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Penning trap mass measurements to test three-body forces in atomic nuclei Gallant, Aaron T. 2015

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Penning trap mass measurements to test three-body forces inatomic nucleibyAaron T. GallantB.Sc., Saint Mary’s University, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)The University of British Columbia(Vancouver)November 2015c© Aaron T. Gallant, 2015AbstractRecent theoretical studies have shown that three-nucleon forces are important for understand-ing neutron-rich nuclei, and for the formation of nuclear shell structure. In particular, theorycan not reproduce the N = 28 magic number in 48Ca using two-body interactions. This magicnumber is only reproduced with the inclusion of three-nucleon forces. Further along the cal-cium isotopic chain, the three-nucleon interaction predicts new magic numbers at N = 32 and34, while calculations with phenomenological interactions predict a magic number at N = 32,but disagree on the magicity of N = 34. An other theoretical tool that has been under significantpressure since the advent of precision mass measurements is the isobaric multiplet mass equa-tion, in which the binding energies of states in an isobaric multiplet should vary quadraticallywith the z-projection of the isospin. This is a consequence of the isospin dependent com-ponent of the nuclear Hamiltonian and Coulomb interactions. We test the predictions of phe-nomenological and three-nucleon interactions through mass measurements of 20,21Mg, 51,52Ca,and 51K with the TITAN Penning trap mass spectrometer. The measured mass excesses wereME(20Mg)= 17477.7(18) keV, ME(21Mg)= 10903.85(74) keV, ME(51Ca)= 36339(23) keV,ME(52Ca) = 34245(61) keV, and ME(51K) = 22516(13) keV. With the calcium and potassiummass measurements, we show that the calculations with three-nucleon forces are able to cor-rectly predict the two-neutron separation energies. In the A = 20 and 21 isobaric multiplets,neither the phenomenological nor the three-nucleon based interactions are able to reproducethe measured behaviour.iiPrefaceThe experimental work presented here was performed at the Tri-University Meson Facility(TRIUMF), Vancouver, BC, using TRIUMF’s Ion Trap for Atomic and Nuclear science (TI-TAN) Penning trap mass spectrometer.The main part of this thesis, the mass measurements of 50,51Ca, 51K and 20,21Mg, are pub-lished in the works• A. T. Gallant, M. Brodeur, C. Andreoiu, et al.. Breakdown of the Isobaric Multiplet MassEquation for the A = 20 and 21 Multiplets. Phys. Rev. Lett., 113, 082501 (2014).• A. T. Gallant, J. C. Bale, T. Brunner et al.. New precision mass measurements of neutron-rich calcium and potassium isotopes and three-nucleon forces, Phys. Rev. Lett., 109,032506 (2012).For both publications I lead and organized the experimental team during the experiment withthe help of A. A. Kwiatkowski and A. Chaudhuri. The data collection was performed withthe help of C. Andreoiu, A. Bader, J. C. Bale, M. Brodeur, T. Brunner, A. Chaudhuri, U.Chowdhury, P. Delheij, E. Mane´, S. Ettenauer, A. Grossheim, R. Klawitter, A. A. Kwiatkowski,K. G. Leach, A. Lennarz, T. D. Macdonald, D. Robertson, H. Savajols, B. E. Schultz, M.C. Simon, V. V. Simon, and M. R. Pearson. For both experiments the use of the TRIUMFResonant Laser Ionization Source (TRILIS), and the later addition of the Ion Guide Laser IonSource (IG-LIS) were crucial in producing the Ca and Mg beams. TRILIS and IG-LIS wererun by J. Lassen, H. Heggen, S. Raeder, and A. Teigelho¨fer. I was responsible for analyzingthe data and preparing the manuscripts. The theoretical calculations presented in these paperswere done by B. A. Brown and A. Magilligan, and by J. D. Holt, J. Mene´ndez, J. Simonis, A.Schwenk.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Precision potentials and the need for three-body forces . . . . . . . . . . . . . 91.3 Mass measurement techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 The importance of Penning-trap mass measurements . . . . . . . . . . . . . . . 151.4.1 CKM unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.2 Nuclear astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4.3 Nuclear halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4.4 Shell structure evolution . . . . . . . . . . . . . . . . . . . . . . . . . 191.5 Penning-trap mass measurements to test 3N forces . . . . . . . . . . . . . . . . 212 Nuclear theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1 Chiral effective field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Effective interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Modern χEFT based calculations . . . . . . . . . . . . . . . . . . . . . . . . . 292.4 Testing χEFT forces: The N = 32,34 sub-shell closures . . . . . . . . . . . . . 