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UBC Theses and Dissertations

Dawning of nuclear magicity in N = 32 seen through precision mass spectrometry Leistenschneider, Erich 2019

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DAWNING OF NUCLEAR MAGICITY IN N = 32 SEENTHROUGH PRECISION MASS SPECTROMETRYbyErich LeistenschneiderA THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinThe Faculty of Graduate and Postdoctoral Studies(Physics)The University of British Columbia(Vancouver)October 2019c© Erich Leistenschneider, 2019The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the dissertation entitled:Dawning of Nuclear Magicity in N=32 Seen Through Precision Mass Spectrometrysubmitted by Erich Leistenschneider in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin Physics.Examining Committee:Jens Dilling, Physics and AstronomySupervisorGordon W. Semenoff, Physics and AstronomySupervisory Committee MemberJohn Behr, Physics and AstronomySupervisory Committee MemberTakamasa Momose, ChemistryUniversity ExaminerAlison Lister, Physics and AstronomyUniversity ExaminerKumar S. Sharma, The University of ManitobaExternal ExaminerAdditional Supervisory Committee Member:Reiner Kruecken, Physics and AstronomySupervisory Committee MemberiiABSTRACTIn the early days of nuclear science, physicists were astounded that specific ”magic”combinations of neutrons or protons within nuclei seemed to bind together moretightly than other combinations. This phenomenon was related to the formation ofshell structures in nuclei. More recently, nuclear shells were observed to emerge orvanish as we inspect nuclei further from stability. The structural evolution of thesechanging shells has been the object of intense experimental investigation, and theirbehavior has become a standard ruler to benchmark theoretical predictions.In this work, we investigated the emergence of shell effects in systems with 32neutrons (N = 32) using mass spectrometry techniques. Evidence for ”magicity”was observed in potassium (with 19 protons, or Z = 19), calcium (Z = 20) andscandium (Z = 21), but not in vanadium (Z = 23) and higher-Z elements. Inbetween, the picture at titanium (Z = 22) was unclear.We produced neutron-rich isotopes of titanium and vanadium through nuclearreactions at the ISAC facility and measured their atomic masses at the TITAN facil-ity, in the TRIUMF Laboratory in Vancouver. These measurements were performedwith the newly commissioned Multiple-Reflection Time-of-Flight Mass Spectrome-ter at TITAN facility and were substantiated by independent measurements fromthe Penning trap mass spectrometer. The atomic masses of 52−55Ti and 52−55Visotopes were measured with high precision, right at the expected emergence ofN = 32 shell effects. Our results conclusively establish the existence of weak shelleffects in titanium and confirm their absence in vanadium.Calculations of the N = 32 nuclear shell are within reach of the so-called abinitio theories. In these, complex atomic nuclei are described theoretically fromfundamental principles, by applying principles of Quantum Chromodynamics tomany-body quantum methods. Our data were compared with a few state-of-the-artab initio calculations which, despite very successfully describing the N = 32 shelleffects in Ca and Sc isotopes, overpredict its strength in Ti and erroneously assignV as its point of appearance. We hope the deficiencies revealed by our work willguide the development of the next generation of ab initio theories.iiiLAY SUMMARYProtons and neutrons inside the atomic nucleus organize themselves in structuresthat resemble the shells of an onion. Specific combinations of protons or neutronsare known to form closed shells, which grant remarkable stability to the nucleusand leave imprints on its mass and other properties. However, these shells mayharden or soften if the number of neutrons is too different from the number ofprotons.We observed the ”hardening” of a nuclear shell formed by 32 neutrons by pre-cisely measuring the masses of rare nuclei. We discovered that this shell does notform when the nucleus has 23 protons or more but is weakly present with 22 pro-tons. The extreme imbalance between protons and neutrons required for shell for-mation makes this case a good test bench for nuclear theories. We tested the predic-tions of a few modern theories, which slightly differ from our results. We hope ourwork will guide future theoretical developments.ivPREFACEThe material presented in this thesis is the result of the collaborative work involv-ing many people. The TITAN facility at TRIUMF Laboratory has been in operationsince 2007. It has been maintained by the TITAN Scientific Collaboration which hasmembers from several Canadian universities and has partners from many interna-tional institutions. Many of the techniques and devices employed in this thesis weredeveloped over the years by several members of the collaboration. In the following,I list all the individual contributions relevant to the work presented in this thesis:• From Summer 2016 to Summer 2019, I lead the working group of the Mea-surement Penning Trap (MPET) mass spectrometer. I was responsible for itsmaintenance, operation, preparation for experiments, and upgrades.• The planning and coordination of the experiment herein described was doneby me and M. P. Reiter.• The preparation and operation of the MPET mass spectrometer before and dur-ing the experiment was done by me, R. Steinbru¨gge and A. A. Kwiatkowski.• The online commissioning of the Multiple Reflection Time-of-Flight Mass Spec-trometer (MR-ToF-MS), as well as its preparation and operation before and dur-ing the experiment, was done by M. P. Reiter, S. Ayet San Andre´s, C. Hornung,C. Will and A. Finlay.• The data collection was performed with the support of B. Kootte, C. Babcock,B. R. Barquest, J. Bollig, E. Dunling, L. Graham, R. Klawitter, Y. Lan, D. Lascar,J. E. McKay and S. F. Paul.• Laser ionization of titanium using the TRILIS ion source was set up by J.Lassen.• The theoretical calculations were performed by J. D. Holt, P. Navra´til, C. Barbi-eri, H. Hergert, A. Schwenk, J. Simonis, V. Soma` and S. R. Stroberg.• The analysis of the MPET data was done independently by me and B. Kootte.The analysis of the MR-ToF-MS data was done independently by M. P. Reiterand S. Ayet San Andre´s.• I organized the scientific discussions following the experimental and theoreti-cal results with all the collaborators.A letter describing the main part of the work, containing the experimental resultswith titanium isotopes, the theoretical calculations and the discussion of scientificimplications, was prepared by me and published inE. Leistenschneider et al.,- Dawning of the N = 32 Shell Closure Seen through PrecisionMass Measurements of Neutron-Rich Titanium Isotopes - Physical Review Letters120, 062503 (2018).vA second article containing further experimental details and results with vana-dium isotopes was published inM. P. Reiter et al.,- Quenching of the N = 32 neutron shell closure studied via preci-sion mass measurements of neutron-rich vanadium isotopes - Physical Review C 98,024310 (2018).Appendix A describes a proof-of-principle experiment on the Decay and Recap-ture Ion Trapping technique and subsequent data analysis and simulations. Theexperiment was performed with R. Klawitter, M. Alanssari, J. C. Bale, B. R. Bar-quest, U. Chowdhury, A. Finlay, A. T. Gallant, B. Kootte, D. Lascar, K. G. Leach,A. Lennarz, A. J. Mayer, D. Short and A. A. Kwiatkowski. The data analysis wasperformed by me and R. Klawitter. The simulations presented were performed byme. The content of the appendix will form a future publication.Appendix B presents the motivation and the design of the cryogenic upgrade toTITAN’s Penning trap mass spectrometer. The motivation material presented in thefirst two sections is my original work and was previously published inE. Leistenschneider et al.,- Vacuum requirements for Penning trap mass spectrometrywith highly charged ions - Nuclear Instruments and Methods in Physics ResearchSection B: Beam Interactions with Materials and Atoms 0168-583X (2019).The simulations and preliminary concept studies are also my original work. Thedesign of the upgraded system was made by me and M. Good based on the originalMPET design by V.L. Ryjkov. Performance tests and assembly of the system wereperformed together with M. Lykiardopoulou, C. Izzo, R. Steinbru¨gge, J. L. Tracy Jr.,J. E. McKay, D. Fusco and M. Vansteenkiste. The unpublished part of this appendixwill also form a future publication.viTABLE OF CONTENTSAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 A Brief History of the Nuclear Force . . . . . . . . . . . . . . . . . . . . 21.2 Atomic Nuclei and Mass Observables . . . . . . . . . . . . . . . . . . . 41.3 The Nuclear Shell Model in a Nutshell . . . . . . . . . . . . . . . . . . 91.4 Not-so-Magic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.1 The New Magic Number 32 . . . . . . . . . . . . . . . . . . . . . 111.4.2 A Test Bench for Nuclear Theories . . . . . . . . . . . . . . . . . 141.5 Scientific Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Ab Initio Nuclear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Construction of a Microscopic Nuclear Theory . . . . . . . . . . . . . . 182.2 Many-Body Quantum Methods . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Harmonic Oscillator Basis . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Reference States and Valence Space . . . . . . . . . . . . . . . . 212.3 Nuclear Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.1 Nuclear Forces from Phenomenological Approaches . . . . . . 222.3.2 Nuclear Forces from Chiral Effective Field Theory . . . . . . . 232.3.3 Interaction Softening . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Tests of Ab Initio Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.1 Tested Forces and Methods . . . . . . . . . . . . . . . . . . . . . 293 Principles of Mass Spectrometry . . . . . . . . . . . . . . . . . . . . . . . 303.1 International Standards and the Atomic Mass Evaluation . . . . . . . . 333.2 Penning Trap Mass Spectrometry . . . . . . . . . . . . . . . . . . . . . . 343.2.1 Confinement and Ion Motion in a Penning Trap . . . . . . . . . 35vii3.2.2 Manipulation of Ion Motion in a Penning Trap . . . . . . . . . . 37Dipole Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . 38Quadrupole Excitations . . . . . . . . . . . . . . . . . . . . . . . 403.2.3 Frequency Measurement Techniques in Penning traps . . . . . 41ToF-ICR: Initial Magnetron Motion . . . . . . . . . . . . . . . . 42ToF-ICR: Quadrupole Excitation . . . . . . . . . . . . . . . . . . 42ToF-ICR: Extraction and Detection . . . . . . . . . . . . . . . . . 453.2.4 Practical Considerations of the ToF-ICR Technique . . . . . . . 473.3 Multiple Reflection Time-of-Flight Mass Spectrometry . . . . . . . . . 483.3.1 Concept of the MR-ToF-MS . . . . . . . . . . . . . . . . . . . . . 49Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Isochronicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.2 Operation as a Mass Spectrometer . . . . . . . . . . . . . . . . . 52Time Focus Matching . . . . . . . . . . . . . . . . . . . . . . . . 54Temporal Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 56Mass Range Selector . . . . . . . . . . . . . . . . . . . . . . . . . 563.3.3 Practical Considerations of the MR-ToF-MS Technique . . . . . 574 Experimental Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1 ISAC Facility and Isotope Production . . . . . . . . . . . . . . . . . . . 59Production of Radioisotopes . . . . . . . . . . . . . . . . . . . . 59Release and Ionization . . . . . . . . . . . . . . . . . . . . . . . . 61Mass Separation and Delivery to Experiments . . . . . . . . . . 624.2 The TITAN Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.1 Preparation of Ion Beams with the TITAN RFQ . . . . . . . . . 64Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Bunching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2.2 The Measurement Penning Trap . . . . . . . . . . . . . . . . . . 664.2.3 The TITAN MR-ToF-MS Spectrometer . . . . . . . . . . . . . . . 68Transport System . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Mass Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2.4 Transport and Optimization of Ion Beams . . . . . . . . . . . . 714.3 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3.1 Initial Beam Assessment with the MR-ToF-MS . . . . . . . . . . 744.3.2 Mass Measurement Procedure: MR-ToF-MS . . . . . . . . . . . 754.3.3 Mass Measurement Procedure: MPET . . . . . . . . . . . . . . . 775 Measurements and Data Analysis . . . . . . . . . . . . . . . . . . . . . . 785.1 Mass Measurements with the MR-TOF-MS . . . . . . . . . . . . . . . . 785.1.1 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1.2 Evaluation of Systematic Errors . . . . . . . . . . . . . . . . . . 825.1.3 Relativistic Corrections . . . . . . . . . . . . . . . . . . . . . . . 835.1.4 Final Mass Values . . . . . . . . . . . . . . . . . . . . . . . . . . 84viii5.1.5 The case of 56Ti . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.2 Mass Measurements with the MPET . . . . . . . . . . . . . . . . . . . . 865.2.1 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2.2 Evaluation of Systematic Errors . . . . . . . . . . . . . . . . . . 895.2.3 Relativistic Corrections . . . . . . . . . . . . . . . . . . . . . . . 905.2.4 Final Mass Values . . . . . . . . . . . . . . . . . . . . . . . . . . 915.3 Low-Lying Isomers and Ground-State Assignment . . . . . . . . . . . 916 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.1 Comparison Between Spectrometers . . . . . . . . . . . . . . . . . . . . 936.2 Comparison with Previous Measurements . . . . . . . . . . . . . . . . 946.2.1 Titanium Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2.2 Vanadium Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . 976.2.3 Other Isotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.3 Updated Isotopic Chains and Evolution of the N=32 Shell Closure . . 996.3.1 Signatures of N=32 Shell Effects in Titanium . . . . . . . . . . . 1026.3.2 Signatures of N=32 Shell Effects in Vanadium . . . . . . . . . . 1026.3.3 Evolution of the N=32 Shell Closure . . . . . . . . . . . . . . . . 1036.4 Tests of Ab Initio Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142A Decay and Recapture Ion Trapping . . . . . . . . . . . . . . . . . . . . . 143a.1 In-EBIT DRIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144a.2 Simulations of Daughter Beam Creation . . . . . . . . . . . . . . . . . . 145a.3 Experiment Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149a.4 Population Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149a.4.1 Optimal Storage Time for DRIT . . . . . . . . . . . . . . . . . . 150a.4.2 Charge State Evolution . . . . . . . . . . . . . . . . . . . . . . . 151a.5 Identification of Daughter Beam through PTMS . . . . . . . . . . . . . 152a.5.1 Demonstrative Experiment . . . . . . . . . . . . . . . . . . . . . 153a.6 Conclusions and Applications . . . . . . . . . . . . . . . . . . . . . . . . 154B The Cryogenic Upgrade to the TITAN MPET . . . . . . . . . . . . . . . 156b.1 Vacuum Requirements for PTMS of HCI . . . . . . . . . . . . . . . . . 158b.2 HCI as Vacuum Probes and the Pressure at MPET . . . . . . . . . . . . 160b.3 Vacuum Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162b.3.1 Monte-Carlo Algorithm . . . . . . . . . . . . . . . . . . . . . . . 162b.3.2 Results: Current System . . . . . . . . . . . . . . . . . . . . . . . 165b.3.3 Results: Upgraded System . . . . . . . . . . . . . . . . . . . . . 166ixb.4 The Cryogenic Measurement Penning Trap . . . . . . . . . . . . . . . . 167b.4.1 Concept of Cryopumping . . . . . . . . . . . . . . . . . . . . . . 167b.4.2 Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . 171b.4.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175xL IST OF TABLESTable 4.1 Characteristic dimensions of the MPET. . . . . . . . . . . . . . 66Table 5.1 Mass excesses and half-lives of the isotopes measured by theTITAN MR-ToF-MS. . . . . . . . . . . . . . . . . . . . . . . . . . 85Table 5.2 Frequency ratios, atomic mass excesses and half-lives of thespecies measured by the TITAN MPET. . . . . . . . . . . . . . 91Table 6.1 Mass differences between atomic masses of Ti isotopes andtheir MR-ToF-MS calibrants, using MPET, MR-ToF-MS andAME16 values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Table 6.2 Combined TITAN measurements of the atomic masses of Tiand V isotopes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Table A.1 Atomic masses of 30Al and 30Mg measured with the MPET. . 154Table B.1 Characteristic dimensions of the CryoMPET. . . . . . . . . . . 173xiL IST OF F IGURESFigure 1.1 Chart of nuclides showing the observed nuclei and thosewhose properties have been calculated through ab initio nu-clear methods up to 2017. . . . . . . . . . . . . . . . . . . . . . 4Figure 1.2 The binding energy per nucleon calculated through the Bethe-Weizsa¨cker formula. . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.3 Two-neutron separation energies around N = 50. . . . . . . . 7Figure 1.4 Empirical shell gaps across the nuclear chart. . . . . . . . . . . 8Figure 1.5 Signatures of magic numbers seen in charge radii and excita-tion energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 1.6 Shell structure obtained through the IPM. . . . . . . . . . . . . 10Figure 1.7 Excitation energies showing signatures of magicity in N =20, N = 28 and N = 32. . . . . . . . . . . . . . . . . . . . . . . 12Figure 1.8 Two-neutron separation energies showing signatures of magic-ity in N = 28 and N = 32. . . . . . . . . . . . . . . . . . . . . . 13Figure 1.9 Shell evolution of N = 32 isotones showing the appearanceof magicity due to the weakening of a proton-neutron resid-ual interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.1 Matrix dimensionality required for the computation of a fewN = Z nuclei using the NCSM method and the limits ofmodern computers. . . . . . . . . . . . . . . . . . . . . . . . . . 20Figure 2.2 Microscopic construction of a closed-shell reference state andof an excited state. . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 2.3 The 1S0 nucleon-nucleon potential derived from Lattice QCD,the AV18 potential and the one-pion exchange potential. . . . 23Figure 2.4 Diagram representation of terms in chiral expansion of nu-clear forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 2.5 Representation and example of an SRG evolution of 2N po-tentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.6 Comparisons of ab initio nuclear theories on the descriptionof binding energies of oxygen isotopes. . . . . . . . . . . . . . 28Figure 2.7 Two-neutron separation energy of neutron-rich Ca isotopes,comparing experimental data to theories that include 3N forces. 28Figure 3.1 A mass spectrum showing resolved and unresolved species. . 31Figure 3.2 Ion motion in a hyperbolic Penning trap. . . . . . . . . . . . . 35Figure 3.3 Ion motion in a Penning trap projected in the radial plane. . . 38Figure 3.4 A four-way segmented ring electrode and its configurationsto create dipole and quadrupole fields in a Penning trap. . . . 39Figure 3.5 Evolution of magnetron radius during a dipole excitation. . . 39xiiFigure 3.6 Conversion of a magnetron motion into a reduced cyclotronmotion by a quadrupole excitation. . . . . . . . . . . . . . . . . 40Figure 3.7 Periodicity of the interconversion between magnetron andred. cyclotron motions by a quadrupole excitation. . . . . . . 41Figure 3.8 Off-axis ion injection in a Penning trap using a Lorentz Steerer. 42Figure 3.9 The RF signal in a standard quadrupole excitation and thegain in kinetic energy introduced to the ion motion. . . . . . 44Figure 3.10 The RF signal in a two-pulse Ramsey quadrupole excitationand the gain in kinetic energy introduced to the ion motion. 44Figure 3.11 Depiction of the ion’s time-of-flight dependency on the en-ergy of the radial motion in the trap. . . . . . . . . . . . . . . . 45Figure 3.12 Examples of standard and Ramsey ToF-ICR cyclotron fre-quency measurements. . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.13 The electrode structure of a typical MR-ToF-MS mass ana-lyzer and its electrostatic potential. . . . . . . . . . . . . . . . . 50Figure 3.14 Scheme of an accelerating Einzel lens. . . . . . . . . . . . . . . 51Figure 3.15 Simulated ion trajectory in an MR-ToF-MS analyzer. . . . . . . 51Figure 3.16 Depiction of the creation of a time focus. . . . . . . . . . . . . 52Figure 3.17 Simulated time-energy relation of an ion cloud through oneturn in a mass analyzer. . . . . . . . . . . . . . . . . . . . . . . 53Figure 3.18 An example of an MR-ToF-MS spectrum. . . . . . . . . . . . . 54Figure 3.19 Depiction of a measurement cycle in an MR-ToF-MS. . . . . . 55Figure 3.20 Example of the impact of a time-resolved calibration on thequality of a time-of-flight spectrum. . . . . . . . . . . . . . . . 55Figure 3.21 Illustration of the ambiguity in the mass range of an MR-ToF-MS and the impact of a mass range selector. . . . . . . . . . . 57Figure 3.22 MR-ToF-MS and ToF-ICR PTMS techniques compared regard-ing resolution and measurement times. . . . . . . . . . . . . . 58Figure 4.1 Representation of the RIB production facility at ISAC and thetarget and ion source module employed in this experiment. . 60Figure 4.2 Simulated yields of radioisotopes produced by ISAC in theconditions of this experiment. . . . . . . . . . . . . . . . . . . . 60Figure 4.3 Schematic overview of the TITAN facility. . . . . . . . . . . . . 63Figure 4.4 The electrode structure of the TITAN RFQ and its generatedelectrical potential. . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 4.5 The MPET electrode structure. . . . . . . . . . . . . . . . . . . 66Figure 4.6 The electrode setup of MPET and its associated ion optics, aswell as a map of the axial magnetic field strength. . . . . . . . 67Figure 4.7 Scheme of the TITAN MR-ToF-MS. . . . . . . . . . . . . . . . . 68Figure 4.8 Electrode structure of the TITAN MR-ToF-MS mass analyzer. 70Figure 4.9 Sketch of the electric potential and beam energy along thebeam transport path to MPET and MR-ToF-MS. . . . . . . . . 72Figure 4.10 Optics map of TITAN beamline. . . . . . . . . . . . . . . . . . 73xiiiFigure 4.11 Automatic optimization of beam transport to MPET. . . . . . 74Figure 4.12 Laser-identification of titanium peaks in MR-ToF-MS spectra. 75Figure 4.13 Optimization of interest-to-contaminant ratio using the ISACMass Separator aided by the TITAN MR-ToF-MS. . . . . . . . 76Figure 5.1 A sample MR-ToF-MS spectrum is shown for each of the mea-sured RIBs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 5.2 Example of the peak-fitting procedure with Gaussian andLorentzian peak shapes. . . . . . . . . . . . . . . . . . . . . . . 81Figure 5.3 Relative mass differences between the values measured bythe MR-ToF-MS and in the AME16. . . . . . . . . . . . . . . . . 83Figure 5.4 Reduction in count-rate of titanium observed after turningTRILIS laser off. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 5.5 Sample ToF-ICR resonances for each of the species measuredwith the MPET. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 5.6 Linear interpolation of reference measurements to determinethe cyclotron frequency of the ion of interest. . . . . . . . . . . 89Figure 5.7 Relative mass differences between the values measured bythe MPET and in the AME16. . . . . . . . . . . . . . . . . . . . 90Figure 6.1 MPET and MR-ToF-MS measurements compared. . . . . . . . 94Figure 6.2 All mass measurements of 51Ti compared. . . . . . . . . . . . . 95Figure 6.3 All mass measurements of 52Ti compared. . . . . . . . . . . . . 95Figure 6.4 All mass measurements of 53Ti compared. . . . . . . . . . . . . 96Figure 6.5 All mass measurements of 54Ti compared. . . . . . . . . . . . . 96Figure 6.6 All mass measurements of 55Ti compared. . . . . . . . . . . . . 96Figure 6.7 Recent mass measurements of 51V compared. . . . . . . . . . . 96Figure 6.8 Recent mass measurements of 52V compared. . . . . . . . . . . 98Figure 6.9 All mass measurements of 53V compared. . . . . . . . . . . . . 98Figure 6.10 All mass measurements of 54V compared. . . . . . . . . . . . . 98Figure 6.11 All mass measurements of 55V compared. . . . . . . . . . . . . 98Figure 6.12 MPET measurements of other species compared to AME16. . 99Figure 6.13 MR-ToF-MS mass measurements of other species comparedto the AME16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Figure 6.14 The mass landscape around Ti and V isotopes before andafter TITAN measurements. . . . . . . . . . . . . . . . . . . . . 100Figure 6.15 Two-neutron separation energies around N = 32 after TITANmeasurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 6.16 Empirical neutron-shell gaps around N = 28 and N = 32after TITAN measurements. . . . . . . . . . . . . . . . . . . . . 105Figure 6.17 Evolution of empirical neutron-shell gaps across isotonic chainsof N = 28 and N = 32. . . . . . . . . . . . . . . . . . . . . . . . 105Figure 6.18 Results of ab initio calculations compared to experimental val-ues of the mass landscape around Ti and V isotopes. . . . . . 107xivFigure 6.19 Results of ab initio calculations compared to experimental val-ues of empirical neutron-shell gaps around N = 28 and N = 32.109Figure 6.20 Evolution of ∆2n across isotonic chains of N = 28 and N = 32as calculated by the VS-IMSRG technique. . . . . . . . . . . . . 109Figure A.1 Schematic overview of the TITAN EBIT. . . . . . . . . . . . . . 145Figure A.2 Flow diagram of the algorithm to simulate daughter beamcreation and its confinement in the TITAN EBIT. . . . . . . . . 147Figure A.3 The decay scheme of the 30Mg → 30Al → 30Si chain and itsrecoiling energy distributions. . . . . . . . . . . . . . . . . . . . 148Figure A.4 Simulated population evolution of radioactive ions trappedin EBIT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Figure A.5 Time-of-flight spectra of ions released from EBIT towards theMCP, which allow m/q identification of trapped species. . . . 150Figure A.6 RIB population count rates reaching the MCP as a functionof storage time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Figure A.7 Charge state evolution of the RIB population in EBIT as afunction of storage time. . . . . . . . . . . . . . . . . . . . . . . 152Figure A.8 A ToF-ICR measurement of 30Al11+ measured with MPET. . . 153Figure A.9 Radioactive ion beams available at the ISAC facility at TRI-UMF, either developed or potentially accessible through DRIT. 155Figure B.1 Degradation of quality in a ToF-ICR measurement of HCIsdue to ion-gas interactions. . . . . . . . . . . . . . . . . . . . . 157Figure B.2 Path lengths of ion motion during ToF-ICR measurements. . . 159Figure B.3 Collision and charge exchange cross sections between Rb+qion and N2 neutral molecule. . . . . . . . . . . . . . . . . . . . 159Figure B.4 Gas pressure required for η < 0.1 in a ToF-ICR measurementof 74Rb+37. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Figure B.5 Ratio of charge-exchanged 133Cs+13 ions during ToF-ICR mea-surements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Figure B.6 RGA analysis of the vacuum in the MPET beamline. . . . . . 162Figure B.7 Survival time of simulated particles in the vacuum system. . . 164Figure B.8 Flow diagram of the algorithm employed to simulate the vac-uum in the MPET system. . . . . . . . . . . . . . . . . . . . . . 164Figure B.9 Simulated pressure in the MPET beamline. . . . . . . . . . . . 165Figure B.10 Simulated pressure in the MPET beamline with the additionof ideal cryopumps. . . . . . . . . . . . . . . . . . . . . . . . . . 166Figure B.11 A model of the CryoMPET upgrade. . . . . . . . . . . . . . . . 167Figure B.12 Saturation pressure curves of common gases. . . . . . . . . . . 169Figure B.13 Scheme of CryoMPET’s pumping by cryosorption. . . . . . . . 170Figure B.14 Scheme of CryoMPET’s trap construction. . . . . . . . . . . . . 174Figure B.15 Pictures of CryoMPET’s electrode structure. . . . . . . . . . . 176Figure B.16 Pictures of the trap and shield assemblies. . . . . . . . . . . . 176Figure B.17 Pictures of the assembled CryoMPET system. . . . . . . . . . . 176xvACRONYMS2N or NN 2-body (or Nucleon-Nucleon) interactions3N 3-body Nuclear interactionsAME16 Atomic Mass Evaluation - 2016 editionARIEL Advanced Rare Isotope LaboratoryBNG Bradbury-Nielsen GateCERN European Organization for Nuclear ResearchχEFT Chiral Effective Field TheoryChPT Chiral Perturbation TheoryCPET Cooler Penning TrapCryoMPET Cryogenic Measurement Penning TrapCSRe Cooler Storage RingDRIT Decay and Recapture Ion Trapping techniqueEBIT Electron Beam Ion TrapEFT Effective Field TheoryFT-ICR Fourier Transform Ion Cyclotron Resonance techniqueFWHM Full Width at Half MaximumGGF self-consistent Gorkov-Green’s FunctionHCI Highly Charged IonsHO Harmonic OscillatorIMS Isochronous Mass SpectrometryIMSRG In-Medium Similarity Renormalization GroupIPM Independent Particle Model (also known as Non-Interacting Shell Model)ISAC Isotope Separator and ACcelerator facilityISOL Isotope Separation On-Line methodISOLDE Isotope Separator On Line DEtector facilityITER International Thermonuclear Experimental ReactorLEBIT Low-Energy Beam and Ion Trap facilityLO Leading OrderMCP Micro-Channel Plates detectorMPET mass Measurement PEnning TrapxviMR-IMSRG Multi-Reference In-Medium Similarity Renormalization GroupMR-ToF-MS Multiple-Reflection Time-of-Flight Mass SpectrometerMRS Mass Range SelectorNCSM No-Core Shell ModelNLO Next-to-Leading OrderNNLO or N2LO Next-to-Next-to-Leading OrderOPEP One-Pion Exchange PotentialPI-ICR Phase-Image Ion Cyclotron Resonance techniquePLT Pulsed Drift TubePTMS Penning Trap Mass SpectrometryQCD Quantum ChromodynamicsQED Quantum ElectrodynamicsRF Radio-FrequencyRFQ Radio-Frequency Quadrupole cooler and buncherRG Renormalization GroupRGA Residual Gas AnalyzerRIB Radioactive Ion BeamRRR Residual Resistive RatioSCI Singly Charged IonsSMILETRAP Stockholm-Mainz Ion Levitation TrapSRG Similarity Renormalization GroupTFS Time Focus ShiftTITAN TRIUMF’s Ion Trap for Atomic and Nuclear scienceToF-ICR Time-of-Flight Ion Cyclotron Resonance techniqueToF-MS Time-of-Flight Mass SpectrometryTOFI Time-of-Flight Isochronous spectrometerTRC Time-Resolved CalibrationTRILIS TRIUMF’s Resonance Ionization Laser Ion SourceUHV Ultra-High VacuumVS-IMSRG Valence Space In-Medium Similarity Renormalization GroupxviiACKNOWLEDGEMENTSIn Portuguese, the word we use to express gratitude is obrigado. It is cognate tothe English obligated: it implies recognition but also the establishment of a moreprofound commitment between the two parts. It is a graceful, warm, and thoughtfulway to thank. And with that word, I sway the tone of my next few words.In the preface, I have formally recognized the academic input of each personinvolved in the work of this thesis. But it is not enough. It ignores personal nuancesand bonds required to forge such work. It disregards the friendship, the leadership,the conflict, the celebration, the anguish, the apprenticeship inherent of any humanteam. And, more than ever, scientific research is a collective business. We mustnever fail to acknowledge the importance of healthy teams to the success of Science.Through those years, our team was led by Jens and Ania. Each at their ownstyle, both advised me through the program — Jens as my academic advisor, Aniaas the captain of the group’s daily operations. I thank them for the guidance, theinspiration, the trust, and the freedom. Above all, I must acknowledge their willand of other TITAN principal investigators to foster a strong team, a safe learningenvironment, and a friendly atmosphere.Aaron and Ania taught me their art of taming ions and performing mass spec-trometry with Penning traps. And, with Julia, Rene´, Marilena, James, and Chris,we were able to run the MPET mass spectrometer smoothly through all these years.I also acknowledge Renee, Pascal, Carla, Kyle, Dan, and all those smart and ex-perienced young researchers who shared their expertise on the other ion traps atTITAN. I thank Mel for sharing his wisdom and providing creative solutions toour problems. Working alongside all those brilliant people deeply refined me as aprofessional.I thank the companionship of all my group mates, Brian, Stefan, Eli, Ruben,Annika, the Andrews, Brad, Yang, Matt, Leigh, Jeff, Usman, Devin, Ish, Roshani,Abhilash, Killian, Zach, Tobias, Jon, Rio, and everyone mentioned previously. Ithank you for all the mundane and scientific discussions, the laughs, the good andbad coffee, the unconditional support, and the joys of being surrounded by you all.I especially thank John, Danny, and Mike for trusting my guidance on their projects.I have learned a lot from our experience.I also acknowledge TRIUMF and UBC staff and scientists for their support. Inparticular, I thank Jason, Sonia, and the members of my advisory committee: John,Reiner, and Gordon. They have shared their time, their expertise, taught me lessons,and had a genuine interest in my success. I also thank my colleagues back home,who encouraged me to pursue this program and kept encouraging me at a distance;with special thanks to Alinka, my former supervisor and friend.xviiiI thank the affection shared with Bruno, Mateus, Carol, Acza, and Rogerio.Through these years, we sew our fabrics into one piece, taking caring of each otherand fighting against homesickness. And above all, I must thank the love and sup-port of my family, specially Taı´s and my parents, Mario and Emili.I must also acknowledge the invisible work of those who invest in Science andadvocate for it. This work was partially funded by Brazil’s Conselho Nacionalde Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq) through the Science WithoutBorders Program (grant 249121/2013− 1).Obrigado implies the acknowledgment of a moral bond between the two parts. Iexpect this bond will be translated into the commitment to, wherever our careerpaths go, support research to our best extents, advocate for Science in our commu-nities, help to build better academia, and foster healthy minds among our peers.But for now, to all of those that made this research possible, please, feel embraced.xix1 INTRODUCTIONPhysical Sciences’ ultimate ambition is to understand nature through the preciselanguage of mathematics. Therefore, increments on the predictive power of theoriesthrough enhanced calculation tools and refined models is the fundamental productof physicists’ work.One remarkable example of the outcomes of such endeavor is the theory referredto as Quantum Electrodynamics (QED), the quantum field theory of the electro-magnetic force. It is currently known as the most successful theory in the field ofphysics, providing accurate predictions of humankind’s most precise measurementsof physical phenomena to date [1, 2, 3].The outstanding predictive power of QED is a result of earlier developments ofelectromagnetism and quantum mechanics, but it is not the only result. Nowadays,the modeling of complex molecular processes derived from the very first princi-ples of the same scientific framework is widely and routinely applied in chemistry,molecular and solid-state physics, pharmaceutical sciences, material developmentand nanotechnology [4, 5, 6].Nuclear science, the field describing phenomena involving the ”strong” force, stilldoes not have the same fortune. After more than one hundred years of studying theatomic nucleus, we are still seeking a theory that explains the behavior of all nuclearmatter. Theoretical frameworks derived from fundamental principles, also knownas ab initio (i.e. ”from the beginning”, in Latin), are already a reality in nuclear sci-ences. However, they are far from having a similar success as in electromagnetism.If developed, they could have an enormous scientific and technological impact. Notonly by enabling the development of reliable models for applications, such as inmedical imaging, radiopharmaceuticals, energy generation and radioactive wastemanagement, but also accurate understanding of the behavior of nuclear mattermay fill critical missing pieces in our understanding of the origin of elements [7], ofthe origins of life [8, 9] and of other fundamental subatomic interactions [10].The story of this thesis is the story of another piece in our ongoing collective questto paint such an unerring mathematical portrait of the atomic nucleus. I describeour efforts to provide precision mass spectrometry data of very rare isotopes thatare of high value to understand particular nuclear structure behaviors. The newdata is used to evaluate the performance of a few state-of-the-art nuclear ab initiotheories. Ultimately, we give guidance on which approaches are best suited toexplain the data and where they must be improved.1In this introductory chapter, I start with a short overview of the research onnuclear interactions and how we arrive at the current methods of describing nuclearmatter from first principles. Then, I motivate the need for mass spectrometry dataand its intimate relationship with the study of the nuclear structure, focusing onthe specific behaviors that are relevant to this study. Finally, I conclude the chapterstating our research goals and achievements, and by giving an overview of thestructure of the remainder of the thesis.1.1 A Brief History of the Nuclear ForceIn 1911, E. Rutherford and his team impinged a gold foil with alpha particles andobserved a tiny fraction of them bouncing back. This observation led to an impor-tant interpretation of the structure of an atom: that most of its mass and all of itspositive charge is concentrated in a diminute fraction of its volume [11]. This struc-ture located at the core is called the nucleus of an atom, and this experiment wasconsidered the birth of nuclear science.Concurrently, studies in chemistry and on the nature of radioactivity had alreadyshown that several chemical elements would present themselves in different atomicweights [12], always approximately a multiple of the mass of a hydrogen atom. Thedifferent mass species of the same chemical element were called different isotopes.By 1920, it had been postulated that the nucleus inside of all isotopes would beformed by combinations of two ”fundamental” nuclear particles [13], or nucleons.One was the nucleus of the hydrogen atom, called proton, which was a positivelycharged particle that defined the chemical properties of the atom. The other wasa neutral counterpart of similar mass: the neutron. The discovery of the neutronin 1932 [14] marked the first triumph of this hypothesis and, nowadays, a proton-neutron model [15] of the atomic nucleus forms the basis of nuclear science.Within such a model, however, the system consisting of protons and neutronscould not be stable based only on electrical forces. Protons would repel each otherwhile the neutrons, transparent to electric fields, would drift away. Another forcewas needed to describe the interactions among nucleons. It needed to have thefollowing features: be attractive, independent of electric charge, stronger than theelectromagnetic force and short ranged, since no influence of it could be felt outsideof the nucleus.One of the first postulations for such nuclear force was made by H. Yukawa in 1934[16]. He proposed that the nuclear potential (VYukawa) arising from a nucleon had asimilar form to the Coulomb electric potential, but it was strictly attractive with anexponential “screening” factor to modulate its range:VYukawa(r) = −g2 e−µ rr, (1.1)2where r is the distance from the generating particle, g is a coupling constant and µis a screening factor known as Yukawa mass. The addition of this screening factorimplied the existence of a “mediating” particle of mass around 100 MeV/c2, laterdiscovered by C. Lattes and colleagues [17] and named pion.Subsequent studies, mainly arising from nucleon-nucleon scattering experiments,revealed further details about the properties of the nuclear force and proposed up-dated versions for the nucleon-nucleon potential [18]. Some of the found remark-able behaviors are the dependence on the intrinsic angular momentum (or spin)of the nucleons [19, 20] (e.g. spin-orbit coupling and pairing), the dependence onthe spin orientation [21] (tensor nature) and even a slight breaking of the chargeindependence [22, 23].The inner mechanisms of the nuclear force were finally unveiled in 1969. A seriesof electron-proton scattering experiments revealed clear evidence of internal struc-ture in the proton at energies above 1 GeV [24, 25]. The proton lost its status as afundamental particle, giving way to a quark model [26, 27] to explain the structure ofprotons, neutrons, pions and many other particles that had already been discoveredby that time.The quark model described protons and neutrons as composite particles of threequarks of types up (u) and down (d): uud for protons and udd for neutrons. Thepion, on the other hand, is formed by a ud pair. Within the theory of QuantumChromodynamics (QCD), those particles are bound together by the action of theso-called strong interaction. In very general terms, each quark carries one of threetypes of color charge (hence chromo), an analog of the electric charge (which has twotypes). The interaction between particles of different color charges is mediated by amassless particle called gluon [28], which has a similar role as the photon in QED.Over the last decades, significant experimental evidence has shown support of thequark model and QCD theory. Nowadays, they form an essential part of what isnow accepted as the Standard Model of particle physics [29]. Although the nature ofthe strong force is still subject to a very active research field, here we are interestedin how its inner workings affect the structure of nuclear matter.After the emergence of QCD, the nuclear force has been interpreted as a resid-ual strong force, similarly as the Van der Waals forces are residual electromagneticforces acting among neutral molecules. However, a direct derivation of nuclearforces from QCD is complicated since the techniques typically employed in quan-tum field theory (mainly from