elements of these dynamics to describe the over-all time evolution of a nitric oxide molecular ultracold plasma, and in particular,characterize the role of dissociative recombination in shaping its properties.2.1 ExperimentalTo measure the rate at which the charge density of the plasma decays, we detect theelectron signal produced on a multichannel plate detector when the plasma volumepasses through the (x,y) vertical plane of a moveable grid system. The position ofthis grid along the axis (z) of the molecular beam determines the distance travelledby the plasma from its point of origin to this detection plane. The Multi-ChannelPlate (MCP) waveform, observed as a function of the selected grid position, then9gauges the number of electrons of the plasma as a function of elapsed time.G1 G2 G3 MCPMolecular Beam12Figure 2.1: Experimental apparatus. A skimmed molecular beam passesthrough grid, G1, where it intersects counter-propagating laser beamsindicated as ω1 and ω2. As the excitation volume transits the plane de-fined by G2, an MCP detector situated behind G3 collects the signal ofextracted plasma electrons. The components surrounded by the light-grey border translate together on a moving carriage.Figure 2.1 diagrams the apparatus. A molecular beam, formed by skimminga pulsed free-jet expansion, passes through the first of three perpendicular grids(G1) to enter a field-free region. There, it crosses a pair of collimated, counter-propagating laser beams. The first dye laser pulse (ω1, frequency doubled to226 nm) drives an electronic transition from the ground rovibronic state of NO(X2Π1/2 v = 0,J = 1/2) to the ground rovibronic level of the A 2Σ+ state. Thesecond frequency-doubled dye laser pulse (ω2) tuned over an interval from 327to 329 nm, excites transitions from this intermediate A-state to Rydberg states ofprincipal quantum number, n, from 35 to the ionization limit.The nozzle diameter (0.5 mm) and total backing pressure (4 atm) of our beamsource predict a centreline NO density of 3.6× 1013 cm−3 at the point of illumi-nation. At a measured rotational temperature of 2.5 K, 89% of these molecules10populate the two parity components of the rotational ground state. We adjust thepower of ω1 and ω2 to saturate both steps of excitation, driving 50 percent of thepopulation in the minus parity component of the ground state to the A state, and50 percent of the A-state population to a selected high-Rydberg state. For ω1 andω2 pulses overlapped optimally in time on the centreline of the beam, stepwise sat-urated excitation of NO under our expansion conditions produces a population ofhigh-Rydberg molecules with a peak density of about 4×1012 cm−3.The A 2Σ+ v = 0,J = 1/2 state decays with a lifetime of 209 ns.[35] Thus,delaying the ω2 pulse with respect to ω1 yields a reduced number of A-state NOmolecules for further excitation. We cannot directly measure the fraction of NO∗Rydberg states converted to free NO+ ions and electrons, but the low observedelectron temperature suggests that roughly half of the NO∗ - e− collisions drivepopulation upward toward ionization. Simple rate-equation models, employingtheoretical rate coefficients for collisional excitation and de-excitation of Rydbergstates, support this estimate. We therefore assume that, with zero delay, saturatedsteps of double resonance ultimately yield a plasma with cations and electrons inapproximately equal peak densities of 2×1012 cm−3. Systematically delaying ω2with respect to ω1 yields a set of well-defined relative reduced charged-particledensities.Our excitation geometry produces a prolate ellipsoid excitation volume with anestimated aspect ratio of 4.8:1. The long axis of this ellipsoid, which is determinedby the width of the molecular beam, lies transverse to the axis of propagation. Theshort axis conforms with the Gaussian intensity profile of the spatially filtered ω1laser beam, collimated to a full-width at half-maximum of 500 µm. The plasmaproduced in this volume travels with the velocity of the molecular beam over anadjustable distance to pass through the imaging grid, G2.The carriage supporting this grid, together with G3 and the MCP detector, trav-els on linear bearings supported by four 0.5 inch diameter stainless steel rods. Abellows-isolated motorized actuator controls the position of this carriage to within0.01 mm over a range of 10 cm.We set G2 to have the same potential as G1 (nominally apparatus ground),creating the field-free region in which we vary the plasma time-of-flight. After G2,the plasma encounters an electrostatic field determined by the potential applied11to a third grid, G3, spaced downstream by a fixed distance of 16 mm. For thepresent experiments, we apply 145 V to G3. As the plasma traverses G2, the fieldbetween G2 and G3 extracts electrons which, on the timescale of the measurement,appear instantaneously as signal at the multichannel plate detector. This signalrepresents the extracted electron density integrated in the transverse dimensionsof the plasma, providing a time-dependent trace of this changing density as theplasma volume passes through G2. We assume that the electrons extracted by G2supply a representative measure of the plasma width and its relative charge density.2.2 ResultsWe have shown in appendix A that a Rydberg gas of nitric oxide entrained in amolecular beam, evolves to form a quasineutral ultracold plasma of NO+ ions andelectrons. [39, 40] The present chapter focuses on the long-time rate of change ofthe charged particle density of such a plasma.Figures 2.2 shows series of electron signal traces recorded following excitationto the Rydberg state with n = 50 in the n f series converging to N+ = 2 (50 f (2)),obtained for ω1-ω2 delays of 50 and 100 ns. With increasing G2 displacement,we see that the position of the late signal advances, and that the arrival time ofthis feature corresponds in each case to the flight time of a neutral volume in themolecular beam from the excitation region to the perpendicular plane of G2.The signal of electrons extracted as the illuminated volume passes through G2broadens and decreases in amplitude with increasing flight time, reflecting the pro-cesses of expansion and decay of the plasma with time. Each trace fits well to aGaussian function, and we use the parameters of such fits to extract the plasma ar-rival time, width and relative measure of the total number of electrons. Figure 2.3plots the fitted width along the axis of propagation (the short axis of the ellipsoidexcitation volume) measured as a function of flight time for plasmas formed withdensities determined by ω1-ω2 delays of 0, 50 and 100 ns.For the purposes of this study, we have chosen to analyze a data set that exhibitsa higher rate of plasma expansion. We have previously reported Gaussian radii thatgrow from 200 µm to 1 mm in a period of 25 µs corresponding to a nominal elec-tron temperature of 5 K.[52, 63] Over the same period, the radius measured here12Figure 2.2: Waveforms captured by the MCP detector for different positionsof the moving carriage. The upper collection of signal waveforms isobtained for a fixed ω1-ω2 delay of 50 ns. The lower waveforms showthe decay and expansion of the plasma signal at an initial density re-duced by increasing the ω1-ω2 delay to 100 ns. Traces have been offsetvertically for clarity.grows to exceed 1.5 mm. Note that the plasma prepared with the lowest density ex-pands to a discernibly larger volume over the flight-time interval of measurement.This particular experiment used an alignment that required a higher intensity ω1pulse, and direct one-colour two-photon ionization appears to have added hot elec-trons to the plasma increasing its quasi-equilibrium electron temperature.For any given flight time, we can gauge the relative number of electrons re-maining in the plasma by integrating the area of the corresponding Gaussian fit. Toexpress this in terms of the absolute concentrations of ions and electrons, nNO+(t)≈ne(t), we extrapolate the earliest-measured data points to the known initial plasmadensity. For present experiments, the signal extrapolated to t = 0 for the densest13plasma we can create defines the starting point for decay from an estimated initialdensity, ne(0)≈ 2×1012 cm−3. The introduction of a delay between the ω1 and ω2excitation pulses systematically reduces the initial charged-particle density. Figure2.4 shows a collection of plasma decays for initial densities varied in this way.2.3 DiscussionOver a time period of 30 microseconds, the volume of the plasma described aboveexpands by a factor of 100, while the total number of charged particles decreasesby a factor of 10. The expansion of a neutral plasma such as this occurs as aconsequence of its electron pressure, while the irreversible decay seen here reflectsthe dissociation of excited neutrals formed by the recombination of electrons withNO+ ions. In the sections to follow, we discuss models for these processes, fromwhich we extract estimates of the electron temperature and rate coefficients forelectron-ion dissociative recombination reactions.2.3.1 Evolution to plasma and the development of space chargeThe formation of this plasma begins with the double-resonant preparation of a 1mm3 sample of 109 NO∗ molecules excited to a single selected Rydberg state,50 f (2), which has a binding energy of 40 cm−1. Strong dipole-dipole forces driveinteractions between these molecules.[2–4] Approximately 10 percent of the NOmolecules excited under these conditions have a nearest-neighbour closer than oneclassical orbital diameter. This leads to rapid Penning ionization, which yields adistribution of free electrons with a peak density of about 2× 1011 cm−3 and aninitial temperature in the range of 20 K.[55]These electrons escape until the space charge created by a growing excess ofNO+ ions rises to trap the remaining population. The distribution of electron en-ergies produced by the Penning ionization of n = 50 supports an excess charge ofapproximately 0.1 %.Initially, the density of NO∗ Rydberg molecules greatly exceeds that of ions.So, at first, electrons collide mainly with neutrals. These collisions drive transitionsup and down the manifold of Rydberg states and ionize NO∗ to form NO+. Thegrowing density of electrons initiates an avalanche of ionization,[34] accompanied14by recombination, Rydberg energy redistribution and some further electron escape.This avalanche quickly relaxes to a quasi-equilibrium of trapped ions and electronsand Rydberg molecules.The distribution of particles at this point has thermal properties, including elec-tron and ion temperatures, Te and Ti, and Rydberg level distribution, that aredetermined by the initial state and density selected by Rydberg molecule photo-preparation. Rate equation calculations, using realistic rate coefficients for electroncollision-induced Rydberg transitions, electron-impact ionization and three-bodyrecombination[48], show that an initial Rydberg density of 1012 cm−3 relaxes to aquasi-equilibrium in 100 ns or less. Thereafter, the distribution of electrons trappedby the space charge can only expand, and decay by recombination with NO+ ions.A previous paper has focused on the short-time evolution to a plasma quasi-steadystate.[63] Here, we wish to describe the long-term dissipation of this plasma.2.3.2 Expansion and electron temperatureThe plasma volume increases as the expanding gas of electrons drives radial hydro-dynamic ion expansion with a local mean velocity, γkrk. Our experiment measuresthis expansion along the propagation axis of the molecular beam, z. At t = 0,the Gaussian intensity distribution of the ω1 laser beam intersects the core of themolecular beam to produce a cylindrically symmetric Gaussian excitation distri-bution in the y,z plane. The peak of this density distribution falls off as a broaderGaussian along the cross-beam axis of laser propagation. We assume that the ini-tial distribution of ions produced in the moving frame of this illumination volumetracks this excitation density. We proceed now to characterize the expansion of thisdistribution, as driven by the motion of the electrons in the field of the ions.Assumming a quadratic spatial dependence of the plasma electronic potentialin the coordinate directions, k= x,y,z, we can describe the ion phase space density,fi, in terms of cartesian rms radii, σk, and local hydrodynamic ion velocities, γk:fi(r,v, t) ∝ exp(−∑kr2k2σ2k)exp(−∑kmi(vk− γkrk)22kBTi,k), (2.1)The ansatz above describes a non-spherical three-dimensional Gaussian distribu-tion. Throughout the evolution of such distribution, it is assumed that the ion and15electron velocities follow a Boltzmann distribution. While ions have different tem-peratures along three dimensions, owing to the velocity field of the nitric oxidein the molecular beam, electrons are assumed to equilibrate to exhibit only onetemperature.The real plasma has a non-spherical shape, and this plays an important role indetermining its hydrodynamics and the rate of expansion. For instance, a cylindri-cal plasma, similar to that produced by experiment, will have a slower expansionin the long axis. At any given time, this slows the consumption of electron thermalenergy by the ambipolar expansion, which leaves more electron thermal energybehind for the expansion in the short axis. Consequently, the expansion rate of theshort axis becomes faster than a similar spherical neutral plasma.In cases where a simple self-similar spherical Vlasov model is used to estimatean electron temperature from expansion data, the extracted parameter does not re-flect the actual but rather an upper bound on the electron temperature. Whetherwe consider a spherical, cylindrical, or more sophisticated shape, the low electrontemperatures extracted from a Vlasov picture of the expansion contradicts the basicassumption that electrons are thermal and non-correlated.For expansion in the z-direction, probability function 2.1 yields a time-dependencefor the radius σz, given by:[63]σz(t) = σz(0)[1+ t2/τ2z ]12 (2.2)where τz describes a characteristic time for z−axis expansion determined by thecoupling of ion hydrodynamic motion to the electrostatic potential in this dimen-sion.τ2z =miσz(0)2kB[Te(0)+Ti(0)](2.3)Here, Te(0) and Ti(0) refer to the quasi-equlibrium temperature of the electronsand the temperature of the ions as determined by the velocity distribution of theNO molecules in the molecular beam.Extrapolating to determine the initial width of the plasma along the z-axis,σz(0), we fit Eq. (2.2) to Gaussian radii, σz(t), measured as a function of flight16time. From this fit, we obtain τz, which yields the sum of initial electron and iontemperatures. Note that the time-dependent width displayed in Figure 2.3 showsevidence of acceleration that is characteristic of an electron-charge-driven ambipo-lar expansion. A Vlasov fit to these measured values of σz(t) yields initial electrontemperatures ranging from 12 K, at the highest density studied in this experiment,to nearly 16 K, attained upon reducing the initial Rydberg density by delaying ω2.A plasma with an electron density of 2× 1012 cm−3 has a Wigner-Seitz ra-dius, a = (3/4piρ)1/3 (average spacing), of 500 nm. At Te = 12 K, this plasmawould have a correlation parameter, Γe = e2/4piε0akBTe, of nearly 3, represent-ing a significant degree of correlation. The Vlasov equations do not accuratelydescribe the expansion of a charged-particle system under such conditions. Us-ing a Debye-Hu¨ckle potential to linearize the Poisson-Boltzmann equation for aone-component plasma, Murillo shows that in the limit of an electron potentialattenuated by exp(−r/λD), the electron pressure (PDH) vanishes for Γ> 2.3.[44]Thus, analyzing our experimental expansion in the simple terms of Eq. (2.2),we predict a degree of electron correlation under which we can expect a suppres-sion of electron pressure. This would suggest that our expansion rate underesti-mates the electron temperature. However, in order for this condition of suppressedexpansion to apply, our plasma must still be in the correlated regime. Faster expan-sions observed for longerω1-ω2 delay times suggest that our lower density plasmasare less correlated.2.3.3 Plasma decayThe escape of electrons in the first few hundred nanoseconds of plasma formationcreates a cationic space charge that acts as an attractive potential for the electronsthat remain. Electrons trapped by this potential can decay only by electron-ionrecombination. Subsequent plasma expansion, however, acts to reduce the depthof the electrostatic well, and this allows a small number of additional electrons toevaporatively escape.A potential sufficient to trap the plasma electrons under our conditions of Terequires an excess cation charge of less than one percent. The decaying plasmathus remains quasi-neutral, and we can assume that the number density of NO+17ions over time, nNO+(t), equals that of electrons, ne(t).Ions capture electrons to bound orbitals by the mechanism of three-body re-combination. Rydberg states populated in this way can collisionally ionize or dis-sociate, in an overall mechanism:NO++ e−+ e− kTBR−→kionNO∗+ e− (2.4)NO∗ kPD−→ N+O (2.5)Electrons can also combine with molecular NO+ to yield neutral products by two-body dissociative recombination:NO++ e− kDR−→ N+O (2.6)Rydberg levels populated by three-body recombination exist in quasi-equilibriumwith ions and free electrons. The rate at which these levels predissociate determinesboth the importance of three-body recombination as a channel for irreversible de-cay and the kinetic order in charged-particle density by which it occurs.On the timescale of expansion and plasma dissipation, we can safely assumethat both the distribution over Rydberg levels and the electron temperature reacha slowly evolving steady state. Accordingly, we represent the rates of three-bodyrecombination, collisional ionization and predissociation in terms of thermal ratecoefficients averaged over the Rydberg level distribution. We then describe thekinetics of electron/ion consumption to form neutral atoms in terms of a steady-state concentration of NO∗:− dne(t)dt=kTBRkPDn3e(t)kionne(t)+ kPD(2.7)Experiments have measured predissociaton rates for many Rydberg states ofNO.[51] Even the fastest of these unimolecular reactions proceeds on a timescalethat is orders of magnitude slower than the rate of collisional ionization determinedfor our electron density from well-established rate theory estimates of kion.[48]Thus, only the kPD kionne limit of Eq. (2.7) applies, and we can confidently ex-18pect three-body recombination to give rise to a dissociative relaxation that proceedsat second order in electron density.Adding terms to account for the electron density change simply associated withthe expansion we observe, and introducing the possibility of some further electronevaporation, we obtain a complete rate law for this channel of electron densitydecay.− dne(t)dt=kTBRkionkPDn2e(t)+ne(t)V (t)dV (t)dt+ kevne. (2.8)Direct dissociative recombination proceeds via binary electron-ion collisions,and the electron density in this case decays according to an elementary bimolecularrate law, modified for expansion and evaporation:− dne(t)dt= kDRn2e(t)+ne(t)V (t)dV (t)dt+ kevne. (2.9)To interpret the current results, we recognize that plasma NO+ ions and elec-trons may decay to neutral atoms by either or both of these mechanisms. Thus,we consider recombination-decay proceeding in parallel with direct dissociativerecombination, Eq. (2.9).To model the overall rate at which ne(t) changes in time, we represent theplasma by a cylinder of concentric shells with radial coordinate, r. The radial Gaus-sian density distribution measured experimentally determines the relative numberof electrons in each shell. Assuming a peak density of 2× 1012 cm−3 puts theserelative densities on an absolute scale. We numerically integrate rate equationsconforming to the dissociation mechanisms discussed above to describe the varia-tion of the number density within each shell as a function of time for a selected setof rate coefficients.As this calculation proceeds, we expand the volume of each shell according to,dVdt=dVdrdrdt, (2.10)using the radial coordinate hydrodynamic mean expansion velocity to define thetime dependence of r, dr/dt = γr, where we determine γ as a function of timefrom the experimentally determined value of τz. We integrate the calculated lo-19cal electron densities over the volume of the model to obtain the total number ofplasma electrons as a function of time, which we compare with the integrated areaof the plasma signal.Solid lines through the experimental decays in Figure 2.4 show such a compari-son for recombinative dissociation in the second order limits of Eq. (2.8) and (2.9).Here, the relative initial reactant density scales according to the known variation inplasma signal with ω1−ω2 delay, and our fits conform with this scaling. We findthat neither third-order nor first-order rate laws describe these decays.Table 2.1 summarizes the parameters determined by least-squares regression,where we define a phenomenological second-order rate constant, k2, that containscontributions from both recombinative dissociation mechanisms, (2.4) and (3.1),viz:k2 =kTBRkionkPD+ kDR . (2.11)Table 2.1: Second order rate coefficients obtained by fitting the plasma signaldecay for a set of initial charged particle densities systematically variedby selecting the ω1−ω2 delay. kev describes the slow first order decreasein plasma signal at long times.∆t(ω1−ω2)/ns ne(0)/cm−3 k2/cm3s−1 kev/s−10 2.0×1012 4.2±0.1×10−7 1.6±0.1×10450 1.4×1012 7.3±0.3×10−7 1.0±0.1×104100 1.0×1012 1.8±0.1×10−6 0.5±0.1×104200 5.0×1011 3.8±0.2×10−6 0.4±0.4×104Experiments in afterglows and accelerators have measured rate constants forthe elementary two-body dissociative recombination of NO+ at temperatures aslow as 200 K.[1, 42, 68] These observations conform well with ab initio R-matrix-multichannel quantum defect theory calculations of kDR for electron temperaturesfrom 1 to 5000 K.[64] In the low-temperature range around 10 K, theory predicts atwo-body rate constant for dissociative recombination of about 1×10−6 cm3 s−1.At the highest initial density of our experiment, we observe a total second-order rate coefficient that is about half of this value. Note however, that the value20we determine for this rate constant depends on the value we choose for the initialcharged particle density. We have confidence in our calculation of the molecularbeam density and in the fact that we saturate the two electronic transitions that pre-pare our Rydberg gas. But, variations caused by incomplete overlap of ω1 and ω2or their slight misalignment with the molecular beam could reduce the real densityby a factor of two or more. If we assume that the present alignment reduces thepeak density at optimum ω1-ω2 delay to 1× 1012 cm−3, the top fit shown in Fig-ure 2.4 scales to yield k2 = 1.3×10−6 cm3 s−1, in good agreement with scatteringtheory estimates.More significant than the correspondence of our absolute rate constants withtheory is the relative variation in k2 we see with density. The observed rate constantfor second-order decay increases to a measurable degree with increasing density.We can certainly anticipate additive contributions to the overall decay rate fromkDR and the mechanism that begins with three-body recombination. And, the phe-nomenological rate constant for electron loss by three-body recombination varieswith the concentration of electrons. But, for all values of ne, Eq. (2.7) predictsan apparent second-order rate coefficient that remains constant or decreases withdecreasing electron density.Thus, again with reference to Eqs. (2.7) and (2.11), we can account for a phe-nomenological second-order rate coefficient that increases with decreasing ne onlyin terms of ne-dependent variations in the rate constants themselves.The expansion data presented in Figure 2.3 suggest that at the highest density,this plasma exists in a regime of electron correlation. The limiting initial tempera-ture derived from a Vlasov fit to the rate of expansion yields Γe = 2.8. Conditionsof electron correlation have been associated with suppressed three-body recombi-nation and enhanced collisional ionization. Plasmas formed at lower density showhigher expansion rates suggestive of reduced correlation. Perhaps the rate coef-ficient for three-body recombination, suppressed by electron correlation at higherdensity, increases under conditions of reduced correlation, the effect of which isseen in the overall rate of plasma decay.212.4 ConclusionsWe have developed a kinetic scheme by which to model the long-time decay of anultracold plasma of electrons and molecular NO+ ions to neutral N and O atoms.From an analysis of this decay as a function of initial density, we extract a second-order rate constant that accords with the elementary rate constant for thermal two-body dissociative recombination, estimated from scattering theory, as calibrated byconventional gas-phase measurements.The rate constant under plasma conditions, however, shows a distinct depen-dence on charged particle density. The steady-state kinetics of recombination be-tween the third-order and second-order limiting regimes predicts a reduction in theapparent second-order rate constant with decreasing reactant density. But the vari-ation observed opposes this. Instead, the apparent rate coefficient extracted fromplasma decay grows with decreasing density.The conditions chosen for this experiment span a range of apparent electrontemperature and density ranging from conditions of moderate correlation (e2/4piε0a> kBTe) to those in which the average Coulomb potential energy equals the ther-mal kinetic energy. To the extent that we can link conditions of correlation with asuppression of three-body recombination, we can associate the increase in the ap-parent dissociative recombination rate coefficient at lower density with a growingimportance of step-wise three-body collisional dissociative recombination in themechanism for plasma dissipation.22Figure 2.3: Fitted plasma widths as a function of centre arrival time at G2for ω1-ω2 delay times of 0, 50 and 100 ns (top to bottom). Solid linesrepresent fits of Eq. 2.2 to the experimental data, yielding a fitted initialtemperatures Te(0)+Ti(0) = 12.1,14.3 and 15.8 K, respectively.23Figure 2.4: Integrated area of the plasma signal as a function of arrival timeat the imaging grid G2 for ω1-ω2 delay times of 0, 50, 100 and 200ns (top to bottom).. The lines are integrated rate equations fitted to thedecay data. The calculations consider second-order decay processes inan expanding plasma with first-order electron evaporation (c.f. Eqs. 2.8and 2.9)24Chapter 3Dissociative Recombination Slowsthe Expansion of a MolecularUltracold Plasma1Ultracold plasmas represent an important new laboratory for plasma physics ina regime of moderate density and very low temperature. These simple, freelyexpanding systems can readily approach conditions of strong Coulomb coupling[28, 57].Many examples have been studied using atomic systems under conditions oflaser cooling in MOT. Here, the control of photoionization energy, combined withprobes of ion spatial and velocity distributions, characterize plasma evolution forcomparison with elementary theoretical models [27, 31, 56].The prerequisite of optical trapping limits MOT studies to a relatively smallnumber of alkali, alkali earth and metastable rare gas atoms. Chemical reactionscan play a role in plasmas of many kinds, and worthwhile opportunities exist in thegeneralization of ultracold studies to include systems of molecules.In recent experiments, we have excited diatomic nitric oxide in a supersonicmolecular beam to form a Rydberg gas with a temperature less than 1 K in themoving frame [40]. Simulation models describe a sequence of collisional pro-1H. Sadeghi and E. R. Grant, Dissociative recombination slows the expansion of a molecularultracold plasma, Phys. Rev. A 86, 052701 (2012)25cesses in which this system relaxes to form a molecular ultracold plasma that hasproperties very comparable to plasmas formed by Rydberg excitation or thresholdphotoionization of atoms in a MOT [39, 41, 61–63].Both MOT and molecular beam plasmas expand on a timescale of microsec-onds. Free electrons drive this expansion at a velocity determined hydrodynam-ically by the temperature of the electrons, Te, and the mass of the positive ionsforming the space charge that contains them. A measurement of the rate of expan-sion provides one means to estimate the temperature of the electrons [56].Atomic plasmas, prepared to have very low electron temperatures, expandfaster than models predict using collisionless hydrodynamics alone. This is nowwell understood to occur as a consequence of electron heating that accompaniesplasma relaxation.Molecular beam plasmas expand slowly by comparison [39, 63]. These beamplasmas, with densities more than two orders of magnitude higher than MOT plas-mas, occupy an interesting regime of interparticle interactions. Their slow expan-sion challenges the hydrodynamic description as conventionally applied [33]. It isimportant to understand how the experimentally observed dynamical behaviour ofsuch plasmas relates to their energy state, particularly their electron temperature.The nitric oxide plasma differs from MOT plasmas in one very important funda-mental respect. Molecular cations carry the positive charge, and when a diatomicNO+ ion recombines with an electron, it can dissociate to neutral atoms.NO++ e− kDR−→ N+O (3.1)We have interpreted the effect of this process on the dissipation of a nitric oxideplasma in terms of the conventional kinetics of two-body dissociative recombina-tion (DR) [61], the rate constant for which has been thoroughly characterized byexperiment and theory [64].kDR varies inversely with electron temperature, and we observe a fast rateof plasma dissipation that serves to place an upper limit on Te. This evolutionof plasma density over time conforms with a second-order rate process that isquenched as the plasma expands. Kinetic fits accord with the observed rate ofexpansion, and this provides a measure of charged-particle density [61].26The DR rate varies with the product of the ion and electron concentrations. Re-combination occurs fastest in the core of the plasma, and this flattens the charged-particle density distribution in the centre.The spatial distribution of ions and electrons in a quasi-neutral plasma deter-mines the driving force for expansion. A solution of the Boltzmann equation forthe self-similar expansion of a spherical Gaussian plasma in a MOT serves well todescribe a characteristic radial acceleration of its ions, which can be analyzed toestimate the electron temperature [33].For a molecular plasma, the alteration of plasma shape with DR changes thedynamics of expansion. In the present chapter, we develop a fluid model confirm-ing that this is the case. We show for a plasma formed to have an initial Gaussianspatial distribution in the binary collision regime with Dissociative Recombina-tion (DR), that the decay of ions and electrons to neutral fragments has a significanteffect on the expansion dynamics.In particular, we find that the flattening of the radial gradient in charged particledensity suppresses the expansion of the core. As a consequence, a comparablereservoir of electron energy produces a much slower rate of ion expansion in thedensest part of the plasma, an effect which could have important implications forultracold plasmas used as bright electron sources [70].3.1 Hydrodynamic model for a plasma undergoingdissociative recombinationWe assume a plasma phase space described by electrons in thermal equilibriumand an ion distribution function fi(v,r, t) that solves a Boltzmann equation (3.2)formulated to include the dissociative recombination of electrons and ions of mass,mi [12].