322.5 Testing χEFT forces: The isobaric multiplet mass equation . . . . . . . . . . . 36ivTABLE OF CONTENTS2.5.1 Two-level mixing and the d term . . . . . . . . . . . . . . . . . . . . . 422.5.2 Testing the IMME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1 Beam production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2 ISAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.2.1 TRILIS and IG-LIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 TITAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.1 RFQ cooling and bunching . . . . . . . . . . . . . . . . . . . . . . . . 513.3.2 EBIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3.3 CPET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3.4 MPET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3.4.1 Ion motion in a Penning trap . . . . . . . . . . . . . . . . . . 593.3.4.2 Sideband quadrupole excitation . . . . . . . . . . . . . . . . 613.3.4.3 Ramsey excitation . . . . . . . . . . . . . . . . . . . . . . . 683.3.4.4 Dipole cleaning . . . . . . . . . . . . . . . . . . . . . . . . . 693.3.4.5 Time-of-flight ion cyclotron resonance . . . . . . . . . . . . 703.3.4.6 Measuring the axial frequency of the Penning trap . . . . . . 703.3.5 Technical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.3.6 Determining the mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.3.7 Systematic shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.3.7.1 Relativistic effect . . . . . . . . . . . . . . . . . . . . . . . . 783.3.7.2 Spatial magnetic field inhomogeneities . . . . . . . . . . . . 783.3.7.3 Non-harmonic imperfections of the trapping potential . . . . 793.3.7.4 Harmonic distortion and magnetic field misalignment . . . . 793.3.7.5 Ion-ion interactions . . . . . . . . . . . . . . . . . . . . . . . 803.3.7.6 Non-linear magnetic field fluctuations . . . . . . . . . . . . . 814 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.1 Existing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.1.1 51Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.1.2 52Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.1.3 53,54Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.1.4 51K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.1.5 20,21Mg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.1.5.1 Isospin multiplet energy levels . . . . . . . . . . . . . . . . . 87vTABLE OF CONTENTS4.2 Discussion and measurements from this study . . . . . . . . . . . . . . . . . . 874.2.1 Calcium and Potassium at N = 32 . . . . . . . . . . . . . . . . . . . . 884.2.2 A = 20, 21 isobaric multiplet mass equation . . . . . . . . . . . . . . . 935 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102A Contributions to TITAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122A.1 Axial Frequency Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 122A.2 Arbitrary function generator programming . . . . . . . . . . . . . . . . . . . . 122A.3 SortEVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.4 Correlations between adjacent frequency ratios . . . . . . . . . . . . . . . . . 124viList of TablesTable 3.1 Characteristic trap dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 59Table 3.2 MPET eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Table 4.1 Ion yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Table 4.2 Measured mass values for 51,52Ca, 51K, and 20,21Mg . . . . . . . . . . . . . 92Table 4.3 A = 20, 21 IMME fit results . . . . . . . . . . . . . . . . . . . . . . . . . . 93Table 4.4 Experimental and calculated ground-state energies of 20,21Mg with respectto 16O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98viiList of FiguresFigure 1.1 Serge` chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Figure 1.2 Nuclear binding energy per nucleon . . . . . . . . . . . . . . . . . . . . . 3Figure 1.3 Shell model energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Figure 1.4 Woods-Saxon potential and density profile . . . . . . . . . . . . . . . . . . 6Figure 1.5 S2n near N = 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 1.6 ∆n surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 1.7 S-wave NN potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Figure 1.8 AV18 + Illinois-7 binding energies . . . . . . . . . . . . . . . . . . . . . . 11Figure 1.9 Lattice QCD derived NN-potential . . . . . . . . . . . . . . . . . . . . . . 12Figure 1.10 Model dependence of r-process abundances . . . . . . . . . . . . . . . . . 18Figure 1.11 Disappearance of N = 20,28 magic numbers . . . . . . . . . . . . . . . . 19Figure 1.12 51,52Ca historical mass excesses . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.1 χEFT diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 2.2 RG evolved χEFT potentials . . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 2.3 Valence space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.4 Comparison of Ca ground states calculated in CC and valence space MBPT 31Figure 2.5 Oxygen binding energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 2.6 E(2+) and B(E2) values for Cr and Ti isotopic chains near N = 32,34 . . . 33Figure 2.7 E(2+) values in the Ca isotopic chain . . . . . . . . . . . . . . . . . . . . . 35Figure 2.8 Ca S2n near N = 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.9 A = 9 isobaric multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 2.10 Energy levels in mirror nuclei . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 2.11 IMME b-coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 2.12 IMME d coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 3.1 TITAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45viiiLIST OF FIGURESFigure 3.2 ISAC production and separation . . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.3 Mg RILIS Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.4 IG-LIS schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 3.5 IG-LIS yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 3.6 RFQ trapping and ejection . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.7 Linear Paul trap stable region . . . . . . . . . . . . . . . . . . . . . . . . . 55Figure 3.8 EBIT schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 3.9 Resolving isomers with highly charged ions . . . . . . . . . . . . . . . . . 57Figure 3.10 A Penning trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 3.11 RF application schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 3.12 Resonance conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 3.13 Quadrupole excitation line shape . . . . . . . . . . . . . . . . . . . . . . . 66Figure 3.14 Quadrupole excitation ion motion . . . . . . . . . . . . . . . . . . . . . . 67Figure 3.15 Excitation pulses for quadrupole and Ramsey excitations . . . . . . . . . . 67Figure 3.16 Ramsey excitation line shape . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 3.17 TOF-ICR resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 3.18 Axial oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 3.19 Axial frequency measurement . . . . . . . . . . . . . . . . . . . . . . . . 73Figure 3.20 Lorentz steerer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 3.21 Ion energy during injection . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 3.22 MPET trap electrode schematic . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 3.23 Count class analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 4.1 51Ca measurement comparisons . . . . . . . . . . . . . . . . . . . . . . . 85Figure 4.2 52Ca measurement comparisons . . . . . . . . . . . . . . . . . . . . . . . 86Figure 4.3 51,52Ca and 51K resonances . . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 4.4 S2n near N = 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Figure 4.5 Ca S2n theory comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Figure 4.6 Ca S2n difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 4.7 20,21Mg resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure 4.8 A = 20 residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Figure 4.9 A = 21 residuals for Jpi = 5/2+ . . . . . . . . . . . . . . . . . . . . . . . 96Figure 4.10 A = 21 residuals for Jpi = 1/2+ . . . . . . . . . . . . . . . . . . . . . . . 96Figure A.1 SortEVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Figure A.2 Frequency Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126ixGlossaryAME03 Atomic Mass Evaluation 2003AME11 Atomic Mass Evaluation 2011AME12 Atomic Mass Evaluation 2012χEFT Chiral Effective Field TheoryCCSD Coupled-Cluster Singles-DoublesEBIT Electron Beam Ion TrapFT-ICR Fourier-Transform Ion Cyclotron ResonanceGXPF1A G-matrix, experimentally fit interactionHCI Highly Charged IonsIG-LIS Ion Guide Laser Ion SourceIMME Isobaric Multiplet Mass EquationISAC Isotope Separator AcceleratorISOL Isotope Separation on-lineKB3G mass-dependent Kuo-Brown interactionLS Lorentz SteererMCP Micro-Channel PlateMPET Measurement Penning TrapMBPT Many-Body Perturbation TheoryxNN nucleon-nucleonPI-ICR Phase-Imaging Ion Cyclotron ResonancePTMS Penning Trap Mass SpectrometryQCD Quantum ChromodynamicsRILIS Resonance Ionization Laser Ion SourceRFQ Radio-Frequency QuadrupoleSCI Singly Charged IonsTOF-ICR Time-of-Flight Ion Cyclotron ResonanceTITAN TRIUMF’s Ion Trap for Atomic and Nuclear scienceTRILIS TRIUMF’s Resonance Ionization Laser Ion SourceUSD Universial sdxiChapter 1IntroductionAtomic masses are fundamental quantites that provides important insight into the inner work-ings of the nuclear interaction through the binding energy, or