perturbation theory) cannot be applied in the energyregime of nuclear phenomena. The description of nuclei from QCD is even harderdue to its many-body nature.In the late 1980’s, effective field theories (EFT) started to be applied to QCD inthe energy regimes of nuclear physics. In those approaches, approximations areperformed to compute only the physical phenomena relevant to the energy scalesof the system of interest. The simplification brought by those techniques finallyopened the doors for nuclear ab initio calculations.30 20 40 60 80 100 120 140020406080100Neutron NumberProton NumberStable nuclidesObserved nuclidesAb initio calculations Not observed, but predicted to existFigure 1.1: Chart of nuclides showing the nuclei observed so far (green, and black for thestables). Nuclei whose properties have been calculated through ab initio methodsup to 2017 are marked in blue (from [30]). The regions where nuclei are believedto exist within the drip lines are marked in yellow.Since then, calculation techniques and EFT-based descriptions for nuclear inter-actions have been evolving. At the current stage, properties from simple and lightnuclear systems (up to a few tens of nucleons) are currently well reproduced. Figure1.1 shows all nuclei whose properties were calculated through those ab initio meth-ods up to 2017 [30]. However, those methods are computationally costly; thus theinterplay between theory and experiment is crucial to pinpoint key behaviors, se-lect successful approaches and benchmark whether the embedded assumptions andapproximations, when performed, are appropriate. As nuclear science advancesside-by-side to technological improvements in computing, larger and more complexsystems are being tackled [31].1.2 Atomic Nuclei and Mass ObservablesThe atomic nucleus is a particular form of nuclear matter. It is a quantum many-body system about which we want to answer questions like: What combinationsof protons and neutrons can form a nucleus, and where are the limits? How doprotons and neutrons behave collectively? How do the properties of the nuclearforce influence those behaviors? Or, can we accurately predict the properties ofnuclei that are barely accessible through experiments or even beyond our reach?4To answer those questions, we need to inspect observables1 of nuclei and studyhow do they evolve as we look at different nuclear systems. Our body of knowledgerevolves around the observations of about 3000 different nuclei (see figure 1.1) [32].About 300 of them are available on Earth’s crust, while the remaining must beproduced in specialized facilities. Meanwhile, another few thousands of nuclei arealso predicted to exist according to some models, but many of them may never beexperimentally accessible.Among all properties, the atomic mass is perhaps the most fundamental observ-able. It reflects the net energy content of the system, including not only the mass ofall individual constituents but also the effects of the bonding agents acting amongthem. These effects result in what we call the binding energy (EB). The mass of aneutral atom (ma) whose nucleus contains N neutrons and Z protons is expressedin this form2:ma(N, Z) = Z ·mp + N ·mn + Z ·me + EB(N, Z)c2 , (1.2)where mp, mn and me are the masses of the proton, the neutron, and the electron,respectively (c is the speed of light).The Bethe–Weizsa¨cker mass formula [33] beautifully illustrates the concept ofbinding energy. It was one of the earliest attempts to model the effects that con-tribute to the binding energy and, consequentially, to infer properties of nuclearmatter. It is based on a very simple model of the nucleus:EB(A, Z) = −aV A + aS A2/3 + aC Z2A1/3+ aAEA(A, Z) + aPEP(A, Z) . (1.3)Here, it is expressed in terms of the atomic (Z) and mass (A = Z + N) numbers.The coefficients aV , aS, aC, aA, aP are determined through a fit to the available data.The first term accounts for the bulk effect of the nuclear force and is the onlyexclusively binding term. It incorporates the same characteristics of the nuclearforce that inspired the Yukawa potential: charge independence and short-rangednature. It assumes that the total binding from nuclear forces will be proportional tothe total number of nucleons (A) because each nucleon will only interact throughnuclear force with its nearest neighbors. The second term, however, accounts fornucleons at the surface of the nucleus, which have fewer neighbors to interact with.The binding is decreased by a factor proportional to the number of nucleons in thesurface (≈ A2/3).These very first two terms make one remarkable assumption about the overallstructure of a nucleus: it behaves like a drop of an incompressible fluid, and theterms may be interpreted as volume and surface tension contributions, respectively.Experimentally, it is known that nuclear matter has a constant density of about 0.17nucleons/fm3 (called nuclear saturation density), which supports this assumption.1 Observables are physical properties that can be measured, such as mass, spin, radius, etc.2 The binding energy, as presented in equation 1.2, accounts for both the binding energy of the nucleonsto the nucleus and the electrons to the atom, although the latter is much more weakly bound than theformer.50 20 40 60 80 100 120 140020406080100012345678Binding Energy per Nucleon  [MeV]Neutron NumberProton NumberFigure 1.2: Nuclear chart showing the binding energy per nucleon calculated through theBethe–Weizsa¨cker formula (eq. 1.3). The white curve connects the predictions formost bound nucleon for every mass number, which has overall good agreementto the actual location of the stable nuclides, marked in light blue.The third term accounts for the Coulomb repulsion between protons and there-fore causes a loss in binding energy. Quantum mechanical effects associated withPauli’s exclusion principle, which states that identical half-integer spin particlescannot occupy the same quantum state, are accounted for in the last two terms (EAand EP). Their interpretation and detailed form were modified and adjusted overthe decades since their first inception.The approach of the Bethe–Weizsa¨cker formula is surprisingly successful givenits simplicity. It correctly predicts EB/A within 0.5 MeV/u for the vast majorityof the known nuclides. It reproduces overall trends and bulk properties of nuclearmatter such as the quadratic behavior of masses of isobars3, the overall shape ofthe stability line4 (see figure 1.2) and the location of the most bound nucleus in thenuclear chart.Albeit this liquid drop model brings an enlightening understanding of the natureof nuclear matter, it fails to describe some details. It strictly considers the nucleusas spherical without accounting for possible deformations, neither it is capable ofgiving any insight into other essential observables, such as nuclear spins and ra-dioactive half-lives. Yet, the most noteworthy limitation of this model is its failureto explain the emergence of magic numbers.Magic numbers are special numbers of nucleons that are common to nuclei withexceptional stability. One way of seeing them is by inspecting the energy necessary3 A set of isobars is a set of nuclides with the same number of nucleons, also referred to as with samemass number (A).4 The stability line is a reference line that virtually connects the stable nuclei at the nuclear chart, markedas light blue in figure 1.2.6Neutron number NS 2n(MeV)40 45 50 55 60 655101520253072Mn67Mn75Fe68Fe77Co69Co80Ni70Ni82Cu71Cu85Zn72Zn87Ga73Ga90Ge74Ge92As75As95Se76Se98Br77Br101Kr78Kr102Rb79Rb103Sr80Sr104Y 81Y105Zr82Zr106Nb83Nb107Mo84Mo85Tc109Ru87Ru90Rh111Pd92Pd94Ag113Cd96Cd98In115Sn101Sn105Sb117Te107Te109I119Xe111Xe120Cs113Cs121Ba115Ba122La118La123Ce121CeFigure 1.3: The two-neutron separation energies between N = 42 and N = 65, adapted from[34] (full circles are based on measured data, while open circles include massextrapolations). The magic number N = 50 is evident from a sudden break ofthe smooth trend for all isotopic break nucleus apart in two or more nuclear systems. Here, we look at the two-particle separation energies for neutrons (S2n) or protons (S2p):S2n(N, Z) = ma(Z, N − 2) + 2mn −ma(N, Z) and (1.4)S2p(N, Z) = ma(Z− 2, N) + 2mp −ma(N, Z) , (1.5)which are simply given by mass differences between atoms. Evidences of magicnumbers can also be noticed in several other separation energies, such as one-protonseparation energy or one-alpha separation energy, but are clearer and more promi-nent in S2n and S2p.An example of how S2n evolves across isotopic chains is shown in figure 1.3. Ascan be seen, S2n exhibit a general smooth trend, until a sudden decrease is observedright after crossing the N = 50 mark, consistently found in all isotopic chains. Thismeans that additional pairs of neutrons to any nucleus with 50 neutrons will beconsiderably less bound.Another way of looking at those effects is through the “derivatives” of the two-particle separation energies (∆2n for neutrons, ∆2p for protons), often called Empiri-cal Shell Gaps, through which special patterns are brought into relief:∆2n(N, Z) = S2n(N, Z)− S2n(N + 2, Z) and (1.6)∆2p(N, Z) = S2p(N, Z)− S2p(N, Z + 2) . (1.7)7204060801000123456780 20 40 60 80 100 120 140Proton NumberN=820 20 40 60 80 100 120 140020406080100012345678Neutron NumberProton NumberN=ZN=50N=126N=28N=20N=814121086420-2Empirical Neutron Shell Gap [MeV]N=ZZ=8Z=20Z=28Z=50Z=821412108642Empirical Proton Shell Gap [MeV]Proton Magic NumbersNeutron Magic NumbersFigure 1.4: Empirical shell gaps across the nuclear chart for protons (top) and neutrons (bot-tom), calculated from the mass data of [34]. This representation makes the loca-tions of magic numbers evident as a few specific isotonic and isotopic chains getmuch “brighter” than their neighbors.Figure 1.4 shows ∆2n and ∆2p for all known nuclei. It makes evident the specialpatterns occurring at nucleon numbers 8, 20, 28, 50, 82 and 126, for either protonsand neutrons.Nuclei that lie in those magic regions are not only more bound than their neigh-bors; they also exhibit a collection of other special properties. They are generallymore compact nuclei (have smaller charge radii, for example) [37], they tend tooffer more resistance to the absorption of nucleons (smaller neutron absorptioncross-section, for example) [38] and typically require much higher energy to be pro-moted to an excited state [36]. Figure 1.5 gives a few examples of the behavior ofsuch observables around the same magic numbers identified previously.Historically, the term ”magic” referred to the lack of plausible explanation (up tomid-1940s) on why particular numbers of protons or neutrons would grant nucleisuch unique properties. To understand the origin of such particular behaviors, it isnecessary to take a more detailed look at the individual interactions between nucle-8+SnZrNbMoRuRhPdKrRbSrYAgCdInSbTeIXeCsBa40  50  60  70  80  90Neutron NumberR (fm) 1.5: Magic properties seen in other nuclear observables: (left) charge radii acrossisotopic chains of Kr to Ba, “kinks” are seen in N = 50 and N = 82 (adaptedfrom [35]); and (right) the excitation energy to the first 2+ state (E(2+1 )) of even-even nuclei for Sn to Sm isotopes (adapted from [36]). A sharp increase is seen atN = 82, also Sn isotopes are all proton-magic (Z = 50) and exhibit greater overallE(2+1 ). Experimental details can be found in the original references.ons than what the liquid drop model can provide. For this purpose, several othertheoretical approaches have been developed. The earliest was the very successfulNuclear Shell Model [39], which is briefly introduced in the next section.1.3 The Nuclear Shell Model in a NutshellSince its initial formulation in the late 1940s [39], the Nuclear Shell Model has beenvery successful in providing intuitive explanations to complex nuclear structure phe-nomena. It has proved to be an extremely powerful theoretical tool and is still beingemployed and further developed. In this section, a simplified version is presented.It is closer to the earlier variants of the model, which is currently called IndependentParticle Model (IPM) or Non-Interacting Shell Model. Introductions to more modernviews can be found in [40] and, in more detail, in [41].The inspiration for the Nuclear Shell Model came from an analogy to the atomicshell model, which provides rules for the structure of electrons in an atom. Likeatoms, nuclei are also many-body fermionic systems, which means their constituentsmust follow the Pauli exclusion principle. Experimentally, atoms of ”closed” elec-tron shells share many properties with magic nuclei: compactness, boundness, andinertness, suggesting that magic nuclei may correspond to closed nuclear shells.The IPM supposes every nucleon is moving independently through the nucleusunder a mean-field potential (V) created by the remaining nucleons. The shape ofsuch potential may be, for example, square wells, harmonic oscillators or the so-called Wood-Saxon [42] potential. They will all generate a shell-like structure. How-ever, the correct reproduction of features corresponding to magic numbers couldonly be achieved by the introduction of a spin-orbit term ~` ·~s , where ~` and ~s are9angular momentum and spin operators, respectively. One example form for thispotential is:V(r) =m2ω2r2 −Voo~` 2 −Vso~` ·~s , (1.8)12682502820822g 9/21i 13/23p 1/23p 3/22f 5/22f 7/21h 9/21h 11/23s 1/22d 3/22d 5/21g 7/21g 9/22p 1/21f 5/22p 3/21f 7/21d 3/22s 1/21d 5/21p 1/21p 3/21s 1/2614163238405864687092100106110112136Figure 1.6: Shell struc-ture obtained throughthe IPM. Each level con-tains the total numberof nucleons occupyingthe nucleus up to thatfull level. The verticalarrows mark large en-ergy separations, theiroccupation number (cir-cled) is associated withmagic numbers.where r is the distance from the nucleon to the centerof the nucleus, m is the mass of the nucleon, and ω isthe harmonic oscillator’s frequency. Voo and Vso are thepotential strengths of their respective terms.Solving the Schro¨dinger equation using this potentialresults in discrete eigenstates that are degenerate with re-spect to the set of quantum numbers [n, `, j ], where n isa principal quantum number, ` is an angular momentum(or orbital) quantum number and j is a total angular mo-mentum number. The latter comes from the spin-orbitinteraction and comprises the sum of angular momen-tum between the nucleon spin and its orbital angularmomentum. The degree of degeneracy (or how manyparticles may occupy the same energy level) is given by(2j + 1).The energy level structure obtained is shown in fig-ure 1.6. Every level is labeled according to the spectro-scopic notation: the first number corresponds to n, thefollowing letter indicates the orbital angular momentum(s corresponds to ` = 0, p to ` = 1, d to ` = 2, andso on...), and the last number indicates the total angu-lar momentum j. Indexes pi and ν are often added tothe spectroscopic notation to differentiate proton shellsfrom neutron shells, respectively.To get the ground-state configuration of a particularnuclide, one can fill those levels according to the require-ments of the Pauli exclusion principle. As one level is”full” (it has the maximum number of nucleons allowedby its degree of degeneracy), it should form a ”closed”configuration, a ”closed shell”.In some cases the energy separation between somelevels is minimal, forming clusters of quasi-degeneratestates. However, the gap between a few particular lev-els happens to be quite large, and they can be associ-ated with magic nuclei. For example, a nucleus with 50neutrons will have every level up to the 1g9/2 full. Anadditional neutron to the system would be placed in amuch higher energy level, 1g7/2, and therefore would beconsiderably less bound to the ”core” nucleus. This neu-10tron would be lying in a ”valence shell” (or non-closed shell). This binding energyargument is consistent with the separation energy signatures of magic numbers.Protons and neutrons fill their shell structures independently. Also, as nuclearforces act in a very similar fashion for both, their level structures are expected to benearly identical too. As can be seen in figure 1.4, magic numbers are the same forprotons and neutrons.The IPM can correctly predict many nuclear properties of nuclei with configura-tions near those with closed shells but fails to describe properties far from them.The base assumption of the IPM (that nucleons move through the nucleus nearlyindependently) may only hold near closed shells.In modern approaches of the Nuclear Shell Model, nucleons in valence shells areallowed to interact, while nucleons in the core (”inner” levels) remain inert. Residualinteractions, originating from nuclear forces between nucleons in the valence space,also play a role in the nuclear potential. This Interacting Shell Model approach hasclose ties to development of ab initio methods, as presented in chapter 2.1.4 Not-so-Magic NumbersAt a glance, “magic” properties seem to be sturdy, consistent, and indifferent to thecounterpart nucleon number. It seems to make sense since protons and neutronsare distinguishable fermions; therefore, they are expected to have their own internalarrangement rules without affecting each other.Nevertheless, this picture only holds near stable nuclei. A closer inspection offigure 1.4 reveals interesting effects around the boundaries of the known mass sur-face. For example, the ∆2p strength of Z = 82 slowly fades away as the neutronnumber decreases. The ∆2n strength of N = 20 and N = 28 abruptly quenches atlow proton numbers. Meanwhile, a weaker but consistent effect seems to emerge atZ = 40 as the neutron number increases, and stronger effects appear to be emergingat N = 16 and Z = 14.As experiments are able to access increasingly unstable species, we find largedeviations from the “canonical” (well established) magic numbers. Some magicnumbers seem to vanish, with evidences for that observed at nucleon numbers 8[43], 20 [44], 28 [45] and 82 [46]. Meanwhile, the appearance of magic-like featuresis seen at nucleon numbers 14 [47], 16 [48], 32 [49], 34 [50] and 40 [51].1.4.1 The New Magic Number 32Particular attention has been given to the emergence of strong magic propertiesamong nuclides with 32 neutrons, on the neutron-rich side of the stability line.The region of interest encompasses nuclei like 56Cr, 55V, 54Ti, 53Sc, 52Ca, 51K and50Ar. Evidence of magicity starts appearing at 56Cr (Z = 24), and it builds up withdecreasing proton number. The strength of the effects appears to peak at Ca, which1116 18 20 22 24 26 28 30 32 34 36121. Neutron Number  E(21+ )   [MeV]Ar (Z=18)Ca (Z=20)Ti (Z=22)Cr (Z=24)Fe (Z=26) Ni (Z=28)   Figure 1.7: Excitation energy to the first 2+ state of even-even nuclei for Ar to Ni isotopes.Data shows clear enhancement pattern of E(2+1 ) in the canonical magic numbers20 and 28, and the emergence of similar behavior in N = 32 is also seen in Z 6 24.Data from [54].is also proton-magic (Z = 20), although very little data exist in this region beyondpotassium and argon (Z < 19).The first theoretical [52] and experimental [53] hints that N = 32 could be a magicnumber date back to the early 1980s, with the discovery of a larger than expectedfirst excitation energy of 52Ca. However, just recently experimental facilities wereable to perform detailed studies in the region with the increase in isotope produc-tion yields.The current spectroscopic data on first excitation energies E(2+1 ) of even-evennuclei are shown in figure 1.7 for the region of interest. The signatures of magicnumbers 20 and 28 are clearly seen as the E(2+1 ) peaks in those neutron numbers.Meanwhile, the data also show a relative, but systematic, increase in N = 32 belowproton number Z = 24, being completely absent otherwise.Since 2012, mass spectrometry facilities have been able to study this region withhigh precision. The current mass data indicate behaviour associated with magicsignatures at N = 32 in the K [55], Ca [56, 49] and Sc [57] isotopic chains. It isevident in the S2n systematics shown in figure 1.8. In contrast, the S2n surface1226 27 28 29 30 31 32 33 345101520253035Two-Neutron Separation Energy [MeV]Neutron NumberKCaSc TiVCrMnFeCoNi30 31 32 33 34101214CrVCaScTwo-Neutron SeparationEnergy [MeV]Neutron NumberTiFigure 1.8: S2n surface: left panel provides a broader view in the surroundings of the interestregion, showing the presence of magic properties at the canonical N = 28 andthe appearance of magicity at N = 32, both marked in blue; right panel showsa zoomed in view of the N = 32 interest region, with linear fits (red) showingthe compatibility of smooth trend (absent magicity) in Ti and V chains. The massdata was taken from [34].is smooth in this region for V and beyond, indicating the absence of magicity, inagreement with spectroscopic data.However, there are still many missing pieces on our knowledge about the evolu-tion of magic properties in N = 32. For instance, the picture at the intermediateTi chain is unclear; presently available mass data point towards the existence ofmodest magicity, but the uncertainties are not sufficiently small to reveal detailedinformation. The data is compatible with the absence of any magic character, asevidenced by the fit shown in the right panel of figure 1.8.Also, most of the data on Ti and V isotopes in the region comes from low-resolution or indirect mass measurement techniques, and large deviations havebeen observed in their vicinity after measurements were performed using high-resolution techniques [56, 57, 58, 34]. These issues make this region very appealingto be further studied through mass spectrometry.Meanwhile, a recent laser spectroscopy measurement of 54Ca revealed its unex-pectedly large charge radius [59]. If N = 32 is a true magic number in this region,54Ca (Z = 20) is expected to be a doubly-magic nucleus (magic in both neutronand proton numbers), which typically have greatly enhanced closed-shell features5.The result of this laser spectroscopy experiment challenges its doubly-magic naturesince it is expected to be a more compact nucleus. However, this is the only chargeradius measurement in the N = 32 isotonic chain so far.5 See, for example, 132Sn (N = 82, Z = 50) in right the panel of figure 1.5.131f5/22p1/22p3/21f7/21d3/22s1/21d5/21p1/21p3/21s1/2Protons Neutrons60Fe  (Z=26)1f5/22p1/22p3/21f7/21d3/22s1/21d5/21p1/21p3/21s1/2Protons Neutrons58Cr  (Z=24)1f5/22p1/22p3/21f7/21d3/22s1/21d5/21p1/21p3/21s1/2Protons Neutrons56Ti  (Z=22)1f5/22p1/22p3/21f7/21d3/22s1/21d5/21p1/21p3/21s1/2Protons Neutrons54Ca  (Z=20)Figure 1.9: Shell evolution of N = 32 isotones showing the appearance of magicity at thisneutron number as the proton number decreases. The strength of the proton-neutron residualinteraction between ν1 f5/2 and pi1 f7/2 orbitals is represented by the thickness of the arrowsconnecting them. The weakening of this interaction shifts the ν1 f5/2 level and increases theenergy gap (in green) from the ν1 f7/2 level.1.4.2 A Test Bench for Nuclear TheoriesSince most of the nuclear models were originally conceived and constrained withdata around the stability line, the behaviors that emerge at the extremes of the nu-clear chart pose a challenge to nuclear theory. Structural evolution around canonicaland non-canonical magic numbers motivated several important updates to nuclearinteractions [60, 61, 62, 63].For example, the magicity in N = 32 and N = 34 has been investigated in a ShellModel framework (see [50] and references therein). The N = 32 is considered asa full valence ν2p3/2 orbital, which is energetically close to the ν1 f5/2 orbital. Thisquasi-degeneracy among orbitals prevents the appearance of shell signatures. Theemergence of magicity at Z 6 24 has been attributed to the weakening of attractiveproton-neutron residual interactions between the ν1 f5/2 and pi1 f7/2 orbitals. Aspi1 f7/2 empties, the neutrons in ν1 f5/2 become less bound. Due to that, the ν1 f5/2and ν2p1/2 orbitals would change their energy order between V and Sc, which hasbeen supported by spectroscopic data [64]. A large energy gap between ν2p3/2orbital and the next level would emerge, breaking the local quasi-degeneracy oflevels and causing the appearance of magic-like features. A representation of suchshell structure evolution is shown in figure 1.9. The mass and spectroscopy data inthe region allowed refinements of the proton-neutron residual interactions betweenthose orbitals.The N = 32 region is especially interesting in the context of nuclear ab initiotheories. Not only due to the interesting emerging phenomena, but also becausethey occur near the current limit of the reach of these techniques. Although a fewab initio methods were able to calculate properties of systems as heavy as 100−110Sn[65] and 132Sn [66], those lie along magic numbers which permit certain controlled14approximations (see discussion in chapter 2), but may not be applied to all systemsin the region6. On the other hand, nuclides up to mass A ≈ 50 are among theheaviest systems that have been accessible and that are well within reach of severalab initio theories [30, 68], as can be seen in figure Scientific GoalsThere is a high demand for precision data to further explore the evolution of themagicity in N = 32. At first glance, charge radii data on other isotones are requiredto inspect trends from the anomalous behavior seen in calcium. Also, pushingmass and spectroscopic information further in the low-Z direction, starting fromargon, is essential to determine whether the magicity vanishes or persists. Likewise,to finely understand the emergence of such effects at the high-Z frontier, preciseexperimental determination of the mass surface around titanium is required.However, except for Ca and K, isotopes of other elements in the N = 32 region arenot easily produced or experimentally accessible in many facilities (see discussionin chapter 4). Therefore these studies may greatly benefit from the development ofhigh sensitivity experimental techniques and new isotope production methods.In this study, our team was capable of creating samples of very neutron-richtitanium isotopes for the first time at the TRIUMF laboratory in Vancouver, Canada,and of performing precision mass spectrometry in the N = 32 region.Although identifying shell effects in mass observables can be typically achievedwith a precision of hundreds of keV/c2, titanium is at a transition point where sucheffects may be very small. Thus, pinpointing detailed magic signatures may requireprecisions better than 50 keV/c2. Additionally, the studied isotopes were expectedto be produced as rarely as a few per minute, accompanied by large amounts ofco-produced contamination.Fortunately, the studies were possible due to the availability of a novel mass spec-trometry technique: the Multiple-Reflection Time-of-Flight Mass Spectrometry (MR-ToF-MS), which had been recently commissioned at TRIUMF’s Ion Trap for Atomicand Nuclear science (TITAN) facility [69]. This technology, employed only in a fewlaboratories so far, is very sensitive and can distinguish isotopes of interest fromcontamination species while providing enough precision to resolve the ambiguitiesof the existence of magic effects in titanium.Masses of titanium isotopes between 51Ti and 55Ti were successfully measuredwith the MR-ToF-MS. The technique was also used to determine vanadium massesbetween 51V and 55V that were also present in the sample. The results confirm theexistence of mild shell effects in titanium and their absence in vanadium, narrowingdown the exact emergence of magicity of N = 32 from mass observables.As this was the first time the MR-ToF-MS technique was employed at TRIUMF, weconfirmed these results by performing the same mass measurements independently6 For example, attempts to perform calculations of properties of 125,127,129Cd did not succeed [67]15at a well established mass spectrometer: TITAN’s Mass measurement PEnning Trap(MPET). Although the MPET is a more precise mass spectrometer, its sensitivity isreduced, and its tolerance to contaminants in the sample is more restrictive. There-fore, only the masses of 51−53Ti were measured. Nevertheless, measurements per-formed with both spectrometers agreed very well and allowed us to benchmark theMR-ToF-MS technique.With a more complete picture of the evolution of magicity of N = 32 in hand, wechallenged four state-of-the-art nuclear ab initio theories. Overall, all tested theoriesperform well reproducing the mass surface in this region, but our work reveals afew systematic deficiencies, such as an overprediction of the ∆2n. This new set ofdata may help in guiding the development of the next generation of ab initio theoriesand nuclear forces.In the next chapter, I will give a more in-depth overview of nuclear ab initio meth-ods, focusing on the theories that were tested against our data. An overview ofmass spectrometry is given in chapter 3, with emphasis on the two mass spectrome-try methods used. In chapter 4, I will present the experimental procedure employed,from the production and manipulation of the isotopes to the measurement proce-dures. The results of the studies with each mass spectrometer are presented inchapter 5, while their comparison and interpretation, as well as the updated pictureon the evolution of shell effects in N = 32 and its comparison with the ab initiotheories are presented in chapter 6. Finally, my conclusions and outlook are givenin chapter 7.162 AB IN IT IO NUCLEAR THEORYAb initio is a Latin expression for ”from the beginning”. In physics, it is used todesignate theoretical approaches that describe phenomena ”from the ground up”,based solely on the most fundamental principles. It differs from phenomenology,whose goal is to describe phenomena as accurately as possible without necessarilyhaving ties to fundamental grounds.In nuclear physics, the Liquid Drop Model and the Independent Particle Model(presented in chapter 1) are good examples of phenomenological models. In bothcases, a model for the nucleus is made from a set of hypotheses about the behaviorof nuclear matter, and parameters of the model are adjusted to fit experimentalproperties. The nucleus is treated in a ”mean-field” approach, where the wholedynamics inside the nucleus is averaged and simplified.Phenomenological models are very useful providing intuitive explanations forcomplex phenomena, and they can be very accurate within the data range that isused to fit them. However, one has very little control over the limits of validity ofthese models. Typically their results significantly diverge in regions where data isscarce and cannot provide enough constraints. There is no means to control or toestimate their accuracy passing these regions.Thereon lies the strength of nuclear ab initio approaches. Ideally, if one can buildup atomic nuclei from QCD without parameters fitted to experimental nuclear quan-tities, properties of any nucleus could be accurately calculated or predicted indepen-dently of the nuclear data available.Instead of a mean-field treatment, ab initio theories study nuclei microscopically.They account for the contribution of every constituent and their interactions. Ideally,these constituents and their interactions should be handled on the most fundamen-tal quark-gluon level. However, as mentioned earlier, QCD is non-perturbative inthe energy ranges of nuclear physics, and only a few simple nuclear physics prob-lems have been tackled at that level (for example, [70]).Therefore, modern-day nuclear microscopic methods still build nuclei from thenucleon-meson level. The interactions employed, however, are built keeping theirties to QCD through an Effective Field Theory (EFT). Truncations and approxima-tions are often necessary, but the theoretical framework offers means to control andestimate their impact on accuracy. Although they are not directly derived fromQCD, we still refer to such theoretical treatments as ab initio. This chapter is ded-icated to providing an overview of how they are constructed and how they canbenefit from experimental input.172.1 Construction of a Microscopic Nuclear TheoryA generic microscopic description of a nucleus allows one to compute the inter-actions among nucleons considering them as the fundamental constituents of themany-body system. It searches for the solution of the A-body Schro¨dinger equation:(A∑ipi22m+ V)Ψ = εΨ , (2.1)where p is the momentum operator, m is the nucleon mass, Ψ is the many-bodywave function, V is the nuclear potential and ε is the eigenenergy associated withΨ. The eigenvalues ε are the objects of interest here since they are connected to themass of the system, but other nuclear observables may also be extracted from Ψ.The nuclear potential V has to reflect all the interactions among participating nu-cleons, including the strong interaction and the Coulomb interaction for protons7. Itmay be constructed using phenomenological approaches or, as has become commonrecently, using an effective field theory of the QCD.Generically, the potential may be written asV =A∑i<jV2Ni,j +A∑i<j<kV3Ni,j,k +A∑i<j<k<lV4Ni,j,k,l + ...+ VANi,j... , (2.2)which explicits the different n-body components of the potential. In most ap-proaches nowadays, this potential is truncated to include up to 3-body (3N) forces.A brief introduction to how nuclear interactions are built is given in section 2.3.It is worth noting that 1-body potentials, like the one presented in equation 1.8,are not included. In phenomenological approaches, such potentials are often usedto describe a ”mean-field”: a simplified potential that represents the average of theinteractions with the other particles in the system. This approach diverges from theab initio philosophy.We also require a method to construct the many-body wave functions and tosolve the many-body system correspondingly. For simplicity, many-body states aretypically constructed as a product of the single-particle states φi (characterized byposition, spin and isospin, here only the position vector ri is shown for brevity):Ψ(r1, r2...rA) = φ1(r1) φ2(r2) φ3(r3) ... φA(rA) =A∏i=1φi(ri) , (2.3)7 The weak interaction is typically neglected as it is irrelevant for the nuclear structure phenomena inves-tigated here. Naturally, the gravitational interaction is also neglected given its much smaller couplingthan the other interactions.18but it needs to be antisymmetrized since nucleons are indistinguishable fermions.The antisymmetrized product state can be written in the form of a Slater determi-nant:Ψ(r1, r2...rA) = A[A∏i=1φi(ri)](2.4)=1√A!∣∣∣∣∣∣∣∣∣∣φ1(r1) φ1(r2) · · · φ1(rA)φ2(r1) φ2(r2) · · · φ2(rA)....... . ....φA(r1) φA(r2) · · · φA(rA)∣∣∣∣∣∣∣∣∣∣. (2.5)Solving the many-body equation is no simple task. Since interactions among allnucleons and all possible configurations need to be considered, the size of modelspaces required grows very quickly (factorially, in some techniques) as the size ofthe system grows. Hence, methods need to incorporate effective ways to reduce thesize of the model space and to perform adequate approximations, without divergingfrom the ab initio philosophy. General concepts on many-body quantum methodsand some of their associated strategies are presented in section 2.2.The interplay between ab initio theories and experiments is crucial to test themethods employed both on developing the interactions and solving the many-bodyequation. A few recent examples that illustrate the importance of these tests arepresented in section 2.4, as well as the specific interactions and methods that wereput under scrutiny in this work.2.2 Many-Body Quantum MethodsMany methods were developed to solve the many-body Schro¨dinger equation (2.1)for systems in several regions of the nuclear chart. The Coupled-Cluster methods[71], the No-Core Shell Model [72], the Green’s Function Monte Carlo [73], and theHyperspherical Harmonics method [74] are a few examples of such techniques.Some are very simple to understand. For example, the No-Core Shell Model(NCSM) [72] can be understood as an extension of the Interacting Shell Model, pre-sented briefly in section 1.3. The valence space, where nucleons are allowed tointeract, is extended to all orbitals, eliminating the roles of an inert core and themean-field potential. The solution can be found simply through a matrix diagonal-ization and is numerically exact.However, the NCSM is only computationally feasible with very light systems.The calculation of systems as simple as 16O already reaches the limit of modernmachines. This is evident in figure 2.1, where the current limit of present-daysupercomputers is compared to the matrix dimensionality required to computea few light nuclei through NCSM. Although undesirable, approaches that aim atthe construction of heavier systems must employ controlled approximations. Some19Matrix Dimension1012101110101091081071061051041031021011000 2 4 6 8 10 12 14NmaxPetascale limit with 2N forcesPetascale limit with 3N forcesFigure 2.1: Matrix dimensionality required for the computation of a few N = Z nuclei usingthe NCSM method as a function of the model space size parameter Nmax. Forcomparison, the calculation of 16O is expected to require Nmax = 10 to get con-verged results. Dashed lines show the computing limits of Petascale machines if2N and 3N forces are included. Figure adapted from [72].strategies involve resorting to known nuclear properties (such as magic numbers)to truncate the model space or applying transformations to reduce computationalcosts.Next, in this section, we explore two general concepts and procedures that arecommonly shared among modern techniques, mainly among those employed inthis work. They provide a perspective on common challenges and opportunitiesfor improvement, but they are far from providing a comprehensive overview of thestate-of-the-art spectrum of these methods.2.2.1 Harmonic Oscillator BasisThe many-body states of the nuclear system need to be constructed using an appro-priate choice of basis. Single-particle states are typically formulated in a harmonicoscillator (HO) basis [75]. Calculations are performed at a fixed oscillator frequencyω, while the model space must be of a finite number of major shells Nmax. Thisimplies a model space truncation with cut-off energy of (Nmax + 3/2)h¯ω.HO basis is preferred due to its properties that facilitate certain calculations andits straightforward correspondence to the Nuclear Shell Model [75]. Nevertheless,the use of other bases can be advantageous for specific methods (see [76] and [77],for example).Despite its benefits, working on HO basis requires that the nuclear interactions,typically formulated in momentum space, get transformed to HO basis. One side20effect is the need for a large HO model space (large Nmax) to accommodate thehigh-momentum contributions of the interaction (see sec. 2.3.3). Furthermore, thecalculations must be performed at different Nmax and ω to ensure convergence ofcalculations and independence of results regarding model space parameters. Ulti-mately, it greatly contributes to the method’s computational costs [75].2.2.2 Reference States and Valence SpaceIn the medium mass region (40 . A . 100), calculations can only be carried out inexceptional circumstances. Properties of closed-shell nuclei, such as their sphericalsymmetry, can be explored to improve their computability by greatly simplifyingthe Hamiltonians [71]. For example, if only nuclei of spin zero are considered, theparts of the Hamiltonian associated with the dynamics of higher spin states can beneglected.Naturally, those simplifications make the range of calculations very limited: onlyground states of even-even closed-shell nuclei and a few particular excited statesare accessible. To extend calculations to open-shell neighbors, the closed-shell Slaterdeterminant can be used as a reference state upon where other states will be built.In this context, the language of second quantization is very convenient: creation(α†i ) and annihilation (αi) operators are introduced to add or remove, respectively,a single-particle state φi from the many-body system in an antisymmetrized way.Hence, a Slater determinant can be written as a string of creation operators actingon a vacuum state Ψ0 (defined as αi Ψ0 = 0 for any i):Ψ(r1, r2...rA) = α†1 α†2 ...