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Integration of a multi reflection time of flight isobar separator into the TITAN experiment at TRIUMF Finlay, Andrew 2017

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Integration of a Multi Reflection Timeof Flight Isobar Separator into theTITAN Experiment at TRIUMFbyAndrew FinlayB.Sc., University of Guelph, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017c© Andrew Finlay 2017AbstractThe TITAN experiment at TRIUMF performs high-precision mass measurements on rareisotopes using Penning trap mass spectrometry. A major challenge is the presence of isobariccontamination introduced by the rare isotope production process. To remove these a newMulti-Reflection Time-of-Flight (MR-ToF) mass spectrometer and isobar separator has beenintegrated into the TITAN system.To facilitate the integration of the MR-ToF the TITAN beamline has been studied using acombination of computer based ion optics simulations and experimental measurements. Simula-tions were benchmarked against a measurement of the ion beam’s transverse emittance using anAllison meter, and beam time profiles measured with microchannel plate detectors. Computermodels were found to be able to reproduce experimental results within a factor of 2. The keysource of differences appears to be the modeling of the cooler buncher TITAN uses for beampreparation.Simulations were used to identify optimal settings on ion optical elements to facilitate themaximum efficiency of ion transport into the MR-ToF and from the MR-ToF to the mea-surement Penning trap. Additional tests of the impact of new optics on the beamline whenbypassing the MR-ToF show beam properties before and after the changes to be identical withinuncertainty. Suggested settings have successfully been used to guide the injection of ions intothe MR-ToF.Once the MR-ToF was installed in the TITAN system, tests were performed to demonstratethe functionality of the MR-ToF using externally produced beam. The ISAC Off-Line IonsSource was used to produce a 40Ar+ ion beam for testing. This was merged with 40K+ ionsfrom the MR-ToF internal ion source to demonstrate the resolving power of the MR-ToF.Mass measurements were performed at a resolving power of 200 000, exceeding performanceexpectations by a factor of 2. Isobar separation was used to remove either Ar or K, requiring amass resolving power ≥ 25 000. The MR-ToF is now a functioning part of the TITAN system,and has already been used to perform mass measurements of rare isotopes.iiLay SummaryThe TITAN experiment at TRIUMF uses ion traps to very quickly perform mass measure-ments of short-lived rare isotopes, providing insights into nuclear structure and astrophysics.A major challenge in this work is the presence of contaminants mixed with the ionized iso-topes being measured. To filter these out we have added a device called a Multi-ReflectionTime-of-Flight (MR-ToF) mass spectrometer.Preparing for this integration, computer simulations were used to study the passage of ionsinside the TITAN system. These were validated with experimental measurements, then usedto calculate settings which would best allow the efficient movement of ions into and out of theMR-ToF. Finally the MR-ToF was tested with ion beams produced externally from TITANto demonstrate the operating abilities of the MR-ToF. Here the MR-ToF was shown able toidentify and remove contaminants at a resolution exceeding design expectations.iiiPrefaceThe experimental work of this thesis was carried out at TRIUMF’s Ion Trap for Atomicand Nuclear science (TITAN) experimental facility in Vancouver, BC.Simulations of the TITAN cooler buncher described in Chapter 3 are based on a reproductionof code developed by M. Smith with buffer gas code rewritten by D. Short. Experimental workin Chapter 3 and 4 was lead by C. Babcock and M.P. Reiter respectively. I was a contributingmember in this during the preparation and data collection process. Data analysis presented ismy own.Figures 1.4, 1.5, 1.6, 2.16, 4.1 and 4.2 have been reproduced with permission.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Penning traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Ion motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Mass measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Preparation traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Paul traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Multi-Reflection Time-of-Flight mass spectrometer and isobar separator . . . . . 41.4 Beam purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 TITAN Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5.1 ISAC Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5.2 Cooler Buncher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5.3 EBIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5.4 Measurement Penning trap . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5.5 TITAN MR-ToF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Principles of ion transport needed for MR-ToF integration . . . . . . . . . . 112.1 Understanding beam dynamics using emittance . . . . . . . . . . . . . . . . . . 112.1.1 Transverse emittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Longitudinal emittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.3 Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Ion optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 Electrostatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Radio Frequency Quadrupole Optics . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 RF Ion Guides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25vTable of Contents2.3.2 Linear Paul Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Beamline studies for MR-ToF integration . . . . . . . . . . . . . . . . . . . . . 343.1 The TITAN beamline prior to integrating the MR-ToF . . . . . . . . . . . . . . 343.1.1 Beamline simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.2 Allison meter emittance measurements . . . . . . . . . . . . . . . . . . . 383.1.3 Beam time profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 MR-ToF integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.1 Beamline modification for MR-ToF . . . . . . . . . . . . . . . . . . . . . 493.2.2 MR-ToF acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2.3 Beam profiles out of the MR-ToF . . . . . . . . . . . . . . . . . . . . . . 553.2.4 Bypassing the MR-ToF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 First tests of MR-ToF integrated into TITAN . . . . . . . . . . . . . . . . . . 614.1 Tests of MR-ToF with OLIS beam . . . . . . . . . . . . . . . . . . . . . . . . . . 614.1.1 Ion transport efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.1.2 Mass measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.1.3 Isobar separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73AppendicesA Introduction to SIMION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79B Experimental operating parameters for emittance and time profile measure-ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80C Allison meter data analysis methodology . . . . . . . . . . . . . . . . . . . . . . 82D Allison meter data analysis Lua code . . . . . . . . . . . . . . . . . . . . . . . . 85viList of Tables3.1 A summary of the labels and corresponding purpose for the ion optics and aper-tures in the TITAN beamline. Positions of these are indicated in Figure 3.1. . . . 363.2 Comparison of CB simulation results produced for this thesis to results obtainedin [68]. Differences between the simulations range from 1% to 35%. Unit con-vention for reporting transverse emittance follows that of [68]. . . . . . . . . . . . 374.1 Literature masses [82] for isotopes of K and Ar used for calibration and comparison. 654.2 Comparison of the measured mass of 40K+ to the literature mass reported inReference [82]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66B.1 A summary of the voltages applied to the TITAN beamline electrodes for thetime-profile measurements discussed in Section 3.1.3. Positions of the electrodesin the TITAN beamline are indicated in Figure 3.1. . . . . . . . . . . . . . . . . . 81viiList of Figures1.1 An overview of a Penning trap structure and the motion of a trapped ion. a) Adiagram of a Penning trap showing a typical electrode structure with the appliedvoltages and magnetic field. b) Calculated motion of an ion in a Penning traphighlighting the components of the ion motion: cyclotron ω+ (yellow), magnetronω− (blue), axial ωz (green), and combined (black). . . . . . . . . . . . . . . . . . 21.2 Diagrams of two Paul trap designs showing the electrode structure and an ex-ample of typical driving voltages. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Outline of the principle of time-of-flight mass spectrometry showing three ionsof different masses becoming separated over a given path length. Top: Ionstravel over a long, linear distance. Middle: Ions are reflected between a pairof electrostatic mirrors, leading to the same effective path length as above in asmaller physical space. Bottom: Time-of-flight spectrum at start and finish ofthese mass separations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 A schematic overview of the TITAN experiment a) before the MR-ToF and b)with the MR-ToF installed. Arrowed lines are used to indicate paths the ionbeam may take to TITAN from the ISAC ion sources, inside TITAN, or fromTITAN to the ISAC collinear laser spectroscopy experiment [35]. . . . . . . . . . 71.5 Diagrams of TITAN EBIT showing: a) Components of the trap along the beamaxis, with the path of ions indicated by a black double arrow. b) The arrangementof SiLi gamma-ray detectors around the trapping region. Figures found in [43]. . 81.6 Schematic of the MR-ToF and its interface with the TITAN beamline. Ion opticalelectrodes are indicated in yellow; the MR-ToF vacuum chamber is highlighted ingreen; detectors are shown in blue; red arrowed lines are used to indicate possiblebeam paths through the TITAN beamline and MR-ToF. Adapted from [47]. . . . 102.1 Example of a) an elliptical emittance and b) a filamented emittance. . . . . . . . 122.2 Top: Visualisation of transverse emittance at different positions of a beam, forwhere the beam is converging, at a focus point (beam waist), and when it isdiverging. Bottom: Example plots of the transverse emittance for the positionsindicated by the blue dashed line over each plot. The left plot includes a di-rect illustration of the relation between the spatial distribution ∆x and beamdivergence ∆x′ to the trace space plot. As shown, the orientation of the ellipsechanges and in real scenarios it may change shape, but the total area is invariantfor constant beam energy in a vacuum. . . . . . . . . . . . . . . . . . . . . . . . . 13viiiList of Figures2.3 An example of a time focus for a distribution of particles. Top: Illustratedhere are a set of particles moving with different velocities represented by rightpointing arrows. Represented are three moments in time as the faster particlesstart behind the slower particles then move to overtake the slower particles.The moment where the faster particles are passing the slower is known as thetime focus. Bottom: For each moment in time, an example of a correspondinglongitudinal emittance plot is shown. . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 An example phase space plot where device acceptance is greater than beam emit-tance but their orientations cause them not to match. The unmatched portionof the beam will be lost, resulting in a lower beam transport efficiency. Such amismatch may be eliminated through the use of ion optical lenses to re-orientthe beam emittance into alignment with the acceptance. . . . . . . . . . . . . . . 152.5 Simulations of ion trajectories (red lines) and ion optical elements (brown) foran Einzel lens showing a) an isometric view of the lens and the ion trajectories.b) and c) show the Einzel lens operating with a decelerating and acceleratingpotential respectively, along with contour lines of the electric potential (blackcurves). The example here shows a parallel beam being focused to a point, theposition of which is determined by the potential VLens. Ion trajectories portrayedhere are from ions flying from left to right. Simulations produced in SIMION [54]. 172.6 Examples of parallel plate bender designs. a) A simple rectangular plate design,used in TITAN for small beam path corrections, up to 9◦. b) A 90◦ cylindricalbender, and c) a 90◦ spherical bender. The direction of the electric field betweenthe bender plates is indicated by the blue arrows. . . . . . . . . . . . . . . . . . . 192.7 An illustration of how the time focus is shifted by an electrostatic mirror. Dashedlines indicate the trajectories followed by individual ions. Ellipses are drawnaround the ions to indicate snapshots in time, showing the process of the timefocus shift. The direction of the electric field is indicated with the blue lines,though this field would be present through the mirror. . . . . . . . . . . . . . . . 202.8 Simulation of an electrostatic mirror using a grid electrode. Ion trajectories (red)start on a) the left, b) the top. Ion collisions with the grid electrode (brown)or the boundary of the simulation area are marked in green. Electric potentialcontour lines are shown in black in b). Simulation produced in SIMION [54]. . . 212.9 Schematic of the TITAN MR-ToF ion optics. . . . . . . . . . . . . . . . . . . . . 212.10 An illustration of the time focus shift (TFS) in the TITAN MR-ToF from injec-tion into the MR-ToF to setting a final time focus on a detector. . . . . . . . . . 222.11 Cross-sectional diagram of a typical quadrupole lens with a potential focusingin the x-direction. a) The arrangement of physical rods with the applied elec-tric potentials. b) A simulation of the resulting electric potential distributionproduced in SIMION [54]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.12 Different views of a simulation of ion trajectories (red) through a quadrupolequadruplet (brown). Trajectories are shown for ions flying left to right parallelto the z-axis. Black contour lines indicate the electric potential around theelectrodes. Simulation produced in SIMION [54]. . . . . . . . . . . . . . . . . . . 232.13 Schematic of radio frequency quadrupole (RFQ) rods and their applied bias. Alsoshown is the radius between rods r0. . . . . . . . . . . . . . . . . . . . . . . . . . 24ixList of Figures2.14 Simulated radial ion motion in an RFQ as a function of time for different q-values.Here an ion of mass 133 u is simulated in a trap with r0 = 10 mm operated at afrequency of 1 MHz with the voltage varied to change the q-value. Simulationsproduced in SIMION [54]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.15 Two schematics and corresponding linear potentials of RFQ ion guides. . . . . . 272.16 Images of RFQ switchyard. a) Photo of the switchyard partially assembled. Thecarbon-doped plastic electrodes are visible here. Photo credit [47]. b) Renderedmodel showing the phases of the applied biases on the switchyard in red and blue. 282.17 Diagrams of the RF switchyard ion guide indicating relative strengths of the DCpotentials to allow ions to pass through the switchyard in: a) a straight path, b)a 90◦ path, or c) merge beams from two directions into a single path. . . . . . . . 292.18 TITAN beamline electrostatic switchyard. . . . . . . . . . . . . . . . . . . . . . . 292.19 Top: Schematic of a linear Paul trap based on segmented rods. Bottom: Exampleof linear potential used to trap ions. . . . . . . . . . . . . . . . . . . . . . . . . . 302.20 Schematic of the TITAN cooler buncher [39] electrode configuration. a) Thearrangement of 24 rod segments which can each be given a different DC poten-tial. Ions are trapped in the region of segment 23, then segments 22 and 24are switched to allow ions to escape the trap and be directed into the TITANbeamline. b) A SIMION [54] calculation of the DC potential for cooling (blue)and ejection (red) as a function of linear position in the CB. . . . . . . . . . . . . 323.1 Schematic overview of the TITAN beamline optics between the Cooler Buncherand MPET as relevant for simulations presented herein. Electrodes are labelledfirst with a section label, then the optics type and assigned number. Electrodetypes are summarized in Table 3.1. Black lines appearing before and after manyelectrostatic optics are “skimmer” plates. Red arrows indicate the beam pathand direction. The MR-ToF and EBIT optics are not shown. . . . . . . . . . . . 353.2 Two examples of how a 5 mm diameter circular aperture is modelled in SIMIONat different resolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3 Schematic diagrams of an Allison meter used for measuring beam transverseemittance. Red lines indicate example paths of individual ions passing throughthe meter; the solid line shows a transmitted ion and the dashed line indicatesa) an ion entering along a path which will not pass through the meter, or b) anion which passes through the detector at the very edge of the detector slit. . . . . 393.4 Detected signals at MCP-1 and the Allison MCP as a function of time. Variouspeaks are indicated. Highlighted are the interpretations of specific peaks inthe time-of-flight spectrum as well as the sections of data used to produce theemittance measurement. This example is an integration of all the time profilesfor a specific measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.5 Detected MCP signal as a function of time as function of Measurement Index fora complete Allison meter emittance measurement. Individual time profiles arepresented as density plots parallel to the y-axis. The time profiles are arrangedhorizontally in the order they were taken. Integrated projections of the dataalong the x- and y-axes are shown above and to the right of the density plot.Allison meter MCP signal can be seen as dots at approximately 68 µs. . . . . . . 41xList of Figures3.6 Emittance plots from one measurement of a 133Cs+ beam out of the TITAN CBwith standard operating parameters (described in text), showing different levelsof processing of the number of ions detected at each position and divergence. a)An emittance plot with no processing of the ion counts, showing a large biasingeffect due to drift in the MCS. b) An emittance plot with ion count normalizationand time profile background removed, but before a threshold was set to removebackground counts for rms emittance calculation. c) The same emittance plot asb), but scaled in the z-axis to highlight the degree of variation of the backgroundcounts. Background variations on the level of 2% of the peak ion counts can beseen. d) An emittance plot after the complete ion-count processing described inthe text. A white ellipse calculated from the Twiss parameters is overlaid onthe plot with the size set by the equivalent emittance 4rms. Calculated rmsemittance and Twiss parameters printed below. . . . . . . . . . . . . . . . . . . . 423.7 A simulation of the Allison meter emittance measurement of the TITAN CBbeam shown in Figure 3.6. Left: SIMION simulation of ion trajectories (red) withcollision points (green) and ion optical elements (brown). Right: Emittance plotwith the ellipse calculated from the Twiss parameters for equivalent emittance(4rms) overlaid in white. Calculated rms emittance and Twiss parameters areprinted below. For this simulation, frequency was set at 480 kHz with a peak-to-peak voltage of 170 V, a gas pressure of 10−2 mbar, and temperature of 300K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.8 A comparison of simulated and experimental time profiles obtained at MCP-1(top) and MCP-0 (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.9 A SIMION simulation of the TITAN beamline between the CB and MPET.Beam trajectories are shown in red and points where ions collide with electrodesor reach the position of MCP-0 are marked in green with notable loss pointscircled in green. Most losses occur at TSYBL:DPA. . . . . . . . . . . . . . . . . . 473.10 A plot of the change in transverse rms emittance as a function of beam time offlight from the CB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.11 Simulated trajectories (red) of ions passing through the differential pumpingaperture TSYBL:DPA, and the points at which some ions collide with the aper-ture structure (green). Trajectories are going right to left here. Simulationsproduced in SIMION [54]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.12 Photos of the split Einzel lens added to the TITAN beamline to facilitate high iontransport efficiency into the MR-ToF. Top: SEL prior to installation. Bottom:SEL installed in its location in the TITAN beamline. . . . . . . . . . . . . . . . . 503.13 An example of the portion of trace-space tested (test area) and acceptance sim-ulated in the investigation of the acceptance of the MR-ToF input optics. . . . . 523.14 SolidWorks model of the MR-ToF input optics indicating the electrode nomen-clature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.15 Samples of the acceptances simulated for the MR-ToF input optics across a rangeof chosen voltages. Numerical estimates of the acceptance are shown in the upperright corner of each plot. For this investigation the voltages on S-In-A1 and S-In-Lens were varied. Other voltage settings are indicated in the text. . . . . . . . 543.16 Simulated phase space of beam emittance (green) from the CB overlapped withthe device acceptance (blue) of the MR-ToF input optics. This was achievedwith EL5 set at -1650 V, SEL at -400 V, In-Lens at -1050 V and In-A1 at 800 V. 553.17 SolidWorks model of the MR-ToF output optics indicating electrode nomenclature. 56xiList of Figures3.18 Simulations of, Left: expected time-of-flight peak (blue) and, Right: energyspread (green) from the MR-ToF output optics detected at the position of MCP-0 for a beam energy of 1.3 keV. Gaussian fits of the data are shown with keyresults on each plot. Results with Out-Lens at -1900V, B4-OUT at 1225 V,Out-A1 at 1100 V and Out-A2 at 1000 V. . . . . . . . . . . . . . . . . . . . . . . 573.19 Simulated longitudinal emittance of beam ejected from the MR-ToF and detectedat MCP-0 for a beam energy of 1.3 keV. This for the ion optical settings: Out-Lens at -1900V, B4-OUT at 1225 V, Out-A1 at 1100 V and Out-A2 at 1000V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.20 A comparison of transverse emittance simulated after the 90◦ bend in the TI-TAN beamline and before TSYBL:DPA without and with the new optics for iontransmission into the MR-ToF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.1 A schematic overview of the TITAN experiment a) before the MR-ToF and b)with the MR-ToF installed. Arrowed lines are used to indicate paths the ionbeam may take to TITAN from the ISAC ion sources, inside TITAN, or fromTITAN to the ISAC collinear laser spectroscopy experiment [35]. . . . . . . . . . 624.2 Schematic of the MR-ToF and its interface with the TITAN beamline. Ion opticalelectrodes are indicated in yellow; the MR-ToF vacuum chamber is highlighted ingreen; detectors are shown in blue; red arrowed lines are used to indicate possiblebeam paths through the TITAN beamline and MR-ToF. Adapted from [47]. . . . 634.3 Photo of the MR-ToF installed and connected to the TITAN beamline. . . . . . 644.4 Time-of-flight spectrum of potassium isotopes used to calibrate the MR-ToFmass measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.5 Three MR-ToF time-of-flight spectra used for mass measurements. Detected ionsare marked in the plots; these ions were identified based on their mass. Time-of-flight mass measurements for each spectrum were calibrated use the 40Ar+peak. For the bottom spectrum 40K+ was added to the beam using the MR-ToFinternal ion source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.6 An example of mass-selective re-trapping performed in the TITAN MR-ToF using40Ar+ and 40K+ ions. a) Number of ions detected for 40Ar+ (blue) and 40K+(red) time-of-flight peaks for different re-trapping times. b) Time-of-flight spectrashowing no re-trapping (black), then with different re-trapping times to captureeither 40Ar+ (blue) or 40K+ (red) while removing the other isobar. . . . . . . . . 684.7 An example of ion motion in relation to axial potential during the process ofisobar separation through re-trapping. Left: Retarding field slowing ions priorto trapping. Right: Shallow potential well applied to trap ion of interest. Ionsare represented in red with the ion of interest and contaminant are representedas m1 and m2 respectively. Axial potentials are represented as blue lines. . . . . 