the difference between the sumof the mass of the constituents and the mass of the whole. Not only is the atomic mass crucialin understanding nuclear physics, it is also required for studying astrophysics, determining theorigin of nuclei heavier than iron in the universe, and in weak-interaction studies. Figure 1.1summarizes our current knowledge of the limits of existence of atomic nuclei. The nuclides inblack are stable against decay, having infinite half-lives, while the nuclides in yellow are unsta-ble against decay, decaying by emitting either β -particles (electrons or positrons), α-particlesor by spontaneous fission. Currently there are 288 known stable nuclides, and approximately3000 known unstable nuclides, while there are predicted to be ≈ 7000 bound nuclei with pro-ton number less than 120 [1]. The path along the stable nuclides in figure 1.1 is known asthe “valley of stability”. Moving away from stability by adding either protons or neutrons,one eventually reaches the proton or neutron driplines. The driplines are defined to be thepoint where the separation energy, the energy required to remove a nucleon from the nucleus,changes from positive to negative. As seen in figure 1.1, the predicted two-neutron driplines,for most of the nuclear chart, lie far beyond the current limits of experimental knowledge.Since the discovery of the nucleus, many seemingly contradictory models have been suc-cessfully used to describe nuclear structure. The earliest models considered the nucleus to bea “liquid drop”, in which protons and neutrons only interact with their nearest neighbours. Asfigure 1.2 shows, this can be seen in the nearly constant binding energy per nucleon for thestable nuclides. While this model was able to describe several features of nuclei, such as gen-eral binding energies and fission energies of heavy nuclei, it was unable to describe the highbinding energies of light nuclei, and the stability of nuclei with specific numbers of protonsand neutrons. A slight improvement on this model is the cluster model, in which clusters of11.1. THE SHELL MODELN=8N=20N=28N=50N=82N=126N=184N=258Z = 8Z = 20Z = 28Z = 50Z = 82two-proton dripline two-neutrondripline020406080100120AtomicNumber,Z0 50 100 150 200 250 300Neutron Number, NFigure 1.1: Serge` chart showing the location of known stable nuclides (black) and un-stable nuclides (yellow). Also shown are theoretical calculations [1] giving thepredicted location of the two-proton and two-neutron driplines. The dashed linesare the “magic numbers” (section 1.1).nucleons inside the nucleus would generally bind as α-particles; however, again, this clustermodel does not describe the entirety of known nuclear properties. Many of these problemswere resolved with the introduction of the nuclear shell model, in which individual nucleonsmove in a mean field and is analogous to the shell model in atomic physics.A pressing question in nuclear physics is how the behaviour of the nuclear interactionchanges as one moves away from the stable nuclei. In particular, it is important to investigate,both experimentally and theoretically, how nuclear structure, or the properties of individual nu-clei, varies towards the driplines. In this thesis precision mass measurements were performed,and the results are compared to shell model calculations.1.1 The shell modelThe atomic nucleus is a complex many-body quantum system, and its understanding has beenan active field of research for more than a century. Atomic nuclei are composed of two nearlyidentical constituent nucleons: protons and neutrons. Their relative mass difference is only≈ 0.14% [2], both are Fermions, and have spins and parities of Jpi = 1/2+. The largest dif-ference between them is their charge—the proton being positively charged q = +1e, and theneutron being neutral, where e is the charge of the electron. The aim of nuclear physics isto understand the interactions between quarks and gluons to form neutrons and protons, and21.1. THE SHELL MODEL0200040006000800010000BAHkeVL0 50 100 150 200Mass Number, AFigure 1.2: Nuclear binding energy per nucleon for the stable nuclides.from there to describe the interaction between ensembles of nucleons, ultimately connectingthese to the fundamental interactions in the Standard Model of particle physics. The nuclearproblem occupies an interesting space, as, except in the lightest nuclei, there are too manyparticles to calculate exact results from first principles, but, even for the heaviest nuclei, thereare not enough particles for a purely statistical approach [3]. The nuclear shell model is usedto describe the region of nuclei that lie between these two extremes, as a mean central potentialis formed through the mutual interaction between the nucleons, but only the valence nucleons,the nucleons near