α†A Ψ0 . (2.6)Alternatively, an existing A-body Slater determinant may be used as a referencestate, so arbitrary Slater determinants may be built from it though particle-holeexcitations, applying a series of creation-annihilation operators.Reference states are used in many different ways to simplify calculations. Rela-tive matrix elements are easier and more efficient to compute than absolute ones.In some methods, the model space can be truncated to a few HO shells aroundthe outermost shells, creating a valence space in a similar prescription as in theInteracting Shell Model. With reduced degrees of freedom, the diagonalization iscomputationally easier, and many more properties are accessible. An illustrativeexample of this procedure can be found in figure 2.2.21Reference Slater Determinant 1p-1h ExcitationValence SpaceInert Core∘ Full Model-Space∘ Simplified Hamiltonian∘ Access to limited properties∘ Truncated Model-Space∘ Full Hamiltonian∘ Access to all propertiesFigure 2.2: Depiction of the construction of a closed-shell reference state from the vacuumusing full model space (left). Then, an excited state can be built from the referenceusing one particle-hole excitation in a truncated model space (right).2.3 Nuclear Interactions2.3.1 Nuclear Forces from Phenomenological ApproachesThe development of phenomenological potentials dates back to the post-Yukawaera, and they have been heavily employed in nuclear physics. For example, theyare used in structure calculations using the Interacting Shell Model (for example in[41]) and in the prediction of scattering cross sections in nucleus-nucleus collisions(as in [78]).Although phenomenological potentials were inspired by QCD, they are not con-structed from it. Their development was historically guided by fitting basic proper-ties of nuclear force between two nucleons (2N), with an underlying ansatz regard-ing multi-meson mediation8. Experimental input to nucleon-nucleon and meson-nucleon interactions is relatively straightforward to be obtained from scatteringexperiments. One example of such potentials is the AV18 [79] (see figure 2.3). Itincluded 18 terms to account for all the clearly disentangled nuclear properties (e.g.,spin-orbit, tensor, spin-spin, etc.).Later on, the refinements of precision potentials lead to the need to include three-nucleon (3N) components [73]. Differently from 2N components, the properties and8 Besides the pion, some potentials also include other mesons, such as the ρ, ω and σ mesons. Theyinfluence shorter ranges in the potential due to their heavier masses.22V(r) [MeV]r [fm]-500501000.0 0.5 1.0 1.5 2.0OPEP100-Lattice QCDAV18Figure 2.3: The 1S0 central component of the nucleon-nucleon potential derived from LatticeQCD [70] and from the AV18 phenomenological potential [79]. The one-pionexchange potential (OPEP), based on Yukawa’s theory, is shown for comparison.underlying effects of 3N interactions are not intuitive and are hardly disentangledfrom experimental data. Even today, the conditions for the emergence of attractiveor repulsive characters in 3N forces are unclear [80]. This lead to a nearly-blindsearch for the form of higher order components of phenomenological potentials,weakening their already fading connection to QCD.Moreover, since the construction of phenomenological forces is based on an ansatzof the inner workings of nuclear interactions, the quantification and the controlof the accuracy of the theories is difficult. Such limitations have motivated manygroups to search for a more consistent route to construct interactions based on QCD.2.3.2 Nuclear Forces from Chiral Effective Field TheoryDirect construction of atomic nuclei from QCD should naturally include higher or-der effects. As mentioned, however, QCD is practically intractable in the low-energyregimes of nuclear physics due to the non-linear nature of the strong force. Studiesusing Lattice QCD, a well-established non-perturbative approach to solve QCD sys-tems, were able to derive simple nucleon-nucleon potentials [70] (see figure 2.3, forexample). However, higher-order interactions built through similar techniques arestill not on the horizon [81].Recently, the description of nuclear forces has enormously advanced using ChiralEffective Field Theory (χEFT), about which a complete description can be found in[82]. In an EFT, a low energy approximation is made by freezing out a few degreesof freedom while working up to a certain energy scale, ignoring the structures thatmay emerge at higher energies. χEFT explores the spontaneous breakdown of thechiral symmetries of QCD to identify the energy scale in which the effective de-grees of freedom of nuclear systems become pions and nucleons rather than quarks23+... +... +...+... +... +...+... +... +... +...2N Force 3N Force 4N Force 5N ForceLO(Q/Λχ )0NLO(Q/Λχ )2NNLO(Q/Λχ )3N3 LO(Q/Λχ )4N4 LO(Q/Λχ )5N5 LO(Q/Λχ )6Figure 2.4: Feynman diagram representation of terms in chiral expansion of nuclear forces,classified by their order hierarchy and N-body components. Solid lines representnucleon propagators while dashed lines represent pion propagators. Reproducedfrom [82].and gluons. Chiral symmetry is approximately valid in regimes of high momentacomparatively to the masses of u and d quarks. The energy scale of χEFT (Λχ)is usually chosen to be near the nucleon mass scale (/ 1 GeV), so the degrees offreedom associated with their excitations are explicitly frozen.Then, within the chiral scale, such forces can be constructed at the nucleon-pionlevel instead of the quark-gluon level. The resulting effective Lagrangian, that con-serves properties of the QCD Lagrangian, is expanded in momentum powers of(Q/Λχ) and can be treated perturbatively with Chiral Perturbation Theory (ChPT)[82]. Figure 2.4 shows the Feynman diagrams that represent terms in this expansion.As can be seen, in the leading order (LO), the interaction includes a one-pion ex-change term, resembling the early Yukawa-like approaches. In the next-to-leadingorder (NLO) more complex terms, that involve two-pion exchange, start to appear.24At the next-to-next-to-leading order (NNLO or N2LO), terms involving three nucle-ons naturally emerge, giving rise to 3N forces.As higher orders are included, their contributions have been shown theoreticallyto diminish. Thus, this order hierarchization allows one to control the errors gener-ated in the truncations, which are proportional to the next order left out. Nowadays,all terms involving 2N interactions were calculated up to N5LO, while 3N interac-tions were evaluated up to N4LO and 4N were only evaluated at N3LO [82].Although χEFT is expected to fail beyond its cut-off scale Λχ, nuclear forcesdo not vanish within the chiral limit. Important features above Λχ, such as ∆ reso-nances, play relevant roles in shaping the nuclear potential at very short ranges. Theproperties beyond this scale are typically computed as ”contact” terms (in figure 2.4,vertices where nucleon propagators meet, without coupling to a pion propagator)and are present in all orders.Coupling constants in 2N forces are typically deduced from nucleon-nucleon andpion-nucleon scattering data. Meanwhile, contact terms and 3N coupling constantsare fitted to match observables of a few light nuclear systems, such as 2H, 3H, 3Heand 4He. The number of coupling constants increases as higher orders are included;therefore, the ambiguities generated may counter-act the purported increase ontheoretical accuracy.The χEFT approach is considered to be more robust than phenomenological ap-proaches. Although χEFT-derived potentials are not purely derived from first prin-ciples, the experimental input comes only from very simple nuclear systems, andthe results could, in principle, be applied to the whole chart of nuclides. Moreover,χEFT permits consistent derivation of many-body forces, allows control of theoreti-cal accuracies and establishes an adequate connection to QCD.Still, their increasing complexity must also be compatible with limitations stem-ming from computing. Strategies to derive those potentials and facilitate their usein many-body calculations form a currently very active area of research.2.3.3 Interaction SofteningOne common trait of nucleon-nucleon potentials is the presence of the ”hard core”,where it becomes strongly repulsive at very short distances. See, for example, thefeatures present at r < 1 fm at the realistic potentials in figure 2.3. Its effect is astrong coupling between low and high momentum modes, and can be seen as non-zero off-diagonal elements in the momentum-space matrix. In turn, calculationsrequire a large model space which, once again, complicates their computation.Modern methods employ Renormalization Group (RG) techniques to ”soften”the hard core of the nuclear potentials. In the class of techniques referred to asSimilarity Renormalization Group (SRG) [83], series of unitary transformations areapplied to the chiral Hamiltonian to drive its matrix elements to a diagonal bandwithin a regulator λ, which act as a cut-off criterion for the off-diagonal matrixelements. The transformations adapt the interaction to a truncated model space25k2(fm-2)k 2 (fm-2)′ k 2 (fm-2)′ k 2 (fm-2)′(a) (b)=5.0 fm-1 =2.0 fm-1 =1.0 fm-1Figure 2.5: SRG evolution aims to drive off-diagonal matrix elements in momentum spacetowards a diagonal band within a regulator λ, as represented in the schematicillustration shown in (a). An example is given in (b): the momentum-space ma-trix elements of a chiral N3LO 2N potential in the 0S1 channel is SRG-evolvedbetween λ = 5.0 and 1.0 fm−1 (in natural units system). Figures adapted from[83] and [72], respectively.while preserving the physical properties contained in the original Hamiltonian. Aschematic representation of such SRG evolution and one example are shown infigure 2.5.Other RG approaches also exist, such as the Vlow k that imposes a sharp cut-off regulator Λ directly onto momentum space9. Thus the choice of a suitablemethod may depend on the forces and the many-body method. In any case, RG-evolved χEFT potentials applied to many-body methods dramatically improve theircomputability [84].2.4 Tests of Ab Initio TheoriesIn recent years impressive progress has been achieved towards the description ofnuclei from first principles. Advances in algorithms, computer performance, andunderstanding of nuclear forces have generated a sudden jump in the number ofnuclei that can be calculated through those techniques [68].Nuclear potentials, however, are not observables. In principle, there can existinfinitely many nuclear potential constructions with nearly identical performance,and as many ways to derive them using the techniques described. Similarly, acollection of many-body quantum methods is also available in the market, eachcovering a specific range in masses and properties that they are able to calculate.The challenge in this field is to narrow down successful approaches against qual-ity, performance, computability, and accuracy. Ideally, one benchmarks the variousapproaches against well established experimental data. The highest sensitivity todifferentiate methods is often found at extreme regions of the nuclear chart, wheresome behaviors are particularly enhanced.One example can be found in figure 2.6. Several theoretical approaches describingground state binding energies of oxygen isotopes are compared. The top two pan-els show different approaches regarding the interaction applied to the many-body9 Note that the cut-off regulators λ and Λ differ from the χEFT scale Λχ.26methods. In panel (a), two phenomenological interactions are employed. As theyare fit to reproduce the known mass surface, they have a remarkable agreement todata closer to stability. As they are used to describe regions where data is scarce,their accuracy is reduced. In panel (b), three χEFT-derived forces are presented: onethat includes only a 2N component and two that also include different 3N forces.Their accuracy does not depend on data, but their results are sensitive to the forceprescription. It highlights the importance of 3N effects for the appropriate nuclearstructure description at extremes of proton-neutron imbalance [85].The two bottom panels in figure 2.6 compare different many-body quantum meth-ods. Panel (c) shows methods that employ valence space truncations. There, it canbe seen that there is a high variability regarding the method. This does not happento methods that employ the full model space, shown at panel (d), which are veryconsistent with each other and compatible with data. Oxygen is proton-magic, andits isotopes are relatively light systems; thus, large scale approaches are feasible. Itworths noting, however, that many of the methods cannot access all isotopes in thischain.As previously mentioned, the identification of shell effects at N = 32 has alsobecome an important test bench of such ab initio theories. This region lies closeto the high mass boundary of current techniques. One of such tests is presentedin figure 2.7, where the S2n data of neutron-rich Ca isotopes are shown. At thetime, this was the first evidence of shell effects at N = 32 seen in mass observables[56, 49].Among the confronted theories, there is a wide variation of many-body methodsand χEFT force prescriptions, although all include 3N interactions. Hence, the dis-parities among results may be interpreted as an overall ”error” of the current stageof the body of techniques. There is, however, a consistency in correctly predictingthe existence of the observed shell effects.2716 16 18 20 22 24 26 28Mass number (A)–60–50–40–30–20–100Energy (MeV)Mass number (A)–180–170–160–150–140–130Energy (MeV)MR-IM-SRGIT-NCSMSCGFLattice EFTCCAME 201218 20 22 24 26 28AME 2012MBPTIM-SRGCCEI−60−40−20−10−30−500Energy (MeV)SDPF-MUSDBExperimentNNNN + 3N (N2LO)NN + 3N (Δ)Experiment16Mass number (A)18 20 22 24 26 28−60−40−20−10−30−500Energy (MeV)16Mass number (A)18 20 22 24 26 28(a)  Calculated from phenomenological forces (b)  Calculated from χEFT forces (c)  Obtained through valence-space methods (d)  Obtained through large model space methodsFigure 2.6: Comparisons of ab initio nuclear theories on the description of ground state bind-ing energies in the oxygen isotope chain between 16O and 26O. Panel (a) comparesresults using two phenomenological forces, while panel (b) compares forces de-rived from χEFT. Many-body quantum methods, with truncated and full modelspaces, are compared in panels (c) and (d), respectively. Panels (a), (b) and (c)show the energies relative to 16O, used as reference nucleus; panel (d) shows ab-solute energies. Adapted from [85], where further details on methods and forcescan be found.28 29 30 31 32 33 34Neutron Number N0246810121416182022S2n(MeV)Shell modelCCSCGFMR-IM-SRGTITAN+ISOLTRAPFigure 2.7: Two-neutron separation energy of neutron-rich Ca isotopes, comparing experi-mental data to theories that include 3N forces. Reproduced from [40] from theworks of [56] and [49].282.4.1 Tested Forces and MethodsIn this work, we compared our data to state-of-the-art ab initio nuclear structure cal-culations. All are based on successful nuclear interactions and many-body methodsfrom recent literature.All calculations were performed with 2N and 3N interactions derived from χEFT,with parameters adjusted typically to light nuclear systems as the only input. Theused interactions were the following:1.8/2.0(EM): this interaction [86, 87, 88] combines an N3LO 2N potential from[89], evolved by SRG techniques with λ = 1.8 fm−1, with a N2LO 3N forcewith momentum cut-off of 2.0 fm−1.N2LOsat : in this interaction [90], 2N and 3N terms are fitted simultaneously toproperties of A = 2, 3, 4 nuclei as well as to heavier selected systems up to 24Oto finely adjust nuclear density saturation10.NN+3N(lnl): this interaction is a variant of the NN+3N(400) interaction [91]. Itapplies regulators to both local and non-local (LNL) 3N forces11 and, subse-quently, refits 3N parameters to A = 2, 3, 4 nuclei under a constraint that thecontact interactions remain repulsive.Complementary, two classes of many-body methods were employed:IMSRG: the In-Medium Similarity Renormalization Group method imports con-cepts of the SRG methods to efficiently solve the many-body equation, decou-pling correlations between the reference state and its particle-hole excitations.Two extensions of this method that allow the computation of open-shell nucleiwere applied: the Multi-Reference (MR-) IMSRG [92, 93, 94] and the ValenceSpace (VS-) IMSRG [95, 96, 97, 98].GGF: the self-consistent Gorkov-Green’s Function method [99, 100, 101, 102] usesGreen’s functions to describe particle-hole excitations in the many-body sys-tem and calculates energy states through sum rules. Its ”self-consistency”resides in iteratively feeding results back to input of calculations until con-vergence [103], making the use of a reference state unnecessary. The Gorkovformalism allows the computation of open-shell nuclei.In all cases, the many-body calculations were performed in a HO basis of 14 majorshells, with 3N interactions restricted to basis states with e1 + e2 + e3 ≤ e3max = 16,where e = 2n + l.The calculations were performed by our collaborators at the Ju¨lich Supercomput-ing Center (JURECA) in Germany, the Tre`s Grand Centre de Calcul du CEA inFrance, the Institute for Cyber-Enabled Research of Michigan State University inthe U.S.A. and the DiRAC Complexity system in the United Kingdom.10 As discussed in sec. 1.2, nuclear matter saturates at a constant density of about 0.17 nucleons/fm3.11 Non-local forces are those that do not depend only on the relative position of the bodies but also dependon other parameters, such as their relative momentum, for example. They pose additional computationalchallenges and may also be treated using similar techniques as presented in subsection PR INCIPLES OF MASSSPECTROMETRYSince the birth of mass spectrometry in the early 20th century [12], a wide varietyof techniques have been developed. When applied to nuclear sciences, two classesof mass measurement approaches exist: the direct and the indirect methods.Indirect or ”calorimetric” methods consists in obtaining the mass of a particle byanalyzing the energy balance of an associated decay or reaction. The binding energy(thus the mass, see eq. 1.2) is calculated by comparing masses of initial and finalsystems and their energy differences. Some of those methods achieve very highprecision, providing relative uncertainties better than 10−8. For example, neutroncapture reactions have provided mass values with relative mass uncertainties in thelevel of 10−10 [104, 34]. However, the kinematic reconstruction may be challeng-ing in some situations, as in three-body decays or in cases that involve multi-levelde-excitations, for example. Consequently, indirect methods are prone to large sys-tematic deviations [34].Direct methods, on the other hand, measure motional properties of the particleto obtain its mass directly. They observe the evolution in time of quantities suchas frequency, displacement or momentum of the particle through a well-controlledregion in space.In most direct techniques, particles are charged and move through tunable elec-tromagnetic fields. The motion of a classical non-relativistic charged particle in aregion of electric field E and magnetic field B is governed by the Lorentz force (F):F(r, v, t) = mdvdt= q e [E(r, t) + v× B(r, t)] , (3.1)where m is the particle’s mass, r and v are the particle’s position and velocity vectors,respectively, t is time, q is an dimensionless integer representing the charge stateof the particle and e is the elementary charge. In this case, mass spectrometersmeasure the mass-over-charge ratio (m/q) of the particle. Thus the charge (or thecharge state) must be known in order to determine the mass.Among the associated variables, time (and time-related quantities, like frequency)is the one that can be most accurately accessed. The measurement of time intervalsand frequencies is done by simply counting the number of cycles of a frequencystandard, like an atomic clock. This procedure is immune to many sources of errorsand is limited to the accuracy of the employed clock. This source of error is ordersof magnitude smaller than the ones relevant for nuclear structure investigations[105, 106].30m# abcΔmFigure 3.1: A mass spectrum: species a is well resolved, while b and c are overlapping. Thegray bands represent the precision δm of the mass centroids, and the mass spread∆m is shown by the arrows only for species a.However, other variables are required to calculate the mass, such as field strength,displacement, speed or energy, which cannot be determined to such level of preci-sion. For this reason, atomic mass measurements are typically done relatively to areference mass. The same measurement procedure is repeated with a calibrant par-ticle, whose mass is known at similar or better precision than the targeted precisionof the experiment. The calibrant probes the same fields and regions of the experi-mental setup as the particle of interest. Ultimately, these variables can be canceledout in the data analysis.The challenge then is to keep the other variables under sufficient control. Sta-bility of the electromagnetic fields throughout the measurement, mainly to ensurethat calibrant and particle of interest experience the same conditions, is a key re-quirement for accuracy of those mass spectrometry techniques. Control over initialconditions of the particles is also crucial to assure reproducibility and to narrow thespread of final results. In recent decades, advances in ion sample preparation andstabilization of power supplies enable some direct techniques to greatly improvetheir relative mass precision, some reaching well beyond the 10−9 level (1 part-per-billion) [107].Besides the mass precision (δm), two other important parameters to characterize amass spectrometer are its resolution (or resolving power, Rm) and its mass range. Theyare related to the performance of the device when dealing with admixtures in thesample. The mass range is the interval of masses a spectrometer can analyze ina single measurement procedure. Meanwhile, the resolving power quantifies thecapability to distinguish species of different masses (or mass-over-charge ratios),and is associated to the statistical dispersion (or spread, ∆m) of the mass measure-ments12:Rm =m∆m. (3.2)These concepts are illustrated in figure 3.1. It shows the resulting spectrum ofa mass measurement, where three species were measured. One species is well12 The definition of the spread ∆m varies according to the measurement technique. For example, in the caseof ToF-ICR technique (see sec. 3.2) it is the resonance width, while in TOF-MS and related techniques(see sec. 3.3) it is typically the Full Width at Half Maximum (FWHM) of a fully resolved mass peak.31separated from the others, whether the two others are not resolved among eachother. This means the mass difference between the two unresolved species is smallerthan the spread of the measurements.Usually, the relative mass precision (δm/m) and resolving power (Rm) are relatedbyδmm=CRm√N, (3.3)where N is the number of measurements registered, which gives the statisticalweight to the precision, and C is a factor dependent on the technique. However,unresolved species may considerably deviate from this relation or impact the accu-racy of the measurement. In nuclear science, it is desirable that the employed massspectrometer is able to resolve between isobars, which typically requires a resolvingpower above 105.Furthermore, modern-day experiments in the field of nuclear physics routinelymust fulfill tougher requirements. As rarer isotopes are probed, techniques needto be faster for shorter lifetimes, more sensitive for lower sample yields and largercontamination levels, and still sufficiently precise for scientific interest. Samplepreparation, from the creation of isotopes to their transport to mass analyzers, mustalso be fast and efficient. To put this in perspective, 56Ti has a half-life of 0.200(5)s [108]. Although it is the most unstable among the cases of interest, even 55Ti and54Ti have half-lives around one second. The complete procedure, from creation tomeasurement, must accommodate this timescale.These requirements are typically fulfilled in some mass spectrometry techniquesbased on ion trapping. Ion traps employ electromagnetic fields to confine an ionisolated from the external environment. The motion of the ion is contained ina small volume with well-controlled electromagnetic fields, which is key to highaccuracy.In this experiment, we employed two direct mass spectrometry techniques basedon ion trapping. One is the Penning Trap Mass Spectrometry (PTMS), which isconsidered the most precise and reliable technique to date. The principles of PTMSare detailed in section 3.2. The other is the Multiple Reflection Time-of-Flight MassSpectrometry (MR-ToF-MS), which is considered an emerging and promising tech-nology. The principles of MR-ToF-MS are described in section 3.3. As in equation3.1, the presented techniques are described in a classical non-relativistic framework.This approach is accurate in most situations except in a few special cases. The possi-ble errors emerging from performing this approximation are discussed case-by-casein chapter 5 in the context of this experiment.It is also worth noting that we must follow international standards while report-ing spectroscopic data. This is briefly discussed in section 3.1.323.1 International Standards and the Atomic MassEvaluationThe body of experimental data on atomic masses is reviewed by the Atomic MassEvaluation group. The group compiles all reported measurements from all differ-ent methods (direct or indirect), performs a global fit taking into account all massrelationships between different nuclides and provides recommended values for themasses of every known isotope. In this work, we use the mass values of the 2016edition of the Atomic Mass Evaluation (AME16) [34].The group also provides guidelines for the standards to be followed while report-ing mass spectroscopic data, which can also be found in [34]. The ones that arepertinent to this work are presented in the following:• For consistency, atomic masses are always published in their neutral form(meaning with all electrons included) in the atomic ground state, even if themeasurement was done using an ionized state. Therefore, results must beadjusted accordingly.• As all atomic mass measurements are relative measurements, the employedreference species must be provided.• The 12C atom serves as a standard to which all masses are connected to: the12th part of its mass defines the unified atomic mass unit (u) [34]. Alternatively,atomic masses (ma) may also be expressed in electron-Volts/c2 (eV/c2) whenshown in more compact forms such as binding energy (see definition in eq.1.2) or mass excess (ME):ME(A, Z) = ma(A, Z)− A mu , (3.4)where mu = 1 u = 9.314940954(57) 108 eV/c2 [34]. Throughout this thesis, themass excess format is preferred since it displays masses in the most compactform.333.2 Penning Trap Mass SpectrometryA Penning trap is a type of ion trap that employs only static electromagnetic fields toconfine ions. It was conceived in the late 1950s [109] and granted its inventor, HansGeorg Dehmelt13, a share of the Nobel prize in Physics of 1989. In this section,I will explain the principles of a Penning trap and how the motion of a chargedparticle confined in such a device can be understood, manipulated, measured and,ultimately, employed in measuring the mass of the particle.In PTMS, the mass measurement of an ion or any charged particle occurs insidean homogeneous magnetic field. The ion (of mass m and charge qe) revolves aboutthe magnetic field (of strength B) in a circular and periodic motion called cyclotronmotion, whose frequency (νc) is given by:νc =12piqemB . (3.5)The measurement of νc is central to PTMS. As can be seen, νc is inversely propor-tional to the mass, q is typically easily determined and B can be calibrated by doinga measurement of νc,re f using a well-known reference ion. The atomic mass (ma) ofthe species of interest can be obtained from the atomic mass (ma,re f ) of the referenceion in charge state qre f and the ratio (Rν) between their cyclotron frequencies:Rν =νc,re fνc=ma − q mema,re f − qre f meqre fq, (3.6)where me is the mass of the electron14. The advantage of this relative measurementis that the absolute knowledge of the magnetic field strength is not required as itcancels out in the equation.The measurement of the cyclotron frequencies is not simple. First of all, the ionsmust not probe inhomogeneities of the magnetic field; otherwise, it will hamperthe measurement of νc. Therefore, they need to be confined in a well-controlledvolume. Such confinement of the ions is done by the magnetic field itself and by anadditional weak electrostatic field, which completes the Penning trap setup.Next, I present how the ion can be confined using static electromagnetic fieldsand how such fields impact the motion of ions inside the trap. Ultimately, theknowledge of how charged particles move inside the Penning trap and how thesemotions can be manipulated enable us to formulate techniques to measure the cy-clotron frequency.34Figure 3.2: In a hyperbolic Penning trap (a), ions are confined in the ”radial” plane by theaction of a magnetic field. Three main electrodes generate an electrostatic field:a central ring (red) and two end caps (blue). The resulting potential well (b) con-fines the ion in the axial direction. Within such fields, the final ion motion (c) isquite complex (green trajectories) and can be broken down in three independenteigenmotions (details in text). Panel (c) adapted from [107].3.2.1 Confinement and Ion Motion in a Penning TrapThe cyclotron motion is confined to an orbit contained in a plane (xy) perpendic-ular to the magnetic field’s direction. It does not provide confinement along thisdirection, called axial (z) direction.To provide confinement in this axial direction, an electrostatic field is superim-posed to the magnetic field, creating a trapping potential well. A natural choice forthe shape of this potential U is a quadrupole, since it generates harmonic oscillatorymotion:U(r, z) =U04 d20(2z2 − r2). (3.7)Here we chose cylindrical coordinates: r is the distance to the z axis (r =√x2 + y2).The parameters U0 and d0 describe the magnitude of the potential and depend onthe shape of the electrodes that generate it. One common choice of electrode geom-etry is a pair of finite hyperboloids of revolution as illustrated in panel (a) of figure3.2. In this case, the surface of the electrodes are equipotentials of eq. 3.7, and theparameters are easily interpreted: U0 is the potential difference between the central(”ring”) and the outermost (”end caps”) electrodes, and d0 is a characteristic trap13 Dehmelt named his invention after Frans Michel Penning, whose work on vacuum gauges inspired theion trap [109].14 Note that eq. 3.6 assumes negligible the electron binding energies.35dimension determined by d0 =√z20/2+ r20/4. Parameters z0 and r0 are distancesfrom the center of the trap to each of the electrodes (see fig. 3.2).Real electrodes, however, cannot be perfect equipotentials of eq. 3.7. Their sur-faces are finite, and apertures are necessary to introduce and remove ions from thetrap volume. For this reason, additional correction electrodes are added to the as-sembly to approximate the true equipotential performance. The system is tunedto have a potential as harmonic as possible in the region where ions are trapped(see panel (b) of fig. 3.2 - some tuning strategies can be found in [110]). Once thisis achieved, ions in the trap will undergo a harmonic oscillatory motion along theaxial direction with frequency (νz):νz =12pi√qemU0d20. (3.8)The motion most often used in mass spectrometry is the one perpendicular to theaxial direction, called radial motion15. In a Penning trap, the magnetic field is not theonly driver of motion in this direction since an electrostatic field is superimposed toit. According to Laplace’s equation (∇2U = 0), there cannot be a global minimumin the electrostatic potential, and a saddle point should exist in the center of thetrap. Therefore, the confining field in the axial direction implies the existence of adeconfining electrostatic field in the radial direction too.This deconfining radial field disturbs the ”true” cyclotron motion and creates anE× B drift, which is called magnetron motion. It precesses about the magnetic fieldaxis and the center of the trap with a magnetron frequency ν−. The frequency ofthe cyclotron motion is then reduced by ν− and is therefore called reduced cyclotronfrequency16, denoted by ν+. The frequencies of the two radial motions are given by:ν± =νc2± νc2√1− 2(νzνc)2. (3.9)This relationship also provides the confinement condition for a Penning trap: νz <νc/√2, which in terms of the fields translates toU0B2<( qem) d202. (3.10)If this condition is not met, the radial deconfinement promoted by the electrostaticfield outweighs the confinement of the magnetic field and the ion escapes radially.Therefore, the electrostatic field in Penning traps are typically much weaker thanthe magnetic field.The total motion of the ion in the trap becomes quite complex, as exemplifiedon panel (c) of figure 3.2. Ultimately, it can be decomposed into three indepen-dent motions (or eigenmotions): axial, magnetron and modified cyclotron, whose15 Radial motion is named as such despite the fact it occurs both in the radial and azimuthal directions.16 Also known as modified cyclotron or trap cyclotron frequency.36characteristic eigenfrequencies follow this hierarchy: νc > ν+  νz  ν−. In a typ-ical Penning trap for ions, νc (and ν+) are on the order of MHz, νz of tens or fewhundreds of kHz and ν− of a few kHz. The eigenfrequencies obey the followingidentities:ν2c = ν2z + ν2+ + ν2− , (3.11)ν2z = 2 ν+ ν− , (3.12)νc = ν+ + ν− . (3.13)Finally, as we verified earlier, mass spectrometry requires the measurement of thetrue cyclotron frequency. Identities in equations 3.11 and 3.13 relate eigenfrequen-cies to the true cyclotron frequency, but equation 3.13 demands only the measure-ment of the frequencies of radial eigenmotions. These identities also make PTMSvery robust.In the next sections, angular frequencies (ωi = 2piνi) are used instead of ordinaryfrequencies for compactness.3.2.2 Manipulation of Ion Motion in a Penning TrapManipulating or exciting eigenmotion modes of an ion in a Penning trap is fun-damental to the measurement principle of ωc. Most frequency measurement tech-niques in Penning traps involve preparing the ion in one pure eigenmotion, dampother modes, or resonantly give energy to a mode. While specific measurement tech-niques are discussed in the following subsection, some associated ion manipulationtechniques are discussed here.Manipulating the ion motion means altering the total energy it carries and howthe energy is distributed among its eigenmodes. The total energy (Ei, summingkinetic and potential energies) stored in each eigenmotion will be a function of theamplitudes ri of each motion [111]:Ez =m2r2z ω2z , (3.14)E+ =m2r2+(ω2+ −ω+ω−), (3.15)E− =m2r2−(ω2− −ω+ω−). (3.16)In the radial motion, the average kinetic energy〈Er,kin〉is given by〈Er,kin〉=m2(r2+ω2+ + r2−ω2−). (3.17)The amplitudes of the radial motions (r+, r−) are the radius of each trajectory (seefig. 3.3). Note that, given the hierarchy of eigenfrequencies, the reduced cyclotronmotion dominates in energy contribution.37r++r-r--r+Figure 3.3: Example of trajectory of the ion in a Penning trap, projected in the radial plane.Enhancing or reducing the total energy of the system can be done by changingthe orbits of the eigenmotions. An ion in a pure magnetron motion (r+ = 0, r− > 0)will store much less mechanical energy than an ion in a pure reduced cyclotronorbit (r+ > 0, r− = 0), for example.The orbit can be changed by applying an external time-varying driving field to theion to transfer energy to its eigenmotions. Radial motions can be manipulated usingazimuthally varying fields. To introduce such fields, a trap electrode (normally thecentral ring) can be split in two or four ways (see fig. 3.4). An additional radio-frequency (RF) signal is fed to each segment of the electrode, superimposed to thetrapping bias. Such RF signal (of angular frequency ωRF and amplitude URF,0,varying in time t) can be of the formURF (t) = URF,0 cos(ωRF t + φRF) . (3.18)The phase φRF of the signal is applied differently in each electrode depending onthe desired field configuration, adjusted to their purposes. Here we briefly discusstwo common excitation modes: dipole excitations and quadrupole excitations of radialmotions, but analogue procedures also exist involving the axial motion. A moreformal and complete description of such procedures can be found in [112].Dipole ExcitationsDipole excitations are commonly used in ion preparation in Penning traps. It simplyalters the amplitude of the desired eigenmotion. The phase φRF is applied with 180◦difference between two parts of the segmented electrode (such as in (b) of fig. 3.4),and a dipole field URF,d is generated in the center of the trap:URF,d (t, x, y) =URF,0r0cos(ωRF t + φRF) y , (3.19)where r0 is the distance between the center of the trap and the segmented ringelectrode.If ωRF matches ω− or ω+, this field will resonantly drive the amplitude of thecorresponding radial eigenmotion of the ion. Depending on the relative phase ∆φbetween the driving field and the ion’s motion, it may initially damp this motion38Figure 3.4: (a) An example of a four-way radially segmented ring electrode (disassembled)to drive RF excitations in the radial motion. On top of their trapping bias, eachelectrode is subject to a signal in the form of eq. 3.18, whose phase φRF is ei-ther 0 (blue) or pi (red) depending on the desired type of field. Examples ofconfigurations to create dipole (b) and quadrupole (c) fields are given.0 2 4 6 8 100.0∆φ_ = 0∆φ_ = π/2∆φ_ = 3π/21 3 5 7 9Td ν_0. [   -1]r - (t) [mm]Figure 3.5: Evolution of magnetron radius as a function of the duration of the dipole excita-tion for three values of the phase difference between the ion’s magnetron motionand the RF field. Although this example is constructed for the magnetron motion,it also applies to the reduced cyclotron motion. Figure adapted from [111].or excite it, as can be seen in figure 3.5. After enough dipole excitation time td, theorbit radius will inevitably be incremented independently of ∆φ.Dipole excitations are very useful in PTMS. They may be applied in preparing theion into a very specific orbit or a pure eigenmotion, by exciting one and dampingthe remaining. One of its most common uses, however, is in the dipole cleaningtechnique [112, 113], which removes contaminant ions from the trap volume. Dipoleexcitations of the reduced cyclotron motion can be used to drive a known intruderto such a high orbit that it lands on the surface of an electrode, effectively removingit from the trap. Since ω+ is as mass-dependent as ωc, a dipole field is tuned to beonly in resonance with contaminant ions, leaving ions of interest unaffected.39x / mmy / mmx / mmy / mmPureMagnetronr- = 1 mmr+ = 0(a) First Half of Quadrupole Excitation  [tconv/2] (b) Second Half of Quadrupole ExcitationPure Red. Cyclotronr- = 0r+ = 1 mm[      ] [      ][      ][      ]Figure 3.6: One full conversion of a pure magnetron motion (green) in a pure reduced cy-clotron motion (red) through a quadruple excitation in the resonance conditionωRF = (ω+ + ω−). The ion trajectory through the procedure is divided in twohalves for clarity: (a) for the first half and (b) for the second. Figure adapted from[111].Quadrupole ExcitationsIn quadrupole excitations, the RF field couples two eigenmotions and, thus, con-nects two eigenfrequencies. Its most common use is to convert a pure magnetronmotion into a pure cyclotron motion through the frequency ω++ω−, which directlyaccesses ωc in one single procedure.The field is generated by applying the signal of eq. 3.18 with 180◦ phase shifts onsets of electrode segments perpendicular to each other, as shown in (c) of figure 3.4.The resulting field is described by a quadrupole:URF,q (t, x, y) = 2URF,0r20cos(ωRF t + φRF) x y . (3.20)If the resonant condition ωRF = (ω+ + ω−) is met, a full conversion betweenr+ and r− can be obtained. This conversion is illustrated in figure 3.6 in whichan initially pure magnetron motion is converted to a pure reduced cyclotron. Theprocess is periodic, and an example of such evolution is shown in figure 3.7. Fullconversions are obtained at excitation times tRF that are multiple of the conversiontime tconv:tconv = 4pir20URF,0B , (3.21)which is dependent on the magnetic field strength B and on the amplitude of theRF field, but not dependent on the mass of the particle. One important propertycan be seen: the product tconv ·URF,0 is a constant of the trap system, which greatlyfacilitates tuning and parameter scaling when different tRF are desired.40r+ r- Amplitude of radial motion [mm]Excitation time  [tRF/tconv] 1 2 3 Figure 3.7: Interconversion of magnetron (green) and red. cyclotron (red) motions as aquadrupole excitation is applied indefinitely. Full conversions occur periodicallyat multiples of tconv. Figure adapted from [111].Quadrupole excitations are the preferred process of measurement of ωc in PTMS.It allows the measurement to be done in one single step instead of measuring ω+and ω− independently.3.2.3 Frequency Measurement Techniques in Penning trapsThere are three techniques to measure characteristic frequencies of ions in a Penningtrap. All methods employ the manipulation techniques described above to enhancetheir signals. Here is a brief description of them:FT-ICR: in the Fourier Transform Ion Cyclotron Resonance technique [114], thesignal that the ions induce in the trap electrodes is amplified by low-noisecryogenic electronics, and a Fourier analysis of the signal generates a frequencyspectrum.PI-ICR: in the Phase-Image Ion Cyclotron Resonance technique [115], the ion’sradial position in the trap is measured by its careful extraction onto a position-sensitive detector. Positions measured at different times enable the reconstruc-tion of the phase evolution and, consequently, the frequency of the motion.ToF-ICR: the Time-of-Flight Ion Cyclotron Resonance technique [116] consists ofapplying an RF excitation and subsequent ejecting of the ion from the traptowards a detector. The time-of-flight from the trap to the detector is measured.In the case when the resonance condition is met, the gain in kinetic energywill be translated into a shorter flight time to the detector. The procedure isrepeated using a different ωRF. The scan over the frequency range reveals thecyclotron frequency of the ion motion.In this work, we employed the ToF-ICR technique. The complete ToF-ICR proce-dure to measure ωc using quadrupole excitations is described in the following.41EBIonFigure 3.8: Off-axis ion injection in a Penning trap using a Lorentz Steerer (blue).ToF-ICR: Initial Magnetron MotionThe first step in ToF-ICR for the measurement of ωc is to prepare the sampled ionin a pure magnetron motion. The ion is normally injected in the center of the trap17,where r+ = r− = 0. Then, a dipole excitation prepares the ion at a magnetronorbit (at TITAN, r− ≈ 1 mm). Since ω− has a weak mass dependence, it may bepreviously measured with a calibrant ion of known mass and easily calculated forthe ion of interest.Alternatively, a technique called Lorentz steering [117] is used for fast ion prepara-tion into a pure magnetron motion. A pair of electrostatic steerer plates is put rightbefore the entrance of the trap, well inside the strong magnetic field. The E × Bdrift drives the ion off-axis prior to its injection into the trap and results in an initialmagnetron motion (see fig. 3.8).This procedure allows one to skip the dipole excitation step, which typically takesa few tens of milliseconds. However, it is also known to induce a small reducedcyclotron motion [117], which is undesirable. Therefore, Lorentz steering is typicallyonly used for mass measurements of very short-lived species (half-life . 