69xiiList of FiguresC.1 Emittance plots from one measurement of a 133Cs+ beam out of the TITAN CBwith standard operating parameters (described in text), showing different levelsof processing of the number of ions detected at each position and divergence. a)An emittance plot with no processing of the ion counts, showing a large biasingeffect due to drift in the MCS. b) An emittance plot with ion count normalizationand time profile background removed, but before a threshold was set to removebackground counts for rms emittance calculation. c) The same emittance plot asb), but scaled in the z-axis to highlight the degree of variation of the backgroundcounts. Background variations on the level of 2% of the peak ion counts can beseen. d) An emittance plot after the complete ion-count processing described inthe text. A white ellipse calculated from the Twiss parameters is overlaid onthe plot with the size set by the equivalent emittance 4rms. Calculated rmsemittance and Twiss parameters printed below. . . . . . . . . . . . . . . . . . . . 83C.2 A plot of the calculated rms emittance as a function of the threshold for settingbackground noise to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84xiiiAcknowledgementsMy thanks to Pascal Reiter, Jens Dilling, Ania Kwiatkowski, Kyle Leach, Carla Babcock,Devin Short and all the other people in TITAN, TRIUMF, and Giessen who I forgot to mentionhere.xivFor my daughter Sonya whose birth gave me a rather compelling motivation to finish thisthesis and get on with things.xvChapter 1IntroductionIon traps at rare-isotope-beam (RIB) facilities are increasingly popular for beam preparationand measurements [1]. Ion traps are either directly used for or assist in making measurementsof atomic masses, decay properties, and nuclear moments [2]. Such information is essential inprobing nuclear structure of exotic species and understanding the nuclear decays and reactionswhich fuel stars. By utilizing the comprehensive theory of electricity and magnetism it ispossible to carefully study the interactions of RIBs with electromagnetic fields to extract moredetails of the properties of individual isotopes.There are three broad categories of ion traps used in experiments at modern RIB facilities:Penning traps use a combination of electric and magnetic fields to trap ions and performprecision measurements or for beam preparation [3][4][5]. Paul or radio frequency (RF) trapsuse dynamic fields to trap ions and are primarily used at RIB facilities for beam preparation[6][7][8]. Multi-Reflection Time-of-Flight (MR-ToF) mass spectrometers and isobar separatorsuse static electric fields to distinguish ion masses based on their time of flight [9][10]. The lastof these is the focus of this thesis.1.1 Penning trapsIn RIB facilities Penning traps are primarily used for mass measurements to a relativeprecision of δmm ∼ 10−9 [11]. The requirements for ion confinement in a Penning trap are anelectric field providing axial confinement and a magnetic field confining ions radially. Typicallythe electric field is produced by a set of three electrodes in a radially symmetric quadrupolararrangement where two end caps are biased higher than a middle ring electrode (Figure 1.1a). Alternatively, a series of cylindrical electrodes may be used to approximate the necessaryquadrupolar electric field [12]. To prevent ions from escaping in the radial direction an axialmagnetic field is applied to curve ion paths away from the trap edges. Together the electricand magnetic field can ideally confine ions indefinitely in static fields [11].1.1.1 Ion motionTo describe ion motion within a Penning trap we start with the cyclotron motion due tothe magnetic field. This motion occurs in a magnetic field with a frequency νc (or expressed asan angular frequency ωc = 2piνc). The cyclotron frequency is determined as,ωc =QemB, (1.1)in which Q is the charge state of the ion, e is the elementary charge, B is the magnetic fieldstrength, and m is the mass of the charged particle.The cyclotron motion of ions in a Penning trap is modified by the presence of the electricfield. The interaction of ions in the presence of magnetic and electric fields within a Penningtrap may be described analytically if the electric potential takes the form of a quadratic saddle11.1. Penning trapsV LowDCVHighDCB-Field(a) Electrode structure.Magnetron (-) Cyclotron (+)Axial (z)Combined(b) Ion motion.Figure 1.1: An overview of a Penning trap structure and the motion of a trapped ion. a) Adiagram of a Penning trap showing a typical electrode structure with the applied voltages andmagnetic field. b) Calculated motion of an ion in a Penning trap highlighting the componentsof the ion motion: cyclotron ω+ (yellow), magnetron ω− (blue), axial ωz (green), and combined(black).potential [11]. Calculated with the origin at the centre of the trap volume this is proportionalto V (r, z) = 2z2 − r2. Such an electric potential will cause the cyclotron motion of the ion inthe trap to split into two radial eigenmotions: small reduced cyclotron rotations of frequencyω+ and a larger magnetron rotation of frequency ω− (Figure 1.1 b). In addition to the radialmotion, ions will have an axial oscillation frequency of ωz. The frequencies of these motionswill follow the invariance relations [13],ω+ + ω− = ωc, (1.2a)ω+ω− =ω2z2, (1.2b)ω2+ + ω2− + ω2z = ω2c . (1.2c)1.1.2 Mass measurementHigh-precision mass measurements are achieved with Penning traps by determining ωc. Theprimary means of measuring the cyclotron frequency at RIB facilities is the time-of-flight ion-cyclotron-resonance (TOF-ICR) technique [3][14]. The precision of such measurements can beshown to be inversely proportional to the charge state of the ion [15]. Therefore, one means toimprove the precision is charge breeding, which refers to the rapid removal of electrons fromthe atomic shell and generating of higher charge states.Beam purity is an important consideration in Penning trap mass measurements. Contam-inant ions simultaneously trapped with the desired species negatively affect the accuracy andprecision of its mass determination. This impact on the measurement quality is due to Coulombinteractions between the ions in the trap. For ions of the same mass there is an equal drivingfield acting on the centre of the cloud, resulting in no frequency shift; however, unequal ionmasses will cause frequency shifts [16]. The removal of contaminants is a key motivation to thework of this thesis.21.2. Paul traps1.1.3 Preparation trapsPenning traps have also found applications in improving beam properties prior to injectioninto a mass measurement Penning trap. For Penning trap mass spectrometry, a number of beampurification techniques have been developed and are discussed in [17] or [18]. These techniquescan be broadband with low resolving power or for a specific nuclear state with resolving powersR ∼ 107 [4]. These traps and techniques are typically used to prepare the beam for TOF-ICRmass determinations.Penning traps are not however the only ion trap solution available in mass spectrometryand beam preparation. Particularly for beam preparation, Paul traps are an essential tool forion trapping in RIB facilities.1.2 Paul trapsIn contrast to the use of static electric and magnetic fields to trap ions in a Penning trap,purely electric fields may be used to trap ions if they are dynamically changing. An ion traprelying on dynamic electric fields is sometimes referred to as a Paul trap after Wolfgang Paul whoinvented the first such device [19][20]. Most Paul traps use quadrupole electric fields oscillatingat radio frequencies to provide ion confinement. An important property of Paul traps is thatonly a limited range of ion mass-to-charge ratios can be stably trapped, the precise range beingdetermined by the trap geometry, electrode voltages, and RF frequency. By adjusting thetrapping voltages it is possible for a Paul trap to act as a mass filter or mass spectrometer [21].It is often useful to introduce a neutral buffer gas to a Paul trap to keep ions close to thecentre of the trap, reduce the spread of momentum and position (or phase space distribution) ofions ejected from the trap, and thereby improve transport efficiency for ions which are injectedand eventually ejected from the trap [21]. These effects are accomplished through collisionsbetween the ions and the buffer gas which can transfer some of the energy from the ions to thegas. Through these collisions the ions tend towards thermal equilibrium with the gas. If thegas is in an open system, maintaining a lower temperature than the ions injected into the trap,the ions become cooled, reducing the phase space of an ion bunch.Here we shall discuss two examples of Paul trap electrode geometries: 3D, and linear (Figure1.2). A 3D Paul trap uses the same electrode geometry as a Penning trap, but achieves 3Dconfinement with an oscillating potential between the ring and endcaps instead of static electricand magnetic fields. However, at RIB facilities the preferred Paul trap design is the linear Paultrap.A linear Paul trap is composed of four elongated parallel and segmented electrodes, asshown in Figure 1.2 b. The outermost electrodes are biased to axially confine the ions whilethe quadrupolar RF field traps them radially. With this trap design it is possible to traplarger quantities of ions than in a 3D Paul trap, which can be useful for the accumulation of acontinuous charged particle beam into bunches [21]. If we introduce a buffer gas to the trap wenow have the basis for a cooler buncher which is an important use of Paul traps at RIB facilities.A cooled and bunched beam improves, for example, the measurement quality in Penning trapmass spectrometry.31.3. Multi-Reflection Time-of-Flight mass spectrometer and isobar separatorV cos(ωt)−V cos(ωt)zr(a) 3D Paul trap.VDC4VDC3VDC2VDC1−V cos(ωt)V cos(ωt)xzy(b) Linear Paul trap.Figure 1.2: Diagrams of two Paul trap designs showing the electrode structure and an exampleof typical driving voltages.1.3 Multi-Reflection Time-of-Flight mass spectrometer andisobar separatorAnother type of ion trap which has quickly been proving its utility for RIB facilities is theMR-ToF mass spectrometer and isobar separator [22][23][24][25][26][27][28]. The core principleof a ToF mass spectrometer is that ions of different masses, moving in the same direction withthe same kinetic energy may be distinguished by the time of flight over which they traverse agiven distance (Figure 1.3). From the classical kinetic energy equation we may derive a massresolving power R measuring mass m with uncertainty ∆m based on time of flight t with anuncertainty ∆t to be [29],R =m∆m=t2∆t. (1.3)However, it may be necessary for ions to traverse hundreds of meters to resolve similarmasses, requiring an impractically large amount of space. Instead, electrostatic mirrors areused to confine the path of the ions to a small space. By this method, ions of similar mass(such as isobars—nuclides of the same atomic mass number A = # protons + # neutrons)may be rapidly separated based on their time of flight. If ions ejected from the MR-ToFafter a desired time then hit a time-sensitive detector it is possible to measure ion masses, amethod which has been shown to be able to achieve mass resolving powers on the order of 105[10]. Alternatively, combining the MR-ToF with a fast deflector allows unwanted species to bedeflected out of the beam once spatially separated after a sufficient time of flight. This way theMR-ToF can act as a mass filter capable even of removing isobaric contamination.1.4 Beam purityMany nuclear-physics experiments require high beam purity although most beam productiontechniques produce a multitude of species rather than the desired species alone. Decay productsfrom contaminant species can hide the signal from the desired species or cause undesirable space-charge effects leading to efficiency losses. In the context of Penning trap mass spectrometry,interactions with contaminant ions can shift the measured cyclotron frequency compared to thetrue cyclotron frequency of the ion of interest [16].The most common class of beam contaminant to confound measurements at RIB facilitiesis isobars. Isobars are nuclides with the same total number of protons Z and neutrons N ,expressed as the atomic mass number A = Z + N . Because protons and neutrons are very41.4. Beam purityPositionlong flight distanceDetectorm1m2m3K1 = K2 = K3m1 < m2 < m3PositionDetectorElectrostatic mirrorsMultiple Reflections,small spacem1m2m3K1 = K2 = K3Time of Flight (t)∆tDetectedIonsTimeFigure 1.3: Outline of the principle of time-of-flight mass spectrometry showing three ions ofdifferent masses becoming separated over a given path length. Top: Ions travel over a long,linear distance. Middle: Ions are reflected between a pair of electrostatic mirrors, leading tothe same effective path length as above in a smaller physical space. Bottom: Time-of-flightspectrum at start and finish of these mass separations.51.5. TITAN Experimentsimilar in mass, isobars will have very similar masses and become difficult to remove froma RIB, thus contaminating the beam being measured. Non-isobaric contaminants are easilyremoved, only requiring resolving powers ∼ 100 which is easily provided by magnetic massseparators.Dipole magnetic mass separators are one solution to beam purification essential in anyRIB facility. These allow near instantaneous filtering of practically unlimited beam currents,removing the vast majority of contaminants produced during the initial RIB production process.In principle this method can achieve resolving powers more than sufficient to remove isobars;however, the necessary magnets are both large and expensive. For example, the high-resolutionmagnetic mass separator planned for the new ARIEL facility at TRIUMF is expected to beable to achieve resolving powers of R ≈ 20 000 [30]. This resolving power is still insufficientto separate many isobars, such as 40Ar+ and 40K+ which have a relative mass difference ofapproximately 25 000.Processes of beam purification within a Penning trap, such as dipole cleaning [17][18], havea different set of strengths and limitations. An attractive aspect of dipole cleaning is that itprovides a high resolving power, up to ∼ 107 [4], sufficient even for the removal of nuclearisomers. However, dipole cleaning requires that ions be individually identified and removed,beam with a small space charge, and at least ≈ 10 ms.The performance of a typical MR-ToF is situated between these two types of purificationtechniques. MR-ToFs have achieved R ≈ 105 in 5 ms [24] and are well suited for low-energy RIBproduction and experiments, such as TITAN [31] at ISAC-TRIUMF [32]. The implementationof an MR-ToF for TITAN designed and assembled by collaborators at the University of Giessenis the subject of this thesis.1.5 TITAN ExperimentTRIUMF’s Ion Trap for Atomic and Nuclear science (TITAN) [31][33] (Figure 1.4) is locatedat the Isotope Separator and ACcelerator (ISAC) [32] experimental facility at TRIUMF. Itsprimary function is the study of rare-isotopes using a number of ion traps. The cornerstoneof the TITAN experimental work has been Penning trap mass spectrometry [33]. TITAN hasalso demonstrated the ability to perform in-trap decay spectroscopy [34]. In the following, theISAC facility and the individual components of TITAN will be introduced.1.5.1 ISAC FacilityThe ISAC facility at TRIUMF produces and delivers rare isotopes to experiments whichfocus on studying fundamental physics in areas such as nuclear structure and nuclear astro-physics. ISAC employs the production of RIBs through the Isotope Separation On-Line (ISOL)method. Rare isotopes are produced by impinging a 480 MeV proton beam accelerated in TRI-UMF’s main cyclotron [36] onto a target [37], causing a vast range of isotopes to be producedthrough nuclear fission, spallation, and fragmentation reactions. The resulting isotopes areionized with an ion source so that they may be transported with electric and magnetic fields[38]. The ions are accelerated to 20–60 keV and passed through a magnetic mass separatorwith R ≈ 3000. The isotopes of the selected mass may then be sent to various low-energyexperiments—including TITAN—or may undergo acceleration to higher energies through a setof linear accelerators to meet the needs of medium- (0.15–1.8 MeV/u) to high-energy (≥ 6MeV/u) experiments.61.5. TITAN Experimentfrom ISACion sourceto collinearlaser spectroscopyCoolerBuncherEBITCPET MPET(a) Before the MR-ToFfrom ISACion sourceto collinearlaser spectroscopyCoolerBuncherMR-ToF EBITCPET MPETContinuous ISAC beamSingly charged, bunchedHighly chargedIsobar separated(b) With the MR-ToFFigure 1.4: A schematic overview of the TITAN experiment a) before the MR-ToF and b) withthe MR-ToF installed. Arrowed lines are used to indicate paths the ion beam may take toTITAN from the ISAC ion sources, inside TITAN, or from TITAN to the ISAC collinear laserspectroscopy experiment [35].1.5.2 Cooler BuncherThe first component of the TITAN system is a radio frequency quadrupole (RFQ) coolerbuncher (CB) [39]. The CB is a linear Paul trap which converts the continuous ISAC beam tobunched beam of lower emittance. It ejects the ion bunches into the TITAN beamline, wherethe beam energy is lowered to around 2.4 keV using a pulsed drift tube. The low-energy, low-emittance bunched beam is transported to the various other ion traps in the TITAN system forbeam preparation and measurements.1.5.3 EBITTITAN uses an electron beam ion trap (EBIT) [40] charge breeder to increase the chargestate of ions undergoing Penning trap mass measurement (Figure 1.5 a). An EBIT superimposesan electron beam axially upon a Penning trap providing radial confinement. Successive electronimpacts remove electrons, increasing the ion charge states. A recent high-voltage upgrade [41]to the TITAN EBIT brings the electron beam energy up to 62 kV which is expected to be able tostrip ions bare for elements up to tellurium (Z=65); as well, the electron gun has been designedto be able to reach electron beam currents up to 5 A. This EBIT is also equipped with SiLigamma-ray detectors (Figure 1.5 b) for in-trap decay spectroscopy [34]. The decay spectroscopybenefits from the effect of positrons being guided away from the decaying ion bunch therebysuppressing background noise in gamma-ray spectra from electron-positron annihilation. Anadditional benefit of decay spectroscopy in the EBIT is the ability to investigate decay propertiesof highly charged ions [42].1.5.4 Measurement Penning trapHigh-precision mass measurements of rare isotopes are performed in the TITAN Measure-ment PEnning Trap (MPET). MPET uses a 3.7 T magnetic field to provide radial confinement71.5. TITAN Experiment(a) EBIT.(b) SiLi gamma-ray detectors.Figure 1.5: Diagrams of TITAN EBIT showing: a) Components of the trap along the beamaxis, with the path of ions indicated by a black double arrow. b) The arrangement of SiLigamma-ray detectors around the trapping region. Figures found in [43].81.6. Outline of this thesisof ions within the trap and then measures ion masses through the TOF-ICR method. Precisionson the order of δm/m = 10−9 have been achieved in MPET [44]. MPET has also been able toachieve very rapid mass measurements of short-lived nuclides having successfully performed amass measurement of 11Li, with a half-life of 8.8 ms [45].1.5.5 TITAN MR-ToFThe most recent addition to the TITAN system has been an MR-ToF mass spectrometer andisobar separator (Figure 1.6). As outlined in Section 1.3, an MR-ToF provides an intermediatespeed and resolving power for the removal of contaminants from a beam while also providinga broad-band mass spectrometer. The TITAN MR-ToF was designed to be able to performmass measurement with a resolving power of 100 000 and also to be able to remove isobariccontaminants with a resolving power of 20 000 within 10 ms, using the mass-selective re-trappingmethod [46].1.6 Outline of this thesisThe focus of this thesis work has been the integration of the TITAN MR-ToF into theTITAN beamline, showing that it performs mass measurement and isobar separations as de-signed and that it can be bypassed without impacting performance of the established TITANsystem. Previous work with the TITAN MR-ToF includes its construction which is discussedin Reference [47] and offline commissioning, outlined in Reference [48].In this thesis we shall first discuss some of the core principles of ion optics and the ionoptical devices used in the transport and confinement of charged-particle beams.These ion optical principles will then be applied to an investigation of the TITAN beamlineand the performance of the beam within. To maximize beam transport efficiency into the MR-ToF, new optics were added into the existing TITAN beamline. With the introduction of thenew ion optics, simulations were performed to examine how the optics transport ions into theMR-ToF or bypass it. Simulations were performed of the TITAN beamline from the TITAN CBto MPET, from the CB to the MR-ToF, and from the MR-ToF to the CB to provide insightsinto the TITAN beam properties where diagnostic components are not available and to predictoperating parameters.Once the MR-ToF was installed in the TITAN beamline, a commissioning experiment wasperformed showing the capability of the MR-ToF to perform mass measurements and isobarseparation as part of the TITAN facility. Studies of ion transport efficiency into the MR-ToFwere also performed.We end by discussing an outlook for how the TITAN MR-ToF may be used to expand thescientific program being pursued with TITAN.91.6. Outline of this thesisMCP DetectorTime-of-Flight AnalyzerMass Range SelectorTrap SystemTransfer RFQChanneltron DetectorThermal Ion SourceRFQ SwitchyardInput RFQInjection OpticsGate ValveSplit Einzel LensDifferential Pumping SectionEinzel LensPulsed Drift TubeCooler Buncherlocation ofSEL MCPElectrostaticBeamlineGate ValveFaraday CupEjection OpticsOutputRFQDifferential PumpingSectionsFigure 1.6: Schematic of the MR-ToF and its interface with the TITAN beamline. Ion opticalelectrodes are indicated in yellow; the MR-ToF vacuum chamber is highlighted in green; detec-tors are shown in blue; red arrowed lines are used to indicate possible beam paths through theTITAN beamline and MR-ToF. Adapted from [47].10Chapter 2Principles of ion transport neededfor MR-ToF integrationThe TITAN experiment seeks to provide valuable insights into nuclear astrophysical pro-cesses and nuclear structure by means of precise mass measurements of rare isotopes usingPenning trap mass spectrometry. The presence of contaminant ions in a Penning trap massmeasurement can cause shifts in the measured cyclotron frequency of the ion of interest, im-peding both the accuracy and precision of the measurement. This reduced quality of massmeasurement motivates the need for a pure ion beam entering the trap. The most commoncontaminants entering the trap are isobars. In order to remove these isobaric contaminantsfrom the beam quickly (< 10 ms) and without prior knowledge of the specific contaminantsa Multi-Reflection Time-of-Flight (MR-ToF) mass spectrometer and isobar separator was de-signed, built, commissioned, and installed in the TITAN beamline.An essential part of integrating the MR-ToF into TITAN was ensuring fast and efficienttransport of beam into the MR-ToF. The purified beam needs be transferred to the Measure-ment PEnning Trap (MPET). To accomplish this beam transport it was important to have anunderstanding of the principles of charged particle transportation.2.1 Understanding beam dynamics using emittanceA key tool used in the study of charged particle optics is the concept of emittance. Emittanceprovides a means of quantifying beam quality which—for example—sets constraints on theability to deliver the beam within given parameters. The definition of emittance is directlyrelated to the 6-dimensional phase space volume occupied by a set of particles.According to Liouville’s theorem this phase space volume is invariant under the conservativeforces which can be defined by a Hamiltonian [49]. Because of this invariance, by measuring thephase space volume in one location, it is known in all other locations the volume may occupy atdifferent times. To introduce the concept of emittance we look at the 2-dimensional phase spaceplanes of the total phase space volume. One definition of emittance () is the phase space areaoccupied by the particle distribution, expressed in the x-axis case for positions x and momentapx as,x =xdxdpx. (2.1)This quantity is invariant for constant beam energy, similar to the invariance of the total phasespace volume. The beam energy dependence however behaves in a predictable fashion and inthe non-relativistic approximation the emittance at a particular energy E may be compared tosome reference emittance ref at energy Eref by the relation, =√ErefEref . (2.2)112.1. Understanding beam dynamics using emittancexpx(a) Elliptical emittancexpx(b) Filamented emittanceFigure 2.1: Example of a) an elliptical emittance and b) a filamented emittance.An alternative definition of emittance is known as the root mean square (rms) emittance,rms =√〈xx〉 〈pxpx〉 − 〈xpx〉2. (2.3)Here we use angle brackets to denote mean values of contained variables. This is a statisticaldescription which is