∂ fi∂ t+ v ∇ fi− emi∇φ ∇v fi =(∂ fi∂ t)DR(3.2)In the equation above, φ is the Coulomb potential of the ion distribution de-scribed by fi.The Boltzmann equation might also include other collisional terms owing tothree-body inelastic processes, such as three-body recombination (TBR), and electron-27Rydberg energy transfer. We assume for the purposes of the present model thatthese processes reach a condition of quasi equilibrium on a timescale much fasterthan dissociative recombination and plasma expansion, and that the realization ofthis quasi equilibrium defines an initial electron temperature, Te(0), which drivesexpansion.We note as well that the term, (∂ fi/∂ t)DR may include a contribution from thespontaneous predissociation of Rydberg molecules formed by TBR. We show in[61] that, under quasi-equilibrium conditions, this loss channel proceeds at a ratethat is second order in charged-particle density, and so simply makes an additivecontribution to what we might term as the total rate of direct plus indirect DR.In Eq (3.2), e is the charge and mi is the ion mass. ∇φ(r, t) represents theelectric potential gradient, which in the well-known quasi-neutral approximation,relates to the density gradient by [27]:e∇φ = kBTen−1i ∇ni (3.3)The right-hand side of Eq (3.2) refers to the rate of change of fi as a result ofdissociative recombination. We can take the time derivative of fi independent ofits integral over velocity, which yields:∂Ni∂ t=∫r∫v(∂ fi∂ t)DRd3vd3r =∫r(∂ni∂ t)DRd3r (3.4)The dissociative recombination of ions and electrons is a second order kineticprocess, which depends on the density of both species. The continuity equationcalculated from Eq (3.2) describes the flow of density and its change as a result ofDR,∂ni∂ t+∇.(niu)+ kDRn2i = 0, (3.5)where u =∫v fid3v/ni is the hydrodynamic velocity of ions, ni =∫fid3v is thedensity of ions, which is assumed to be equal to electron density ni ≈ ne, and kDRis the electron-ion dissociative recombination rate constant.The first moment of the Boltzmann equation (3.2) determines a potential gra-28dient that forms the driving force of the expansion [16, 17]:dudt=− emi∇φ (3.6)With an ion temperature of the order of 1 K, the ion pressure has only a small effecton the expansion of a ultracold plasma, which we neglect. We assume an initialspatial distribution of charged particles described by a spherical Gaussian.We model the effect of electron pressure on the expansion of this charge distri-bution by considering a set of spherical shells [61]. As the plasma expands, eachshell grows to maintain a density of ions that changes due only to expansion anddissociative recombination. The continuity equation (3.7) represents the density ofeach shell in the moving frame by:∂n j(t)∂ t+ kDRn j(t)2+n j(t)Vj(t)∂Vj(t)∂ t= 0, (3.7)where the index, j, represents the shell number, andVj(t), its volume. The last termin Eq (3.7) describes the decrease in density of shell j caused by the expansion ofVj(t).The radius and volume of each shell evolves in time according to the hydrody-namic velocity of the ions.∂ r j(t)∂ t= u j(t) = γ j(t)r j(t) (3.8)Vj =43pi(r3j+1− r3j ) (3.9)∂Vj(t)∂ t≈ 3γ j(t)Vj(t) (3.10)If we choose a large enough number of shells, the density inside each one willbe uniform, and we can integrate Eq (3.7) to find:n j(t) =n0e−r j(0)2/2σ20Vj(0)/Vj(t)1+n0e−r j(0)2/2σ20∫ t0 kDRVj(0)/Vj(τ)dτ(3.11)where τ is a dummy variable for time integration, n0 represents the peak density of29the initial Gaussian spatial distribution, σ0 is its initial rms width, and r j(0) denotesthe initial radius of each shell.The variation of density with j, described by Eq (3.11), directly determines thepotential gradient of charge separation:− e∇φ j(t) =− kBTen j(t)n j+1(t)−n j(t)r j+1− r j (3.12)For the case of no dissociative recombination (kDR = 0), the hydrodynamicvelocity factor reduces to a constant value for all shells, γ = t/(t2 + τ2), and Eqn.(3.12) becomes−e∇φ = kBTer/σ(t)2, which equals the force calculated for a self-similar Gaussian distribution [27]. For this case, a test of the shell model givesresults identical to the analytic Vlasov solution of this problem.At the centre of the plasma with dissociative recombination, the conversionof molecular ions to neutral atoms flattens the charge density profile, causingn j(t)≈ n j+1(t), which suppresses the potential gradient (3.12) and diminishes theexpansion force. Much slower dissociative recombination at larger distances leavesdensity gradients unaltered, and the force varies linearly with r. In general, γ j(t)for a given shell in a system with nonlinear forces depends on its position, j. As aresult, no analytical solution exists for the force (3.12).We can use energy conservation to describe the evolution of electron temper-ature in an expanding plasma with dissociative recombination. Neglecting the iontemperature, the total energy of the system at time t depends on the thermal energyof the electron gas and the kinetic energy transferred to the radial motion of theions:E =32NekBTe+12mi∑N jγ2j r2j (3.13)where Ne and N j refer to the number of electrons and ions.These quantities change over time owing to DR. The total energy changes dueonly to a change in the number of electrons and ions. Neglecting any adjustment inthe quasi equilibrium population of Rydberg molecules, the electron temperaturedecrease exactly balances the ion velocity increase, such that dE = (∂E/∂N)dN,or explicitly:30dEdt=32dNedtkBTe+12mi∑jdN jdtγ2j r2j (3.14)Equating Eq (3.14) with the time derivative of Eq (3.13), we obtain an expres-sion for the time derivative of temperature.dTedt=−2Te× ∑N jγ jr j(−∂rn j)/n j3∑N j (3.15)In the self-similar expansion of a Gaussian plasma without dissociation, γ changesin time, but is independent of j, and can be taken out of the summation. The term,(−∂rn j)/n j, then simplifies to r/σ2, reducing Eq (3.15) to ∂tTe =−2γTe.Eqs. (3.7), (3.6), and (3.12), together with the temperature variation of thesystem (3.15) form a complete set for numerical integration.3.2 Results and discussion3.2.1 Time evolution of charged-particle densityWe illustrate the effect of DR on the expansion of a molecular ultracold plasmaby showing results calculated for a quasi-neutral system of NO+ and electrons.The spatial distribution starts as a spherical Gaussian with a peak ion density ofn0 = 2× 1012 cm−3, initial width of 200 µm, and initial electron temperatureTe=20 K. To represent the rate constant for dissociative recombination, we use aform established by experimental measurements and scattering theory calculations,kDR = 6× 10−6 T−1/2e cm3 s−1 [64]. We have found that this expression returnsa second-order rate constant that is consistent with experimental measurements ofoverall plasma decay [61].Figure 3.1 compares the evolving density profile of a dissociating plasma withthat of one without dissociation. Both representations begin with spherical Gaus-sian charged particle distributions. The left frame shows the evolution of densityin the first microsecond. The right frame details the plasma shape after 10 and 13µs.Initially, on a timescale of τ ≈ 1/kDRn0 = 0.4 µs, during which little expansion31Figure 3.1: Density of electrons in the presence (black curves) and absence(red curves) of DR for expanding spherical shells. Densities are plottedat times (from top to bottom), t = 0,0.4 and 0.8 µs (left) and with ex-panded scale at later times t = 10 and 13 µs (right). Results obtainednumerically for the non-dissociating case agree precisely with the ana-lytical solution of the Vlasov equations for conditions given in the text.occurs, fast dissociation in the core flattens the density profile of the plasma withDR. This flatness in the centre reduces the charge-density gradient which reducesthe local rate of expansion.In the outer shells of the plasma volume, where the charged particle density islower, dissociative recombination proceeds at a much slower rate, and the plasmaretains a Gaussian charge-density gradient. Accordingly, the electron-cation po-tential at 4σ more resembles that of the self-similar Vlasov expansion.3.2.2 Time evolution of ion velocityFigure 3.2 plots the hydrodynamic velocity of ions as a function of time for shellsof different radius, r, for a plasma with dissociative recombination compared with acollisionless plasma. We see that the innermost ions experience much less accelera-32Figure 3.2: Radial velocities of ions as a function of time in shells with initialradii at: 1σ (top-left), 2σ (top-right), 3σ (bottom-left), and 4σ (bottom-right) positions of the t = 0 Gaussian. Conditions as in Fig. 3.1. Redcurves describe the self-similar expansion of a non-dissociating plasma.Black curves show the effect of dissociative recombination.tion in the presence of DR. Ion velocities in the outermost shells of the dissociatingplasma rise to conform closely with those of the self-similar Vlasov expansion.Figure 3.3 similarly shows the expansion velocities of selected shells in the disso-ciating plasma expressed in terms of the parameter γ j(t), compared with γ(t) for aVlasov expansion with the same Te(0).Curves in Figure 3.4, use the hydrodynamic acceleration of a non-dissociatingplasma, to scale the acceleration of ions in a dissociating plasma. At t = 0, bothplasmas have Gaussian density distributions, and calculated quantities match forall values of r. But, after about 1 µs, DR flattening of the plasma shape reduces theacceleration of core ions by about a factor of 3. In this initial phase, the non-lineareffect of DR extends to about three times the RMS radius of plasma, at which pointthe acceleration rises to meet that of the non-dissociating plasma.A balance of factors determines the comparative acceleration of these periph-33Figure 3.3: Plot points (black) marking γi as a function of time for shellswith initial radii, reading from bottom to top, of 1σ , 2σ , 3σ ), and 4σpositions of the t = 0 Gaussian. Solid red line gives γ(t) for all radii ofthe non-dissociating self-similar plasmaeral ions. Electron energy ineffectively coupled to the core ions remains availableto drive the expansion of the outer shells. The loss of particle density removesenergy from the system, which tends to diminish acceleration globally.After a few microseconds, the drop in charged particle density owing to disso-ciation and expansion slows the rate of DR. The retarded velocity of the core ionsallows them to remain in a spatial distribution that is relatively sharp compared tothe regions of collisionless plasma expanding rapidly at larger r. In other words,during this interval, γ j(t) varies strongly with j.This non-linearity acts to cause the density gradient over inner shells to steepen,increasing the relative expansion force at small r. But, the effect of this developinggradient moderates quickly with the increase in plasma volume. The absolute valueof the hydrodynamic force at 10 µs is smaller than the initial radial force by a factorof 100.3.2.3 Time evolution of electron temperatureIn hydrodynamics models for plasmas with and without dissociative recombina-tion, the electron temperature drops, as expansion transfers thermal kinetic energy34Figure 3.4: Acceleration of ions scaled by the linear acceleration in a colli-sionless plasma as a function of scaled distance at different times. Blackcurves show the scaled acceleration in a dissociating system and red dot-ted line represents the constant scaled acceleration in the collisionlesssystem.of the electrons to radial kinetic energy of the ions. The flattening of the charge-density distribution in the core dramatically effects the properties of this conversionfor the innermost shells.But, the volume occupied by the core shells is relatively small. 80 percentof the particles in the initial Gaussian charge distribution occupy shells of radiuslarger than r/rrms = 1. By t = 10µs, this proportion rises to more than 95 percent.At the lower density of these outer shells, dissociative recombination proceeds ona timescale that is slow compared with expansion. The spatial distribution beyond2σ thus retains its Gaussian shape, and collisionless Vlasov hydrodynamics governthe expansion of a large proportion of the plasma mass.This self-similar expansion of the outer shells serves to regulate the plasmatemperature, which is equilibrated throughout the plasma by rapid electron colli-sions. Even for ballistic energies of a few tens of millivolts, as might be transferredby inelastic collisions with low-n Rydberg molecules, the high total charge densityof this plasma permits the escape of only a few thousand electrons, which insures35that energy of those that remain is well described by a Maxwell-Boltzmann distri-bution.In a Vlasov limit, the electron temperature evolves according to:Te(t) =Te(0)1+ t2/τ2(3.16)where τ defines a characteristic plasma expansion time,τ =√miσ20 /kB[Te(0)+Ti(0)] (3.17)Eq (3.16) succeeds well in describing the evolution of electron temperature inMOT plasmas prepared to have initial electron temperatures approaching 100 K [27,33]. For electron temperatures lower than 30 K, a rapidly growing cross section forthree-body recombination with decreasing temperature leads to the formation ofRydberg atoms, which relax by collisions with electrons that transfer energy andretard the cooling predicted by Eq (3.16).3.2.4 Collisional processes and the evolution of quasi equilibriumThe conditions of a molecular beam yield a plasma with three orders of magnitudehigher density than that of a MOT. As a result, electron-ion and electron-Rydbergcollisions drive the beam plasma to a state of quasi equilibrium on a timescalemuch faster than expansion.This relaxation outpaces dissociative recombination and Rydberg predissocia-tion as well. For example, in a plasma with an electron temperature of 15 K andcore density of 2× 1012cm−3, the detailed rate calculated for three-body recom-bination into the single highest thermally bound Rydberg level (n = 89) exceedsthe rate of dissociative recombination by a factor of more than 103 [48]. However,under these conditions, the initial rate of dissociative loss of Rydberg moleculesin quasi equilibrium makes a minor contribution. At steady-state, predissociationproceeds as a second-order process with a rate constant five times smaller than kDR[62].Coupled rate-equation simulations confirm that, at its core density, the molec-ular beam plasma forms this quasi equilibrium on a timescale much faster than36dissociative recombination and expansion [62]. This rapid approach to quasi equi-librium releases electron energy that redistributes over the plasma volume, estab-lishing an initial electron temperature, Te(0), for the subsequent hydrodynamicsand dissociation to neutral fragments.As expansion proceeds, conditions change in the plasma, and the NO+–NO∗quasi equilibrium adjusts to these changes. A more complete model for the evolu-tion of Te(t) must look beyond Eq (3.15) to consider the effects of this adjustment.For example, electrons do work on the ions causing Te to fall globally. Thisincreases the cross section for three-body recombination. The formation of neutralsenables the plasma to oppose cooling by releasing Rydberg binding energy.Expansion, however, reduces density as well, and the quasi equilibrium op-poses this by promoting Rydberg molecules to the ionization continuum, whichabsorbs thermal energy. This competing factor moderates the adjustment of thequasi equilibrium and enables the electron temperature to fall with decreasing den-sity.Looking locally, the expansion velocity of the flattened core substantially lagsthat of the Gaussian wings (cf. Figure 3.2). Electron temperature equilibrateseverywhere, and density in the innermost shells decreases over time owing to dis-sociative recombination. This serves to diminish the collisional loss of ions, stabi-lizing a charge density in the frozen core sufficient for ion correlation. Such a stateof ion correlation could act as a drag on overall expansion [44].The outer shells occupy a much larger volume and account for most of theplasma mass. Here the plasma retains its Gaussian density gradient and undergoesa Vlasov expansion that acts over time to cool the electrons in the frozen core.3.2.5 Relation to experimental measurementsModel calculations presented above show the distinct effect of dissociative recom-bination on the evolution of plasma density within a radius of r/σ = 1. The molec-ular beam experiment, however, does not sample this density distribution directly,but, rather measures the profile along a propagation axis, z, of the density projectedonto the x,y plane as the plasma transits a perpendicular grid. Figure 3.5 illustratesthis detection scheme. Figure 3.6 compares the widths of hypothetical electron37jj-1 j+1Figure 3.5: Schematic cross section of plasma shells illustrating the volumessampled by projection onto an x,y grid by plasma propagation in the zdirection.(top waveform) Distribution of density as a function of shellradius along z. (bottom waveform) Distribution of number per shell asa function of shell radius along z.waveforms obtained by sampling such projections for hydrodynamic model plas-mas with and without dissociative recombination.For each value of z, this projection combines plasma electrons sampled fromouter shells with a varying contribution from shells of smaller radius. As notedabove, shells with radii larger than 2σ contain more than 90 percent of the plasmamass, and the ions in these shells experience an acceleration force little differentfrom that of the Gaussian self-similar expansion of a non-dissociating plasma withthe same initial conditions. Thus, a typical experimental measurement using thismethod would find it difficult to distinguish the expansion of a dissociating plasmafrom a non-dissociating one.38Figure 3.6: Widths of electron waveforms sampled by projection of themodel plasmas of Fig. 3.1, expanding hydrodynamically with (black)and without (red) dissociative recombination. The solid red dot-curveplots the measured width of the density distribution σ(t) calculatedanalytically for the Vlasov expansion of a non-dissociating Gaussianplasma. Red dots give the same quantity obtained numerically allow-ing a shell model plasma without dissociation to expand from an initialGaussian density distribution.3.3 ConclusionsA fluid model for the expansion of a molecular ultracold plasma composed of NO+ions and electrons shows that fast dissociative recombination flattens the charged-particle density distribution, effectively freezing the expansion of the plasma core.In the wings of the plasma, a far greater number of particles evolve at lowdensity. Here, dissociative recombination has little effect, and ion motion is drivenby a Gaussian charge-density gradient. The energy balance in these outer shells isgoverned by the collisionless acceleration of the ions under the radial pressure ofthe expanding electron gas. Neglecting collisional effects, the electron temperatureevolves much like that of a Vlasov expansion.At its peak density, the molecular-beam plasma relaxes to a quasi equilibriumof ions, electrons and high-Rydberg molecules on a timescale that is fast comparedwith expansion. A globally falling electron temperature causes this quasi equilib-rium to adjust. In the core, where expansion is limited, this adjustment consumes39charged particles and releases energy to the electron bath. The resulting retardationin the rate of temperature decrease occurs over an extended period of time with aneffect that diminishes as the plasma increases in size.Even though DR-flattening dramatically suppresses the radial velocity of ionsin the densest part of the plasma, the evolution of the overall plasma width as mea-sured by a sequence of planar sections remains dominated by the expansion of theouter shells. Thus, to conventional molecular beam experiments, a dissociatingplasma with an initial electron temperature Te(0) exhibits a time evolving widththat differs only slightly from a Vlasov expansion with the same Te(0). Obser-vation of the dramatically different hydrodynamics of the flattened core awaits anexperiment able to image the evolution of core charge density in three dimensions.40Chapter 4Penning Lattice14.1 IntroductionWhen a gas of charged particles reaches limits of low temperature and/or highdensity, interactions governed by long-range interparticle potentials grow to exceedthe average energy of thermal motion. Such coupling gives rise to a condition ofmany-body correlation that profoundly alters the collision physics and collectiveproperties of a system, causing conventional gas-kinetic and fluid models to fail[25]. The experimental and theoretical investigation of strong coupling representsan important contemporary frontier in condensed matter and plasma physics.One conventionally gauges the strength of ion or electron coupling in a plasmaby the parameter, Γα = e2/4piε0awskBTα , where e represents the electron charge,aws is the Wigner-Seitz radius, determined by the charge density ρ as 3√3/4piρ ,and Tα denotes the corresponding ion or electron temperature.Ultracold plasmas, in particular, offer a novel environment in which to studythe properties of Coulomb systems in or near states of charged particle correlation.Formed under laboratory conditions in magneto-optical traps (MOT) [27] or super-sonic molecular beams [23, 40], they directly display the influence of long-rangeinteractions on collective properties, such as density oscillations [10], ambipolarexpansion [33] and recombination [7, 30, 48].1H. Sadeghi et. al., Dissociation and the Development of Spatial Correlation in a MolecularUltracold Plasma, Phys. Rev. Lett. 112, 075001 (2014)41Spatially correlated ultracold plasmas can manifest liquid-like or even crys-talline properties. But, interparticle forces arising from strong coupling causedisorder-induced or correlation heating, which acts intrinsically to limit the sus-tained development of strong coupling as a dense but randomly ordered neutral gasevolves to a plasma [45].It has been recognized for some time that spatially pre-correlating the excitedgas can offer a means to overcome this limitation. For example, theorists haveproposed to impose order on a gas before ionization by evolving to plasma fromthe spatially correlated initial state of a Fermi degenerate gas, or confining theinitial atom distribution to an optical lattice [21].Dipole blockade also forms strong correlations in the spatial positions of Ry-dberg atoms created by narrow-band laser excitation [14]. Recent models and ex-perimental tests suggest that the ionization of a blockaded Rydberg gas could offera means to reduce disorder-induced heating [6, 53].Here we describe a novel, highly robust method for the introduction of spatialcorrelations that occurs naturally in the evolution from a molecular Rydberg gas toan ultracold plasma. Recognizing the spatial selectivity of Penning ionization, wedevelop a model, supported by experimental results, that defines conditions for theformation of a lattice-like spatial distribution of ions, which leads to a state of ioncorrelation that observably affects the free space expansion of the plasma.4.2 Experiment and methodOur experiment [40] creates a plasma by exciting nitric oxide entrained in a seededsupersonic expansion through a 0.5 mm nozzle from a stagnation pressure of 5 atm.A hydrodynamic model predicts that the 10 percent NO seeded in this expansionrelaxes to a translational temperature on axis of T‖ ≈ 500 mK. Geometric coolingreduces the transverse average velocity to a value at least ten times lower [40].Pulsed dye laser double-resonant excitation via NO A 2Σ+ v= 0, N′ = 0 yieldsa Rydberg gas of NO molecules in a single rovibrational quantum state with aselected initial principal quantum number n0 from 30 to 80 in the long-lived seriesof f Rydberg states that converges to the N+ = 2 level of NO+, v = 0, n f (2).Saturating the first (ω1) and second (ω2) one-photon resonant transitions, forms an42initial density of Rydberg molecules as high as 5×1012 cm−3.The Rydberg gas prepared in this way includes a fraction of excited moleculesthat have nearest-neighbour distances within a critical radius for prompt Penningionization. The size of this fraction varies with the density of the Rydberg gas andthe principal quantum number to which it has been promoted.The Penning interaction between two excited NO molecules with principalquantum n0 yields a molecular NO+ ion and a free electron, together with a deac-tivated NO molecule that has the internal energy of a Rydberg state with principalquantum number n′ ≤ n0/√2 [56].Prompt electrons produced by Penning ionization escape the Rydberg gas vol-ume until the excess of ions grows to no more than one percent, at which point thespace charge traps the electrons that remain. The collisions of trapped electronswith frozen Rydberg molecules initiates an electron-impact ionization avalanche[69].The interaction volume moves downstream with the velocity of the molecu-lar beam to encounter the detection plane created by a perpendicularly mounted,moveable grid, G2. This grid, together with an upstream entrance aperture, G1,define a field-free region. As this volume of gas transits G2, it emits electrons,a fraction of which are collected by a weak field and conveyed to a multichannelplate detector.The ionized volume sampled by the detection grid grows with increasing flighttime at a rate faster than the thermal spread of our molecular beam. In a freelyexpanding plasma, the ambipolar pressure of the electron gas causes a radial accel-eration of the ions [33]. The expansion we observe accords with this mechanism[39, 61], and we find that the ions reach a terminal velocity that varies with theinitial quantum number selected in the preparation of the Rydberg gas.4.3 ResultsFigure 4.1 plots terminal velocities of plasma expansion as a function of n0. Plas-mas initiated from Rydberg gases prepared by excitation to principal quantumnumbers in the range from n0 = 45 to 60 show the slowest rates of expansion.What characteristic of the plasma varies with Rydberg gas n0 to alter its expansion43velocity to such a degree?35 37 39 41 43 45 47 49 51 53 30 35 40 45 50 55 60 65 70 75 80 Terminal Velocity (m/s) Principal Quantum Number  Figure 4.1: Terminal radial velocities of ions measured in a long-time limitof ion expansion for ultracold plasmas created from molecular Rydberggases of NO with initial principal quantum number, n0. Error bars de-termined from the residuals of Gaussian fits to experimental waveformsproduced by plasma volumes transiting G2. Grey points represent radialexpansion velocities predicted for Te = 9 K by a hydrodynamical modelaccounting for spatial correlation calculated stochastically (see text).Two properties do vary strongly with n0 over this range: Rydberg binding en-ergy and orbital radius. But electron-impact avalanche and evolution to plasmaquasi-equilibrium act quickly to redistribute population and erase memory of theinitial state of the Rydberg gas.For this reason, a selected value of n0 can affect long-time plasma propertiesonly by altering initiation dynamics. To explore the possibility of any such effect,we consider the initial steps of plasma formation in a molecular Rydberg gas ofparticular n0 and density.The second step of pulsed dye laser excitation from the NO A state to then0 f (2) Rydberg state creates a Rydberg gas in which kth nearest-neighbour dis-44tances between excited molecules, r, conform with the Erlang distribution [14]:Pk(r) =4pir2ρ(1− k)!