100 ms),when long ion preparation times cannot be afforded.ToF-ICR: Quadrupole ExcitationWith the ion in a pure magnetron motion, a conversion to a pure reduced cyclotronis done through quadrupolar excitation. The RF excitation frequency is scannedthrough many measurement cycles. The detuning frequency (∆ωRF) is defined as:∆ωRF = ωRF −ωc . (3.22)Naturally, the conversion is maximal when ∆ωRF = 0. Since ω+  ω−, the ion’sradial kinetic energy is also maximum (see eq. 3.17), so it can be employed as a”gauge” of achieving resonant conditions.If ∆ωRF 6= 0, the complete conversion cannot occur. However, some incompleteconversion may still occur if the detuning is sufficiently small. This can be seen in17 Tuning procedures to achieve that are described in section 3.9, where the radial kinetic energy of the ion after the quadrupole excitationis shown in function of the detuning frequency.The width of this frequency window (∆ν) is an approximate measure of the fullwidth at half maximum (FWHM) of the conversion line shape as a function of ∆ωRF(see panel (b) of fig. 3.9). It depends on the conversion time (tconv):∆ν ≈ 0.8tconv, (3.23)which implies that a ”sharper” (i.e. more precise and better resolving) measurementof ωc can occur with a longer measurement time.In fact, the conversion line shape is very similar to the Fourier transform of theexcitation signal, which is a sine wave modulated by a square pulse of duration tRF(see fig. 3.9). As tRF → ∞, the signal tends to a pure sine wave, whose Fouriertransform is a delta function.Alternatively, a different excitation scheme, invented by Norman Ramsey (NobelPrize 1989), may be used to reduce ∆ν using the same measurement time. It con-sists of splitting the RF signal in multiple shorter pulses, separated by a time ofexcitation-free evolution [118, 119].In a two-pulse Ramsey scheme, as depicted in panel (a) of figure 3.10, an initialquadrupole excitation pulse of duration ton is applied to achieve 50% conversion.Then, both radial eigenmotions are allowed to evolve freely for a time to f f , creatinga phase difference between them. Finally, another quadrupole excitation pulse ofduration ton completes the conversion to pure reduced cyclotron motion. The totalprocedure time is tRF = 2 ton + to f f , although the conversion occurs for a durationtconv = 2 ton. The amplitude of the signal must be adjusted according to eq. 3.21.Using this scheme, the conversion is still maximal at the resonance conditionωRF = ωc, but it creates multiple fringes that spread across a large detuning range18(see (b) of fig. 3.10). This is an effect of the phase interference created as a resultof the excitation-free evolution. The big advantage of this scheme is that the centralfringe is narrower than the line width obtained using the standard scheme at thesame tRF. The Ramsey excitation improves the precision of the measurement by afactor 2 or 3.One drawback of the Ramsey excitation scheme is the need to perform an initialmeasurement using a standard quadrupole scheme. Since many fringes may gen-erate conversion levels very close to 100%, even at large ∆ωRF, the position of thecentral fringe must be unambiguously identified for accurate frequency measure-ment.18 Equation 3.23 is still valid, but for Ramsey excitations tconv < tRF , hence the worse overall resolvingpower.43∆νRF tRF-3  -2  -1  0  1  2  3E r  /  Er(∆ν RF = 0) Signal [arb. un.] 210-1-2-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2t / tRFStandard Excitation Scheme(a) (b)∆ν tRFFigure 3.9: (a) The RF signal (red) in a standard quadrupole excitation can be seen as thesuperposition of a sine wave (yellow) and a square pulse of duration tRF (black).(b) The degree of conversion achieved can be measured by the kinetic energy ofthe radial motion, which is maximal when ωRF = ωc.∆νRF tRF-3  -2  -1  0  1  2  3E r  /  Er(∆ν RF = 0) Signal [arb. un.] 3210-1-2-3-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2t / tRFRamsey Excitation Scheme(a) (b)∆ν tRFFigure 3.10: (a) The RF signal (red) in a two-pulse Ramsey quadrupole excitation is the super-position of a sine wave (yellow) and two square pulses of duration ton separatedby an interval of duration to f f (black). (b) The kinetic energy of the radial mo-tion (red) is also maximal when ωRF = ωc. However, multiple fringes are seenand the line width in the central fringe is sharper than in the standard scheme(dashed gray), which improves the precision of the measurement.44+BzDet.++Fμμ ωRF = ωcωRF ≠ ωcFigure 3.11: Illustrated example on the ion’s time-of-flight dependency on the energy of theradial motion in the trap. The radial energy is converted into axial energy as theion passes through the gradient of the magnetic field; an ion with larger radialenergy (by means of an excitation for example) will arrive sooner at the detector.The bottom graph shows the axial map of the magnetic field strength (red) andthe magnitude of the force (green, see eq. 3.25) along the path to the detector.ToF-ICR: Extraction and DetectionIn this last step, the ”gauge” of achieving resonant condition (ion’s radial kineticenergy) is translated to a detectable signal. After the quadrupole excitation, the ionis extracted from the trap by lowering the potential of one end cap electrode. Itflies towards a detector, and the time between the trap opening and the detectorregistering a hit is measured. The ion flies faster towards the detector if ωRF = ωc,as explained below.The ion in the trap, with its radial motion about the magnetic field (of strengthB), generates a magnetic dipole whose momentum (µ) is dependent on its radialkinetic energy (Er,kin):µ = −Er,kinBzˆ , (3.24)which should be also a function of the applied excitation frequency ωRF.The detector lies outside the magnet. Therefore, the ion must pass through aregion of a gradient magnetic field. Here, it is subject to a force (Fµ):Fµ = ∇(µ · B) . (3.25)The ion’s radial energy is converted to the axial direction and boosts the ion’s ve-locity towards the detector. The time-of-flight from the trap to the detector, passingthrough the electric potential Uz(z) and the magnetic field strength Bz(z) mappedthe axial direction, is then given by:to f (ωRF) =∫ zdzt√m2 [Ez(zt)− qe ·Uz(z) + µ(ωRF) · Bz(z)] dz , (3.26)45626466687072744 6 8 10 12 14 6466687072744 6 8 10 12 14 16 2 νRF - 1386700 Hz  [Hz]νRF - 1386700 Hz  [Hz]Time-of-Flight [μs]Time-of-Flight [μs]tRF = 500 mstON  = 80 mstOFF = 240 ms41K+41K+(a)(b)Figure 3.12: Two ToF-ICR measurements of the cyclotron frequency of 41K+, measured atTITAN facility: (a) with standard quadrupole excitation for 500 ms and (b) withRamsey quadrupole excitation for 400 ms (ton − to f f − ton = 80− 240− 80 ms).Red curves are analytical fits to the data based on eq. 3.26.where zt and zd are the positions in the axial direction of the center of the trapand the detector, respectively. Ez(zt) is the initial kinetic energy of the ion. Sincethe kinetic energy is boosted by the conversion, the time-of-flight is considerablyshorter (see figure 3.11).Examples of typical ToF-ICR resonances where νc is measured are shown in figure3.12, using both standard and Ramsey excitation schemes. Note the resemblancebetween the line shapes and the corresponding radial kinetic energy profile seen infigures 3.9 and 3.10.463.2.4 Practical Considerations of the ToF-ICR TechniqueThe ToF-ICR technique is a reliable technique that is extensively employed in massspectrometry. The technique routinely achieves relative mass precisions as low asa few parts per billion and enabled the measurement of nuclides as short-lived as11Li, with a half-life of only 8.8 ms [120]. However, it does have limitations regardingmore challenging experiments in nuclear sciences, as explained next.Although it is a very precise technique, its resolving power is proportionallyworse than in other techniques. In equation 3.3, CToF-ICR is on the order of unity,while in the MR-ToF-MS it can be over an order of magnitude larger. The resolutionin ToF-ICR is essentially given by νc/∆ν, which according to eqs. 3.23 and 3.5 isRToF-ICRm ≈ 0.2qe Bmtconv . (3.27)Therefore, it has a hard limit bound by the measurement time and cannot be fur-ther improved using better preparation techniques, as most mass spectrometry tech-niques do. It is considerably worse in the Ramsey excitation case, which sacrificesresolution for large gains in precision, as panel (b) of figure 3.12 illustrates.Isotopes of interest are often co-produced with many other isotopes that may bepresent in the sample. In a ToF-ICR measurement, such contaminant species mayoverlap with the species of interest due to the reduced resolution. In addition tothat, the sensitivity of the technique is reduced in the presence of contaminants.Clear resonances are typically obtained with a contaminant-to-interest ratio betterthan 1:5, otherwise the ions of interest would be hardly distinguishable from thebackground. The dipole cleaning technique can successfully remove contaminantspecies at ratios of about 1:200. However, it is not infrequent that contaminantspecies are million times more abundant than the species of interest.For the same reason, it is rare that the species of interest and the calibrant can bemeasured in the same frequency spectrum, which is called internal calibration. In-stead, it requires measurements of νc and νc,re f to be done in separate measurements.This procedure demands extra caution regarding time-dependent fluctuations in thesetup and may lead to additional systematic errors.ToF-ICR is also a scanning technique. It is based on a search for a matchingcondition, which requires the measurement of many ions to trace a clear baseline.In the measurement shown in panel (a) of figure 3.12, about 75% of investigatedions were not in resonant condition, but are necessary to identify the position ofthe resonance accurately. A typical measurement requires at least a few hundredions to be successful. It is a concerning factor when dealing with species that areproduced in minimal amounts. Moreover, although the mass range can be of almostarbitrary length, it is very costly to perform broad frequency scans. In practice, themass range of ToF-ICR is very narrow, surrounding only closely lying species.Next, another mass spectrometry technique is presented. Although not as preciseas PTMS, it addresses many of the issues presented here and enables one to extendmass measurements to rarer species.473.3 Multiple Reflection Time-of-Flight MassSpectrometryTime-of-Flight Mass Spectrometers (ToF-MS) also makes up a class of instruments ofwidespread use in many fields [121], such as nuclear science and analytical chem-istry. It is based on the time separation between two particles, distinct in mass,while traveling the same distance L, starting together at same kinetic energy Ekin.The time-of-flight (to f ) for each particle (of mass m) is given byto f = L√m2 Ekin, (3.28)thus the time separation between particles with masses m1 and m2 is proportionalto L(√m1 −√m2).In such devices ionized particles are sent, in bunches19, through an acceleratingpotential Ua, which defines the initial kinetic energy. The ions enter a field-freeregion where they drift towards a detector. In such case, as Ekin = qe Ua (where qeis the charge of the ion), the time separation between different particles arriving atthe detector scales with L(√m1/q1 −√m2/q2).Similarly to PTMS, the atomic mass (ma) of the species of interest can be obtainedfrom the atomic mass of a reference ion (ma,re f ) and the ratio Rto f between theirtime-of-flights:Rto f =to fto fre f=√ma − q mema,re f − qre f meqre fq, (3.29)where q and qre f are the charge states of the ion of interest and the reference ion,respectively; and me is the mass of the electron.In experiments, the true time-of-flight through the drift region is systematicallyshifted from the measured time-of-flight (tmeas) by a constant time offset (t0):to f = tmeas − t0 . (3.30)The nature of this offset is mostly from ion’s flight outside the drift region, such asduring the acceleration stage, but has also contributions from the detectors and theelectronic processing of the signal. The overall offset is typically a constant in theexperiment and needs to be evaluated.Measurements in ToF-MS are fast compared to PTMS. Cycle times can be onthe order of a tens of µs, permitting high measurement cycle frequency. Largeion samples can be distributed over many cycles, so systematic errors from ion-ioninteraction are reduced. Moreover, ToF-MS is a non-scanning20 technique with awide mass range, which allows the analysis of complex admixtures in the sample19 The bunching feature is necessary to correlate in time a detected event to the start of the flight.20 A scanning technique, as defined in sec. 3.2.4, is based on a search for a matching condition, like thefrequency in ToF-ICR.48in a single measurement. Advanced ToF-MS spectrometers can reach relative massprecisions of a few parts-per-million (10−6) and resolutions of 104.The resolving power (Rm) of such technique depends on the initial energy spreadand initial spatial spread of the ions, which translates to a measured time-of-flightdispersion ∆t in the detected spectrum:RTOF-MSm =to f2∆t. (3.31)The overall effect of ∆t diminishes as the time-of-flight increases. Therefore theresolution of such technique greatly benefits from an extended flight path. Yet, thereare clear practical limitations that prevent the drift region from being expanded toarbitrary lengths. One elegant way of circumventing this is to recycle the flight path,circulating the ions through the same analyzer multiple times.This is the concept behind the Multiple-Reflection Time-of-Flight Mass Spectrom-eter (MR-ToF-MS) [122, 123]. The field-free drift region is placed between a pair ofelectrostatic mirrors. The ion sample is reflected back for as many passes throughthe analyzer region as desired, effectively making the device in a spectrometer withtunable length. Flight paths on the order of kilometers are achievable using a com-pact system of about a meter long.Next, the concept of the MR-ToF-MS and its operation as a mass spectrometer areexplained in detail in subsections 3.3.1 and Concept of the MR-ToF-MSA typical MR-ToF-MS setup includes an ion preparation device, a mass analyzer,and a fast-timing detector. The preparation device is required to appropriatelyinject ions in the mass analyzer. As already mentioned, the ion bunches’ initial dis-persions in energy and position should be as low as possible for a greater resolution[123]. In many MR-ToF-MS devices a dedicated preparation ion trap, with capabili-ties of cooling and bunching ion samples, is placed right before the entrance of theanalyzer. Further details on ion preparation are given in sec. 4.2.4.The mass analyzer of an MR-ToF-MS is composed of a drift section between twoelectrostatic reflectors. A typical electrode structure concept is shown in figure 3.13.During the measurement procedure, the electric potential in the mirrors exceeds thekinetic energy of the ions. Therefore the ions are confined inside the device, movingback and forth between mirrors. The outermost electrodes can be switched to lowerpotential to allow ion bunches to be injected and extracted through apertures placedin the end electrodes. After the ions fly a certain number of turns inside the analyzer,they are extracted to the detector. The time-of-flight through the whole procedure,from the moment the ions are injected in the analyzer to when they hit the detector,is registered.In the mass analyzer, the drift section is a field-free region where the ions fly at aconstant speed. It is specially designed to minimize electric field penetrations that49DetectorU(z)Ion injectionFigure 3.13: Scheme of the electrode structure of a typical MR-ToF-MS mass analyzer (top)and the correspondent generated electrostatic potential along the axis of thespectrometer (bottom). Adapted from [124].could interfere in the ion’s trajectory. Optionally, some systems include a mass rangeselector in the middle of the drift region to aid in the selection of a specific massrange [123]. More details are given in sec. 3.3.2.The design of the mirrors is a critical step to the appropriate working of the massspectrometer. They are usually a stack of a few electrodes that shape the electricfield of the mirror. The mirrors must be gridless; otherwise, the multiple passesof an ion bunch through the grids would lead to scattering and changes in theion energy and trajectory, and considerably reduce the efficiency of the device. Agridless design requires extra care in the construction of the mirrors to minimizeaberrations, i.e. imperfections on the shape of the mirror potential that cause animperfect reflection.Most importantly, the mirrors must be tuned to ensure two properties: confine-ment and isochronicity, described next.ConfinementIn order to extend ion flight paths to long lengths (typically hundreds of meters orkilometers), ions need to have stable orbits inside the device. Per design, the axialconfinement is provided by the potential well created by the mirrors. However, ionbunches also expand radially. As the flight path grows, this expansion may causesevere ion losses due to collisions with electrodes.To account for such expansion, one electrode in each mirror also works as anelectrostatic device called Einzel lens [125] to refocus the ion trajectories in everyturn. An Einzel lens, as in fig. 3.14, creates a small region of fast acceleration anddeceleration, which changes the ion velocity in the radial direction. Parallel ionbeam trajectories should converge in a focal point after the lens. The position of thefocal point is tunable according to the potential difference (UEL) between the lens’electrodes and is independent of the ion’s mass [126].50UELU(z)zFigure 3.14: Parallel ion trajec-tories are focused as they passthrough an Einzel lens. Below, theelectrostatic potential in the axisof the lens is shown.Figure 3.15: Example of a simulated trajectory of an ion beinginjected, making one full turn in the analyzer and being ejected.Adapted from [129].In the MR-ToF-MS analyzer, the Einzel lenses in the mirrors can be identified infig. 3.13 as the two deep potential wells. With appropriate tuning of the lens’ poten-tial, radial ion confinement is achievable. Hence, both axial and radial confinementare provided by the mirrors, and trajectories inside the analyzer can be made stablewith adequate tuning. One example of such a stable orbit is given in figure 3.15.Consequentially, the MR-ToF-MS can be considered as a form of electrostatic iontrap. The achievable storing time, however, is typically short compared to mostion traps. The current generation of MR-ToF-MS can confine ions for tens of mil-liseconds. Meanwhile Penning traps can store a sample for several seconds, oreven months in a few devices [127]. Although this remains to be further investi-gated, losses in MR-ToF-MS are generally attributed to ion interactions with thebackground gas, and transverse expansion of the beam due to optical aberrations[124, 128, 123].IsochronicityIsochronicity is the property that time-of-flight differences in any ToF-MS systemonly depend on the ion’s mass-over-charge ratio, independent of their initial posi-tions, incident angles and energies. Only then eq. 3.29 can be true. Yet, time spreadsregarding initial conditions are unavoidable, although they can be minimized.Ultimately, it is required that the timing detector is positioned in a time focus ofthe analyzer. The concept of time focus is illustrated in an example in figure 3.16.Ions that start their flights at different positions and energies are subject to slightlydifferent works by the accelerating potential. Ions initially closer to the drift sectionenter it with lower speed than ions initially further; eventually, trajectories cross thesame plane at the same time. Such accelerating potentials can be tuned to generatea time focus in a specific point of the setup.The field gradients of the mirrors are tuned so that faster ions fly further insidethe mirror and, thus, take more time to turn. The right balance between the timespent in the analyzer and the time spent in the mirror yields a time focus for everyturn. Consequentially isochronous orbits are obtained in the analyzer.51zTime FocusU(z)Ion EnergyFigure 3.16: The energy and position spread of the ions (with same m/q), under an accel-erating potential U(z), causes differences in speed and path length in the driftsection. This effect cancels out in the ”time focus” position, where ions simul-taneously arrive. Changing the shape of U(z) shifts the position of the timefocus.The tuning of the reflectors is done for an ion bunch with specific energy dis-tribution. If one parameter is changed, the system loses its overall isochronicity.Therefore, the stability of the overall system is important, which includes tempera-ture stability, mechanical stability, and stability of power supplies.Also, a finite number of mirror electrodes means that the ”ideal” reflecting po-tential is truncated up to some order. It generates optical aberrations that affection orbit and, thus, the isochronicity of the system. This is illustrated in figure3.17, which shows the time-of-flight of several simulated particles (with a Gaussianenergy distribution) after one isochronous turn in an analyzer. The effect of aber-rations due to the truncated potential is seen in the tails of the energy distribution.On the other hand, adding more mirror electrodes increases the degrees of freedomof the system, which makes it harder to tune and more prone to instabilities. Anoptimal configuration balances aberration effects and ease of tuning and operation.3.3.2 Operation as a Mass SpectrometerThe isochronous operation of the mass analyzer turns the MR-ToF-MS into a fastand versatile high-resolution mass separator. However, challenges appear when thesystem is used to reach the highest possible resolving power.In ToF-MS techniques, the final mass resolution depends on the time spread ∆t0acquired by initial conditions. MR-ToF-MS systems minimize the influence of initialconditions by extending the flight path to a large number of turns Na inside a massanalyzer. However, as seen in fig. 3.17, the isochronicity is not perfect, and a smalltime-of-flight error ∆ta is added at each turn. The resolving power in such systemis then [124]RMR-ToF-MSm =t0 + ta Na2√(∆t0)2 + (∆ta Na)2, (3.32)520. dispersion [ns]Ion kinetic energy [eV]Figure 3.17: Simulated time-energy relation of an ion cloud through one turn in a mass an-alyzer. All particles departed simultaneously from the middle plane of the ana-lyzer with an energy spread of 22.5 eV around 1.5 keV. The top panel showsthe histogram of the simulated events projected in the time axis. The non-isochronous patterns seen in the scattered events are created by the time-of-flight aberrations. Adapted from [129].where ta is the time-of-flight through a single turn in the analyzer. As the numberof turns increases, the resolution reaches saturation:limNa→∞(RMR-ToF-MSm ) =ta2∆ta, (3.33)which can be orders of magnitude larger than a standard ToF-MS system.In figure 3.18, a sample time-of-flight spectrum is shown, which was taken nearthe saturation resolving power of the used spectrometer. The achieved resolutionexceeded 2 · 105 and enabled clear separation of the isobar doublet 40Ar+ and 40K+.Naturally, ∆ta can be reduced with careful design and construction of the ana-lyzer: aberrations and field penetrations need to be minimized; pieces need to beprecisely machined and aligned; good vacuum need to be reached; materials withlow thermal expansion coefficients need to be selected in order to minimize lengthsfluctuations due to thermal expansion; power supplies need to be stable, and theelectronics for signal processing need to have their noise minimized.However, a few operational strategies and techniques considerably improve theresolution and the mass spectrometry capabilities of the MR-ToF-MS. Some of those,which are employed at the TITAN MR-ToF-MS system, are discussed next.5340Ar+40K+600 Turns (+1 TFS)Rm = 202 000Time-of-Flight - 7477.0 μs   [μs] Figure 3.18: A time-of-flight spectrum from a sample containing 40Ar+ and 40K+ analyzedat TITAN’s MR-ToF-MS using 600 isochronous turns. Reproduced from [130].Time Focus MatchingAs mentioned, a typical mass measurement cycle in an MR-ToF-MS consists of theinjection of ions from a preparation device followed by an arbitrary number ofisochronous turns inside the analyzer. Subsequent extraction of ions to a timingdetector follows. However, the time focus at the isochronous turns is tuned for aspecific flight path that does not comprise of the injection and extraction procedures.In such cases, the ideal time focus lies outside the analyzer.This can be corrected: the time focus matching from injection and extraction isdone by time focus shift (TFS) reflections [131]. In those cases, one mirror potentialis temporarily switched to a different potential shape than the isochronous one. A”harder” potential (steeper derivative) brings the time focus closer, while a ”softer”drives it further away. The switch to (or from) a TFS configuration is done whenions are in the other mirror to minimize the influence of ringing and switchingnoises.In figure 3.19, one operation scheme is shown. It includes one TFS in the firstturn to drive the time focus to the center of the drift region, and another TFS in thelast turn to shift the time focus to the detector. In fact, many operations schemesare possible as long as the time focus remains at the detector. Studies reported thatthe same resolving power was obtained with 3 times fewer turns with the use ofTFS turns [131].54zTFS toAnalyzerTFS toDetectorDetectorPrep. trapTimeDrift regionMirror MirrorTimeFociFigure 3.19: Example of a measurement cycle in an MR-ToF-MS. Ions (black trajectories) areinjected into the analyzer, where they performed four isochronous turns beforegetting extracted to a detector. Two time focus shift reflections are performedto focus-match the injection and extraction of ions. Note that more energeticions (solid line) have a larger amplitude, entering further inside the mirrors; yet,trajectories inside the analyzer are isochronous.CountsMeasurement CycleTime-of-Flight [arb.] Time-of-Flight [arb.](a) no TRC (b) with TRCFigure 3.20: Time-of-flight histograms over many measurement cycles of the same sample,measured at the TITAN MR-ToF-MS. On top are the integrated histogram of allmeasurement cycles. In (a), the raw spectra shows large time fluctuations. In(b), a time-resolved calibration was applied to the same data, and the resolutionwas improved in 75%. The sample was analyzed for 20 minutes.55Temporal FluctuationsA typical mass spectrometry measurement for one isotope may range in time from afew minutes to an hour, depending on the desired precision and the available yieldof the ion of interest. The longer the measurements take, the more the system isprone to instabilities of power supplies or thermal swings that may alter the overalllength of the analyzer. Such temporal variations slightly change the time-of-flightof ions and cause broadening of the peaks.Such effect can be seen in panel (a) of figure 3.20: ion samples from the samesource were repeatedly measured for 20 minutes, and the position of its time-of-flight peak significantly varies in time.To first order, fluctuations affect all ions proportionally to their√m/q, keepingthe isochronicity of the system. Using this fact, a time resolved calibration (TRC) canbe applied to correct the spectrum for time-dependent drifts, as a post-measurementprocedure [132].In this procedure, the data is partitioned in time, and one intense peak in thespectrum is taken as a reference (as an internal calibrant). Every time partition isshifted in time-of-flight so that the centroid of the reference peak coincides with itscentroid in the first partition. The result can be seen in panel (b) of figure 3.20, andmay significantly improve the resolving power.This procedure requires that an intense and well-resolved peak is present in thesample. If such species are not present, one artificially introduces an ion to thesample, preferentially with the same mass number as the species of interest, froman external ion source. Such a procedure is described in [133].Mass Range SelectorA limitation of the MR-ToF-MS technique is an ambiguous mass range: ions withdifferent masses may undergo a very different number of turns in the analyzer, butmay appear very closely, or even overlapping, in the spectrum. This problem isillustrated in panel (a) of figure 3.21.The ”unambiguous” mass range can be defined as the mass range that containsonly species that went through the same number of turns (Na). It can be character-ized by the ratio between the maximum and minimum m/q:(m/q)max(m/q)min<(Na + 1Na)2. (3.34)As the number of turns increase, the unambiguous mass range shrinks and thespectrum is more likely to have mixed-turn species.In principle, it is possible to disentangle mixed-turn spectra by taking multiplespectra at different numbers of turns. However, an elegant solution to this is toinclude a mass range selector (MRS) [134, 123, 135] in the middle of the mass analyzer.It consists of a pair of deflecting electrodes with fast-switching capabilities. An RFsignal is applied to one of the electrodes, momentarily switching the MRS from56zDetectorPrep. trapTimeDrift regionMirror MirrorMRSzDetectorPrep. trapTimeDrift regionMirror MirrorMRS DeflectingMRS OpenIon outside mass range deflected4 turns4 turns5 turns tof tof(a) (b)Figure 3.21: (a) The ambiguous mass range of MR-ToF-MS illustrated: a sample containingthree different species are analyzer by an MR-ToF-MS, one (red) is considerablylighter and undergo more turns than the other two. In the spectrum (insert) itappears like an intermediate mass between the other two species. (b) A solutionto this is to have a mass range selector tuned to only allow species within acertain mass range, deflecting the unwanted species and making the spectrumunambiguous. The green band shows the spatial dimensions of the MRS in theanalyzer, while grey bands show the times when the MRS is open state (through which ions pass unaffected) to a deflecting state, where atransverse electric field drives ions well out of their isochronous orbit, effectivelyremoving them from the analyzer. The frequency of the signal is tuned to therevolution frequency of a specific desired mass range inside the analyzer, and ionswithin the range always pass through the MRS while it is in the open state.The principle of the MRS is illustrated in panel (b) of figure 3.21. It must benoted that the design of the MRS must take into account minimal fringe fields inthe analyzer section, and only low noise electronics should be employed.3.3.3 Practical Considerations of the MR-ToF-MS TechniqueMR-ToF-MS is an emerging technology in nuclear sciences. Its use in nuclear sciencewas first proposed less than two decades ago [136]. The current generation of suchspectrometers is able to achieve resolutions of 105 and relative mass precisions of10−7 [123] which, combined with the advantages of a typical ToF-MS spectrometer,makes it a very competitive tool for nuclear sciences. Due to its performance, suchsystems are now in use or planned in every major radioactive ion beam facility[137].As mentioned, MR-ToF-MS are fast and non-scanning spectrometers. A typicalmeasurement cycle analyzes all species contained in the sample with single ionsensitivity and may only take a few milliseconds. If compared to the typical cycleof hundreds of milliseconds of the ToF-ICR PTMS technique, its performance is5710-5 10-4 10-3 10-2 10-1 100104105106107ToF-ICR PTMS    m/q:            2    6   20   60          200  Mass Resolving PowerMeasurement Time [s]MR-ToF-MSFigure 3.22: Performance of two mass spectrometry techniques (for several m/q) regardingtheir resolution as a function of their measurement times: MR-ToF-MS in redand ToF-ICR PTMS in blue. Values are based on the two spectrometers of TITANfacility (see sec. 4.2). It worth noting that ToF-ICR PTMS typically yields overone order of magnitude larger precision than MR-ToF-MS for the same resolvingpower and accumulated statistics.very appealing for measuring very unstable and rare species. In figure 3.22 themass resolution of the two techniques is compared as a function of the time scale oftheir measurements.Because of its advantages, MR-ToF-MS systems have also been used for other ap-plications in rare ion beam facilities. They have been employed as a high-resolutionmass separator to deliver isobarically clean samples to other experiments [49] andas a beam diagnosis tool [138], which is also done in this work. The broadbandoperation and high sensitivity enable quick identification and quantification of un-known ion samples, which in turn allows fine-tuning of parameters involved in ionsample production and transport.584 EXPER IMENTAL OVERVIEWThe experiment at the core of this thesis was performed at TRIUMF’s Ion Trap forAtomic and Nuclear science (TITAN) facility [69]. Both PTMS and MR-TOF-MS sys-tems were employed. TITAN’s Penning trap mass spectrometer, the MeasurementPenning Trap (MPET), has been in operation for over a decade. It is a very wellestablished spectrometer for the mass measurement of radioisotopes and holds theWorld record for the fastest PTMS measurement [120]. TITAN’s MR-ToF-MS was in-stalled at the facility in April 2017. The results described here are from the inauguralexperiment of the device; therefore, parallel and independent measurements of thesame samples were performed using the two spectrometers. In this way, the resultsof the MR-ToF-MS could be compared with the results of the MPET spectrometer.The TITAN facility itself is located at the Isotope Separator and ACcelerator(ISAC) [139] facility of TRIUMF laboratory in Vancouver, Canada. In the follow-ing section, an overview of the ISAC facility and its equipment employed in thisexperiment is presented. The TITAN facility is introduced in detail in section 4.2.Finally, the experimental procedure is outlined in section ISAC Facility and Isotope ProductionISAC uses the well established Isotope Separation On-Line (ISOL) method [139,140] to produce Radioactive Ion Beams (RIBs). The RIBs are produced throughnuclear reactions induced by a highly energetic (typically between 50 MeV to 1.5GeV) driver beam on a thick target, inside which the reaction products stop. Thetarget is coupled to an ion source, and the ionized isotopes are formed into a beamand sent through a mass separator. The selected isotope beam is then available forexperiments.The ISAC RIB production facility is depicted in figure 4.1. The main stages ofthe ISOL technique - production, ionization, and separation - are described in thefollowing in the context of this experiment.Production of RadioisotopesThe neutron-rich titanium isotopes were produced in an ISAC target made of nat-ural tantalum impinged by a 480 MeV proton beam at 40 µA. The driver beamwas provided by TRIUMF’s main accelerator, an 18-meter cyclotron that can deliverproton beams up to 520 MeV.59laserbeamslaserbeamsFigure 4.1: The RIB production facility at ISAC receives a high-energy proton beam (blue)from TRIUMF’s main cyclotron and produces radioisotopes in a hot thick target,whose module is depicted in the inset. Ions are produced, in the case of thisexperiment, through resonant photo-ionization by shining lasers (green) onto thetarget module. An ion beam (red) is created, sent through a mass separator anddelivered to experiments. Figure adapted from [141].Neutron Number0 20 40 60 80 100 120Proton Number010203040506070803104105106107108109101010N=32Z=22 (Ti)In-Target Production [isotopes/s/(mmol/cm2)/μA] ISAC simulation ID: geant4-10-02-QGSP_INCLXX_HP-ABLA_V3 Target: natural Ta (Z=73), 0.05 mol/cm2Incident Particle: proton, 480 MeVFigure 4.2: Simulated normalized production yields of radioisotopes using a tantalum ISACtarget, obtained through the ISAC Yield Database [142]. Both stable isotopes (inblack) and the region of interest (titanium isotopes around N = 32) are markedfor reference.60The protons induce nuclear reactions in the target. Tantalum nuclei undergospallation and fragmentation reactions, and the product particles stop inside thetarget medium. In this process, a large number of isotopes are produced [143].Figure 4.2 shows simulated radioisotope production under the conditions of theexperiment, highlighting the region of interest in this experiment.Release and IonizationIn the ISOL technique, to form an ion beam, the isotopes of interest need to bereleased from the target and reach an ion source. The release process requiresisotopes to diffuse out to the surface of the target material, undergo desorption andeffuse out of the target container towards the ion source.ISOL targets are heated to high temperatures (T ≈ 2000 ◦C) to enhance the releaseprocesses. However, chemical and physical processes that influence the releaseefficiency, such as diffusibility, volatility, and decay half-life, affect the availabilityof certain beams [140, 144]. As a consequence, for example, the production ofion beams of refractory metals and reactive elements, such as iron and boron, iscurrently very challenging [145].After the release, the isotopes are formed into an ion beam by an ion source.The most common and simple ionization process found in ISOL facilities is surfaceionization. In many cases, the produced ion beam will have surface ionized com-ponents (unless the source is specially designed to suppress them, such as in [146]).The surface ionization efficiency (εion) strongly depends on the temperature (T) andon the species’ first ionization potential (Ei):εion ∝[1+ e−(W−Ei)/kBT]−1, (4.1)where W is the work function of the surface material and kB is the Boltzmannconstant.For example, alkali species, whose ionization potential is typically below 5 eV, arevery easy to be surface ionized. Titanium, however, has a relatively high ionizationpotential (∼ 6.8 eV, see fig. 10 of [147]). Moreover, its target release properties aresuch that it was even included among particularly difficult (or almost impossible)ISOL beams [145]. Therefore, a different technique was employed to efficientlyionize titanium atoms released from the target: the resonant laser ionization [148].The technique photo-ionizes atoms by using high-power lasers that resonantlypromote atomic transitions. In a multi-step process, a valence electron is excitedto a high-lying energy level and then detached from the atom by another transition(via an auto-ionizing state, for example) [148]. This method is element selective, andmost atoms that reach the laser interaction region are successfully ionized, makingit extremely powerful for rare isotope research.At ISAC, the TRIUMF’s Resonance Ionization Laser Ion Source (TRILIS) [149] wasemployed in the ionization of titanium isotopes, using a two-step laser ionization61scheme [150]. The two laser beams were merged into the ionization region of thetarget module.Mass Separation and Delivery to ExperimentsAfter ionization, the ions were formed into a continuum beam from the target. Thetarget and ion source modules were biased at 20 kV, which electrostatically acceler-ated ions as they leave the modules. The beam was sent through a two-stage dipolemagnet separator that includes a pre-separator and a high-resolution magnet sep-arator [151]. This setup has a mass resolution of about Rm = 2000 (see eq. 3.2),which is enough to select the desired mass number but does not allow to entirelyseparate isobars. The ion beam was transported through electrostatic optics in ahigh vacuum environment and delivered to the TITAN facility.4.2 The TITAN FacilityThe TITAN facility is an ion trap system developed to perform mass spectrometryof rare and short-lived isotopes [152]. It has been in operation since 2007, with thecommissioning of its Penning trap mass spectrometer [120, 153]. It distinguishesitself from other facilities by its ability to employ fast measurement cycles and byits capability of increasing the charge state of the analyzed ions. Its diverse combi-nation of ion traps, coupled to the intense isotope production sources available atISAC, makes TITAN a unique facility in the world.The TITAN facility is depicted in figure 4.3, highlighting the main components.The two mentioned mass spectrometers are described in detail in sections 4.2.2 and4.2.3. The following devices are also central to the operation of TITAN experiments:RFQ: the ion beam delivered from ISAC is accumulated in a Radio-FrequencyQuadrupole cooler and buncher (RFQ) [154]. The RFQ is a preparation iontrap filled with He gas for beam cooling. The RFQ delivers bunched beamwith a small energy spread to the other ion traps at TITAN. Further details onthis device are presented in sec. 4.2.1.Stable Ion Source: the RFQ can also receive stable beams from a surface ioniza-tion alkali source, typically employed for tuning and optimizing the systemand as a source of calibrant ions. For both applications, it is desired to use ionswith a mass close to the species of interest, A = 50− 56 in this case. A sourcecontaining natural potassium and rubidium was installed, so ion species of39K, 41K, 85Rb and 87Rb were available.EBIT: an Electron Beam Ion Trap (EBIT) is typically used as an ion charge breeder,further ionizing ions through electron impact ionization [155]. Employing ionsat high charge states is known to boost precision and resolving power in theToF-ICR technique (see eq. 3.27) [156]. However, it requires additional ionpreparation time, which can be prohibitive for very short-lived species, and62RFQ Cooler & BuncherEBITMPETMR-ToF-MSCPET(future)Stable Ion SourceBeam from ISACFigure 4.3: Schematic overview of the TITAN facility, depicting its ion traps and the beamtransport paths employed in this experiment. Continuous ion beams (blue lines)are received from ISAC or TITAN’s stable ion source in the RFQ cooler andbuncher. The RFQ delivers cold ion bunches (red dashes) to other traps. Inthis experiment, ion bunches were independently sent to the MR-ToF-MS and tothe MPET.may reduce the quality of resonances. A Cooler Penning Trap (CPET), de-signed to provide further cooling of highly charged ions