very useful for describing beams which lack clearly defined boundaries,such as Gaussian or similar distributions typical of experimental measurements. rms is not aninvariant quantity, it will grow if the area of the emittance becomes distorted. It is commonfor the emittance to be elliptically distributed (Figure 2.1 a), but the beam evolution mayundergo non-uniform shifts in the emittance distribution. Such a beam evolution can leadto arm-like patterns in the emittance distribution referred to as filaments. An example of afilamented emmitance is shown in Figure 2.1 b where the filaments diverge from an otherwiseelliptical distribution. The examples in Figure 2.1 both fill the same phase space area; howeverin the filamented case the rms emittance will be larger. Emittance filamentation also createschallenges for beam matching which may require complex ion optical lens arrangements orlarger acceptance (as indicated with the dashed line). Another difference with rms is that ittypically gives a smaller value than the emittance defined as the phase space area. In the caseof a uniform elliptical distribution, rms =4pi [50].The rms emittance provides a basis for further description of the emittance distributionthrough what are known as Twiss parameters. These parameters are used to describe anemittance ellipse and may be used to calculate how it evolves through various ion optics. TheTwiss parameters are defined as,β =〈xx〉rms(2.4a)γ =〈pxpx〉rms(2.4b)α = −〈xpx〉rms. (2.4c)122.1. Understanding beam dynamics using emittance−→pBeam DirectionBeam Directionxx′Converging∆x′∆x∆x∆x′ xx′Beam Waistxx′DivergingFigure 2.2: Top: Visualisation of transverse emittance at different positions of a beam, forwhere the beam is converging, at a focus point (beam waist), and when it is diverging. Bottom:Example plots of the transverse emittance for the positions indicated by the blue dashed lineover each plot. The left plot includes a direct illustration of the relation between the spatialdistribution ∆x and beam divergence ∆x′ to the trace space plot. As shown, the orientationof the ellipse changes and in real scenarios it may change shape, but the total area is invariantfor constant beam energy in a vacuum.From this we may calculate an ellipse as,γx2 + 2αxpx + βp2x = rms. (2.5)2.1.1 Transverse emittanceThe convention used in this thesis is to define the z-direction to be the average directionof beam propagation and to refer to the associated emittance as the longitudinal emittance;emittance in the transverse directions x and y are then known as the transverse emittances.Experimentally it is more practical for measuring and computing transverse emittance to de-scribe the transverse momentum px with its derivative in the longitudinal direction x′ = dpxd|−→p | .If pz  px we may use the small angle approximation to relate x′ to the ion trajectory’s angleaway from the beam axis—its divergence—by x′ = tan(θ) ≈ θ, measured in radians. This x–x′description of the phase space is often referred to as the trace space [50]. An example of thespatial evolution and the corresponding trace space evolution of a beam can be seen in Figure2.2.For the scales typical of ion optics it is convenient to express measurements of x in mm andx′ in mrad. Using these units, emittance may be expressed in units of mm mrad; since radiansare not formally an SI unit this may be expressed more concisely as µm. Some communitiesprefer to divide the trace-space area by pi and express the emittance in units of pi mm mrad. Inthe case of an elliptical trace space distribution removing the factor of pi makes the emittancesimply the product of the semimajor and semiminor axes, it also creates a more natural link torms. For this thesis the division by pi is eschewed in favour of expressing transverse emittancesin units of µm.132.1. Understanding beam dynamics using emittance−→vtEtETime FocustEFigure 2.3: An example of a time focus for a distribution of particles. Top: Illustrated hereare a set of particles moving with different velocities represented by right pointing arrows.Represented are three moments in time as the faster particles start behind the slower particlesthen move to overtake the slower particles. The moment where the faster particles are passingthe slower is known as the time focus. Bottom: For each moment in time, an example of acorresponding longitudinal emittance plot is shown.2.1.2 Longitudinal emittanceTo discuss longitudinal emittance we must first note that this concept is only useful in thecase of bunched beam; such beams consist of packets of ions with a finite longitudinal positionand momentum spread, able to be described with emittance. Similar to the approximationof the transverse momentum px by the divergence x′, if the longitudinal momentum spread issmall compared to the beam central momentum we may represent the longitudinal emittanceby its spread in energy E and time t. As with x′, expressing longitudinal emittance in terms ofenergy and time has the added benefit of being more practical to measure experimentally.An application of longitudinal emittance is the concept of a time focus. The time focus isthe longitudinal analogy to the beam waist seen in Figure 2.2. Let us imagine a set of ionsare distributed longitudinally as in the left-hand side of Figure 2.3. In the figure we start withthe higher energy ions located behind lower energy ions in the direction of propagation. Asthe faster ions overtake the slower ions there is a moment where the time distribution is at itsnarrowest, this moment is the time focus. In time-of-flight mass spectrometry the time focusis extremely important as performing measurements away from this focus will unnecessarilyincrease the time width and reduce resolving power.2.1.3 AcceptanceHaving quantified the quality of a beam using emittance, a key application is to match thebeam emittance to the so-called acceptance (α) of a given device. Acceptance corresponds tothe maximum emittance which can be tranported into the device without losses for which it isdefined (α = maximum). In matching emittance to acceptance it is important that maximum ≤α and that the phase space volume described by the emittance be contained within that of theacceptance.If the phase space orientation of the beam emittance does not match that of the device142.2. Ion opticsxpxAcceptanceMatched emittanceUnmatched emittanceFigure 2.4: An example phase space plot where device acceptance is greater than beam emit-tance but their orientations cause them not to match. The unmatched portion of the beamwill be lost, resulting in a lower beam transport efficiency. Such a mismatch may be eliminatedthrough the use of ion optical lenses to re-orient the beam emittance into alignment with theacceptance.acceptance as is, there exist a number of ion optical elements which may be used to match thetwo. If  ≤ α but the beam emittance shape and/or orientation do not match as in Figure 2.4,then lenses such as electrostatic lenses may be used to match the beam (Section 2.2.1). In caseswhere  > α the emittance may be reduced without increasing the beam energy by using buffergas cooling in a Paul trap (Section 2.3.2).2.2 Ion opticsIon optics are an essential part of any rare-isotope-beam (RIB) facility as they allow theRIB to be transported quickly and efficiently, while having appropriate emittance for matchingthe beam to the acceptance of various experiments. Ion optics also play a crucial role in thedesign and operation of ion traps. This control of RIBs is accomplished through the carefulshaping of electric and magnetic fields.An important consideration in the design of ion optical systems is the creation of aberrationsin the beam properties. These aberrations refer to deviations in ion motion from an ideal ionpath with no beam width. We describe these ion motions by their position and momentum,and deviations in flight time, kinetic energy, and mass-to-charge ratio. When applying Taylorseries expansions to these descriptions the first order terms are known as the paraxial (linear)coefficients which form the basis of a linear approximation. Higher order terms are referredto as aberration coefficients and describe the non-linear evolution of a beam passing throughgiven optics. For this thesis we will limit ourselves to referring to the order on which beamaberrations occur for a sense of scale, however a deeper discussion of this formalism and otherion optical topics may be found in References [50], [51] and [52].The focus of this thesis has been on ion transport within a low energy beam transport(LEBT) beamline (E ≤ 60 keV) and ion confinement within a Paul trap or MR-ToF. Thisrequires an understanding of electrostatic and radio frequency (RF) ion optics.152.2. Ion optics2.2.1 ElectrostaticElectric fields are an attractive option for ion optics due to their high field energy den-sity which allows them to be produced with compact devices. Static electric fields are limitedhowever by discharges which begin happening for fields around 10–20 kV/mm in high vacuum(P < 10−3 mbar). The precise field strength at which breakdown occurs is substantially influ-enced by the material and structure of the surfaces between which the discharges would occur[53]. This limitation makes purely electrostatic ion optics for beam transport only appropriatefor LEBT; magnetic optics become essential for higher energy beams. When electrostatic opticsare a viable option, their design benefits from their independence from mass-to-charge ratios,allowing the same optics and settings to be used for a wide range of ion masses.The types of electrostatic ion optics most important for this thesis are Einzel lenses andquadrupole lenses for focusing beams and matching emittance, and parallel plate benders toredirect ion paths. Another type of electrostatic ion optic—the electrostatic mirror—will bediscussed in the context of MR-ToF optics.Einzel lensAn Einzel lens (from the German Einzellinse for single lens) consists of three hollow cylin-drical electrodes with the outer two electrodes typically at an electric potential of zero and thecentre electrode having some non-zero potential (Vlens) (Figure 2.5 a). So long as Vlens is nothigh enough to prevent passage of ions, the resulting electric field will have an overall focusingeffect on the beam. The radial component of the electric field becomes stronger further fromthe beam axis causing ions at various radial positions to converge to a focus point, the posi-tion of which is determined by Vlens. There are two basic modes in which an Einzel lens maybe operated: decelerating, where Vlens > 0 (Figure 2.5 b), and accelerating, where Vlens < 0(Figure 2.5 c)). Due to restrictions incurred by Laplace’s equation, the electric fields cannot beshaped in an arbitrary fashion, this leads to aberrations in the beam properties. Third orderangular aberrations (specifically referred to as spherical aberrations); as well as second orderaberrations in the correlation between ion flight time and position, and ion divergence andenergy. These effects will normally be larger with decelerating lenses as the ions will tend todiverge more from the beam axis before being focused [52].For TITAN, Einzel lenses are spread along the beamline to maintain beam confinementthrough long straight sections. They are also used to focus the beam on narrow differentialpumping apertures which separate beamline sections with different gas pressures.Parallel plate deflectorA simple but important type of ion optic is the parallel plate deflector; the plates of thedeflector are given different potentials which ideally produces a constant electric field betweenthe two. The most basic parallel plate deflector consists of two straight rectangular plates(Figure 2.6 a). Such an arrangement is commonly used for applying small corrections (≤ 9◦) tobeam trajectories to ensure a straight beam path. Rectangular parallel plates have the capacityto apply much larger angular shifts in the beam path, but this does not maintain the beamfocus which would be desirable where a well defined beam emittance is important.For a beam being bent in the y-direction we can maintain a well defined beam emittancefirst by curving the bender electrodes in the y-direction as in Figure 2.6 b. Such a bender designis known as a cylindrical bender. The variation in the radial electric field strength suggestedby the field lines in Figure 2.6 b has a focusing effect on the beam in the y-direction. The162.2. Ion optics(a) Isometric image of Einzel lens with focused beam.V = 0VLens > 0V = 0(b) Decelerating Einzel lens.V = 0VLens < 0V = 0(c) Acclerating Einzel lens.Figure 2.5: Simulations of ion trajectories (red lines) and ion optical elements (brown) for anEinzel lens showing a) an isometric view of the lens and the ion trajectories. b) and c) showthe Einzel lens operating with a decelerating and accelerating potential respectively, along withcontour lines of the electric potential (black curves). The example here shows a parallel beambeing focused to a point, the position of which is determined by the potential VLens. Iontrajectories portrayed here are from ions flying from left to right. Simulations produced inSIMION [54].172.2. Ion opticsassociated radial dependence of the electric potential is described as,V (r) =1ln(r2/r1)[(V2 − V1) ln r (V1 ln r2 − V2 ln r1)] , (2.6)in which r1 and r2 are the radii of the inner and outer plates with V1 and V2 being the corre-sponding potentials. The effect of this radial dependence of the potential is that faster movingions will travel through the bender at a larger radius and experience a slight retarding fieldrelative to a beam axis at r0 = (r2− r1)/2. Slower moving ions will have a smaller bend radius,causing them to experience an accelerating potential towards r0. Together these have the effectof focusing the beam towards the beam axis along r0. However, in the x-direction there is nofocusing effect, so the beam behaves as though it were in a field-free region in the x-direction.To provide equal focusing in the x- and y-directions through a bend the cylindrical benderdesign may be modified to have an electrode curvature of equal radius in the x- and y-directions.Such a design is known as a spherical bender, a diagram of which is shown in Figure 2.6c. Spherical benders are used as a standard optic in the TRIUMF ISAC Low Energy BeamTransport (LEBT) area [55] and are also used in the TITAN beamline. Two designs of sphericalbender are used: a 45◦ bender which bends beam from one direction; and a 36◦ bender whichcan bend beam from two perpendicular directions and allow transport straight through from athird direction. The 36◦ bender is coupled with a rectangular parallel plate which bends thebeam by an additional 9◦, in total bending the beam by 45◦.These parallel plate designs all seek to produce electric fields perpendicular to the ion pathto redirect the ions without changing the average beam energy. However, there are cases whereit is useful to redirect an ion beam with electric fields near antiparallel to the beam path usingelectrostatic mirrors.Electrostatic Mirrors and MR-ToFsThe invention of the electrostatic mirror was a key advance in the development of time-of-flight mass spectrometry. Prior to their invention, time-of-flight mass spectrometry could onlyprovide resolving powers of several hundred [56]. In contrast, modern time-of-flight based massspectrometers are able to achieve resolving powers three orders of magnitude greater [10].An electrostatic mirror requires a region with a higher potential than the beam energy whichreflects the beam path. The electric field is primarily perpendicular to the initial motion of theions, causing them to slow and reverse direction at some turning point. The depth to whichions reach in the mirror field is energy dependent such that higher energy ions will penetratedeeper into the field and thus take longer to be reflected than lower energy ions. In the case ofbunched beams, the net effect is to shift the time focus of a reflected beam (Figure 2.7). Theelectric field strength may then be modified to change the location of the time focus in orderto set it on a time-sensitive detector or another desired location.Early versions of electrostatic mirrors used parallel plate grid electrodes to produce constantelectric fields for reflecting ions [56]. Such grids would cause ions to be lost through collisionswith the grid, reducing the efficiency of transmission through the mirror. Figure 2.8 shows anexample of an electrostatic mirror using a grid electrode and the resulting ion losses at thegrid. If ions are to undergo multiple reflections on an electrostatic mirror—as is the case inan MR-ToF—the ion losses to grid electrodes would become very large. Such ion losses wouldbe unacceptably large in the context of RIB experiments where efficiency is very important,thus a gridless mirror needed to be developed. The design of a gridless mirror has the beampath surrounded by one or more electrodes, such as a series of hollow cylindrical electrodes;182.2. Ion opticsV2V1E-fieldIonPathyz(a) Rectangular parallel plate bender.V2V1E-fieldIonPath r1r0r2y zyz(b) 90◦ cylindrical bender.V2V1E-fieldr1r0r2r2IonPathy zyz(c) 90◦ spherical bender cross-section.Figure 2.6: Examples of parallel plate bender designs. a) A simple rectangular plate design,used in TITAN for small beam path corrections, up to 9◦. b) A 90◦ cylindrical bender, and c) a90◦ spherical bender. The direction of the electric field between the bender plates is indicatedby the blue arrows.192.2. Ion opticsElectrostatic MirrorE-FieldInitialTime focusFinalTime focusFigure 2.7: An illustration of how the time focus is shifted by an electrostatic mirror. Dashedlines indicate the trajectories followed by individual ions. Ellipses are drawn around the ionsto indicate snapshots in time, showing the process of the time focus shift. The direction of theelectric field is indicated with the blue lines, though this field would be present through themirror.however, these introduce problematic field inhomogeneities [52]. These inhomogeneities werefirst overcome in a rotationally symmetric mirror by having an additional cylindrical electrodewith an accelerating potential before the reflecting portion of the mirror, effectively acting asan Einzel lens [57].The decelerating portion of a gridless electrostatic mirror will typically be made of multipleelectrodes in order to remove time-of-flight aberrations. These aberrations need to be reducedbecause they diminish the precision and resolving power of a time-of-flight mass measurement.Increasing the number of electrodes used increases the degrees of freedom for the removal ofaberrations, meaning that higher order aberrations can be removed by increasing the numberof electrodes. The TITAN MR-ToF uses three mirror electrodes on either end (Figure 2.9) toprovide third-order time-of-flight focusing. The gains in mass resolution from adding mirrorelectrodes are tempered by voltage instabilities in the power supplies (currently ∼ 10 mV)which set the potential on each electrode. Such instabilities are a major limiting factor in theresolution achievable in contemporary MR-ToFs [58].In the TITAN MR-ToF the mirrors are tuned to provide a time focus in the centre of theanalyzer during the multiple-reflection phase of its operation (Figure 2.10). Ions entering theanalyzer do not initially have this desired time focus, so they must undergo a time focus shift(TFS) in their first reflection. After the TFS the mirrors are set to maintain a constant timefocus until it is time for the ions to be ejected. Ejected ions undergo a second TFS to give atime focus on the detector. This focusing scheme allows the number of “turns” (ions reflectingoff both mirrors and returning to an arbitrary start point) to be changed without additionaltuning of electrode potentials. Further discussion of this concept may be found in Reference[59].One challenge which arises in MR-ToF mass spectrometry is that as ions undergo many turns202.2. Ion optics(a) 3D view. (b) 2D view.Figure 2.8: Simulation of an electrostatic mirror using a grid electrode. Ion trajectories (red)start on a) the left, b) the top. Ion collisions with the grid electrode (brown) or the boundaryof the simulation area are marked in green. Electric potential contour lines are shown in blackin b). Simulation produced in SIMION [54].TrapSteeringEinzelLensMirror electrodesAcceleratingPotentialMass Range SelectorAcceleratingPotentialMirror electrodesDetectorElectrostatic optics RF optics Detector Grounded elementsFigure 2.9: Schematic of the TITAN MR-ToF ion optics.212.2. Ion opticsTFS reflectionTFS reflectionInjection trapEntrance reflector Exit reflectorDetectorPrimarytime focus Intermediatetime focusFinaltime focusFigure 2.10: An illustration of the time focus shift (TFS) in the TITAN MR-ToF from injectioninto the MR-ToF to setting a final time focus on a detector.the lighter ions will experience more turns than heavier ions. This difference in turn numbers atbest causes the time-of-flight mass spectrum to be more difficult to read and analyze; at worstit causes interference between a signal of interest and that of a contaminant. The solution forthis problem used in the TITAN MR-ToF is a Mass Range Selector (MRS). The MRS is a setof electrodes in the centre of the MR-ToF analyzer (Figure 2.9) which can be given a dipolepotential to deflect unwanted ions out of the beam. The MRS is pulsed on and off so that onlya limited mass range is measured at the detector [60].Quadrupole lensesAn additional type of ion optical lens uses a transverse quadrupolar field to focus the beam.This field is typically produced by a quadrupolar arrangement of cylindrical electrodes as inFigure 2.11. This radial potential variation contrasts with the mostly longitudinal potentialchange of an Einzel lens. For the Einzel lens most of the potential gradient is longitudinal,yet it is the small radial component which provides the focusing effect. The result is thata quadrupole lens may achieve similar focusing power with a much lower electric potentialon its electrodes. Quadrupole lenses also benefit from allowing separate control of the beamproperties in the two transverse directions. However quadrupole lenses require a higher degreeof complexity for their design and operation due to the larger number of electrodes and relatedpower supplies.First we must consider that a single electrostatic quadrupole lens can only focus in onetransverse direction and necessarily defocuses the beam in the other. For this thesis we willrefer to electrostatic quadrupole lenses focusing in the x-direction as focusing lenses (F) andthose focusing in the y-direction as de-focusing (D). In order for quadrupole lenses to focus inboth transverse directions, sets of individual lenses must be arranged into quadrupole multiplets.Two lenses are the minimum needed for focusing in both transverse directions [51], but largernumbers of quadrupoles can be used to provide greater control of the beam focus as well asfor reducing aberrations in the beam. A minimum number of four quadrupole lenses would beneeded to mimic the action of an Einzel lens to first order by arranging their focusing actionin an axial symmetric fashion, such as DFFD [52]. Such an arrangement is referred to as aquadrupole quadruplet, a simulated example of which may be seen in Figure 2.12.In the TITAN beamline DFFD quadrupole multiplets are used in 90◦ bend sections toprovide focusing between 45◦ bends from spherical and rectangular benders. This provides the222.2. Ion optics−VVyx(a) Cross section of physical rods.xyPotentialLowHigh(b) Transverse potentials.Figure 2.11: Cross-sectional diagram of a typical quadrupole lens with a potential focusing inthe x-direction. a) The arrangement of physical rods with the applied electric potentials. b) Asimulation of the resulting electric potential distribution produced in SIMION [54].