(43pir3ρ)k−1e−43pir3ρ (4.1)where, ρ represents the number density of the Rydberg gas.Strong dipole-dipole interactions cause Rydberg molecule pairs with nearest-neighbour distances that fall within a critical radius, rc, to undergo Penning ioniza-tion.NO(n0)+NO(n0)−→ NO(n′ < n0)+NO++ e− (4.2)Molecular dynamics simulations estimate that semiclassical ionization occurs witha 90 percent probability after 800 Rydberg periods for rc = 1.8×2n20a0. This timeincreases with distance beyond rc as r5/2 [56].Electrons released by Penning ionization initiate an electron impact avalanche.Conditions in the Rydberg gas yield a plasma with a spatial distribution of quasi-stationary ions that falls between two limiting cases. An avalanche that ionizes allRydberg molecules remaining in level n0, as well as all Penning partners distributedover deactivated levels n′, produces a plasma with a completely random distributionof ion-ion distances.In the opposite limit, where none of the neutral Penning partners ionize, theplasma distribution of ion-ion distances contains few members spaced closer thanrc. When the conditions of n0 and density cause the semi-classical orbital radius,n20a0 to approach half the Wigner-Seitz radius, aws, such a loss of Penning partnerssignificantly depletes the leading half of the ion-ion nearest-neighbour distancedistribution function. This produces a highly correlated spatial distribution, whichwe term a Penning lattice.The degree to which the system achieves such a state of spatial correlation hasimportant consequences for ultracold plasma physics in this density range. Ne-glecting the effects of electron screening, the ion density (1012 cm−3) and initialtemperature (< 500 mK), afforded by plasma formation in our molecular beam,yields a Γi greater than 50, substantially in the regime of liquid-like behavior.However, the Coulomb repulsion in a plasma of randomly distributed ions withΓi > 1 represents a significant store of potential energy. In a random plasma, dis-45order induced heating to the correlation temperature inevitably limits Γi to a valueof 2 or less [36, 47].Spatial correlation of the ions dramatically reduces ion-ion electrostatic corre-lation heating. To illustrate this, we present a simple model for spatial distributionsin the plasma produced by Penning ionization and avalanche in a molecular Ryd-berg gas as a function of n0 and density. In this model, Penning ionization occursinstantaneously, and avalanche follows on a ns to sub-ns timescale.Over time intervals as short as this, supersonically cooled ions and Rydbergmolecules move a negligible fraction of the Wigner-Seitz radius. Thus, we canreasonably model the spatial consequences of Penning ionization and unimoleculardissociation by assuming a frozen Rydberg gas composed of excited moleculespositioned randomly in a box with sides that are much longer than the Wigner-Seitz radius for our density. A typical simulation starts with N = 106 Rydbergmolecules.We consider the coordinates of each molecule, measuring the distance fromeach of its neighbours to identify pairs with nearest-neighbour spacings that fallwithin rc for Penning ionization. At higher values of n0, Rydberg molecules in-creasingly find multiple partners within r< rc. In such cases, we choose interactingpairs at random.For each selected pair, one molecule instantaneously ionizes, and the otherrelaxes to join a distribution of deactivated Penning partners in states with lowerprincipal quantum numbers, n′. The probability of a given n′ falls from a maximumat the highest accessible level, n0/√2, as n′5 [56]. The calculation assigns a valueof n′ to each primary Penning partner by sampling from this distribution.At higher values of n0, deactivated molecules can undergo secondary Penninginteractions with n0 molecules within the original rc sphere. The model assumesthat these also occur instantaneously whenever this secondary pair falls within thesmaller critical radius figured for the combination of n0 and a principal quantumnumber sampled from the distribution of n′. This process ionizes the n0 moleculeand further deactivates the n′ energy donor.The distribution of Rydberg molecules and free electrons formed by this Pen-ning cascade undergoes an avalanche of electron-impact excitation and ionization- in competition with dissociation. The time scales for these competing processes46depend on n0, the electron density and the final distribution over n′.Coupled rate equation simulations show that this avalanche of molecular Ry-dberg states to ions builds in time [63]. The time period over which avalancheoccurs for a Rydberg gas density of 1012 cm−3 shortens with increasing n0, from50 ns for a Rydberg gas initial n0 = 30, to 100 ps for n0 = 80. The deactivation ofPenning partners to states of lower principal quantum number retards the climb toionization, and increases the fraction driven down by electron collisions. Resultsderived from our coupled rate equation calculations accord with avalanche onsetrates extrapolated from measurements made in a rubidium atom MOT at a seriesof lower densities by Robert-de-Saint-Vincent, et al. [53]Rydberg states of NO predissociate with a rate that falls with increasing n. Awidely used model developed by Bixon and Jortner [8, 43, 51] explains this vari-ation, and accounts well for long-time plasma decay rates that we have observedexperimentally in our system [61, 63]. Taking predissociation rates as a function ofn from this model, the calculation computes the fraction of excited molecules lostto neutral N + O products, by integrating n-detailed decay rates over the avalancheinduction period appropriate to the selected value of n0.Generally speaking, for low-n0 conditions, where rc is much smaller than aws,Penning ionization occurs for a very small fraction of molecules in the Rydberggas. For example, in a model Rydberg gas at n0 = 30, fewer than 1 percent ofthe molecules undergo a Penning interaction. All of the neutral Penning partnersdissociate within the 50 ns induction period for avalanche ionization. This depletesthe leading edge of the nearest-neighbour distribution slightly, but otherwise yieldsa plasma with a random distribution of ion-ion distances.At very high principal quantum number, where rc is greater than aws, all Ry-dberg molecules undergo a Penning interaction. Avalanche proceeds on a sub-nstimescale, so nearly all Rydberg molecules ionize. Any dissociation specific toPenning partners affects molecules with interaction distances distributed over theentire range of r, and thus causes no change in the final ion-ion distributionBetween these limits, a significant fraction of molecules Penning ionize and,on the avalanche timescale, a significant fraction of these dissociate. Figure 4.2shows simulation results describing final distributions of ion-ion distances for ini-tial Rydberg gas densities of 1012 cm−3 with initial principal quantum numbers47from n0 = 30 to 80.0 2 4300 1350 0.5 1400 0.5 1population density (arb.) 450 0.5 1500 0.5 1550 0.5 1600 0.5 1650 0.5 1700 0.5 1750 0.5 180Ion−Ion nearest neighbour distance (µm)Figure 4.2: Distributions of ion-ion nearest neighbours following Penningionization and electron-impact avalanche in a predissociating molecu-lar Rydberg gas of initial principal quantum number, n0, from 30 to 80.Dashed lines mark corresponding values of aws. Calculated by countingion distances after relaxation to plasma in 106-particle stochastic sim-ulations. Integrated areas proportional to populations surviving neutraldissociation.For n= 30, we see that very little Penning ionization occurs, while dissociationof initial Rydberg molecules yields a plasma with only 3% of the starting density ofthe Rydberg gas. The resulting ion-ion distances thus conform with a low-densityErlang distribution. Dissociation similarly reduces the overall density of the plasmaformed by a Rydberg gas with n= 35. For initial principal quantum numbers fromn = 40 to 60, deactivated Penning partners dissociate disproportionately, and this48produces a striking deficiency of smaller nearest-neighbour distances. Above n0 =60, the increased range of Penning ionization combines with slower dissociationand faster avalanche to produce a plasma with random ion-ion nearest-neighbourdistances.Following avalanche ionization, we gauge the effect of spatial correlation bycalculating the average ion-ion pairwise electrostatic repulsion energy as,∑E(r)/2N,where E(r) = e2/4piε0rIn a perfect lattice of ions, all distances, r, equal aws, and this unscreened av-erage repulsion energy is Eaws = e2/4piε0aws. For a random ionized gas under thesame conditions, we compute the average repulsion energy, as the integral of E(r)over Eq. (4.1) for k = 1. The result, Erandom, equals the pairwise repulsion en-ergy of the perfect lattice multiplied by the gamma function, Γ(2/3). Figure 4.3compares these limits to the average ion-ion repulsion energy, calculated stochas-tically for the Penning lattice, and plotted as a fraction of the difference betweenthe repulsion energy of a random ion gas and that of a perfect lattice.Inspection of Figure 4.3 shows that the distribution of ion-ion distances shownin Figure 4.2 for n0 = 55 reduces the average ion-ion repulsion energy by anamount equal to more than 80 percent of the potential energy removed in a re-arrangement from the random gas to a perfect lattice. We take this quantity as arelative measure of the ion spatial correlation immediately after avalanche.This plasma evolves on the timescale of the ion plasma frequency to approach alocal equilibrium in which the relaxation of ion-ion repulsive forces drives disorder-induced heating. Here, a state variable, Uii, describes the stabilization affordedby the electron background in a system of locally repelling ions. Charbrier andPotekhin (CP) have developed an analytic expression for the correlation energy ofsuch an electron-screened ionic fluid that extends to Γ of 100 or more [11].Pohl, Pattard and Rost (PPR) show that the thermal energy added by the re-laxation of ions from an initially random spatial distribution gives rise to a localequilibrium in which little correlation energy remains [47]. A perfectly orderedion gas has no such initial displacement from local equilibrium. Between theselimits, we consider the effect of the spatial distribution of ions on correlation en-ergy, by defining an effective initial Uii(0) = α Ueqii (0), where, α = (Erandom−EPenning)/(Erandom−Eaws), andUeqii (0) is a reference correlation energy simply cal-4930 40 50 60 70 80−1−0.8−0.6−0.4−0.20Principal Quantum NumberProportional stabilizationFigure 4.3: Reduction of average ion-ion repulsion energy in Penning lat-tices formed by Rydberg gases of initial density, 1012 cm−3 and princi-pal quantum number from 30 to 80, plotted as a fraction of the pairwiserepulsion energy difference between a gas of random ions and a perfectlattice. Data points at each value of n0 taken from the results of foursimulations.culated from the CP expression assuming local equilibrium at the ion temperatureand density of the initially formed (unrelaxed) plasma.To predict terminal expansion velocities as a function of n0, we apply the hy-drodynamical model developed by PPR [47], using an electron temperature of 9 K,determined experimentally from a measured rate of expansion in the uncorrelatedregime of n0. This yields a result shown by grey points plotted in Figure 4.1.The model presented here presumes a quasi-neutral plasma that undergoes anormal ambipolar expansion driven by the thermal pressure of an electron gas.We regard this plasma as unconventional only by virtue of the spatial correlationcreated in the distribution of ion-ion nearest-neighbour distances by Penning ion-ization in Rydberg gases of certain selected initial principal quantum numbers and50density. At the maximum Penning lattice effect, this spatial correlation reduces theelectrostatic potential energy of the ions to a value near that of ions in a perfectlattice. This stabilization dissipates with falling density, giving rise to the force weobserve opposing expansion.Thus, much like the evolution suggested for an ion distribution that has beenspatially ordered by dipole blockade [6, 53], spatial correlation produced by theshort-range interaction of Rydberg states that forms a Penning lattice enables theultracold plasma to sustain a low ion temperature and reveal the effects of Γi on thehydrodynamics of expansion.51Chapter 5Selective Field Ionization Studyof Rydberg Gas and Plasma5.1 IntroductionUltracold plasmas offer a means to observe strong coupling under laboratory condi-tions. Theory predicts that interesting and important phenomena such as modifiedtransport, and suppressed three body recombination happen in this regime. Thesesystems have been studied extensively in magneto-optical traps [27] and supersonicmolecular beams [40]. The implications of studies of these laboratory systems canbe extended to the field of astrophysics as analogues for star formation and plane-tary evolution [25] using classical Coulomb scaling [41].The first experiments in this field studied the plasma formed by the Rydberg ex-citation [58] and the photoionization [29] of metastable xenon atoms in a magneto-optical trap. The density of these systems puts them into a regime of moderate ioncoupling with a coupling factor close to 1, or Γi = e2/(4piε0awskBT )≈ 1.A Rydberg gas in a supersonic beam can evolve to form a plasma [39, 40].This experimental condition can create much higher density. Experimental resultsfrom such systems show evidence of stronger coupling [60, 63]. Experiments inbeams are particularly versatile in that supersonic expansion can be configured toentrain many atomic and molecular species. Hydrodynamic equations describe theproperties of supersonic beams such as their spatial dimensions, density, and tem-52perature. Therefore, these systems facilitate systematic measurements of velocitydistributions and decay dynamics for many different initial quantum states of aRydberg gas [65].The ultracold plasma of xenon that forms in a molecular beam has been studiedusing delayed pulsed-field-ionization [24]. Here we use the similar methods tostudy the ultracold plasma of nitric oxide molecule. A study of the dynamics offield ionization of a Rydberg gas of NO as a function of delay time and density fora broad range of initial quantum states serves to characterize the n-dependence anddensity dependence of Rydberg-Rydberg collisional energy redistribution, Penningionization, and electron impact avalanche to plasma.We find that the evolution dynamics of NO Rydberg gas observed experimen-tally agree with a coupled-rate-equation simulation picture, if we account for thedensity distribution of excited NO molecules. This variation in the density of Ry-dberg molecules, a common feature of laser photo excitation, causes the electronimpact avalanche to propagate from the dense core of the Rydberg gas outward toits periphery. This has the effect of slowing the electron impact avalanche, chang-ing the composition of the system and its quasi-equilibrium. And, it retards theambipolar expansion of the ions that form. The present work and its demonstratedrelevance to the body of NO plasma experiments provide a useful point of referencein the interpretation of these systems.5.1.1 Selective field ionization (SFI) spectroscopyThis chapter explores the effect of an external electric field on a nitric oxide Ry-dberg gas. Selective Field Ionization (SFI) spectroscopy is used as a method formeasuring the composition of a Rydberg gas/plasma mixture by using a time de-pendent electric field to ionize it. As a result, it produces a time dependent electronsignal, that after converting to a field domain signal, can give us insights about thenature of the system.A Rydberg gas in presence of an external field ionizes at a threshold that de-pends on its principal quantum number. The Rydberg gas can evolve to a plasmaunder field free conditions. By using SFI, we can probe the real time evolution ofa Rydberg to a plasma. The resulting plasma can interact with an external electric53field and produce an electron signal that depends on its properties: the number ofparticles, density, and temperature.The set of experiments in this chapter uses the SFI method in a number of ways,each targeting a different property of the evolving Rydberg gas.The first method examines the principal quantum number dependence of theproperties of Rydberg gas/ plasma. Scanning ω2, the frequency of the laser usedto excite the NO molecules to a high Rydberg state, a ramped electric field appliedimmediately after photo-excitation yields in a high resolution spectrum of NO inwhich lines converging to different rotational levels of NO separate into groups.This occurs because the position of NO lines in the photon-frequency domain,depend on the energy difference between the first excited state and the Rydbergstate, while, the position of a NO peak in the field domain depends on the bindingenergy of this Rydberg state.We then apply the SFI method to investigate the composition-density phasediagram of the Rydberg gas. As mentioned in previous chapters, a sequence oftwo pulsed lasers excite NO molecules to a selected Rydberg state. The time de-lay between the two laser pulses controls the surviving population of excited NOmolecules, and consequently the density of the Rydberg gas. We vary this time de-lay to produce Rydberg gases of different densities. A pulsed electric field appliedafter excitation interrogates the state of these Rydberg gases with correspondingdensities.5.1.2 Evolution of Rydberg gas to plasmaRydberg molecules in a dense gas can interact by dipole-dipole coupling to producean ion-electron pair and a deactivated state. The pairs of Rydberg states that arecloser than a critical distance of an orbital diameter of NO, interact instantly ona nanosecond timescale. The fraction of such close pairs depends on the averagespacing of particles which varies inversely with density. The electrons producedin this process drive the ionizations of the remaining Rydberg molecules that leadto avalanche ionization [46]. The rate of this process scales with the energy of thefree electrons, and the inverse of the binding energy of the Rydberg state. The rateof this second order process also scales with the density of the electrons and the54Rydberg states.The conversion of Rydberg gas to plasma advances by the two processes men-tioned above. Electron-Rydberg impact also causes Rydberg populations to relaxto other energy states. Mansbach and Keck calculated electron-Rydberg inelasticcollision cross sections in 1967 [37]. Pohl and coworkers improved these crosssections for small energy transfers [48]. As Rydberg population flows to otherprincipal quantum number, response to an external electric field changes accord-ingly.Electron impact ionization is balanced by three-body recombination. In thisprocess one electron carries the excess energy of recombined electron-ion pair.This process involves three particles and its rate scales with the cube of electrondensity (owing to quasi-neutrality electron and ion densities are equal.)All these processes shape the dynamics of Rydberg gas evolution to a plasma.Given the density of a Rydberg gas, coupled-rate simulations provide a means topredict the phase of the evolution and the composition of the gas at any time. SFImethod enables us to determine the composition of the system and observe densitydependent properties of the evolving Rydberg gas.Properties of a plasma, its rate of expansion, binding energy of electrons tothe plasma volume, and its collective behaviour strongly depend on charge density.The SFI method provides a means to probe the density of the plasma, and measurethe binding energy of electron to its volume. For instance, an electron signal pro-duced by SFI, can supply information about the energy spectrum of an expandingplasma. During the expansion of a plasma, the density drops. The electrons movefurther from positive ions, such that they can be extracted by a smaller externalelectric field.5.1.3 Factors contributing to the density of NO Rydberg gasA Rydberg gas is created from a molecular beam of ground state NO by inter-cepting two laser beams. The density of the molecular beam is determined by thediameter of the nozzle and the skimmer and the back pressure of the gas. With a10% mixture of NO in helium, the molecular beam has a ground state density inthe order of 1013 cm−3.55The first UV laser excites the NO molecules to the 2Σ+1/2 excited state. Thecross section for this transition is very large, such that a pulse laser with only a fewmicro Joules of energy can saturate the transition. Before reaching saturation, thefirst transition produces an excited state density of ρexρex = σρgsIdt (5.1)where σ is the photon-atom dipole transition cross section, ρgs is the ground stateNO density, and Idt is the number of photons in a single pulse of the laser. Asimilar equation holds for the second excitation which yields a Rydberg density ofρryd = σ2ρexI2dt (5.2)The estimated density of the Rydberg gas when both excitations are saturated isabout 2×1012 cm−3 [40].If the second laser is fired with a delay, t, after the first excitation, only thesurviving fraction of the excited state NO populates the Rydberg states.ρryd = ρmaxe−t/τ (5.3)In the equation above τ = 209ns [9], is the lifetime of NO and t is the time delaybetween the two lasers.5.2 Field ionization of Rydberg statesIn the presence of an electric field, the energy landscape of a hydrogenic atom ormolecule changes in the following wayE =−kz+ eFz (5.4)where k is the Coulomb factor, z is distance from origin, e is the electric charge,and F is the magnitude of the applied electric field. The potential energy curve tiltsin presence of a static field. Consequently, the potential energy for electrons willhave a saddle point. The energy of the saddle point equals the binding energy of a56Rydberg state for a field magnitude:F =116n4(5.5)This is called classical or adiabatic ionization threshold. This simple formula ig-nores the energy gained by the electron in presence of an external field and treatsthe system semi-classically. As it happens the two approximations cancel out incertain conditions and adiabatic ionization can be observed for, i.e., highly excitedstates of the sodium atom [20].In reality, the electron gains or loses energy when a field is applied dependingon the direction of the motion of the electron. Quantum mechanically, energylevels of various |ml| split and form a Stark manifold. The Stark shift along withcontinuum lowering due to the external field results in diabatic ionization thresholdof [19]F =19n4(5.6)For a molecule with a finite-sized ionic core, neither zero-field levels nor finitefield levels are spherical. However, in the limit of higher principal quantum num-bers, we can assume that zero field levels are spherical with well-defined quantumnumbers nlm.Nevertheless, in presence of an external field, parabolic quantum numbersnn1n2m describe a quantum level efficiently. In the regime of electric field withF>1/3n5, the Stark manifolds of different n mix which form avoided crossings.An energy level optically pumped initially, follows the adiabatic path and stays inthe same energy level with a constant parabolic quantum number n1. However, ifthe external electric field grows too fast, the energy level cannot follow the adia-batic path and will make a transition through many or all avoided crossing. Thisresults in the observation of field ionization at the higher diabatic threshold fornon-Hydrogenic atoms or molecules [19].5.3 Experiment and methodA pulsed supersonic beam releases a 400µs jet of NO from a stagnation pressureof 5 bar. The beam passes through a 1 mm skimmer positioned 3 cm from the57nozzle. A simple hydrodynamic model predicts that the molecular beam of NOhas a longitudinal temperature of 0.5 K and a divergence velocity of 5 m/s. Theback pressure and nozzle diameter of our beam predicts an on-axis ground stateNO density of 1013 cm−3 in the excitation region [40].A 500 l/s turbo pump maintains an operating pressure of 10−4 mbar in thesource chamber. A second turbo pump holds the excitation and detection chamberto a pressure of 10−8 mbar. In the excitation region, a collimated beam of a Sirahdye laser pumped with a Nd:YAG laser, excites NO from ground state to the excitedstate with a UV light wavelength of 225 nm. A second laser beam excites NOto a Rydberg level with principal quantum number n= 30-80 (f), to a series thatconverges to N+ = 2.We obtain selective-field ionization spectra by connecting the output of a Behlkehigh-voltage switch to one of the two plates that isolates the travel path of theplasma. The parity of the field is chosen to accelerate electrons to the detector.Using a 10 kΩ resistor we slow the skew rate of the electric field to about 1(V/ns).The pulsed electric field extracts electrons and accelerates them toward an MCP de-tector. This forward bias potential sweeps electrons off of the plasma and ionizesRydberg molecules. We initiate this pulse immediately or some time after photo-selection, and it ends before the illuminated volume reaches the opposing grid (asillustrated in figure 5.1).We focus on an experiment in which we probe the composition of the systemby an external electric field at a fixed time. The frequency of the second laser (ω2)is set to excite the NO molecule to a Rydberg level n= 49 f (N+ = 2). The energyof the first and second UV beams are 4µJ and 5mJ per pulse, respectively. Wecontrol the density by delaying the second UV laser which allows a fraction of theexcited NO A state molecules to decay. The relative density falls with delay time,according to,Ln(ρrydρmax)=−tτ(5.7)where τ = 209ns [9] is the lifetime of NO.If we saturate both excitations, the resulting density of the NO Rydberg gas is2×1012 cm−3. Using a 4µJ pulse for the first UV laser, we can reach a density of1012 cm−3.58G1 G2 G3 MCPMolecular Beam12Figure 5.1: Experimental apparatus. A skimmed molecular beam passesthrough grid, G1, where it intersects counter-propagating laser beamsindicated as ω1 and ω2. Before the excitation volume transits the planedefined by G2, a ramped electric field is applied to G1 with a forwardbias. An MCP detector situated behind G3 collects the signal of extractedplasma and Rydberg electrons.The pulsed field is applied 20 ns after the photo-selection. The field risesvery quickly and extracts a large number of electrons, such that all the processesthat depend on electron impact freeze upon application of the field. We thereforeassume the rising field gives us a measure of the composition of the system at thestarting time of the pulse.5.4 Results5.4.1 Rydberg spectra with selective field ionizationWe tune the first UV laser to excite NO into the excited 2Σ1/2 K = 0 state, followedby a second UV laser that excites the molecule to a level with high quantum numberin the range of n= 30−80. The energy of the first and second UV beams are 2µJ59and 5mJ per pulse, respectively. The excited volume is then exposed to a fieldramp that rises to 550 V/cm. Varying the second laser frequency produces anexcitation spectrum of high Rydberg states. At every frequency, the electric pulserises to yield a field dependent trace. Stacking these traces together forms a two-dimensional spectrum (Fig. 5.2). A second spectrum is obtained by applying thepulsed field 200 ns after laser excitation.Converting time in the ramp to instantaneous voltage for a known plate spacingyields the traces in figure 5.2 and 5.3. We obtain the y-axis of the figures by vacuumcorrecting the wavelength.Figure 5.2: Spectrum of NO Rydberg states obtained by applying a time-varying electric field right after two-photon excitation. Each horizontalline is a field ionization trace and contains two or more peaks appearingat different fields. The features appear at higher field when the wave-length of the second photon is increased. The series of peaks in theupper left corner belongs to an excited rotational level, K = 4, of 2Σ+1/2.60Figure 5.3: Spectrum of NO Rydberg states obtained by applying a time-varying electric field 200 ns after two-photon excitation. Each hori-zontal line is a field ionization trace and contains two or more peaksappearing at different fields. The features appear at higher field whenthe wavelength of the second photon is increase. The series of peaksin the upper left corner are the result of overlap between two transitionlines of ground state NO.Figure 5.2 shows that at longer wavelengths of the second photon, which ex-cites NO molecules to a lower principal quantum number, the position of the peaksmoves to higher fields and the separation grows. The position and the spacing ofeach peak on the y-axis depends on the energy level and the energy spacing ofRydberg levels.The set of stronger peaks in the bottom-right corner of the figure 5.2, belongsto Rydberg series that is originated from the intermediate excited state, 2Σ+1/2, inthe ground state rotation with K = 0. The group of peaks in the upper-left corner,61belong an excited rotational level, K = 4, of 2Σ+1/2.A closer look at figures 5.2 and 5.3 reveals that some of the distinct peaks inthe first figure, smears to softer, broader peaks. Furthermore, for the traces at thebottom of the first figure, lower wavelengths, the maximum intensity is located atfields of 20 V/cm and higher, while the same traces in the second figure have amaximum at zero field.If we take a closer look at some of these traces, e.g. figure 5.4 for n = 59, wesee that the Rydberg lines indicating field ionization at early times, disappear after200 ns and form an exponentially decaying electron signal with a maximum at zerofield.Figure 5.4: Field ionization traces at a fixed density at time delays t = 0 and200 ns for n=59. The blue solid trace is taken at t=0 and the dashedorange traces is taken at 200 ns.The same set of traces at t = 0 and 200 ns for a Rydberg gas with n = 43 (figure5.5), shows a different evolution. The blue trace shows a signature of Rydbergmolecule field ionization at fields above 100 V/cm. Under the same conditions asn = 59, after 200 ns, there is no conversion of this high field signal to low field.However, there is a significant drop of Rydberg signal amplitude and broadeningof the two peaks. Also the maximum of the