from EBIT, is beingprepared to be installed in the TITAN beamline in the near future [157].With the installation of the MR-ToF-MS, it is essential to compare the two spec-trometers accurately. Therefore, it is desirable that both spectrometers measure thesame ions in similar conditions to avoid systematic effects. The titanium ions weremeasured in a singly charged state (q = 1) in both spectrometers. The EBIT wasbypassed in this experiment.All TITAN ion traps are connected by about 12 meters of ion transport beam lineskept under ultra-high vacuum (UHV) conditions (≈ 10−8 Pa). The transport of ionsbetween devices is done by electrostatic ion optical devices such as Einzel lenses,correction steerers, benders and quadrupoles. The transport and optimization ofion beams through TITAN is described in section 3 2422+ ++++continuous beambunched beamcapture potential20 10 0 -10 -20+++extraction potentialcooling and accumulationAxial Potential [V]zzxyRadial Potential [V]x y(a) (b)(c) (d)Figure 4.4: The electrode structure of the TITAN RFQ viewed from the (a) axial and (b) radialorientations, the color code indicates the phases of the RF applied to the rods andpanel (b) also features a simplified electronic scheme used to bias each electrodewith RF and DC voltages. (c) The electrical potential in the axial direction, as usedin this experiment, is shaped to guide the ion as it cools towards a potential wellby the extraction end. (d) In the radial direction, the alternating saddle potentialcreated between the rods keeps ions confined. Adapted from [154].4.2.1 Preparation of Ion Beams with the TITAN RFQThe ISAC facility delivers continuum ion beams to TITAN at a kinetic energy oftypically 20 keV. The radioisotope production process introduces energy spreads ofa few tens of eV. Such conditions are not favorable for precision experiments in iontraps. In ion traps, ions are brought to a quasi-rest state, with a low energy spread.Moreover, it is desirable to delivered ions in bunches for optimized capture into atrap.The TITAN RFQ provides the interface between ISAC and TITAN [154]. It is agas-filled Paul trap (introduced below) that is used to accumulate the delivered ionsfor a certain period, reduce their energy and energy spread, and send them to thesubsequent traps in bunches.The overall structure of the RFQ is shown in figure 4.4. It is mainly composedof four parallel rods of 70 cm length, and each rod is segmented into 24 electrodes.It is contained in a chamber filled with high-purity He gas at a pressure of ≈ 1Pa. Differential pumping apertures on each side allow ion injection and extractionbetween the gas-filled chamber and the UHV environments of ISAC and TITAN.In the following, the principle of trapping, cooling, and bunching of the RFQ areexplained.64TrappingAs in any ion trapping device, ions must be confined in three dimensions. In a linearPaul trap such as the TITAN RFQ, the confinement in one of the directions (axial)is given by an electrostatic potential well as in Penning traps. Such configurationcan be generated by segmenting the rods, here into 24 electrodes. The very first 22electrodes generate a slowly-varying drag field that guides ions towards one endof the RFQ. The last few electrodes create a deeper potential well where ions areaccumulated as they cool down. This is illustrated in panel (c) of figure 4.4.A dynamic electric field generates the confinement in the radial (or transverse)direction. In each rod, an RF signal is superimposed on the static field that generatesthe axial potential. The signal is applied with 180◦ phase shifts on sets of rodsperpendicular to each other, as shown in (b) of figure 4.4. The resulting field isdescribed by a time-varying quadrupole. The potential surface at a given time isdescribed by a saddle, as in panel (d) of figure 4.4. It is metastable. With the correcttuning of the frequency and amplitude of the signal to the m/q of the trappedspecies, an average confinement is created. Further details can be found in [154].CoolingAs mentioned, the incoming beam needs to be decelerated from 20 keV to quasi-rest, with minimal energy spread. The bulk of the deceleration is done by floatingthe RFQ on a high-voltage platform with bias slightly below the beam energy. Thebeam is injected into the RFQ with a few tens of eV, just enough to surpass theentrance potential barrier.The rest of the energy is dissipated by collisions with the helium buffer gas insidethe RFQ. Multiple collisions occur until the ions are in thermal equilibrium with thegas. After thermalization, the energy distribution is Boltzmann-like with a spreadof a few tens of meV. The complete cooling process takes a few milliseconds [154].Helium is an appropriate choice of a buffer gas21. It is a light gas, so it efficientlyabsorbs momentum from almost any (heavier) incoming beam. Also, He is chem-ically inert, with a high ionization energy of 24 eV. This is essential to prevent theinjected ion from changing its ionization state by electron exchange with the gasparticles. This is further facilitated by high purity. Small amounts of contaminationin the buffer gas may significantly reduce the efficiency of the device.BunchingAfter a few milliseconds of continuous beam injection and cooling, the ions areaccumulated in the potential well. This bunch is then ready to be sent to the otherion traps at TITAN. This is done by switching the last few electrodes to an opentrap configuration, as shown in panel (c) of figure 4.4 (dashed green curve). Thebunch is released to the TITAN system and can be transported to the other devices.21 Occasionally, different buffer gases may be more suitable to cool the incoming beam. For example, withlighter species (A < 12), H2 has shown to increase the efficiency by a factor 2 compared to He.65CentralRingSegmented CorrectionGuardsEnd CapsCorrection Tubes(a) (b)Figure 4.5: (a) Schematic of the MPET electrode structure. On the guard electrodes, the blue-red color code express the opposite phases of quadrupolar excitation. Figureadapted from [110]. (b) Photography of the MPET assembly for scale, courtesyof Stuart Shepherd.4.2.2 The Measurement Penning TrapThe TITAN MPET is a Penning trap mass spectrometer designed to provide fastmeasurements of short-lived species, and it is the only in the world to be able toperform mass spectrometry of unstable highly-charged ions. It currently employsthe ToF-ICR method and can provide mass precision and accuracy as good as onepart-per-billion [110, 158]. The characteristics of the MPET mass spectrometer aredescribed below.Electrode configuration: the MPET uses a hyperbolic Penning trap system suchas described in chapter 3. A schematic of its electrode structure is available infigure 4.5 and its characteristic dimensions are presented in table 4.1. The cen-tral ring electrode and the end caps are shaped as surfaces of two hyperboloidsof revolution. Besides them, additional electrodes are included to correct thepotential for truncations of those surfaces: ”tube” electrodes are added afterthe end caps to compensate for the 4 mm apertures used to inject and extractions, ”guard” electrodes are added between ring and caps to provide higherorder field corrections. The trap ”depth” of MPET, defined as the potentialdifference between the ring and end caps, is typically of 35.75 V. The electrodecompensation procedure to shape the adequate trapping potential is describedin [110].Table 4.1: Characteristic dimensions of the MPET, according to eq. 3.7.r0 15.000 mmz0 11.785 mmd0 11.210 mm66Ions  MCPMPET Injection OpticsExtraction Optics-600  -500  -400  -300  -200  -100     100  200  300  400  500   0Distance from trap center  [mm]43210Magnetic Field  [T]Extr. Drift Cone Extr. Drift TubeEinzel Lens Set Pulsed Drift Tube (PLT)Lorentz Steerer Inj. Drift Cone Inj. Drift Tube(b)(a)(c)Figure 4.6: (a) Drawing of the MPET setup enclosed inside the magnet bore, showing thetrap, its injection and extraction systems and the support and wiring structures.(b) The electrode setup of MPET and its associated ion optics. (c) Map of the axialmagnetic field strength across the position inside the bore.Magnetic field: the trap configuration sits in a 3.7 T magnetic field generated bya superconducting solenoid (see fig. 4.6). The used magnet performs stablyover time, which reduces systematic uncertainties. The MPET magnetic fielddecays at a relative rate on the order of 10−11 per hour.RF excitations: the RF signal for ion manipulations is applied to the guard elec-trodes, which are appropriately segmented (see fig. 4.5.a).Fast ion preparation: MPET is one of the few PTMS systems to employ Lorentzsteerers in ion preparation into an initial magnetron orbit [117] (see fig. 4.6).As explained in section 3.2, this technique allows for rapid magnetron radiuspreparation of the sample and achieves a high level of reproducibility. The useof Lorentz steering enabled, for example, the measurement of 11Li at MPET,with a half-life of 8.75(14) ms [120].Detector: the ToF-ICR method requires detection with single-ion sensitivity. Forthis purpose, a Micro-Channel Plates (MCP) [159] detector was installed aboutone meter downstream of the trap ejection. The efficiency of MCP detectorstypically ranges between 40% to 60%, depending on the ion’s velocity, chargestate, age and conditioning of the detector.The sources of systematic deviations in mass measurements due to the construc-tion of MPET, such as imperfections of trapping electrodes, field misalignments,instabilities of the trapping potential and magnetic field and anharmonicities of thetrapping potential, were evaluated. Combined, they yield deviations on the orderof 2 · 10−10 per unit of m/q [110]. However, other systematic sources such as ion-ioninteractions and relativistic effects should be evaluated on a case-by-case basis.MPET sits in a UHV environment at a pressure on the order of 10−7 Pa, which en-ables mass measurements of ions at high charge states up to q ≈ 18. The experiment67MassAnalyzerTransportSystemPreparation Paul trapMCP DetectorMRSIons fromTITAN RFQEinzel LensIon SourceSwitchyardTransport RFQsAccumulation RFQsIons toMPET or EBITFigure 4.7: Scheme of the TITAN MR-ToF-MS, highlighting its main components. Figureadapted from [160].described here is the last one performed with this trap configuration after a decadeof operation. A new Penning trap system that will replace MPET is under com-missioning. The new system, described in appendix B, has an improved cryogenicvacuum system to be able to access charge states well beyond +20 and is beingprepared to enable the PI-ICR technique [115] of cyclotron frequency measurement(briefly described in section 3.2.3).4.2.3 The TITAN MR-ToF-MS SpectrometerThe TITAN MR-ToF-MS system [160] was installed and commissioned at TITANfacility in April 2017 [130]. It was developed by the IONAS group at the Universityof Gießen [161, 123] based on the established concept of the MR-ToF-MS installeddownstream of the fragment separator at the GSI laboratory [124] in Darmstadt,Germany. It can provide mass values with 10−7 relative precision with great ionsensitivity. In addition, the system is able to tolerate contaminant ions with ratesup to 106 times higher than the rate of ions of interest. Hence, it acts as a comple-mentary mass spectrometer to MPET.Besides being a research station itself, the TITAN MR-ToF-MS was also designedto serve as a high-resolution mass separator to couple to other traps at TITAN. Witha mass resolution beyond 2 · 105, overall efficiency on the order of 50% and capacity68to hold more than 104 ions per cycle, the TITAN MR-ToF-MS is capable of perform-ing sample purification up to the isobar level. Therefore, beyond its operation as amass spectrometer, its setup is also able to separate specific isobars from a highlycontaminated sample in the mass analyzer, recapture only the desired species, andsend ions back to the TITAN facility to be used in other experiments. This sectionfocuses on the operation of the TITAN MR-ToF-MS as a mass spectrometer.The MR-ToF-MS was designed to be a stand-alone system. It has its own internalion source and ion preparation apparatus to allow independence with the rest ofTITAN beamline. It enables its tuning and operation simultaneously with otherTITAN systems without conflicts. The MR-ToF-MS system is schematically depictedin figure 4.7. It can be divided into two sectors: the ion transport system and themass analyzer, described next.Transport SystemAs a stand-alone system, the TITAN MR-ToF-MS has its own ion preparation setup.Ions are received from the TITAN RFQ and accumulated, cooled and bunched be-fore their injection into the mass analyzer. The preparation and transport systemallows the TITAN MR-ToF-MS flexibility. Beyond the aforementioned advantage ofindependent tuning from the rest of the TITAN system, the MR-ToF-MS can keepits own measurement cycle independent of the rest of the TITAN setup.The transport system is schematically depicted in figure 4.7. All transport com-ponents are based on RF quadrupole optics in a He buffer gas environment, usingthe same concept of the TITAN RFQ22. Ions are efficiently transported through thesystem at low kinetic energy, on the order of a few eV. The main components arebriefly described in the following, while a more complete description can be foundin [162].Accumulation RFQs: the interface between TITAN and its MR-ToF-MS is donethrough a pair of RFQs. One receives ions from the TITAN RFQ while theother sends isobarically clean beams towards MPET or EBIT. The latter is doneafter a separation cycle in the mass analyzer.RFQ Switchyard: the RFQ switchyard is a cube-shaped RF optics device with 6RFQs overlapping on its center. It is capable of receiving and transmitting ionsat any side. It may receive ions from multiple sides and merge them into onesingle output. Technical aspects of the switchyard are discussed in [163, 162].Internal Ion Source: the MR-ToF-MS has a surface ionization source to providealkali ions. It is employed for offline tuning of the system and to provide cal-ibrant ions for online measurement when convenient. The ion source streamsions into one of the inputs of the switchyard, which can merge it with thesample coming from TITAN RFQ.22 One important technical difference between the TITAN RFQ and the RFQs in the MR-ToF-MS is thegeneration of the potential gradient that guides ions from one side to the other. They are created byresistive plastic rods, which provides a much smoother gradient than the discrete segmentation of therods.69Upstreammirror (E1-E4)Downstreammirror (E6-E9)MCPguardDrift tube(E5)Mass rangeselectorBafflesDistance between reflection points ~ 46 cmPreparation Paul trapFigure 4.8: Electrode structure of the mass analyzer of the TITAN MR-ToF-MS, including thepreparation Paul trap upstream. Figure adapted from [162].Transfer RFQ: this RFQ connects the switchyard to the preparation Paul trap atthe entrance of the analyzer. It can work in both directions, bringing ions to theanalyzer or returning purified samples from the analyzer to the switchyard.Preparation Paul trap: prior to their injection into the analyzer, ions are furthercooled in a Paul trap. It provides well-defined starting conditions of the ionsample, so ions enter the analyzer with a small time spread.Additional to the main RF optics, electrostatic optic elements are included throughthe transport system for steering and focusing of ions. Similarly to the TITAN RFQ,it is also filled with high-purity He gas at a pressure on the order of ≈ 1 Pa. Dif-ferential pumping apertures separate the transport system from the ≈ 10−5 Paenvironment of the mass analyzer.Mass AnalyzerThe mass analyzer of the TITAN MR-ToF-MS is a scaled version of the mass analyzerof the MR-ToF-MS at GSI [160]. Its electrode structure is depicted in picture 4.8. Ionsfly roughly one meter each turn with a kinetic energy of about 1.3 keV. The centraldrift region is grounded, while the mirrors can be switched to load or eject ions, orto perform TFS turns (see section 3.3).The analyzer is symmetric, having identical Einzel lensing mirrors in both ends.The mirrors have a four-electrode design, which is a compromise between tolerableaberrations and ease of tuning and stability. The electric potential generated at themass analyzer during isochronous reflections is also similar to the one depicted infigure 3.13. The existing aberrations introduced by the mirrors allow symmetricpeak shapes, with roughly identical tails in both sides. This is advantageous duringdata analysis, reducing potential systematic errors.A mass range selector (MRS) system, as described in sec. 3.3.2, is installed at thecenter of the drift region, and its performance is characterized in [135]. A pair ofbaffles are mounted in the analyzer to block the particles deflected by the MRS andto limit the field penetration between mirrors.As in the MPET, the time-of-flight detector placed after the analyzer is an MCPdetector.704.2.4 Transport and Optimization of Ion BeamsISAC delivers ion beams typically at a kinetic energy of 20 keV. Such energies allowbeam transport through long and complex paths with high efficiency. To simulatesimilar conditions of beam entering the TITAN RFQ, the TITAN ion source for stableisotopes is also biased at 20 kV.The beam is decelerated and cooled at the TITAN RFQ, which is biased slightlybelow 20 kV to allow for ions to enter. As ions are bunched out of the RFQ, theyare accelerated again. The next unit is a pulsed drift tube (PB5). The PB5 has itsbias pulsed down while ions are flying through it. This allows one to adjust thetransport energy through the TITAN beamline.The ion transport energy through TITAN is typically a few keV, which allows ef-ficient beam transport without the need for advanced high-voltage instrumentation.The beam transport to the MPET and the MR-ToF-MS are sketched in figure 4.9.At the MR-ToF-MS, the transport system is biased to 1.3 keV and ions are decel-erated and transported at a few eV. Then, they are accelerated back to 1.3 keV whenentering the analyzer. The use of its own independent transport system and high-stability power supplies makes the beam properties and the tune of MR-ToF-MSvery stable and independent of the tune of the TITAN RFQ.The beam is sent to the MPET with 2.2 keV transport energy. Before entering thetrap, the beam is decelerated and pulsed down by another pulsed drift tube (PLT).The energy is regulated so that ions enter the trap and reach its center at quasi-rest,when the trap is closed. The long flight path to the trap makes the transport toMPET very sensitive to instabilities in power supplies. One volt fluctuation in thebiases of RFQ or PB5 (∼ 0.005%) alters considerably the properties of the ion cap-tured. Therefore, beam energies (biases of PB5 and PLT) and the capture timing arefinely tuned to yield the ToF-ICR resonance with the best quality and ion transportefficiency, and those must be readjusted once or twice daily to account for potentialdrifts. The procedure for optimizing injection, mainly regarding beam energy andtiming, is discussed in [141].The beam is guided through the TITAN beam line using electrostatic optical de-vices. Benders, pairs of parallel plates to which a potential difference is applied,are used to steer the beam path, either to transfer the beam to a different sectionof the beamline or to perform a small correction. In addition, the beam focus isadjusted by Einzel lenses (already explained in ch. 3, see fig. 3.14) and electrostaticquadrupoles. The map of the TITAN beam line showing its optical devices betweenthe RFQ and the MR-ToF-MS and the MPET is shown in figure 4.10.Since the MR-ToF-MS is closer to the TITAN RFQ and has its own preparationsystem, tuning the beam transport to it is relatively straightforward. The transportto the MPET, however, requires the optimization of a few tens of parameters, andthis may be a very complex procedure. Recently, an algorithm was implemented toperform the optimization of beam transport once an initial tune is found.71Ion SourceRFQ PB5 PLT MPET MCP2.2 keV200 eV2.2 keV beam transport to MPET20 keV beam transport to TITANSwitchSwitchPotential   (arbitrary scale)cMeasurementezc    Ion capturee    Ion extractionBeam energyElectric potentialPotential after switchRFQ opens0.0Ion SourceRFQ PB5 MCP1.3 keV1.3 keV beam trans-port to MR-ToF-MS20 keV beam transport to TITANSwitchPotential   (arbitrary scale)zRFQ opens0.0MR-ToF-MSMass AnalyzerMR-ToF-MSTransport SystemMerging beamwith ion source'sInput RFQSwitchyardTransfer RFQPrep.PaulTrapBeam Transport to TITAN MR-ToF-MSBeam Transport to TITAN MPETFigure 4.9: Sketch of the electric potential and total beam energy along the beam transportpath to send ions from the used ion source (either TITAN’s or ISAC’s) to theMPET (top) and the MR-ToF-MS (bottom). The contributions to the electric po-tential generated by electrostatic optic elements are not shown. Shaded regionsrepresent the placement of specific equipment. Distances and energies are not toscale.72EBITT R FC :R FQ,T R FC :B IA S(Cooler Buncherupper portion)T R FC :P B5T R FC :E L 5T R FCBL :DPASE L -M CPT R FCBL :XCB0,T R FCBL :CCB0(now split E inzel lens)T R FCBL :B1-IN / OUTHole to M R -ToF-MST R FCBL :Q1T R FCBL :Q2M CP -1T R FCBL :B4-IN / OUTT R FCBL :Y CB4,T R FCBL :XCB4,T R FCBL :CCB0T SY BL :DPAT SY BL :Y CB0,T SY BL :XCB0T SY BL :B1-IN / OUTT SY BL :E L 1T SY BL :E L 3T SY BL :B8-IN / OUTT SY BL :Y CB8,T SY BL :XCB8T SY BL :Y CB9,T SY BL :XCB9,T SY BL :CCB2T SY BL :E L 4M P ET BL :E L 2M P ET BL :Y CB3,M P E T BL :XCB3,M P E T BL :CCB3M CP -0to MPETElectrostatic opticsRF opticsMCPsGrounded elementsMR-ToF-MSBeam PathFigure 4.10: Optics map of TITAN beamline to transfer beam from the RFQ to the MR-ToF-MS or the MPET. The beam line towards EBIT is not shown. Figure adaptedfrom [130].The optimizer is a genetic-type algorithm that searches for the optimal value ofa given criterion. Generically, the criteria combine transport efficiency (number ofion counts per cycle), ToF-ICR resonance quality and proximity of the convergedparameters to the ones found by theory23. For each element in the beamline, theoptimizer searches for the value that optimizes the criteria. Then it moves to thenext element in the list and keeps performing this procedure until convergence.In figure 4.11, an example of such optimization is shown. Initially, a reasonableion count (black line) is observed, but a large injection steering (red line) is present.This means that the incoming beam is not being injected in a straight path and,thus, the quality of its injection is non-optimal. Injection steering can be measuredby the time-of-flight effect when the Lorenz steerers are off. Time-of-flight effectis defined by by ToFe f f = [to f (0) − to f (ωRF)]/to f (ω0) (see eq. 3.26) and is anindirect measurement of initial magnetron radius. Ideally, with the Lorenz steerersoff, ion should be injected in the center and ToFe f f = 0. After about 80 iterations, thesteering was removed and the transmission improved by 40%. The whole procedureran for about 12 hours without human interference.Typically, beam transport optimization is done using stable ions from the TITANion source. Once optimal settings are found, it is only necessary to scale the timingsfor the m/q of the isotopes of interest from the time values obtained for the stableions. Ion optical devices are kept constant.23 For example, an ideal transmission tune would not require correction benders to be used. Therefore theoptimizer shall search for a configuration in which such elements are performing minimal steering.73Figure 4.11: An example of automatic optimization of beam transport to MPET. The algo-rithm scanned a parameter space of 16 optical elements to remove an initialinjection steering, seen on its effects on the the time-of-flight effect (red line),and improve the count-rate (black line).4.3 Experimental ProcedureThe experimental procedure for the presented studies was divided into two phasesfor each ion beam received from ISAC. Firstly, the beam was sent to the MR-ToF-MS for a preliminary characterization and optimization, taking advantage of thefast diagnosing capabilities of the spectrometer. Then, a mass measurement wasperformed with the MR-ToF-MS and, subsequently, sent to the MPET also for amass measurement whenever available isotope yields allowed. The two phases, ofcharacterization and measurement, are described next.4.3.1 Initial Beam Assessment with the MR-ToF-MSEach beam delivered was initially characterized at the MR-ToF-MS. A typical spec-trum obtained is shown in figure 4.12, through which the beam composition canbe assessed. In a preliminary analysis, the peaks in the spectra were fitted andtheir masses calculated through a previous calibration using ions from the stableion source. The preliminary masses were compared to the values reported in theliterature [34], and each peak was assigned.Besides Ti, the delivered beam typically contained surface-ionized V, Cr, Mn, andother lesser produced isobars. The identification of the titanium species was con-firmed by blocking one of the laser ionization transitions in the TRILIS ion source.In this way, only surface-ionized species were transmitted, and a reduction in the Tiyields could be observed. This can be seen in figure 4.12.In some mass numbers, a few peaks present in the spectra could not be identified.Although they are probably from ionized molecular species, little effort was taken7411 01 0 01 0 0 07 . 2 7 . 4 7 . 6 7 . 8 8 . 0 8 . 211 01 0 01 0 0 0 Measurement time: 31 minMeasurement time: 12 min  Counts / 1.6 ns5 4 C r +5 4 M n + 5 4 T i +TRILIS Lasers OFF5 4 V +TRILIS Lasers ON ( T i m e  o f  F l i g h t  -  7 . 4 2  m s )   [ µ s ]5 4 T i +5 4 V +5 4 M n +5 4 F e +5 4 C r +Figure 4.12: This typical MR-ToF-MS spectrum shows how the identification of titaniumpeaks was confirmed by blocking one of the TRILIS lasers. Then, only surfaceionized species were delivered to TITAN, causing a reduction only in Ti yields.Red curves are fits to the data. Figure published first in [164].to identify them. Since they are not overlapping with peaks of species of interest,they were considered not to influence the relevant measurements.The fast measurement provided by the MR-ToF-MS allowed, for the first time,the fine optimization of the ISAC Mass Separator for the species of interest. Al-though the resolution of the mass separator is not high enough to fully separatemost isobars, it can favor the transmission of certain species over others within thatrange.Guided by the MR-ToF-MS, the ratio between titanium species over contaminantspecies was optimized by changing the parameters of the ISAC mass separator[138]. The impact of such a procedure can be seen in figure 4.13. This is particularlyuseful for very exotic species, where beams are typically highly contaminated bystable isobars but also share a significant mass difference with them.4.3.2 Mass Measurement Procedure: MR-ToF-MSDuring the online studies, the first mass measurement at every mass number wasdone at the MR-ToF-MS. All were done with 512 isochronous turns plus one TFSturn inside the analyzer for the ions of interest. The total length of the measurementcycle was ≈ 20 ms. 13 ms of those were spent on cooling and preparation in thetransport system.In the analyzer, the ions spent between 7.2 ms to 7.6 ms (depending on theirmass number) before being sent to the MCP. Peaks in the time-of-flight spectra hada width of about 17 ns, yielding a mass resolving power of Rm ≈ 220 000. For7549 50 51 52 53 54 55 5610-410-310-210-1100 Ratio of Ti in beam compositionMass Number Delivered After OptimizationFigure 4.13: Ratio of titanium isotopes present in the beam delivered from ISAC before(black) and after (red) optimization of the ISAC Mass Separator aided by theTITAN MR-ToF-MS. As can be seen, up to two orders of magnitude in interest-to-contaminant ratio was recovered from this procedure in the most exotic cases.In the A = 55 case, no 55Ti could be seen initially. Figure first published in [138].isotopes of each mass number, data were taken until at least a few hundreds ofcounts were acquired at the corresponding titanium peak.Chromium ions, mostly stable or close to stability, were widely present and wereidentified as appropriate calibrants for all spectra except of mass numbers A = 51and 56, in which 51V and 56Fe were chosen as calibrants. The atomic masses of thesecalibrants would also be measured with the MPET for an independent verification(see sec. 4.3.3).Measurements were performed with less than one ion per cycle on average. Thisis well below the threshold where ion-ion interactions inside the analyzer start caus-ing a relevant systematic effect. The MRS was used to deflect any particle outsidethe desired mass window.A few supporting measurements were also taken. One lower statistics spectrumwas acquired with the MRS switched off in order to allow calibrant ions with a dif-ferent mass number to reach the detector. Thus 39K+ and 41K+ from MR-TOF-MS’sinternal ion source could be measured simultaneously to the radioactive ion beamand could be used to validate the mass accuracy. Moreover, a single turn spectrumusing 39K+ and 41K+ was acquired to calibrate time offsets in the electronics andacquisition system.The total data taking time with the MR-ToF-MS was 12 hours, excluding prepara-tion and characterization times. Results of the measurements with the MR-ToF-MS,as well as the data analysis procedure, are presented in section 5.1.764.3.3 Mass Measurement Procedure: MPETFollowing the mass measurement at MR-ToF-MS, the beam was sent directly fromthe TITAN RFQ to the MPET. Mass measurements of both the titanium isotopesand of the chosen MR-ToF-MS calibrants were performed whenever possible.Although the MPET has a Lorentz steerer, the half-lives of the species of interestwere well above 100 ms, and ion preparation through dipole excitation was possible.Therefore, ions were injected into the center of the trap, one ion per bunch onaverage, and prepared in a pure initial magnetron motion (of radius ≈ 1.5 mm) bydipole excitation. Subsequently, the major contaminant ions, previously identifiedthrough the MR-ToF-MS spectra, were removed through dipole excitation of theirreduced cyclotron motion. The total ion preparation time in the MPET was between60 ms and 70 ms.After preparation, the ions were subjected to the ToF-ICR measurement of thecyclotron frequency of the interest ions. A two-pulse Ramsey excitation scheme(see sec. 3.2.3) was employed for measurements of all ions with A ≤ 53 while astandard scheme was used for A = 54. Total ToF-ICR excitation times ranged from100 to 250 ms, depending on the species.Every νc measurement of the ions of interest was interleaved by a νc,re f measure-ment of a reference 39K+ ion from TITAN’s ion source, to calibrate the magneticfield and to account for other possible time-dependent variations during the mea-surement. Measurements of νc of 85Rb+ ion, also obtained from TITAN’s ion source,were also interleaved to study potential systematic mass-dependent shifts.Complementary, systematic mass-dependent shifts were studied in more detailin dedicated mass measurements of stable species before and after the experiment.Measurements of 41K+ and 85Rb+ were performed in the same conditions of theonline measurements, also using 39K+ as reference ion.The total measurement time with the MPET was of 14 hours, excluding prepara-tion times. Results of the measurements with the MPET, as well as the data analysisprocedure, are presented in section 5.2.775 MEASUREMENTS AND DATAANALYS ISIn this chapter, I present the characteristics of the data acquired during the exper-iment (described in sec. 4.3) and the analysis procedure employed to obtain themass values of the studied nuclides. Section 5.1 presents the data pertaining to theMR-ToF-MS, while section 5.2 discusses the data acquired with the MPET. Finally,a brief discussion on the assignment of the nuclear state of the measured species ispresented in section Mass Measurements with the MR-TOF-MSA sample MR-ToF-MS spectrum of each mass number is shown in figure 5.1. In eachspectrum, a calibrant was chosen among the identified species (see sec. 4.3.1 for adescription of the identification procedure). The criteria for calibrant choice werespecies whose mass was known in literature [34] with a precision better than whatthe MR-ToF-MS can provide (mass uncertainty . 1 keV), preferably with at least1000 counts in a non-overlapping peak. The characteristics of the spectra acquiredat each mass number are described next.A = 51: for this mass number, measurements with the MR-ToF-MS cannot con-tribute to improve the mass precision of the isotopes of interest, since they arereported to sub-keV/c2 levels in the literature [34]. The measurements wereperformed to validate the accuracy of the technique employed. The MR-TOF-MS acquired time-of-flight spectra for about half an hour, and peaks of 51Ti+,51V+ and 51Cr+ were identified, as well as 10 counts compatible with 51Sc+.The peak corresponding to 51Cr+ was found overlapping with the much moreintense 51V+ peak. Although the masses of both species are known to sub-keV/c2 levels [34], the position of the 51V+ peak is much less affected by itsoverlap with the 51Cr+ peak due to its much higher intensity. Therefore, 51V+was chosen as a suitable calibrant for the spectrum.A = 52: as in the A = 51 case, the MR-ToF-MS measurements in this mass numberalso served to validate the technique. Data were acquired for about one hour.52Cr+, stable and whose mass is known to 0.34 keV/c2 precision [34], was themost intense component of the beam, making it a suitable calibrant. 52Ti+ wasalso largely present, while a few tens of events corresponding to 52V+ and52Mn+ were registered.787.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4110100100010000  Counts / 1.6 nsTime-of-Flight - 7.550 ms   (µs)7.2 7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0110100100010000  Counts / 1.6 nsTime-of-Flight - 7.484 ms   (µs)7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2110100100010000  Counts / 1.6 nsTime-of-Flight - 7.416 ms   (µs)7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2110100100010000  Counts / 1.6 nsTime-of-Flight - 7.347 ms   (µs)7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8110100100010000  Counts / 1.6 nsTime-of-Flight - 7.280 ms   (µs)7.2 7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.011010010001000040Ca16O+?56Ti+?56Cr+56Mn+56Fe+112Sn2+55Ti+110Sn2+55V+55Cr+55Mn+54Ti+54V+54Mn+54Fe+54Cr+53Ti+53V+53Cr+52Ti+52Mn+52Cr+52V+51Sc+51Ti+51Cr+A=56A=55A=54A=53A=52  Counts / 1.6 nsTime-of-Flight - 7.207 ms   (µs)A=5151V+Figure 5.1: A sample MR-ToF-MS spectrum is shown for each of the measured radioactivebeams delivered from ISAC (with TRILIS lasers on). Identified peaks are marked.For further details of each measurement, see text.79A = 53: this was the first mass number where the MR-ToF-MS could provide im-provements on literature, since the mass of 53Ti+ is reported to have an uncer-tainty of 100 keV/c2 [34]. Data were acquired for about one and a half hours.53Cr+ was the most intense peak and taken as calibrant. 53Ti+ and 53V+ werealso present in the spectra.A = 54: as the probed titanium isotopes become more neutron-rich, their yieldssignificantly decrease, and the measurements require more time to gatherenough statistics. The measurements on this mass number required about2.5 hours, and 54Ti+ was a minor component of the beam. 54Cr+ was the dom-inant component and chosen as calibrant. 54Mn+, 54V+ and 54Fe+ were alsopresent.A = 55: the MR-ToF-MS acquired data for about 2 hours in this mass number.55Mn+ and 55Cr+ were the dominant species, and the chromium isotope waschosen as calibrant. 55Ti+ and 55V+ appear in the spectra but represent to-gether less than 2% of the counts. The doubly-charged 110Sn2+ also appearedin large quantities in the spectra. Since it has the same m/q of the interestspecies, its behavior under the influence of electrostatic ion optics is identical.Therefore, once it is produced in the ISAC target and doubly-ionized in the ionsource, it will be present in the sample.A = 56: the beam at this mass number was composed by 56Fe+, 56Mn+, 56Cr+and 112Sn2+ However, after 4 hours of data acquisition, it is uncertain whether56Ti+ was present. The observed peak may correspond to ionized 40Ca16O+,a common molecule that may be released from the ISAC target. A completediscussion about this issue is given in sec. 5.1.5. 56V+ was not found in thebeam. 56Fe+ was chosen as a suitable calibrant for the spectra on this massnumber.5.1.1 Data AnalysisThe data analysis procedure for the MR-ToF-MS spectra followed very similar guide-lines as for the MR-ToF-MS system at GSI, presented in [132], but adapted to thespecifications of the TITAN system.First, all time-of-flight spectra were corrected for temperature drifts and instabili-ties in the power supplies by using a time-resolved calibration (see sec. 3.3.2) withrespect to the calibrant peak.To obtain the time-of-flight of each species, their peaks in the spectra must be fitto an appropriate peak function. The shape of the peaks is a function that reflectsseveral processes, such as ion optical parameters of the mass analyzer (see fig. 3.17)and ion-gas collisions [129, 165]. In some MR-ToF-MS systems, time-of-flight peakscan be Gaussian-like or present large asymmetric tails (see for example [55]). Nev-ertheless, the peak shape is identical for all isobars in the same spectrum, since theygo through the same processes.806 . 9 7 . 0 7 . 1 7 . 2 7 . 3 7 . 4 7 . 51 01 0 01 0 0 01 0 0 0 05 2 C r +  L o r e n t z G a u s s Counts / 1.6 nsT i m e - o f - f l i g h t  -  7 . 2 8  m s  ( µs )7 . 2 2 7 . 2 3 7 . 2 4 01 0 0 02 0 0 0 Figure 5.2: Fits of a 52Cr+ time-of-flight peak with Gaussian (blue) and Lorentzian (red) peakshapes. The vertical axis is in logarithmic scale to evidence the shape of the tails.The insert contains the same information, but it has its vertical axis in linear scaleto evidence the centroid.Several approaches exist in the literature on how to obtain the correspondingpeak shape. Some authors obtain a peak shape from computer simulations of theion optics of their device (such as in [165]); while others perform one high-statisticsmeasurement of a single-species sample and fit the data to a complex function withmany degrees of freedom (as done in [166]).The TITAN MR-ToF-MS has the advantage of generating symmetric peaks, whichsimplifies this analysis. This is a feature of the optical aberrations of the mirrors and,ultimately, of how they are tuned. We opt to fit the data to two standard spectralline shapes: Gaussian ( fGauss) and Lorentzian ( fLorentz), whose functions are:fGauss(t) = Y e−12 [(t−tc)/σ]2 and (5.1)fLorentz(t) = Yσ(t− tc)2 + σ24. (5.2)In both cases, Y, tc and σ are free parameters of the fit. Y is the height of the peak,related to the number of counts measured; tc is the centroid, from where the species’time-of-flight is obtained; and σ is a parameter related to the width of the peak.A sample peak fit with each of the functions is shown in figure 5.2. The Gaussianpeak shape reproduces very well the data around the centroid, but not the tails ofthe peak. Meanwhile, the Lorentzian reproduces the tails well but has a reducedperformance around the centroid.We performed an independent analysis using both peak shapes. All peaks in ev-ery spectrum were fitted by a Least-Square method [167] using a multi-peak fittingroutine, which accounts for overlapping peaks. Parameters Y and tc were adjustedfor every peak, the width parameter σ was fitted under the constraint that it wasthe same to every peak in the spectrum (σ ≈ 17 ns). Doubly-charged peaks had81slightly larger widths24 and were excluded from the analysis. The fits using the twopeak shapes resulted in values for tc with a relative difference within < 5 · 10−8 andcompatible uncertainties. The final tc (and respective uncertainties) for each peakwas obtained through an unweighted average of the two values.The time-of-flight of every species was calibrated in m/q using (see eq. 3.29):m/q = C (tc − t0)2 , (5.3)where C is a calibration factor and t0 is an offset that mostly comes from delaysin the electronic signal processing, which is nearly constant for a given experiment.The calibration factor C was obtained by the atomic mass (ma,re f - taken from [34]),charge state (qre f ) and measured time-of-flight (tc,re f ) of the calibrant ion:C = ma,re f − qre f meqre f (tc,re f − t0)2, (5.4)where me is the mass of the electron. The offset t0 was measured from a single turnspectrum containing 39K+ and 41K+ ions and evaluated in t0 = 111(4) ns.The obtained ionic masses were converted to atomic form (see ch. 3). In allcases, ions of interest and calibrant ions were in a singly charged state (q = 1);therefore, atomic mass calculations account for one electron removed. Electronbinding energies (on the order of a few eV [168]) are negligible for the studiedcases.Contributions to the statistical errors came from the peak fits and the calibration.The evaluation of systematic errors is discussed in the following.5.1.2 Evaluation of Systematic ErrorsSystematic contributions to uncertainties in MR-ToF-MS can come from many sources,such as effects from voltage fluctuations during ejection of the ions from the ana-lyzer, from the time-resolved calibration procedure, from ion-ion interaction duringthe flight, from the use of the mass-range selector, or from the presence of overlap-ping peaks when applicable [132, 164].The upper limit of systematic errors of the TITAN MR-ToF-MS was evaluated tocontribute up to 3 · 10−7 to the relative mass uncertainty [135]. It was determinedfrom offline accuracy measurements performed before and after the experimentwith 39K+ and 41K+ ions. This can also be verified by comparing the online massmeasurements to well known mass values in the Atomic Mass Evaluation of 2016(AME16) [34]. We inspected the relative atomic mass differences (∆m) between the24 Although it remains to be further investigated, peaks of doubly-charged species may have larger widthdue to differences in the ion preparation step, before entering the analyzer.8251Cr 51Ti 52Ti 52V 53V 54Fe 54Mn 55Mn 56Mn-8x10-7-6x10-7-4x10-7-2x10-702x10-74x10-76x10-78x10-7  ∆mFigure 5.3: Relative atomic mass differences (eq. 5.5) between the values measured by theMR-ToF-MS and the mass values in the AME16 [34]. Only species with wellknown masses in the literature are shown. Error bars represent the statisticalerror from the MR-ToF-MS measurement only.values obtained by the MR-ToF-MS (ma(MR-ToF-MS)) and the mass values recom-mended by the AME16 (ma(AME16)):∆m =ma(MR-ToF-MS)−ma(AME16)ma(AME16). (5.5)Figure 5.3 shows ∆m for the 9 species whose mass uncertainty reported in theliterature was smaller than the obtained with the MR-ToF-MS. On average, the MR-ToF-MS mass values leaned towards the heavier side of the AME16 values by aboutone part in ten million, compatible with the offline measurements [135]. This is nottrue for 51Cr only, which was affected by a more intense overlapping species (seefig. 