(a) Isometric view of quadruplet.zx(b) x-z plane of quadruplet.zy(c) y-z plane of quadruplet.Figure 2.12: Different views of a simulation of ion trajectories (red) through a quadrupolequadruplet (brown). Trajectories are shown for ions flying left to right parallel to the z-axis.Black contour lines indicate the electric potential around the electrodes. Simulation producedin SIMION [54].232.3. Radio Frequency Quadrupole Opticsr0−V cos(ωt)V cos(ωt)−V cos(ωt)yxFigure 2.13: Schematic of radio frequency quadrupole (RFQ) rods and their applied bias. Alsoshown is the radius between rods r0.beam confinement in both transverse direction similar to an Einzel lens while providing theability to adjust x and y focusing independently.A limitation faced by all electrostatic optics is that at very low beam energies (10s of eV)ion trajectories become unstable. This instability is due to slow moving ions staying longerwithin the influence of a given ion optic and any small inhomogeneities in its field [52]. Toaddress this limitation we now introduce dynamic fields to our ion optics.2.3 Radio Frequency Quadrupole OpticsA popular and robust solution for maintaining confinement of low energy ions is to intro-duce a time-dependent oscillation to the polarities of a quadrupole lens (Figure 2.13). Suchoscillations are typically driven in the radio frequency range, creating a category of ion opticsknown as radio frequency quadrupole or RFQ. RFQ ion optics provide a basis for some massspectrometer designs, mass filters, ion guides, as well as cooler bunchers and other preparationtraps [12][21].We know from the LaPlace equation in electrostatics (∇2V = 0) that a static electricpotential must have its maxima and minima at physical boundaries, which in practice areelectrodes. However, in RF optics it is possible to have a time-averaged field which guides ionsaway from boundaries. We describe the approximation of these average fields by introducingthe concept of a pseudopotential (U (PS)) [21][52]. Just like in real electric potentials, ions ina pseudopotential well experience a force towards minima in the potential. This provides aconvenient conceptual tool for understanding and describing the average force experienced byions in RF fields. In the case of RFQ optics, the minimum in the transverse direction is in thecentre of the rod arrangement and the pseudopotential is described as [52],U(PS)quad =QeV 2mωr20(r2r20). (2.7)Here Q is the charge state of particle, e the elementary charge, r the radial position of theparticle, r0 the radius to the inner edge of the quadrupole rods (Figure 2.13), V the 0-to-peaktime varying electric potential on the rods, m is the particle mass and ω is the RF angularfrequency.We see in Figure 2.14 simulations of the transverse motion of an ion relative to the axis ofa linear Paul trap. Shown is the combination of a larger macro motion corresponding to the242.3. Radio Frequency Quadrupole OpticsTransverse positionTimeq=0.05q=0.3q=0.6q=0.7Figure 2.14: Simulated radial ion motion in an RFQ as a function of time for different q-values.Here an ion of mass 133 u is simulated in a trap with r0 = 10 mm operated at a frequency of1 MHz with the voltage varied to change the q-value. Simulations produced in SIMION [54].time-averaged fields described by the pseudopotential as well as smaller micro motions arisingfrom the instantaneous states of the RF fields.Though the pseudopotential is useful for developing an intuition of ion confinement, itdoes not provide a complete description of the conditions necessary for RFQ ion confinement.An indispensable tool for understanding ion confinement within an RF field is the stabilityparameter (q)[21], which is calculated as,q =4QeVmω2r20. (2.8)In the absence of additional factors such as buffer gas (discussed in the next subsection) or aDC offset in the quadrupole potential [21] ions will be stably confined in an RFQ field if q isin the interval 0 < q . 0.908 [52]. We see in Figure 2.14 how ion motion becomes larger andmore chaotic as it approaches the upper limit of this stable range.2.3.1 RF Ion GuidesIf an RF potential is only applied transverse to the beam direction to provide radial con-finement we may now introduce a static (DC) axial potential gradient to guide the motion oflow energy ions. Methods of creating this gradient include segmenting the RFQ rods [61][62] orrods made of a resistive material which allows a continuous variation in potential [63] (Figure2.15). Frequently RF ion guides are also filled with a neutral buffer gas to reduce the ion phasespace [12][21][52].252.3. Radio Frequency Quadrupole OpticsGas-filled ion guidesWe now introduce ion interactions with a buffer gas; this causes ions to tend towardsfilling a smaller space in the bottom of a pseudopotential well with a smaller emittance. Ionsinteracting with an ideal gas will evolve towards establishing thermal equilibrium with the gasthrough collisions between molecules. If an ion beam is interacting with a buffer gas with alower average kinetic energy than the beam, then the beam will typically establish a lowerenergy thermal equilibrium through a process known as collisional cooling. If the gas is inan open system (constant gas exchange) the beam will be cooled to a consistent temperaturewhich causes a reduction in the energy and spatial spread of the ion beam. To prevent ionlosses through charge exchange with the gas we typically use a noble gas such as helium, thusenabling a higher transport efficiency.The combination of RF fields with buffer gas creates complications in the theoretical be-haviour of particles in both gas and RF fields which must be accounted for. RF heating happenswhen a collision kicks the charged particle out of phase of the RF, and into a different phase.This change in trajectory and connection to the RF phase on average leads to a gain in energy,called “heating” [12]. RF heating limits the range of values q may take on to provide stable ionconfinement. In the presence of a buffer gas we typically have an upper limit of q ≈ 0.8, thoughlarge increases in average kinetic energy may already be observed for q > 0.6 [52]. However, wecan typically keep q < 0.6 to safely benefit from the smaller phase space afforded by gas-filledRF ion guides and traps.Segmented rodsAn option for guiding ions in the desired direction through an RF ion guide is to segmentthe RFQ rods (Figure 2.15 a). Segmented rods are often connected electrically with resistorchains to create a desired DC potential gradient along the length of the rod. Alternatively thesegments of such RFQ rods may have independently powered DC potentials to allow greaterflexibility in defining the shape and direction of the axial fields, as well as giving the optionto trap ions. A key limitation of segmented rods stems from the finite length of the rodswhich creates flat regions in the axial potential as seen in Figure 2.15 a, which has in somecases inhibited ion transport [63]. Furthermore, segmented rods are more complex to build andoperate than resistive rods.Resistive rodsAnother approach to producing an axial electric field in RFQs is to build the rods from aplastic doped with carbon such as carbon-fibre-reinforced plastic (CFRP) [63]. These rods givea constant resistance per length, allowing an even axial gradient along the rod and are usedfor ion transport in the TITAN MR-ToF. A photo of an RF ion optical device built with sucha material can be seen in Figure 2.16. To control this axial gradient, conductive electrodesmust be placed at either end of the resistive rod. The principal disadvantage of these rodsstems from the plastic used in their construction. Plastics typically have higher outgassingrates than metals [64] which tends to increase the gas pressure, as well as adding contaminationto the beam in the form of organic molecules. As a result, it may be necessary to have highpumping capacity to maintain a high vacuum, also there may be greater ion losses throughcharge exchange with the organic molecules. Low pressures can be very important in contextssuch as an MR-ToF where large losses in efficiency have been observed at pressures greater262.3. Radio Frequency Quadrupole OpticsVDC4VDC3VDC2VDC1BeamdirectionDCPotentialPosition(a) Segmented RFQ rod.BeamdirectionResistive materialConductiveelectrodeVDC = HighConductiveelectrodeVDC = LowDCPotentialPosition(b) Resistive RFQ rod.Figure 2.15: Two schematics and corresponding linear potentials of RFQ ion guides.272.3. Radio Frequency Quadrupole Optics(a) Photo of switchyard.−V cos(ωt)V cos(ωt)(b) RF arrangement of electrodes.Figure 2.16: Images of RFQ switchyard. a) Photo of the switchyard partially assembled. Thecarbon-doped plastic electrodes are visible here. Photo credit [47]. b) Rendered model showingthe phases of the applied biases on the switchyard in red and blue.than 10−7 mbar [58]. One benefit of building RF optics out of plastic is that it becomes easierto construct different shapes, creating curved paths or other complex arrangements [52].RF switchyard (cube)One of the complex RF ion guide constructions which benefits from resistive carbon dopedplastics is the RF switchyard or “cube” designed for the TITAN MR-ToF [65]. The design ofthis switchyard is reminiscent of six quadrupole ion guides converging to a single intersectionfrom perpendicular directions. There are eight electrodes built in a three dimensional “L” shapewith conductors at the ends to set the electric potentials along the electrodes (Figure 2.16). Aswith other RF ion guides, radial confinement is maintained through the RF switching potentialsand ions are moved in the desired direction through application of DC potentials. An overviewof the DC potentials used for its operational modes is shown in Figure 2.17. This design isused in the TITAN MR-ToF transport system to allow a robust transport of ions into or outof the switchyard from any of the six directions as well as merging beams from two directionswhich is then extracted in a third direction. The RF switchyard provides greater flexibility thanelectrostatic counterparts, such as the switchyard in the TITAN beamline for transport to theEBIT (Figure 2.18). The EBIT switchyard lacks the capacity to merge two beams and is also amuch larger structure (∼1 m2 compared to ∼10 cm2). A diagram of the TITAN MR-ToF andits transport system, including the various RF optics described here is shown in Figure 1.6.2.3.2 Linear Paul TrapsThere are a variety of cases in which one may want to use the principles of RF ion con-finement for the trapping of ions. Applications of RF ion traps include mass spectrometry, ionstorage [21], and cooling and bunching continuous ion beams [6] to prepare them for injectioninto another device. To accomplish these goals, we may for example take an RFQ ion guidebuilt with segmented rods then set a high potential on the end segments of the rods and lowpotentials on the middle segments (Figure 2.19). The result of such an arrangement is a classof ion trap known as a linear Paul trap or simply a linear ion trap [21].282.3. Radio Frequency Quadrupole OpticsV HighDCV MidDCV MidDCV LowDCIonpath(a) Straight ion path.IonpathV MidDCV LowDCV HighDCV MidDC(b) 90◦ path.IoninputIoninputMergedbeamV MidDCV HighDCV MidDCV LowDC(c) Merging ion beams.Figure 2.17: Diagrams of the RF switchyard ion guide indicating relative strengths of the DCpotentials to allow ions to pass through the switchyard in: a) a straight path, b) a 90◦ path, orc) merge beams from two directions into a single path.Spherical bendersEinzel lensesQuadrupolelensesRectangularx,y steerersRectangularx,y steerersSphericalbenderBeam fromCB or MR-ToFBeam toMPETBeam to/fromEBITFigure 2.18: TITAN beamline electrostatic switchyard.292.3. Radio Frequency Quadrupole OpticsVDC4VDC3VDC2VDC1DCPotentialPositionFigure 2.19: Top: Schematic of a linear Paul trap based on segmented rods. Bottom: Exampleof linear potential used to trap ions.RFQ Cooler BuncherTrapping can typically be achieved with fewer ion losses and higher mass separation resolvingpower with bunched beams, yet often ion sources produce continuous beam. In addition, acontinuous high-energy beam of modest emittance will have a much larger emittance whenslowed to very low energies for trapping due to the relation described by Equation 2.2. Forexample, in the LEBT section of the ISAC facility a typical beam might be at an energy of 60keV with a transverse emittance of 150 µm [39]; electrostatically slowing this beam to 4 keVwe would expect the transverse emittance to increase to 580 µm. To avoid this, TITAN uses aclass of linear Paul trap called an RFQ cooler buncher (CB) to achieve an emittance of 12.5 µmat 4 keV [66]. Thus using a CB we are able to convert the higher energy continuous ISAC beamto a low-energy, low-emittance bunched beam to facilitate Penning trap mass measurements.As with gas-filled RF ion guides a CB is filled with a neutral buffer gas which reduces thephase space of incoming ions through collisional cooling. Collisional cooling is only effectivefor cooling beams with energy on the order of 10 eV; at higher energies the beam would bemoving too quickly to be cooled in the trap. To compensate for this cooling limitation theentire trap will typically be given a DC bias potential slightly below the incoming beam energyto slow the beam electrostatically before entering the trap. A DC potential gradient is thenset such that ions will be guided to the extraction end of the CB while remaining trappedprior to ejection. The quantity of ions in a given bunch will be determined by the continuousbeam current entering the trap, the collection time in the trap, and the limit at which Coulombrepulsion will force ions out of the trap (space charge limit).Ions cooled in the CB are gathered to an axial DC potential well near the end of the traplike that shown in Figure 2.20 b. Ions are gathered in this well to form a bunch prior to beingejected from the CB. Typically extraction is accomplished by lowering the DC potential on theoutput end of the trap to allow ions to escape, creating a field which “pulls” the ions out of thetrap. Sometimes this pull may be accompanied by a raising of the potential behind the bunch302.4. Summaryto add a “push”. Various electrode arrangements on the extraction end may be used to shapethe electric field extracting the ions to control the longitudinal emittance of the bunch leavingthe trap [6].A small linear Paul trap is used for injection of ions into the TITAN MR-ToF (“TrapSystem” in Figure 1.6). This trap also is used to recapture ions which have been separatedin the MR-ToF mass analyzer and remove isobaric contamination through mass-selective re-trapping, described in Reference [46].The main CB in the TITAN beamline provides the entry point for ions into the TITANbeamline, applying the cooling and bunching process to continuous 20 keV ISAC beam. Theimport of this CB to TITAN is such that we adopt the short-hand of referring to it as theTITAN CB. This CB uses segmented RFQ rods to control axial potentials for ion trappingand ejection. The rods each have 24 segments, with shorter segments near the ends of the CBto allow greater control of the electric fields near the ejection point. An overview of the rodsegments and their numbering is shown in Figure 2.20 a). During ion cooling and bunchingthe segments have DC potentials set for trapping, with segment 23 having the lowest potential,defining where ions will gather. For ion ejection, segments 22 and 24 are switched to allow theions to escape the CB. These DC potentials are shown in Figure 2.20 b.This particular CB uses square-waves to drive the RF switching of electrodes [39] insteadof the more common sinusoidal waves. The advantage of the square-wave driven RFQ is inthe flexibility of frequency achievable. Most sinusoidal driven RFQs have a specific optimaloperational frequency, effectively limiting the RFQ to a single frequency thus limiting the rangeof masses which may be stably confined. The electronics necessary for the square-wave driveallow a range of frequencies to be used, making a larger range of masses accessible using thisCB.Ions ejected from the CB enter a pulsed drift tube which is set with a potential below thatof the CB so as to accelerate ions to the chosen beam energy. Pulsing of the drift tube is set sothat when ions are in the middle of the tube, the drift tube potential will be lowered to ground.This allows the potential energy of the ions to be lowered without subjecting the ions to anyaccelerating field. Once the ions have left the drift tube they may then be transported throughan electrostatic beamline towards the other ion traps in TITAN for further preparation and/ormeasurement.2.4 SummaryThrough this chapter we have discussed the ion optics which are relevant for the work ofthis thesis. The main focus of the following is: transporting ions into the TITAN MR-ToF formass measurement and isobar separation; transport of isobar separated ions from the MR-ToFto MPET; and bypass of the MR-ToF when not in use. To accomplish this goal a variety ofelectrostatic ion optics have been discussed including Einzel lenses and quadrupole multipletsfor beam confinement and focusing. Parallel plate benders are used to curve and correct ionpaths, of these, rectangular and spherical benders are used in the TITAN beamline. In additionto the electrostatic optics used for ion transport we also discussed the electrostatic mirrors usedto perform mass measurement and isobar separation in the MR-ToF mass analyzer.Various RFQ ion optics are used for beam preparation and as ion guides. 20 keV continuousISAC beam entering TITAN is cooled and bunched in an RFQ CB before injection into theTITAN electrostatic beamline. The TITAN CB uses segmented RFQ rods to gather ions nearthe end of the trap before ejection. The TITAN MR-ToF has a set of gas-filled RF ion guides312.4. Summary20 keVcontinuousISAC beam1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 192021222324Ejectedbunchedbeam(a) Cooler Buncher segments and their numbering.DC PotentialPositionEjectionCooling(b) Cooler Buncher potentials.Figure 2.20: Schematic of the TITAN cooler buncher [39] electrode configuration. a) Thearrangement of 24 rod segments which can each be given a different DC potential. Ions aretrapped in the region of segment 23, then segments 22 and 24 are switched to allow ions toescape the trap and be directed into the TITAN beamline. b) A SIMION [54] calculation of theDC potential for cooling (blue) and ejection (red) as a function of linear position in the CB.322.4. Summarybuilt with resistive RFQ rods to transport ions in and out of the MR-ToF mass analyzer. TheMR-ToF transport section includes a novel RF switchyard which enables ions to enter and exita cube-like structure from any of six directions as well as allowing the merging of two ion beams.With an understanding of the principles of these ion optics we are now ready to apply thisknowledge to an investigation of the TITAN beamline, particularly for transport of ions into,out of, and around the MR-ToF.33Chapter 3Beamline studies for MR-ToFintegrationThe value of the MR-ToF to the scientific program at TITAN depends on the ability ofthe MR-ToF to function as an integrated part of the whole TITAN system. Previous work hasincluded the design, construction and initial testing of the device [47]; then offline commissioningwhich showed the TITAN MR-ToF is capable of performing mass measurements and isobarseparation at or above required performance [48]. For offline commissioning the MR-ToF andits transport system were operated as a standalone device, performing tests with an ion sourcelocated just below the input optics seen in Figure 1.6. With the completion of this previouswork, the next step was to integrate the MR-ToF into the TITAN beamline. Simulationswere performed to examine beam behaviour when being transported into, out of, or bypassingthe MR-ToF. As a result, a new split Einzel lens was introduced into the TITAN beamlineto improve transmission into the MR-ToF. The impact of this lens on the TITAN beamlinehas been examined. Simulations were validated against experimental studies, looking at timeprofiles of the beam at two locations as well as measurements of the transverse emittance of abeam transported towards the MR-ToF.3.1 The TITAN beamline prior to integrating the MR-ToFA schematic overview of the TITAN beamline prior to the installation of the MR-ToF isshown in Figure 3.1, with clarification on notation found in Table 3.1. This portion of beamlinehas been the focus of study for this chapter. Also indicated is where the MR-ToF was installedand locations of modifications made to the beamline optics to improve the efficiency withwhich ions can be transported into the MR-ToF from the CB. The function of most of the ionoptics outlined in Figure 3.1 have been explained in Chapter 2 with a few exceptions: Theindicated skimmer plates are grounded under normal operation and used to limit the range ofthe electrostatic fields to near the electrodes producing the fields. Most of the spherical bendersare 36◦ benders with an additional 9◦ provided by neighbouring rectangular benders, with theexception of the B1 bender which is a 45◦ spherical bender. Differential pumping apertures areused to limit gas flow between regions of the beamline at different pressure levels. For example,the right side of the TSYBL:DPA aperature is typically at 10−7 mbar, whereas to the left apressure of 10−10 mbar is maintained.343.1.TheTITANbeamlinepriortointegratingtheMR-ToFEBITTRFC:RFQ,TRFC:BIAS(Cooler Buncherupper portion)TRFC:PB5TRFC:EL5TRFCBL:DPASEL-MCPTRFCBL:XCB0,TRFCBL:CCB0(now split Einzel lens)TRFCBL:B1-IN/OUTHole to MR-ToFMR-ToFTRFCBL:Q1TRFCBL:Q2MCP-1TRFCBL:B4-IN/OUTTRFCBL:YCB4,TRFCBL:XCB4,TRFCBL:CCB0TSYBL:DPATSYBL:YCB0,TSYBL:XCB0TSYBL:B1-IN/OUTTSYBL:EL1TSYBL:EL3TSYBL:B8-IN/OUTTSYBL:YCB8,TSYBL:XCB8TSYBL:YCB9,TSYBL:XCB9,TSYBL:CCB2TSYBL:EL4MPETBL:EL2MPETBL:YCB3,MPETBL:XCB3,MPETBL:CCB3MCP-0to MPETzyz yzyElectrostatic opticsRF opticsMCPsGrounded elementsFigure 3.1: Schematic overview of the TITAN beamline optics between the Cooler Buncher and MPET as relevant for simulationspresented herein. Electrodes are labelled first with a section label, then the optics type and assigned number. Electrode types aresummarized in Table 3.1. Black lines appearing before and after many electrostatic optics are “skimmer” plates. Red arrows indicatethe beam path and direction. The MR-ToF and EBIT optics are not shown.353.1. The TITAN beamline prior to integrating the MR-ToFTable 3.1: A summary of the labels and corresponding purpose for the ion optics and aperturesin the TITAN beamline. Positions of these are indicated in Figure 3.1.Label PurposeRFQ CB RFQ rod segment DC voltageBIAS DC bias on CBPB Pulsed drift tubeEL Einzel lensXCB Rectangular parallel plate benders for x-directionYCB Rectangular parallel plate benders for y-directionCCB Power supply shared between XCB and YCB bendersB-IN Spherical bender inside electrodeB-OUT Spherical bender outside electrodeDPA Differential pumping aperture3.1.1 Beamline simulationsAs a preparation for integrating the MR-ToF the TITAN beamline was modelled in thesimulation software SIMION 8.1 [54]. SIMION allows numerical calculations of electric andmagnetic fields and resulting trajectories of charged particles through these fields [67]. Furtherdiscussion of SIMION may be found in Appendix A. These simulations were used to betterunderstand ion transport in the beamline and to predict optimal settings on the ion optics tofacilitate ion transport into, out of, and bypassing the MR-ToF.Cooler BuncherTo define initial conditions of ions, simulations of the TITAN CB were created based onprevious work described in [68]. Past simulations focused on the cooling of 20 keV ISAC beamand extraction into optics planned for offline testing of the CB. For this thesis work the focuswas not on the cooling process—as it is well established—but instead, the TITAN beamlineand the role of the CB therein. As such, only segments 18–24 of the CB were simulated. Byneglecting the other 17 segments, sufficient computing resources were available to increase thePA resolution from 1 grid unit per mm to 0.5 grid unit per mm. This allowed a more accuratemodelling of a 5 mm diameter circular aperture at the exit of the CB. We can see in Figure 3.2differences in how the aperture edge is modelled at different resolutions. 