dashed orange trace has moved tohigher fields.62Figure 5.5: Field ionization traces at a fixed density at time delays t = 0 and200 ns for n=43. The blue solid trace is taken at t=0 and the dashedorange traces is taken at 200 ns.5.4.2 Density variationWe focus on a set of experiments that set the frequency of the second laser (ω2) toexcite the NO molecule to a Rydberg level n = 49 f (N+ = 2). The energy of thefirst and second UV beams are 4µJ and 5mJ per pulse, respectively.We vary the delay of the second laser pulse in order to control the density ofthe final Rydberg gas. During the delay of the second laser, the NO moleculesexcited to the A 2Σ21/2 state decay, reducing this population in a well defined way.The density at zero delay can be calculated using the properties of the molecularbeam and the cross sections of the two photo-excitation processes. Using a 4 µJω1 energy pulse, this density approximately equals to 1012 cm−3 [40].63Figure 5.6: Field ionization traces at various densities determined by the timedelay between the two exciting laser beams. The y axis is the strengthof the signal. The base line of each signal is offset by the logarithm ofthe density relative to the maximum (zero delay between the two lasers).From top to bottom, the density becomes smaller. The traces have lowfield signal at higher densities. At lower densities, each curve has twopeaks at high fields.The ramped electric field is applied 20 ns after the second laser excitation inorder to probe the excited volume. Figure 5.6 shows a set of field ionization tracestaken 20 ns after the formation of the Rydberg gas. Each trace is offset by thelogarithm of its relative density on the y-axis. The waveform at the top with thehighest density, 1012 cm−3, shows broad, strong peak that starts at low field. Whenthe second laser pulse is delayed 600 ns, we obtain the bottom trace with the peakdensity of 5×1010 cm−3 (with the baseline at Ln(ρ/ρ0) = −3). This trace showstwo peaks at 90 and 140 V/cm. The clear distinction between the group of traces atthe bottom and the top of the plot disappear at around 2-3×1011 cm−3 (traces withthe baseline at Ln(ρ/ρ0) =−1).The traces in figure 5.6 are normalized. The raw traces for few densities areshown in figure 5.7. The total signal grows in amplitude as the density grows. No-tice how the electron signal at the highest density starts from low field and extendsto higher fields. The signal for this density at high field could originate from ion-ization of the Rydberg molecules, or simply the extension of the decaying signal64that peaks at around 40 V/cm. Nonetheless, a distinct Rydberg signal cannot beseen.Figure 5.7: Field ionization traces at various densities determined by the timedelay between the two exciting laser beams for a Rydberg gas with theprincipal quantum number n = 49. The y axis is the amplitude of thesignal in an arbitrary unit. The solid line is taken at the maximum den-sity (zero delay between the two lasers). The signal is strongest and itextends to higher fields. The dashed line has a density 30% of the max-imum. This trace has features of the strong signal and two peaks at 90and 140 V/cm. The dot-dashed curve has a density 5% of the maximum,and only shows small low field and slightly stronger signal at high field.We quantify this density dependence of the composition of the Rydberg gas byintegrating the area under the curves in figure 5.6 to obtain a measure of number ofelectrons and Rydberg molecules (figure 5.8). The integration is done in the timedomain, where the spacing is uniform, however the limits of integration are set bythe field ionization threshold of a Rydberg state with n = 49. We define electronsignal as the area under the curve between 0 and 75 V/cm. The signal in this rangecan be caused by plasma electrons and Rydberg molecules with n>49. The areaunder the curve above 75 V/cm is defined as the number of Rydberg molecules.Rydberg molecules with principal quantum number equal or smaller than 49, anddeeply bound electrons in a plasma can contribute to this signal.65The black dots in figure 5.8 show that the number of Rydberg molecules (thebottom plot) grow with density linearly at densities below 2.5×1011cm−3. Abovethis density, the rate of growth reduces significantly. This signal, defined as Ry-dberg states of n≤49, is contaminated by the long tail of electron signal at highdensities that extends to higher fields.The electron signal is very insignificant for densities below 2.5×1011cm−3. Athigher densities, electron signal grows exponentially. A transition in the behaviourof the system happens at densities about 2-3×1011 cm−3. This is the same point atwhich the qualitative behaviour of these SFI traces changed in figure 5.6.66Density #10111 2 3 4 5 6 7 8 9 10Number of Electorns#107012345ExperimentalShell modelUniform modelDensity #10111 2 3 4 5 6 7 8 9 10Number of Rydberg molecules #1070510Figure 5.8: Number of electrons (top graph) and surviving Rydberg states(bottom graph) after 20 ns as a function of density determined by thedelay of the two laser pulses. Black dots are experimental points ob-tained by integrating the area under the curves in figure 5.6. In the topplot, the black dots are the integral of the curves from zero field to 75V/cm, and in the bottom plot, from 75 V/cm to 250 V/cm. The greendashed line represent the coupled rate equation model for uniform dis-tribution Rydberg gases. Solid red line is calculated by a similar modelwith a Gaussian density distribution.5.5 DiscussionAn external electric field alters the potential energy surface seen by electrons trappedin a plasma volume or bound to atoms. The field at which an electron signal ap-pears reveals vital information about its environment. Rydberg states diabaticallyionize in presence of an electric field larger than 1/(9n4). Therefore field ionization67can be used to detect Rydberg states. For a high density gas of Rydberg states, thedisappearance of such a threshold signal and the rise of low-field, quantum num-ber independent electron waveform indicates a production of free surface electronswhich signifies the evolution of the gas to a plasma.5.5.1 Dependence of Rydberg gas evolution on principal quantumnumberAs explained in section 5.2, plasma and Rydberg molecules have a signature ap-pearance in a SFI trace. For instance, a SFI trace that moves to higher voltages as theUV laser is tuned to lower frequency (lower principal quantum number) signifiesthe field ionization of a Rydberg gas. While a signal that appears at low voltagesand declines exponentially with field and is independent of laser tuning signifies aplasma.Figure 5.2 shows the quantum number dependence of the field ionization sig-nal. As the wavelength of the second photon increases (lower quantum numbers),the required field for ionization increases. This field dependence fits well with dia-batic field ionization of Rydberg molecules, as the binding energy scales with fieldaccording toE(cm−1)= 4.12√F (V/cm) (5.8)where the factor 4.12 is associated with diabatic ionization, E is the binding energyin cm−1 and F is the field in V/cm. Appearance of multiple peaks at one photonfrequency signifies multiple values of the binding energy with respect to differentionization limits set by the rotational energy of the NO+ core. Each group of peaksis separated from the next by a constant energy equal to the separation of rotationallevels in the NO+ core. The energy of each peak can be formulated based on theionization field according toE = I.P.(N+ = 0)−2×107/λ +2BN+ (N++1) (5.9)In the equation above the binding energy of a Rydberg state is related to the wave-length of the photon, the ionization potential (I.P.) of NO in the ground core ratio-68nal level, and the quantum number of the rotational level of the NO core. The thirdterm in the right hand side is responsible for the appearance of multiple peaks inthe field domain at the same frequency. (Note that the separation of peaks grows inthe field domain, because the field scales with the square of the binding energy.)A Rydberg gas with a low principal quantum number, e.g. n = 43, evolvesvery slowly, such that after 200 ns there is little or no plasma formation under theconditions of this experiment (Fig. 5.5). However, under the same condition ofω1 laser energy, which determines the density, a Rydberg gas with n = 59 evolvesto a plasma (Fig. 5.4). This picture matches the coupled-rate equation model [48]and the ionization due Rydberg-Rydberg interaction [55]. For a Rydberg state witha principal quantum number n, the radius of the molecules, and consequently, thecritical radius of Rydberg-Rydberg interaction grows as square of n. Therefore,at the same density, the fraction of pairs of molecules within the critical distancegrows with the principal quantum number. This Penning fraction determines theinitial electron density and the rate of electron-ionization avalanche. As a result,for example, the Rydberg gas with n=59 evolves to a plasma faster than the onewith n = 43.The fast evolution of the Rydberg gas with n = 59, has a secondary effect on itsproperties. The total area under the curve in figure 5.4 has not changed significantlyafter 200 ns. The same is not true about n = 43. The amplitude of the signal hasdropped by a factor of two. The origin of this drop is the dissociative decay ofRydberg molecules to neutral atoms. The rate of this decays grows inversely withthe radius of the molecule [8]. Furthermore, fast conversion of the Rydberg gaswith n = 59 to plasma, deprives these molecules of this dissociative channel. Thelifetime of the NO with n = 59 for dissociative channel is 265 ns [8], while thetime of avalanche is less than 28 ns for densities above 1011 cm−3 according tocoupled-rate equation model. Therefore, the rate of avalanche is much faster thanthe rate of decay. The opposite is true for a Rydberg gas of n = 43, for which thedissociation life time is 62 ns and the electron-ionization avalanche time is 113 nsfor the same density.The broadening and the shift of field ionization signal to higher fields in figure5.5 for n = 43, is due to the l-mixing of the highly excited molecules. The l-mixing creates high angular momentum states that ionize at a slightly higher field.69L-mixing that occurs on the timescale of tens of microseconds for a low densityxenon Rydberg gas [24], occurs on a nanosecond time scale for high density NORydberg gas.5.5.2 Density dependence of electron avalancheThe rate of electron avalanche ionization strongly depends on the density of theRydberg gas. At a higher density, a larger fraction of molecules ionize due toRydberg-Rydberg interactions [55]. The resulting electrons from this Penning ion-ization, seeds an electron-Rydberg reaction that scales with the density of bothspecies. Therefore, at higher densities, a Rydberg gas of n = 49 evolves to aplasma faster. At lower densities, a smaller fraction of highly excited moleculesinteract. The result is a low electron density that takes much longer to initiateelectron-ionization avalanche.We can use a coupled rate equation model (as described in details in appendixA) to find Rydberg-gas/plasma compositions for a range of densities at a giventime. We use a coupled-rate equation model that accounts for the inelastic collisionof electrons, ions, and Rydberg states first developed by N. Saquet and J. Morrison[63] using cross sections calculated by Pohl and co-workers [48].The previous models treat the excited volume as a uniform entity. In this workwe have built on this model by introducing a more realistic system in which par-ticles are distributed by a Gaussian profile, mimicking the appearance of the laserbeams and molecular beam used. The cylindrical volume is sliced into disks, eachwith a uniform density determined by the profileρ = ρ0e−x2/2σ2 (5.10)where ρ0 is the peak density, and σ = 1213 µm is the standard width along the xaxis, and x is the distance of the centre of the disk from the middle of the cylinder.The radius of the cylinder is determined by the width of the lasers, σy = 200µm.The peak density is given as an input parameter.Model calculations with various peak densities are run and compared with ex-periment. The solid red curve in figure 5.8 is the results of shell model calculations.Each simulation starts with a pure Rydberg gas of n = 49. The cylindrical volume70is divided into 20 disks with a uniform thickness. We assume a uniform densitywithin each disk.The coupled rate equation model is run in each disk independently. Using theinstantaneous temperature of the electrons, we calculate the rate of electron-impactionization, electron-Rydberg inelastic collision, and three-body recombination. Weassume that a finite number of Rydberg states around n = 49 are accessible. Thelower limit on the Rydberg manifold is n = 10 and the upper limit is n= 89. Thelower limit is defined based on the work of Morrison and coworkers [41] to ensureconsistency with the results of molecular dynamics simulations. The upper limit issafely above the limit at which Rydberg molecules are thermally unstable.We include dissociative channels for NO, including the electron-ion dissocia-tive recombination and the pre-dissociation of Rydberg molecules.The temperature is calculated from the total energy of the electrons in all disksat thermal equilibrium. For densities in the range of 1012 to 109 cm−3, we integratethe set of equations for 20 ns. Then we construct a calculated electron signal thatis the number of electrons plues the number of Rydberg molecules with n>49.Similarly we define Rydberg molecules as the number of molecules with n≤49.The result of calculations (Fig. 5.8) shows that the population of survivingRydberg molecules scales linearly with the initial density as long as the densityis lower than 8×1010 cm−3. After that, the Rydberg population grows slowerwith density until it reaches a plateau. At higher densities there are more Ryd-berg molecules to begin with, but the rate of electron-impact ionization avalancheis higher as well. The shell model predicts that in a Gaussian distribution of parti-cles, no matter how high the peak density, there is a low density region of the gasvolume in which electron-avalanche is insignificant. The population of Rydbergmolecules in such regions contributes to the plateau in the bottom plot.The dashed green curve (Fig. 5.8) is computed using an ordinary coupled-rate equation model with a uniform density. The parameters of the coupled-rateequation model are similar to the ones used in the shell model, except that the setof equations is integrated for a single volume of the initial density.The uniform coupled-rate equation model does not predict a balance in therate of avalanche. As the density grows, the rate of avalanche increases, such thata smaller number of Rydberg molecules survive. When particles are distributed71uniformly, with the increase of the initial density, the rate of avalanche can growout of proportion, such that only small fraction of initial Rydberg molecules cansurvive.The shell model predicts that the rate of avalanche grows with density. Atdensities above 8×1010 cm−3, the number of produced electrons grows rapidlywith density. The rate of ionization avalanche at densities much lower than 8×1010cm−3 is slow enough that after 20 ns there is no considerable amount of plasmaformation. This behaviour is very similar to what we observe experimentally withthe exception that the point of transition in the behaviour happens at a differentdensity, 2-3×1011cm−3.Shell model calculations predict a very similar behaviour to that observed inthe experiment. However, its predictions are off in terms of initial peak density bya factor of 0.27. We can calculate the peak density of Rydberg gases, producedby delaying the second laser, precisely relative to the maximum. We can calculatethe maximum density using the properties of the molecular beam and cross sec-tions of the two photo-selection processes. We find that the maximum density isabout 1012cm−3. Consequently the density of the transition falls in the range of2-3×1011cm−3.The disagreement between the experiment and the theoretical calculations canbe a result of number of reasons.First, our estimate of density can be inaccurate. If we correct the calculatedmaximum density by a factor of 0.27, the point of transition for experimental datain figure 5.8 will overlay with the calculated values from the shell model. However,the properties of a molecular beam are well defined and our knowledge of the crosssection of the two photo-selection processes is very accurate. Therefore, we willseek to understand our models based on the relatively accurate experimental data,as oppose to correct for the density.The model calculations are constructed on a set of parameters and assumptions.In the shell model, we assume a uniform density throughout the cross section of thecylinder. However, a laser beam with a Gaussian profile will create a Rydberg gaswith the same distribution. Electron evaporation is known to happen in ultracoldplasmas, which will influence the dynamics of the system. We ignore this processin our simulation.72The coupled-rate equation model, whether implemented in the form of ordi-nary coupled-rate equation model, or in the form of shell model, is bound to failfor a system in the regime of strong Coulomb coupling. The underlying assump-tion in the calculation of the cross sections for electron-Rydberg impact ionizationand inelastic n-changing collisions, is the predominance of long-range small-anglecollisions [37]. At high densities, the mean free path of electrons is very small.Before an electron can travel the length of average spacing, it is deflected by theragged potential energy surface of the surrounding ions. This many-body effectcan influence the rate of electron-impact ionization.Furthermore, the interaction of two Rydberg states that initiates the plasmaformation process is also described in terms of two-body interaction of atoms [55].However, in a high density gas, this process must be treated as a many-body pro-cess. Also, the production of ions and electrons as a result of Penning ionizationmay have an influence on the dynamics of interacting Rydberg molecules. Thiseffect has not been studied.The coupled-rate equation model predicts fast conversion of Rydberg moleculesof n=49 to states with lower and higher n. Such Rydberg states respond differentlyto an external electric field. For instance, a Rydberg molecule with a lower valueof n=48, has a field ionization threshold at a slightly higher field. From figure 5.6,we see no evidence of formation of lower Rydberg states.5.6 ConclusionWe used a time varying electric field to interrogate the state of matter of a dense ex-cited volume of nitric oxide molecules. Photo-excitation to highly excited Rydbergstates followed by field ionization reveals vital information about the dynamics ofthe illuminated volume.The Rydberg gas of NO evolves on a time scale of 20 nanosecond to forma plasma that is then detected by time varying electric field. If we reduce thedensity by introducing a time delay between the two exciting photons, the resultingRydberg gas does not evolve to a plasma on a time-scale of 200 ns. However, thesehighly excited molecules either decay, or due to dipole coupling, they interact toform Rydberg states of higher angular momentum. A field ionization trace for73such a low density Rydberg gas shows decay of the field ionization signal andbroadening and shift to higher fields as a result of l-mixing.We studied the evolution of a Rydberg gas of NO to a plasma for a range ofdensities using experiment and model calculations. The parameters of the molecu-lar beam and the dipole transition cross section of the laser beam interaction withNO gives us a relatively accurate account of the density of the NO Rydberg gas.We have developed a coupled-rate equation model using the current theoreticalunderstanding of such systems to find that theory predicts faster electron-ionizationavalanche than what we observe experimentally. This may be the result of strongCoulomb coupling in the ultracold plasma. As oppose to an ideal low densityplasma in which long-range small-angle collisions are predominant, in an ultracoldplasma electrons scatter off of a ragged potential energy surface. Our experimentalfindings suggest that a more refined theory is required to more accurately describesuch systems.Due to interference of electron signal with the detection of Rydberg states, wecan draw little conclusion about the lifetime and the count of surviving Rydbergmolecules. However, we see no evidence of fast n-changing electron-Rydberg col-lisions predicted by Mansbach and Keck [37]. It appears that the evolution of theRydberg gas to a plasma happen by electron-Rydberg impact ionization, but non-changing collision occurs.74Chapter 6ConclusionsWe have detailed the formation of an ultracold molecular plasma from a cold Ryd-berg gas of NO. Using field-free experiments and selective field ionization method,we have characterized the properties of our system. Expansion and decay of theplasma has been observed by translation of the detector along the axis of propaga-tion of the plasma, using the moveable grid machine. The result shows very slowexpansion of the ultracold plasma into vacuum indicating low electron temperaturein the order of 5-15 K. This slow expansion suggests higher Coulomb correlationthan plasmas created in MOTs (2).We have developed a kinetic model to simulate the long-time decay of an ultra-cold plasma of electrons and NO+ ions to neutral N and O atoms (2). By analysingthe decay, we obtain a second order rate constant that agrees with dissociative re-combination cross section estimated from scattering theory [64].We have developed a fluid model for expansion of molecular ultracold plasmasthat includes the dissociation channel. It shows that fast decay in the centre of theplasma flattens a the charged particles density distribution. The rate of expansionin the core of the plasma, which is proportional to the density gradient, retards asa result of DR-flattening. However, this process slows the consumption of electronenergy that drives the plasma expansion. This reservoir of electron energy in thecentre, propagates to the outer regions and causes a faster expansion. Observationon these findings requires an imaging system that is capable of probing the chargedensity of the plasma in two or three dimensions (3).75In chapter 4, we have described a set of experiments that hints the existence ofspatial correlation that reduces the rate of expansion. We have developed a kineticmodel to describe this system of correlated ions. We have described a model thatpresumes a quasi-neutral plasma that undergoes an ambipolar expansion drivenby the thermal pressure of an electron gas. We assume this plasma is spatiallycorrelated as a result of Penning ionization that alters the distribution of ion-ionnearest-neighbour distances. This spatial correlation reduces the electrostatic po-tential energy of the ions to a value near that of ions in a perfect lattice. The lowerenergy of the relaxed ion configuration opposes the expansion.Similar to the evolution suggested for ion distribution with spatial order formedby dipole blockade [6, 54], spatial correlation produced by the short-range interac-tion of Rydberg states enables the ultracold plasma to sustain a low ion temperatureand reveal the effect of ion-ion correlation on the hydrodynamics of expansion.Lastly, we have developed a selective-field ionization method to probe the stateof the evolving Rydberg gas to a plasma. The Rydberg gas of NO evolves on a timescale of 20 nanosecond to form a plasma that is then detected by a time varyingelectric field. If we reduce the density by introducing a time delay between thetwo exciting photons, the resulting low density Rydberg gas does not evolve to aplasma on a time-scale of 200 ns. However, these highly excited molecules eitherdecay, or due to dipole coupling, they interact to form Rydberg states of higherangular momentum. A field ionization trace for such a low density Rydberg gasshows decay of the field ionization signal and broadening and shift to higher fieldsas a result of l-mixing (5).We have developed a coupled-rate equation model using the current theoreti-cal understanding of electron-Rydberg dynamics to find that theory predicts fasterelectron-impact ionization avalanche than what we observe experimentally. Thismay be the result of strong Coulomb coupling in the ultracold plasma. As op-pose to an ideal low density plasma in which long-range small-angle collisions arepredominant, in an ultracold plasma electrons scatter off of a ragged potential en-ergy surface. Our experimental findings suggest that a more refined theory may berequired to more accurately describe such systems (5).In Rydberg gases with a high density, the evolution to a plasma is accompaniedby fast n-changing electron-Rydberg collisions. The cross section of this process76suggest that the Rydberg gas forgets its initial principal quantum number in thetime scale of nanoseconds. While the cross section for this process is well es-tablished for large and small energy transfers [37, 48], we do not experimentallyobserve such fast flow of population of Rydberg states to neighbouring states. Amolecular dynamics simulation of Rydberg gas that includes many-body effects ofthe surrounding environment of charge particles may serve to better characterizethe evolution of such systems. Monte Carlo study of electron-ion recombinationperformed by Mansbach and Keck, and works that followed the same basis, ne-glects many body effects.The SFI experiment as described in the previous chapter is at its early stages.During the writing stage many new results have been obtained that requires fur-ther analysis and discussion. We aim to combine this method with millimetre wavespectroscopy of Rydberg states to further investigate the evolution of a Rydberg gasin this novel physical state. A photon with a frequency equal to the energy differ-ence of two Rydberg states can induce a transition between the two level. Presenceor absence of a molecule in certain energy levels can help us better understandthe evolution of a Rydberg gas. An electro magnetic wave with a frequency closeto electron plasma frequency, ωp,e =√ρe2/meε0, resonates with the oscillationof electrons and raises temperature. Observation of strong absorption at this fre-quency can give us vital information about the real-time density of the evolvingRydberg gas and the expanding plasma.To complement these experiments, we have built an imaging apparatus thatmaps the density profile of the plasma in the direction of the propagations. In arecent publication, Schulz-Weiling has reported results of an imaging experiment[66]. 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Raimond, M. Gross, and S. Haroche. Rydberg to plasmaevolution in a dense gas of very excited atoms. J. Phys. B, 15:L49–L55,1982.