5.1). Finally, a systematic contribution of 3 · 10−7 was added to the relative massuncertainty of every species.5.1.3 Relativistic CorrectionsIt is also important to consider relativistic effects during the measurement. In high-precision measurements, a small relativistic effect may be introduced by the spec-trometer when operated at sufficiently high kinetic energies. In chapter 3, the prin-ciples of the MR-ToF-MS were presented using a classical non-relativistic formal-ism. In the following, the error in approximating the problem with non-relativisticphysics is quantified.83An MR-ToF-MS measures the time (t) passed as the particle travels a fixed dis-tance (L), or equivalently, its speed (v = L/t). At fixed kinetic energy (Ekin), thespeed of the particle in the appropriate relativistic description is given byv = c√1−(mc2Ekin + mc2)2, (5.6)where m is the mass of the particle and c is the speed of light. The non-relativisticlimit is taken when Ekin  mc2, so the expression can be approximated by a Taylorexpansion around Ekin/(mc2) = 0 asv = c√2(Ekinmc2)− 3c√8(Ekinmc2)3/2+ . . . (5.7)The first term in the right-hand side is the non-relativistic expression, and the sec-ond can be used to estimate the error in approximating the problem within a non-relativistic framework. The relative error (δv) in velocity by calculating it using theclassical expression (vcl) is then given byδv =v− vclvcl≈ 34(Ekinmc2). (5.8)For the values of this experiment (Ekin = 1.3 keV, m ≈ 50 u), the relative error is onthe order of δv = 2 · 10−8. Since the fight path is fixed, the relative error in velocitydirectly translates into the relative error in time-of-flight, which is much smallerthan the resolution of our equipment. It is then justifiable to neglect relativisticcorrections.5.1.4 Final Mass ValuesThe mass values obtained with the MR-ToF-MS are shown in table 5.1 as massexcesses (ME - see eq. 3.4). The errors presented show both statistic and system-atic contributions. The mass differences (∆MCal-IoI) between calibrants and ionsof interest measured by the MR-ToF-MS are also shown to allow the independentreconstruction of the mass relationship between the pair of isobars. In total, themasses of 14 isotopes were measured.5.1.5 The case of 56TiIn the A = 56 spectrum, a total of 264 counts were registered in a peak compatiblewith previous mass measurements of 56Ti. However, the measurement with TRILISlasers off showed too little reduction of the count rate. While most isotopes showeda reduction between 80% to 90%, the peak corresponding to 56Ti reduced by only25(15)% (see fig. 5.4).84Table 5.1: Mass excesses (ME) and half-lives of the isotopes measured by the TITAN MR-ToF-MS. Half-life data were taken from [108]. The mass difference (∆MCal-IoI)between the pair of isobars measured by the MR-ToF-MS are also shown to allowthe reconstruction of their mass relationship.Species Half-life [s] Reference ∆MCal-IoI [keV/c2] ME [keV/c2]51Cr 2.3934(1) · 106 51V −729 (17) −51474 (8.8)stat (14)sys51Ti 345.6(6) 51V −2482 (16) −49722 (5.5)stat (14)sys52V 224.6(3) 52Cr −4002 (25) −51417 (21)stat (14)sys52Ti 102(6) 52Cr −5953 (17) −49466 (7.5)stat (14)sys53V 92.6(8) 53Cr −3436 (20) −51851 (13)stat (15)sys53Ti 32.7(9) 53Cr −8410 (18) −46877 (9.6)stat (15)sys54Fe (stable) 54Cr −683 (16) −56252 (6.6)stat (15)sys54Mn 2.697(2) · 107 54Cr −1384 (15) −55550 (3.9)stat (15)sys54V 49.8(5) 54Cr −7031 (17) −49904 (6.7)stat (15)sys54Ti 2.1(1.0) 54Cr −11191 (16) −45744 (4.7)stat (15)sys55Mn (stable) 55Cr +2594 (15) −57704 (3.8)stat (15)sys55V 6.54(15) 55Cr −5985 (27) −49125 (22)stat (15)sys55Ti 1.3(1) 55Cr −13277 (29) −41832 (24)stat (15)sys56Mn 9.2840(4) · 103 56Fe −3697 (16) −56910 (4.2)stat (16)sys52 53 54 55 56020406080100 Count-rate reduction after turning laser off (%)Mass NumberFigure 5.4: Reduction in count-rate of titanium observed after turning TRILIS laser off. Mostisotopes show a reduction between 80− 90 %.Moreover, the peak is also compatible in mass with the ionized molecule of40Ca16O+. It is unclear whether the observed peak is purely 56Ti or an admixture ofthe isotope of interest and the contaminant molecule. Therefore, the identificationof the isotope of interest was inconclusive in this case.855.2 Mass Measurements with the MPETMeasurements with the Penning trap targeted the titanium isotopes and the speciesused as calibrant of the MR-ToF-MS. A sample ToF-ICR resonance obtained for eachspecies of interest is shown in figure 5.5.As mentioned in sec. 4.3.3, 39K+ was the reference ion for all measurements andmeasurements of 85Rb+ were also taken to evaluate the accuracy of the procedure.Two-pulse Ramsey excitation schemes were preferred as they yield higher precision,but only if the species of interest were expected to be well resolved from the otherisobars present in the sample.The characteristics of the measurements performed at each mass number are de-scribed in the following.A = 51: the masses of the species of interest (51Ti and 51V) are already known withhigh precision [34]. Thus, measurements were performed to validate furtherthe accuracy of the procedure. Ramsey resonances with ton = 40 ms andto f f = 120 ms (see sec. 3.2.3) were employed, preceded by a dipole excitationof the magnetron motion for 50 ms and a dipole cleaning for 20 ms. Thecleaning procedure was set to remove 51V when 51Ti was being measured andvice-versa. In total 5 ToF-ICR resonances were acquired for each species, takingabout 5 minutes each. The complete measurement took about one hour.A = 52: the species of interest for this mass number are 52Ti and 52Cr. The char-acteristics of the cycle remained unchanged from the previous mass number.5 ToF-ICR resonances were acquired for 52Cr, taking about 5 minutes each; 4resonances were measured for 52Ti, taking about 50 minutes each. The mea-surements in this mass number took about 6 hours in total.A = 53: in this mass number, the yield of titanium in the beam decreased consid-erably compared to the previous masses. To compensate for the lower rates, afew adjustments were made to keep the same precision of the measurement.The Ramsey excitation time was increased to ton = 50 ms and to f f = 150ms and the magnetron preparation time was reduced to 45 ms. 3 ToF-ICRresonances were acquired for 53Ti, taking about one and a half hours each; 4resonances were measured for 53Cr, taking about 10 minutes each. The mea-surements took about 6 hours in total.A = 54: as can be seen in the MR-ToF-spectrum (fig. 5.1), many isobars were moreabundant than 54Ti, such as 54Cr, 54Fe and 54Mn. The dipole cleaning of allcontaminant species became challenging, and obtaining a ToF-ICR resonanceof 54Ti was not possible. The measurement focused only on the calibrant of theMR-ToF-MS spectrum 54Cr. The isobars 54Fe and 54Mn are also very close inmass to 54Cr, so a standard ToF-ICR resonance (tRF = 100 ms) was preferredover Ramsey. In total 5 ToF-ICR resonances were acquired, which took about5 minutes each. The procedure took about one hour.8670727476788082νRF - 1114900 Hz  [Hz]80 85 90 95 10051V+Time-of-Flight [μs]6870727476788082νRF - 1114900 Hz  [Hz]20 25 30 35 40 51Ti+Time-of-Flight [μs]727476788082νRF - 1093500 Hz  [Hz]85 90 95 100 105 52Cr+Time-of-Flight [μs]727476788082νRF - 1093400 Hz  [Hz]52 54 56 58 60 62 64 66 68 70 72 7452Ti+Time-of-Flight [μs]727476788082νRF - 1072900 Hz  [Hz]25 30 35 40 45 53Cr+Time-of-Flight [μs]707274767880828486νRF - 1072700 Hz  [Hz]42 44 46 48 50 52 54 56 58 60 62 6453Ti+Time-of-Flight [μs]74767880828486νRF - 1053000 Hz  [Hz]30 40 50 60 70 80 90 100 54Cr+54Mn+?Single-species fitTwo-species fitTime-of-Flight [μs]νcνcνcνcνc νcFigure 5.5: A sample ToF-ICR resonance is shown for each of the species measured with theMPET in this campaign. Red curves are analytical fits to the data and, in the caseof Ramsey resonances, the center fringe is indicated by the arrow and labelled νc.For further details of each measurement, see text.87At masses A = 55 and A = 56, the ratio of species of interest over contaminantswas even worse than in the A = 54 case. Therefore, MPET measurements in thosemasses were not attempted.5.2.1 Data AnalysisAs discussed in sec. 3.2, obtaining the atomic masses from the ToF-ICR methodrequires obtaining the ratio (Rν) between cyclotron frequencies of the reference ionand the ion of interest (see eq. 3.6). Thus it requires the fitting of the ToF-ICRresonances to obtain the cyclotron frequencies.First, a preliminary selection was performed to the ToF-ICR resonance data. Eachtime-of-flight event was accepted if two criteria were met:1. Its measurement cycle had a maximum of two ions detected. This is to min-imize effects from ion-ion interactions that may result in shifts from the truecyclotron frequency [169].2. The events were inside a time-of-flight window predetermined by its m/q. Forexample, for 54Cr+ this window was between 63 µs and 91 µs. Events regis-tered outside this window are considered dark counts. They can be triggered,for example, by cosmic rays passing through the detector or by decays of ra-dioactive ions deposited in the MCP detector.After the data cuts, all ToF-ICR resonances were fit to a function in the form of eq.3.26 to obtain the cyclotron frequency νc of the ion of interest. The fits are shownin fig. 5.5 (red curves). However, the measurements were not done simultaneouslyand the magnetic field may have varied over the course of the procedure. Therefore,the cyclotron frequency of the reference ion is interpolated to the mid-time of themeasurement of the ion of interest. This is depicted at figure 5.6. The cyclotronfrequency ratio was obtained with the interpolated ν˜c,re f :Rν =ν˜c,re fνc. (5.9)A weighted average was performed with all measurements of Rν of the same ionof interest, and its atomic mass (ma) was calculated throughma = Rν(ma,re f − qre f me) qqre f+ q me , (5.10)where ma,re f is the atomic mass of the reference ion 39K, taken from [34], and meis the mass of the electron. The charge states q and qre f of the ions of interest andreference, respectively, were both +1. Note that this formula does not account forelectron binding energies to the atom. In the cases tackled in this experiment, theelectron binding energies are on the order of a few eV [168], which are negligiblecompared to the precision achieved in the measurements.88Cyclotron freq.Timet1 t2 t3νc,ref(t1)νc,ref(t3)νc,ref(t2)~νc(t2)Figure 5.6: A linear interpolation of the reference measurements (black) is done to determinethe reference cyclotron frequency (blue) at the time of the measurement of the ionof interest (red).The statistical uncertainties come from the fit and the frequency interpolation.Systematic errors are discussed in the following.5.2.2 Evaluation of Systematic ErrorsCommon sources of systematic errors are well studied in Penning traps [170, 110].They come from the construction of the trap, such as imperfections of trappingelectrodes and field misalignments, from instabilities of the trapping potential andmagnetic field and from ion-ion or ion-atom interactions.The effect of ion-ion interactions was minimized in this experiment due to thedata selection, as explained in the previous section. In a previous analysis, thesources of systematic deviations in MPET related to its construction and field stabil-ity were evaluated to 2 · 10−10 per unit of m/q [110]. This yields deviations on theorder of 10−8 in relative mass uncertainty for this experiment.The magnitude of the systematic deviations in this experiment was evaluated us-ing the measurements of 85Rb taken during the experiment and the measurementsof 85Rb and 41K performed before and after it. In total, 90 mass measurements of41K were done and 107 of 85Rb, all using 39K as a reference ion. Since they weremeasured many more times than the ions with masses between A = 51 and 54,their higher precision provides a better determination of systematic deviations.In a similar way as done for the MR-ToF-MS, we inspected the relative atomicmass differences (∆m) between the values obtained by the MPET (ma(MPET)) andthe mass values recommended by the AME16 (ma(AME16)):∆m =ma(MPET)−ma(AME16)ma(AME16). (5.11)Figure 5.3 shows ∆m for 85Rb and 41K. Measurements for some species of interestwere also included, given that their mass uncertainty reported in the literature [34]was smaller than the obtained with the MPET. A mass-dependent systematic trendis observed. It was evaluated in 6.5 · 10−10 per m/q. This is larger than obtained8940 50 60 70 80 90-8x10-8-6x10-8-4x10-8-2x10-802x10-84x10-8 Mass Number  ∆m39K Figure 5.7: Relative atomic mass differences (eq. 5.11) between the values measured by theMPET and the mass values in the AME16 [34]. Only species with well knownmasses in the literature were considered. Error bars represent the statistical errorfrom the MPET measurement. The green line is a linear fit to the data (lightgreen region represents a 95% confidence band) to get the systematic trend fromthe reference mass (39K, marked in blue).by [110], but on the same order of magnitude. The systematic contribution to therelative mass uncertainty was evaluated in < 1.5 · 10−8 among the masses of interest.It is worth noting that this analysis significantly differs from the one performedfor the MR-ToF-MS, shown in figure 5.3. In the MR-ToF-MS case, all ions of interestwere calibrated using an isobaric reference ion. Therefore, mass-dependent system-atic effects are expected to be negligible. In the case of the measurements done withMPET, the reference particle (39K) is several mass units away from the masses ofinterest. For this reason, mass-dependent systematic effects are expected to play amore relevant role.5.2.3 Relativistic CorrectionsDifferently than the MR-ToF-MS case (sec. 5.1.3), relativistic corrections in Penningtraps are more frequently needed since they are more precise. In the MPET, rela-tivistic shifts were observed when measuring light ions such as 6Li+ [158].The cyclotron frequency (νc) in the relativistic formulation is simplyνc =12piqe Bγm, (5.12)where qe and m are the charge and mass of the particle, B the magnetic field strengthand γ is the Lorentz factor. The relative error in the mass (δm) in approximatingthe cyclotron frequency using the classical expression (γ = 1) can be estimated byδm =m−mclmcl=1γ− 1 =√1−(vc)2 − 1 , (5.13)90where mcl is the mass obtained from the classical approximation, v is the ion mo-tion’s tangential velocity and c is the speed of light. The velocity v can be estimatedby v ≈ 2pi r νc, where r is the radius of the radial motion of the ion in the trap.Considering r = 1.5 mm, δm for 39K+ (νc ≈ 1.45 MHz) is on the order of 1 · 10−9.Meanwhile, δm for the heaviest species measured in this experiment, 85Rb+ (νc ≈0.66 MHz), is on the order of 2 · 10−10. Since these values are much smaller thanthe precision and accuracy of this experiment, relativistic effects were considerednegligible.5.2.4 Final Mass ValuesThe frequency ratios and mass values obtained with the MPET are shown in table5.1, masses are presented as mass excesses (ME - see eq. 3.4). The errors presentedshow both statistic and systematic contributions to the final mass value. In total,the masses of 7 species were measured.Table 5.2: Frequency ratios (Rν), atomic mass excesses (ME) and half-lives of the speciesmeasured by the TITAN MPET. Half-life data were taken from [108].Species Half-life [s] Reference Rν † ME [keV/c2]51V (stable) 39K 1.307476380 (50) −52203.5 (1.2)stat (0.65)sys51Ti 345.6(6) 39K 1.307544491 (58) −49731.5 (1.5)stat (0.65)sys52Cr (stable) 39K 1.333052991 (55) −55421.3 (1.3)stat (0.70)sys52Ti 102(6) 39K 1.333216716 (83) −49479.1 (2.3)stat (0.70)sys53Cr (stable) 39K 1.358721924 (52) −55288.4 (1.1)stat (0.75)sys53Ti 32.7(9) 39K 1.358953560 (80) −46881.4 (2.2)stat (0.75)sys54Cr (stable) 39K 1.384341984 (130) −56929.3 (3.8)stat (0.81)sys† The cyclotron frequency ratios are relative to the measured ionic species.5.3 Low-Lying Isomers and Ground-StateAssignmentIsomers are long-lived excited nuclear states. They may be co-produced with theground-state configurations and may live long enough to be delivered and partici-pate in the experimental procedure. They play a relevant role in several studies ofnuclear structure. However, in the case of this experiment, the scientific interest liesin the ground-state configurations only. Therefore, the impact of the presence ofisomeric states in the sample must be considered.With the achieved resolving power of 220 000, the MR-ToF-MS would be able toresolve isomeric states with excitation energies above ∼ 250 keV. In the case of theMPET, the resolving power depends on the excitation time and on the excitation91scheme employed, as discussed in section 3.2 (see eq. 3.23). In this experiment, theMPET would be able to resolve isomeric states with minimum excitation energiesbetween 150 keV and 500 keV, depending on the case.The presence of unresolved low-lying isomers in the sample may cause several un-desired effects. For example, they can cause a broadening of the MR-ToF-MS peak[132] or of the ToF-ICR resonance [171], which may also lead to a systematic shiftin the position of the peak’s or resonance’s centroid. On some occasions, isomericstates may be much more abundant in the sample than the ground state [67, 172],which may lead to incorrect assignment of the state of the observed species.However, no broadening was observed in any of the acquired data sets. Moreover,this particular region of the nuclear chart has little known or reported isomers [54].Among the nuclides of interest, the only known isomer is a 5+ state of 54V, whichhas excitation energy of 108(1) keV and half-life of 0.9(5) µs [54]. It does not livelong enough to take part in the experiments and influence our results, since the ionpreparation state takes tens of ms.Given the lack of evidence of isomeric states in the data, the systematic absenceof known low-lying and long-lived isomers in the region, and the good agreementbetween our data and the ground-state properties reported in the literature (as dis-cussed in the following chapter), all the measured nuclides of interest were assignedto be in their ground-state configuration.926 DISCUSS IONThe impact and the implications of our measurements are discussed in this chapter.First, in section 6.1, the performance of the MR-ToF-MS is compared to the MPET.In section 6.2, the measurements are compared with the existing measurementsreported in the literature. With the new data, the mass surface in the isotopicchains of Ti and V is updated, and its implications on the evolution of the N = 32shell closure are discussed in sec. 6.3. Finally, in sec. 6.4, the predictions of the abinitio theories presented in chapter 2 are tested against the data.6.1 Comparison Between SpectrometersIn order to provide further validation of MR-ToF-MS measurements, the differences∆MCal-IoI between atomic masses (ma) of ions of interest and their MR-ToF-MS cali-brants were evaluated for both spectrometers:∆MCal-IoI = ma(MR-ToF-MS calibrant) − ma(Ion of Interest) . (6.1)This is a more robust form of comparison than comparing directly mass values. Itis essentially insensitive to choices of calibration and to mass-dependent systematiceffects. In table 6.1 and figure 6.1, this quantity is presented using values from theMPET, the MR-ToF-MS and the AME16.Table 6.1: Mass differences ∆MCal-IoI (in keV/c2 ) between atomic masses of Ti isotopes andtheir MR-ToF-MS calibrants, using MPET, MR-ToF-MS and AME16 [34] values.Species MPET MR-ToF-MS AME1651V - 51Ti −2472.0 (2.8) −2482 (16) −2471.01 (64)52Cr - 52Ti −5942.2 (3.7) −5953 (17) −5949 (7)53Cr - 53Ti −8407.0 (3.5) −8410 (18) −8456 (100)54Cr - 54Ti −11191 (16) −11313 (82)55Cr - 55Ti −13277 (29) −13440 (160)The three cases that were measured by both spectrometers (51V-51Ti, 52Cr-52Tiand 53Cr-53Ti) confirm the agreement between the two techniques. Figure 6.1 alsoillustrates well the strengths and weaknesses of each technique. The MR-ToF-MShas higher sensitivity and could measure isotopes with lower production rate, whilethe MPET could provide one order of magnitude better precision in the current935 1 V  - 5 1 T i 5 2 C r  - 5 2 T i 5 3 C r  - 5 3 T i 5 4 C r  - 5 4 T i 5 5 C r  - 5 5 T i-200-150-100-50050100  AME16 MPET MR-ToF-MS∆ΜCal-IoI (AME16 - TITAN) [keV]Figure 6.1: The comparison between mass differences ∆MCal-IoI (between Ti isotopes andtheir respective MR-ToF-MS calibrant) measured by both spectrometers show theagreement between them. Here, ∆MCal-Ti is plotted against the AME16 values fora cleaner comparison. Grey bands represent the uncertainties of the AME16.configuration of both systems. The results of both spectrometers were also in goodagreement (within one sigma) with the values in the literature, which is discussedfurther in the next section.6.2 Comparison with Previous MeasurementsThe agreement between the two spectrometers is essential to assess the reliabilityof our resulting mass values, particularly in the case of the newly commissionedMR-ToF-MS system. It is also essential that the results are compared with the mea-surements previously reported in literature, especially with well-measured cases.In this section, a compilation of all the measurements of each isotope measured inthis campaign is presented. In all cases, the results of both TITAN spectrometersare generally in good agreement with past measurements and within 1.5 σ againstthe AME16 recommended values.6.2.1 Titanium Isotopes51Ti: In 51Ti, shown in fig. 6.2, all previous measurements are indirect. AME16includes two early measurements from the β-decay of 51Ti [173, 174] and fourreaction-based measurements: three one-neutron transfer reactions [175, 176]and a neutron capture reaction [177]. The two most recent reaction experiments[176, 177] provided measurements with precision on the order of one keV. Thetwo TITAN measurements agree within 1σ with the AME16.941 2 3 4 5 6 7 8-30-20-10010205 1 T iM P E TT I T A NT I T A NM R - T o F - M S1 9 7 15 0 T i ( n , γ) 5 1 T i1 9 7 61 9 6 71 9 6 71 9 5 55 0 T i ( d , p ) 5 1 T i5 1 T i ( β - ) 5 1 V ME (Meas. - AME16)  [keV]Measurement1 9 5 5Figure 6.2: Difference between AME16 value [34]and all individual mass measurementsof 51Ti [173, 174, 175, 176, 177, 164].Open symbols are indirect measure-ments and the gray band shows the errorof the AME16 value.1 2 3 4 5 6-50050100150800100012001 9 7 11 9 6 65 2 T iM P E TT I T A NT I T A NM R - T o F - M S1 9 6 75 0 T i ( t , p ) 5 2 T i5 2 T i ( β - ) 5 2 V ME (Meas. - AME16)  [keV]Measurement1 9 8 55 2 S c ( β - ) 5 2 T iFigure 6.3: Difference between AME16 value [34]and all individual mass measurementsof 52Ti [178, 53, 179, 180, 164]. Open sym-bols are indirect measurements and thegray band shows the error of the AME16value.52Ti: All previous measurements of the mass of 52Ti are indirect, as can be seenin fig. 6.3. Two measurements were obtained through the β-decays of 52Ti[178] and of 52Sc [53]. The latter significantly disagrees with all other measure-ments. The lowest uncertainties were provided by two measurements of the50Ti(t,p)52Ti reaction [179, 180]. The two TITAN measurements agree within1σ with the AME16.53Ti: Only one measurement of the mass of 53Ti is reported in literature, comingfrom the analysis of its β-decay [181]. As typical for this type of measurement,its uncertainty is large (100 keV). The two TITAN measurements are easily inagreement, as seen in figure 6.4.54Ti and 55Ti: the knowledge of the masses of the 54Ti and 55Ti isotopes was ob-tained by the same experiments. The only indirect information comes froman experiment dedicated to studying the β-decays of these isotopes [182]. Theremaining data come from direct measurements, mostly from three ToF-MS ex-periments performed at the Time-of-Flight Isochronous (TOFI) spectrometer inLos Alamos [183, 184, 185]. Additionally, an Isochronous Mass Spectrometry(IMS) measurement of 54Ti was recently performed at the experimental CoolerStorage Ring (CSRe) in Lanzhou [57]. Overall, TITAN MR-ToF-MS values aresystematically lighter than the previous measurements, but still in good agree-ment with them. As shown in figures 6.5 and 6.6, MR-ToF-MS values arewithin 1σ against most measurements, being barely over 1σ only against a fewTOFI results.951 2 3-100-50050100 5 3 T iM P E TT I T A NT I T A NM R - T o F - M S5 3 T i ( β - ) 5 3 V ME (Meas. - AME16)  [keV]Measurement1 9 7 7Figure 6.4: Difference between AME16 value [34]and TITAN mass measurements of 53Ti[164]. The AME16 value is based on asingle indirect measurement [181] andthe gray band shows its error.1 2 3 4 5 6-400-300-200-10001002003004005 4 T iT I T A NM R - T o F - M S1 9 9 65 4 T i ( β - ) 5 4 V ME (Meas. - AME16)  [keV]MeasurementT o F - M S1 9 9 01 9 9 41 9 9 82 0 1 5I M SFigure 6.5: Difference between AME16 value [34]and all individual mass measurementsof 54Ti [182, 183, 184, 185, 57, 164]. Theopen symbol is an indirect measurementand the gray band shows the error of theAME16 value.1 2 3 4 5-500-400-300-200-10001002003004005005 5 T iT I T A NM R - T o F - M S1 9 9 65 5 T i ( β - ) 5 5 V ME (Meas. - AME16)  [keV]MeasurementT o F - M S1 9 9 01 9 9 41 9 9 8Figure 6.6: Difference between AME16 value [34]and all individual mass measurementsof 55Ti [182, 183, 184, 185, 164]. Theopen symbol is an indirect measurementand the gray band shows the error of theAME16 value.1 2-3.0-2.5-2.0-1.5-1.0- 1 VM P E TT I T A N2 0 1 7 ME (Meas. - AME16)  [keV]MeasurementL E B I T  P T M S*  A M E 1 6  v a l u e  i s  b a s e d     o n  4 1  m e a s u r e m e n t s  Figure 6.7: Difference between AME16 value [34]and the recent mass measurements of51V [164, 186]. The AME16 value isbased on 41 different measurements (notshown) and the gray band shows its er-ror.966.2.2 Vanadium Isotopes51V: the mass of 51V is very well determined in the literature, with 41 direct andindirect measurements included in the AME16 evaluation [34]. They contributeto an uncertainty below 0.5 keV of this quantity. Also, recent results fromthe LEBIT PTMS system at Michigan State University provided updated massvalues for Cr and V isotopes near stability [186] but are not included in theAME16. The LEBIT value for 51V has an uncertainty of only 0.13 keV, butis about 2.5σ from the AME16 value. Nevertheless, the results of the TITANMPET agree with both, as shown in fig. 6.7.52V: the mass of 52V is also well measured, with 11 measurements included in theAME16 [34]. However, they are all from indirect techniques. The TITAN MR-ToF-MS measurement is the first to assess this quantity directly. It is 1σ fromthe AME16, as shown in fig. 6.8.53V: similarly to 52V, all previous measurements of the mass of 53V are indirect, ascan be seen in fig. 6.9. Two measurements were obtained through the β-decaysof 53V [187] and of 53Ti [181]. The remaining data were obtained throughnuclear reactions: one from the 51V(t,p)53V reaction [188] and another fromthe 54Cr(d,3He)53V reaction [189], which provides the lowest uncertainty. TheTITAN MR-ToF-MS measurement is in good agreement with all measurements,except with the one obtained through the β-decay of 53V.54V: in the literature, three indirect measurements were reported for the mass of54V, as shown in fig. 6.10. Two are from β-decays: of 54V [190] and of the parent54Ti [182]. The lowest uncertainty is provided by a 54Cr(t,3He)54V reactionexperiment [191]. The TITAN MR-ToF-MS measurement agrees with all ofthem.55V: Only one measurement of the mass of 55V is reported in the literature. Itcomes from the analysis of its β-decay [192] and has a large uncertainty (∼ 100keV). The TITAN MR-ToF-MS measurement is in good agreement with it, asseen in figure Other IsotopesThe other isotopes measured in this campaign are all well studied and documentedin literature, with a few tens of different measurements reported for each. Sincethere is a high degree of reproducibility and small uncertainties among those previ-ous measurements, they provide a great test to our data.In figure 6.12, mass measurements of chromium isotopes 52−54Cr performed withthe MPET are compared to AME16 values. In total, the AME16 congregates 33measurements of 52Cr, 28 of 53Cr and 38 of 54Cr [34]. Results from the LEBIT PTMSspectrometer [186] published after the AME16 evaluation provided better precisionand are also shown. The TITAN MPET measurements are in good agreement withall presented values.97101020304050605 2 VT I T A NM R - T o F - M S ME (Meas. - AME16)  [keV]Measurement*  A M E 1 6  v a l u e  i s     b a s e d  o n  1 1  i n d i r e c t     m e a s u r e m e n t s  Figure 6.8: Difference between AME16 value [34]and the recent first direct mass measure-ment of 52V [193]. The AME16 value isbased on 11 different indirect measure-ments (not shown) and the gray bandshows its error.1 2 3 4 5-100-50050100 5 3 VT I T A NM R - T o F - M S1 9 5 65 1 V ( t , p ) 5 3 V5 3 V ( β - ) 5 3 C r ME (Meas. - AME16)  [keV]Measurement5 3 T i ( β - ) 5 3 V1 9 7 71 9 6 71 9 7 95 4 C r ( d , 3 H e ) 5 3 VFigure 6.9: Difference between AME16 value [34]and all individual mass measurementsof 53V [187, 181, 188, 189, 193]. Opensymbols are indirect measurements andthe gray band shows the error of theAME16 value.1 2 3 4-150-100-50050100150 5 4 VT I T A NM R - T o F - M S1 9 7 05 4 V ( β - ) 5 4 C r ME (Meas. - AME16)  [keV]Measurement5 4 T i ( β - ) 5 4 V1 9 9 61 9 7 75 4 C r ( t , 3 H e ) 5 4 VFigure 6.10: Difference between AME16 value [34]and all individual mass measurementsof 54V [190, 182, 191, 193]. Open sym-bols are indirect measurements andthe gray band shows the error of theAME16 value.1 2-100-500501001505 5 VT I T A NM R - T o F - M S1 9 7 75 5 V ( β - ) 5 5 C r ME (Meas. - AME16)  [keV]MeasurementFigure 6.11: Difference between AME16 value [34]and TITAN mass measurements of 55V[193]. The AME16 value is based on asingle indirect measurement [192] andthe gray band shows its error.98-4-2024681012  TITAN MPET LEBIT PTMS5 2 C r ME (Meas. - AME16)  [keV]Isotope5 3 C r 5 4 C rFigure 6.12: Difference between AME16 value [34]and the recent PTMS mass measure-ments of 52−54Cr [164, 186]. The grayband shows the errors of the AME16values.-30-25-20-15-10-5051015202530 TITAN MR-ToF-MS ME (Meas. - AME16)  [keV]Isotope5 1 C r 5 4 F e 5 4 M n 5 5 M n 5 6 M nFigure 6.13: Difference between AME16 value [34]and the recent MR-ToF-MS mass mea-surements of 51Cr, 54Fe, 54Mn, 55Mnand 56Mn. The gray band shows theerrors of the AME16 values.The measurements performed with the MR-ToF-MS are in good agreement withthe AME16. The comparison is shown in figure 6.13 for 51Cr, 54Fe, 54Mn, 55Mnand 56Mn. The body of knowledge involves 13 measurements of 51Cr, 49 of 54Fe,39 of 55Mn and 9 of 56Mn [34]. In the case of 54Mn, 13 indirect measurements arereported. However, none is reported from direct techniques, making the one doneat the MR-ToF-MS the first for this isotope.6.3 Updated Isotopic Chains and Evolution of theN=32 Shell ClosureGiven the good agreement between the results of our spectrometers and with the lit-erature, the masses of the measured isotopes were updated by taking the weightedaverage of all measurements of each of them. In table 6.2, the weighted averageof the measurements of the two TITAN spectrometers are shown. As can be seen,the combined TITAN values are dominated by MPET measurements where avail-able, since it provided lower uncertainties than MR-TOF-MS. Also in table 6.2 theweighted average between TITAN measurements and the values recommended byAME16 are shown. With TITAN data, the mass uncertainties were considerablyreduced in 6 of the isotopes: of 52−55Ti and 54,55V.The new measurements bring the structure of the nuclear mass landscape aroundN = 32 of the Ti and V isotopic chains to the scale of a few tens of keV. Since therewere no deviations from the literature, the data did not change the shape of the mass99- 4 8 0- 4 6 0- 4 4 0- 4 2 091 21 51 82 14 8 5 0 5 2 5 4 5 612345 ( I . c )( I . b )( I . a )T i t a n i u m  I s o t o p e s  ( Z = 2 2 ) 2 8 Binding Energy [MeV]N e u t r o n  N u m b e r2 6 3 2 3 43 0 AME16 AME16 + TITAN No-shell Hypothesis  S 2n [MeV]N o - s h e l l  h y p .  r e s i d u a l : ∆ 2n [MeV]M a s s  N u m b e r52 54 56-  - 4 8 0- 4 6 0- 4 4 0- 4 2 01 21 51 82 14 9 5 1 5 3 5 5 5 712345 ( I I . c )( I I . b )( I I . a )V a n a d i u m  I s o t o p e s  ( Z = 2 3 ) 2 8 Binding Energy [MeV]N e u t r o n  N u m b e r2 6 3 2 3 43 0 AME16 AME16 + TITAN No-shell Hypothesis  S 2n [MeV]N o - s h e l l  h y p .  r e s i d u a l : ∆ 2n [MeV]M a s s  N u m b e r53 55 570.00.4  Figure 6.14: The mass landscape around titanium (I) and vanadium (II) isotopes is shownfrom three perspectives: (a) absolute masses (shown in binding energy format),(b) its first “derivative” as two-neutron separation energies (S2n), and (c) itssecond “derivative” as empirical neutron-shell gaps (∆2n). Data prior to thisexperiment are shown in black, while updated data are shown in red. The no-shell hypothesis on N = 32 is presented in the (b) panels as smooth linear fitsto S2n AME16 data between 52−56Ti and 53−57V, and their residuals are shownin the inserts.100Table 6.2: Mass excesses (in keV/c2) of Ti and V isotopes recommended by AME16 and mea-sured with the TITAN MR-ToF-MS and the TITAN MPET. Weighted averages ofcombined TITAN measurements and of all values are presented. Neighboringisotopes are also shown for reference. TITAN measurements performed in otherisotopic chains did not provide significant improvements to the existing mass un-certainties, so they were omitted.Species AME16 MR-ToF-MS MPET TITAN (combined) AME16 + TITAN49Ti −48563.79 (11) −48563.79 (11)50Ti −51431.66 (12) −51431.66 (12)51Ti −49732.84 (51) −49722 (15) −49731.5 (2.1) −49731.3 (2.1) −49732.75 (49)52Ti −49470 (7) −49466 (16) −49479.1 (3.0) −49478.7 (3.0) −49477.3 (2.8)53Ti −46831 (100) −46877 (18) −46881.4 (2.9) −46881.3 (2.9) −46881.2 (2.9)54Ti −45622 (82) −45744 (16) −45744 (16) −45740 (15)55Ti −41668 (162) −41832 (29) −41832 (29) −41827 (29)56Ti −39320 (121) −39320 (121)57Ti −33916 (256) −33916 (256)49V −47961.93 (83) −47961.93 (83)50V −49224.01 (41) −49224.01 (41)51V −52203.84 (40) −52203.5 (1.8) −52203.5 (1.8) −52203.83 (39)52V −51443.77 (42) −51417 (26) −51417 (26) −51443.76 (42)53V −51851 (3) −51851 (19) −51851 (19) −51851 (3)54V −49893 (15) −49904 (17) −49904 (17) −49898 (11)55V −49145 (95) −49125 (27) −49125 (27) −49126 (26)56V −46155 (177) −46155 (177)57V −44413 (80) −44413 (80)surface in the region. However, the refined precision solved ambiguities regardingthe existence of particular shell effects. The remaining of this section explores thisin detail.The impact of the measurements can be seen in figure 6.14, which presents thebinding energies (EB) and their “derivatives”: the two-neutron separation energies(S2n) and the empirical neutron-shell gaps (∆2n). These quantities were definedin chapter 1 in equations 1.2, 1.4 and 1.6, respectively. For convenience, they arereproduced here:EB(N, Z) =[ma(N, Z)− Z ·mp − N ·mn − Z ·me]c2 ,S2n(N, Z) = [ma(Z, N − 2) + 2mn −ma(N, Z)] c2 and∆2n(N, Z) = S2n(N, Z)− S2n(N + 2, Z) ;where Z and N are the atomic and neutron numbers, ma, mp, mn and me are themasses of the atom, the proton, the neutron and the electron, respectively, and c isthe speed of light.In figure 6.14, the canonical N = 28 shell closure is easily recognized through thesharp features at S2n and ∆2n around 50Ti and 51V. In the S2n, it can be recognized bya downwards ”kink”. In the ∆2n, it is evidenced by a peak arising from the baseline,which in the region is around 2 MeV. A search for these signatures in N = 32 was101performed among titanium isotopes (described in sec. 6.3.1) and vanadium isotopes(shown in sec. 6.3.2).As pointed out in section 1.2, these signatures must be consistent across severalisotopic chains to characterize a shell closure. Therefore the evolution of shell effectsin N = 32 is also analyzed across the neighboring isotopic chains in sec. Signatures of N=32 Shell Effects in TitaniumAlthough less pronounced, similar features seen at N = 28 are also present around54Ti. They possibly correspond to the effects of a shell closure in N = 32. To explorethese effects in the S2n surface with more precision, the hypothesis of absence of anyshell effect (No-Shell Hypothesis) was analyzed. This hypothesis assumes a smoothand linear behavior of S2n around N = 32, with no observed discontinuities orkinks.To test this hypothesis, a linear function was fit to the S2n data between 52Ti and56Ti. The result is shown as the green line in panel (I.b) of figure 6.14, and the resid-ual of the fit is shown in the insert of the same panel. Using only data from AME16,the fit produces25 a χ2red of 1.94. This result is not statistically significant to rule outany interpretation. With the inclusion of the data from this experiment, the fit pro-duces a χ2red of 51.13. Therefore, the No-Shell Hypothesis is completely ruled out bythe new mass measurements. Then, with the observation of a downwards ”kink”in S2n, the data presented here conclusively establish the existence of signatures ofshell effects at N = 32 in the Ti chain.The empirical neutron-shell gap at 54Ti (panel (I.c) of figure 6.14) has changedfrom 2.45(17) MeV to 2.70(12) MeV, with the mass of 56Ti now the largest source ofuncertainty. The new measurements also better define the peak at N = 32 risingfrom the ∼ 2 MeV baseline, confirming the signature of shell effect seen at S2n.However, the shell features observed in titanium are very weak and well belownormally expected for shell closures. A comparison with the traits seen at N = 28 infigure 6.14 illustrates this discrepancy. The existence of a special pattern at titaniumcan be identified by looking at the systematics with the nearby elements, which isdone in sec. Signatures of N=32 Shell Effects in VanadiumThe signatures of a shell closure in N = 32 are entirely absent in vanadium. Thiswas evident from the previous measurements and confirmed by the addition of thenew measurements with higher precision.In the ∆2n data (panel (II.c) of fig. 6.14), there is clearly no peak or raise abovethe baseline seen in 55Ti. In the S2n data (panel (II.b) of the same figure), a smoothtrend is seen between 53Ti and 57Ti.25 χ2red is the χ2 divided by the number of degrees of freedom of the fitting procedure.102A similar analysis of a No-shell Hypothesis reveals it to be incompatible with theS2n data. It reveals a subtle upwards ”kink” at 55Ti, which does not characterize ashell effect. Possible interpretations for this effect are discussed in the following.6.3.3 Evolution of the N=32 Shell ClosureIt is important to analyse the features seen in titanium and in vanadium in thecontext of the neighboring nuclides. In figures 6.15 and 6.16, the S2n and ∆2n dataare shown between K (Z = 19) and Cr (Z = 24) isotopic chains around N = 32. Thesame shell signatures seen in Ti are reproduced, with more intensity, towards lowerZ. It is evident that titanium is at a transition point between V and Cr, which showsno sign of a N = 32 shell closure, and the strong closure seen in Sc, Ca, and K.Towards higher Z, starting from vanadium, different phenomena occur. The S2nsurface subtly bends upwards, systematically through many isotopic chains. Thisbehavior is commonly associated with the emergence of nuclear shape deformation[194, 195]. V, Cr, and Mn are located in a mid-shell region between Z = 20 andZ = 28, where the effects of deformation are more likely to occur. Moreover, as wasmentioned in sec. 1.4.2, there is spectroscopic evidence that there is an energy levelordering change between Sc and V [64]. Nevertheless, it is possible to affirm thatvanadium is also a transition point where a particular systematic behavior startsto occur towards higher Z. However, it requires a more in-depth analysis to stateanything about the nature of this behavior.To better inspect the evolution of the N = 32 shell closure, figure 6.17 showsthe empirical neutron-shell gap across the N = 32 isotonic chain and compares itto the data across the canonical N = 28 chain. Overall, ∆2n(N = 32) mirrors thebehavior of ∆2n(N = 28), being systematically lower by about 2 MeV. It suggeststhat the mechanism that governs the evolution of the N = 28 shell might be thesame governing the evolution of the emerging N = 32 shell. This hypothesis couldbe explored in future shell model studies.Figure 6.17 also includes ∆2n data across the N = 30 isotonic chain. Since shellfeatures are not recognized in this neutron number, this data can be used as anestimation of the ∆2n baseline. It is interesting to point out the crossing between∆2n(N = 32) and ∆2n(N = 30) occurring amid Ti (Z = 22) and V (Z = 23), revealingthe point of emergence of shell signatures in N = 32 among mass observables.The trends in ∆2n also reveal how abrupt the onset of magicity in N = 32 is. Itstarts from weak effects in titanium to its peak in scandium. However, peaks inthe shell gap usually occur in doubly-magic nuclei, as illustrated by the N = 28case shown in 6.17. It is expected that the peak of ∆2n(N = 32) occurs in 52Ca,which should be extra bound due to the closure of the proton shell in Z = 20.After the mass measurements described here, the only isotopic chain not studiedwith high-resolution mass spectrometry techniques is of scandium. Therefore, newmass measurements between 51Sc and 55Sc are required to confirm the evolution ofmagicity in N = 32.1032 9 3 0 3 1 3 2 3 3 3 4891 01 11 21 31 41 51 61 71 8M nKC rVC aS c  S 2n [MeV]N e u t r o n  N u m b e rT iFigure 6.15: The two-neutron separation energies between K (Z = 19) and Mn (Z = 25)isotopic chains around N = 32. Data is taken from the AME16 [34], chainsupdated with TITAN data are shown in red.1042 4 2 6 2 8 3 0 3 2 3 4- 1 2- 1 0- 8- 6- 4- 2024681 01 2 C rVT iS cC aK ∆ 2n [MeV]N e u t r o n  N u m b e r( - 1 2  M e V )( - 8  M e V )( - 4  M e V )( + 4  M e V )( + 8  M e V )Figure 6.16: Empirical neutron-shell gaps between K (Z = 19) and Cr (Z = 24) isotopicchains showing the effect of the N = 28 shell and the rise of shell effects inN = 32. Data is taken from the AME16 [34], chains updated with TITAN dataare shown in red. Each isotopic chain was shifted by a multiple of 4 MeV forclarity.1 8 2 0 2 2 2 4 2 6 2 80123456 N = 2 8 N = 3 0 N = 3 2 ∆ 2n  [MeV]A t o m i c  N u m b e rFigure 6.17: Empirical neutron-shell gaps across isotonic chains of N = 28 (black) and N =32 (red), where shell effects have been recognized, and of N = 30 (gray) whereno shell effects exist and serve as a reference line. The crossing between N = 30and N = 32 ∆2n data in Ti (Z = 22) and V (Z = 23) reveals the point ofemergence of shell signatures in N = 32 among mass observables.1056.4 Tests of Ab Initio TheoriesThe clearer picture of the evolution of the N = 32 shell allows us to investigatehow well our knowledge of nuclear forces and many-body methods describes theobserved behaviors. The mass data in the N = 32 region was compared to state-of-the-art ab initio nuclear structure calculations. The tested methods and interactionswere described in sec. 2.4.In particular, the Multi-Reference In-Medium Similarity Renormalization Group(MR-IMSRG) [92, 93, 94], the Valence-Space (VS-) IMSRG [95, 96, 97, 98], and theself-consistent Gorkov-Green’s Function (GGF) [99, 100, 101, 102] approaches wereapplied. All calculations were performed with interactions based on the chiral ef-fective field theory: 1.8/2.0(EM) [86, 88], the N2LOsat [90] and the NN+3N(lnl)interactions. They all included two- (NN) and three-nucleon (3N) interactions withparameters adjusted typically to the lightest systems (A = 2, 3, 4) as the only input26[85, 196, 197].To get an overall picture of the current stage of techniques, four large-scale calcu-lations were performed using different permutations among the available methodsand interactions. IMSRG-based methods employed two different interactions: the1.8/2.0(EM) interaction was used in a VS-IMSRG calculation, and the N2LOsat wasused in an MR-IMSRG calculation. The GGF method was also employed in calcu-lations with two different interactions: N2LOsat and NN+3N(lnl) interactions. TheN2LOsat interaction was employed in calculations using the GGF method and MR-IMSRG method. In this way, the breadth of the variations seen among the resultscan be interpreted as an overall ”error” of the employed techniques.All calculations were performed for all Ti isotopes in the region of interest. Oneexception is the calculation performed with the MR-IMSRG method, which can onlyprovide results for even-mass titanium isotopes. Additionally, since the VS-IMSRGcan access all nuclei in this region, the calculations employing the 1.8/2.0(EM) +VS-IMSRG combination were extended to neighboring isotopes as well. Some ofthe VS-IMSRG calculations had already been published for the Cr [195], Sc [198]and Ca [88] chains. Calculations employing the NN+3N(lnl) + GGF combinationwere also available for odd-mass scandium isotopes.6.4.1 ResultsThe results of the calculations are compared to experimental mass data for Ti and Vin figure 6.18. All approaches predicted signatures of shell closures at N = 28 andN = 32, although the strengths of the neutron-shell gaps in the magic numbers aresystematically overpredicted in all cases.Calculations with the N2LOsat interaction typically performs well for radii andcharge distributions in this mass range [59], but here are found to be the least ac-26 Exception given to the N2LOsat interaction that also uses selected heavier systems to adjust the nuclearsaturation density, as explained in sec. 2.4.106- 4 8 0- 4 6 0- 4 4 0- 4 2 0- 4 0 081 21 62 02 44 8 5 0 5 2 5 4 5 61234567( I . a )A M E 1 6T I T A N  +        2 8 Binding Energy [MeV]N e u t r o n  N u m b e r2 6 3 2 3 41 . 