2–3% of ions ejectedfrom the CB in simulation were lost at this aperture, thus it is important to ensure that theportion of ion phase space being removed reflects what is lost in the real aperture.133Cs+ ions were created in simulation in a random spherical Gaussian phase space distri-bution determined from previous CB simulations of cooled ions. Specifically, with a standarddeviation in position of 0.62 mm and velocity of 0.1 mm/µs. This was centred on the z-axis inthe middle of the length of segment 23 (see Figure 2.20). To protect against any possible errorsin this initial phase space distribution, simulations allowed ions to cool for 100 µs in the buffergas before ejection into the TITAN beamline.An initial benchmark used for this CB simulation was a comparison to results reported inReference [68] shown in Table 3.2. ∆V22 and ∆V24 refer to the DC potentials on segments 22and 24 relative to segment 23 (see figure 2.20), then the rms emittance values for these settingswere simulated and recorded. Reference [68] reports using a q-value of 0.4 and Vpp of 400 Vfor simulating 133Cs+ ions, requiring a simulated RF frequency of 606 kHz. The reported gas363.1. The TITAN beamline prior to integrating the MR-ToF(a) 1 grid unit per mm. (b) 0.5 grid unit per mm.Figure 3.2: Two examples of how a 5 mm diameter circular aperture is modelled in SIMIONat different resolutions.Table 3.2: Comparison of CB simulation results produced for this thesis to results obtained in[68]. Differences between the simulations range from 1% to 35%. Unit convention for reportingtransverse emittance follows that of [68].This Thesis Reference [68] Difference∆V22 ∆V24 trans (pi mm mrad) long (eV µs) trans (pi mm mrad) long (eV µs) trans long0 -30 3.52 ± 0.11 5.81 ± 0.17 3.3 ± 0.3 4.7 ± 0.2 6.7% 23.7%0 -60 4.16 ± 0.12 7.5 ± 0.2 4.2 ± 0.1 7.4 ± 0.3 1.0% 1.0%30 -30 3.31 ± 0.10 1.75 ± 0.05 3.8 ± 0.2 1.3 ± 0.1 12.8% 34.6%60 -60 3.39 ± 0.10 1.79 ± 0.05 3.6 ± 0.1 1.4 ± 0.1 5.7% 27.7%500 -500 3.27 ± 0.10 1.95 ± 0.06 4.8 ± 0.2 1.8 ± 0.1 31.9% 8.5%pressure is 2.5× 10−2 mbar with a temperature of 300 K. In addition, a rise time of 10 µs wasused for the switching of segments 22 and 24 from trapping to extraction mode.The comparison in Table 3.2 shows emittance values differing between 1% and 35%. Theknown differences between these simulations and those in [68] are the difference in extractionoptics, the initial phase space distribution, and the change in simulation resolution. Ion ex-traction from the CB was done with optics for a test stand used in offline testing, whereas forthis thesis the extraction optics of the TITAN system were used. Inconsistencies may also arisedue to any non-linear evolution of the rms emittance through the different optics. As noted onpage 12, the calculated value for rms emittance depends on how ions are distributed in phasespace and will, for example, be inflated by filamentation of the emittance. Another concernis any unanticipated sensitivity to the initial ion phase space distribution. The precise initialdistribution used in [68] is unknown, thus this could not be exactly reproduced. Finally, theincreased simulation resolution may have had an impact, though this has not been tested. Itwas observed that 2–3% of ions simulated in the CB are lost on extraction; the phase spaceproperties of these will be affected by the shape and size of the aperture where the ion lossesoccur. As seen in Figure 3.2, this is impacted by the simulation resolution which could in turnhave a measurable effect on the emittance. However, further exploration of these differenceswas eschewed in favour of the more crucial matter of reproducing experimental results. To thisend we shall compare CB simulation results to measurements of transverse emittance and beamtime profiles in the Sections 3.1.2 and 3.1.3 respectively.373.1. The TITAN beamline prior to integrating the MR-ToF3.1.2 Allison meter emittance measurementsTo further our understanding of the CB performance and to experimentally test the validityof these CB simulations emittance measurements of the beam from the CB were performed.The tool used to measure the beam emittance was an Allison meter [69]. The principle of anAllison meter is to sample the number of ions detected across a grid of points in trace spaceone at a time. An outline of the components used in this sampling is shown in Figure 3.3. Themeter is moved between x-positions using an actuator. At each position the divergence x′ ismeasured by varying the voltage difference on the plates V such that only ions of a particulardivergence will reach a MicroChannel Plate detector (MCP) [70] on the opposite end of theAllison meter. The divergence is calculated as,x′ =Leff4gEV, (3.1)where Leff is the length of the parallel plates in the direction of beam propagation, g is the gapbetween plates, and E is the beam energy. At each position and voltage setting the number ofions successfully passing through the detector are counted with an MCP from which a densityplot of the emittance may be determined. As explained earlier, because the edges of suchdistributions are not clearly defined we use rms emittance to quantify the emittance.The Allison meter was placed at the top of the B1 bender box in the TITAN beamline(Figure 3.1) to measure beam emitted from the CB to the location where the MR-ToF inputoptics would eventually be located.These measurements used 133Cs+ ions produced in the TITAN internal ion source (locatedbelow the CB [71]). The CB was operated at current standard operating parameters with afrequency of 480 kHz with a peak-to-peak voltage of 170 V (q = 0.27) and gas pressure of 10−2mbar at room temperature. DC voltages are listed in Appendix B.Analysis of raw dataRaw data from the Allison meter was analyzed using a script in the programming languageLua 5.1 (shown in Appendix D). The methodology is adapted from MATLAB code developedfor beam analysis use at TRIUMF [72].To convert the MCP signal into ion counts a Multi-Channel Scalar (MCS) was used. On theMCS a discriminator is used to set a voltage threshold which must be overcome for electricalpulses to be treated as real counts from the MCP. For each cycle the MCS would acquire countsover a time of 81 µs, placing counts in bins 40 ns wide. The signal from the MCP could thenbe identified by the corresponding peak in the time profile recorded at the MCS.Fluctuations were observed in the threshold set on the MCS discriminator which causedchanges in the number of MCP ion detections counted. This effect on the counted ions isdue to variations in the energy deposited on the detector by a given ion; some ions will depositmore energy, producing voltage spikes above the discriminator threshold, others of lower energy,falling below the threshold. The result is that a drifting threshold will change the average ionscounted for a constant ion current. To minimize the uncertainty created by this drift, anotherMCP (MCP-1) was used to provide a constant beam current measurement for normalizationof the Allison meter MCP signal. To produce this signal, a switch was used to turn theTRFCBL:B1 bender on and off so that ions would alternate between being sent to the Allisonmeter and being sent to MCP-1. By assuming a constant ion current to MCP-1 the signaldetected by the Allison meter was normalized.383.1. The TITAN beamline prior to integrating the MR-ToFAveragebeam direction(energy E)gVLeffActuatorMCPxz(a) Cross-section of Allison meter.Averagebeam directionwLLmax MCPyz(b) Top view of Allison meter lower plate and housing.Figure 3.3: Schematic diagrams of an Allison meter used for measuring beam transverse emit-tance. Red lines indicate example paths of individual ions passing through the meter; the solidline shows a transmitted ion and the dashed line indicates a) an ion entering along a path whichwill not pass through the meter, or b) an ion which passes through the detector at the veryedge of the detector slit.393.1. The TITAN beamline prior to integrating the MR-ToF 0 10000 20000 30000 40000 50000 60000 0  10  20  30  40  50  60  70  80Integrated CountsTime (µs)Switching noise MCP-1(normalization)Allison MCPSubtractedbackgroundFigure 3.4: Detected signals at MCP-1 and the Allison MCP as a function of time. Variouspeaks are indicated. Highlighted are the interpretations of specific peaks in the time-of-flightspectrum as well as the sections of data used to produce the emittance measurement. Thisexample is an integration of all the time profiles for a specific measurement.An additional step in determining ion counts was accounting for background noise. A sectionof the detected time profile separate from any signal was selected to calculate the mean averageof the background. This background was normalized to MCP-1. The mean background wassubtracted from each bin counted by the Allison meter MCP.An example of all time profiles for a complete emittance measurement is shown in Figure3.4. It highlights the peak from the MCPs and the region used for background subtraction.Alternatively we may see a summary of all the MCS data recorded for an emittance measure-ment in Figure 3.5. This plot presents each time profile vertically as a density plot. Eachsuccessive time profile is numbered with a Measurement Index. The order in which the profileswere produced was to cycle through all voltage settings from most positive to most negativeat a set Allison meter position. Then the meter would be moved to the next position and theprocess of cycling through the voltage settings would be repeated. We can see the effect of onlysome voltage and position settings allowing transmission of ions to the Allison meter MCP bywhat appear to be dots in the density plot at approximately 68 µs.Combining this Allison meter position, divergence (from voltage, using Equation 3.1), andion count data we may produce a plot such as those in Figure 3.6. Figure 3.6 a and b show theimpact of normalizing with MCP-1. In Figure 3.6 c the z-axis has been rescaled to make visiblethe background noise which is . 2% of the maximum ion count. This noise must be excludedfor rms emittance calculations, otherwise false counts will distort our results. To remove thiseffect we set a threshold which ion counts must surpass to be counted. This threshold was setas three standard deviations of the ion counts in a background region of the emittance plot,far from the real data. By setting ion counts below the threshold to zero, some real countswill be removed along with the noise, thereby artificially lowering the calculated emittance. Toaccount for this effect we introduce an inflation factor ifac equal to the ratio total of ion countsto ion counts removed by the threshold. Assuming a bi-Gaussian ion phase space distribution403.1. The TITAN beamline prior to integrating the MR-ToFFigure 3.5: Detected MCP signal as a function of time as function of Measurement Index fora complete Allison meter emittance measurement. Individual time profiles are presented asdensity plots parallel to the y-axis. The time profiles are arranged horizontally in the orderthey were taken. Integrated projections of the data along the x- and y-axes are shown above andto the right of the density plot. Allison meter MCP signal can be seen as dots at approximately68 µs.we find,ifac =pfacpfac− (1 + log (pfac)) , (3.2)where we define pfac is the ratio of the peak number of counts in the emittance distributionand thresh is the level of the threshold in counts:pfac =peakthresh. (3.3)We expect the real emittance value to be somewhere between rms and ifac · rms.The result of this analysis for the described CB operating parameters is shown in Figure3.6 d. Further details of this analysis methodology are outlined in Appendix C.Measurement uncertaintyThere are three quantities which are required for the measurement of an emittance value,each introducing its own uncertainty the: ion position, divergence, and counts per parameterset. The Allsion meter was positioned using a Thermionics Northwest FLMR-275-50-4/MSactuator driven by a Lin Engineering 5718X-18DE-01 stepper motor. The actuator is specifiedas having 20 turns/inch and the stepper motor to have 200 steps per turn, together giving6.35 × 10−3 mm/step, which was experimentally verified. From this we estimate the accuracyof the actuator precision to half a step, thus we assign an uncertainty of δx ≈ 3.2× 10−3 mm.The process of estimating the uncertainty of x′ was a more complex process due to themultiple components in the measurement indicated in Equation 3.1. The energy spread δE hadbeen estimated using a Retarding Field Analyzer (RFA) [73] and found to be 4 eV. The RFA isdescribed in Reference [74], and these specific measurement are from diagnostics of the TITANsystem [75].The voltage V applied to the Allison meter was measured using an Agilent 34401A DigitalMultimeter. The accuracy of this multimeter is specified as±(0.0002% of the reading + 0.0001%413.1. The TITAN beamline prior to integrating the MR-ToF(a) (b)(c)Emittance:rms: 13.24± 0.9 µmInflation factor: 1.09Twiss Parameters:α: −4.65± 0.10 pi−1β: 3.57± 0.08 mpi−1γ: 6.34± 0.14 (pim)−1(d)Figure 3.6: Emittance plots from one measurement of a 133Cs+ beam out of the TITAN CBwith standard operating parameters (described in text), showing different levels of processingof the number of ions detected at each position and divergence. a) An emittance plot withno processing of the ion counts, showing a large biasing effect due to drift in the MCS. b)An emittance plot with ion count normalization and time profile background removed, butbefore a threshold was set to remove background counts for rms emittance calculation. c) Thesame emittance plot as b), but scaled in the z-axis to highlight the degree of variation of thebackground counts. Background variations on the level of 2% of the peak ion counts can beseen. d) An emittance plot after the complete ion-count processing described in the text. Awhite ellipse calculated from the Twiss parameters is overlaid on the plot with the size set bythe equivalent emittance 4rms. Calculated rms emittance and Twiss parameters printed below.423.1. The TITAN beamline prior to integrating the MR-ToFof the range). For the range of voltages measured the highest relative uncertainty had a valueof δV/ |V | = 5.5× 10−5. Given that this value is one or more orders of magnitude smaller thanother uncertainties relevant to calculating x′, this maximum δV/ |V | was applied to all voltagemeasurements to be conservative.The accuracy of g and Leff were subject to the uncertainty determined by the tolerancesspecified for their manufacture. δg was estimated as 0.04 mm; however the process of determin-ing δLeff was complicated by the width of the slit through which ions enter the Allison meter(denoted as w in Figure 3.3 b). Because of the slit width, it is possible for ions to be crossingover the plate on a diagonal path, thus longer than if they had passed straight over the plate,creating a larger Leff . We estimate an upper limit on this divergence from the length betweenslits in the Allison meter L and the slit width. We can calculate the maximum path lengthbetween slits as Lmax =√L2 + (w/2)2. This gives a divergence > 2000 mrad, which is an orderof magnitude greater than that measured in the y-axis. Without having directly measured thex-axis emittance we take this as an acceptably pessimistic estimate of the uncertainty and useit to assign a relative uncertainty to Leff of 0.021, or δLeff of 1.5 mm. Direct measurements ofthe x-axis emittance were not pursued due to the x-y symmetry of the CB, by which we wouldpredict identical emittance in either transverse direction.There were two primary considerations in the measurements of ion counts: the normalizationdiscussed in the previous section and the consistency of beam current. The individual datapoints N were estimated to have an uncertainty of√N . Standard error propagation techniqueswere used to incorporate these uncertainties. The uncertainty in the average background countswas estimated as the standard deviation of the mean. To account for variations in the beamcurrent, the current was measured over ≈ 24 hours to see how it shifted and was found to varyby less than 1% per hour.An additional concern here is whether the distribution can be accurately described as Gaus-sian. Fitting the distribution to a Gaussian function it was found that it follows a Gaussiandistribution well except for a small cut off observable in the upper right corner of the phasespace distribution shown in Figure 3.6. The apparent cut-off of beam is equivalent to 2% ofthe total ion counts. This is likely due to the edge of the beam being cut at an aperture in thebeamline. The most likely position for these ion losses is the hole in the B1 bender plate. Insimulation (see Figure 3.7) it was found to be the most common cause of ion losses.Together, the uncertainties in the emittance measurements resulted in uncertainties of ap-proximately 1% on the measurements performed.Results and comparison to simulationThe key results for the beamline studies discussed here were from measuring the emittancefrom the CB under standard operating conditions. This result is plotted in Figure 3.6 d), whichshows an rms emittance of 13.24± 0.09 µm with an inflation factor of 1.09 for a bunched beamof 133Cs+ ions at an energy of 2.4 keV. The ion current out of the CB was set to release singleion bunches. This is in agreement with previous measurements of the CB emittance describedin Reference [66] which reports an emittance of 4± 1 pi mm mrad at a beam energy of 4 keV.Scaling this previous measurement to 2.4 keV we find it to be 16± 4 µm.This measurement was reproduced in simulation with the same operating parameters anda temperature set at 300 K, the result of which is shown in Figure 3.7. We can see here thatthe experimentally measured best estimate emittance is approximately 1.7 larger than thatsimulated. This same difference was seen in comparing experimental results in Reference [66]to simulations in Reference [68] on which these simulations were based. This suggests that433.1. The TITAN beamline prior to integrating the MR-ToFEmittance:rms: 7.5± 0.4 µmTwiss Parameters:α: −4.2± 0.3 pi−1β: 2.93± 0.18 mpi−1γ: 6.5± 0.4 (pim)−1Figure 3.7: A simulation of the Allison meter emittance measurement of the TITAN CB beamshown in Figure 3.6. Left: SIMION simulation of ion trajectories (red) with collision points(green) and ion optical elements (brown). Right: Emittance plot with the ellipse calculatedfrom the Twiss parameters for equivalent emittance (4rms) overlaid in white. Calculated rmsemittance and Twiss parameters are printed below. For this simulation, frequency was set at480 kHz with a peak-to-peak voltage of 170 V, a gas pressure of 10−2 mbar, and temperatureof 300 K.these CB simulations are consistently simulating the beam having a smaller phase space andthus being more cooled than what is achieved experimentally. A possible source of the largerexperimental phase space is space charge effects within the CB. A significant portion of theions cooled in the CB (∼ 10%) are lost before extraction. This means that when single ions arebeing extracted from the CB experimentally there may still be enough ions to inflate the phasespace through Coulomb interactions. Such an effect is not currently being simulated. If theCB is to continue to be incorporated into future beamline simulations it may be necessary touse a more empirically defined initial phase space and extract ions immediately. Alternatively,the simulations of the cooling process may undergo further development to incorporate spacecharge effects.3.1.3 Beam time profilesOne of the beam diagnostic devices permanently in the TITAN beamline is a set of MCPswhose locations are noted in Figure 3.1. MCPs are able to provide ion time-of-flight (ToF) data443.1. The TITAN beamline prior to integrating the MR-ToFby recording how long after ejection from the CB ions are detected. MCPs are also sensitiveto single-ion counts, useful for detecting low beam currents. The resulting beam time profileshave been used as a point of comparison between simulation and experiment.To compare simulated and experimental time profiles, the experimentally chosen timingtriggers and electrode voltages were used as input data into the simulation. Ions were createdone at a time inside the CB, then cooled, then transported to the location of the relevant MCPwhere the ion position, velocity and ToF would be recorded for analysis. Two MCPs were usedfor this comparison which we will refer to by TITAN nomenclature as MCP-0 (immediatelybefore MPET), and MCP-1 (in the midpoint of the 90◦ bend section of the TITAN beamline),both shown in Figure 3.1.Prior to transport through the TITAN beamline 133Cs+ ions were trapped in the TITANCB with an RF frequency of 480 kHz, peak-to-peak voltage of 170 V, and helium buffer-gaspressure of approximately 10−2 mbar. Voltages used on the electrostatic beamline elementsare summarized in Appendix B. We note that the experimental voltages on the electrostaticion optical elements appear to give a relatively high ion transmission in simulation without anymodification as seen in Figure 3.9. Bender voltage differences of a few volts (< 1%) are sufficientto completely block transmission through TSYBL:DPA, thus suggesting that the electrostaticportion of the simulation provides a reliable model of the real beamline.A comparison of experimental and simulated time profiles at MCP-1 and MCP-0 may be seenin Figure 3.8, including the results of Gaussian fits to the data. Figure 3.9 shows the simulatedion trajectories from the CB to MCP-0. The same settings were used for transporting beam toMCP-1 and MCP-0; thus the trajectories seen in Figure 3.9 would be the same in both cases.At both MCPs the simulated time of flight is approximately 4 µs longer than in experiment andin both cases the Full Width at Half Maximum (FWHM) is larger. In the case of MCP-1 thesimulated FWHM is 62% larger than in experiment and for MCP-0 the FWHM is 13% larger.These differences between simulation and experiment require some further discussion.Examining the ion trajectories shown in Figure 3.9 we can see that there are multiple regionsin the beamline where the beam passes very close to the ion optics, in some places collidingwith electrodes. Since the experimental CB emittance is approximately 1.7 times larger than insimulation, this indicates that the probability of ions passing close to or colliding with electrodesis greater in the physical beamline. Experimentally, ion losses along the beamline have beendetected by measuring radioactivity during online experiments. In particular, a large amountof ion loss was observed at the differential pumping aperture after the 90◦ bend (TSYBL:DPA),consistent with losses seen in simulation.For the case of ions passing close to electrodes but not colliding, we expect a high sensitivityto fringe-field effects and misalignments. Close to the electrodes, small imperfections in themachining or alignment of electrodes will have a larger effect on ion trajectories. Similarly, thesimulation of the electrodes and resulting electric fields has a finite resolution, which can alsointroduce field aberrations, particularly close to the electrodes. Over long distances the effectsof sampling these field aberrations will become increasingly pronounced. This effect may beobserved through an increase in rms emittance, though we could not expect this increase to beidentical between simulation and experiment. In Figure 3.10 we can see in simulation this effectof the rms emittance growing as the beam travels through the beamline. A notable exceptionto this trend towards increasing emittance comes where the beam passes through TSYBL:DPA.A simulation of these ion losses is shown in Figure 3.11.The large drop we see in the emittance after TSYBL:DPA results from the aperture actingas a collimator. TSYBL:DPA is 10 cm long with a 4 mm diameter which results in only alimited range of ion trace space properties being allowed to pass through. All other ions which453.1. The TITAN beamline prior to integrating the MR-ToF 0 100 200 300 400 500 600 41 41.5 42 42.5 43 43.5 44 44.5 45FWHM:	0.689 ± 0.009 µsToF:	42.809 ± 0.005 µsCountsTime (µs)(a) Experimental time-of-flight measurement at MCP-1. 0 20 40 60 80 100 44.5 45 45.5 46 46.5 47 47.5 48 48.5FWHM:	1.119 ± 0.013 µsToF:	46.486 ± 0.008 µsCountsTime of Flight (µs)(b) Simulated time-of-flight measurement at MCP-1. 0 10 20 30 40 50 60 70 80 130  131  132  133  134  135  136FWHM:	0.94 ± 0.02 µsToF:	132.497 ± 0.011 µsCountsTime (µs)(c) Experimental time-of-flight measurement at MCP-0. 