[70] E. Vredenbregt and J. Luiten. Electron diffraction: Cool beams in greatshape. Nat. Phys., 7:747–748, 2011.84Appendix ACoupled-Rate Equation ModelThe coupled rate equation model is a method that is broadly used in many areas ofscience and engineering. In this method the rate of change of density (population)of species is related to the density (population) of others and some macroscopicparameters such as temperature and volume. In particular, the model for Rydberggas evolution to a plasma is concerned with the density of electrons, Rydberg statesof different quantum numbers n, and ion density. In the course of the evolution, theelectron-impact ionization is the dominant process,A∗(n)+ e−→ A++ e−+ e− (A.1)where A∗ is a highly excited molecule and A+ is a resulting ion. Electron impactcan also drive the population of a Rydberg state to a manifold of states with variousquantum numbers n.A(n)+ e−→ A(n′)+ e− (A.2)For an ultra cold plasma the reverse of the process in equation A.1, known as three-body recombination, is relatively fast as the rate of the process scales with T−9/2[37].A++ e−+ e−→ A∗(n)+ e− (A.3)A molecular plasma has additional decay channels. The recombination of an85electron and an ion can channel the excess energy into very high vibrational and ro-tational modes that breaks the molecule apart. For example, nitric oxide dissociatesupon two body recombination.NO++ e−→ N+O (A.4)The energy of a highly excited electronic state of NO molecule is above disso-ciation limit, therefore these Rydberg states can dissociate to neutral atoms.NO∗(n)→ N+O (A.5)The life time of the NO molecule for this process is very short. However, itis shown that due to stray and external fields, and dipole interactions, the Pre-Dissociation (PD) process can slow down significantly. It is proposed that themechanism for this change of rate is the m,l-mixing of Rydberg levels in pres-ence of an external field [43]. Our numerical integration of coupled-rate equationcan include or exclude PD.We use the rate constants for ionization, three-body recombination, and electron-Rydberg impact collision calculated by Pohl and co-workers [48].The transition rate for electron-Rydberg inelastic collision from ni to n j isk (ni,n j) = k0ε5/2i ε3/2fε5/2>e−(εi−ε<)[22(ε>+0.9)7/3+9/2ε5/2∆ε4/3>](A.6)where k0 is a constant defined ask0 =e4kBT√mR(A.7)where R is the Rydberg constant. εi, ε f , ε<, ε> are the binding energy of the initial,final, lower state, and higher state, respectively. ∆ε is the absolute of the energydifference between two states. The ionization and three-body recombination rate86constants are related as followkION (ni) =kTBR (ni)n2i Λ3ρeeεi=11(R/kBT )1/2 k0 e−εiε7/3i +4.38ε1.72i +1.32εi(A.8)where Λ is thermal de Broglie wavelength defined asΛ=√h2/2pimkBT (A.9)PD rates are calculated by [8, 51] and DR is calculated by [64].Given the processes in this section the full set of equations that governs theevolution dynamics of Rydberg gas is as follow:dρed t=nmax∑n=nminkION (T,n)ρeρn−nmax∑n=nminkTBR (T,n)ρ3e (A.10)dρn jd t=nmax∑n=nmink (T,ni,n j)ρeρni−nmax∑n=nmink (T,n j,ni)ρeρn j−kION (T,n j)ρeρn+ kTBR (T,n j)ρ3e(A.11)Quasi-neutrality is assumed through out that ion and electron density are equallocally ρe = ρi. An equation for temperature evolution can be derived by startingfrom an expression for conservation of energy. If we neglect expansion for earlydynamics, the energy is stored solely in kinetic energy of electrons and bindingenergy of Rydberg states.E =32kBρeT −nmax∑n=nminRρnn2(A.12)We set the derivative of the energy conservation expression equal to zero to find anexpression of time derivative of the temperatured Td t=nmax∑n=nmin2R3kBρe n2dρnd t− Tρedρed t(A.13)We study the dynamics in the presence of DR and PD to compare the effect. Equa-87tion A.10 and A.11 with DR and PD are modified the following waydρed t=nmax∑n=nminkION (T,n)ρeρn−nmax∑n=nminkTBR (T,n)ρ3e − kDR (T )ρ2e (A.14)dρn jd t=nmax∑n=nmink (T,ni,n j)ρeρni−nmax∑n=nmink (T,n j,ni)ρeρn j−kION (T,n j)ρeρn+ kTBR (T,n j)ρ3e − kPD (n j)ρn j(A.15)For longer evolution time we consider the expansion of the system and it’s effecton the density and reaction rates. Given the density ρ =N/V is number per volumewe can write an equation for the evolution of density with expansion as followdρd t=dN/Vd t=−ρVdVd t+1VdNd t(A.16)The second term in the right hand side of equation A.16 can be substituted byequation A.10 and A.11. The first term serves as a correction for density due toexpansion. For a freely evolving plasma, electron pressure causes an expansionresulting only in a negative first term that slows down all non-linear processes.A.1 Shell model for coupled-rate equationIn a realistic experiment the distribution of particles is not uniform. The molecularbeam that intersects with two laser beams will have higher density at the centreof beams intersection than other regions. For simplicity we assume only densityvariation along the propagation of the laser beam (due to gradient of density pro-duce by molecular beam and sampled by laser beams). This is true for many ofthe experiments performed for this thesis in which the laser beam is collimatedand passed through a relatively large iris. Therefore the laser profile is relativelyflat across its radius. For experiments in which a small diameter pinhole is usedto filter the laser beam, it is necessary to assume a Gaussian profile. Nevertheless,the molecular beam passes through a small diameter skimmer and downstream itdiverges.88For the above reasons, we model the plasma by assuming a cylindrical volumewith uniform density along the cross section of the cylinder and a Gaussian profileacross the length of the cylinder with the maximum at the middle the cylinder. Wethen slice the cylinder into disks with uniform densities. The coupled-rate equationmodel applies independently to each disk without any mass exchange between dif-ferent sections. However, the electron temperature is assumed to equilibrate acrossthe entire cylinder. The coupled rate equations for the shell index, k aredρe,kd t=nmax∑n=nminkION (T,n)ρe,kρn,k−nmax∑n=nminkTBR (T,n)ρ3e,k−ρe,kVkdVkd t(A.17)dρn j,kd t=nmax∑n=nmink (T,ni,n j)ρe,kρni,k−nmax∑n=nmink (T,n j,ni)ρe,kρn j,k−kION (T,n j)ρe,kρn j,k+ kTBR (T,n j)ρ3e,k−ρe,kVkdVkd t(A.18)Each disk is located at a distance rk from the middle of the cylinder with vol-ume piR2 (rk+1− rk). We pick a disk radius of R = 200µm that slowly varies ac-cording to phenomenological Vlasov fit described in chapter 3 corresponding toan apparent electron temperature of 5K. Disks are formed with the same width tobegin with, but are not restricted to evolve in a same way. Due to non-linear natureof avalanche process the final plasma charge density profile will not be the sameas the Rydberg density profile. The avalanche process scales quadratically withthe density and Penning fraction that seeds the avalanche scales exponentially withdensity. Therefore the plasma density profile will be much sharper than the Ryd-berg gas density. Ultimately, the expansion force, proportional to density gradient,will be greater close to the centre pushing some of the disk closer and stretchingthe others. The acceleration of ions is given byd ukd t=−2kBTmi1ρ∂ρ∂ r=−kBTmiρe,k+1−ρe,k(ρe,k+1+ρe,k)(rk+1− rk) (A.19)Additionally, a fast charge exchange between ions and Rydberg moleculessmudges their identity, such that the force created by an electron is felt by both89species. This means that the ion mass in equation A.19 must be replaced by theeffective mass which is m∗ = mi×Ne/(Ne+NRyd). As a result a fraction of thehydrodynamic force is transferred to each ion and Rydberg molecules.The energy conservation for the entire volume isE =Nshell∑k=1[3/2kBTNe,k+1/2miNe,ku2k−Rnmax∑n=nminNn,kn2](A.20)By setting the time derivative equal to zero we find an expression for the timeevolution of the temperature as followd Td t=∑Nshellk=1[3/2kBT∂Ne,k/∂ t+1/2mi∂(Ne,ku2k)/∂ t−R∑nmaxn=nmin ∂Nn,k/∂ t/n2]∑Nshellk=1 3/2kBNe,k(A.21)A.2 Non-uniform model resultsFor the purpose explained in chapter 5 we run many model calculations with vari-ous densities to simulate a Rydberg gas prepared by exciting nitric oxide to n= 49to fit Rydberg and electron signal.Figure A.1 show the density profile of the Rydberg gas and the resulting profileof the electron density created by Penning ionization.90Position normal to propagation (mm)-4 -3 -2 -1 0 1 2 3 4Density (scaled to maximum)00.20.40.60.811.2 Initial Rydberg densityPenning electron densityFigure A.1: Density of Rydberg gas versus Penning electron. The blackcurve is a Gaussian profile with a standard width of 800 µm. The pro-file is scaled by its values at the centre. The dashed red line, is theelectron density after Penning ionization which is equal to density ofthe Rydberg gas times Penning fraction.To obtain this figure, or equally to setup the initial value problem for numeri-cal integration, we calculate the Penning fraction for each disk at distance r withdensity ρ (r)f (r) = 1− e4piR3ρ(r)/3 (A.22)where R = 3.6a0 n2 is the critical distance of two Rydberg state (with quantumnumber n) under which Penning ionization is instantaneous [55], and ρ (r) is givenby a Gaussian distribution. The initial electron density in the shell k is given byρe,k (t = 0) =12fkρ (rk) (A.23)the factor of 1/2 is introduce because as a result of Penning ionization, one Ryd-berg state ionizes and the other decays into a state with lower quantum number.NO∗(n)+NO∗(n)→ NO++ e−+NO∗(n′) (A.24)91In the process describe by the equation above, n′ is a distribution of quantumnumbers with no lower bound (except the ground state) and the upper bound de-termined by the conservation of energy. If the lowest energy pumped to the freeelectron is only enough to ionize it we can write the following expression for theupper bound of n′2×(− Rn2)= 0+(− Rn′2)(A.25)resulting in n′ = n/√2. The distribution of the resulting bound state is proportionalto (n′/n)5.As a result of the imbalance in the initial electron density, avalanche takes placeon different time scales throughout the excited volume. Figure A.2 shows howelectron density evolves in each shell. The inner shells with higher densities evolveto a plasma on a times scale of few nanosecond, and outer shells with lower densitydo not evolve within up to a microseconds. When there is only small number ofelectrons in a shell (e.g. outer shells) compared to the number of Rydberg states,the force calculated by A.19 is very small due to high effective mass. In effect, ionsappear heavier due fast charge exchange with Rydberg molecules by a factor of(Ne+Nn)/Ne. This factor is included in the model calculation. The fast avalanchein the inner shells causes fast expansion because of high ion to Rydberg populationratio and slower expansion in the outer shells. This will cause the layers to movecloser in the middle to create shocks.92Time (7s)0 0.02 0.04 0.06 0.08 0.1# e/shell#10700.511.522.5Figure A.2: Number of electrons in each disk (total of 15 disks) as a functiontime. Initially each disk is populated with a Rydberg gas with a densitydetermined by the between distance between its centre and cylinder’s.The shell with the highest number of electron (top curve) initiates afast avalanche. Shell number 7 and above have a very small number ofelectrons.Figure A.3 shows how shells get closer together as the expansion occurs. Thefirst few shells are largely populated with free electrons (due to fast plasma for-mation) and they expand relatively fast even though expansion rate is proportionalto the distance from the centre. The outer shells are almost pure Rydberg gas andthey do not expand at all on this time scale. In this simple model with 15 shells,shell number 10 and 11 get very close and shell 11 and 12 collide. Alternatively,the figure A.4 shows the distance of two adjacent shells or equally the thickness ofeach shell.93Time (7s)0 1 2 3 4 5Radius (7m)010002000300040005000Figure A.3: The distance of each disk from the middle of the cylinder. Thecylinder is divided into 15 disk each place at uniform interval. The rateof the expansion is faster as the disk is place at longer distances, how-ever, the rate of avalanche act in the opposite way. Since the outermostdisk has not evolved to a plasma, it dissipates the force generated byelectron pressure to a point that it acts as a solid wall in front of otherexpanding shells.94Time (7s)0 1 2 3 4 5Distnace of adjacent shells (7m)0100200300400Figure A.4: Similar to Fig. A.2, but instead of the distance of each disk, thethickness of it is plotted. The thickness is defined as the distance be-tween the centre of two adjacent disks. Some disks expand as predictedby ambipolar expansion. However, some disk are squeezed due to fastexpansion of the inner ones.Figure A.5 illustrates this unusual phenomena. The shell number 12 is nearstationary in the first 3µs due to absence of electron population. It is howeverpushed by the inner shell and exponentially accelerates. Figure A.6 presents thesame quantity versus the shell index. It is evident that the rate of expansion ismaximum for some middle shells.95Time (7s)0 1 2 3 4 5Shell Velocity (m/s)050100150200250Figure A.5: The velocity of each disk as a function time. Disks with thehigher Rydberg gas density, avalanche to a plasma quickly, and accel-erate proportional to their distance after that. Some disks with very lowelectron density, are stationary at first, but after a late avalanche (e.g. 3µs), they are pushed forward.Shell index0 5 10 15Shell Velocity (m/s)050100150200250Figure A.6: The velocity of each disk as a function of their index as a functionof time. The velocity grow in time and with index number, but dropsvery fast in regions where no plasma is formed.96A.2.1 ResultsWe have run model calculations with many initial Rydberg gas densities and quan-tum numbers to find a phenomenological expression for the avalanche time.The avalanche time is going to be very different for Rydberg gases of variousdensities. The avalanche time is shorter for higher quantum number, because largerfraction of molecules fall within the critical radius, rc = 3.6a0 n2. This radiusdetermines a distance under which two Rydberg states Penning ionize instantly. Allpairs with distance rc and lower constitute the fraction of pairs, Penning fraction,that Penning ionize very fast and it is give by A.22.A determining time scale that determines the avalanche time can be definedsimilar to plasma frequency:ω =√ρe2meε0(A.26)where ρ is the Rydberg gas density. This quantity is interpreted as the frequencyof bumping into a Rydberg state, therefore at higher density, an electron morefrequently collides with Rydberg states and therefore the avalanche time scalesinversely with this quantity.97#10-60 1 2 3 4 5 6Probability Density (m-1 ) #1050246Erlang Distribution ; = 1010 cm-3Distance (m) #10-60 1 2 3 4 5 6Probability Density (m-1 ) #106012Erlang Distribution ; = 1012 cm-3rc  n = 50rc  n = 70Figure A.7: Plot of nearest neighbour distribution for two densities 1010 cm−3and 1012 cm−3. At the lower density the distances are larger. Two linesindicate the critical distance for Rydberg states with n= 50 and n= 70.The critical distance (rc) scales with square of quantum number. Fora higher n, a larger fraction of pairs are within rc, the same applies tohigher density.Figure A.8 presents the evolution of scaled density ρ/ρ0 as a function of scaledtime, tωe f3/4p . The plot shows electron density over initial Rydberg density as afunction of scaled time. Red traces are computed for n=30, blue for n=50, andblack for n=70. For each n, computation for four different densities is performed.For small Penning fractions fp << 0.5, the family of curves can be described asone curve as a function of the dimensionless quantity tωe f3/4p . For any set ofparameters (n and ρ) that Penning fraction is small and plasma formation goesthrough a transition time, that transition time happens at around 100 units of scaledtime.98t we pf3/40 100 200 300 400 500 600scaled density00.10.20.30.40.50.60.70.80.9n=30n=50n=70Figure A.8: Plot of electron density over initial Rydberg density as a functionof scaled time. Red traces are computed for n=30, blue for n=50, andblack for n=70. For each n, computation for four different densities isperformed.A.3 Shell model codes for rate equation model withexpansion for non-uniform densityA.3.1 Non-uniform coupled rate equation: setting up the parametersand initial value problemfunction [ y0 , param ,Ne,N,Nn]= i n i t g u ( n0 , t f , den )% setup de f au l t i npu tparam . s0=1213;param . nshe l l s =20;param . n f i r s t =20;param . n0=n0 ;99param . numlev=80;param . t f = t f ; % usparam . t b i n =50;param . den=den ;param . r e l t o l =1e−10;param . PD f lag = f a l s e ;param . type = 1;kB = cons . kB ;Rydhc = cons . Rydhc ;% setup she l l sdr = 5∗param . s0 / param . nshe l l s ;r1 = dr : dr :5∗param . s0 ; r1=r1 ( : ) ;r0 = r1 − dr ;r = ( r0+r1 ) / 2 ;dens i t y = param . den ∗ exp(− r . ˆ 2 / 2 / param . s0 ˆ 2 ) ;nshe l l s= param . nshe l l s ;% V = 4∗p i /3∗ ( r1 .ˆ3− r0 . ˆ 3 ) ;V =pi ∗200ˆ2∗( r1−r0 ) ;N= dens i t y .∗V;% bu i l d n ln l =param . n f i r s t : param . n f i r s t +param . numlev−1; n l =n l ( : ) ;% ca l cu l a t e e lec t r on dens i t y and rydberg dens i t y i n each she l lf =@( x ) x . ˆ 5 ;np=param . n f i r s t : f i x ( param . n0 / sqrt ( 2 ) ) ;y0 = zeros ( param . nshe l l s , param . numlev +2) ;Ne = zeros ( nshe l l s ) ;Nn = zeros ( nshe l l s , param . numlev ) ;for she l l = 1 : param . nshe l l saden=0;[ ˜ , eden , rden ]= cons . penn ing f rac t i on ( param . n0 , dens i t y ( s he l l ) ) ;nden=n l ∗0;nden ( n l ==param . n0)= rden ;nden ( np−param . n f i r s t +1)=eden∗ f ( np / param . n0 ) /sum( f ( np / param . n0 ) ) ;Ne( s he l l ) = eden∗V( she l l ) ;Nn( she l l , : ) = nden∗V( she l l ) ;y0 ( she l l , : ) = [ Nn( she l l , : ) ’ ; Ne( s he l l ) ; aden ] ;end[ a , b ] = size ( y0 ) ;% ca l cu l a t e the e lec t r on temperature o f the en t i r e plasmaNe=y0 ( : , param . numlev +1) ;E0 = −Rydhc∗sum(N ) / param . n0 ˆ 2 ;E = −Rydhc∗sum(sum(Nn ) ’ . / n l . ˆ 2 ) ;T = −(E−E0 ) / ( 1 . 5∗ kB∗sum(Ne ) ) ;% E = E + 1.5∗kB∗sum(Ne)∗T ;100% setup up hydrodynamicsu = 0∗ r ;EDR=0;% update y0 , an ar ray o f i n i t i a l valuesy0 = reshape ( y0 , [ a∗b , 1 ] ) ;y0 = [ y0 ; r1 ; u ; T ;EDR] ;endA.3.2 Non-uniform coupled rate equation: numerical integrationfunction [ t , y , t i c t o c ]=gu ( y0 , param )nshe l l s = param . nshe l l s ;numlev = param . numlev ;type = param . type ;t f =param . t f ;t b i n =param . t b i n ;t = l inspace (0 , t f , t b i n ) ;n f=param . n f i r s t ;n l =( n f : n f+numlev−1) ’ ;Rydhc = cons . Rydhc ;mi = cons . mi ;kB = cons . kB ;[ n i , nf , I I , minn ,maxn , dn ]= bu i l dns ( n l ) ;op t ions=odeset ( ’ RelTol ’ , param . r e l t o l ) ;t i c t o c = t i c ;i f param . PD f lag[ t , y ]=ode113 (@internalodePD , t , y0 , op t ions ) ;else[ t , y ]=ode113 ( @internalode , t , y0 , op t ions ) ;endt i c t o c =toc ( t i c t o c ) ;function dy= in te rna lode ( t , y )f p r i n t f ( ’%.2e\n ’ , t ) ;T = y (end−1);y = reshape ( y ( 1 :end−2) , [ nshe l l s , numlev +4 ] ) ;%r1 = y ( : , numlev +3) ;r0 = [ 0 ; r1 ( 1 :end−1) ] ;u = y ( : , numlev +4) ;% V = 4∗p i /3∗ ( r1 .ˆ3− r0 . ˆ 3 ) ;% s = 200∗ sq r t (1+ t ˆ 2 / tau ˆ2 )V =pi ∗200ˆ2∗( r1−r0 ) ;dy = 0∗y ;k t b r n T = kTBR( nl , T ) ;kION n T = kION ( nl , T ) ;kn np=knnp ( ni , nf , I I , minn ,maxn , dn , T ) ;101Nn to ta l = zeros ( nshe l l s , 1 ) ;dNe2=zeros ( nshe l l s , 1 ) ;dEDR=0;for she l l = 1 : nshe l l sNn=y ( she l l , 1 : numlev ) ’ ;Nnm=repmat (Nn ( : ) ’ , [ numlev , 1 ] ) ;Ne=y ( she l l , numlev +1) ;k t b r =k tb r n T∗Neˆ 3 /V( s he l l ) ˆ 2 ;k ion=kION n T .∗Nn∗Ne/V( s he l l ) ;knp n=kn np ’ . ∗Nnm;dNe=sum( kion−k t b r ) ;dNn=ktbr−kion−sum( kn np , 2 ) .∗Nn∗Ne/V( s he l l ) . . .+sum( knp n ,2 )∗Ne/V( s he l l ) ;dNe2 ( she l l , : ) = dNe ;dNe=dNe−kDR(T)∗Ne . ˆ 2 / V( s he l l ) ;daden=kDR(T)∗Ne . ˆ 2 / V( s he l l ) ;dEDR=dEDR+kDR(T)∗Ne . ˆ 2 / V( s he l l ) ∗ . . .(1 .5∗kB∗T+0.5∗mi∗u ( s he l l ) ˆ 2 ) ;dy ( she l l , 1 : numlev+2) = [ dNn ; dNe ; daden ] ;Nn to t a l ( s h e l l ) = sum(Nn ) ;end% update some valuesNe = y ( : , numlev +1) ;Nn = y ( : , 1 : numlev ) ;dNe = dy ( : , numlev +1) ;dNn = dy ( : , 1 : numlev ) ;% she l l expansiondr = u ;% force eva lua t i onswi tch typecase 1f o r c e f a c t o r = sum(Ne ) . / ( sum(Ne)+sum(sum(Nn , 2 ) , 1 ) ) ;sigma = sqrt (sum( r1 . ˆ 2 .∗Ne ) /sum(Ne ) ) ;f o rce = kB∗T / ( mi∗sigma ˆ2 )∗ ( r1+r0 ) / 2 ;du = f o r c e f a c t o r ∗ f o rce ;case 2f o r c e f a c t o r = Ne . / ( Ne+sum(Nn , 2 ) ) ;sigma = sqrt (sum( r1 . ˆ 2 .∗Ne ) /sum(Ne ) ) ;fo rce1 = kB∗T / ( mi∗sigma ˆ2 )∗ ( r1+r0 ) / 2 ;fo rce = kB∗T / mi ∗ dens i t y g rad i en t (Ne,V, r0 , r1 ) ;f o rce = ( fo rce+ force1 ) / 2 ;du = f o r c e f a c t o r .∗ f o rce ;case 4f o r c e f a c t o r = Ne . / ( Ne+sum(Nn , 2 ) ) ;sigma = sqrt (sum( r1 . ˆ 2 .∗Ne ) /sum(Ne ) ) ;fo rce1 = kB∗T / ( mi∗sigma ˆ2 )∗ ( r1+r0 ) / 2 ;102f o rce = kB∗T / mi ∗ dens i t y g r ad i en t t 4 (Ne,V, r0 , r1 ) ;f o rce = ( fo rce+ force1 ) / 2 ;du = f o r c e f a c t o r .∗ f o rce ;i f min ( r1−r0)<25for she l l =1: nshe l l s−1i f r1 ( s he l l )− r0 ( s he l l )<25du ( s he l l +1)=du ( s he l l ) ;i f u ( s he l l +1)< 0.2∗u ( s he l l )du ( s he l l +1)=10∗du ( s he l l ) ;endendendendcase 3f o r c e f a c t o r = Ne . / ( Ne+sum(Nn , 2 ) ) ;f o rce = kB∗T / mi ∗ dens i t y g rad i en t2 (Ne,V, r0 , r1 ) ;du = f o r c e f a c t o r .∗ f o rce ;end% temperature evo l u t i onNRi = Ne + sum(Nn , 2 ) ;dNRi = dNe + sum(dNn , 2 ) ;dT= (−0.5∗mi∗sum(2∗u .∗du .∗NRi+u . ˆ 2 .∗ dNRi ) + . . .Rydhc∗sum(sum(dNn ) ’ . / n l . ˆ 2 ) − . . .1.5∗kB∗T∗sum( dNe2 ) ) / ( 1 . 5∗ kB∗sum(Ne ) ) ;dy ( : , numlev+3) = dr ;dy ( : , numlev+4) = du ;dy = reshape ( dy , [ nshe l l s ∗( numlev +4 ) , 1 ] ) ;dy =[ dy ; dT ;dEDR ] ;endfunction f = dens i t y g rad i en t (Ne,V, r0 , r1 )rho = Ne . / V ;rhop1 = [ rho ( 2 :end ) ; rho (end)∗2− rho (end−1) ] ;rhom1 = [ rho (1)∗2− rho (2 ) ; rho ( 1 :end−1) ] ;f = −(rhop1−rhom1 ) . / ( r1−r0 ) . / rho ;endfunction f = dens i t y g r ad i en t t 4 (Ne,V, r0 , r1 )rho = Ne . / V ;rhop1 = [ rho ( 2 :end ) ; rho (end)∗2− rho (end−1) ] ;rhom1 = [ rho (1)∗2− rho (2 ) ; rho ( 1 :end−1) ] ;f = −(rhop1−rhom1 ) . / ( r1−r0 ) . / rho ;ind =( r1−r0 )<5;f ( ind ) = −(rhop1 ( ind)−rhom1 ( ind ) ) . / 5 . / rho ( ind ) ;endfunction f = dens i t y g rad i en t2 (Ne,V, r0 , r1 )rho = Ne . / V ;r = ( r0+r1 ) / 2 ;c f = f i t ( r ( : ) , log ( rho ( : ) ) , f i t t y p e ( ’ smooth ingspl ine ’ ) ) ;103f = −d i f f e r e n t i a t e ( cf , r ) ;endfunction k ion=kION (n , T )% output u n i t i s umˆ3 us−1% f i n d reduced i n i t i a l energyeps i =1.578875091189012e+05 . / ( power ( n ,2 )∗T ) ;% knot= 2.735776154492852/Tk ion =1.195770111325817e+07/power (T , 3 / 2 ) . ∗ . . .exp(−eps i ) . / ( power ( epsi , 7 / 3 ) + . . .4.38∗power ( epsi ,1 .72)+1.32∗ eps i ) ;endfunction kdr=kDR(T)kdr =5.623∗power (T,−0.528);endfunction k t b r =kTBR(n , T )% TBR ra tes output un i t s i n umˆ6 us−1% f i n d reduced i n i t i a l energyeps i =1.578875091189012e+05 . / ( power ( n ,2 )∗T ) ;% knot= 2.735776154492852/Tk t b r =power ( n ,2.0).∗4.952079396744990e3 . / . . .power (T , 3 ) . / ( power ( epsi , 7 / 3 ) + 4 . 3 8∗ . . .power ( epsi ,1 .72)+1.32∗ eps i ) ; % un i t umˆ6 ns−1endfunction out=knnp ( ni , nf , I I , minn ,maxn , dn , T )% ra te f o r t r a n s f e r from n to n ’% Rates ca l cu la ted using PVS PRL 2008% output un i t s i s umˆ3 usˆ−1% use in con junc t ion wi th [ n i , nf , I I , minn ,maxn , dn ]= bu i l dns ( n l ) ;e i =1.578875091189012e+05 . / ( power ( n i , 2 . 0 )∗T ) ;e f =1.578875091189012e+05 . / ( power ( nf , 2 . 0 )∗T ) ;eg=1.578875091189012e+05 . / ( power (minn ,2 )∗T ) ;es=1.578875091189012e+05 . / ( power (maxn,2 )∗T ) ;de=1.578875091189012e+05.∗dn . / T ; % scale dn p rope r l yout= I I .∗(2.735776154492852e3 . / T.∗power ( e i , 5 / 2 ) . . ..∗power ( ef , 3 / 2 ) . / power ( eg , 5 / 2 ) ) . . ..∗exp(−( e i−es ) ) . ∗ ( ( 2 2 . / ( power ( eg + 0 . 9 , 7 / 3 ) ) ) . . .+ ( 4 . 5 . / ( power ( eg , 2 . 5 ) .∗ power ( de+1− I I , 4 / 3 ) ) ) ) ;endfunction [ n i , nf , I I , minn ,maxn , d i f f s n ]= bu i l dns ( n l )a= length ( n l ) ;I I =ones ( a , a)−eye ( a ) ; % not suren i =zeros ( a , a ) ;n f=zeros ( a , a ) ;minn=zeros ( a , a ) ;maxn=zeros ( a , a ) ;for i =1:afor j =1:a104n i ( i , j )= n l ( i ) ; % an ar ray o f i n i t i a l s ta tesnf ( i , j )= n l ( j ) ; % an ar ray o f f i n a l s t a t e%f i n d min o f i n i t and f i n a l s t a t e p o t e n t i a l problemminn ( i , j )=min ( n i ( i , j ) , n f ( i , j ) ) ;%f i n d max of i n i t and f i n a l s t a t emaxn( i , j )=max( n i ( i , j ) , n f ( i , j ) ) ;endend% d i f f e r e n t i n energy between the 2 sta tes , no un i t sd i f f s n =abs ( 1 . / n i . ˆ2−1. / n f . ˆ 2 ) ;endfunction y=kPD( n i )y=zeros ( size ( n i ) ) ;for qq=1: length ( n i )nn=n i ( qq ) ;au = 1 . / ( 2 . 4 2 e−17);tau=ones (1 , nn )∗0.00003;gvalue = 0 . ;gsum = 0 . ;tau (1 ) = 0.014;tau (2 ) = 0.046;tau (3 ) = 0.029;tau (4 ) = 0.0012;for kk = 1: nngsum = gsum + (2 .0∗ ( kk−1) + 1 . 0 ) ;endfor i i = 1 : nngvalue = gvalue + (2 .0∗ ( i i −1) + 1.0)∗ tau ( i i ) ;endy ( qq)= gvalue ∗ ( 1 . / gsum )∗ ( 1 . / ( 2 .∗ pi∗power ( nn , 3 . ) ) ) ∗ au ;endy=y∗1e−9;y=y ( : ) ;endendA.3.3 Non-uniform coupled rate equation: testing energy andparticle conservationfunction d = gudata ( y , param )d . T = y ( : , end−1);d .EDR = y ( : , end ) ;y = y ( : , 1 : end−2);numlev = param . numlev ;nshe l l s = param . nshe l l s ;t b i n = min ( [ param . t b i n size ( y , 1 ) ] ) ;105n f i r s t = param . n f i r s t ;n l = n f i r s t : n f i r s t +numlev−1;kB= cons . kB ;mi = cons . mi ;Rydhc = cons . Rydhc ;for i =1: t b i nytemp = reshape ( y ( i , : ) , [ nshe l l s , numlev +4 ] ) ;d . r ( i , : ) = ytemp ( : , numlev +3) ;d . u ( i , : ) = ytemp ( : , numlev +4) ;r0 = [0 d . r ( i , 1 : end−1) ] ;d .V( i , : ) = pi ∗200ˆ2∗(d . r ( i , : )− r0 ) ;d .Ne( i , : ) = ytemp ( : , numlev +1) ;d . eden ( i , : ) = d .Ne( i , : ) . / d .V( i , : ) ;d .Na( i , : ) = ytemp ( : , numlev +2) ;d .Nn( i , : , : ) = ytemp ( : , 1 : numlev ) ;Nn = ytemp ( : , 1 : numlev ) ;d . Nn pern ( i , : ) = squeeze (sum( d .Nn( i , : , : ) , 2 ) ) ;d . Nn pershe l l ( i , : ) = squeeze (sum( d .Nn( i , : , : ) , 3 ) ) ;d . N pershe l l ( i , : ) = d . Nn pershe l l ( i , : ) + d .Ne( i , : ) + d .Na( i , : ) ;d . N to t ( i ) = sum( d . N pershe l l ( i , : ) , 2 ) ;d . N to t2 ( i ) = sum( d . Nn pern ( i , : ) , 2 ) + . . .sum( d .Ne( i , : ) , 2 ) +sum( d .Na( i , : ) , 2 ) ;d .E2( i ) = 1.5∗kB∗d . T ( i )∗sum( d .Ne( i , : ) ) − . . .Rydhc∗sum(sum(Nn , 1 ) . / n l . ˆ 2 ) + 0.5∗mi ∗ . . .sum( d . u ( i , : ) . ˆ 2 . ∗ ( d .Ne( i , : ) +sum(Nn , 2 ) ’ ) ) ;d .E( i ) = 1.5∗kB∗d . T ( i )∗sum( d .Ne( i , : ) ) − . . .Rydhc∗sum(sum(Nn , 1 ) . / n l . ˆ 2 ) + 0.5∗mi ∗ . . .sum( d . u ( i , : ) . ˆ 2 . ∗ ( d .Ne( i , : ) +sum(Nn , 2 ) ’ ) ) + d .EDR( i ) ;endt = l inspace (0 , param . t f , t b i n ) ;d . y=y ;d . y0=y ( 1 , : ) ;d . param=param ;d . t = t ;function t e s t p l o t ( d )n=4;m=4; f s =10; fw= ’ normal ’ ;t =d . t ;p=d . param ;index = 1:p . nshe l l s ;c l f ;% t o t a l number o f p a r t i c l e ssubplot ( n ,m, 1 ) ; hold on ;plot ( t , d . N to t ) ;plot ( t , d . N to t2 ) ;xlabel ( ’ Time (\mus) ’ )106ylabel ( ’Number o f p a r t i c l e s ’ ) ;N Error = (max( d . N to t )−min ( d . N to t ) ) /mean( d . N to t )∗100;t i t l e (num2str ( N Error , ’ E r ro r = %.4 f %%’ ) , . . .’ f o n t s i z e ’ , fs , ’ f on twe igh t ’ , fw )% t o t a l number o f p a r t i c l e s i n each she l lsubplot ( n ,m, 2 ) ;plot ( t , d . N pershe l l ) ;xlabel ( ’ Time (\mus) ’ )ylabel ( ’ # p t c l s / s h e l l ’ ) ;% t o t a l number o f p a r t i c l e s i n each she l lsubplot ( n ,m, 3 ) ;plot ( index , d . N pershe l l ) ;xlabel ( ’ She l l index ’ )ylabel ( ’ # p t c l s / s h e l l ’ ) ;%subplot ( n ,m, 4 ) ;plot ( t , d . Nn pershe l l )xlabel ( ’ Time (\mus) ’ )ylabel ( ’ # R { t o t a l } / s h e l l ’ ) ;subplot ( n ,m, 5 ) ;plot ( t , d .Ne)xlabel ( ’ Time (\mus) ’ )ylabel ( ’ # e / s he l l ’ ) ;subplot ( n ,m, 6 ) ;plot ( t , d .Na)xlabel ( ’ Time (\mus) ’ )ylabel ( ’ # a / s he l l ’ ) ;subplot ( n ,m, 7 ) ;plot ( index , d . Nn pershe l l )xlabel ( ’ She l l index ’ )ylabel ( ’ # R { t o t a l } / s h e l l ’ ) ;subplot ( n ,m, 8 ) ;plot ( index , d .Ne)xlabel ( ’ She l l index ’ )ylabel ( ’ # e / s he l l ’ ) ;subplot ( n ,m, 9 ) ;plot ( index , d .Na)xlabel ( ’ She l l index ’ )ylabel ( ’ # a / s he l l ’ ) ;subplot ( n ,m, 1 0 ) ;plot ( t , d . T , ’ . ’ ) ;xlabel ( ’ Time (\mus) ’ )107ylabel ( ’T (K) ’ ) ;subplot ( n ,m, 1 1 ) ; hold on ;plot ( t , d .E ) ;plot ( t , d . E2 ) ;xlabel ( ’ Time (\mus) ’ )ylabel ( ’ Energy ’ ) ;E Er ror = abs (max( d .E)−min ( d .E ) ) / abs (mean( d .E) )∗100 ;t i t l e (num2str ( E Error , ’ E r ro r = %.4 f %%’ ) , . . .’ f o n t s i z e ’ , fs , ’ f on twe igh t ’ , fw )subplot ( n ,m, 1 2 ) ; hold on ;plot ( t , d . u ) ;xlabel ( ’ Time (\mus) ’ )ylabel ( ’ Hydrodynamic v e l o c i t y (m/ s ) ’ ) ;subplot ( n ,m, 1 3 ) ; hold on ;plot ( index , d . u ) ;xlabel ( ’ She l l index ’ )ylabel ( ’ Hydrodynamic v e l o c i t y (m/ s ) ’ ) ;subplot ( n ,m, 1 4 ) ; hold on ;plot ( t , d . r ) ;xlabel ( ’ Time (\mus) ’ )ylabel ( ’ Radius (\mum) ’ ) ;subplot ( n ,m, 1 5 ) ; hold on ;plot ( index , d . r ) ;xlabel ( ’ She l l index ’ )ylabel ( ’ Radius (\mum) ’ ) ;subplot ( n ,m, 1 6 ) ;plot ( index , d . eden )xlabel ( ’ She l l index ’ )ylabel ( ’ # eden / s he l l ’ ) ;A.4 Coupled rate equations with GPUA coupled-rate equation model is not a very expensive computation, however, anhour of computation for each density and principal quantum number for a set ofthese parameters, adds up to very long times. Using a massively parallel algo-rithm, the computation for integration of the set of equation to 2 times pass theavalanche time takes about 60 seconds. This model ignores non-homogeneity ofthe Rydberg gas volume and expansion. A parallel OpenCL kernel evaluates theelectron-Rydberg inelastic collision rate constants matrix at every time step.108A.4.1 Rate constants” ” ”@author : hossein” ” ”from math import exp , fabsfrom numpy import array , arange , append , onesfrom sc ipy . constants import constants as consimport sc ipyphyscons = sc ipy . constants . phys i ca l cons tan tskB = cons . k #m2 kg s−2 K−1;J K−1; 8.62e−5 #eV K−1,Ryd = cons . Rydberg #m−1 #13.6 #eV ,NOmass = 30.01∗cons .m p #4.98e−26 #kgemass = cons .m e #kgechg = cons . e #Ch = cons . h ; #m2 kg / s 4.13e−15 #eV seps i l on = cons . eps i l on 0 #C2 J−1 cm−1c = cons . c #m s−1Rydhc = cons . Rydberg∗cons . h∗cons . c #J : Ryd [cm−1] −−> [ J ]a0 = physcons [ ” Bohr rad ius ” ] [ 0 ]PI = cons . p iRydhckB = cons . Rydberg∗cons . h∗cons . c / cons . kdef kION (n , T ) : # ra te constant f o r A + e −−> A+ + e wi th A at QN neps i = RydhckB / (pow( n , 2 . 0 )∗T)return 1.195770111325817e+07/pow(T , 1 . 5 )∗ exp(−eps i ) /\(pow( epsi , 7 . 0 / 3 . 0 )+4 .38∗pow( epsi ,1 .72)+1.32∗ eps i ) ;def kTBR(n , T ) : # ra te constant f o r A+ + e− + e−−−> A + e− ( prop . to k ion )eps i = RydhckB / (pow( n ,2 )∗T)k t b r = 4.952079396744990e+03∗pow( n , 2 . 0 ) /pow(T , 3 . 0 ) /\(pow( epsi , 7 . 0 / 3 . 0 )+4 .38∗pow( epsi ,1 .72)+1.32∗ eps i ) ;return k t b rdef kn np ( ni , nf , T ) :i f n i == nf :knn = 0.else :e i = RydhckB / (pow( n i , 2 . 0 )∗T)e f = RydhckB / (pow( nf , 2 . 0 )∗T)eg = max( e i , e f )es = min ( e i , e f )de = fabs ( ef−e i )knn = (2.735776154492852e+03/T∗pow( e i , 2 . 5 )∗pow( ef , 1 . 5 ) /pow( eg , 2 . 5 ) ) \∗exp(−( e i−es ) ) ∗ ( ( 2 2 . 0 / (pow( eg +0 .9 , 7 . 0 / 3 . 0 ) ) ) +\( 4 . 5 / (pow( eg , 2 . 5 )∗pow( de , 4 . 0 / 3 . 0 ) ) ) )109return knn ;def kPD( n i ) :au = 1 . / ( 2 . 4 2 e−17) #s# tau = ones (5)∗0.046tau = ar ray ( [0 .014 ,0 .046 ,0 .029 ,0 .0012 ,0 .00003 ] )extendtau = 0.00003∗ones ( ni−len ( tau ) )tau = append ( tau , extendtau )l l i s t = arange ( f l oa t ( n i ) )gsum = n i ∗∗2.gvalue = ar ray ( ( 2 .∗ l l i s t [ : ] + 1 . )∗ tau [ : ] )return 1.0e−6∗(sum( gvalue ) / gsum )∗ ( 1 . / ( 2 .∗ PI∗ f l oa t ( n i )∗∗3 . ) )∗audef kIONep ( epsi , T , term ) :# ra te constant f o r A + e −−> A+ + e wi th A at QN n# term = (pow( epsi , 7 . 0 / 3 . 0 )+4 .38∗pow( epsi ,1 .72)+1.32∗ eps i )return 1.195770111325817e+07/pow(T , 1 . 5 )∗ exp(−eps i ) / term ;def kTBRep(n , epsi , T , term ) :# ra te constant f o r A+ + e− + e−−−> A + e− ( prop . to k ion )k t b r = 4.952079396744990e+03∗pow( n , 2 . 0 ) /pow(T , 3 . 0 ) / term ;return k t b rdef kn npep ( ei , ef , T ) :i f e i == ef :knnp = 0.0knpn = 0.0else :eg = max( e i , e f )es = min ( e i , e f )de = fabs ( ef−e i )knnterm = 2.735776154492852e+03/T /pow( eg ,2 .5 )∗\( ( 2 2 . 