8 / 2 . 0 ( E M )V S - I M S R GN N + 3 N ( l n l )G G FN 2 L O s a tG G FN 2 L O s a tM R - I M S R G3 0T i t a n i u m  I s o t o p e s  ( Z = 2 2 )( I . c )( I . b ) S 2n [MeV] ∆ 2n [MeV]M a s s  N u m b e r- 4 8 0- 4 6 0- 4 4 0- 4 2 01 21 51 82 14 9 5 1 5 3 5 5 5 712345( I I . a )A M E 1 6T I T A N  +     2 8 Binding Energy [MeV]N e u t r o n  N u m b e r2 6 3 2 3 41 . 8 / 2 . 0 ( E M )V S - I M S R G3 0V a n a d i u m  I s o t o p e s  ( Z = 2 3 )( I I . c )( I I . b ) S 2n [MeV] ∆ 2n [MeV]M a s s  N u m b e rFigure 6.18: Results of ab initio calculations (lines) are compared to experimental values(points) of the mass landscape around titanium (I) and vanadium (II) isotopes.Mass observables are shown from three perspectives: (a) binding energy, (b) two-neutron separation energies (S2n), and (c) empirical neutron-shell gaps (∆2n).Experimental data combines measurements incorporated in AME16 and thisTITAN experiment.107curate in describing mass observables. They predict underbound titanium isotopesby about 30 MeV, independently of the many-body method employed. Also, this in-teraction highly overpredicts the strength of the neutron-shell gaps both at N = 28and N = 32, compared to the other studied interactions.The calculations with the 1.8/2.0(EM) interaction provide the best description ofthe Ti data. Masses are overbound by only ≈ 3.0 MeV, and the neutron shell gapsare the closest to the experimentally observed values. In the vanadium chain it hadsimilar performance; however, it wrongly predicted the existence of shell effects inN = 32.The results with the NN+3N(lnl) interaction are also in good agreement withdata, but titanium isotopes are underbound by about 20 MeV. Truncation schemescurrently employed in GGF calculations are known to result in less total bindingenergy (typically 10− 15 MeV for mid-mass nuclei) compared to more advancedtruncation schemes [199]. This would bring NN+3N(lnl) in better agreement withthe experimental binding energies.Results of ∆2n calculations for all isotopic chains between Ca (Z = 19) and Cr(Z = 24) with the 1.8/2.0(EM) + VS-IMSRG combination are shown in figure 6.19.Calculations with the NN+3N(lnl) + GGF combination are also included for Ti andSc chains. The calculations provide an excellent description of neutron shell evolu-tion at N = 28 and of the N = 32 where it is strong, in Ca and Sc. Also, whilethere is a general overprediction of the neutron shell gap at N = 32, the trendsfrom N = 28 to N = 32 are mostly reproduced. In contrast, calculated shell gaps intitanium rise too steeply from N = 30 to N = 32 compared to experiment and evenpredict modest shell effects in the vanadium chain. This indicates that the N = 32closure is predicted to arise too early towards Ca.The evolution of ∆2n across isotonic chains is shown in figure 6.20 with resultsfrom the 1.8/2.0(EM) + VS-IMSRG calculation. The comparison with experimentaldata makes evident the overestimation of shell gaps by the theoretical calculation.Both in the N = 28 and the N = 32 shells, titanium (Z = 22) is the isotone withlargest overprediction of ∆2n. While the origin of this discrepancy is not completelyclear, it is known that signatures of shell closures, such as first excited 2+ ener-gies and shell gaps, are often modestly overestimated by the VS-IMSRG [88]. Thissystematic overprediction might also explain the early rising of shell effects in vana-dium.As also seen in experimental data, the 1.8/2.0(EM) + VS-IMSRG calculation pre-dicts that the N = 32 shell mirrors the evolution of the N = 28 shell but systemati-cally lower by about 2 MeV. This reinforces the idea that similar mechanisms governthe evolution of both shells. However, as discussed in chapter 2, ab initio approachesare not the best tool to investigate the nature of specific effects.Unlike the experimental data, the calculations predict that the peak of N = 32shell effects happen in calcium (Z = 20) and not in scandium (Z = 21). It furthermotivates the need for new high-resolution mass measurements in the scandiumchain.1082 4 2 6 2 8 3 0 3 2 3 4- 1 0- 8- 6- 4- 2024681 01 2 A M E 1 6  +  T I T A NC rVT iS cC a   1 . 8 / 2 . 0 ( E M )    V S - I M S R G  N N + 3 N ( l n l )    G G F∆ 2n [MeV]N e u t r o n  N u m b e r( - 8  M e V )( - 4  M e V )( + 4  M e V )( + 8  M e V )Figure 6.19: Results of ab initio calculations (lines) are compared to experimental values(points) of empirical neutron-shell gaps between K (Z = 19) and Cr (Z = 24)isotopic chains around N = 28 and N = 32. Experimental data combines mea-surements incorporated in AME16 and this TITAN experiment.1 8 2 0 2 2 2 4 2 6 2 81234567 N = 3 2    A M E 1 6  +  T I T A N N = 3 2    V S - I M S R G N = 2 8    A M E 1 6 N = 2 8    V S - I M S R G ∆ 2n  [MeV]A t o m i c  N u m b e rFigure 6.20: Evolution of empirical neutron-shell gaps across isotonic chains of N = 28(black) and N = 32 (red), as calculated by the 1.8/2.0(EM) + VS-IMSRG combi-nation (lines). Calculations are also compared with experimental data (points).1097 CONCLUS IONS AND OUTLOOKIn the last decades, we witnessed unprecedented progress in our understanding ofthe atomic nucleus. Novel high-sensitivity experimental techniques have opened upthe pathway to inspect properties of very rare nuclei with high precision. Throughthem, surprising new phenomena were discovered, which challenge our nucleartheories. Also, for the first time, powerful computers enabled theories to describecomplex atomic nuclei from fundamental principles, applying features of QuantumChromodynamics to many-body quantum methods. These advances may finallytrace the right route towards the long-sought understanding of nuclear matter. Ifsuccessful they may also fill critical missing pieces in related fields, such as on theorigin of chemical elements and the evolution of astrophysical objects.These two fronts of research walk side-by-side, as advances in one guide advancesin the other. The study presented at the core of this thesis is a state-of-the-artexample of the interplay between experiment and theory in the field of nuclearstructure. In this case, the emergence of shell effects at the non-canonical magicnumber N = 32 was investigated though precision mass measurements. Samples ofneutron-rich titanium isotopes were produced by the ISAC facility through the ISOLmethod and ionized through the laser ionization technique. Vanadium isotopeswere co-produced and were also present in the sample. The ions were transported tothe TITAN’s Penning trap and multiple-reflection time-of-flight mass spectrometers,where measurements were performed independently.The atomic masses of 51−55Ti and 51−55V were successfully measured. The newmeasurements agree with the previous measurements in the literature and bringthe structure of the nuclear mass landscape to the scale of a few tens of keV aroundN = 32. The results conclusively establish the existence of weak shell effects atN = 32 among titanium isotopes and confirm their absence among vanadium iso-topes. It narrows down the evolution of this nuclear shell closure, mainly regardingits abrupt emergence. The N = 32 shell evolution outlined from inspecting massobservables agrees very well with conclusions drawn from other experimental ap-proaches, such as from γ-spectroscopy (see fig. 1.7, for example).The measurements were compared with the results of several ab initio nucleartheories. Overall, all presented theories perform well in this region, but the studyreveals deficiencies in the description of the N = 32 shell. Specifically, the strengthof shell effects are systematically overestimated, and they are wrongly predicted tostart emerging in vanadium. This work provides fine information for the develop-ment of the next generation of nuclear ab initio techniques.110As demonstrated by this and other studies, the new magic number N = 32 is afascinating phenomenon in nuclear structure and requires further experimental andtheoretical exploration. As pointed out in our analysis, new mass measurements ofscandium isotopes in the region are needed to better understand the peak strengthof shell effects at N = 32. Moreover, the similarities between the evolution ofthe N = 28 and the N = 32 shells are remarkable and should be investigatedtheoretically, especially through shell model approaches.Naturally, further investigation is required towards the low-Z boundary of N =32 shell closure where data is most scarce. Trends from experimental data andresults from theoretical models suggest the magicity of N = 32 should vanish inthe region with Z < 18.Similarly, shell effects emerging at N = 34 are receiving increasing attentionfrom the scientific community [50, 200]. Shell model calculations predict similarmechanisms that drive the evolution of N = 32 and N = 34 shells. Meanwhile,recent experiments suggest that its effects start appearing at much lower Z than theN = 32, but become much stronger right in the argon chain [201].Nevertheless, experiments pushing towards N = 34 and the low-Z boundarywill probe increasingly rare isotopes. Consequently, techniques need to be able toovercome the challenges associated with them: shorter lifetimes, lower intensities,and more substantial contamination levels.In face of those challenges, the results of the TITAN MR-ToF-MS for Ti and V pro-vide an essential milestone on precision mass spectroscopy of rare isotopes. Theyhighlight the scientific capabilities of this new type of spectrometer and illustratehow its sensitivity enables probing of much rarer species with competitive preci-sion. Now successfully benchmarked against the ”standard” mass measurementtechnique - Penning trap mass spectrometry - the TITAN MR-ToF-MS is an idealcandidate to probe outer regions of the nuclear chart for nuclear structure investi-gations.In the near future, the TRIUMF Laboratory will commission its Advanced RareIsotope Laboratory (ARIEL) [202] to expand its scientific capabilities beyond theISAC production facility. It will diversify the radioisotope production through dif-ferent target and driver beam combinations and deliver more exotic isotope speciesat high intensities to experiments. The MR-ToF-MS together with ongoing upgradesto the EBIT [203], the MPET (see appendix B) and the CPET [157] will leverageTITAN’s capacity to perform experiments in the ARIEL era. They will place theTITAN facility in a prime position to push the precision boundary of the nuclearchart and answer the outstanding questions in nuclear structure.111B IBL IOGRAPHY[1] D. Hanneke, S. Fogwell, and G. Gabrielse. 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Triumf uhv cleaning and assembling procedures.TRIUMF Design Notes, 2012.141APPENDICESIn the following appendices, I describe a few of my contributions to further lever-age the experimental capabilities of the TITAN facility, and that may allow futureextensions of the work presented at the core of this thesis. In appendix A, I describea novel technique to enable experimental investigation of some radioactive speciesthat the ISAC facility cannot provide directly. In appendix B, I give an overview ofthe cryogenic upgrade of the MPET mass spectrometer.142A DECAY AND RECAPTURE IONTRAPP INGAccess to exotic radioactive nuclei enables a broad range of scientific investigationsfrom fundamental to applied sciences. The improvement of their production tech-niques, focusing on quality, intensity, purity, efficiency, and selectivity, has been ahighly active area of research [140]. The ISOL method of production of radionu-clides, employed at ISAC, is known for producing high-quality and high-intensityRIBs. However, as explained in section 4.1, chemical and physical processes thatinfluence the extraction of species of interest from the production target (such asionization potential and volatility) limit the availability of certain beams [140, 144].Consequently, ion beams of refractory metals and reactive elements, such as ironand boron, are challenging to produce.One way to circumvent these limitations is the Decay and Recapture Ion Trapping(DRIT) technique. It allows the production of particular beams that the RIB facilityis unable to provide directly but can provide a parent species. It consists of storinga cloud of the parent ion in an ion trap for a time comparable to its half-life orlonger. Then, a cloud of the ion of interest can be created if the trapping potentialis deep enough to recapture the recoiling particles. It permits the creation of beamsof not only refractory and reactive elements, in the case of ISOL facilities, but alsoof isomeric beams if a suitable parent is available.The technique was first employed at CERN-ISOLDE for the creation of 37Ar [204]and 61−63Fe [205] ions for mass spectrometry experiments. In both cases, parent ionswere stored in a buffer-gas-filled Penning trap, which reportedly re-trapped about50% of recoils. Two other experiments [206, 207] performed in the same facilityemployed Electron Beam Ion Traps (EBIT) [208, 209] as storage device. Evidencefrom these experiments suggests that EBITs have high re-trapping efficiencies andare ideal storing devices for this technique.Here, we further explore the use of such ion traps and report the in-EBIT produc-tion of 30Al from a parent beam of 30Mg, performed for the first time at TITAN. Wecharacterized the evolution of the trapped contents and adjusted operating param-eters to optimize the creation of daughter beam. The 30Al ions were extracted anddelivered to MPET, where its mass was successfully measured.In the following section, we discuss the technique in detail and the advantagesand challenges regarding using an EBIT as the storage device. In section A.2, wepresent a simple model to describe the evolution of stored radioactive contents inEBIT, applied to the proof-of-principle experiment herein. In section A.3, we de-scribe our apparatus and of the experimental procedures employed. In section A.4,143we present the results of the systematic investigations on the creation of the daugh-ter beam using EBIT. In section A.5, we demonstrate the production of daughterbeam through unambiguous identification using the mass spectrometer and de-scribe the subsequent mass measurements performed. We conclude with possibleapplications of the technique.a.1 In-EBIT DRITConceptually, the DRIT technique is simple. If a suitable parent species is available,it is stored for the timescale of its half-life and its daughter is recaptured. The keyquestion to the success of this method lies in the confinement capabilities of the trap:first, can the cloud be stored for such a period? Second, how many of the decaydaughters will be retained in the trapping volume? These questions are central toselecting the most suitable trapping setup and, more indirectly, potential candidatespecies.Ion traps, such as Penning or Paul traps, typically create potential wells of a feweV to a few hundreds of eV deep. The three-body β-decay, for example, generatesa recoil energy spectrum that can span a few hundreds of eV. Therefore, β-decayproducts can be recaptured in typical ion traps. In contrast, α-decays may producerecoils up to a few hundreds of keV, which makes their recapture challenging. Here,we focus on the recapture of weak decay products only.EBITs have storing capabilities that typically outperform other ion traps [208, 209].They have an electrode structure that provides axial ion confinement overlappedwith a strong magnetic field that provides radial confinement. EBITs also have adense electron beam passing through the trapping region (see figure A.1), whichionizes ions through electron impact. For this reason, they are widely used toprovide charge-bred beams for experiments or post-acceleration [210]. The electronbeam also deepens confinement: as the ion’s charge state increases, it experiencesa deeper trapping potential in all directions. Consequently, the recoil energy hasless effect on re-trapping efficiency. Furthermore, highly charged ions (HCI) remaincharged after decay, whereas singly charged ions (SCI) may be neutralized in gas-filled traps or by β+-decays [204]. Therefore, EBITs offer a nearly ideal environmentto re-trap decay products.However, the use of EBITs poses particular challenges that must also be consid-ered: (a) Electron-impact ionization generates a distribution of charge states. If thegenerated beam is transported out of the trap, only the fraction in a charge state ofchoice will be used, typically around 20% [210, 211]. (b) The electron beam also ion-izes residual gases that are present in the trapping volume, which may contaminatethe beam. Last (c), EBITs may be “hot” environments due to complex thermody-namical processes, such as heating of the ion cloud by the electron beam and heatexchange by ion-ion collisions. A large thermal input to the ion of interest may“boil” them out of the trap and reduce the efficiency of the technique.144--Trap drift tubesMagnetic fieldElectroncollectorViewport HPGeIon cloudγCathodee-beamIon injection& extractionFigure A.1: Schematic overview of the TITAN EBIT. Trapping of ions is achieved axially byan electrostatic potential and radially by both a magnetic field and an intenseelectron beam passing through the trapping region.Therefore, EBIT operating parameters must be carefully tuned in order to bal-ance re-trapping efficiency, charge breeding, ion-cloud thermodynamics, and otherphysical processes that influence the final quality of the created beam. In the nextsection, the evolution of stored radioactive contents in EBIT is described using asimple model, which illustrates some of these challenges and provides a clearerpicture of the variables that play a relevant role in the technique.a.2 Simulations of Daughter Beam CreationTo understand the daughter beam creation and its confinement, we simulated thein-trap decay of a cloud of a parent ion in the TITAN EBIT, including populationevolution and a simple EBIT thermodynamical model. 30Mg was chosen as a suit-able parent to explore as a proof-of-principle case, both from its decay propertiesand from its experimental availability.The decay chain of 30Mg → 30Al → 30Si provides representative examples oftypical β− decays. First, the half-lives of 30Mg and 30Al, respectively 0.335(10) s[212] and 3.62(6) s [213], allow us to probe two different timescales of interest, eachone order of magnitude apart. Second the Qβ values of the decays (both above 6MeV) are higher than average [54], allowing for a higher recoil energy contribution.Their decay schemes are also complex, with many intermediate γ de-excitationsteps, but are well understood [212, 213]. Moreover, 30Mg+ can be provided toTITAN as a pure beam by the ISAC facility. Hence, this decay chain allows a robustproof-of-principle experiment to explore the DRIT technique at TITAN.In a Monte Carlo approach, particles in an initially pure 30Mgq+ cloud, where qis the charge state of the ion, were randomly assigned a starting energy following aMaxwell-Boltzmann distribution. Particles were evolved in time (t, iterated by timestep ∆t) and were subject to three physical processes described below: Spitzer heat-ing, radioactive decay, and trap escape. A flow diagram of the employed algorithmis shown in figure A.2.145Spitzer heating [208]: Trapped ions constantly gain energy from collisions withthe intense electron beam. This is known to be a dominant process in EBITthermodynamics and, if no active cooling mechanism is employed, ions willeventually gain enough energy to escape the trap barriers. The energy inputper particle from Spitzer heating is proportional to q2 and was calculated bythe prescription outlined in ref. [208], assuming a perfect overlap between theelectron beam and the ion cloud.Radioactive decay: The population change from decay was modeled through sim-ple decay laws, while the heating contribution from recoiling energy requiredconsideration of the three-body nature of β-decay. We followed the same kine-matic procedure outlined in detail in ref. [205] to calculate the recoil energydistributions. The calculation included not only the β-decay itself but alsothe intermediate de-excitation through γ-emission and all possible known de-cay branches. The β-decays schemes of 30Mg and 30Al are well described inrefs. [212] and [213], respectively. These are almost pure Gamow-Teller transi-tions, for which the spectral shape of recoil energy distribution is well known[214, 215]. In the case of γ-emission, there are no long-lived isomeric states inthe cases studied; therefore, we assumed that all de-excitations occur instan-taneously. The final recoiling energy distributions are shown in fig. A.3. TheQβ-value of the decay of 30Al (8561 keV) is higher than that of the 30Mg (6990keV). However, the average recoil energy gained by the daughter is higherin the 30Mg decay (355 eV) than in the 30Al decay (214 eV). The 30Al decaymainly decays to much higher excited states of 30Si, thus in some cases photonemission can revert the orientation of the recoiling momentum gained in theβ-decay.Trap escape: The escape of ions from the trap was incorporated at the end of everytime step by removing particles from the simulation if their energy exceededthe trapping potential barriers. The escape also cools the trapped ion cloudthrough evaporative cooling. It was assumed that ion losses occur predomi-nantly through axial potential barriers, as prescribed in [208].For simplicity, other typical EBIT processes [208] such as radiative recombination,charge exchange, ionization heating, and effects from ion-ion interaction were ne-glected. The simulation also did not include the whole charge breeding dynamics,and the population was assumed to be in the +11 charge state through the wholecalculation. The choice of charge state was guided by an EBIT charge-state evolu-tion calculation (such as in [208]) that showed that the ion cloud should be predomi-nantly in charge states between +10 to +12 for the timescales of the half-lives of theradioactive species. In section A.4.2, we verified this assumption experimentally.All the parameters needed for the calculation are also displayed in fig. A.2. Thetrap parameters were chosen according to the characteristics of the TITAN EBIT.The chosen electron beam energy was twice the threshold energy to completelyionize Mg ions (1.962 keV [168]). Most parameters are typically well known oreasily calculable, making this approach almost clear of free parameters. However,146No(30Si)NoCalculate energy input from Spitzer heating given ΔtUpdate particle energy (from recoil and Spitzer)Does particle energy exceeds trap barrier?NoYesIterate tIterate NInicialize new particle:30Mg+qSample initial energy fromBoltzmann distributionIs the particleRadioactive?Yes(30Mg or 30Al)Calculate decay probability and sample decay given ΔtDid the particle decay?YesUpdate species and samplerecoil energy from distribution Ion cloud properties:Ion mass Charge state Initial temperatureT1/2 (30Mg) T1/2 (30Al) Trap parameters:Electron beam energy Electron beam current Electron beam radius  Magnetic field Axial trapping barrier  30 a.m.u.+110.1 - 20 eV0.335 s3.62 s3940 eV110 mA225(75) μm4.5 T170 VSimulation Parameters:Figure A.2: A simplified flow diagram of the Monte Carlo algorithm to calculate daughterbeam creation and its confinement in the TITAN EBIT. Only three physical pro-cesses are included: Spitzer heating [208], radioactive decay and trap escape. Therelevant parameters on these processes are described in the table. Nuclear prop-erties were taken from [212, 213], and typical TITAN EBIT operating parameterswere chosen.the radius of the electron beam of the TITAN EBIT is known to be between 150and 300 µm, which does not sufficiently constrain the calculation of Spitzer heatingrate [208]. This uncertainty was accounted for in the calculation by running thealgorithm with different values of electron beam radius inside the range.Results of the simulation are displayed in figure A.4. The total population, de-fined as the sum of populations of 30Mg, 30Al and 30Si, remained constant untilabout t ≈ 2.0 s, when it sharply decreased. According to these results, created 30Aldominates the population after ≈ 300 ms or roughly T1/2(30Mg), but the popula-tion of 30Si could not become dominant before the cloud vanished from the trap(2 s < T1/2(30Al)).The same calculation was performed without any additional recoiling energy inorder to understand the contribution of decay energy input to ion losses. The totalstoring time was prolonged by only 20%. This result indicates that the recoilingenergy contributes little to the losses mechanisms in an EBIT as a storage device andthat losses from Spitzer heating dominate. This finding contrasts with observationsof DRIT in Penning traps [204, 205], where ion losses were attributed to recoil energyexceeding trap barriers.The results of these simulations were confirmed by an experiment at the TITANfacility, described in the next section.147-200 0 200 400 600 8000. Energy [eV]30Mg 30Al* 30Al30Al 30Si* 30Si0+30335 ms1+687.52+243.93+3.6 s30 β−β−2+34982+22350+030Qβ= 6990 keVMgAlSi8561 keVQβ=keVkeVkeVkeV(a) (b)Figure A.3: (a) The decay scheme of the 30Mg→ 30Al→ 30Si chain [212, 213] shows the en-ergy levels and major transitions (arrows). (b) The recoiling energy distributionsfollowing the β-decays and γ de-excitations of 30Mg and 30Al in the center-of-mass reference frame. Negative energies refer to a final recoil momentum direc-tion opposite to that initially gained in the β-decay, meaning an antiparallel andhigher-energetic γ-recoil.0.01 0.1 1  Total 30Mg 30Al 30Si t  (storage time) [s]  Normalized PopulationFigure A.4: Simulated population evolution of radioactive ions trapped in EBIT. Total pop-ulation (black curve - normalized to initial) is shown alongside its constituents:30Mg (red), 30Al (blue) and 30Si (green) populations. The grey band representthe uncertainty of the total population evolution due to electron beam radius.148a.3 Experiment OverviewIn the experimental procedure described herein, we investigated RIB populationevolution inside EBIT, tracked ion losses, and unambiguously identified the daugh-ter species after storage. In addition, we performed an experiment with the createdbeam outside the EBIT to substantiate the technique.The RIB was produced at ISAC by impinging a 480 MeV proton beam onto a low-power uranium-carbide target. Magnesium isotopes were selectively ionized usingTRILIS [149] (see sec. 4.1). The beam was extracted, separated by ISAC’s massseparator [151], and sent initially to ISAC’s Yield Station [216] for a compositionmeasurement, which revealed a purity of ≥99.4(1)% of 30Mg+.The 30Mg+ beam was then delivered to the TITAN facility, presented in sec. 4.2(see fig. 4.3). The beam was accumulated in the TITAN RFQ and sent in bunches tothe EBIT [155], where DRIT was performed.The TITAN EBIT is designed to provide charge-bred ions for mass measurementswith MPET [156] and to perform in-trap decay spectroscopy [217, 218]. In thisexperiment, its trapping and electron beam parameters were identical to those listedin figure A.2. For mass spectrometry, as beams are extracted from the EBIT and senttowards MPET, they pass through a Bradbury-Nielsen Gate (BNG) [219], whichselects the bunched beam by its time-of-flight and thus mass-to-charge ratio (m/q).Additionally, an MCP detector can be moved into the beamline before the MPET,about 10 meters downstream of EBIT. It yields a time-of-flight spectrum that allowsthe m/q identification of the constituents of the beam extracted from the EBIT.a.4 Population EvolutionIn this section, we describe the systematic studies of the evolution of the trappedradioactive ion cloud over a range of storage time. To ensure a constant amountof RIB was employed every cycle, the RFQ accumulated RIB for 100 ms. Then thebeam was sent to the EBIT, where the storage time was varied between t = 15 msand 8 s, covering one order of magnitude above and below the half-life of 30Mg.After the storage, the composition of the ion cloud was inspected by extracting itonto the MCP detector.The time-of-flight spectrum allowed m/q identification of each species as well astheir count rates. A typical time-of-flight spectrum can be seen in fig. A.5 (toppanel), where the average spectra of measurements with t ≥ 100 ms is shown. Asionized residual gases could have the same m/q as the RIB, spectra were measuredwith and without the injection of RIB into EBIT under identical conditions. The sub-tracted spectrum contains only the RIB species, and an example can be seen in fig.A.5 (bottom panel). The spectra obtained at each storage time were analyzed with amultiple-peak fitting routine, through which the count rate for each peak could be149RIB (A=30)+12 +11 +10 +8+916O +7 +6 +5 +414N +6 +5 +412C +5 +4 +340Ar +16 +14 +11+15 +12+13138Ba (evaporated from e-gun):+36 +35 +34 +33 +32 +31 +30Species frombackground gaseswith RIB injectionwithout RIB injectionRIB only Time of Flight to MCP detector [μs]15  16  17  18  19  20  21 Normalized CountsFigure A.5: Spectra of time-of-flight measurements between EBIT and the MCP (top panel),which allow m/q identification of trapped species. Spectra were obtained with(top, red) and without (top, green) RIB injection into EBIT. RIB peaks can beidentified by subtracting the two spectra (bottom, blue). RIB charge states be-tween +9 and +12 of A = 30 beam can be clearly identified. Other identifiedspecies are also marked for reference. The shown spectra are averaged among allmeasurements with t ≥ 0.1 s.determined, including from overlapping peaks. With these data, we monitored theevolution of both the charge state distribution and the absolute population of RIB.A.4.1 Optimal Storage Time for DRITWe verified if the ion cloud could be stored as long as needed to create a sample of30Al or 30Si through DRIT. At each storage time, we analyzed the total count rate ofRIB by summing the counts of all RIB peaks present on the spectrum. The result isshown in figure A.6.The count rate was nearly constant and independent of t up to t = 2.0 s, muchlonger than T1/2(30Mg). This suggests a high re-trapping efficiency of the decaydaughters, especially at t & 300 ms ≈ T1/2(30Mg). After t ≈ 2 s, the RIB count ratedropped. This is shorter than T1/2(30Al), thus the creation of a 30Si beam may notbe feasible.The simulated evolution of the total RIB population (sec. A.2, fig. A.4) is overlaidon the data in figure A.6 and agrees well with it. This result substantiates the simpleEBIT thermodynamical model and confirms Spitzer heating as the dominant sourceof ion losses.1500.01 0.1 1 100.11 Simulated Observed t  (storage time) [s]  Count rate [counts/cycle] T1/2(30Mg) T1/2(30Al)Figure A.6: Integrated RIB count rates on the MCP as a function of storage time, obtainedfrom the spectral analysis of time-of-flight data. The simulated RIB populationevolution from sec. A.2 is shown normalized to the measured count rate data.A.4.2 Charge State EvolutionIn EBITs, the storing time governs the charge state distribution of the ion cloud and;thus, it influences confinement and Spitzer heating. It also influences how muchbackground gas gets ionized, which might contaminate the produced beam. Fur-thermore, if the beam were used outside the EBIT, it needs to match the subsequentdevice’s acceptance. Therefore, the storing time and the desired charge state mustbe chosen to balance the quality of the created beam and the amount of daughterspecies it contains.To maximize the production of daughter beam, optimize its purity, and minimizedecay losses for its mass measurement, we analyzed the time-of-flight data lookingfor charge state evolution of the stored RIB. In fig. A.5, the RIB peaks correspondingto q = +9 to +12 can be seen, as well as the series of peaks of typical backgroundgas species (12Cq+, 14Nq+, 16Oq+ and 40Arq+) and of 138Baq+ originating from evap-orated material from the electron source cathode.The relative populations of each of the RIB charge states are shown in figure A.7as function of storage time. During the analyzed storage time interval, the RIBcloud evolved from charge state q = +5 to +12.The charge state q = +11 dominates the RIB population for t ' 300 ms, whichconfirms the charge state chosen to perform the simulations in sec. A.2. It is alsoclear from fig. A.5 that the q = +11 RIB peak is the most separated charge statefrom any background species. Thus it was the best choice to be employed for furtherstudy outside of the EBIT.151In-EBIT relative charge state population of A=30 beam0.01 0.1 1 (storage time) [s]T1/2(30Mg) T1/2(30Al)Figure A.7: Evolution of charge state population of A = 30 beam in EBIT as a function ofstorage time. Relative populations are stacked by charge state. Darkened regionsare the error of the higher charge-state population. The increase in uncertain-ties after 3 s is due to the drop in RIB count rate and to overlap with intensebackground peaks.a.5 Identification of Daughter Beam through PTMSAlthough the systematic measurements performed with EBIT reveal strong evi-dence of in-trap decay and creation of daughter beam, it still did not provide defini-tive proof that the daughter species was present in the beam extracted from theEBIT. The time-of-flight to the MCP cannot resolve isobars. For this reason, theidentification of the daughter beam was performed with the MPET high-precisionmass spectrometer, which also served as a demonstrative experiment.High-precision mass spectrometry provided unambiguous identification of eachspecies. The mass precision required to resolve 30Al+q from 30Mg+q (in the samecharge state) is 3 · 10−4 [34], which is well within MPET’s capabilities [110].In the first identification measurement, the RIB was stored in EBIT for t = 50 ms,which is much shorter than the half-life of 30Mg (T1/2 = 335(10) ms). The BNGselected the +8 charge state of RIB, which was the most populated (see Fig. A.7).The selected beam was loaded to the MPET, where a search for 30Mg8+, 30Al8+ and30Si8+ was done. The analysis revealed the presence of 30Mg8+, but no detectableamount of 30Al8+ or 30Si8+ was found in the beam.Likewise, the RIB was stored in EBIT for t = 300 ms, close to 30Mg half-life,which mostly populated the +11 charge state. This time, 30Al11+ was successfullyidentified in the MPET, confirming the production of a daughter beam in EBIT.Figure A.8 shows a sample ToF-ICR resonance of 30Al11+ acquired. Once again,30Si was not observed. However, measurements with a longer storing time at EBIT,which could enable the creation of 30Si, were not attempted.152Figure A.8: A typical ToF-ICR resonance of 30Al11+ measured with MPET. The beam usedwas created by storing the parent beam for t = 300 ms in the EBIT. The red curveis an analytical fit to the data.A.5.1 Demonstrative ExperimentThe last goal of this proof-of-principle experimental campaign was to use the beamcreated through DRIT in an experiment outside the storage device. High-precisionmass measurements using MPET are susceptible to the incoming beam quality. Asuccessful precision mass spectrometry measurement reveals that the beams pro-duced using the technique meet high-quality criteria.High-precision mass measurements of 30Al11+ and 30Mg8+ were performed. Themeasurement and analysis procedure followed the same as described in section 5.2.However, differently than the measurements described in the core of this thesis,PTMS measurements employing HCI may have a non-negligible contribution fromthe binding energy (Be) of the electrons removed from the atomic form of the ion ofinterest [67]. The ratio (Rν, see eq. 3.6) between their cyclotron frequencies (νc) ofthe ion of interest and the reference ion is updated to include the electron bindingenergies:Rν =νc,re fνc=(ma − q me + Be) qre f(ma,re f − qre f me + Be,re f ) q , (A.1)where me is the mass of the electron, ma is the atomic mass and q is the charge stateof the ion. The subscript re f refers to the calibrant ion, otherwise it refers to theion of interest. In the case of this experiment, 16O6+ (Be = 0.43 keV) was used asa reference for 30Al11+ (Be = 2.22 keV), while 39K10+ (Be = 1.28 keV) was used for30Mg8+ (Be = 1.03 keV). Atomic masses of the reference ions were taken from [34],while electron binding energies were taken from [168].153The results of the mass measurements performed are shown in table A.1. Ourvalues agree with the Atomic Mass Evaluation of 2016 [34] and provide a modestimprovement on their precision.Table A.1: Atomic masses of 30Al and 30Mg, given as mass excesses. Values presented inthis work are compared to [34]. The weighted average of the measured frequencyratios between the ion of interest and of reference are also given.Ion of interest Reference Rν Atomic mass excess [keV]This work Literature30Al11+ 16O6+ 1.022476354(95) −15863.5(2.6) −15864.8(2.9)30Mg8+ 39K10+ 0.962122910(48) −8880.9(1.4) −8883.8(3.4)a.6 Conclusions and ApplicationsWe successfully demonstrated the Decay and Recapture Ion Trapping techniqueusing the EBIT at TITAN facility. A cloud of 30Mgq+ ions was stored in the EBIT,and the creation of a 30Alq+ daughter beam was identified through Penning trapmass spectrometry. The extraction for mass measurement in the Penning trap alsodemonstrated the capability of DRIT to produce beams compatible with subsequentexperiments.We performed EBIT simulations to understand losses mechanisms related to re-trapping efficiency. Simulated results agree with the observations made in the ex-periment and indicate that Spitzer heating is the primary source of observed losses,and not the recoil energy from β-decay. Our results are in line with findings ofother experiments performed using DRIT with EBITs as storage media [206, 207],which indicated high re-trapping efficiency of decay products. However, they con-trast with those performed in Penning traps [204, 205], which suggest a significantinfluence of the recoil energy in the observed efficiencies.EBITs are reliable storage devices for DRIT. They have demonstrated