0 20 40 60 80 100 120 134  135  136  137  138  139  140FWHM:	0.985 ± 0.016 µsToF:	136.716 ± 0.009 µsCountsTime of Flight (µs)(d) Simulated time-of-flight measurement at MCP-0.Figure 3.8: A comparison of simulated and experimental time profiles obtained at MCP-1 (top)and MCP-0 (bottom).463.1. The TITAN beamline prior to integrating the MR-ToFto MPETMCP-0 TSYBL:DPACBFigure 3.9: A SIMION simulation of the TITAN beamline between the CB and MPET. Beamtrajectories are shown in red and points where ions collide with electrodes or reach the positionof MCP-0 are marked in green with notable loss points circled in green. Most losses occur atTSYBL:DPA.MCP-1TSYBL:DPAMCP-0Figure 3.10: A plot of the change in transverse rms emittance as a function of beam time offlight from the CB.473.1. The TITAN beamline prior to integrating the MR-ToFFigure 3.11: Simulated trajectories (red) of ions passing through the differential pumping aper-ture TSYBL:DPA, and the points at which some ions collide with the aperture structure (green).Trajectories are going right to left here. Simulations produced in SIMION [54].would give the beam a larger emittance are removed. This beam collimation likely producesthe closer agreement between simulation and experiment by removing many of the ions out ofthe CB which cause the experimental emittance to be larger than that simulated.In Figure 3.9 we see that there are ion losses around the drift tube which sits at the locationof MCP-1 when the MCP is not in the beamline. When MCP-1 is in the beamline it has asmall 1.3 cm long section of drift tube before it which for a larger emittance beam may havesome collimation effect. It is feasible that this is a factor in the smaller ToF FWHM measuredexperimentally.We therefore conclude that these simulations can reproduce experimental measurementswithin a factor of 2. Inconsistencies appear to arise primarily from differences between thesimulated and actual CB and the resulting sampling of fringe-field effects and ion losses as ionspass close to the beamline electrodes. Agreement between simulation and experiment appearsto become closer for beam which has passed through TSYBL:DPA due to collimation effectsfrom the aperture removing inconsistencies arising from limitations in the CB simulation.A number of simulations were run to see if other effects could account for the disagreementsbetween simulation and experiment, looking primarily at the output of the CB. Considerationwas given to possible differences between experimental and simulated electrode voltages, andrise times.For voltages, the possibility that power supplies had not delivered the requested voltages tosome CB electrodes was investigated. Simulations showed that to bring simulated ToF peaksinto closer agreement (a few percent different), voltage differences would need to be increasedby tens of volts. Such differences are much larger than anything which has been experimentallymeasured (< 1 V), and thus this explanation was rejected.No record was available of the electrode rise time for switching the CB from cooling toextraction, only that it could be as long as 10 µs [76]. Within this constraint simulationswere run for a range of rise times ranging from 10 ns to 10 µs. Shorter rise times can seedecreases in the ToF (∼ 1 µs) and ToF FWHM (∼ 100 ns) which bring simulation closer toexperiment, but these effects were not sufficient to bring simulated ToF measurements intoagreement with experiment. Thus, without any obvious indication that this might be the causeof the disagreement, adjustments in rise time were set aside until such a time as it can bemeasured experimentally.If a closer agreement between simulation and experiment is required for further beamline483.2. MR-ToF integrationstudies and development it may be necessary to apply a more empirical approach in simulatingthe beam. Using the measured transverse emittance and Twiss parameters it is possible toinfer the transverse beam properties out of the CB using methods discussed in [51]. Withmeasurements of the beam energy spread using an RFA, and ToF information from the MCPsan empirically derived longitudinal emittance could also be defined. Such an approach wouldhowever lose the freedom to easily simulate different cooling and extraction parameters in theCB. For this thesis there was not sufficient time to change to an empirically defined beam.3.2 MR-ToF integrationHaving developed simulations of the TITAN beamline we now turn our attention to theintegration of the MR-ToF. The MR-ToF was designed to interface with the TITAN system atthe 90◦ bend section indicated in Figure 3.1. This location allows beam from the CB to be in-jected into the MR-ToF transport system from below, undergo mass separation, then be ejectedfrom the MR-ToF transport system towards MPET for high precision mass measurement, orto other experiments.3.2.1 Beamline modification for MR-ToFWhen the TITAN beamline was first designed the MR-ToF was not foreseen as part of thesystem [71][77]. Critically, there was no path for ions to enter the MR-ToF from the CB asthe outer electrode of the B1 bender would be a barrier to passage. To allow ion transportthrough the B1 bender a new B1-OUT electrode was machined with a 7.9 mm diameter hole onthe beam axis out of the CB to allow ion transmission. However, simulations of the beamlineshowed that the established TITAN optics were only able to provide 33% ion transmission atthe beam energy the MR-ToF was designed to accept (1.4 keV). A key issue was that there wereno focusing optics after the aperture TRFCBL:DPA and before the B1 bender. EL5 neededto focus beam on TRFCBL:DPA, resulting in a significant beam divergence and resulting ionlosses at the position of the hole in B1-OUT.To improve ion transmission it was necessary to introduce a new lens between TRFCBL:DPAand the B1 bender. An additional concern was beam steering to compensate for any possiblebeamline misalignments. Due to the limited space available between TRFCBL:DPA and B1 thedecision was made to combine these steering and lensing properties into a single ion optic. Thislens design was achieved by splitting the middle electrode of an Einzel lens into four quartersable to have independently set voltages providing simultaneous focusing and steering. We shallrefer to this lens design as a split Einzel lens (SEL), photos of which are shown in Figure 3.12.The SEL was placed in the TITAN beamline in the location of the XCB0 benders.Due to time constraints a quantitative analysis of the improvement to ion transmissionafforded by the SEL was not possible. The SEL was installed amidst the process of measuringthe beam energy with an RFA located above the B1 box. Prior to the SEL installation ioncurrents at the RFA were too low to be distinguished from background noise; however, after theSEL was installed, a very clear ion current was immediately detected, indicating a substantialgain in transport efficiency. This improvement in transmission is reflected by simulations whichshowed near 100% ion transmission into the MR-ToF upon introduction of the SEL (comparedto the previous 33%).493.2. MR-ToF integrationFigure 3.12: Photos of the split Einzel lens added to the TITAN beamline to facilitate high iontransport efficiency into the MR-ToF. Top: SEL prior to installation. Bottom: SEL installedin its location in the TITAN beamline.503.2. MR-ToF integration3.2.2 MR-ToF acceptanceOne component in the maximization of the beam transmission into the MR-ToF was anunderstanding of the MR-ToF acceptance. To investigate the acceptance of the MR-ToF,simulations were produced to examine the ion trace space properties which could enter theMR-ToF system input optics.Acceptance simulation principlesThe geometry of the input optics was reproduced in SIMION. Then ions were simulatedas starting at the top of the B1 box and allowed to fly towards the input optics. Trace spaceproperties of ions passed to the MR-ToF were defined in a rectangular grid 2xmax wide by2x′max high. Within these bounds, nx initial x values and nx′ x′ values were defined, coveringa total trace space area of 4xmaxx′max, with a resolution of nx × nx′ . This rectangle could berotated to match more closely the observed acceptance trace space orientation (as in Figure3.13) and limit the portion of trace space tested outside the acceptance, thereby reducing thenumber of ions which needed to be simulated in characterizing the device acceptance.Having defined the portion of trace space to test, each ion with a set initial x and x′ wasflown towards the MR-ToF input optics. The initial properties of ions successfully enteringthe device were recorded to characterize the device acceptance. If it appeared the tested tracespace was too small to cover the device acceptance, the test area was increased until the totalacceptance area was enclosed. An example of a chosen test area and acceptance are shown inFigure 3.13.To numerically define the acceptance, each flown ion was assumed to represent a set fractionof the test trace space area,α1 =4xmaxx′maxnxnx′. (3.4)From this we calculate the device acceptance α from the number of accepted ions Naccepted as,α = Naccepted · α1. (3.5)It should however be noted that this method of acceptance prediction should be considered anupper limit on the device acceptance. The acceptance may be filamented due to aberrationsfrom ion optical elements, making it very difficult (though not impossible) to match beamemittance to. This consideration would be of particular importance in cases where the beamemittance is of a similar magnitude to the device acceptance.Application to MR-ToF acceptanceTo study the MR-ToF acceptance it was important to look for optimal ion optical operatingparameters to facilitate matching to the beam from the CB. A SolidWorks [78] model of theMR-ToF input optics is shown in Figure 3.14. The labels indicate the function of each ionoptical element. Some of the ion optical functions have been discussed in Chapter 2, specificallythe Einzel lens (S-In-Lens), the contacts for the resistive RFQ rods (S-In-RFQBot and S-In-RFQTop), and the RFQ trapping segments (S-In-TrapS). Other electrostatic optics are: Thesteering electrodes (S-In-Steer1-4) which are individually controlled electrodes in a quadrupolararrangement able to provide transverse beam steering. The deceleration stack, which is a set offour electrodes connected with a resistor chain where the bottom electrode is on ground and thetop defined by the voltage S-In-DecTop. The deceleration stack is used to slow 1.4 keV beam to513.2. MR-ToF integrationFigure 3.13: An example of the portion of trace-space tested (test area) and acceptance simu-lated in the investigation of the acceptance of the MR-ToF input optics.somewhere on the order of 10–100 eV. The apertures after the deceleration stack (S-In-A1–3)are a set of small apertures which restrict gas flow from the gas-filled RFQ section above whilealso allowing a small amount of electric field manipulation. For notational brevity in furtherdiscussion of the input optics we will neglect the “S-” prefix as it is present in all MR-ToFoptics discussed in this chapter.In defining the optimized operation parameters for the various electrodes in simulation therewere a few necessary considerations. The deceleration stack was set using In-DecTop at 1380V to reduce the beam energy to ≈ 20 eV before entering the RFQ. The DC voltages on theRFQ rods were set below that of the deceleration stack to guide the low energy ions in thedesired direction. To this end, In-TrapS and In-RFQBot were given voltages of 1347.8 V and1344.5 V respectively. In-RFQTop was set at the same voltage as In-RFQBot because we didnot need to simulate ion movement through the MR-ToF transport section, only ions enteringthe system, thus simulating the DC gradient in the RFQ section was unnecessary. Steerers werepresent to correct for any alignment issues. However, any misalignments in the physical devicehave not been reproduced in simulation, thus we set the steerers to ground. The aperturesIn-A2 and In-A3 were set at 1350 V to follow the choice of voltage used in offline testing of theMR-ToF. For In-A1 it was realized that by setting the voltage a few hundred volts lower thanthe neighbouring electrodes it can act as a small Einzel lens. The small length of the In-A1aperture is still sufficient to have a noticeable effect due to the low energy of the ions passingthrough it. To examine the possible lensing effect of In-A1, a range of voltages were tested forsimulating the MR-ToF acceptance. The final optical element which needed to have a voltageset was the In-Lens. A range of voltages were also examined for the In-Lens to determine anappropriate voltage for optimizing acceptance of the input optics.Having identified two ion optical elements which we wish to examine more closely to simulatethe MR-ToF acceptance we now look at some of the results. As mentioned above, ions werecreated in simulation at a location at the top of the B1 box with set trace space properties,then flown towards the MR-ToF. Ions which successfully entered the gas-filled RFQ section werecounted as accepted by the device. Tests of the 2-dimensional parameter-space of the In-Lens523.2. MR-ToF integrationRFQ Rod Contact (S-In-RFQTop)RFQ RodsRFQ Rod Contact (S-In-RFQBot)RFQ Trap (S-In-TrapS)RFQ Input Aperture 3 (S-In-A3) 1.5 mmRFQ Input Aperture 2 (S-In-A2) 3.0 mmRFQ Input Aperture 1 (S-In-A1) 3.0 mmDeceleration Stack(S-In-DecTop)(GND)Steerers (S-In-Steer1-4)(GND)Einzel Lens (S-In-Lens) 10 mm(GND)(GND)Input beam directionFigure 3.14: SolidWorks model of the MR-ToF input optics indicating the electrode nomencla-ture.and In-A1 voltages were performed to examine how they affect the MR-ToF acceptance. Firsta qualitative analysis was used to evaluate an appropriate range and step size for the voltagesto be tested. It was decided to vary the In-Lens voltages from 0 to -2250 V in 150 V steps,In-A1 was varied from 200 to 1200 V in 200 V steps.A sample of the results from varying the electrodes In-A1 and In-Lens is shown in Figure3.15. First an observation can be made that all cases show that a converging beam wouldbe optimal for matching beam emittance. For observations specific to particular optics, it wasnoted that changes in the In-Lens voltage had very little effect on the acceptance area, typicallycausing changes < 1%. The effect of the In-Lens was primarily to change the focus point towhich the beam is converging. This is reflected in changing trace space width and orientationof the acceptance. Variations of In-A1 had little effect on the focus point, instead increasingthe width of beam which can be accepted, converging to a focus point set by In-Lens; thisreflected by the linearly increasing acceptance coupled with a constant trace space orientation.The width of the trace space acceptance does not change noticeably with In-A1, meaning thatit has little effect on improving acceptance of diverging beams.Both measurements and simulations of the beam from the TITAN CB showed the beamemittance being much smaller than the acceptance values simulated here, but diverging. As aresult, settings for In-A1 and In-Lens which allowed the largest portion of a divergent beam to533.2. MR-ToF integrationFigure 3.15: Samples of the acceptances simulated for the MR-ToF input optics across a rangeof chosen voltages. Numerical estimates of the acceptance are shown in the upper right cornerof each plot. For this investigation the voltages on S-In-A1 and S-In-Lens were varied. Othervoltage settings are indicated in the text.be accepted was desired. It was believed that this could be best achieved with an In-A1 voltageof 800 V and an In-Lens voltage of -1050 V.Matching emittance to acceptanceHaving investigated the MR-ToF acceptance, attention was then shifted to optimizing beamproperties from the CB to match the MR-ToF acceptance. The available lenses for optimizingbeam from the CB into the MR-ToF were EL5 and the new SEL. During normal operationEL5 would be set to focus on TRFCBL:DPA. However, at the time the MR-ToF was beingintegrated into the TITAN system, experiments were being done to test the effects of removingthe narrow part of this aperture, meaning the voltage on EL5 could be changed much morefreely. Thus a large degree of freedom was afforded to optimize EL5 and the SEL to maximizetransmission into the MR-ToF.A range of voltage values were examined for EL5 and the SEL. For EL5, voltages from 0 V to-3600 V were tested and for the SEL, 0 to -3200 V. When a region in the resulting 2-dimensionalparameter space was identified which provided high ion transmission, simulations were refinedto a higher resolution in those areas. In the area of what appeared to be optimal parameters,voltages were adjusted in 50 V steps. The final result was an apparent optimal voltage of -1650V on EL5 and -400 V on SEL for a beam energy of 1.4 keV. The Figure 3.16 shows an overlayof the simulated emittance over the simulated acceptance for these parameters.When the MR-ToF was being commissioned with beam from ISAC these voltages were543.2. MR-ToF integrationFigure 3.16: Simulated phase space of beam emittance (green) from the CB overlapped withthe device acceptance (blue) of the MR-ToF input optics. This was achieved with EL5 set at-1650 V, SEL at -400 V, In-Lens at -1050 V and In-A1 at 800 V.modified slightly, giving EL5 -1400 V and SEL -550 V. For the MR-ToF In-Lens was set to-1080 V and In-A1 was set to 250 V. This change was also accompanied by the decision tolower the voltage on In-DecTop to 1123 V, meaning a higher beam energy and lower emittanceentering the gas-filled region of the MR-ToF input optics. Such a decrease in emittance makesit easier to maximize ion transmission into the MR-ToF. This difference in emittance wouldchange the exact beam parameters which would allow matching beam emittance to deviceacceptance. Changing the operation of the MR-ToF input optics effectively used the inputRFQ of the MR-ToF as a small CB, slowing the beam of a few hundred eV to a few eV fortransport in the gas-filled transport section of the MR-ToF. In simulation this achieved an 84%transmission efficiency into the MR-ToF.3.2.3 Beam profiles out of the MR-ToFHaving optimized beam transport into the MR-ToF the final step in interfacing the transportsystem of the MR-ToF to the rest of the TITAN beamline was to optimize beam transport outof the MR-ToF. A SolidWorks model of the MR-ToF output optics is shown in Figure 3.17.Ions were trapped in the region of the Out-TrapS electrode prior to ejection towards MPET.Segments Out-A1 and Out-A2 were kept at a high potential prior to ejection from the trap, thenpulsed to a low potential to eject ions from the trap. The beam energy was set by the voltage onOut-TrapS, for these simulations this was set to 1300 V to produce a 1.3 keV beam. The 1300 Vpotential is determined by the bias on the MR-ToF transport optics, used to achieve a 1.3 keVbeam energy within the MR-ToF analyzer. A major constraint on possible ion optical settingswas ensuring high transmission though TSYBL:DPA. As discussed above, this aperture has asmall acceptance, greatly restricting the portion of ion trace space which can be transmitted.The main lenses available for focusing beam from the MR-ToF through this differential553.2. MR-ToF integrationRFQ Rod Contact(S-Out-RFQIn)RFQ RodsRFQ Rod Contact(S-Out-RFQOut)RFQ Trap(S-Out-TrapS)RFQ Aperture 1(S-Out-A1)RFQ Aperture 2(S-Out-A2)Steerers(S-Out-Steer1-4)Einzel lens(S-Out-Lens)GroundedelementsExtractedbeam directionFigure 3.17: SolidWorks model of the MR-ToF output optics indicating electrode nomenclature.pumping aperture are Out-Lens and B4-OUT. The B4 bender is built to allow beam to passthrough from three different directions. For ions ejected from the MR-ToF, the central boreof the B4 bender may be used as an Einzel lens. In simulations the B4 bender was given anaccelerating potential to achieve this Einzel lens functionality. It was found in simulation thattransmission through the pumping aperture could be maximized by setting Out-Lens to -1900V and B4-OUT to -1225 V.Investigations of the extraction voltages on Out-A1 and Out-A2 suggested that Out-A1should be set at 1100 V and Out-A2 at 1000 V. Figure 3.18 shows the corresponding beam ToFand energy plots at the position of MCP-0. The spread in ToF appears consistent with what isachieved with beam from the TITAN CB, however the energy spread is much larger at around25 eV.The large energy spread merits further optimization which was not achievable with thesettings on ion optical elements tested here. This investigation could be further expanded byvarying the trap depth in the output RFQ prior to ejection to attempt to reduce the longitudinalemittance. It should also be possible to pulse the voltage on Out-RFQOut to introduce anadded degree of freedom to optimize the longitudinal ion trace space distribution. We see inthe simulated longitudinal emittance shown in Figure 3.19 that the trace space distributionis very long and narrow. This leads to either a wide ToF distribution, energy distribution,or both. For the investigation done thus far, this appears to have been the best achievabledistribution. However, by introducing the aforementioned additional degrees of freedom to ourbeam optimization we may be able to achieve a wider, shorter trace space distribution. Such adistribution would enable a smaller spread in both ToF and energy.3.2.4 Bypassing the MR-ToFOne final concern with the integration of the MR-ToF into the TITAN beamline was toensure that measurements can still be performed using EBIT and MPET when bypassing theMR-ToF. For isolation of the MR-ToF vacuum a gate valve was installed before the MR-ToFinput optics and after the output optics (along the beam path). In terms of ions optics, theonly changes in the previously existing TITAN beamline was the introduction of the opening563.2. MR-ToF integration(a) Simulated time-of-flight measurement at MCP-0. (b) Simulated energy measurement at MCP-0.Figure 3.18: Simulations of, Left: expected time-of-flight peak (blue) and, Right: energy spread(green) from the MR-ToF output optics detected at the position of MCP-0 for a beam energyof 1.3 keV. Gaussian fits of the data are shown with key results on each plot. Results withOut-Lens at -1900V, B4-OUT at 1225 V, Out-A1 at 1100 V and Out-A2 at 1000 V.Figure 3.19: Simulated longitudinal emittance of beam ejected from the MR-ToF and detectedat MCP-0 for a beam energy of 1.3 keV. This for the ion optical settings: Out-Lens at -1900V,B4-OUT at 1225 V, Out-A1 at 1100 V and Out-A2 at 1000 V.573.2. MR-ToF integrationFigure 3.20: A comparison of transverse emittance simulated after the 90◦ bend in the TITANbeamline and before TSYBL:DPA without and with the new optics for ion transmission intothe MR-ToF.in the B1-OUT bender plate and the replacement of XCB0 with the SEL. Naively we wouldexpect these to have a minimal impact on the beam through this section. The opening in B1-OUT is small (roughly an order of magnitude smaller than the bender radius), leaving most ofthe existing structure in place to provide the intended bending and focusing. Replacing XCB0with the SEL should provide the same steering capacity while also introducing steering in they-direction and additional focusing. However, to be thorough, this naive expectation was testedwith simulations.The section of beamline from the CB to just before TSYBL:DPA was simulated to evaluatethe impact of the new optics for the MR-ToF integration. Simulations were done of 133Cs+ions passing through this section at 2 keV without, and with the new optics. This beam energyis typical of operation of MPET and EBIT without the MR-ToF. For these simulations, nopotential was put on the SEL so as to allow a more direct comparison between the two setups.Transverse emittance was simulated in these two situations and the results are overlaid in Figure3.20. Numerically the rms emittance in the two cases agree within uncertainty, indicating thatwe should expect the same transmission efficiency to be achievable with the introduction ofthe SEL. Initial tests bypassing the MR-ToF to move ions directly into MPET support thisconclusion.We also note that the additional focusing of the lens and y-direction steering provide newdegrees of freedom to improve ion transmission further. At present tests have not yet beenperformed to optimize ion optical parameters to maximize transmission using the SEL.583.3. Summary3.3 SummaryThrough this chapter we have examined the transport of ions into and out of the MR-ToFas well as bypassing the MR-ToF. To accomplish this study, simulations were made in SIMIONof the TITAN beamline from the CB to the MR-ToF or to MPET.The simulations of the CB were validated through comparison of previous simulations,reproducing results reported in [68] within 35% or less. The cooler buncher simulations werealso compared to measurements made with an Allison meter in the position where the MR-ToFis located. The cooler buncher simulations predicted an emittance of 7.5 ± 0.2 µm comparedto the measured emittance of 13.24± 0.09 µm, giving a measured to simulated emittance ratioof 1.7. This same ratio was observed through cooler buncher simulations described in [68] andmeasurements described in [66].Further comparisons of simulations to experiments were performed through measuring thebeam time profiles at two MCPs in the TITAN beamline. At the first MCP the ToF FWHMwas found to be 62% larger in simulation than experimentally, whereas at the later MCP therewas a 13% disagreement. One source of disagreement is the different phase space of cooledions simulated in the CB. Another cause is due to the beam passing close to, and sometimescolliding with, the edges of the beamline electrodes. This difference would be made greater bythe discrepancy between simulated and experimental beam emittance. The closer agreement islikely achieved by the collimation effect of the aperture TSYBL:DPA (shown in Figure 3.1).Use of experimental electrostatic ion optical parameters in simulation produces high iontransmission in simulation. This transmission requires ions pass to through TSYBL:DPA forwhich differences in bender voltages of a few volts can completely stop ion transmission. Thusthe electrostatic portion of the simulation appears to reflect experiment well.For future use, the choice will need to be made between defining the initial phase spaceof the ion beam passing through the beamline using the CB simulation or empirically definedproperties. The CB simulation allows users to explore effects of different cooling and extractionproperties, but tests have shown it not able to reliably predict experimental results better thanwithin a factor of 2. Using experimental measurements of the beam it is possible to empiricallydefine initial beam properties in simulations, but this will be limited to simulating the beam asit was produced by the CB at the time of measurement.Acceptance of the MR-ToF input optics was examined and a set of