0 / (pow( eg + 0 . 9 , 7 . 0 / 3 . 0 ) ) ) + ( 4 . 5 / (pow( eg , 2 . 5 )∗pow( de , 4 . 0 / 3 . 0 ) ) ) )knnp = pow( e i , 2 . 5 )∗pow( ef , 1 . 5 )∗ exp(−( e i−es ) )∗ knntermknpn = pow( ef , 2 . 5 )∗pow( e i , 1 . 5 )∗ exp(−( ef−es ) )∗ knntermreturn knnp , knpn” ” ”def kDR(T ) :r e t u rn 5.623e−6∗pow(T , −0.528)#From Schneider e t a l , F ig . 7 : 9.1e−6∗Te∗∗(−0.54)def kNOTds(T ) :110r e t u rn 1e6∗pow( echg , 4 . ) / ( kB∗T∗ sq r t ( emass∗Rydhc)∗pow(4 .∗ PI∗eps i lon , 2 . ) ) ;def kIONds (n , T ) :# ra te constant f o r A + e −−> A+ + e wi th A at QN neps i = Rydhc / ( pow(n , 2 . )∗ kB∗T ) ;r e t u rn (11.∗ sq r t ( Rydhc / ( kB∗T) )∗kNOTds(T)∗exp(−eps i ) ) / ( pow( epsi , ( 7 . / 3 . ) ) \+ 4.38∗pow( epsi , 1 . 72 ) + 1.32∗ eps i ) ;def kTBRds (n , T ) :# ra te constant f o r A+ + e− + e−−−> A + e− ( prop . to k ion )eps i = Rydhc / ( pow(n , 2 . )∗kB∗T ) ;lmbd = 1e2∗h / sq r t (2 .∗ PI∗emass∗kB∗T ) ;# thermal de Brog l i e wavelength quan t i f y i ng average gas# p a r t i c l e wavelength : lmbd << a , c l a s s i c a l M−B gask t b r = kION (n , T)∗pow(n , 2 . )∗pow( lmbd , 3 . )∗ exp ( eps i ) ;r e t u rn k t b rdef kn npds ( ni , nf , T ) :i f n i == nf :knn = 0.e lse :eppsi = Rydhc / ( pow( ni , 2 . )∗ kB∗T ) ;epsf = Rydhc / ( pow( nf , 2 . )∗ kB∗T ) ;knn = (kNOTds(T)∗pow( eppsi , ( 5 . / 2 . ) ) ∗ pow( epsf , ( 3 . / 2 . ) ) / \pow( max( epsf , eppsi ) , ( 5 . / 2 . ) ) )∗exp ( −(eppsi − min ( epsf , eppsi ) ) )∗ \( ( 2 2 . / ( pow( (max( epsf , eppsi ) + 0 . 9 ) , ( 7 . / 3 . ) ) ) ) + ( ( 9 . / 2 . ) \/ (pow( max( epsf , eppsi ) , ( 5 . / 2 . ) )∗ ( pow( fabs ( epsf−eppsi ) , \( 4 . / 3 . ) ) ) ) ) ) ;r e t u rn knn ;def kDRds (T ) :r e t u rn 5.623e−6∗pow(T , −0.528)#From Schneider e t a l , F ig . 7 : 9.1e−6∗Te∗∗(−0.54)def kPDds ( se l f , n i ) :au = 1 . / ( 2 . 4 2 e−17) #s# tau = ones (5)∗0.046tau = ar ray ( [0 .014 ,0 .046 ,0 .029 ,0 .0012 ,0 .00003 ] )extendtau = 0.00003∗ones ( ni−len ( tau ) )tau = append ( tau , extendtau )l l i s t = arange ( f l o a t ( n i ) )gsum = n i ∗∗2.gvalue = ar ray ( ( 2 .∗ l l i s t [ : ] + 1 . )∗ tau [ : ] )r e t u rn (sum( gvalue ) / gsum )∗ ( 1 . / ( 2 .∗ PI∗ f l o a t ( n i )∗∗3 . ) )∗au” ” ”111A.4.2 Coupled rate equation: setting up the parameters and initialvalue problem” ” ”@author : hossein” ” ”import mathfrom re funs . RateConstants import kIONep , kTBRep , kn npep , kPDfrom sc ipy . i n t eg r a t e import ode in tfrom sc ipy . constants import constants as consimport sc ipyimport numpy as npphyscons = sc ipy . constants . phys i ca l cons tan tsa0 = 1.0e6 ∗ physcons [ ” Bohr rad ius ” ] [ 0 ]Rydhc = cons . Rydberg∗cons . h∗cons . ckB = cons . kRydhckB = cons . Rydberg∗cons . h∗cons . c / cons . kclass Rateeq ( object ) :” ” ” parameters f o r ra te equat ion model ” ” ”def i n i t ( se l f , n0 = 50 , numlev = 60 , dens i t y = 0.001 , \ns t a r t = 10 , t f = 0.05 , t b i n = 50 , r e l e r r = 1e−10):s e l f . numlev = numlevs e l f . n0 = n0s e l f . n s t a r t = n s t a r ts e l f . dens i t y = dens i t ys e l f . n0Ind = s e l f . n0 − s e l f . n s t a r ts e l f . nsq r t = i n t ( round ( s e l f . n0 / math . sq r t ( 2 . 0 ) ) )s e l f . nsq r t I nd = s e l f . nsq r t − s e l f . n s t a r ts e l f . n l i s t = np . ar ray ( range ( s e l f . ns ta r t , s e l f . n s t a r t + s e l f . numlev ) )s e l f . neqn = s e l f . numlev + 3s e l f . y0 = np . zeros ( s e l f . neqn , dtype=np . f l o a t 64 )s e l f . t f = t fs e l f . t b i n = t b i ns e l f . r e l e r r = r e l e r rs e l f . b u i l d t ( )s e l f . bu i l dy0 ( )s e l f . bu i l d y ( )def nden ( s e l f ) :return s e l f . y [ : , 0 : s e l f . numlev ]def eden ( s e l f ) :return s e l f . y [ : , s e l f . numlev ]112def aden ( s e l f ) :return s e l f . y [ : , s e l f . numlev+1]def T( s e l f ) :return s e l f . y [ : , s e l f . numlev+2]def n0den ( s e l f ) :return s e l f . y [ s e l f . n0Ind ]def nden0 ( s e l f ) :return s e l f . y0 [ 0 : s e l f . numlev ]def eden0 ( s e l f ) :return s e l f . y0 [ s e l f . numlev ]def aden0 ( s e l f ) :return s e l f . y0 [ s e l f . numlev+1]def T0 ( s e l f ) :return s e l f . y0 [ s e l f . numlev+2]def n0den0 ( s e l f ) :return s e l f . y0 [ s e l f . n0Ind ]def penn i ng f r ac t i on ( s e l f ) :a = 4.0∗np . p i∗ s e l f . dens i t y / 3 . 0Rmax = 2.0 ∗ a0 ∗ pow( f l oa t ( s e l f . n0 ) , 2 . 0 ) ∗ 1.8return 0.9∗(1−math . exp(−a∗Rmax∗∗3))def calc eden0 ( s e l f ) :eden0 = s e l f . penn i ng f r ac t i on ( )∗ s e l f . dens i t y / 2 . 0s e l f . y0 [ s e l f . numlev ] = eden0return eden0def calc n0den0 ( s e l f ) :n0den0 = (1− s e l f . penn i ng f r ac t i on ( ) )∗ s e l f . dens i t ys e l f . y0 [ s e l f . n0Ind ] = n0den0return n0den0def ca lc Penn ing ion ized ( s e l f ) :n p l i s t = np . ar ray ( range ( s e l f . ns ta r t , s e l f . nsq r t +1) )index = np . ar ray ( range ( s e l f . nsq r t I nd +1) )asser t len ( index )== len ( n p l i s t )pop = np . zeros ( len ( index ) , dtype=np . f l o a t 64 )pop = pow( n p l i s t / f l oa t ( s e l f . n0 ) , 5 . 0 )popto t = sum( pop )p f = s e l f . penn i ng f r ac t i on ( )pop = pop / popto ts e l f . y0 [ index ] = pop∗pf /2 .0∗ s e l f . dens i t yreturn popdef bu i ldy0 ( s e l f ) :s e l f . calc n0den0 ( )s e l f . calc eden0 ( )s e l f . ca lc Penn ing ion ized ( )s e l f . ca lc T ( )def bu i l d y ( s e l f ) :s e l f . y = np . ar ray ( [ [ 0 . 0 for j in range ( s e l f . t b i n ) ] \for i in range ( s e l f . neqn ) ] , dtype=np . f l o a t 64 )113def ca lc T ( s e l f ) :FinalRydEnergy = 0for i in range ( s e l f . numlev ) :FinalRydEnergy += s e l f . nden0 ( ) [ i ] /pow( f l oa t ( s e l f . n l i s t [ i ] ) , 2 . 0 )T = −Rydhc∗( s e l f . dens i t y /pow( f l oa t ( s e l f . n0 ) , 2 . 0 ) − FinalRydEnergy )\/ ( 1 . 5∗ kB∗ s e l f . eden0 ( ) )s e l f . y0 [ s e l f . numlev+2] = Treturn Tdef b u i l d t ( s e l f ) :h = s e l f . t f / ( f l oa t ( s e l f . t b i n )−1.0)t a r r a y = np . zeros ( s e l f . t b in , dtype=np . f l o a t 64 )for i in range ( s e l f . t b i n ) :t a r r a y [ i ]= h ∗ is e l f . t = t a r r a yclass RateeqPI ( object ) :” ” ” parameters f o r ra te equat ion model ” ” ”def i n i t ( se l f , T = 100 , numlev = 60 , dens i t y = 0.001 ,\ns t a r t = 10 , t f = 0.05 , t b i n = 50 , r e l e r r = 1e−10):s e l f . numlev = numlevs e l f . T = Ts e l f . n s t a r t = n s t a r ts e l f . dens i t y = dens i t ys e l f . n l i s t = np . ar ray ( range ( s e l f . ns ta r t , s e l f . n s t a r t + s e l f . numlev ) )s e l f . neqn = s e l f . numlev + 3s e l f . y0 = np . zeros ( s e l f . neqn , dtype=np . f l o a t 64 )s e l f . t f = t fs e l f . t b i n = t b i ns e l f . r e l e r r = r e l e r rs e l f . b u i l d t ( )s e l f . bu i l dy0 ( )def nden ( s e l f ) :return s e l f . y [ : , 0 : s e l f . numlev ]def eden ( s e l f ) :return s e l f . y [ : , s e l f . numlev ]def aden ( s e l f ) :return s e l f . y [ : , s e l f . numlev+1]def Temperature ( s e l f ) :return s e l f . y [ : , s e l f . numlev+2]def nden0 ( s e l f ) :return s e l f . y0 [ 0 : s e l f . numlev ]def eden0 ( s e l f ) :return s e l f . y0 [ s e l f . numlev ]def aden0 ( s e l f ) :return s e l f . y0 [ s e l f . numlev+1]114def T0 ( s e l f ) :return s e l f . y0 [ s e l f . numlev+2]def bu i ldy0 ( s e l f ) :s e l f . y0 [ s e l f . numlev ] = s e l f . dens i t ys e l f . y0 [ s e l f . numlev + 2] = s e l f . Tdef b u i l d t ( s e l f ) :h = s e l f . t f / ( f l oa t ( s e l f . t b i n )−1.0)t a r r a y = np . zeros ( s e l f . t b in , dtype=np . f l o a t 64 )for i in range ( s e l f . t b i n ) :t a r r a y [ i ]= h ∗ is e l f . t = t a r r a yclass RateeqAtomic ( Rateeq ) :def i n t e g r a t e ( s e l f ) :numlev = s e l f . numlevn l i s t f = np . zeros ( numlev , dtype=np . f l o a t 64 )for i in range ( numlev ) :n l i s t f [ i ] = f l oa t ( s e l f . n l i s t [ i ] )dy = np . zeros ( numlev+3 , dtype=np . f l o a t 64 )numlevrange = range ( numlev )def dydt ( y , t ) :nden = y [ 0 : numlev ]eden = y [ numlev ]T = y [ numlev+2]deden = 0.0dT = 0.0for i in numlevrange :eps i = RydhckB / (pow( n l i s t f [ i ] , 2 . 0 )∗T)term = pow( epsi , 7 . 0 / 3 . 0 )+4 .38∗pow( epsi ,1 .72)+1.32∗ eps ik t b r = kTBRep( n l i s t f [ i ] , epsi , T , term )∗pow( eden , 3 . 0 )k ion = kIONep ( epsi , T , term )∗eden∗nden [ i ]deden += kion − k t b rtmp knnp = 0.0for j in numlevrange :epsf = RydhckB / (pow( n l i s t f [ j ] , 2 . 0 )∗T)knnp , knpn = kn npep ( epsi , epsf , T )tmp knnp = tmp knnp − knnp∗nden [ i ]∗eden \+ knpn∗nden [ j ]∗edendy [ i ] = tmp knnp + k t b r − k iondT += RydhckB ∗ dy [ i ] /pow( n l i s t f [ i ] , 2 . 0 )dT = (dT − 1.5∗T∗deden ) / ( 1 . 5∗ eden )dy [ numlev ] = dedendy [ numlev+1] = 0.0 # change of atomic species , zero when no PDdy [ numlev+2] = dT115return dys e l f . y = ode in t ( lambda y , t : dydt ( y , t ) ,\s e l f . y0 , s e l f . t , r t o l = s e l f . r e l e r r )s e l f . dy = dydt ( s e l f . y0 , s e l f . t )class RateeqAtomicPI ( RateeqPI ) :def i n t e g r a t e ( s e l f ) :numlev = s e l f . numlevn l i s t f = np . zeros ( numlev , dtype=np . f l o a t 64 )for i in range ( numlev ) :n l i s t f [ i ] = f l oa t ( s e l f . n l i s t [ i ] )dy = np . zeros ( numlev+3 , dtype=np . f l o a t 64 )numlevrange = range ( numlev )def dydt ( y , t ) :nden = y [ 0 : numlev ]eden = y [ numlev ]T = y [ numlev+2]deden = 0.0dT = 0.0for i in numlevrange :eps i = RydhckB / (pow( n l i s t f [ i ] , 2 . 0 )∗T)term = pow( epsi , 7 . 0 / 3 . 0 )+4 .38∗pow( epsi ,1 .72)+1.32∗ eps ik t b r = kTBRep( n l i s t f [ i ] , epsi , T , term )∗pow( eden , 3 . 0 )k ion = kIONep ( epsi , T , term )∗eden∗nden [ i ]deden += kion − k t b rtmp knnp = 0.0for j in numlevrange :epsf = RydhckB / (pow( n l i s t f [ j ] , 2 . 0 )∗T)knnp , knpn = kn npep ( epsi , epsf , T )tmp knnp = tmp knnp − knnp∗nden [ i ]∗eden \+ knpn∗nden [ j ]∗edendy [ i ] = tmp knnp + k t b r − k iondT += RydhckB ∗ dy [ i ] /pow( n l i s t f [ i ] , 2 . 0 )dT = (dT − 1.5∗T∗deden ) / ( 1 . 5∗ eden )dy [ numlev ] = dedendy [ numlev+1] = 0.0 # change of atomic species , zero when no PDdy [ numlev+2] = dTreturn dys e l f . y = ode in t ( lambda y , t : dydt ( y , t ) , s e l f . y0 , s e l f . t ,\r t o l = s e l f . r e l e r r )s e l f . dy = dydt ( s e l f . y0 , s e l f . t )class RateeqMolecular ( Rateeq ) :def i n t e g r a t e ( s e l f ) :numlev = s e l f . numlevn l i s t f = np . zeros ( numlev , dtype=np . f l o a t 64 )116for i in range ( numlev ) :n l i s t f [ i ] = f l oa t ( s e l f . n l i s t [ i ] )dy = np . zeros ( numlev+3 , dtype=np . f l o a t 64 )numlevrange = range ( numlev )def dydt ( y , t ) :nden = y [ 0 : numlev ]eden = y [ numlev ]T = y [ numlev+2]deden = 0.0daden = 0.0dT = 0.0for i in numlevrange :eps i = RydhckB / (pow( n l i s t f [ i ] , 2 . 0 )∗T)term = pow( epsi , 7 . 0 / 3 . 0 )+4 .38∗pow( epsi ,1 .72)+1.32∗ eps ik t b r = kTBRep( n l i s t f [ i ] , epsi , T , term )∗pow( eden , 3 . 0 )k ion = kIONep ( epsi , T , term )∗eden∗nden [ i ]deden += kion − k t b rdadentemp = kPD( n l i s t f [ i ] )∗ nden [ i ]daden += dadentemptmp knnp = 0.0for j in numlevrange :epsf = RydhckB / (pow( n l i s t f [ j ] , 2 . 0 )∗T)knnp , knpn = kn npep ( epsi , epsf , T )tmp knnp = tmp knnp − knnp∗nden [ i ]∗eden\+ knpn∗nden [ j ]∗edendy [ i ] = tmp knnp + k t b r − k iondT += RydhckB ∗ dy [ i ] /pow( n l i s t f [ i ] , 2 . 0 )dy [ i ] = dy [ i ] − dadentempdT = (dT − 1.5∗T∗deden ) / ( 1 . 5∗ eden )dy [ numlev ] = dedendy [ numlev+1] = dadendy [ numlev+2] = dTreturn dys e l f . y = ode in t ( lambda y , t : dydt ( y , t ) , s e l f . y0 , s e l f . t ,\r t o l = s e l f . r e l e r r )s e l f . dy = dydt ( s e l f . y0 , s e l f . t )A.4.3 Coupled rate equation: numerical integrationimport numpy as npimport pyopencl as c limport osimport pyopencl . a r ray as c l a r r a y117from re funs . Rateeq import Rateeq , RateeqPIfrom sc ipy . i n t eg r a t e import ode in tfrom sc ipy . constants import constants as consRydhc = cons . Rydberg∗cons . h∗cons . ckB = cons . kRydhckB = cons . Rydberg∗cons . h∗cons . c / cons . kclass ReKernel ( ) :def i n i t ( se l f , numlev , c t x = None , queue = None ) :s e l f . c t x = c txs e l f . queue = queues e l f . numlev = numlevi f s e l f . c t x == None :s e l f . c t x = c l . c reate some context ( )i f s e l f . queue == None :s e l f . queue = c l .CommandQueue( s e l f . c tx ,p rope r t i e s= c l . command queue properties .PROFILING ENABLE)s e l f . mf = c l . mem flagst ry :absolutePathToKernels = os . path . dirname (os . path . rea lpa th ( f i l e ) )s rc = open ( absolutePathToKernels + ’ / reKerne l . c l ’ ,’ r ’ ) . read ( )except :s rc = open ( r ’C:\Users\User\Dropbox\ ra teeq\ re funs\ reKerne l . c l ’ ,’ r ’ ) . read ( )s e l f . compknnpF = c l . Program ( s e l f . c tx , s rc )t ry :s e l f . compknnpF . bu i l d ( op t ions =[ ’−DNUMLEV= ’+st r ( s e l f . numlev ) ] )except :pr in t ( ” E r ro r : ” )pr in t ( s e l f . compknnpF . g e t b u i l d i n f o ( s e l f . c t x . devices [ 0 ] ,c l . p rog ram bu i l d i n f o .LOG) )raises e l f . compknnpF . reKerne l . se t s ca l a r a rg d t ypes ([ None , None , np . f l oa t32 , np . f l oa t32 , np . f l oa t32 ,np . in t32 , None , None , None ] )s e l f . compknnpD = c l . Program ( s e l f . c tx , s rc )t ry :s e l f . compknnpD . bu i l d ( op t ions =[ ’−DNUMLEV= ’+st r ( s e l f . numlev ) ] )except :118pr in t ( ” E r ro r : ” )pr in t ( s e l f . compknnpD . g e t b u i l d i n f o ( s e l f . c t x . devices [ 0 ] ,c l . p rog ram bu i l d i n f o .LOG) )raises e l f . compknnpD . reKerne l . se t s ca l a r a rg d t ypes ([ None , None , np . f l oa t64 , np . f l oa t64 , np . f l oa t64 ,np . in t32 , None , None , None ] )def computeKnnp ( se l f , nden , n l i s t , T , eden , eden3 , numlev ,dnden , k tb r , k ion ) :prec = nden . dtypei f prec == np . f l o a t 32 :s e l f . compknnpF . reKerne l ( s e l f . queue ,( s e l f . numlev , 1 ) ,( s e l f . numlev , 1 ) ,nden . data , n l i s t . data ,np . f l o a t 32 (T ) , np . f l o a t 32 ( eden ) , np . f l o a t 32 ( eden3 ) ,np . i n t 32 ( numlev ) ,dnden . data , k t b r . data , k ion . data ,g t imes l = False )e l i f prec == np . f l o a t 64 :s e l f . compknnpD . reKerne l ( s e l f . queue ,( s e l f . numlev , 1 ) ,( s e l f . numlev , 1 ) ,nden . data , n l i s t . data ,np . f l o a t 64 (T ) , np . f l o a t 64 ( eden ) , np . f l o a t 64 ( eden3 ) ,np . i n t 32 ( numlev ) ,dnden . data , k t b r . data , k ion . data ,g t imes l = False )else :pr in t ( ”Unknown f l o a t type . ” )class RateeqAtomicGPU ( Rateeq ) :def i n i t ( se l f , n0 = 50 , numlev = 60 , dens i t y = 0.001 , n s t a r t = 10 ,\t f = 0.05 , t b i n = 50 , r e l e r r = 1e−10, c t x = None , queue = None ) :Rateeq . i n i t ( se l f , n0 , numlev , dens i t y , n s t a r t , t f , t b i n\, r e l e r r )i f c tx == None :s e l f . c t x = c l . c reate some context ( i n t e r a c t i v e = True )s e l f . queue = c l .CommandQueue( s e l f . c t x )else :119s e l f . c t x = c txs e l f . queue = queues e l f . computeKnnp = ReKernel ( numlev , s e l f . c tx , s e l f . queue )def i n t e g r a t e ( s e l f ) :numlev = s e l f . numlevn l i s t = s e l f . n l i s tdy = np . zeros ( numlev+3)dnden = np . zeros ( numlev )k t b r k i on = np . zeros ( numlev )dndenn l i s t = np . zeros ( numlev )queue = s e l f . queuecomputeKnnp = s e l f . computeKnnpn l i s t d e v = c l a r r a y . to dev i ce ( queue , n l i s t )nden dev = c l a r r a y . to dev i ce ( queue , dnden )dnden dev = c l a r r a y . z e r o s l i k e ( nden dev )k tb rk ion dev = c l a r r a y . z e r o s l i k e ( nden dev )dndenn l i s t dev = c l a r r a y . z e r o s l i k e ( nden dev )def dydt ( y , t ) :eden = y [ numlev ]eden3 = eden∗eden∗edenT = y [ numlev+2]nden dev = c l a r r a y . to dev i ce ( queue , y [ 0 : numlev ] )computeKnnp . computeKnnp ( nden dev , n l i s t dev , T , eden , eden3 ,numlev , dnden dev , k tb rk ion dev , dndenn l i s t dev )dnden dev . get ( queue , dnden )k tb rk ion dev . get ( queue , k t b r k i on )dndenn l i s t dev . get ( queue , dndenn l i s t )deden = −sum( k t b r k i on )dT = RydhckB ∗ sum( dndenn l i s t )dT = (dT − 1.5∗T∗deden ) / ( 1 . 5∗ eden )dy [ 0 : numlev ] = dndendy [ numlev ] = dedendy [ numlev+1] = 0.0 # change of atomic species , zero when no PDdy [ numlev+2] = dTreturn dys e l f . y = ode in t ( lambda y , t : dydt ( y , t ) , np . ar ray ( s e l f . y0 ) , s e l f . t ,r t o l = s e l f . r e l e r r )s e l f . dy = dydt ( s e l f . y0 , s e l f . t )class RateeqAtomicDRGPU( Rateeq ) :120def i n i t ( se l f , n0 = 50 , numlev = 60 , dens i t y = 0.001 , n s t a r t = 10 ,t f = 0.05 , t b i n = 50 , r e l e r r = 1e−10, c t x = None , queue = None ) :Rateeq . i n i t ( se l f , n0 , numlev , dens i t y , n s t a r t , t f ,t b i n , r e l e r r )i f c tx == None :s e l f . c t x = c l . c reate some context ( i n t e r a c t i v e = True )s e l f . queue = c l .CommandQueue( s e l f . c t x )else :s e l f . c t x = c txs e l f . queue = queues e l f . computeKnnp = ReKernel ( numlev , s e l f . c tx , s e l f . queue )def i n t e g r a t e ( s e l f ) :numlev = s e l f . numlevn l i s t = s e l f . n l i s tdy = np . zeros ( numlev+3)dnden = np . zeros ( numlev )k t b r k i on = np . zeros ( numlev )dndenn l i s t = np . zeros ( numlev )queue = s e l f . queuecomputeKnnp = s e l f . computeKnnpn l i s t d e v = c l a r r a y . to dev i ce ( queue , n l i s t )nden dev = c l a r r a y . to dev i ce ( queue , dnden )dnden dev = c l a r r a y . z e r o s l i k e ( nden dev )k tb rk ion dev = c l a r r a y . z e r o s l i k e ( nden dev )dndenn l i s t dev = c l a r r a y . z e r o s l i k e ( nden dev )def dydt ( y , t ) :eden = y [ numlev ]eden3 = eden∗eden∗edenT = y [ numlev+2]kDR = 5.623∗pow(T , −0.528)nden dev = c l a r r a y . to dev i ce ( queue , y [ 0 : numlev ] )computeKnnp . computeKnnp ( nden dev , n l i s t dev , T , eden , eden3 ,numlev , dnden dev , k tb rk ion dev , dndenn l i s t dev )dnden dev . get ( queue , dnden )k tb rk ion dev . get ( queue , k t b r k i on )dndenn l i s t dev . get ( queue , dndenn l i s t )deden = −sum( k t b r k i on )dedenDR = −kDR∗eden∗edendT = RydhckB ∗ sum( dndenn l i s t )dT = (dT − 1.5∗T∗deden ) / ( 1 . 5∗ eden )121dy [ 0 : numlev ] = dndendy [ numlev ] = deden + dedenDRdy [ numlev+1] = 0.0 # change of atomic species , zero when no PDdy [ numlev+2] = dTreturn dys e l f . y = ode in t ( lambda y , t : dydt ( y , t ) , np . ar ray ( s e l f . y0 ) , s e l f . t ,r t o l = s e l f . r e l e r r )s e l f . dy = dydt ( s e l f . y0 , s e l f . t )class RateeqPIGPU ( RateeqPI ) :def i n i t ( se l f , T = 100 , numlev = 60 , dens i t y = 0.001 , n s t a r t = 10 ,t f = 0.05 , t b i n = 50 , r e l e r r = 1e−10):RateeqPI . i n i t ( se l f , T , numlev , dens i t y , n s t a r t , t f , t b i n ,r e l e r r )s e l f . c t x = c l . c reate some context ( i n t e r a c t i v e = True )s e l f . queue = c l .CommandQueue( s e l f . c t x )s e l f . computeKnnp = ReKernel ( numlev , s e l f . c tx , s e l f . queue )def i n t e g r a t e ( s e l f ) :numlev = s e l f . numlevn l i s t = s e l f . n l i s t ;dy = np . zeros ( numlev+3)dnden = np . zeros ( numlev )k t b r k i on = np . zeros ( numlev )dndenn l i s t = np . zeros ( numlev )queue = s e l f . queuecomputeKnnp = s e l f . computeKnnpn l i s t d e v = c l a r r a y . to dev i ce ( queue , n l i s t )nden dev = c l a r r a y . to dev i ce ( queue , dnden )dnden dev = c l a r r a y . z e r o s l i k e ( nden dev )k tb rk ion dev = c l a r r a y . z e r o s l i k e ( nden dev )dndenn l i s t dev = c l a r r a y . z e r o s l i k e ( nden dev )def dydt ( y , t ) :eden = y [ numlev ]eden3 = eden∗eden∗edenT = y [ numlev+2]nden dev = c l a r r a y . to dev i ce ( queue , y [ 0 : numlev ] )computeKnnp . computeKnnp ( nden dev , n l i s t dev , T , eden , eden3 ,numlev , dnden dev , k tb rk ion dev , dndenn l i s t dev )dnden dev . get ( queue , dnden )k tb rk ion dev . get ( queue , k t b r k i on )122dndenn l i s t dev . get ( queue , dndenn l i s t )deden = −sum( k t b r k i on )dT = RydhckB ∗ sum( dndenn l i s t )dT = (dT − 1.5∗T∗deden ) / ( 1 . 5∗ eden )dy [ 0 : numlev ] = dndendy [ numlev ] = dedendy [ numlev+1] = 0.0 # change of atomic species , zero when no PDdy [ numlev+2] = dTreturn dys e l f . y = ode in t ( lambda y , t : dydt ( y , t ) , np . ar ray ( s e l f . y0 ) , s e l f . t ,r t o l = s e l f . r e l e r r )s e l f . dy = dydt ( s e l f . y0 , s e l f . t )import numpy as npimport pyopencl as c limport osimport pyopencl . a r ray as c l a r r a yfrom re funs . Rateeq import Rateeq , RateeqPIfrom sc ipy . i n t eg r a t e import ode in tfrom sc ipy . constants import constants as consRydhc = cons . Rydberg∗cons . h∗cons . ckB = cons . kRydhckB = cons . Rydberg∗cons . h∗cons . c / cons . kclass ReKernel ( ) :def i n i t ( se l f , numlev , c t x = None , queue = None ) :s e l f . c t x = c txs e l f . queue = queues e l f . numlev = numlevi f s e l f . c t x == None :s e l f . c t x = c l . c reate some context ( )i f s e l f . queue == None :s e l f . queue = c l .CommandQueue( s e l f . c tx ,p rope r t i e s= c l . command queue properties .PROFILING ENABLE)s e l f . mf = c l . mem flagsabsolutePathToKernels = os . path . dirname (os . path . rea lpa th ( f i l e ) )s rc = open ( absolutePathToKernels + ’ / reKernelMol . c l ’ ,’ r ’ ) . read ( )s e l f . compknnpF = c l . Program ( s e l f . c tx , s rc )t ry :s e l f . compknnpF . bu i l d ( op t ions =[ ’−DNUMLEV= ’+st r ( s e l f . numlev ) ] )123except :pr in t ( ” E r ro r : ” )pr in t ( s e l f . compknnpF . g e t b u i l d i n f o ( s e l f . c t x . devices [ 0 ] ,c l . p rog ram bu i l d i n f o .LOG) )raises e l f . compknnpF . reKerne l . se t s ca l a r a rg d t ypes ([ None , None , np . f l oa t32 , np . f l oa t32 , np . f l oa t32 ,np . in t32 , None , None , None , None ] )s e l f . compknnpD = c l . Program ( s e l f . c tx , s rc )t ry :s e l f . compknnpD . bu i l d ( op t ions =[ ’−DNUMLEV= ’+st r ( s e l f . numlev ) ] )except :pr in t ( ” E r ro r : ” )pr in t ( s e l f . compknnpD . g e t b u i l d i n f o ( s e l f . c t x . devices [ 0 ] ,c l . p rog ram bu i l d i n f o .LOG) )raises e l f . compknnpD . reKerne l . se t s ca l a r a rg d t ypes ([ None , None , np . f l oa t64 , np . f l oa t64 , np . f l oa t64 ,np . in t32 , None , None , None , None ] )def computeKnnp ( se l f , nden , n l i s t , T , eden , eden3 , numlev ,dnden , k tb rk ion , dndenn l is t , PD) :prec = nden . dtypei f prec == np . f l o a t 32 :s e l f . compknnpF . reKerne l ( s e l f . queue ,( s e l f . numlev , 1 ) ,( s e l f . numlev , 1 ) ,nden . data , n l i s t . data ,np . f l o a t 32 (T ) , np . f l o a t 32 ( eden ) , np . f l o a t 32 ( eden3 ) ,np . i n t 32 ( numlev ) ,dnden . data , k t b r k i on . data , dndenn l i s t . data , PD. data ,g t imes l = False )e l i f prec == np . f l o a t 64 :s e l f . compknnpD . reKerne l ( s e l f . queue ,( s e l f . numlev , 1 ) ,( s e l f . numlev , 1 ) ,nden . data , n l i s t . data ,np . f l o a t 64 (T ) , np . f l o a t 64 ( eden ) , np . f l o a t 64 ( eden3 ) ,np . i n t 32 ( numlev ) ,dnden . data , k t b r k i on . data , dndenn l i s t . data , PD. data ,124g t imes l = False )else :pr in t ( ”Unknown f l o a t type . ” )class RateeqMolecularGPU ( Rateeq ) :def i n i t ( se l f , n0 = 50 , numlev = 60 , dens i t y = 0.001 , n s t a r t = 10 ,t f = 0.05 , t b i n = 50 , r e l e r r = 1e−10, c t x = None , queue = None ) :Rateeq . i n i t ( se l f , n0 , numlev , dens i t y , n s t a r t , t f , t b i n ,r e l e r r )i f c tx == None :s e l f . c t x = c l . c reate some context ( i n t e r a c t i v e = True )s e l f . queue = c l .CommandQueue( s e l f . c t x )else :s e l f . c t x = c txs e l f . queue = queues e l f . computeKnnp = ReKernel ( numlev , s e l f . c tx , s e l f . queue )def i n t e g r a t e ( s e l f ) :numlev = s e l f . numlevn l i s t = s e l f . n l i s t ;dy = np . zeros ( numlev+3)dnden = np . zeros ( numlev )k t b r k i on = np . zeros ( numlev )dndenn l i s t = np . zeros ( numlev )PD = np . zeros ( numlev )queue = s e l f . queuecomputeKnnp = s e l f . computeKnnpn l i s t d e v = c l a r r a y . to dev i ce ( queue , n l i s t )nden dev = c l a r r a y . to dev i ce ( queue , dnden )dnden dev = c l a r r a y . z e r o s l i k e ( nden dev )k tb rk ion dev = c l a r r a y . z e r o s l i k e ( nden dev )dndenn l i s t dev = c l a r r a y . z e r o s l i k e ( nden dev )PD dev = c l a r r a y . z e r o s l i k e ( nden dev )def dydt ( y , t ) :eden = y [ numlev ]eden3 = eden∗eden∗edenT = y [ numlev+2]nden dev = c l a r r a y . to dev i ce ( queue , y [ 0 : numlev ] )computeKnnp . computeKnnp ( nden dev , n l i s t dev , T , eden , eden3 ,numlev , dnden dev , k tb rk ion dev , dndenn l is t dev , PD dev )dnden dev . get ( queue , dnden )k tb rk ion dev . get ( queue , k t b r k i on )dndenn l i s t dev . get ( queue , dndenn l i s t )125PD dev . get ( queue , PD)deden = −sum( k t b r k i on )dT = RydhckB ∗ sum( dndenn l i s t )dT = (dT − 1.5∗T∗deden ) / ( 1 . 5∗ eden )dy [ 0 : numlev ] = dndendy [ numlev ] = dedendy [ numlev+1] = sum(PD) # change of atomic species , zero when no PDdy [ numlev+2] = dTreturn dys e l f . y = ode in t ( lambda y , t : dydt ( y , t ) , np . ar ray ( s e l f . y0 ) , s e l f . t ,r t o l = s e l f . r e l e r r )s e l f . dy = dydt ( s e l f . y0 , s e l f . t )class RateeqMolecularPIGPU ( RateeqPI ) :def i n i t ( se l f , T = 100 , numlev = 60 , dens i t y = 0.001 , n s t a r t = 10 ,t f = 0.05 , t b i n = 50 , r e l e r r = 1e−10):RateeqPI . i n i t ( se l f , T , numlev , dens i t y , n s t a r t , t f , t b i n ,r e l e r r )s e l f . c t x = c l . c reate some context ( i n t e r a c t i v e = True )s e l f . queue = c l .CommandQueue( s e l f . c t x )s e l f . computeKnnp = ReKernel ( numlev , s e l f . c tx , s e l f . queue )def i n t e g r a t e ( s e l f ) :numlev = s e l f . numlevn l i s t = s e l f . n l i s t ;dy = np . zeros ( numlev+3)dnden = np . zeros ( numlev )k t b r k i on = np . zeros ( numlev )dndenn l i s t = np . zeros ( numlev )PD = np . zeros ( numlev )queue = s e l f . queuecomputeKnnp = s e l f . computeKnnpn l i s t d e v = c l a r r a y . to dev i ce ( queue , n l i s t )nden dev = c l a r r a y . to dev i ce ( queue , dnden )dnden dev = c l a r r a y . z e r o s l i k e ( nden dev )k tb rk ion dev = c l a r r a y . z e r o s l i k e ( nden dev )dndenn l i s t dev = c l a r r a y . z e r o s l i k e ( nden dev )PD dev = c l a r r a y . z e r o s l i k e ( nden dev )def dydt ( y , t ) :eden = y [ numlev ]eden3 = eden∗eden∗edenT = y [ numlev+2]nden dev = c l a r r a y . to dev i ce ( queue , y [ 0 : numlev ] )computeKnnp . computeKnnp ( nden dev , n l i s t dev , T , eden , eden3 ,126numlev , dnden dev , k tb rk ion dev , dndenn l is t dev , PD dev )dnden dev . get ( queue , dnden )k tb rk ion dev . get ( queue , k t b r k i on )dndenn l i s t dev . get ( queue , dndenn l i s t )PD dev . get ( queue , PD)deden = −sum( k t b r k i on )dT = RydhckB ∗ sum( dndenn l i s t )dT = (dT − 1.5∗T∗deden ) / ( 1 . 5∗ eden )dy [ 0 : numlev ] = dndendy [ numlev ] = dedendy [ numlev+1] = sum(PD) # change of atomic species , zero when no PDdy [ numlev+2] = dTreturn dys e l f . y = ode in t ( lambda y , t : dydt ( y , t ) , np . ar ray ( s e l f . y0 ) , s e l f . t ,r t o l = s e l f . r e l e r r )s e l f . dy = dydt ( s e l f . y0 , s e l f . t )i f name == ’ ma in ’ :p = RateeqMolecularGPU ( )p . i n t eg r a t e ( )A.4.4 OpenCL kernels for parallel computing# i f de f c l k h r f p 64#pragma OPENCL EXTENSION c l k h r f p 64 : enable#endif#define RydhckB 157887.52145860967ke r ne l void reKerne l (const g l o ba l double∗ nden ,const g l o ba l i n t∗ n l i s t ,double T , double eden , double eden3 , i n t numlev ,g l o ba l double∗ dnden , g l o ba l double∗ k tb rk ion , g l o ba l double∗ dndenn l i s t ){i n t i = g e t l o c a l i d ( 0 ) ;i f ( i<numlev ){double knnp = 0 .0 ;double knpn = 0 .0 ;double e i = RydhckB / ( pow( n l i s t [ i ] , 2 . 0 )∗T ) ;double ef ;double eg ;double es ;double de ;double t e rmtb r = (pow( ei , 7 . 0 / 3 . 0 )+4 .38∗pow( ei ,1 .72)+1.32∗ e i ) ;127double term = 0 .0 ;l o c a l double eps [NUMLEV] ;l o c a l double ndenl [NUMLEV] ;l o c a l double dndenl [NUMLEV] ;b a r r i e r (CLK LOCAL MEM FENCE ) ;eps [ i ] = e i ;ndenl [ i ] = nden [ i ] ;k t b r k i on [ i ] = ((4.952079396744990e+03∗pow( n l i s t [ i ] , 2 . 0 ) / pow(T , 3 . 0 )∗ eden3)−(1.195770111325817e+07/pow(T , 1 . 5 )∗ exp(−e i )∗eden∗ndenl [ i ] ) ) / t e rmtb r ;dndenl [ i ] = k t b r k i on [ i ] ;b a r r i e r (CLK LOCAL MEM FENCE ) ;for ( i n t j = 0 ; j<numlev ; ++ j ){i f ( i != j ){ef = eps [ j ] ;eg = max( ei , e f ) ;es = min ( ei , e f ) ;de = fabs ( ef−e i ) ;term = 2.735776154492852e+03/T∗ ( ( 2 2 . 0 / ( pow( eg +0 . 9 , 7 . 0 / 3 . 0 ) ) )\+ ( 4 . 5 / ( pow( eg , 2 . 5 )∗pow(de , 4 . 0 / 3 . 0 ) ) ) ) / pow( eg , 2 . 5 )∗ exp ( es ) ;knnp = pow( ei , 2 . 5 )∗pow( ef , 1 . 5 )∗ exp(−e i )∗ term ;knpn = pow( ef , 2 . 5 )∗pow( ei , 1 . 5 )∗ exp(−ef )∗ term ;dndenl [ i ] += eden∗(−knnp ∗ ndenl [ i ] + knpn ∗ ndenl [ j ] ) ;}}ba r r i e r (CLK LOCAL MEM FENCE ) ;dnden [ i ] = dndenl [ i ] ;dndenn l i s t [ i ] = dndenl [ i ] / pow( n l i s t [ i ] , 2 . 0 ) ;}}/ / OpenCL kerne l f o r coupled−r a te equat ion model# i f de f c l k h r f p 64#pragma OPENCL EXTENSION c l k h r f p 64 : enable#endif#define RydhckB 157887.52145860967ke r ne l void reKerne l (const g l o ba l double∗ nden ,const g l o ba l i n t∗ n l i s t ,double T , double eden , double eden3 , i n t numlev ,g l o ba l double∗ dnden , g l o ba l double∗ k tb rk ion ,g l o ba l double∗ dndenn l is t , g l o ba l double∗ PD){128i n t i = g e t l o c a l i d ( 0 ) ;i f ( i<numlev ){double knnp = 0 .0 ;double knpn = 0 .0 ;double e i = RydhckB / ( pow( n l i s t [ i ] , 2 . 