superiorconfinement capabilities, higher charge-space limit, and larger recapture efficienciesthan other ion traps. Moreover, Spitzer heating depends on operating parametersthat can be tuned for specific experiments. Since EBITs are regularly employed inRIB facilities to provide charge-bred beams for experiments and post-acceleration,this technique has the potential to become a regular tool to increase beam availabil-ity.DRIT can allow access to non-ISOL beams at ISOL facilities. For example, 34Si[220] and 88Zr [221], nuclides that were the object of particular interest in recentyears, could be produced from 34Al and 88Y. Using the parameters of this exper-iment only, over 50 new nuclides could be available at ISAC facility using DRIT.These are shown in fig. A.9 together with the currently available RIB yields [142].The technique also can give direct access to certain nuclear isomers. The long-lived154isomeric 1+ state of 34Al, for instance, has been an object of curiosity [222, 223, 224]and is preferentially and cleanly populated only through a 34Mg parent beam.In some cases, better beam properties could be achieved through DRIT even foravailable beams. A clear example is the measurement of the Q-value of the superal-lowed β decay 74Rb→ 74Kr [156, 225]. Noble gases produced in ISOL facilities sufferfrom high levels of contamination co-produced in the ion sources. In contrast, purealkali beams are commonly available. The 74Kr beam can be more cleanly producedthrough the decay of 74Rb.01020304050600 20 40 60 80 100ISAC yields2468101214log 10(Yield [1/s])Available through DRITNeutron NumberProton NumberFigure A.9: Current yields of radioactive ion beams available at the ISAC facility at TRIUMF[142]; purple nuclides are accessible through DRIT. These were selected based onthe availability of a suitable parent with a minimum yield of 50 pps, maximumhalf-life of 2 s and Qβ < 15 MeV. Courtesy of R. Klawitter.155B THE CRYOGENIC UPGRADE TO THETITAN MPETThe mass Measurement Penning Trap (MPET) at TITAN has been successfully per-forming precision mass measurements of radioactive nuclei for over a decade. Asdetailed in sec. 4.2.2, it is designed to probe masses of ions living as short as 10 ms,reaching relative mass precision in the 10−7 − 10−9 range. A powerful way to boostthis precision and its resolving power is to charge breed the inspected ion [156, 171],as can be seen by the dependence with the charge state q in the cyclotron frequency(eq. 3.5). At TITAN charge breeding is achieved through electron impact ionizationin the EBIT, as illustrated in appendix A.The advantage of employing highly charged ions (HCI) in PTMS is illustratedin the top panel of figure B.1. It shows a typical ToF-ICR measurement (see sec.3.2) performed at the MPET. The spectrum shows the presence of two close-lyingspecies: 130Cs+12 and 130Ba+12, whose mass difference is in the order of 3 parts in amillion [34]. Using HCI, in this case, boosted the resolution and enabled the massseparation of the isobar pair in a measurement of only 100 ms.HCIs are widely used in nuclear and atomic sciences, but experimental investiga-tions with them are challenging. They interact more strongly with neutral moleculesthan singly charged ions (SCI), thus keeping their charge states for long measure-ment times requires more stringent criteria for vacuum conditions. In PTMS, in-creasing charge state or measurement time also raises the probability that the ion in-teracts with residual gas species in the trap. Ion-gas viscous interactions are knownto create damping artifacts on ToF-ICR measurements using SCI [227]. When usingHCI, charge-exchange reactions, such asX+q +N2 −→ X+(q−i) +N+i2 ,become dominant.Such electron recombination reactions can have very detrimental effects on thequality of ToF-ICR resonances. One example is shown in the bottom panel of fig-ure B.1. Charge exchange events ionize background gas particles, which have verydifferent m/q than the sampled ion. Such ions will appear in very different posi-tions in the time-of-flight spectrum. Meanwhile, the charge exchanged ions slightlyincrease their m/q, leaving the measured cyclotron frequency range but still appear-ing as a background in a similar time-of-flight range as the undisturbed ions ofinterest. Furthermore, the simultaneous trapping of the produced contaminant ionwill affect the ion motion of the ion of interest that could result in a systematic shiftof the measured cyclotron frequency.15644.2 45.8 47.4 49.0 50.6 52.2Time of Flight [μs]1000 ms measurement N2+1He+1H2+1100 ms measurement130Cs+12 130Ba+12Measured Frequency - 5247200 Hz   [Hz]-6.4 9.6 25.6 41.6 57.6 73.644.2 45.8 47.4 49.0 50.6 52.250443832262075553515Time of Flight [μs]Measured Frequency - 5247200 Hz   [Hz]130Ba+12CountsCountsSpecies of InterestSpecies of InterestFigure B.1: A comparison of ToF-ICR resonances taken at different measurement times il-lustrates the degradation of measurement quality due to ion-gas interactions.On top, the measurement done in 100 ms shows the presence of two species:130Cs+12 and 130Ba+12 (bi-dimensional histogram events is in blue and averagetime-of-flights in black, red curve is an analytical fit [116]). On bottom, the samespecies are measured for 1 s, centered on 130Ba+12, but the resonance pattern canno longer be seen. Also the presence of additional species is now seen in thetime-of-flight spectrum (grey histogram), and are compatible with ionized N+2 ,H+2 and He+. First published in [226].Recently, the TITAN EBIT was upgraded to enable access to bare, H-like or He-like configurations of species up to Z = 70 [203]. With higher charge states available,the demand for better vacuum in the MPET increases. In section B.1, we estimate therequired vacuum in order to take full advantage of EBIT’s improved performance.In section B.2, we evaluate the pressure levels in the MPET, which revealed thatsubstantial improvements to its vacuum system were required. We carried outvacuum simulations, discussed in section B.3, to explore the impact of potentialmodifications to the vacuum system. Finally, in section B.4, the concept of theupgraded system is presented.157b.1 Vacuum Requirements for PTMS of HCIThe maximum pressure in the experimental setup is bound to the maximum tolera-ble level of ion-gas interactions during a measurement. In order to understand thefactors that play a role in such interactions, we model the ion as a particle travelingthrough a region filled with a gas of constant density. We assume the backgroundgas is in the Knudsen’s regime (molecular flow) [228, 229] and thus can be approxi-mated by an ideal gas of temperature T and pressure P.The ion travels a distance s through the background gas and has an interactioncross section σ with the residual gas particles. The expected number of interactionsη between the ion and residual gas particles is given by the number of particlespresent in the ”interaction volume” σs:η = σ sPkB T, (B.1)where kB is the Boltzmann’s constant27. We use this equation to relate an expectednumber of interaction events to the corresponding pressure in the trap volume.Next, we discuss how s and σ were calculated in the context of ToF-ICR measure-ments of HCI, subject to charge-exchange interactions.The path length (s) of the ion motion inside a Penning trap grows with q/m ofthe ion and with measurement time (tRF). Here, we estimated it from the equationsof motion of the ion during a ToF-ICR measurement, as formulated in [116]. Itcan be approximated by the length covered by an ion through a conversion from apure magnetron motion (of angular frequency ω− and initial radius r0) to a pure re-duced cyclotron motion (of angular frequency ω+) by a quadrupole radiofrequencyexcitation:s(tRF) ≈ r0∫ tRF0√ω2+ sin2(pi2ttRF)+ω2− cos2(pi2ttRF)+ 2 sin(pi2ttRF)cos(pi2ttRF)cos[(ω+ −ω−)t] dt .Examples of path lengths for a few ions are shown in figure B.2. As can be seen, itcan easily surpass a kilometer in a typical measurement procedure.Cross-section data for charge-exchange processes are scarce. An estimate for oneelectron transfer can be obtained through a simple scaling rule obtained by Mu¨llerand Salzborn [230]:σq→q−1 = C qα E−β0 [cm2] , (B.2)where E0 is the first ionization potential (in eV) of the neutral gas molecule. AtTITAN, a residual gas analysis revealed that the background gas is mainly com-posed by nearly equal amounts of N2 and H2 (see discussion in the next section),both with E0 ≈ 15.5 eV [231]. The empirical parameters are C = 1.43(76) · 10−12,27 Note that this equation is only valid if the kinetic energy of the ion is much greater than the backgroundgas particles’ kBT (∼ tens of meV). The energy of the ion motion in a typical ToF-ICR measurement is onthe order of tens of eV, so this condition is easily satisfied.1580.2 0.4 0.6 0.8 10.010.1110 7Li+1 74Rb+1 74Rb+37Path Length [km]Measurement time [s]0.1Figure B.2: Calculated path length of the motion of a few ions during a ToF-ICR measure-ment procedure in MPET, as a function of the measurement time. First publishedin [226].α = 1.17(9) and β = 2.76(19) [230]. The charge-exchange cross section is typicallyon the order of 10−14 cm2 (or 1010 b) for q > 10, about an order of magnitude largerthan cross section of ion-gas collisions (see fig. B.3). Multiple-electron transferswere not considered.0 10 20 3010-1510-1410-13σ[cm2 ]Charge State (q)Charge ExchangeCollisionFigure B.3: Calculated interaction cross sections between Rb+q ion and N2 neutral molecule.The cross section for charge exchange process (red) was calculated through eq.B.2 and has a dependence with the charge state q. The collision cross section(blue) was calculated from the radii of the molecule and the singly charged Rb+ion (taken from [231]) and therefore it does not have the appropriate chargedependence built in.Finally, in order to have an undisturbed measurement, we chose η < 0.1, whichcorresponds to less than 10% of ions experiencing any interaction with the residualgas during a measurement. The maximum pressure required to achieve such con-ditions can be calculated through inverting eq. B.1. In figure B.4, this pressure iscalculated for ToF-ICR measurements of the superallowed β-emitter 74Rb+37, fullyionized (m/q = 2).1590.01 0.1 110-1110-1010-9  74Rb+37 Required Pressure [mbar]Measurement time [s]Figure B.4: Maximum background N2 gas pressure required for η < 0.1 calculated for aToF-ICR measurement of 74Rb+37, as a function of measurement time. The grayband represents the error inherent to the model of eq. B.2 [230]. First publishedin [226].The results of this analysis for many different HCI at charges higher than +20reveal that longer measurements (tRF > 100 ms) require pressures at least in the10−11 mbar range. To benchmark our method, we verified that such requirementlevels were satisfied at the SMILETRAP mass spectrometer [232], which was used toperform ToF-ICR of stable HCI up to 1 s. Next, we evaluate the pressure levels at theMPET mass spectrometer at TITAN and discuss its agreement with the establishedcriteria.b.2 HCI as Vacuum Probes and the Pressure at MPETAt TITAN, two ionization vacuum gauges measure the pressure at the ends ofMPET’s superconducting magnet, both yielding ∼ 10−10 mbar. Such pressure val-ues are higher than desired, given the analysis presented in the previous section.However, the gas pumping conductance from the inner volume of the Penning trapis very restricted. Thus, the pressure levels are expected to be higher than wherethe gauges are installed. Yet, in order to address appropriate solutions to improvethe vacuum at MPET, it is imperative that we measure the pressure inside the trap.The inner volume of a precision Penning trap is a tightly enclosed space thatrestricts the placement of any vacuum gauge. Therefore, we resorted to indirectways to determine the pressure of residual gases by observing charge exchangereactions with the sampled ions. ToF-ICR measurements of 133Cs+13 ions wereperformed at several measurement times, ranging from 25 ms to 100 ms. First, wedetermined the ratio of 133Cs ions that underwent charge-exchange during each160measurement. The η was estimated by the ratio between the number of charge-exchange events (Ncx) and the number of 133Cs ions (Nion):η =NcxNion. (B.3)Nion is determined by the number of counts inside the time-of-flight window thatcontains the species of interest (see, for example, the ranges marked in the his-tograms of figure B.1). Similarly, Ncx can be determined from the number of countsoutside the same range, which should correspond to ionized background gasescounts. In figure B.5 the measured η values are presented for all inspected measure-ment times. The expected linear trend from eq. B.1 is clearly followed.0 20 40 60 80 1000. ηMeasurement Time [ms]Figure B.5: Measured ratio of 133Cs+13 ions that went through charge exchange, as a functionof the measurement time. The red line is a linear fit to the data. First publishedin [226].Using the data in figure B.5 and equation B.1, the pressure in MPET was esti-mated to be 8(5) · 10−9 mbar. By comparing this number to the calculated requiredpressures presented in section B.1, it is immediate that the quality of the vacuumin MPET is not good enough for long measurements of HCI with q > +20. In thesituation presented in figure B.4, for example, the Penning trap requires a vacuumupgrade of about two orders of magnitude to be able to perform a mass measure-ment of 74Rb+37 of a few hundreds of milliseconds long.Besides the magnitude of the pressure levels in the trap, it is also important toknow the composition of the residual gas. Therefore, a Residual Gas Analyzer(RGA) was installed in the beamline near MPET. This device works by ionizingthe residual gas particles and sending the ionized particles through a mass filter(further details can be found at [233]). The generated mass spectrum is shown infigure B.6. The analysis revealed the presence mostly of ions of mass 2 u and 28u, assumed to be from ionized H+2 and N+2 or CO molecules. Their presence isalso seen in the charge-exchange products shown in the bottom panel of figure B.1.Smaller amounts of water and carbon dioxide were also seen, but no meaningfulamount of He gas was detected.1610 4 8 12 16 20 24 28 32 36 40 44 4812320%CO2: 10%Partial pressure [ x10-9mbar]Mass [u]H2: 38% N2 / CO: 32 %Figure B.6: RGA analysis of the vacuum in the beamline near MPET, taken with a RGA200module from Stanford Research Systems [233].b.3 Vacuum SimulationsThe need for improvements in the vacuum system of MPET had already been iden-tified in the early experiments employing HCI [156, 225]. Past attempts include”baking” of the system (degassing using high temperatures) [234] and the additionof new vacuum pumps. However, the analysis presented in the previous sectionsrevealed the need for more profound upgrades.Possible upgrades were studied through computer simulations of the MPET vac-uum system. The procedure had two phases. The first consisted of creating a modelof the vacuum profile in the present system. It had to accurately reproduce the pres-sure in the three different locations of the system where measurements exist: insidethe Penning trap (by the method described in sec. B.2), and by two ionizationgauges placed near the two ends of the superconducting magnet. The second phaseconsisted in exploring potential modifications to the vacuum system to verify whichcan produce the required pressure in the trap.B.3.1 Monte-Carlo AlgorithmThe developed algorithm followed a Monte-Carlo approach. It simulated the trans-port of residual gas particles through the vacuum system to analyze how they couldbe more efficiently pumped out and where density pockets could be formed.Outgassing from the internal surfaces was considered the only source of parti-cles, and the system was considered to be in the molecular flow regime. In themolecular flow, the gas particles are non-interacting and behave as in an ideal gas.Therefore, the simulation is only adequate to describe the steady-state and cannotmodel pumping speeds or any gas flow dynamics happening in the transient phase.162First, a geometrical model of the vacuum system was constructed, including vac-uum chambers, pumps, and objects like electrodes and support structures (alsoconsidering their materials and respective outgassing rates). Then, particles weregenerated and transported through the system following the prescription below:1. Outgassing events: a new particle is initiated at a random point of the surfaceof a random object inside the system. The likelihood of an ”outgassing event”happening at a given object is regulated by the outgassing rates of the object’smaterial. The event happens at a random instant (t) between t = 0 and t = tend(see item 5 below). The particle is assumed to be of N2 molecule, but minimalvariation was seen in the results by using other particles.2. Release from surface: the particle is emitted from the surface of the object.The emission angle with respect to the surface follows Knudsen’s Cosine Law[228, 229], while the particle’s velocity (v) follows a Boltzmann distributionwith the temperature of the surface (∼ 300 K).3. Transport through the system: the particle travels through the vacuum sys-tem in fixed distance steps (∆s). The time is iterated in the appropriate timestep ∆t = ∆s/v. The advantage of fixing the ∆s instead of ∆t is the bettercontrol over the spatial resolution of the generated density profile, which mustbe much smaller than the geometrical features of the objects in the system. Inthis case, ∆s = 0.1 mm.4. Interaction with objects: the particle may hit an object as it is transportedthrough the system. If it reaches an ordinary object, it is instantly re-emittedfollowing the prescription of item 2. If it reaches a pump, it is removed fromthe simulation. Pumps are assumed to be ideal, with no backflow to the sys-tem.5. Density snapshot: if the particle survives in the system until the time tend isreached, its position is registered in the density profile. The density profile is asnapshot of the particle distribution inside the system taken at the instant tend.To ensure it reflects an equilibrium condition, tend must be much longer thanthe average lifetime of particles inside the system, and the particles need to beintroduced into the system at random times during the inspected period. Ananalysis of the average lifetime of particles in the system is shown in figure B.7.It reveals that tend = 0.1 s fits the criteria.The procedure described is depicted in the flow diagram of figure B.8. After asufficiently large amount (N, usually on the order of 106) of iterated particles, thegenerated particle density distribution at the end of the simulation was transformedinto a pressure distribution following procedures described in the next section.1631E-4 0.001 0.01 0.1100100010000100000  Number of ParticlesTime inside system [s] Pumped particles Remaining particlesFigure B.7: Survival time of simulated particles in the vacuum system. Almost all particlesare expected to be pumped out after 0.1 s of their release in the system.Inicialize new particle:∙ Random point at internal surfaces∙ Random initial time before tendAssign random velocity vector∙ Boltzmann distribution  ∙ Knudsen's Law∙ Calculate Δt from fixed Δs  Transport particle through system∙ Iterate t  Did the particle hit an object?Pump ora surface?Did it reachtend ?Register particle in density profile∙ Iterate N  SurfaceRemove particle from system∙ Iterate N  NoYesPumpNoYesParticleTemperature tendΔsN2300 K0.1 s0.1 mmSimulation Parameters:Figure B.8: A simplified flow diagram of the Monte-Carlo algorithm employed to simulatethe vacuum in the MPET system.1640 50 100 150 200 250 300012345678BeamPumps Pump9∙10-10  mbarSimulated Pressure  [10-9  mbar]1.2∙10-10  mbarBeam line  [cm]Vacuum in the trap:Simulated:  4∙10-9  mbarMeasured:  8(5)∙10-9  mbarTrap Injection OpticsExtraction OpticsFigure B.9: Simulated pressure along the ion transport axis in the MPET beamline. In thebottom is shown a schematic representation of the elements in the beamline forreference. Points where pressure measurements were taken, either by gauges orby indirect techniques (sec. B.2), are marked with black diamonds.B.3.2 Results: Current SystemSimulations using the geometry of the current system were performed. Since thesimulated density distribution is generated in arbitrary units, it was normalized tofit the ionization gauge measurements in the real system. Outgassing rates of thedifferent materials were varied within an order of magnitude to achieve the bestagreement with the ionization gauge measurements.The resulting simulated pressure profile of the current MPET system is shown infigure B.9. It shows the average pressure along a cylindrical volume of 5 mm radiusalong the beam axis, where the ion transport path is expected to be contained. Thesimulated pressure in the trap and in the ionization gauges are compatible with themeasured pressures in the same locations.16580 100 120 140 160 180 20010-1210-1110-1010-910-8Ideal Cryogenic Pump as: Current (no pump) All trap internal surfaces  Cold finger close to trap   Simulated Pressure [mbar]Beam line [cm]Required pressureTrap regionFigure B.10: Simulated pressure along the ion transport axis in the MPET beamline, withthe addition of ideal cryopumps: a cold finger near the trap (blue) and all trapelectrodes acting as cryopumps (red). The black curve is the simulated pressurewith no modifications (as in fig. B.9). The trap region is marked by the yellowshaded area and the dashed line marks the required pressure in the trap.B.3.3 Results: Upgraded SystemWith the simulation framework successfully benchmarked, it was used to study po-tential upgrades to the vacuum system. The goal was to reach a pressure on theorder of 10−11 mbar in the trap volume. The modeled system included modifica-tions that ranged from the addition of new vacuum pumps, the use of materialswith lower outgassing rates and geometrical changes to the objects (for example,increasing diameters of apertures to improve gas flow). The generated densitydistributions were converted into realistic pressure distributions using the samenormalization factor obtained in the simulations of the current system.Most of the explored modifications produced minimal improvements. Only oneclass was able to produce the desired vacuum levels in the trap: the ones thatincluded cryopumps in the trap region. Cryopumps were simulated as objects thatacted as ideal pumps. Particles that hit them would not be reflected back to thesystem. Figure B.10 shows the simulated pressure profile generated by a few of thestudied modifications including cryopumps.166Figure B.11: Model of the CryoMPET upgrade highlights its main components: a cryogenicPenning trap is coupled to a two-stage cryocooler by a high-purity Cu finger.The remaining of the system is kept at room temperature.b.4 The Cryogenic Measurement Penning TrapThe vacuum simulations detailed in the previous section suggested that a solutionbased on cryopumping can successfully bring the pressure levels in the MPET downto the required levels for PTMS of HCI. This result led to the development of theCryogenic Measurement Penning Trap (CryoMPET): the upgrade system to MPETwhich incorporated a cryopump into the Penning trap.The goal was to address a cryopumping solution to MPET that minimized changesto the current setup, enabled cryogenic operation for extended periods, and couldbe integrated into the system within its restrictive spatial limitations. In the de-signed system a new Penning trap, kept at cryogenic temperatures, has capabilitiesof cryopumping the residual gas in its interior. The trap is coupled to an externalcryocooler by a high-purity Cu finger. A thermal shield encloses the cold pieces, andthe remaining of the system is kept at room temperature. A model of the upgradedsystem is shown in figure B.11. The concept, the design, and the construction ofCryoMPET are detailed in the next sections.B.4.1 Concept of CryopumpingCryoMPET’s vacuum pumping mechanism is based on cryopumping. Backgroundgas particles are removed from the volume of the trap by adhering to cold surfacesthrough weak intermolecular bonds. Particles are not removed from the system,167but they are removed from the region where they could interact with ions during amass measurement. A review article on cryopumping can be found in [235].The trap is essentially a cold enclosed volume with apertures for ion injectionand extraction. The cold surfaces of the trap are required to remove from the vol-ume the room-temperature gas particles that enter the trap through the apertures.To achieve that, two cryopumping mechanisms are employed in the CryoMPET:cryosorption, targeting light and weakly interacting gasses such as He or H2, andcryocondensation, targeting heavier gasses. The concept of these mechanisms is ex-plained in the following in the context of the upgraded system.CryocondensationCondensation is the most elementary form of cryopumping. It consists of the so-lidification of the gas molecules onto the cold surface. The achievable pressure isdetermined by the saturation pressure of the gas at the temperature of the cold sur-face, which is shown in figure B.12 for a few common gases. As can be seen, thesaturation pressure decreases very rapidly as the system is cooled down. This iswhere lies the strength of this type of pumping: a small decrease in temperaturecan lead to orders of magnitude better vacuum quality.The saturation curve is defined in the thermodynamic equilibrium of the solid-gasphase at the given temperature. It means that at this pressure, the flux of particlesleaving the surface is equal to the flux of particles sticking to the surface; thereforeit has zero net pumping. Moreover, the particle adhesion occurs when the particleis at near thermal equilibrium with the surface. With warm gas entering the trapthrough the apertures, its particles will require to bounce in the internal surfacesa few times before adhesion. For these reasons, it is recommended to design thetemperature of the system to achieve two orders of magnitude lower saturationpressure than required [235].In CryoMPET, the background gas composition is expected to be similar to whatwas measured in the MPET (see fig. B.6). Then, according to figure B.12, a tem-perature below 20 K should be sufficient to pump most gases, targeting speciallyN2. However, efficient pumping of light gases such as He and H2 is hardly donethrough cryocondensation.CryosorptionThe cryopumping of gases of difficult condensation is typically done with the useof cryosorbents. They are made of high porosity materials that can have a largesurface area. As the gas particles enter the pores and reach the inner surfaces, theirregular geometry constricts particle motion, and the particles remain confined inthe material [235, 237].Cryosorbents can pump large amounts of gas and considerably reduce the pres-sure of the system. However, they saturate after some time, and their effect becomesnegligible. Therefore, a regeneration procedure that includes the warming of thesystem must be incorporated into the routine operation. In addition, the design of16810 10010-1410-1210-1010-810-610-410-2100102104Saturation Pressure  [mbar]Temperature [K]H2 NeN2 O2 CO2H2ORequired PressureFigure B.12: Saturation pressure curves of several common gases, with data from [236]. Cry-oMPET’s required pressure is marked by the dashed line.the system must ensure that the sorbents are not directly exposed to the inflow ofgas particles. Otherwise, the material may quickly saturate with gases that are easyto condense.Cryosorption is less dependent on the temperature than condensation and moredependent on other factors like chemical properties and construction geometry.Nevertheless, it is still advisable to keep the temperature as low as possible. Typicaltemperatures employed in cryosorption are around 10 K or lower.At CryoMPET, the cryosorbent of choice was the OLC AW 12X40 coconut-basedactivated charcoal from Calgon Carbon. This product is the equivalent as the oneemployed in the vacuum system of the International Thermonuclear ExperimentalReactor (ITER) [237, 238], whose performance was extensively tested against manyother products in the market.FeasibilityAlthough the concept is feasible according to the simulations with ideal cryopumps,a simple model can provide a more analytical look into the involved factors. Cry-oMPET can be modeled as an enclosed volume of internal temperature Ttrap andinternal pressure Ptrap. Two apertures (for ion injection and extraction) connect itsinterior to the exterior environment, whose gas is at pressure Pout at room tem-perature Tout. The apertures can be modeled as cylindrical pipes (like the tubeelectrodes at MPET, see sec. 4.2.2) of diameter Dtube and length Ltube. Figure B.13shows a schematic illustration of this model.To achieve equilibrium, the flux of gas particles to the interior of the trap (Qin)must be equal to the flux of cryopumped particles (Qout) to the internal surfaces.169Figure B.13: Schematic illustration of the variables involved in the concept of CryoMPET’spumping by cryosorption.The flux Qin can be modeled as the flux through the two tubes in molecular flow,considering Pout  Ptrap:Qin = 2pi D3tube3 LtubePout√kBTout2pi m, (B.4)where kB is the Boltzmann constant and m is the mass of the background gasmolecule.If the design temperature is below 20 K, most gases are pumped by condensationin the tubes or inner surfaces of the trap. Therefore their vapor pressure becomes anegligible contribution to the Ptrap. Then, the pumped flux Qout can be modeled aspure cryosorption [235]:Qout = Awalls Ptrap√kBTtrap2pi mα(Ttrap, gas, surf.) , (B.5)where Awalls is the internal surface area available for cryosorption and α is calledthe ”sticking” coefficient, which is the probability of a gas particle adhere to thesurface of the sorbent upon impinging into it. It greatly depends on the specificcombination of gas, sorbent material and temperature.Imposing the equilibrium condition Qin = Qout, we have thatPoutPtrap=(38LtubeDtubeAwallsAaperture) √ToutTtrapα(Ttrap, gas, surf.) , (B.6)where Aaperture = piD2tube/4 is the area of the apertures of the trap. The geometry-dependent factor (in between brackets) is on the order of 300 using the MPET trapdesign. However, Awalls can be greatly increased by the use of cryosorbents, whichhave a large surface area due to their porosity. The sticking coefficient has a mildvariation with temperature. For H2 pumping by the ITER-type charcoal, α = 0.6 at5 K and α = 0.3 at 12 K [238].According to the analysis presented in sections B.1 and B.2, it is desirable thatPout/Ptrap ≈ 500. Given the variables in equation B.6, considering the use of ITER-type cryosorbents and trap temperature on the order of < 20 K, the desired pressurein the trap is achievable.170B.4.2 Design ConsiderationsThe overall design concept of CryoMPET is displayed in figure B.11 and presentedin the beginning of this section. Next, the choices and considerations that wereincorporated into the design to enable the cryopumping concept are described.General designMaterial selection: in general, the materials and components employed in Cry-oMPET have to be non-magnetic to not disturb the homogeneity of the magneticfield in the trap, have low outgassing rates for better vacuum properties, haveknown performance at cryogenic temperatures, and withstand temperatures upto 120◦ C for baking of the system and regeneration of cryosorbents.Cryocooler: we opt for a cryogen-free approach for cooling. Since a typical MPETexperiment requires continuous operation for many weeks, including tuningand preparation, a liquid helium cryostat would require high maintenance costsas it consumes about a liter of the cryogen per hour. With a closed-cycle cry-ocooler, the system can run continuously with minimal human interference orconsumption of expensive goods28. The employed cryocooler in CryoMPET isthe DE-215S model from Advanced Research Systems Corporation. It has twostages, with zero heat load the first stage can reach down to 20 K, while thesecond stage can reach 2.5 K. A critical disadvantage of using a cryocooler isthat the cold tip lies about half a meter away from the trap and requires a veryefficient thermal transport solution.Trap Temperature and Coupling to Cryocooler: the trap is thermally coupledto the second stage of the cryocooler by a rod of solid high-purity copper (alsodenoted by cold finger). The rod is anchored to the ring electrode, which isthermally coupled to the other electrodes of the trap. Given the limited space inthe setup, the finger diameter was limited to only 12.7 mm. Therefore, we resortto special materials in order to have efficient thermal coupling.Some materials have an exceptionally high thermal conductivity at cryogenictemperatures [239]. It is related to the Residual Resistive Ratio (RRR) grading,which is the ratio between the electrical resistance of the material at room tem-perature and 4.2 K. In metals, the RRR grading is mostly governed by its purityand crystalline structure [240].The copper rod was fabricated by Luvata Special Products and had a purityof 99.99995% and RRR ≈ 2000. The expected thermal conductivity is on theorder of 104 W/(m·K) at ∼ 10 K. A Finite Elements Method analysis [241] re-vealed that the trap temperature can reach 8 K using a Cu rod of RRR = 1000,considering the specifications of the cryocooler and heat input only from ther-mal radiation. This result does not consider heat input from conduction fromthe planned points of contact (see below). The trap electrodes were made from99.9995% pure Cu of RRR ≈ 1080 produced by the same company.28 The employed model requires regular maintenance after 10, 000 hours of operation171Thermal Shield: the cold system containing the trap and cold finger was enclosedby a thermal shield to limit the exposure of the cold surfaces to the room tem-perature environment. The shield was attached to the first stage (20 K) of thecryocooler and was polished for high reflectance. Given the surface area of theshield, it could receive up to 3 W from thermal radiation depending on the qual-ity of the polishing. The shield pieces were made of Oxygen-Free ElectronicCopper (OFE Cu, alloy C10100).Thermal contacts: good thermal coupling between different pieces is essentialto have the necessary heat conduction and thus to reach the desired tempera-tures. In most cases, the pieces are screwed or clamped together with a layerof a vacuum-friendly grease in between. In many other cases, good thermalcoupling is desired, but also electrical insulation is required. It is the case ofthe connections between trap electrodes and between the trap and the cold fin-ger. The coupling of these parts is then mediated by a custom piece made ofsapphire, which has a high thermal conductivity at cryogenic temperatures.Fixture and interface: minimal contact must exist between warm and cold struc-tures to avoid heat conduction to the trap. However, they need to be integratedwith each other for appropriate positioning and alignment. In the original MPETdesign, all ion optics elements and the trap were held in place by a support struc-ture made of three parallel titanium rods. In the CryoMPET design, the sametriple rod structure is maintained, but ceramic sleeves of low thermal conductiv-ity cover two of the rods. The shield assembly, which also holds the trap in itsinterior, is held by gravity on these two covered rods touching only a few sup-port points (see fig. B.14). The same idea is reproduced to hold the trap insideits shield. A pair of ceramic rods run across the inside of the thermal shieldstructure; the trap assembly rests on the two rods at two support points each,without touching the shield assembly.Thermal contraction: besides minimizing heat conduction, another advantage ofhaving the trap lying on the ceramic rods is that it can slide freely along themwithout losing alignment as the system cools down. The cold finger, which isabout 70 cm in length, is expected to contract about 2.2 mm from its original sizeat room temperature [242]. This was incorporated into the design: the trap sitsat the point of maximum homogeneity of the magnetic field when cold.Instrumentation: two pairs of high precision thermal sensors (model DT-670B-SDfrom Lakeshore) and resistive heaters were incorporated into the cryogenic sys-tem. The heaters will aid in the regeneration cycles, and the sensors will monitorthe temperature of the system. One instrumentation pair was placed at the tipof the cryocooler while the other was installed at one of the trap electrodes. Theheater was placed as near as possible to the sensor for most accurate readingsduring regeneration.172Safety: gas leaks in cryogenic vacuum systems can become a relevant safety issue.If the leaked gases liquefy and accumulate, they may quickly expand when thesystem is warmed up. Therefore a burst disk was installed at the vacuum systemto generate relief in case of overpressurization. Besides, an interlocking systemwas added to the system; if the pressure inside the vacuum chamber exceeds 5mbar, it will automatically turn the cryocooler off.Trap designThe design of the cryogenic Penning trap followed the design of the precursor trap,adapted to the needs of the new setup. Further modifications unrelated to thecryogenic upgrade were included to improve its overall performance. The designchoices are described in the following. A schematic representation of CryoMPETconstruction is shown in figure B.14.Trap dimensions: the trap hyperbolic shape was slightly modified from the orig-inal MPET version to better agree with the optimal configuration identified in[243]. The characteristic dimensions of the new trap are shown in table B.1. Theouter structure of the electrodes was also designed to be massive and bulkierthan previously to facilitate thermal conduction.Table B.1: Characteristic dimensions of the CryoMPET, according to eq. 3.7.r0 13.670 mmz0 11.785 mmd0 10.778 mmGold plating: copper easily oxides when exposed to air, potentially forming in-sulating patches on the surface of the electrode. These patches may charge anddisturb the transport of ions crossing its vicinities [244] and can be very detri-mental for the correct shaping of the trap potential. A common solution is toplate the piece with an inert metal, such as gold. All CryoMPET trap electrodesreceived gold plating (type I, grade A, 7.5 µm thick) with a thin underlayer ofsilver acting as a diffusion barrier29.Electrodes for RF: in the original MPET design the RF signal for ion motion excita-tion was applied to the guard electrodes, which were appropriately segmented.However, their smaller size and longer distance to the ions compared to thecentral ring electrode resulted in a lower RF power delivered to the ions. Con-sequently, the implementation of broadband RF techniques such as describedin [113] was challenging. In CryoMPET, the guard electrodes are solid, and theRF excitations are delivered by the central ring electrode, which is segmentedaccordingly. This modification is expected to increase the RF power delivered tothe ions by a factor 10. A picture of the new segmented ring electrode (disassem-bled) is shown in picture 3.4.29 A more appropriate diffusion barrier for gold is nickel; however, its use in the Penning trap is undesirabledue to its magnetic properties.173Side view:Front view:← to Cryocooler← to MCPCold finger & trap electrodesThermal ShieldCeramicsSapphireCryosorbentStructure & fastenersIon opticsFigure B.14: Schematic illustration of the CryoMPET trap construction and its surroundingstructures.174Tube electrodes: their lengths were extended out to shape the injection and extrac-tion potential beyond the shield. In addition, they received a 1 mm increment indiameter from the original MPET design, which is expected to increase the ionextraction efficiency.Guard electrodes: two solid ring pieces form the guard electrodes and their shapewas slightly modified from the previous design. Instead of a flat surface, theyare formed by two angled surfaces to avoid Knudsen re-emission towards thecenter of the trap.Wiring: both the trap and the shield have built-in connector assemblies. Wiresfrom the room temperature system are connected to the shield external connec-tors. In the interior of the shield, cryogenic thin wires leave the connectors andare wrapped onto thermal anchoring poles. From there, the cryogenic wires areattached to the trap connectors.Cryosorbents: the activated charcoal granules were applied to six surfaces in theelectrode structure that are not relevant to the shaping of the trapping potential:the back of the endcap electrodes and the two sides of the supporting structureof the tube electrodes. The granules were applied one by one on a layer of about1 mm thick of the MCT 3715− 2SE adhesive from MicroCoat Technologies. Afterthe application, the pieces were baked at 150◦ C for an hour at a vacuum ovenfor curing of the adhesive.B.4.3 ConstructionMPET was removed from the TITAN beamline in December 2017 for the upgrade.The pieces from the CryoMPET were machined at the specialized machine shop atTRIUMF laboratory. Every component went through a rigorous UHV cleaning pro-cedure [245], and the assembly occurred in the Winter of 2018− 2019. The systemwas installed back to the TITAN facility in March 2019 and is now under commis-sioning. Figures B.15, B.16 and B.17 shows pictures of the CryoMPET assemblies.175Figure B.15: Pictures of the assembled electrode structure of the cryogenic Penning trap: (a)without one segment of the central ring electrode showing its interior, courtesyof Stuart Shepherd; (b) fully assembled with cryosorbent, wires, connectors andinstrumentation.Figure B.16: Pictures of the trap and shield assemblies: (a) with open shield viewed from ioninjection side, showing the trap; (b) closed shield viewed from the ion extractionside.Figure B.17: (a) picture of the trap assembly in the together with structural elements andion optics; (b) CryoMPET system installed at the superconducting magnet, thecryocooler and part of the shield around the cold finger are visible.176


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