suggested optic voltageswere determined which provided near 100% ion transmission into the MR-ToF in simulation.These voltages provided a basis for those chosen in experiment, for which similar settings werefound to allow successful ion transport into the MR-ToF.Simulations were done of the beam ejected from the MR-ToF towards MPET to determineoptimal electrode voltages. Suggested settings were determined to optimize beam transportefficiency and quality for measurements of isobar separated beam in MPET and EBIT. Exper-imental verification is pending. Additional ion optical elements are suggested to provide moredegrees of freedom for further optimization of MR-ToF output.Finally, the impact of implementing a split Einzel lens in the TITAN beamline and anaperture in the B1-out electrode was examined. Simulations showed the rms emittance beforeand after the introduction of these optics to agree within uncertainty. This indicates that we canexpect the same ion transmission towards MPET and EBIT to be achievable while bypassingthe MR-ToF as was the case before the addition of the MR-ToF. The lack of impact on iontransmission is reflected experimentally. The addition of the SEL also introduces degrees offreedom for further improvement to ion transmission when bypassing the MR-ToF.With this examination of the TITAN beamline and the interface of the MR-ToF with the593.3. SummaryTITAN beamline, we may now turn our attention to the first online tests of the MR-ToF itselfto see how it has performed as a part of TITAN.60Chapter 4First tests of MR-ToF integratedinto TITANThe MR-TOF was designed and assembled at the University of Giessen. It was shipped toTRIUMF and set up in an off-line testing area in the ISAC-I experimental hall. The devicewas tested and characterized with an additional ion source [48]. In parallel, the simulationsand optics discussed in the previous chapter were prepared. Following the off-line preparationand installation of the additional ion optical elements, the MR-ToF was installed in the TITANbeamline (shown in Figure 4.1).First a pair of gate valves were added to the TITAN beamline at the points of entry toand extraction from the MR-TOF as was the split Einzel lens. These can be seen in Figure4.2. The MR-TOF was installed, then optically aligned vertically with the cooler buncher andhorizontally with the measurement Penning trap. A new high-voltage cage, its 3 kV transformer,and the electrical system were installed. The complete installed MR-TOF system is shown inFigure 4.3.For operation, the control and data acquisition systems had been developed in Giessenand shipped with the device. The control system relied on a software called VCNew, a C-based computer code which allows one to remotely control the optics and vacuum system.The data acquisition system, Massdata-Acquisition (MAc) [79], includes the timing generator,records signals from either an Analog-to-Digital Converter (ADC) or Time-to-Digital Converter(TDC), displays the data in real time, and allows for a simplified on-line analysis. The ADC ispreferred for high count rates as it measures average MCP current rather than individual ions,leading to zero dead time. The TDC is favoured for low count rates ( 1 ion/ns) as it allowsthe background from the detector to be excluded.For this commissioning test, the ISAC OffLine Ion Source (OLIS) [80] was used to producea beam of stable ions for testing the MR-TOF. This provided a reliable source of ions with awell known mass for benchmark purposes. These measurements were intended to demonstratetransport of externally produced beam into the MR-TOF, to estimate transport efficiencies, toperform a mass measurement, and to verify the resolving power. The expected performanceparameters for the MR-ToF prior to construction were that it would achieve R ≥ 20 000 forisobar separtion and R ≥ 100 000 for mass measurements within 10 ms and with a transportefficiency ≥ 20%. These tests and their results are described here.4.1 Tests of MR-ToF with OLIS beamA beam of 40Ar+ was produced in OLIS using the microwave source [81] which produces ionsthrough a microwave discharge plasma. Also ionized are residual gases such as N2. The beamwas delivered at 20 keV to the TITAN cooler buncher (CB). It was then transported to the MR-ToF device at an energy of 1.4 keV. When an isobaric doublet was required, 40K+ was ionizedin the MR-ToF ion source and merged with the OLIS beam in the MR-ToF switchyard. The614.1. Tests of MR-ToF with OLIS beamfrom ISACion sourceto collinearlaser spectroscopyCoolerBuncherEBITCPET MPET(a) Before the MR-ToFfrom ISACion sourceto collinearlaser spectroscopyCoolerBuncherMR-ToF EBITCPET MPETContinuous ISAC beamSingly charged, bunchedHighly chargedIsobar separated(b) With the MR-ToFFigure 4.1: A schematic overview of the TITAN experiment a) before the MR-ToF and b) withthe MR-ToF installed. Arrowed lines are used to indicate paths the ion beam may take toTITAN from the ISAC ion sources, inside TITAN, or from TITAN to the ISAC collinear laserspectroscopy experiment [35].combined beam was then separated and measured in the analyzer. This allowed the performanceof the MR-ToF using beam from ISAC as in online experiments to be studied. The purposewas to measure the device efficiency, to perform mass spectrometry, and to demonstrate theseparation of the 40Ar+ and 40K+ isobars.4.1.1 Ion transport efficiencyThe MR-ToF efficiency was estimated by comparing ion detections on MCPs, one locatedupstream of the buffer-gas-filled RFQ ion guides and another downstream of the analyzer sec-tion. Fluctuations of the ion source varied the count rate up to 30%. Multiple measurementswere performed at different microchannel plate (MCP) bias voltages to ensure that the max-imum detection efficiency was achieved and for a different number of turns in the analyzer.When the beam was transported directly through the analyzer section, that is without anyreflections, the efficiency was 10.08(15)%. This is lower than the expected 20%, however weexpect that greater efficiencies could be achieved when measuring isotopes of elements with ahigher ionization energy than argon. This is because ions of elements with higher ionizationenergies have a larger probability of undergoing charge exchange when interacting with residualimpurities in the helium buffer gas. As the number of turns was increased inside of the ana-lyzer, the efficiency decreased due to collisions with residual gases from 10.08(15)% to 1.31(5)%for 600 turns at maximum resolving power. This loss in efficiency over longer flight times canmost likely be improved by reducing the pressure within the analyzer. Expectations are thata pressure of ∼ 10−8 mbar is a critical pressure at which this loss in efficiency with flight timemostly disappears [58].4.1.2 Mass measurementTo demonstrate the mass measurement capability of the MR-ToF with ISAC beam, mea-surements were performed operating the MR-ToF at a rate of 50 Hz, taking 40Ar+ ions at a624.1. Tests of MR-ToF with OLIS beamMCP DetectorTime-of-Flight AnalyzerMass Range SelectorPreparationTrap SystemTransfer RFQChanneltron DetectorThermal Ion SourceRFQ SwitchyardInput RFQInjection OpticsGate ValveSplit Einzel LensDifferential Pumping SectionEinzel LensPulsed Drift TubeCooler Buncherlocation ofSEL MCPElectrostaticBeamlineGate ValveFaraday CupEjection OpticsOutputRFQDifferential PumpingSectionsFigure 4.2: Schematic of the MR-ToF and its interface with the TITAN beamline. Ion opticalelectrodes are indicated in yellow; the MR-ToF vacuum chamber is highlighted in green; detec-tors are shown in blue; red arrowed lines are used to indicate possible beam paths through theTITAN beamline and MR-ToF. Adapted from [47].634.1. Tests of MR-ToF with OLIS beamFigure 4.3: Photo of the MR-ToF installed and connected to the TITAN beamline.rate ≤ 20 ions/second. This was a sufficiently low rate to record the ion detections with theTDC, meaning measurements were sensitive to single ion events. A series of mass measurementswere performed around mass A = 40, first using the MR-ToF internal ion source to produce Kisotopes 39K+ and 41K+ for calibration. After calibration, mass measurements were performedat increasing ion flight times to demonstrate the ability of the MR-ToF to identify a wide rangeof ions, accurately determine their masses, and resolve isobars at a resolving power greater than100 000.For all mass measurements discussed here time-of-flight (ToF) peaks have been fit withLorentz functions to determine the peak FWHM (∆t) and average ion time of flight measuredat the MR-ToF MCP (t). The full ToF is determined as a combination of the delay after whichToF data starts to be recorded ts and a constant signal delay caused by the electronics andtransit time through the cables t0. From these we determined,ToF = t+ ts + t0. (4.1)With the above we could calculate the mass resolving power as,R =ToF2∆t. (4.2)Mass calibrationIn order to ensure ion identification and mass measurement it is necessary to calibrate theconversion of ToF to mass using known masses. For this purpose, potassium from the MR-ToFinternal ion source was used as a calibrant ion, for which a spectrum is shown in Figure 4.4.In the figure we see ToF peaks for 39K+ and 41K+. 40K+ is not visible due to its low isotopic644.1. Tests of MR-ToF with OLIS beam39K+41K+Figure 4.4: Time-of-flight spectrum of potassium isotopes used to calibrate the MR-ToF massmeasurements.Table 4.1: Literature masses [82] for isotopes of K and Ar used for calibration and comparison.Isotope Mass (u)39K 38.963706487(5)40K 39.96399817(6)41K 40.961825258(4)40Ar 39.9623831238(24)abundance (0.012%). Using these isotopes with known masses a calibration could be definedas,m = C (t− ts − t0)2 . (4.3)Here C is a calibration factor and m is the resulting calculated mass. ts is calculated from thechosen timing signals, but C and t0 must be calculated using two or more known masses. Cneeds to be determined for each ToF spectrum, but t0 can be assumed constant as no cables andelectronics have been changed between measurements. A fit was done of the 39K+ and 41K+peaks shown in the spectrum in Figure 4.4 to determine the corresponding ToF data. Themasses of these isotopes and corresponding uncertainties are found in [82] and shown in Table4.1. Previous measurements of variations in ts have shown it to have an uncertainty ∼ 100 ps[58]. The uncertainty in t was determined as the fit uncertainty of the mean ToF peak value.Thus using the available fit and mass data we calculate a value for t0 of 126.1± 2.4 ns.High resolution mass measurementFigure 4.5 shows a set of mass measurements performed in the MR-ToF ranging from a shortto long ion flight times and the corresponding increase in resolving power. In these exampleswe refer to the process of an ion being reflected by both electrostatic mirrors and returningto an arbitrary start position as a “turn” in the analyzer. For each spectrum the ions wentthrough one time focus shift (TFS) turn to set a constant time focus. After the TFS turn, the654.1. Tests of MR-ToF with OLIS beamTable 4.2: Comparison of the measured mass of 40K+ to the literature mass reported in Refer-ence [82].ToF (ms)40K+ Mass (u) RelativeDifferenceMeasured Literature7.4721(6) 39.964012(36) 39.963998166(6) 4× 10−7ions undergo a number of isochronous turns (IT) to provide the ToF by which the ion massesare resolved and measured.The first spectrum in Figure 4.5 (1 IT) shows a broadband mass measurement performedwith the MR-ToF using a short ToF. We see the 40Ar+ ions from OLIS along with moleculesof ArH+ and a number of contaminant organic molecules. The dominant source of the contam-inants is impurities introduced with the helium buffer gas which are ionized during MR-ToFoperation. Increasing the number of IT and thereby the flight time we see the mass resolutionincreasing. In the 600 IT spectrum the MR-ToF internal ion source had been turned on to add40K+ to the beam, demonstrating the capacity of the MR-ToF to resolve isobars.Ions identified in Figure 4.5 were determined based on their mass calculated using thepreviously found t0 and calibrating C for each spectrum using the40Ar+ peak. The resultsof the mass measurement of 40K+ in the 600 IT spectrum is shown in Table 4.2. This massmeasurement agrees with the literature value within uncertainty[82], with a relative differenceof 4 × 10−7. Here a resolving power of 202 000 has been achieved, which is consistent withresults seen during offline commissioning of the MR-ToF [48] as well as exceeding the expectedperformance by a factor of 2.4.1.3 Isobar separationThe final functionality of the MR-ToF which needed to be tested was its capacity to removeisobaric contaminants from the ion beam through dynamic recapture in the preparation trap,which we call re-trapping here. Again a combination of 40Ar+ from OLIS and 40K+ from theMR-ToF internal ion source were delivered into the MR-ToF analyzer. The ions underwent1 TFS and 256 IT before being ejected from the analyzer for re-trapping. After re-trappinga mass measurement would be performed in the analyzer to see the degree to which isobariccontaminants were removed.To determine an optimal time to re-trap either 40Ar+ or 40K+ a range of re-trapping timeswere tested from 3212.68 to 3212.88 µs. This allowed us to determine the re-trapping time wherethe contaminant isobar is completely removed while still leaving the largest possible amount ofthe ion of interest. For each re-trapping time the absolute number of ions counted by the TDCin each mass peak after 2 minutes was recorded, the results of this are shown in Figure 4.6 a.We can clearly see how changing the re-trapping time allows removal of one mass or the other.Figure 4.6 b shows an overlay of three mass measurement spectra, first with no re-trapping,then with specific re-trapping times to isolate either 40Ar+ or 40K+. This demonstrates thateither of these isobars can be successfully removed from the beam, providing at minimum aresolving power of 25 000, exceeding the expected performance of 20 000.Re-trapping efficiencyFor this example of re-trapping, around 20% of the ions were successfully re-trapped ascompared to a maximum of 35% achieved with ionized caffeine molecules in [46]. This efficiency664.1. Tests of MR-ToF with OLIS beamC3H+340Ar+C3H+4ArH+ C3H+5C3H+340Ar+C3H+4ArH+ C3H+540Ar+40K+Figure 4.5: Three MR-ToF time-of-flight spectra used for mass measurements. Detected ionsare marked in the plots; these ions were identified based on their mass. Time-of-flight massmeasurements for each spectrum were calibrated use the 40Ar+ peak. For the bottom spectrum40K+ was added to the beam using the MR-ToF internal ion source.674.1. Tests of MR-ToF with OLIS beam(a) Testing re-trapping times.(b) Time-of-flight spectra after re-trapping.Figure 4.6: An example of mass-selective re-trapping performed in the TITAN MR-ToF using40Ar+ and 40K+ ions. a) Number of ions detected for 40Ar+ (blue) and 40K+ (red) time-of-flight peaks for different re-trapping times. b) Time-of-flight spectra showing no re-trapping(black), then with different re-trapping times to capture either 40Ar+ (blue) or 40K+ (red) whileremoving the other isobar.684.2. SummaryAxial PositionEnergy m1K1 = 0m2K2 > 0Axial PositionEnergy m1K1 = 0m2K2 > 0Potential WellDepthFigure 4.7: An example of ion motion in relation to axial potential during the process of isobarseparation through re-trapping. Left: Retarding field slowing ions prior to trapping. Right:Shallow potential well applied to trap ion of interest. Ions are represented in red with the ionof interest and contaminant are represented as m1 and m2 respectively. Axial potentials arerepresented as blue lines.will however vary with ion mass and the depth of the potential well used for re-trapping. Inre-trapping a retarding field is formed in the preparation trap until the moment the ion ofinterest has zero kinetic energy at which time the potential is switched to a shallow trappingpotential (Figure 4.7). A shallow well is used so that contaminant ions still have sufficientkinetic energy to escape the trap. However, real ions will have a distribution in phase spacerather than arriving at a single time and energy. The result is that some ions of interest willstill have sufficient kinetic energy to escape a potential well shallow enough to exclude isobariccontaminants, reducing the ion capture efficiency. The well can be made deeper to improvecapture efficiency, but this will also reduce the resolving power of the separation method. Theresult is a trade-off between increased resolving power and capture efficiency [46].Another factor affecting re-trapping efficiency stems from the mass dependence of the ion ofinterest’s speed. Ideally the trap would close instantaneously and ions would experience staticaxial electric fields from either the open or closed potential. However the actual switchingtime is around 100 ns, which results in ions sampling axial electric fields in an intermediatestate between the open and closed state. Faster moving light ions will experience more ofthese intermediate fields as ions moving at higher speed will be further from the trap centrewhen switching begins. Mass A = 40 ions at 1300 eV will move at approximately 80 mm/µs,traversing a distance of 8 mm within 100 ns, comparable to the length of the trap which is 7mm. The ions experience the electric potential dynamically increasing as they are slowed, thusincreasing their kinetic energy in a fashion similar to how a pulsed drift tube decreases the totalion energy. The result is that trapping efficiency of lighter ions will be decreased.4.2 SummaryThe TITAN MR-ToF has now been installed into the TITAN beamline and it was demon-strated to be able to accept ISAC beam, perform mass measurements, and remove isobariccontamination through mass-selective re-trapping. An ion transport efficiency of around 10%was achievable with an 40Ar+ beam transported from the TITAN CB to the MR-ToF. Thisefficiency is below the expected 20%, though we expect higher efficiencies to be achievable whentransporting elements with a lower ionization energy. The MR-ToF demonstrated a mass re-694.2. Summarysolving power of 202 000 while measuring 40Ar+ and 40K+, exceeding expectation by a factorof 2. Also the MR-ToF was shown to be able to remove either of these two ion species throughmass-selective re-trapping, requiring a minimum resolving power of 25 000; above the expectedresolving power of 20 000. The mass measurement and isobar separation resolving powers aregreater than the performance expectations and both were achieved within the required 10 ms.With the MR-ToF functioning as part of the TITAN system, as well as beam transport prop-erties discussed in Chapter 3, the MR-ToF is ready for use with radioactive beam from ISACto contribute to the scientific work of TITAN.70Chapter 5Summary and outlookThe TITAN experiment at TRIUMF has had many years of successful Penning trap use. Theincoming beam was prepared using a linear Paul trap and an EBIT for charge breeding to enablehigh precision mass measurements of rare isotopes. Now an additional type of ion trap hasbeen added to TITAN to expand the possibilities for measurements, a Multi-Reflection Time-of-Flight (MR-ToF) mass spectrometer and isobar separator. Through this thesis the processof integrating an MR-ToF into the TITAN experiment has been discussed. This has involvedstudies of how to maximize ion transport into and out of the MR-ToF and demonstrating thecapacity of the MR-ToF to perform mass measurement and isobar separation as part of theTITAN system.A key component of the MR-ToF integration was understanding how to maximize theefficiency of ion transport through the TITAN beamline to, from, or bypassing the MR-ToF.We also desired to optimize beam emittance out of the MR-ToF for measurements using EBITand MPET. To this end a set of SIMION simulations were produced of the TITAN beamlineto study and optimize the movement of ions through the beamline. The starting point of thesesimulations was the TITAN cooler buncher (CB) which is a linear Paul trap used to preparebeam for measurements in TITAN. CB simulations were validated first through comparison toearlier CB simulations, then through comparison to experimental emittance measurements. AnAllison meter was used to measure the transverse emittance of the beam ejected from the CBto the location of the MR-ToF input optics. Comparing transverse emittance measurements,the experimental measurement was found to be 1.7 times larger than simulation. Furthercomparisons between simulation and experiment were found through measuring and simulatingthe beam time profiles at two MCPs in the TITAN beamline. Disagreements as large as afactor of 2 have been observed in the time profiles. It appears that the ion phase space of thecooled ions simulated in the CB is different than what is achieved in experiment, possibly dueto space charge effects experienced by ions in the CB which were not simulated. For future useof these simulations it may be more effective to empirically define the initial phase space to havecloser agreement between simulated and experimental emittance. An additional impediment toproducing closer agreement between simulation and experiment is the interaction of ions withstructures in the beamline. Ion trajectories passing close to beamline structures are sensitiveto minor differences between simulation and experiment, likely contributing to disagreementbetween simulation and experiment. Again, this may merit a more empirical approach tosimulating the initial ion phase space within the CB for future simulation work. Notablyhowever, settings on electrostatic beamline elements reproduced in simulation show high iontransmission, even where differences in bender voltages < 1% would completely prevent iontransmission. This appears to indicate a useful simulation of the electrostatic beamline insimulation. Thus an understanding was gained of how precisely these simulations could guideexperimental beamline optimizations.Using the beamline simulations, an exploration of optimal electrode settings for ion injectioninto the MR-ToF was performed. The MR-ToF injection ion optics were recreated in simulationto study the acceptance of the optics. The acceptance was simulated for a range of voltages71Chapter 5. Summary and outlookon two key electrodes to find optimal settings for matching the beam emittance from theTITAN CB. To improve efficiency of ion transport into the MR-ToF through matching the CBemittance to the MR-ToF acceptance a new split Einzel lens (SEL) was added to the TITANbeamline. Simulations were done of the TITAN beamline between the CB and MR-ToF inputoptics to find optimal settings of lenses EL5 and SEL for injection of a 1.4 keV beam intothe MR-ToF. Simulations predict 95% efficiency of beam transport into the MR-ToF fromthe CB. Similar voltages to those suggested were used for successful ion transport into theMR-ToF in experiment. In addition to injection into the MR-ToF, simulations were used toprovide suggested voltages for beam transport from the MR-ToF towards MPET. A maximumtransmission of 19% was achieved for the variables adjusted. It may be possible to improvethis transmission and the related phase space distribution by adjusting the depth of the trappotential well. Also we may explore the impact of introducing switching the DC voltage on theoutput RFQ electrodes.Additional simulations were used to investigate the impact of MR-ToF integration on theTITAN beamline when bypassing the MR-ToF. To allow transport of ions into the MR-ToFthe SEL was added to the TITAN beamline and a 7.9 mm diameter hole was created in theB1-OUT electrode; simulations were used to characterize the impact of these modifications.This was accomplished by examining the transverse emittance at a position past the sectionwhere the modifications were made. Comparisons show the calculated emittance before andafter the changes agree within uncertainty and the emittance plots appear nearly identical. Thelack of negative impact from the new optics has been supported by experimental observations.The final step in integrating the MR-ToF into the TITAN system was the physical instal-lation and tests of the MR-ToF using beam delivered from ISAC. These tests were performedusing an 40Ar+ beam produced in the ISAC OffLine Ion Source (OLIS) to measure beam trans-port efficiency and demonstrate mass measurements as well as isobar separation in the MR-ToF.Efficiency measurements found it possible to achieve 10% transport efficiency of 40Ar+ beamfrom the CB to MR-ToF, lower than the anticipated 20%. A major factor in the loss of effi-ciency is due to charge exchange with contaminant ions in the helium buffer gas. We expecthigher efficiencies can be achieved using elements with lower ionization energies which have alower probability of charge exchange.Mass measurements were performed for a range of ion flight times. One function was toperform broadband mass measurements to identify various beam constituents. A high resolutionmass measurement was also performed, demonstrating a resolving power of 202 000, exceedingthe expected 100 000 by a factor or 2. Isobar separation was demonstrated using mass-selectivere-trapping. For this test a mixture of 40Ar+ and 40K+ was fed into the MR-ToF, then variousre-trapping times were studied to show either isobar can be removed. This isobar separationshowed the MR-ToF achieving at minimum a resolving power of 25 000, an improvement over theanticipated 20 000. Thus it was demonstrated that the MR-ToF can function as an integratedcomponent of the TITAN system and work with externally produced beam delivered from ISAC.Since online commissioning the MR-ToF has already begun to show its utility in TITAN.The MR-ToF was used as a complementary mass spectrometer for measurements of neutron-rich titanium measurements done in MPET. The MR-ToF was able to provide diagnostics of thebeam constituents as well as measure isotopes arriving with too high a contamination level formeasurement in MPET. 