0 )∗T ) ;double ef ;double eg ;double es ;double de ;double t e rmtb r = (pow( ei , 7 . 0 / 3 . 0 )+4 .38∗pow( ei ,1 .72)+1.32∗ e i ) ;double term = 0 .0 ;double tau [NUMLEV] = {0.00003} ;const double au = 4.1322314049586776e+10;const double PI = 3.14159265;l o c a l double eps [NUMLEV] ;l o c a l double ndenl [NUMLEV] ;l o c a l double dndenl [NUMLEV] ;l o c a l double PDl [NUMLEV] ;tau [ 0 ] = 0.014;tau [ 1 ] = 0.046;tau [ 2 ] = 0.029;tau [ 3 ] = 0.0012;double gsum ;double gvalue = 0 . 0 ;b a r r i e r (CLK LOCAL MEM FENCE ) ;eps [ i ] = e i ;ndenl [ i ] = nden [ i ] ;k t b r k i on [ i ] = ((4.952079396744990e+03∗pow( n l i s t [ i ] , 2 . 0 ) / pow(T , 3 . 0 )∗ eden3)−(1.195770111325817e+07/pow(T , 1 . 5 )∗ exp(−e i )∗eden∗ndenl [ i ] ) ) / t e rmtb r ;dndenl [ i ] = k t b r k i on [ i ] ;PDl [ i ] = (0.30492+(3.0e−5)∗ n l i s t [ i ]∗ n l i s t [ i ] ) /( ( double ) n l i s t [ i ]∗ n l i s t [ i ] ) / ( 2 . 0 ∗ PI∗pow( n l i s t [ i ] , 3 . 0 ) )∗ au∗ndenl [ i ] ;b a r r i e r (CLK LOCAL MEM FENCE ) ;for ( i n t j = 0 ; j<numlev ; ++ j ){i f ( i != j ){ef = eps [ j ] ;eg = max( ei , e f ) ;es = min ( ei , e f ) ;de = fabs ( ef−e i ) ;term = 2.735776154492852e+03/T∗ ( ( 2 2 . 0 / ( pow( eg +0 . 9 , 7 . 0 / 3 . 0 ) ) )\+ ( 4 . 5 / ( pow( eg , 2 . 5 )∗pow(de , 4 . 0 / 3 . 0 ) ) ) ) / pow( eg , 2 . 5 )∗ exp ( es ) ;knnp = pow( ei , 2 . 5 )∗pow( ef , 1 . 5 )∗ exp(−e i )∗ term ;129knpn = pow( ef , 2 . 5 )∗pow( ei , 1 . 5 )∗ exp(−ef )∗ term ;dndenl [ i ] += eden∗(−knnp ∗ ndenl [ i ] + knpn ∗ ndenl [ j ] ) ;}}ba r r i e r (CLK LOCAL MEM FENCE ) ;dndenn l i s t [ i ] = dndenl [ i ] / pow( n l i s t [ i ] , 2 . 0 ) ;dnden [ i ] = dndenl [ i ] − PDl [ i ] ;PD[ i ] = PDl [ i ] ;}}130Appendix BSpectroscopy of Nitric OxideNitric oxide like any other molecule is not spherically symmetric. As a conse-quence the total orbital angular momentum ~L is not a good quantum number tobe described simultaneously with energy. Instead the projection of the angular mo-mentum on the inner nuclear axis of the diatomic molecule ML is a quantity that canbe used. Due to azimuthal symmetry of the diatomic molecules, states of the samemagnitude of ML are degenerate. Since values of ML are indistinguishable, theenergy of the most general wavefunction only depends on Λ= |ML|= 0,1,2, ...,L.Ground state nitric oxide molecule has an unpaired electron in its pi∗ orbital.This orbital is the mixture of atomic orbitals perpendicular to the internuclear axiswith possible values of ML for it are −1 and 1 which is represented as Π for Λ =ML = 1. The spin of the molecular terms of nitric oxide with one unpaired electron,as long as no core electron is excited, is always Σ = S = 1/2 which correspond toa doublet degeneracy.Ground state NO is best represented as Hund case (a) where the coupling be-tween spin and angular momentum on the internuclear axis is very strong. There-fore energy levels of the ground state electronic level are largely separated by spin-orbit coupling with constant A= 123.16cm−1 [26] as followTe = T0+AΛΣ (B.1)where Te is the corrected energy for Λ−Σ interaction, and T0 is the energy without131the interaction. With the coupling strength of A > 0 the ground state level is Ω =Λ−Σ = 1/2, with total angular momentum J = Ω+N, where N is the rotationalquantum number. The values of J are half-integers J = 1/2,3/2,5/2, ... and theyconstitute the energyF0 (J) = B0(J (J+1)−Ω2) (B.2)where B0 = 1.67cm−1 [22] is the rotational constant of ground state NO.First excited electronic state of NO, 2Σ1/2, has the single electron in a molecularorbital formed from linear combination of atomic 4s orbitals, and core electrons”untouched”. The orbital angular momentum of this orbital is zero, regardlessof any symmetry considerations. This electronic state is best described by Hundcase (b) in which spin is not coupled to the internuclear axis and Ω is not defined.Instead the projection of orbital angular momentum adds with rotational quantumnumber to yield K = Λ+N. For the case of Σ with Λ = 0, the quantum numberK simply indicates the rotation of the molecule N. Total angular momentum istherefore J = K+S, ..., |K−S| with the only value 0−1/2 missing. However, theenergy of rotational levels is given by the value of KF1 (K) = B0K (K+1)+12γK (B.3)F2 (K) = B0K (K+1)− 12γ (K+1) (B.4)where the two energies are for levels with J = K+ 1/2 and J = K− 1/2 respec-tively. The coefficient γ is very small, and for all purposes in this paper, it canbe ignored. The rotational energy constant for the excited NO state is largerB0 = 1.98cm−1 due to smaller bond length of N −O in the excited state. Thisvalue is very similar to that of NO+ and highly excited Rydberg states.Figure B.1 shows an example of a REMPI spectra in which two-photon one-color ionization produces an electron signal detected by an MCP. The frequencyof the photon is scan over a small range to produce the lines. Each line is labeledaccording to the standard rules. The first small letter indicates the change of therotational quantum number, N, with p=−1, q= 0, and r =+1. Similarly, capitalletters, PQR, indicate the change of total angular momentum. The number fol-lowing the letter are related to the fine structure of the excited and ground state,132respectively. The number in parenthesis indicates the total angular momentum ofthe ground state.Figure B.1: Simulated spectrum of one color REMPI on top and experimentalobservation at the bottom. The simulated spectrum with a temperatureof 2.0 K matches the experiment.At a very cold rotational temperature of Trot = 2K only few rotational levels of2Π1/2 are populated, of which, we select the transition J = 1/2 (−)→ K = 0 (+)that only results in K = 0 (total angular momentum regardless of spin). This wayonly one possible transition K = N = 0 (+)→ N = 1 (−) exist from the excited Astate to Rydberg states.A highly excited state of NO with principal quantum number n= 30−80 whichis of our interest in this thesis, can be described as an atomic state. The reason beingthat the electron is on average very far from the NO+ core and the wave functiondescribing the electron in a highly excited states is nearly spherical. Thereforeenergy terms are indicated as nl(N+), where l is the angular momentum of atomicstates and N+ is the rotational quantum number of the NO+ core.133Figure B.2: Simulated and real spectrum of nitric oxide Rydberg excitationvia 2Σ+1/2 in presence of an external electric field. Details of the experi-ment are discussed in chapter ??.Total angular momentum N = 1 (regardless of spin) results in many possibletransitions with N+ = 0,1,2 and l = 0,1,2,3 such that |N+± l| = 1. The possi-bilities are l(N+) = p(0), s(1), p(1), d(1), p(2), d(2), f (2). In Fig. B.2, real andsimulated spectrum of NO are displayed. While the overall resemblance of the twofigures indicates our good understanding of NO rovibronic structure, some linesdo not appear in the real spectrum. This is largely related to the pre-dissociationof NO molecule. Some lines that belong to the same series have a quantum defect134that is different which happens especially when two lines of different series arevery close together.In the next section we will see that excited rotational levels of NO can interferewith the Rydberg spectrum to introduce extra lines.B.0.1 Rotational levels and Rydberg series of nitric oxideThe intensity plot (Fig. 5.2) reads an ω2 laser wavelength scan with field ionizationdetection. Each horizontal line is a field ionization trace which shows at what fieldan electron signal is observed. The vertical axis shows the wavelength of ω2 laser.Given the appearance time of the ionization signal for each trace, we calculate(Eqn. B.5) the required field (V/cm) for the ionization of a Rydberg state.F =V (t−0.043)4.55(B.5)In Eqn. B.5, V (t) is a known function that represents the time dependent potential(V ) applied between G1-G2 with separation distance of 4.55cm. The input time isshifted to account for the arrival time of electrons that is 0.043µs.To calculate the field ionization threshold, Eqn. B.6 is used.E(λ ,N+)cm−1 = IP(N+ = 0)+∆rot(0→ N+)− 2×107λ(B.6)The binding energy is calculated with respect to ionization potential. If the seriesconverge to a different rotational level than N+ = 0, ∆rot corrects for the differencein rotational energy, given in Eqn. B.7.∆rot(0→ N+)= 2BN+ (N++1) (B.7)In the equation above, B = 1.98cm−1 is the rotational constant of NO+. The ion-ization threshold field is related to the binding energy by Eqn. B.8:F =(E (λ ,N+)4.12)2(B.8)135Figure B.3, shows three lines from left to right for N+ = 0, 1, 2, respectively.Three series can be identified. The strongest one is far to the right with N+= 2. Theother two series appear strong or weak depending on the color of ω2. The spacingof each strong trace suggest that Rydberg levels are optically pumped to a purelevel converging to N+ = 2. During the field ramp, however, the level crossingsmake lower continuum levels available to the stark manifold.Figure B.3: Intensity map of laser scan with field ionization, first UV laser istunned to lower rotational level of the excited electronic A state, how-ever, there is an overlap with a transition J− 4.5→ K = 4. Red linesindicated Rydberg series converging to N+ = 0,1,2 resulted from theoriginal rotational level k = 0.136Figure B.4: Intensity map of laser scan with field ionization, first UV laser istunned to lower rotational level of the excited electronic A state, how-ever, there is an overlap with a transition J−4.5→ K = 4. Green linesindicated Rydberg series converging to N+ = 1,2,3,4,5 resulted fromk = 4 level.B.0.2 Observation of a new seriesWe use figure B.3 and B.4 to isolate each series. The integrated signal is shownin figure B.5 and B.6. To get the integrated signal (Fig B.5), the peaks betweenthe two bottom red lines in figureB.3 are integrated to yield the bottom, invertedtrace, and the peaks above the top line in figureB.3 are integrated to produce thetop trace. The peaks at the bottom of figureB.3 are integrated to obtain figure B.6.137Wavenumber (cm-1) #1043.044 3.045 3.046 3.047 3.048 3.049 3.05Figure B.5: Signal integrated over two regions indicated by red lines in B.3.The bottom trace is the integral of peaks between the two bottom linesand the top trace is the integral of peaks above the top red line.Wavenumber (cm-1) #1043.0435 3.044 3.0445 3.045 3.0455 3.046 3.0465 3.047 3.0475Figure B.6: Signal integrated over peaks at the bottom triangle of figure B.3.The peaks correspond to a series of Rydberg states converging to N+ =2 via K=4 A state.To calculate the ionization potential, first I correct for the air refractive indexto get the wavelength in the vacuum. Equation B.9 give the correction when wave-length is in nanometer.λ0 = λ(1+0.05792105238.0185−10−6λ−2 +0.0016791757.362−10−6λ−2)(B.9)where λ0 is the vacuum wavelength. By minimizing the objective function (Eq.138B.10) we find the ionization potential and quantum defect.f (IP,δ ) =∑(λ0−λcalc (IP,δ ))2 (B.10)where calculated wavelength (λcalc) is2×107λcalc (IP,δ )= IP− R(n−δ )2 (B.11)We identify two distinct series of lines one in the top right half of the imagein figure B.3 and one in the bottom left corner. Using the fit, the quantum defectfor both series is δ f = 0.01. The main series has fitted IP = 30535.60cm−1 veryclose to N+ = 2 ionization threshold of nitric oxide. The other series fits IP =30496.00cm−1, about 28.1cm−1 lower than minimum ionization potential of nitricoxide. After careful analysis, we find that this low apparent IP results from aK = 4 series above the excited A state (2Σ1/2). The K = 4 state is populated by thesame UV photon that creates the excited state. Assuming a thermal equilibrationof rotational states, at a temperature of 15K, N = 4 of the ground state NO can bepopulated significantly to be observed in the experiment. Given that the groundstate is described by Hund’s case (a), the angular momentum is J = Ω+N = 4.5.Using the equation B.2, we find the energy of the state to be about 40.08 cm−1above the ground state. The excited state is described by Hund’s case (b) andtherefore the rotational energy is simply computed by the rotational number Kwhich yields an energy of 39.98 cm−1 above the excited state. The difference ofthe two values corresponds to a line that is slightly tuned to the red of the main line.Under certain circumstances, high rotational temperature and slight red detuning,the Rydberg levels corresponding to K = 4 can be observed. We test this theoryby looking at the spectrum produced by a pure transition (Fig. B.7) that populatesthe same rotational level in the excited state. We confirm the interference of theK = 4 by detuning the first UV photon to blue (away from hot rotational lines)which causes the lines in question to disappear.139Figure B.7: Intensity map of laser scan with field ionization, first UV laser istuned to an isolated peak that selects a K = 4 of the excited electronicA state.140Appendix CDetailed Analysis of FieldIonization DataIt is evident from the Fig. C.1 that the prompt signal arrives earlier (relative to theprompt signal for the earliest pulse delay) as the pulse is delayed. This reflects thefact that, after a long delay the plasma volume has travelled closer to the detector.Therefore, applying a late pulse, extracts electrons at a position closer to the de-tector than a pulse applied earlier. Figure C.2 highlights earlier arrival time of thesignal as electric field is delayed. Appendix C offers a simple method to correctfor this issue.141Figure C.1: DC electric field ionization traces at various time delay at fixeddensity.142Time (7s)0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26Pulse Delay (7s)024681012141618Figure C.2: Pulsed field ionization early peak in time domain. Rydberg staten = 49, ω1 power = 2µJ, ω2 power = 6mJ. Only the early part of thePFI is plotted for various delays. The y axis for each traces is offset bythe pulse delay.The estimated distance between the laser interaction region and the first gridcan be calculated from the arrival time of the field free experiment. As shown inFig. C.3 the plasma volume arrives at 20.0 µs at the speed of 1400 m/s measuringa distance of 28 mm. There a fixed distance of 10 mm separates the second gridand the detector.1430 5 10 15 20 25Time (µs)Figure C.3: A typical scope trace shows an early prompt peak and a latepeak. The first one corresponds to arrival of hot electrons boiled offof the plasma volume and the late peak represents the bulk volume ofthe plasma that travels at the speed of molecular beam.Assuming that the difference in the arrival time originates from the movingvolume, we can correct for this time difference and achieve a visualization of theearly peak as shown in Fig. C.4.144Time (7s)-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03Pulse Delay (7s)024681012141618Figure C.4: Time corrected. Pulsed field ionization early peak in time do-main. Rydberg state n= 49, ω1 power = 2µJ, ω2 power = 6mJ. Onlythe early part of the PFI is plotted for various delays. The y axis foreach trace is offset by the pulse delay.This time correction depends both on the pulse delay and the detector position.Multiple trials of this experiment were run at various detector positions. Figure C.5shows how arrival time depends on the detector position. The line fits an electronspeed ve= 2×106 m/s which corresponds to the electric field of 11.4V/cm, whichmeans the skew rate of the pulsed field is much slower than the motion of electrons,therefore, electron escapes the area between the two plates before the pulse canreach a field strength more than 11.4 V/cm. This analysis gives us an estimate ofthe arrival time that matches the distance of the laser interaction region with thefirst plate of the MCP detector. By correcting for the arrival time the required fieldfor extraction of surface electrons becomes zero. At detector position l = 4.55cm,145the calculated delay is 0.193µs. By subtracting the lag time of the high-voltageswitch, 0.150 µs, we obtain the value of 0.043 µs that was previously obtainedindependently to zero field.Figure C.5: The position of the detector as a function of arrival time of theprompt electron signal.Flight time (7 s)0.188 0.19 0.192 0.194 0.196Flight length (mm)404244464850525456146Appendix DMolecular Dynamic Simulation:Parallel ImplementationD.1 OpenCL kernels for molecular dynamics simulationAll codes in the following section are developed by Dominic Meiser (Tech-X cor-poration) and the author./ / OpenCL kerne l f o r computing t o t a l energy i n scaled un i t s# i f de f USE DOUBLE# i f de f c l k h r f p 64#pragma OPENCL EXTENSION c l k h r f p 64 : enable#define REAL double#define REAL3 double3#define REAL4 double4#define EPS2 1.0e−30#endif#else#define REAL f l oa t#define REAL3 f l o a t 3#define REAL4 f l o a t 4#define EPS2 1.0e−18f#endifke r ne l void ca l c po t en t i a l ene r gy (g l o ba l REAL∗ x , g l o ba l REAL∗ y , g l o ba l REAL∗ z ,g l o ba l REAL∗ charge , g l o ba l REAL∗ potEnergy ,g l o ba l REAL∗ Vxx , g l o ba l REAL∗ Vxy ,147i n t nPtc ls ){i n t t i d = ge t g l o b a l i d ( 0 ) + ge t g l o b a l i d ( 1 ) ∗ ge t g l o ba l s i z e ( 0 ) ;REAL energy = 0 . 0 ;REAL Vxx loca l = 0 . 0 ; / / x as i n e l ec t r on or ion ,t he re fo re xx means repu l s i veREAL Vxy loca l = 0 . 0 ; / / and xy means a t t r a c t i v ei f ( t i d < nPtc ls ){REAL3 r1 = (REAL3 ) ( x [ t i d ] , y [ t i d ] , z [ t i d ] ) ;REAL q1 = charge [ t i d ] ;REAL twobody = 0 . 0 ;for ( i n t i =0; i<nPtc ls ;++ i ){i f ( t i d != i ){REAL3 r2 = (REAL3 ) ( x [ i ] , y [ i ] , z [ i ] ) ;REAL q2 = charge [ i ] ;REAL3 dr = r1 − r2 ;REAL r12 = fma ( dr . x , dr . x , fma ( dr . y , dr . y ,fma ( dr . z , dr . z , (REAL)IMPACTFACT ) ) ) ;twobody = q1∗q2 / sq r t ( r12 ) / ( REAL ) ( 6 . 0 ) ;i f ( q1==q2 ){Vxx loca l += twobody ;}else i f ( q1==−q2 ){Vxy loca l += twobody ;}energy += twobody ;}}potEnergy [ t i d ] = energy ; / / ?? d i v i s i o n by 2?Vxx [ t i d ] = Vxx loca l ;Vxy [ t i d ] = Vxy loca l ;}}/ / The OpenCL kerne l f i l e f o r c a l c u l a t i o n o f energy post−i n t e g r a t i o n# i f de f USE DOUBLE# i f de f c l k h r f p 64#pragma OPENCL EXTENSION c l k h r f p 64 : enable#define REAL double148#define REAL3 double3#define REAL4 double4#define EPS2 1.0e−30#endif#else#define REAL f l oa t#define REAL3 f l o a t 3#define REAL4 f l o a t 4#define EPS2 1.0e−18f#endifREAL3 two body i n t e rac t i on (REAL4 ptc l1 ,REAL4 p tc l2 , REAL impactFact ) ;k e r ne l void compute coulomb accelerat ion (/ / pos i t i o ns o f p a r t i c l e sconst g l o ba l REAL∗ xGlob ,const g l o ba l REAL∗ yGlob ,const g l o ba l REAL∗ zGlob ,/ / v e l o c i t i e s o f p a r t i c l e sconst g l o ba l REAL∗ vxGlob ,const g l o ba l REAL∗ vyGlob ,const g l o ba l REAL∗ vzGlob ,/ / charge of p a r t i c l e sconst g l o ba l REAL∗ charge ,/ / mass o f p a r t i c l e sconst g l o ba l REAL∗ mass ,REAL k ,REAL impactFact ,i n t nPtc ls ,/ / acce le ra t i ons ( output )g l o ba l REAL∗ axGlob ,g l o ba l REAL∗ ayGlob ,g l o ba l REAL∗ azGlob) {l o c a l REAL4 xyzqCache [ BLOCK SIZE ] ;p r i v a t e i n t n = ge t g l o b a l i d ( 0 ) +ge t g l o b a l i d ( 1 ) ∗ ge t g l o ba l s i z e ( 0 ) ;p r i v a t e REAL4 p t c l 1 [PTCL UNROLL FACTOR ] ;p r i v a t e REAL m1[PTCL UNROLL FACTOR ] ;p r i v a t e REAL3 acce le ra t i on [PTCL UNROLL FACTOR ] ;for ( i n t i = 0 ; i < PTCL UNROLL FACTOR; ++ i ) {i n t p t c l I d = n + i ∗ ge t g l o ba l s i z e (0 ) ∗ ge t g l o ba l s i z e ( 1 ) ;149p t c l 1 [ i ] . x = ( p t c l I d < nPtc ls ) ? xGlob [ p t c l I d ] : 0 ;p t c l 1 [ i ] . y = ( p t c l I d < nPtc ls ) ? yGlob [ p t c l I d ] : 0 ;p t c l 1 [ i ] . z = ( p t c l I d < nPtc ls ) ? zGlob [ p t c l I d ] : 0 ;p t c l 1 [ i ] . w = ( p t c l I d < nPtc ls ) ? charge [ p t c l I d ] : 0 ;m1[ i ] = ( p t c l I d < nPtc ls ) ? mass [ p t c l I d ] : 0 ;acce le ra t i on [ i ] = (REAL3) 0 ;}i n t numTiles = nPtc ls / BLOCK SIZE ;for ( i n t t i l e = ge t g roup id ( 0 ) ; t i l e< ge t g roup id ( 0 ) + numTiles ; ++ t i l e ) {i n t globInd ;g lobInd = g e t l o c a l i d ( 0 ) + ( t i l e % numTiles ) ∗ BLOCK SIZE ;xyzqCache [ g e t l o c a l i d ( 0 ) ] . x = xGlob [ g lobInd ] ;xyzqCache [ g e t l o c a l i d ( 0 ) ] . y = yGlob [ g lobInd ] ;xyzqCache [ g e t l o c a l i d ( 0 ) ] . z = zGlob [ g lobInd ] ;xyzqCache [ g e t l o c a l i d ( 0 ) ] .w = charge [ g lobInd ] ;b a r r i e r (CLK LOCAL MEM FENCE ) ;#pragma un r o l l 8for ( i n t j j = 0 ; j j < BLOCK SIZE ; ++ j j ) {i n t j jS taggered = ( j j + g e t l o c a l i d ( 0 ) ) & (BLOCK SIZE − 1 ) ;REAL4 p t c l 2 = xyzqCache [ j jS taggered ] ;for ( i n t i = 0 ; i < PTCL UNROLL FACTOR; ++ i ) {acce le ra t i on [ i ] += two body i n t e rac t i on ( p t c l 1 [ i ] ,p tc l2 , impactFact ) ;}}ba r r i e r (CLK LOCAL MEM FENCE ) ;}for ( i n t j = numTiles ∗ BLOCK SIZE ; j < nPtc ls ; ++ j ) {REAL4 p t c l 2 = (REAL4 ) ( xGlob [ j ] , yGlob [ j ] , zGlob [ j ] , charge [ j ] ) ;for ( i n t i = 0 ; i < PTCL UNROLL FACTOR; ++ i ) {acce le ra t i on [ i ] += two body i n t e rac t i on ( p t c l 1 [ i ] ,p tc l2 , impactFact ) ;}}for ( i n t i = 0 ; i < PTCL UNROLL FACTOR; ++ i ) {i n t p t c l I d = n + i ∗ ge t g l o ba l s i z e (0 ) ∗ ge t g l o ba l s i z e ( 1 ) ;i f ( p t c l I d < nPtc ls ) {REAL myFact = k ∗ p t c l 1 [ i ] . w / m1[ i ] ;axGlob [ p t c l I d ] += myFact ∗ acce le ra t i on [ i ] . x ;ayGlob [ p t c l I d ] += myFact ∗ acce le ra t i on [ i ] . y ;azGlob [ p t c l I d ] += myFact ∗ acce le ra t i on [ i ] . z ;}}}150REAL3 two body i n t e rac t i on (REAL4 ptc l1 , REAL4 p tc l2 ,REAL impactFact ) {REAL3 dr = p t c l 2 . xyz − p t c l 1 . xyz ;REAL d i s tSq r = fma ( dr . x , dr . x , fma ( dr . y , dr . y ,fma ( dr . z , dr . z , EPS2 + impactFact ) ) ) ;REAL i n vD i s t ;# i f de f USE DOUBLEi n vD i s t = r s q r t ( d i s tSq r ) ;#elsei n vD i s t = n a t i v e r s q r t ( d i s tSq r ) ;#endifREAL invDistCube = i n vD i s t ∗ i n vD i s t ∗ i n vD i s t ;REAL s = p t c l 2 .w ∗ invDistCube ;REAL3 acc = −s ∗ dr ;return acc ;}D.2 Python routines for setting up the initial valueproblem and numerical integration# Bor is Update f o r i n t e g r a t i o nimport numpyimport uc i . P t c l s as P tc l simport pyopencl as c limport pyopencl . a r ray as c l a r r a yimport sysimport osclass Bor isUpdater ( ) :def i n i t ( se l f , c t x = None , queue = None ) :s e l f . c t x = c txs e l f . queue = queuei f s e l f . c t x == None :s e l f . c t x = c l . c reate some context ( )i f s e l f . queue == None :s e l f . queue = c l .CommandQueue( s e l f . c tx ,p rope r t i e s= c l . command queue properties .PROFILING ENABLE)def update ( se l f , xd , yd , zd , vxd , vyd , vzd , qd , md, forces ,t , dt , num steps ) :151axd = c l a r r a y . z e r o s l i k e ( xd )ayd = c l a r r a y . z e r o s l i k e ( xd )azd = c l a r r a y . z e r o s l i k e ( xd )for i in range ( num steps ) :# F i r s t h a l f o f pos i t i o n advancexd += (0 .5 ∗ dt ) ∗ vxdyd += (0 .5 ∗ dt ) ∗ vydzd += (0 .5 ∗ dt ) ∗ vzdaxd . f i l l ( 0 . 0 , s e l f . queue )ayd . f i l l ( 0 . 0 , s e l f . queue )azd . f i l l ( 0 . 0 , s e l f . queue )for acc in fo rces :acc . computeAcc ( xd , yd , zd , vxd , vyd , vzd , qd , md,axd , ayd , azd , t )vxd += dt ∗ axdvyd += dt ∗ aydvzd += dt ∗ azd# Second ha l f o f pos i t i o n advancexd += (0 .5 ∗ dt ) ∗ vxdyd += (0 .5 ∗ dt ) ∗ vydzd += (0 .5 ∗ dt ) ∗ vzdt += dtreturn t# Python func t i on to c a l l the OpenCL kerne l f o r energy ca l c u l a t i o nimport numpyimport pyopencl as c limport pyopencl . a r ray as c l a r r a yimport osclass ComputePotentialEnergy ( ) :def i n i t ( se l f , c t x = None , queue = None , impactFact = 0 .0001) :s e l f . c t x = c txs e l f . queue = queues e l f . impactFact = impactFacti f s e l f . c t x == None :s e l f . c t x = c l . c reate some context ( )i f s e l f . queue == None :s e l f . queue = c l .CommandQueue( s e l f . c tx ,152p rope r t i e s= c l . command queue properties .PROFILING ENABLE)s e l f . mf = c l . mem flagsabsolutePathToKernels = os . path . dirname (os . path . rea lpa th ( f i l e ) )s rc = open ( absolutePathToKernels + ’ / ca lc energy gpu sca led . c l ’ ,’ r ’ ) . read ( )s e l f . compEnergyF = c l . Program ( s e l f . c tx , s rc )t ry :s e l f . compEnergyF . bu i l d ( op t ions=\[ ’ −DIMPACTFACT= ’+st r ( s e l f . impactFact ) ] )except :pr in t ( ” E r ro r : ” )pr in t ( s e l f . compEnergyF . g e t b u i l d i n f o ( s e l f . c t x . devices [ 0 ] ,c l . p rog ram bu i l d i n f o .LOG) )raises e l f . compEnergyF . ca l c po t en t i a l ene r gy . se t s ca l a r a rg d t ypes ([ None , None , None , None , None , None , None ,numpy . i n t 32 ] )s e l f . compEnergyD = c l . Program ( s e l f . c tx , s rc )t ry :s e l f . compEnergyD . bu i l d ( op t ions=\[ ’ −DUSE DOUBLE=TRUE −DIMPACTFACT= ’+st r ( s e l f . impactFact ) ] )except :pr in t ( ” E r ro r : ” )pr in t ( s e l f . compEnergyD . g e t b u i l d i n f o ( s e l f . c t x . devices [ 0 ] ,c l . p rog ram bu i l d i n f o .LOG) )raises e l f . compEnergyD . ca l c po t en t i a l ene r gy . se t s ca l a r a rg d t ypes ([ None , None , None , None , None , None , None ,numpy . i n t 32 ] )def computeEnergy ( se l f , x , y , z , q ) :xd = c l a r r a y . to dev i ce ( s e l f . queue , x )yd = c l a r r a y . to dev i ce ( s e l f . queue , y )zd = c l a r r a y . to dev i ce ( s e l f . queue , z )qd = c l a r r a y . to dev i ce ( s e l f . queue , q )coulombEnergy = c l a r r a y . z e r o s l i k e ( xd )coulombEnergy . f i l l ( 0 . 0 , s e l f . queue )coulombEnergyxx = c l a r r a y . z e r o s l i k e ( xd )coulombEnergyxx . f i l l ( 0 . 0 , s e l f . queue )coulombEnergyxy = c l a r r a y . z e r o s l i k e ( xd )coulombEnergyxy . f i l l ( 0 . 0 , s e l f . queue )153coulombEnergyHost = numpy . z e r o s l i k e ( x )coulombEnergyxxHost = numpy . z e r o s l i k e ( x )coulombEnergyxyHost = numpy . z e r o s l i k e ( x )prec = x . dtypei f prec == numpy . f l o a t 32 :s e l f . compEnergyF . ca l c po t en t i a l ene r gy ( s e l f . queue ,( x . s ize , ) , None ,xd . data , yd . data , zd . data ,qd . data , coulombEnergy . data , coulombEnergyxx . data ,coulombEnergyxy . data , numpy . i n t 32 ( len ( x ) ) ,g t imes l = False )e l i f prec == numpy . f l o a t 64 :s e l f . compEnergyD . ca l c po t en t i a l ene r gy ( s e l f . queue ,( x . s ize , ) , None ,xd . data , yd . data , zd . data ,qd . data , coulombEnergy . data , coulombEnergyxx . data ,coulombEnergyxy . data , numpy . i n t 32 ( len ( x ) ) ,g t imes l = False )else :pr in t ( ”Unknown f l o a t type . ” )# re t u rn the ar ray o f a l l p o t e n t i a l energ iescoulombEnergy . get ( s e l f . queue , coulombEnergyHost )coulombEnergyxx . get ( s e l f . queue , coulombEnergyxxHost )coulombEnergyxy . get ( s e l f . queue , coulombEnergyxyHost )return coulombEnergyHost , coulombEnergyxxHost , coulombEnergyxyHosti f name == ’ ma in ’ :cpe = ComputePotentialEnergy ( )N = 100;x = numpy . random . random (N)y = numpy . random . random (N)z = numpy . random . random (N)q = numpy . concatenate ( ( numpy . ones (N/2) ,−numpy . ones (N / 2 ) ) )energyd , energyxxd , energyxyd = cpe . computeEnergy ( x , y , z , q )energyh = numpy . z e r o s l i k e ( x )for i in range ( x . s i ze ) :energy = 0.0for j in range ( x . s i ze ) :i f i != j :r = numpy . sq r t ( ( x [ i ]−x [ j ])∗∗2+\( y [ i ]−y [ j ] )∗∗2+( z [ i ]−z [ j ] )∗∗2+ cpe . impactFact )i f q [ i ]∗q [ j ]>0:energy += q [ i ]∗q [ j ] / r / 6 . 0 ;else :energy += q [ i ]∗q [ j ] / r / 6 . 0 ;energyh [ i ] = energyasser t numpy . fabs ( energyh [ i ]−energyd [ i ] ) < 1e−10154import numpyimport uc i . P t c l s as P tc l simport pyopencl as c limport pyopencl . a r ray as c l a r r a yimport sysimport osimport mathclass CoulombAccScaled ( ) :BLOCK SIZE = 256PTCL UNROLL FACTOR = 1maxNumThreadsX = 2∗∗12k = 1 .0 / 3 . 