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In: Chinese Physics C 41.3 (2017), p. 030003.78Appendix AIntroduction to SIMIONSIMION is a software package able to numerically calculate electric and magnetic fieldsand the resulting trajectories of charged particles flying through these fields [67]. By defaultthese fields are simulated as static, however dynamic changes may be simulated through userprogramming implemented via scripts in the programming language Lua 5.1. In this thesis,Lua scripts were used to implement time-varying effects in the simulations such as switching ofRF fields and other electrode potentials, as well as buffer-gas collisions. Scripts were also usedto vary simulation parameters such as electrode voltages, gas pressure, and different timingparameters. The varying of simulation parameters was achieved by inputting a list of values tobe used for a given parameter.Electric fields are calculated through solving the Laplace equation based on a defined elec-trode geometry and potentials. Magnetic fields were not simulated for this thesis, so we shallneglect a discussion of how they are rendered. Geometries are defined by filling in points in athree dimensional grid, wherein each point is either part of an electrode, grounded, or emptyspace. The resolution with which a given geometry is modelled depends on the length per gridunit defined for each linear direction (x, y, and z). Resolutions are expressed as grid units permm (gu/mm). A major limiting factor in the resolutions which can be achieved is the computermemory requirements to produce high resolution simulations. These electrode geometries andthe resulting Laplace solutions are saved in Potential Array (PA) files.Where electrode geometries are simple, measurements could be taken directly from drawingsand models of the beamline to define the geometries within SIMION. The direct definition ofsimulation geometries is done using geometry or GEM files. The commands available to defineGEM files are limited in their scope, making it difficult to render complex geometries. Thus formore complex structures it is desirable to define an electrode model in some external software,such as SolidWorks [78], which can be imported into SIMION.If a geometry has been saved as a STereoLithography (STL) model, it is possible to convertthe geometry into a PA. This conversion of STL models was used—for example—in defining thespherical benders in the TITAN beamline, which were significantly more difficult to reproduceusing the the tools available in SIMION. STL models do however have the limitation thatsymmetry planes cannot be defined to reduce the computational load of simulating the PA.Conversions of STL models are also prone to small aberrations in the geometries which canproduce unphysical inhomogeneities in the resulting electric fields. Such inhomogeneities wereobserved in some beamline simulations where the ion beam was passing straight through sectionsconverted from STL models and experienced steering effects where there should be none. Insuch cases, simplified GEM file versions of the electrodes were defined to simulate scenarioswhere the beam is not being steered.79Appendix BExperimental operating parametersfor emittance and time profilemeasurements80Appendix B. Experimental operating parameters for emittance and time profile measurementsTable B.1: A summary of the voltages applied to the TITAN beamline electrodes for thetime-profile measurements discussed in Section 3.1.3. Positions of the electrodes in the TITANbeamline are indicated in Figure 3.1.Beamline Section Electrode Set Voltage (V)TRFCRFQ18 4.8RFQ19 4RFQ20 3.2RFQ21 2.4RFQ22 1RFQ23 -14RFQ24 20RFQ22 (extraction) 1RFQ24 (extraction) -21Bias 20000PB5 17600EL5 -3600TRFCBLXCB0 100CCB0 100B1-IN 334B1-OUT 388Q1 250Q2 250XCB4 105YCB4T 222YCB4B -222TSYBLEL1 2100EL3 2780CCB2 250XCB9 262YCB9 250EL4 2350MPETBLEL2 1770CCB3 250XCB3 254YCB3 25881Appendix CAllison meter data analysismethodologyTo summarize the process of converting the raw time profile data into ion counts for aparticular Allison meter setting, let us make reference to the summary of data presented inFigure 3.5. We shall refer to the number of counts in an individual time profile bin as Nij ,where i corresponds to the Measurement Index in Figure 3.5 and j refers to the time profilebin. Specific time intervals were chosen to cover time profile data from the Allison meterMCP, MCP-1, and a sample of the background counts. We represent the start and end indicesof the Allison meter MCP data as Al0–Al1. Then, expressing the normalization for a givenmeasurement as νi and the mean of the background counts as〈NBG〉i, we may approximatethe real ion counts at the Allison meter MCP Nˆi as,Nˆi = νiAl1∑j=Al0(Nij −〈NBG〉i). (C.1)With the number of counts for a given Allison meter position and voltage determined, wefinally combine this with the meter position and the divergence calculated with Equation 3.1to produce a plot of the emittance. Without the normalization and background subtractionof ion counts the result is Figure C.1 a, but with this processing we instead get Figure 3.6 b.We see in Figure C.1 c the result of rescaling the z-axis of the emittance plot to highlight thefluctuations in the background noise of the data. The processing of ion counts removes most ofthe noise, but there remains a fluctuation in the background ≤ 2% of the maximum ion count.In order to calculate the rms emittance we need to ensure that we are not changing the resultsby counting of background noise.To correct for the background counts in the rms emittance calculation a threshold wasset under which any counts would be set to zero. The level of this threshold was chosen byexamining how the calculated rms emittance varies as a function of where the threshold levelwas set, as shown in Figure C.2. The threshold was calculated as a multiple of the standarddeviation of a section of background far away from the real ion counts to allow easy adaptationto different levels of background variations. For the example here we can see the effect thatthere are more negative background counts than positive, which reduces the rms emittancecalculation to zero for a threshold below ≈-1.5 (we note that the value would actually beimaginary due to taking a square root of a negative value, but the key point here is that theresult of a low threshold is a calculation which does not reflect physical reality). The calculatedemittance approaches a maximum as the threshold goes to zero due to negative backgroundcounts being removed while the effect of positive background noise remains. We can see wherethe inflating effect from the background is removed when the rms emittance decreases steadilyas a function of threshold. In the example of Figure C.2, this inflating effect disappears at athreshold of ≈ 3 background standard deviations. For higher threshold values we see a slow,but steady decrease in calculated emittance due to real counts being removed by the threshold.82Appendix C. Allison meter data analysis methodology(a) (b)(c)Emittance:rms: 13.24± 0.9 µmInflation factor: 1.09Twiss Parameters:α: −4.65± 0.10 pi−1β: 3.57± 0.08 mpi−1γ: 6.34± 0.14 (pim)−1(d)Figure C.1: Emittance plots from one measurement of a 133Cs+ beam out of the TITAN CBwith standard operating parameters (described in text), showing different levels of processingof the number of ions detected at each position and divergence. a) An emittance plot withno processing of the ion counts, showing a large biasing effect due to drift in the MCS. b)An emittance plot with ion count normalization and time profile background removed, butbefore a threshold was set to remove background counts for rms emittance calculation. c) Thesame emittance plot as b), but scaled in the z-axis to highlight the degree of variation of thebackground counts. Background variations on the level of 2% of the peak ion counts can beseen. d) An emittance plot after the complete ion-count processing described in the text. Awhite ellipse calculated from the Twiss parameters is overlaid on the plot with the size set bythe equivalent emittance 4rms. Calculated rms emittance and Twiss parameters printed below.83Appendix C. Allison meter data analysis methodology;Figure C.2: A plot of the calculated rms emittance as a function of the threshold for settingbackground noise to zero.A complication introduced by zeroing all counts below the threshold is that some quantity ofreal ion counts will be removed along with the noise, thereby artificially lowering the emittancecalculated. To estimate this effect of lowering the calculated emittance we shall introduce aninflation factor, ifac. This inflation factor is the ratio of total ion counts to ion counts removedby the threshold, assuming a bi-Gaussian trace space distribution. To calculate the inflationfactor we will define the following terms,pfac =peakthresh, (C.2)where peak is the peak number of counts in the emittance distribution, and thresh is the setthreshold value in counts. We can then calculate the inflation factor as,ifac =pfacpfac− (1 + log (pfac)) . (C.3)In order to exclude data with excessive noise a standard was set that emittance calculationsyielding an inflation factor greater than 1.5 would be rejected. Above this inflation factor itcould be that more than 33% of the real ion counts have been removed, calling into questionthe reliability of the results. With the inflation factor calculated, we expect that the actualemittance will be somewhere between the calculated rms and ifac · rms.The final result of this analysis for beam from a standard tune of the TITAN CB is shownin Figure C.1 d.The implementation of this analysis was done through a Lua script, shown in Appendix D.The methodology is adapted from that described in Reference [66] which in turn was based onMATLAB code developed for beam analysis use at TRIUMF [72].84Appendix DAllison meter data analysis Lua codelocal inputFilename = "4129"local csvInput = string.format("%s.csv", inputFilename)local outputFilename = inputFilenamelocal calFile = string.format("%scal", inputFilename)cal = require(calFile)local setSecBounds = cal.setSecBounds -- boolean todecide whether or not to use manually defined section boundslocal sectionIndex0, sectionIndex1 = cal.sec0, cal.sec1 -- sets bounds ofwhere in dataset measurement is happening (minimum is 0)local trBound0, trBound1 = 0,2040 -- sets boundson integration of tracelocal alliPk0, alliPk1 = cal.alliPk0, cal.alliPk1 -- sets trace indexboundaries on Allsion MCP peaklocal normPk0, normPk1 = cal.normPk0, cal.normPk1 -- sets trace indexboundaries on MCP-1 normalisation peaklocal bkGnd0, bkGnd1 = cal.bkGnd0, cal.bkGnd1 -- sets trace indexboundaries background-- Conversion and range parameters for interpreting raw Allison meter data to givephysical resultslocal pos0, pos1, posDelta = cal.pos0, cal.pos1, cal.posDelta -- sets position boundsand step sizelocal volt0, volt1 = cal.volt0, cal.volt1 --sets voltage boundslocal ampRatio = cal.ampRatio -- ratio by which soft daq voltage is multipliedby amplifier to give readout voltagelocal KE = cal.KE -- beam energylocal voltCol = cal.voltCol -- sets column to reference for Allisonmeter voltage used (mainly to choose between softdac or readback)local trStart = cal.trStart -- column where trace data starts (in casethere are other variables in the data)local voltInvert = cal.voltInvert -- input 1 or -1 - this is to invert the voltageto account for plates being reversed (can also apply global multiplicative factor,but probably don’t need that...)local posCal = 1.0 -- calibration on position [mm/step]local Leff = 69.85 -- length of Allison meter plates [mm]local gap = 4.0 -- gap between Allison meter plates[mm]local x_err = 0.0032*posCal -- [mm] (stepper motor resolution of 200steps/turn)85Appendix D. Allison meter data analysis Lua codelocal V_errR = 5.5e-5local KE_err = 4.0local Leff_errR = 0.021local gap_errR = 0.01local equivEmit = 4 -- factor determines size of plottedemittance ellipse by calculating equivalent emittance from rms-emittance with thisfactor-- Set which functions to run:local _writeEmitPlotData = true -- Function to write emittance plotdata and produce plotlocal tMinX,tMaxX = cal.tMinX,cal.tMaxX -- x boundaries onbackground region for thresholdinglocal tMinXP,tMaxXP = cal.tMinXP,cal.tMaxXP -- x’ boundaries onbackground region for thresholdinglocal threshold = cal.threshold -- [BG SD] thresholdunder which all data is cut off, set in units of the background standarddeviationlocal lab1x,lab1y = 0.05,0.92 -- Graph position of label givingcalculated rms-emittance resultslocal gpExec = "gnuplot" -- gnuplot executable (needed for migrating code to other computers)local csvLines = {}local parsedData = {}--[[General utility functions]]---- Function to check if two numbers are the same within some tolerancefunction equalish(number,target,tolerance)if tonumber(number) > target-tolerance and tonumber(number) < target+tolerancethenreturn trueelsereturn falseendend-- Function to round a number to the nearest integer valuefunction round(number)if number - math.floor(number) < 0.5 then return math.floor(number)else return math.ceil(number) endend-- Sum all numbers in list from index "start" to index "finish"function integrate(list,start,finish)sum = 0for i = start, finish dosum = sum + tonumber(list[i])endreturn sumend-- Function to calculate average of a listfunction average(list)86Appendix D. Allison meter data analysis Lua coderesult = 0if #list ~= 0 thenfor i=1, #list doresult = result + tonumber(list[i])endreturn result/#listelsereturn 0endend-- Calculates average from a list. Code acquired from rosettacode.org - 2017-02-16function median (numlist)if type(numlist) ~= ’table’ then return numlist endtable.sort(numlist)if #numlist %2 == 0 then return (numlist[#numlist/2] + numlist[#numlist/2+1]) / 2endreturn numlist[math.ceil(#numlist/2)]end-- Function to calculate standard deviation listfunction stdDev(list)mean = average(list)dev2 = {}for i = 1, #list do dev2[i] = (list[i] - mean)*(list[i] - mean) endreturn math.sqrt(average(dev2))end--[[Parsing the csv data]]---- Read csv data into list of stringslocal i = 1local f = assert(io.open(csvInput,’r’))if f thenfor line in f:lines() docsvLines[i] = linei = i+1endelseerror("File went bye bye. :(")endf:close()-- Break up strings at commas to isolate individual elements (probably should be partof previous part, but, ugh)for i=1, #csvLines doparsedData[i] = {}local j=1for token in string.gmatch(csvLines[i], "([^,]+),%s*") doparsedData[i][j] = tokenj = j + 1endend--[[Analysing the parsed data]]---- Calculate rms-emittancefunction rmsEmit(sec0,sec1,trBound0,trBound1,parsedData,thresh,maxPk)87Appendix D. Allison meter data analysis Lua codelocal points = {x = {}, xp = {}, xp_err2 = {}, Nx = {}, Nxp = {}, N = {}, nu ={}, Nnorm = {}, ABG = {}, BGSD = {}, NAl = {}}local norm = 0 -- integrate over first normalization peak for referencelocal BGSD_table = {}-- Take raw parsed data and assign it to variables for calculationslocal n = 1for i=sec0, sec1 dolocal bkGnd,BGSD,normRatio = 0,0,1.0local bkGndList = {}for j=1, bkGnd1-bkGnd0 do bkGndList[j] = parsedData[i][trStart+bkGnd0-1+j] endbkGnd = average(bkGndList)points.ABG[n] = bkGnd -- save average background for error propagation-- calculate list of variations from mean if mean background is beingsubtractedbkGndList = {}for j=1, bkGnd1-bkGnd0+1 do bkGndList[j] = (parsedData[i][trStart+bkGnd0-1+j]-bkGnd)^2 end-- calculate standard deviation from list of variations for errorpropagationpoints.BGSD[n] = math.sqrt(integrate(bkGndList,1,#bkGndList)/#bkGndList)if i == sec0 then norm = integrate(parsedData[sec0],trStart+normPk0,trStart+normPk1) - bkGnd*(normPk1-normPk0+1) endif (integrate(parsedData[i],trStart+normPk0,trStart+normPk1) - bkGnd*(normPk1-normPk0+1)) < 1 thennormRatio = 0 -- norm -- set counts to zero where normalizationfalls below the backgroundelsenormRatio = norm/(integrate(parsedData[i],trStart+normPk0,trStart+normPk1) - bkGnd*(normPk1-normPk0+1))endpoints.nu[n] = 1/(integrate(parsedData[i],trStart+normPk0,trStart+normPk1) - bkGnd*(normPk1-normPk0+1)) -- saving normalization ratiofor error propagationif points.nu[n] ~= points.nu[n] then print(points.nu[n]) end-- shows if normalization = "nan"points.Nnorm[n] = integrate(parsedData[i],trStart+normPk0,trStart+normPk1) -- saving normalization peak counts for error propagationpoints.NAl[n] = integrate(parsedData[i],trStart+alliPk0,trStart+alliPk1) -- saving Allison peak counts for error propagationif points.Nnorm[n]<1 then points.Nnorm[n] = 1 end -- prevent divide byzero errorsbkGnd = normRatio*bkGnd -- apply normalization to averagebackground for foregoing calculations88Appendix D. Allison meter data analysis Lua codepoints.x[n] = parsedData[i][2] * posCalpoints.xp[n] = parsedData[i][voltCol] * 250*Leff/(KE*gap)*voltInvertpoints.xp_err2[n] = points.xp[n]*points.xp[n] * ((V_errR*parsedData[i][5]+4e-6*(volt1-volt0))^2 + (Leff_errR)^2 + (KE_err/KE)^2 + (gap_errR)^2)if normRatio*integrate(parsedData[i],trStart+alliPk0,trStart+alliPk1) -bkGnd*(alliPk1-alliPk0+1) > thresh thenpoints.Nx[n] = parsedData[i][2]*posCal * (normRatio*integrate(parsedData[i],trStart+alliPk0,trStart+alliPk1) - bkGnd*(alliPk1-alliPk0+1))points.Nxp[n] = parsedData[i][voltCol]*250*Leff/(KE*gap)*voltInvert * (normRatio*integrate(parsedData[i],trStart+alliPk0,trStart+alliPk1) - bkGnd*(alliPk1-alliPk0+1))points.N[n] = normRatio*integrate(parsedData[i],trStart+alliPk0,trStart+alliPk1) - bkGnd*(alliPk1-alliPk0+1)elsepoints.Nx[n] = 0points.Nxp[n] = 0points.N[n] = 0endn = n + 1endlocal tot = integrate(points.N,1,#points.N)local avgX = integrate(points.Nx,1,#points.Nx) / totlocal avgXP = integrate(points.Nxp,1,#points.Nxp) / totlocal dX2 = {}; for i=1, #points.x do dX2[i] = points.N[i]*(points.x[i] - avgX)^2 end -- variance of xlocal sdX2,avg_dX2 = integrate(dX2,1,#dX2), integrate(dX2,1,#dX2) / tot-- save sum and averagelocal dXP2 = {}; for i=1, #points.xp do dXP2[i] = points.N[i]*(points.xp[i] -avgXP)^2 end -- variance of x’local sdXP2,avg_dXP2 = integrate(dXP2,1,#dXP2), integrate(dXP2,1,#dXP2) / tot-- save sum and averagelocal dXdXP = {}; for i=1, #points.x do dXdXP[i] = points.N[i]*(points.x[i] -avgX)*(points.xp[i] - avgXP) end -- covariancelocal sdXdXP,avg_dXdXP = integrate(dXdXP,1,#dXdXP), integrate(dXdXP,1,#dXdXP)/ tot -- save sum and average of covariance termslocal temp = avg_dX2*avg_dXP2 - avg_dXdXP^2if temp < 0 then temp = 0 end -- prevent sqrt(negative) errorlocal emittance = math.sqrt(temp) -- pi mm mradlocal emitErr = 0local avg_dX2_err,avg_dXP2_err,avg_dXdXP_err = 0,0,0function errorCalc()function calcN_err(i,N) -- function for calculating error in Nlocal nBGSD = 1 -- multiplicative factor on backgrounderror (not currently in use)if points.N[i] > thresh then --> threshreturn points.nu[i]*math.sqrt( ((normPk1-normPk0+1)*N-(89Appendix D. Allison meter data analysis Lua codealliPk1-alliPk0+1))^2*points.ABG[i]/(bkGnd1-bkGnd0+1)+ points.NAl[i] + N^2*points.Nnorm[i] + (0.02*N/points.nu[i])^2)elsereturn 0endendif emittance > 0 thenlocal sN_term = 0 -- summing over ( Ni_err/(2*N*emit) * [avgX2*(dXP2-avgXP2) + avgXP2*(dX2-avgX2) - 2*avgXXP*(dX*dXP-avgXXP) ] )^2local sx_term = 0 -- summing over ( xi_err /(emit)*Ni/N * [dX*avgXP2 - dXP*avgXXP ] )^2local sxp_term = 0 -- summing over ( xpi_err/(emit)*Ni/N * [dXP*avgX2 - dX*avgXXP ] )^2for i=1, #points.x doif points.nu[i] > 0 then -- and points.N[i] > threshlocal Ni = points.N[i]/normsN_term = sN_term + ( calcN_err(i,Ni)*norm/(2*tot*emittance) * ( avg_dX2*((avgXP-points.xp[i])^2-avg_dXP2) + avg_dXP2*((avgX-points.x[i])^2-avg_dX2) - 2*avg_dXdXP*((avgX-points.x[i])*(avgXP-points.xp[i])-avg_dXdXP) ) )^2sx_term = sx_term + ( Ni*x_err*norm/(tot*emittance) * ( (avgX-points.x[i])*avg_dXP2 - (avgXP-points.xp[i])*avg_dXdXP ) )^2sxp_term = sxp_term + ( Ni*points.xp_err2[i]*norm/(tot*emittance) * ( (avgXP-points.xp[i])*avg_dX2 - (avgX-points.x[i])*avg_dXdXP ) )^2endendemitErr = math.pi*math.sqrt(sN_term + sx_term + sxp_term)elseemitErr = 0endavg_dX2_err,avg_dXP2_err,avg_dXdXP_err = emitErr/emittance*avg_dX2,emitErr/emittance*avg_dXP2,emitErr/emittance*avg_dXdXPenderrorCalc()if thresh < 1e-12 then thresh = 1e-12 endlocal pfac = maxPk/threshlocal ifac = pfac/(pfac-(1+math.log(pfac)))local pfac_err = pfac/math.sqrt(maxPk)local ifac_err = math.abs( math.log(pfac)*pfac_err/(1-pfac+math.log(pfac))^2 )print("rms-emittance: " .. math.pi*emittance .. " +/- " .. emitErr .. " um")print("Inflation factor: " .. ifac .. " +/- " .. ifac_err)return math.pi*emittance,emitErr, ifac, avgX,avgXP, avg_dX2,avg_dXP2,avg_dXdXP, avg_dX2_err,avg_dXP2_err,avg_dXdXP_err -- [um]end90Appendix D. Allison meter data analysis Lua code-- Function to write emittance plot datafunction writeEmitPlotData(sectionIndex0,sectionIndex1,pos0,pos1,posDelta,parsedData,filename)-- Process datalocal sec0,sec1 = 2, #parsedData-- default range ofmeasurements to analyseif setSecBounds then sec0,sec1 = sectionIndex0+2,sectionIndex1+2 end --optional custom range of indices to analyselocal position = pos0local points = {x = {}, xp = {}, N = {}}local BG1 = 0for i=sec0, sec1 dolocal bkGnd,BGSD,normRatio = 0,0,1.0local bkGndList = {}for j=1, bkGnd1-bkGnd0+1 do bkGndList[j] = parsedData[i][trStart+bkGnd0-1+j] endbkGnd = average(bkGndList)if i == sec0 then BG1 = bkGnd endlocal norm = (integrate(parsedData[sec0],trStart+normPk0,trStart+normPk1) - BG1*(normPk1-normPk0+1))-- integrate over firstnormalization peak for referenceif (integrate(parsedData[i],trStart+normPk0,trStart+normPk1) -bkGnd*(normPk1-normPk0+1)) < 1 thennormRatio = 0 --normelsenormRatio = norm/(integrate(parsedData[i],trStart+normPk0,trStart+normPk1) - bkGnd*(normPk1-normPk0+1))endlocal counts = 0counts = round(normRatio*(integrate(parsedData[i],trStart+alliPk0,trStart+alliPk1) - bkGnd*(alliPk1-alliPk0+1)))if equalish(parsedData[i][2],position,0.1) thenpoints.x[i-sec0+1] = position*posCalpoints.xp[i-sec0+1] = parsedData[i][1]*250*Leff/(KE*gap)*ampRatio*voltInvertpoints.N[i-sec0+1] = countselseif tonumber(parsedData[i][2]) < position-0.01*posDelta theni = i+1elseposition = position + posDeltapoints.x[i-sec0+1] = position*posCalpoints.xp[i-sec0+1] = parsedData[i][1]*250*Leff/(KE*gap)*ampRatio*voltInvertpoints.N[i-sec0+1] = countsendend91Appendix D. Allison meter data analysis Lua code-- Calculate standard deviation of selected background region and identifypeak countslocal peakMax = 0local BGList = {}local n=1for i=1, #points.N doif points.x[i] >= tMinX and points.x[i] <= tMaxX and points.xp[i] >=tMinXP and points.xp[i] <= tMaxXP thenBGList[n] = points.N[i] -- gather counts in selectedbackground region for calculating SDn = n+1endif peakMax < points.N[i] then peakMax = points.N[i] endendlocal BGSD = stdDev(BGList)-- Write data to filelocal f = assert(io.open(string.format("%s_emit.txt", filename),’w’))f:write(string.format("#Position (mm)\tDivergence (mrad)\tNormalizedCounts\n"))if points.N[1] < threshold*BGSD thenf:write(string.format("%f\t%f\t%d\n", points.x[1],points.xp[1],0))elsef:write(string.format("%f\t%f\t%d\n", points.x[1],points.xp[1],points.N[1]))endfor i=2, #points.x doif points.N[i] < threshold*BGSD thenif points.x[i] > points.x[i-1]+0.2*posDelta thenf:write(string.format("\n%f\t%f\t%d\n", points.x[i],points.xp[i],0))elsef:write(string.format("%f\t%f\t%d\n", points.x[i],points.xp[i],0))endelseif points.x[i] > points.x[i-1]+0.2*posDelta thenf:write(string.format("\n%f\t%f\t%d\n", points.x[i],points.xp[i],points.N[i]))elsef:write(string.format("%f\t%f\t%d\n", points.x[i],points.xp[i],points.N[i]))endendendf:close()-- Calculate rms-emittancelocal emittance,emitErr, ifac, avgX,avgXP, x2,xp2,xxp, x2_err,xp2_err,xxp_err= rmsEmit(sec0,sec1,alliPk0,alliPk1,parsedData,threshold*BGSD,peakMax)endif _writeEmitPlotData then writeEmitPlotData(sectionIndex0,sectionIndex1,pos0,pos1,92Appendix D. Allison meter data analysis Lua codeposDelta,parsedData,outputFilename) end93

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