0def i n i t ( se l f , c t x = None , queue = None , impactFact = 0 .0001) :s e l f . c t x = c txs e l f . queue = queues e l f . impactFact = impactFacti f s e l f . c t x == None :s e l f . c t x = c l . c reate some context ( )i f s e l f . queue == None :s e l f . queue = c l .CommandQueue( s e l f . c tx ,p rope r t i e s= c l . command queue properties .PROFILING ENABLE)s e l f . mf = c l . mem flagsabsolutePathToKernels = os . path . dirname (os . path . rea lpa th ( f i l e ) )s rc = open ( absolutePathToKernels +\’ / compute coulomb accelerat ion . c l ’ ,’ r ’ ) . read ( )s e l f . compAccF = c l . Program ( s e l f . c tx , s rc )t ry :s e l f . compAccF . bu i l d ( op t ions = [’ −DBLOCK SIZE= ’ + st r ( s e l f . BLOCK SIZE) +’ −DPTCL UNROLL FACTOR= ’ + st r ( s e l f .PTCL UNROLL FACTOR ) ] )except :pr in t ( ” E r ro r : ” )pr in t ( s e l f . compAccF . g e t b u i l d i n f o ( s e l f . c t x . devices [ 0 ] ,c l . p rog ram bu i l d i n f o .LOG) )raises e l f . compAccF . compute coulomb accelerat ion . se t s ca l a r a rg d t ypes ([ None , None , None , None , None , None , None , None ,155numpy . f l oa t32 , numpy . f l oa t32 , numpy . in t32 , None , None ,None ] )s e l f . compAccD = c l . Program ( s e l f . c tx , s rc )t ry :s e l f . compAccD . bu i l d ( op t ions = [’ −DUSE DOUBLE=TRUE −DBLOCK SIZE= ’ + st r ( s e l f . BLOCK SIZE) +’ −DPTCL UNROLL FACTOR= ’ + st r ( s e l f .PTCL UNROLL FACTOR ) ] )except :pr in t ( ” E r ro r : ” )pr in t ( s e l f . compAccD . g e t b u i l d i n f o ( s e l f . c t x . devices [ 0 ] ,c l . p rog ram bu i l d i n f o .LOG) )raises e l f . compAccD . compute coulomb accelerat ion . se t s ca l a r a rg d t ypes ([ None , None , None , None , None , None , None , None ,numpy . f l oa t64 , numpy . f l oa t64 , numpy . in t32 , None , None ,None ] )def computeAcc ( se l f , xd , yd , zd , vxd , vyd , vzd , qd , md, axd , ayd ,azd , t , d t = None ) :” ” ”Compute acce le ra t i on due to coulomb fo rces between p a r t i c l e s .A l l va r i ab l es are device ar rays . They are assumed to be thesame leng th and p rec i s i on . The computed acce le ra t i ons areaccumulated i n t o the a?d ar rays ( i . e . the a?d ar rays are notset to zero ) .” ” ”s e l f . computeLaunchConfig ( xd . s i ze )prec = xd . dtypei f prec == numpy . f l o a t 32 :s e l f . compAccF . compute coulomb accelerat ion ( s e l f . queue ,( s e l f . numThreadsX , s e l f . numThreadsY ) ,( s e l f . BLOCK SIZE , 1 ) ,xd . data , yd . data , zd . data ,vxd . data , vyd . data , vzd . data ,qd . data , md. data ,numpy . f l o a t 32 ( s e l f . k ) , numpy . f l o a t 32 ( s e l f . impactFact ) ,numpy . i n t 32 ( xd . s ize ) ,axd . data , ayd . data , azd . data ,g t imes l = False )e l i f prec == numpy . f l o a t 64 :s e l f . compAccD . compute coulomb accelerat ion ( s e l f . queue ,( s e l f . numThreadsX , s e l f . numThreadsY ) ,( s e l f . BLOCK SIZE , 1 ) ,156xd . data , yd . data , zd . data ,vxd . data , vyd . data , vzd . data ,qd . data , md. data ,numpy . f l o a t 64 ( s e l f . k ) , numpy . f l o a t 64 ( s e l f . impactFact ) ,numpy . i n t 32 ( xd . s ize ) ,axd . data , ayd . data , azd . data ,g t imes l = False )else :pr in t ( ”Unknown f l o a t type . ” )def computeLaunchConfig ( se l f , numPtcls ) :s e l f . numThreads = i n t (math . c e i l ( f l oa t\( numPtcls ) / s e l f .PTCL UNROLL FACTOR) )s e l f . numThreadsX = s e l f . BLOCK SIZE ∗ \i n t (math . c e i l ( f l oa t ( s e l f . numThreads ) /f l oa t ( s e l f . BLOCK SIZE ) ) )i f s e l f . numThreadsX > s e l f .maxNumThreadsX :s e l f . numThreadsX = s e l f .maxNumThreadsXs e l f . numThreadsY = i n t (math . c e i l \( f l oa t ( s e l f . numThreads ) /f l oa t ( s e l f . numThreadsX ) ) )import uc i . BendKickUpdater as BendKickUpdaterimport uc i . TrapAcc as TrapAccimport uc i . CoulombAcc as CoulombAccimport uc i . P t c l s as P tc l simport uc i . T rapConf igura t ion as TrapConf igura t ionimport uc i . F r i c t i onAcc as F r i c t i onAccimport uc i . HeatingAcc as HeatingAccimport numpyimport pyopencl as c limport pyopencl . a r ray as c l a r r a yimport copyclass Sim ( ) :def i n i t ( se l f , c t x = None , queue = None ) :s e l f . accL i s t = [ ]s e l f . p t c l s = P tc l s . P t c l s ( )s e l f . t = 0.0s e l f . c t x = c txs e l f . queue = queuei f s e l f . c t x == None :s e l f . c t x = c l . c reate some context ( )157i f s e l f . queue == None :s e l f . queue = c l .CommandQueue( s e l f . c tx ,p rope r t i e s= c l . command queue properties .PROFILING ENABLE)s e l f . t r apCon f i gu ra t i on = TrapConf igura t ion . TrapConf igura t ion ( )s e l f . updater = BendKickUpdater . BendKickUpdater ( s e l f . c tx , s e l f . queue )s e l f . updater . t r apCon f i gu ra t i on = s e l f . t r apCon f i gu ra t i ondef copy ( s e l f ) :theCopy = Sim ( s e l f . c tx , s e l f . queue )theCopy . accL i s t = s e l f . accL i s ttheCopy . updater = s e l f . updatertheCopy . t = copy . deepcopy ( s e l f . t )theCopy . t r apCon f i gu ra t i on = copy . deepcopy ( s e l f . t r apCon f i gu ra t i on )theCopy . updater . t r apCon f i gu ra t i on = theCopy . t r apCon f i gu ra t i ontheCopy . p t c l s . p t c l L i s t = copy . deepcopy ( s e l f . p t c l s . p t c l L i s t )theCopy . i n i t s i m ( theCopy . t rapCon f i gu ra t i on ,theCopy . updater . ax ia lDampingCoef f i c ien t ,theCopy . updater . angularDampingCoef f ic ient )return theCopydef i n i t i a l i z e ( s e l f ) :s e l f . updater . SetUpAs ( s e l f . p t c l s . x ( ) . shape , s e l f . p t c l s . x ( ) . dtype )def i n i t s i m ( se l f , t rapConf ig , ax ia lDampingCoef f i c ien t ,angularDampingCoef f ic ient , r e c o i l V e l o c i t y = None ,sca t te rRate = None ) :i f r e c o i l V e l o c i t y == None :r e c o i l V e l o c i t y = 0.1i f sca t te rRate == None :sca t te rRate = 0.0s e l f . t r apCon f i gu ra t i on = t rapConf igs e l f . updater . t r apCon f i gu ra t i on = t rapConf igs e l f . updater . ax ia lDampingCoef f i c ien t = ax ia lDampingCoef f i c ien ts e l f . updater . angularDampingCoef f ic ient = angularDampingCoef f ic ients e l f . r e c o i l V e l o c i t y = r e c o i l V e l o c i t ys e l f . sca t te rRate = sca t te rRates e l f . accL i s t = [ ]s e l f . accL i s t . append (CoulombAcc . CoulombAcc ( s e l f . c tx , s e l f . queue ) )s e l f . t rapAcc = TrapAcc . TrapAcc ( s e l f . c tx , s e l f . queue )s e l f . t rapAcc . t r apCon f i gu ra t i on = t rapConf igs e l f . accL i s t . append ( s e l f . t rapAcc )158s e l f . t = 0.0def take s teps ( se l f , dt , numSteps = 1 ) :xd = c l a r r a y . to dev i ce ( s e l f . queue , s e l f . p t c l s . x ( ) , async = True )yd = c l a r r a y . to dev i ce ( s e l f . queue , s e l f . p t c l s . y ( ) , async = True )zd = c l a r r a y . to dev i ce ( s e l f . queue , s e l f . p t c l s . z ( ) , async = True )vxd = c l a r r a y . to dev i ce ( s e l f . queue , s e l f . p t c l s . vx ( ) , async = True )vyd = c l a r r a y . to dev i ce ( s e l f . queue , s e l f . p t c l s . vy ( ) , async = True )vzd = c l a r r a y . to dev i ce ( s e l f . queue , s e l f . p t c l s . vz ( ) , async = True )qd = c l a r r a y . to dev i ce ( s e l f . queue , s e l f . p t c l s . q ( ) , async = True )md = c l a r r a y . to dev i ce ( s e l f . queue , s e l f . p t c l s .m( ) )s e l f . t = s e l f . updater . update ( xd , yd , zd , vxd , vyd , vzd , qd , md,s e l f . accL is t , s e l f . t , dt , numSteps )s e l f . queue . f i n i s h ( )xd . get ( s e l f . queue , s e l f . p t c l s . x ( ) , async = True )yd . get ( s e l f . queue , s e l f . p t c l s . y ( ) , async = True )zd . get ( s e l f . queue , s e l f . p t c l s . z ( ) , async = True )vxd . get ( s e l f . queue , s e l f . p t c l s . vx ( ) , async = True )vyd . get ( s e l f . queue , s e l f . p t c l s . vy ( ) , async = True )vzd . get ( s e l f . queue , s e l f . p t c l s . vz ( ) )def spin up ( s e l f ) :r a d i i = numpy . sq r t ( s e l f . p t c l s . x ( ) ∗∗ 2 + s e l f . p t c l s . y ( ) ∗∗ 2)v e l o c i t i e s = s e l f . updater . t r apCon f i gu ra t i on . omega ∗ r a d i ifor i in range (0 , s e l f . p t c l s . p t c l L i s t . shape [ 1 ] ) :v = numpy . ar ray ([− s e l f . p t c l s . y ( ) [ i ] , s e l f . p t c l s . x ( ) [ i ] ] )v = v / numpy . l i n a l g . norm ( v )v = v e l o c i t i e s [ i ] ∗ vs e l f . p t c l s . vx ( ) [ i ] = v [ 0 ]s e l f . p t c l s . vy ( ) [ i ] = v [ 1 ]def r a d i a l v e l o c i t i e s ( s e l f ) :radVel = [ ]for i in range (0 , s e l f . p t c l s . p t c l L i s t . shape [ 1 ] ) :rad ia lUn i tVec = numpy . ar ray ( [ s e l f . p t c l s . x ( ) [ i ] ,s e l f . p t c l s . y ( ) [ i ] ] )r ad ia lUn i tVec = rad ia lUn i tVec /numpy . l i n a l g . norm ( rad ia lUn i tVec )radVel . append (numpy . inne r ( rad ia lUn i tVec ,numpy . ar ray ( [ s e l f . p t c l s . vx ( ) [ i ] , s e l f . p t c l s . vy ( ) [ i ] ] ) ) )return numpy . ar ray ( radVel )def angu l a r v e l o c i t i e s ( s e l f ) :angVel = [ ]159for i in range (0 , s e l f . p t c l s . p t c l L i s t . shape [ 1 ] ) :angularUni tVec = numpy . ar ray ([− s e l f . p t c l s . y ( ) [ i ], s e l f . p t c l s . x ( ) [ i ] ] )angularUni tVec = angularUni tVec /numpy . l i n a l g . norm ( angularUni tVec )angVel . append (numpy . inne r ( angularUnitVec ,numpy . ar ray ( [ s e l f . p t c l s . vx ( ) [ i ] , s e l f . p t c l s . vy ( ) [ i ] ] ) ) )return numpy . ar ray ( angVel )def r a d i i ( s e l f ) :return numpy . ar ray ( [ numpy . l i n a l g . norm (numpy . ar ray ([ s e l f . p t c l s . x ( ) [ i ] , s e l f . p t c l s . y ( ) [ i ] ] ) ) for i inrange (0 , s e l f . p t c l s . p t c l L i s t . shape [ 1 ] ) ] )def xyInRotat ingFrame ( s e l f ) :t he ta = s e l f . updater . t r apCon f i gu ra t i on . the tareturn numpy . ar ray ([ numpy . cos ( the ta ) ∗ s e l f . p t c l s . x ( )+ numpy . s in ( the ta ) ∗ s e l f . p t c l s . y ( ) ,−numpy . s in ( the ta ) ∗ s e l f . p t c l s . x ( )+ numpy . cos ( the ta ) ∗ s e l f . p t c l s . y ( ) ] )# Routine to set up the ob jec t t ha t de f ines a l l# the parameters o f the plasmaimport randomimport numpyimport mathdef compute k ine t i c energ ies ( vx , vy , vz , m) :” ” ” Compute the k i n e t i c energies o f a set o f p a r t i c l e s . ” ” ”return 0.5 ∗ m ∗ ( vx∗∗2 + vy∗∗2 + vz∗∗2)def compute temperature ( vx , vy , vz , m) :” ” ” Compute temperature o f neu t r a l p a r t i c l e s i n f ree space .This f unc t i on assumes t ha t a l l motion o f the p a r t i c l e s i sthermal , i . e . there i s no center o f mass momentum or angularmomentum.” ” ”kB = 1.3806e−23return ( 2 .0 / 3 .0 ) ∗numpy .mean( compute k ine t i c energ ies ( vx , vy , vz , m) ) / kBclass Ptc l s ( ) :160” ” ” An ensemble o f p a r t i c l e s . ” ” ”def i n i t ( se l f , n = 1 ) :s e l f . numPtcls = 1s e l f . sigma = 1.0e−4s e l f . vbar = 0s e l f . v th = 0s e l f . p t c l L i s t = 0.0 ∗ numpy . ndarray ( [ 8 ,s e l f . numPtcls ] , dtype=numpy . f l o a t 64 )s e l f . prec = numpy . f l o a t 64s e l f . numPtcls = ndef i n i t p t c l s ( se l f , p s t a r t = 0 , pend = 0 , charge = 1.602176565e−19,mass = 8.9465 ∗ 1.673e−27, source = ’ gaussian ’ ) :s e l f . p t c l L i s t = 0 . ∗ numpy . ndarray ( [ 8 ,s e l f . numPtcls ] , dtype = s e l f . prec )i f pend == 0:pend = s e l f . numPtclsi f source== ’ gaussian ’ :for j in range ( ps ta r t , pend ) :r = abs ( random . gauss ( 0 . , s e l f . sigma ) )the ta = numpy . arccos (2 .∗ random . random()−1)ph i = 2.∗numpy . p i∗random . random ( )s e l f . p t c l L i s t [ 0 ] [ j ]= r∗numpy . s in ( the ta )∗numpy . cos ( ph i )s e l f . p t c l L i s t [ 1 ] [ j ]= r∗numpy . s in ( the ta )∗numpy . s in ( ph i )s e l f . p t c l L i s t [ 2 ] [ j ]= r∗numpy . cos ( the ta )v r = random . gauss ( s e l f . vbar , s e l f . v th )the ta = numpy . arccos (2 .∗ random . random()−1)ph i = 2.∗numpy . p i∗random . random ( )s e l f . p t c l L i s t [ 3 ] [ j ]= v r∗numpy . s in ( the ta )∗numpy . cos ( ph i )s e l f . p t c l L i s t [ 4 ] [ j ]= v r∗numpy . s in ( the ta )∗numpy . s in ( ph i )s e l f . p t c l L i s t [ 5 ] [ j ]= v r∗numpy . cos ( the ta )s e l f . p t c l L i s t [ 6 ] [ j ]= charges e l f . p t c l L i s t [ 7 ] [ j ]=masse l i f source :s e l f . p t c l L i s t = numpy . l o ad t x t ( source )s e l f . numPtcls = myarr . shape [ 1 ]def temperature ( s e l f ) :” ” ” Compute temperature o f the ensemble o f p a r t i c l e s .This f unc t i on assumes t ha t a l l motion o f the p a r t i c l e s i sthermal , i . e . there i s no center o f mass momentum or angular161momentum.” ” ”return compute temperature ( s e l f . p t c l L i s t [ 3 ] , s e l f . p t c l L i s t [ 4 ] ,s e l f . p t c l L i s t [ 5 ] , s e l f . p t c l L i s t [ 7 ] )def se t np t c l s ( se l f , np ) :s e l f . numPtcls = nps e l f . p t c l L i s t = numpy . res i ze ( s e l f . p t c l L i s t , [ 8 , s e l f . numPtcls ] )def x ( s e l f ) :return s e l f . p t c l L i s t [ 0 ]def y ( s e l f ) :return s e l f . p t c l L i s t [ 1 ]def z ( s e l f ) :return s e l f . p t c l L i s t [ 2 ]def vx ( s e l f ) :return s e l f . p t c l L i s t [ 3 ]def vy ( s e l f ) :return s e l f . p t c l L i s t [ 4 ]def vz ( s e l f ) :return s e l f . p t c l L i s t [ 5 ]def q ( s e l f ) :return s e l f . p t c l L i s t [ 6 ]def m( s e l f ) :return s e l f . p t c l L i s t [ 7 ]162Appendix EMatlab Routines for SelectiveField Ionization CalculationsE.1 List of constants and conversion functionsc lassde f consp rope r t i e s ( Constant= t rue )Avo=6.02214129e23 ;R=8.314462145468951;F=96485.3365;e l =1.60217657e−19;eps i l on =8.854187817e−12;eps i lon2d=7.323564369075211e−18;kB=1.3806488e−23;mi=4.981733643307871e−26; % mass of NO in kgme=9.10938291e−31;mp=1.67262178e−27;h=6.62606957e−34;hbar=cons . h / (2∗ pi ) ;G=6.67384e−11;c=299792458; %m/ sa0=5.2917721092e−5;kBau=0.012374764324710;Ryd=10973731.6;Rydhc=2.179872000000000e−18;%2.181381270723222e−18;RydhcAU=1.955173475509261e+03;RydkB=1.579968251682269e+05;NOrot=1.67195; %cm−1163NOprot=1.9971945;NOIP=30522.45;NOIPN2=30522.45+11.9;endmethods ( S t a t i c )function y=NN( r , den )y=4∗pi∗den∗ r . ˆ 2 .∗ exp(−4∗pi∗den∗ r . ˆ 3 / 3 ) ;endfunction [ pf , eden , rden ]= penn ing f rac t i on ( n , den )Rn0=n . ˆ2∗ cons . a0 ;% rad ius o f Rydb . by bohr model using semi−c l a s s i c a l methodRmax=1.8∗ (Rn0∗2) ;% Robicheaux paper , w i t h i n t h i s d is tance , 90% penning i on i zepf=1−exp(−4∗pi∗den∗Rmax . ˆ 3 / 3 ) ;% propo r t i on between 0 and Rmaxeden=pf /2∗den ;% the den of e l ec t r on produce i s ha l f the p ropo r t i on% (1e− per par tne r )rden=(1−pf )∗den ;% th i s i s remaining dens i t y o f rydbergsendfunction [ r , rp ]= randonspherex=( rand− .5)∗2;y=( rand−.5)∗2∗sqrt (1−x . ˆ 2 ) ;z=sqrt (1−x.ˆ2−y . ˆ 2 ) ;i f rand>.5z=−z ;endr = [ x ; y ; z ] ;r = r ( randperm ( 3 ) ) ;x=( rand− .5)∗2;y=( rand−.5)∗2∗sqrt (1−x . ˆ 2 ) ;z=sqrt (1−x.ˆ2−y . ˆ 2 ) ;i f rand>.5z=−z ;endrp =[ x ; y ; z ] ;rp=rp ( randperm ( 3 ) ) ;rp=rp−r∗sum( rp .∗ r ) ;rp=rp / sqrt (sum( rp . ˆ 2 ) ) ;endfunction [ r , rp ]= randonc i r c l e% create a vec to r po i n t i ng randomlyx=( rand− .5)∗2;y=sqrt (1−x . ˆ 2 ) ;i f rand>.5164y=−y ;endr = [ x ; y ] ;r = r ( randperm ( 2 ) ) ;% do i t one more t imex=( rand− .5)∗2;y=sqrt (1−x . ˆ 2 ) ;i f rand>.5y=−y ;endrp =[ x ; y ] ;rp=rp ( randperm ( 2 ) ) ;% sub t rac t the common pa r t o f the two vec to r to create normal% ones t ha t are prepend icu la rrp=rp−r∗sum( rp .∗ r ) ;% normal ize the new prepend icu la r vec to rrp=rp / sqrt (sum( rp . ˆ 2 ) ) ;endfunction GHz=cmtoGHz(cm)GHz=cons . c∗cm∗100;endfunction mm=GHztomm(GHz)mm=cons . c / (GHz∗1e9)∗1000;endfunction cm=nmtocm(nm)cm=1e7 . / nm;endfunction y=lambdanm(n1 , n2 )y=1e9 ./(10968800∗abs ( 1 . / n1 . ˆ2−1. / n2 . ˆ 2 ) ) ;endfunction y=yukawa ( r , l )y=cons . e l . ˆ 2 . / ( 4 ∗ pi∗cons . eps i l on∗ r ) .∗ exp(− r / l ) ;endfunction y=aws ( den )y = (3 . / ( 4∗ pi∗den ) ) . ˆ ( 1 / 3 ) ;endfunction y=a2d ( den )y = (1 . / ( 4∗ pi∗den ) ) . ˆ ( 1 / 2 ) ;endfunction y=awsden (aws )y =1 . / (4∗ pi∗aws . ˆ 3 / 3 ) ;endfunction y=debye (Te , ne )y=sqrt ( cons . eps i l on∗cons . kB .∗Te . / ( ne∗cons . e l ˆ 2 ) ) ;endfunction y=debnum(Te , ne )165y=4∗pi∗ne∗cons . debye (Te , ne ) ˆ 3 / 3 ;endfunction y=yukawaTene ( r , Te , ne )y=cons . yukawa ( r , cons . debye (Te , ne ) ) ;endfunction y= n c r i t i c a l (T )y=round ( sqrt ( cons . Rydhc / cons . kB . / T ) ) ;endfunction y=scaledT ( Ti , Te , ne )y=cons . kB∗4∗pi∗cons . eps i l on∗Ti∗cons . debye (Te , ne ) / cons . e l ˆ 2 ;endfunction y=scaledn ( ni , Te , ne )y=4∗pi∗n i∗cons . debye (Te , ne ) . ˆ 3 / 3 ;endfunction y=g ( den , Te )y=cons . e l ˆ 2 . / ( 4∗ pi∗cons . eps i l on∗cons . kB∗Te∗cons . aws ( den ) ) ;endfunction y=we( den )y=sqrt ( den∗cons . e l ˆ 2 / ( cons .me∗cons . eps i l on ) ) ;endfunction y=wpi ( den )y=sqrt ( den∗cons . e l ˆ 2 / ( cons . mi∗cons . eps i l on ) ) ;endfunction y=av ( r ,N)y=sqrt (sum( r . ˆ 2 .∗N, 2 ) . /sum(N, 2 ) / 3 ) ;endfunction y=scaledtoEn ( scEn , den )y=scEn∗cons .me∗cons . aws ( den ) . ˆ 2 . ∗ cons .we( den ) . ˆ 2 / cons . kB ;endfunction y=sca led to t ( sct , den )y=sc t / cons .we( den ) ;endfunction y=encm(n )y=cons .Ryd /100 / n ˆ 2 ;endfunction y=dErot ( J1 , J2 ,B)y=B∗( J1∗( J1+1)−J2∗( J2 +1 ) ) ;endfunction y=EF(n , J , a )y = ( ( cons . encm(n)−cons . dErot (2 , J , 2 ) ) / a ) ˆ 2 ;endfunction n=boundton (En , den )n=(−cons . scaledtoEn (En , den)∗ cons . kB / cons . Rydhc ) . ˆ ( −1 / 2 ) ;endfunction s t r =get t imedate ( )s t r =da tes t r (now ) ;s t r =strrep ( s t r , ’ : ’ , ’ ’ ) ;166s t r =strrep ( s t r , ’− ’ , ’ ’ ) ;s t r =strrep ( s t r , ’ . ’ , ’ ’ ) ;s t r =strrep ( s t r , ’ ’ , ’ ’ ) ;endfunction cc= f i t g auss ( t , data , t l im , tag )[ a , ˜ ] = size ( data ) ;ind= t l i m ( 1 ) : t l i m ( 2 ) ;t = t ( ind ) ’ ; t = t ( : ) ;data=data ( : , ind ) ;f t = f i t t y p e ( ’ gauss1 ’ ) ;cc=zeros ( a , 3 ) ;pa r f o r i =1:ay=data ( i , : ) ; y=y ( : ) ;fo ( i )= f i t o p t i o n s ( f t ) ;fo ( i ) . Lower=[0 min ( t ) 0 ] ;ub=max( y)−min ( y )+eps ;fo ( i ) . Upper =[ub∗1.2 max( t ) max( t )−min ( t ) ] ;i f strcmp ( tag , ’ ramp ’ )[ ˜ , b ]= f indpeaks ( y , ’ minpeakdistance ’ , 1 0 0 , . . .’ minpeakheight ’ ,mean( y ) / 2 , ’ s o r t s t r ’ , ’ descend ’ ) ;i f length ( b)>=1b=b ( 1 ) ;fo ( i ) . S t a r tPo i n t =[ y ( b ) t ( b ) 1 ] ;elsefo ( i ) . S t a r tPo i n t =[mean( y ) / 2 mean( t ) 1 ] ;ende l se i f strcmp ( tag , ’ f i x ed ’ )fo ( i ) . S t a r tPo i n t =[mean( y ) / 2 mean( t ) max( t )−min ( t ) ] ;endc f = f i t ( t , y , f t , fo ( i ) ) ;cc ( i , : ) = coe f f va lues ( c f ) ;endendfunction EF=ntoEF (n , tp )% n=(5.14E+9. / (16∗EF ) ) . ˆ ( 1 / 4 ) −> EF=5.14E+9. / (16∗n . ˆ 4 )i f strcmp ( tp , ’ a ’ )EF=5.14E+9. / (16∗n . ˆ 4 ) ;e l se i f strcmp ( tp , ’ d ’ ) ;EF=5.14E+9 . / (9∗n . ˆ 4 ) ;endendfunction y=hgauss ( x , cc )a=cc ( 1 ) ;b=cc ( 2 ) ;c=cc ( 3 ) ;y=a∗exp(−(x−b ) . ˆ 2 / c ˆ 2 ) ;167endfunction tau = rydper iod ( n )%\ tau ˆ2 = {4\p i ˆ2\mu \over kZeˆ2}aˆ3tau = sqrt (16∗pi ˆ3∗cons . eps i l on∗cons .me / ( cons . e l ˆ 2 ) . . .∗( cons . a0∗1e−6∗n ˆ 2 ) ˆ 3 ) ;endfunction tau = rydper iodsca led ( n , den )%\ tau ˆ2 = {4\p i ˆ2\mu \over kZeˆ2}aˆ3tau = cons . rydper iod ( n)∗ cons .we( den ) ;endendendfunction wl = vacuumcorrect ion ( wl )% co r rec t f o r the change of r e f r a c t i v e index . Make sure the wavelength i s% f o r the frequency double l i g h twl=wl .∗(1+0.05792105./(238.0185−1e6 . / wl . ˆ 2 ) . . .+0.00167917./(57.362−1e6 . / wl . ˆ 2 ) ) ;%re f r a c t i v e index minus 1E.2 Binary data and text file handling% DA class stands f o r data ana lys i s done f o r NO experiments a t Prof .% Grant ’ s l abo ra t o r y . This c lass c a l l e s sen t i a l f unc t i ons t ha t loads data% from a se lec ted d i r e c t o r y . There are some func t i ons t ha t f a c i l i t a t e% p l o t t i n g and some other tasks .c lassde f DAmethods ( S t a t i c )function d=readspec ( path )% readspec take path as i npu t . The path should i nd i c a t e the% loca t i o n o f the f o l d e r t ha t conta ins p f i , w2 , or w1 spectrum% f i n d f i l e s t ha t end wi th . wff =di r ( f u l l f i l e (path , ’ ∗ . wf ’ ) ) ;% i n i t i a t e a data s t r u c t u r ed= s t r u c t ;% loop over a l l . wf f i l e sfor i =1: length ( f )% p r i n t the f i l e name in the command windowf p r i n t f ( ’%s\n ’ , f ( i ) . name ) ;% read dataset from a set o f f i l e s% the inpu ts are the main path to the f o l d e r and the name168% of each f i l e w i th . wf i n the f o l d e r path% the outputs are 2D data , s p e c i f i c a t i o n s o f the waveform ,% time array , and the scanning va r i ab l e ( delay , wavelength ,% . . . )[ d ( i ) . data , d ( i ) . wf , d ( i ) . t , d ( i ) . wl ] = . . .DAIO . readdat (path , f ( i ) . name ) ;% add a new f i e l d ca l l ed name which i s the same as the f i l e% name except the date a t the beginning and . wf a t the endd ( i ) . name= f ( i ) . name(13 :end−3);endendfunction d=readPS (path , l i n e s )% readPS take path and l i n e s as i npu t and re tu rns pulse shape% data . Path i s the f o l d e r l o ca t i o n o f the pulse shapes and% l i n e s i s the number o f comment l i n e s on top of the data . The% code w i l l e l im ina te the top pa r t o f the f i l e as spec i f i ed by% number o f l i n e s and reads the res t as data% f i n d f i l e s t ha t end wi th . wff =di r ( f u l l f i l e (path , ’ ∗ . wf ’ ) ) ;% i n i t i a t e a data s t r u c t u r ed= s t r u c t ;for i =1: length ( f )% readwl i s the core f unc t i on t ha t read a pulse shape f i l e .% i t take the o r i g i n a l path and the name of each pulseshape% in the f o l d e r and the number o f l i n e s o f comments and% processes the data[ d ( i ) . t , d ( i ) . v , d ( i ) . wf ]=DAIO . readwl (path , f ( i ) . name, l i n e s ) ;% make sure t ha t there i s no o f f s e t to the vo l taged ( i ) . v=d ( i ) . v−d ( i ) . v ( 1 ) ;% add a new f i e l d name from the f i l e named ( i ) . name= f ( i ) . name ;% EF ( e l e c t r i c f i e l d ) used to be vo l tage d iv ided by% dis tance . Right now EF i s redundant as i t i s the same as% v . Let ’ s keep i t t ha t way f o r now .d ( i ) . EF=d ( i ) . v ;% f i t a smooth f unc t i on to the shape of the e l e c t r i c f i e l d% wi thou t worry ing much about i t s f u n c t i o na l formd ( i ) . c f = f i t ( d ( i ) . t ( : ) , sort ( d ( i ) . EF ( : ) ) , . . .f i t t y p e ( ’ smooth ingspl ine ’ ) ) ;end169endfunction x= f i t n s ( wl , g )% th i s f unc t i on take a set o f wavelength a t maxiumum absorb t ion% and an i n i t i a l guess o f g f o r the n0 = quantum number and% de l t a = quantum defec t and at tempts to f i t IP , de l ta , and% quantum number% wavelengths must be consequt ive . I t can be ascending or% descending . Wavelength i s o f the doubled frequency and vacuum% cor rec ted% no matter what or wl is , make i s ascendingwl=sort ( wl ) ;% compute wavenumber from double frequency wavelength i n vacuumwavenum=1e7 . / wl ; % inpu t i s i n nm. output i s i n cmˆ−1% Rydberg constant cmˆ−1Ryd=109737.316;% an ob j ec t i v e f unc t i on t ha t re tu rns the e r r o r o f f i tfunction o=obj ( x )% quantum defec tde l t a=x ( 3 ) ;% bu i l d a range of numbers t ha t when added to n0 = g (1 )% w i l l g ive the range of quantum numbers .n= length ( wl )−1:−1:0;% conver t the range of numbers to range of quantum numbersn=n+ f i x ( x ( 2 ) ) ;% compute d i f f e r ence of exper imenta l wavenumbers and the% computed oneo=wavenum’−( x(1)−Ryd . / ( n−de l t a ) . ˆ 2 ) ;end% minimize e r r o r o f the ob j e c t i v e f unc t i on to f i n d the best f i t% IP n0 de l t a% x i s 3 by 1 ar ray : IP , n0 , and de l t ax= l sqnon l i n (@( x ) ob j ( x ) , [30544.2333070219 ,g ( 1 ) , g ( 2 ) ] ) ;endfunction x= f i t n s q d f i x e d ( wl , g )% th i s f unc t i on take a set o f wavelength a t maxiumum absorb t ion% and an i n i t i a l guess o f g f o r the n0 = quantum number and% de l t a = quantum defec t and at tempts to f i t IP , de l ta , and% quantum number170% wavelengths must be consequt ive . I t can be ascending or% descending . Wavelength i s o f the doubled frequency and vacuum% cor rec ted% no matter what or wl is , make i s ascendingwl=sort ( wl ) ;% compute wavenumber from double frequency wavelength i n vacuumwavenum=1e7 . / wl ; % inpu t i s i n nm. output i s i n cmˆ−1% Rydberg constant cmˆ−1Ryd=109737.316;% quantum defec tde l t a=g ( 2 ) ;% an ob j ec t i v e f unc t i on t ha t re tu rns the e r r o r o f f i tfunction o=obj ( x )% bu i l d a range of numbers t ha t when added to n0 = g (1 )% w i l l g ive the range of quantum numbers .n= length ( wl )−1:−1:0;% conver t the range of numbers to range of quantum numbersn=n+ f i x ( x ( 2 ) ) ;% compute d i f f e r ence of exper imenta l wavenumbers and the% computed oneo=wavenum’−( x(1)−Ryd . / ( n−de l t a ) . ˆ 2 ) ;end% minimize e r r o r o f the ob j e c t i v e f unc t i on to f i n d the best f i t% IP n0 de l t a% x i s 3 by 1 ar ray : IP , n0 , and de l t ax= l sqnon l i n (@( x ) ob j ( x ) , [30544.2333070219 ,g ( 1 ) ] ) ;endfunction wl = ntowl ( ns , x )% compute the wavelength o f the double frequency l i g h t i n% vacuume (make sure mu l t i p l y by 2 i f you ’ re working wi th the% fundamental wavelength out o f ND6000wl = 1e7 . / ( x(1)−cons .Ryd / 1 0 0 . / ( ns−x ( 2 ) ) . ˆ 2 ) ;endendend171c lassde f DAIO%DAIO Summary o f t h i s c lass goes here% DAIO stands f o r data ana lys i s i npu t output% t h i s c lass conta ins core func t i ons t ha t are ca l l ed by DA c lass . I t i s% seldom necessary to c a l l DAIO d i r e c t l y from your s c r i p tmethods ( S t a t i c = t rue )function [ dat , wf , t , wl ] = readdat (path , fname )% th i s f unc t i on read the . dat f i l e con ta in ing the 2D i n t e n s i t y% p l o t% read the . wf f i l e s to get the parameters o f the waveform% f i lename should be imported as something . wf ( handled by DA% class )wf = DAIO . readwf (path , fname ) ;% the 3rd parameter from . wf f i l e i s the # of po in t s i n a% s ing l e t race , same as the leng th o f the t ime ar raynum points = wf ( 3 ) ;% now keep the same f i l e name but swap the extens ion . The new% f i lename w i l l look f o r a . dat f i l e w i th the same name as . wffname = regexprep ( fname , ’ . wf ’ , ’ . dat ’ ) ;% the f u l l f i l e name i s path + namef i l e = [ path , fname ] ;% open a f i l e as read−only and re tu rn an ID to the f i l e handlef i d = fopen ( f i l e ) ;% read from the f i l e w i th the ID = f i d . I n f s pec i f i e s the% number o f bytes to read , i n t h i s case every th ing . ’ u in t16 ’% spec i f i e s the format o f the b inary data which i s unsigned 16% b i t i n t ege r ( the number o f po in t s on the y axes of the scope% s t a r t i n g from zero . ’ b ’ stands f o r b igendian . Labview uses% IEEE standard to record numbers as binary , we t e l l matlab how% to read t ha t number as an i n t ege rrawdat = fread ( f i d , i n f , ’ u in t16 ’ , ’ b ’ ) ;% close the connect ion to the f i l e ( a good p rac t i c e to always% do tha t )fclose ( f i d ) ;% rawdata w i l l be a long ar ray o f i n t ege rs . The processdat% func t i on i n t h i s c lass w i l l reshape and scale the data to% give vo l tages i n mV. I have e l im ina ted the need to o f f s e t172% data i n the core f unc t i on .dat=DAIO . processdat ( rawdat , num points , wf ) ;% bu i l d t ime ar ray based on the number o f po in ts , d i v i s i o n per% po in t and the o f f s e tt =DAIO . b u i l d t ( wf , num points ) ;% th i s f unc t i on w i l l read the wl f i l e assoc ia ted wi th o ther% f i l e s t ha t have the same name but ends i n . wlwl=DAIO . w l f i l e u n i (path , fname ) ;% make sure a l l the connect ion to f i l e s are closedfclose ( ’ a l l ’ ) ;endfunction t = b u i l d t ( wf , num points )t0=wf ( 6 ) ; % i n i t i a l scope t imet i n c =wf ( 5 ) ; % time incrementt f = t i n c ∗num points+ t0 ; % f i n a l t ime% generate an ar ray w i th spec i f i ed s t a r t , end and number o f% po in t st =1e6∗ l inspace ( t0 , t f , num points ) ;endfunction dat=processdat ( rawdat , num points , wf )% reshape the rawdata based on the number o f po in t i n each% s ing l e t racedat = reshape ( rawdat , num points , max( size ( rawdat ) ) / . . .num points ) ;%j u s t reshape i t% conver t i n t ege rs to doubledat = double ( dat ’ ) ; %conver t to double% conver t raw values to mVdat=−dat∗wf ( 8 ) ; %conver t to mVendfunction wf=readwf (path , fname )% th i s f unc t i on w i l l read the waveform f i l e t ha t con ta i ns t some% in fo rma t i on about the% bu i l d f u l l path to the f i l efname = [ path fname ] ;% open f i l e as read−onlyf ID=fopen ( fname ) ;173% read one un i t o f b inary data i n the format spec i f i ed as ’ b ’ =% bigendian . The un i t w i l l depend on the format . i n t 16 w i l read% 2 bytes , wh i le double w i l l read 8 byteswf (1)= double ( fread ( f ID ,1 , ’ i n t 16=> i n t 16 ’ , ’ b ’ ) ) ;wf (2 )= fread ( f ID ,1 , ’ i n t 16=> i n t 16 ’ , ’ b ’ ) ;wf (3 )= fread ( f ID ,1 , ’ i n t 32=> i n t 32 ’ , ’ b ’ ) ;wf (4 )= fread ( f ID ,1 , ’ i n t 32=> i n t 32 ’ , ’ b ’ ) ;wf (5 )= fread ( f ID ,1 , ’ double=>double ’ , ’ b ’ ) ;wf (6 )= fread ( f ID ,1 , ’ double=>double ’ , ’ b ’ ) ;wf (7 )= fread ( f ID ,1 , ’ i n t 32=> i n t 32 ’ , ’ b ’ ) ;wf (8 )= fread ( f ID ,1 , ’ double=>double ’ , ’ b ’ ) ;wf (9 )= fread ( f ID ,1 , ’ double=>double ’ , ’ b ’ ) ;wf (10)= fread ( f ID ,1 , ’ i n t 32=> i n t 32 ’ , ’ b ’ ) ;% close connect ion to a l l f i l e sfclose ( ’ a l l ’ ) ;endfunction [ t , d , wf ]= readwl (path , fname , l i n e s )fname2=regexprep ( fname , ’ . wf ’ , ’ . wl ’ ) ;d= impor tdata ( [ path fname2 ] , ’\ t ’ , l i n e s ) ;wf=DAIO . readwf (path , fname ) ’ ;d=d . data ;t =(d ( : , 1 ) ’ + wf (6 ) )∗1e6 ;d=d ( : , 2 ) ’∗ wf ( 8 ) ;fclose ( ’ a l l ’ ) ;endfunction [ t , d , wf ]= readwlraw (path , fname , l i n e s )fname2=regexprep ( fname , ’ . wf ’ , ’ . wl ’ ) ;d= impor tdata ( [ path fname2 ] , ’\ t ’ , l i n e s ) ;wf=DAIO . readwf (path , fname ) ’ ;d=d . data ;t =(d ( : , 1 ) ’ + wf (6 ) )∗1e6 ;d=d ( : , 2 ) ’ ;fclose ( ’ a l l ’ ) ;endfunction wl= w l f i l e u n i (path , fname )% th i s f unc t i on does the same th i ng as% readwl . However , i t w i l l not requ i re you to spec i f y the% number o f l i n e s o f commnets% prepare the name of the f i l e w i th the same name as the . dat% f i l e but w i th . wl extens ionfname2=regexprep ( fname , ’ . dat ’ , ’ . wl ’ ) ;% fname2=[ fname ( 1 : end−3) ’ . wl ’ ] ;% open the f i l e w i th read−only permiss ionf i d =fopen ( [ path fname2 ] ) ;174% [ path fname2 ]% get the f i r s t l i n et l i n e = f ge t l ( f i d ) ;% whi le the f i r s t l i n e or the next one are not #START cont inue% reading l i n e swhile ˜ strcmp ( t l i n e , ’ #START ’ )t l i n e = f ge t l ( f i d ) ;i f t l i n e ==−1break ;endendq=1;% the curser to the f i d must be r i g h t a f t e r #START, where% comments end and data beginswhile ˜ feof ( f i d )% read a new l i n et l i n e = f ge t l ( f i d ) ;% s p l i t the l i n e by \ t = tab between the numberst l i n e = s t r s p l i t ( t l i n e , ’\ t ’ ) ;% conver t the ar ray o f t e x t c e l l s to an ar ray o f doubleswl ( q , : ) = s t r2doub le ( t l i n e ) ;% advance the indexq=q+1;end% close connect ion to a l l f i l e sfclose ( f i d ) ;fclose ( ’ a l l ’ ) ;endendend175

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