the Fermi surface, play an active role in determining nuclear structure andproperties: structure such as shape (is the nucleus spherical, prolate or oblate), and propertiessuch as the spins, parities, and half-lives of the ground and excited states.One early approach to solving the nuclear problem was the independent particle model [4],in which nucleons move in a mean attractive potential well with no interactions with othernucleons. The Hamiltonian is formed by a spin-independent central potential plus a spin-orbitpotential and an orbit-orbit term,H(r) =−V0+T + 12mω2r2−VSO~` ·~s−VB~`2, (1.1)where V0 is the central depth of the potential (typically≈−51 MeV), T is the kinetic energy ofthe nucleon, VSO is the spin-orbit potential which can depend on r or derivatives of the central31.1. THE SHELL MODELpotential, and VB is the orbit-orbit potential. A harmonic oscillator (HO) basis is often usedin nuclear theory, as it greatly simplifies the mathematics involved; however, quite often largesets of basis states are needed to accurately describe nuclear wavefunctions. The spin-orbitpotential is similar to the spin-orbit coupling of electrons in the atomic potential, but in nucleithe coupling is much stronger and has an opposite sign to the atomic case. Further, while thespin-orbit potential in atomic systems arise from the magnetic field generated by the movementof the electron, the spin-orbit term in atomic nuclei is a property of the strong force and is notof a magnetic origin.The energy levels of a three-dimensional harmonic oscillator in spherical coordinates areE = h¯ω(N + 3/2) = h¯ω(2n+ `+ 3/2), where N = 2n+ ` is the major quantum number, n isthe radial quantum number, and ` is the angular momentum quantum number. For even-N,only even `-values are allowed, and for odd-N, only odd-` values are allowed. As a result eachmajor shell alternates the parity of the angular momentum wavefunction. The orbit-orbit termbreaks the degeneracy of the harmonic oscillator; however the states can still be grouped bytheir major oscillator number. The shift in energy is given by VB`(`+1). The spin-orbit termfurther breaks the degeneracy in the energy levels, splitting each state depending on the totalangular momentum. The energy splitting is given by the expectation value of ~` ·~s, which canbe found using the total angular momentumj2 = (~`+~s)2 = `2+ s2+2~` ·~s. (1.2)This leads to a spin-orbit energy shift ofE(SO) =−Vso2( j( j+1)− `(`+1)− s(s+1)) . (1.3)States with higher j are lowered in energy, while states with lower j are raised in energy.Figure 1.3: (Continued on following page) Single particle energies in the shell model.The left column shows the harmonic oscillator levels, the second column shows theeffect of the orbit-orbit VB`(`+ 1) term, the third column shows the effect of thespin-orbit Vso~` ·~s term, and the last column shows the energy levels for a Woods-Saxon potential suitable for 208Pb, calculated with the program wspot [5]. Onlybound states are shown. The states are labelled n` j, where n is the radial quantumnumber, `, the angular momentum quantum number, is labelled s, p,d, f ,g,h, i for` = 0,1,2,3,4,5,6, and j is the total angular momentum ~l +~s. The numbers inbrackets denotes the maximum occupation for a given orbit. The magic numbersare also labelled.41.1. THE SHELL MODELN = 0N = 1N = 2N = 3N = 4N = 5N = 60s0p0d0f0g0h0i1s1p1d1f1g2s2p2d3s0s12 @2D0p12 @2D0p32 @4D0d32 @4D0d52 @6D0f52 @6D0f72 @8D0g72 @8D0g92 @10D0h92 @10D0h112 @12D0i112 @12D0i132 @14D0j152 @16D1s12 @2D1p12 @2D1p32 @4D1d32 @4D1d52 @6D1f52 @6D1f72 @8D1g72 @8D1g92 @10D2s12 @2D2p12 @2D2p32 @4D2d32 @4D2d52 @6D3s12 @2DH.O. Orbit-Orbit Spin-Orbit Woods-Saxon2820285082126Figure 1.3: (Continued from previous page.)51.1. THE SHELL MODELrVHrLΡHrLHaL HbLFigure 1.4: (a) Schematic plot of the Woods-Saxon potential, and density, for 208Pb. (b)Top: Experimentally extracted density profile for 208Pb. Bottom: Central Woods-Saxon potential during the calculation. Figure (b) reproduced with permission from[6].Instead of a harmonic oscillator potential, there are several other central potentials that canbe used. A commonly used potential is the Woods-Saxon potential [7] with the formV (r) =−V01+ exp((r−R)/a) (1.4)where R is the mean radius of the nucleus, and a is the mean skin thickness, typically chosento be R = 1.25A1/3 fm and a = 0.524 fm. A Woods-Saxon form is quite natural, as it is a closeapproximation to the same form that the nuclear density takes. As seen in figure 1.4, this isconfirmed by the measurement of the charge density in 208Pb through the use of elastic electronscattering [6].The ordering of the states depends on the values chosen for Vso and VB, as well as on theform of the central potential, evidenced by the re-arrangement of states between the HO andWoods-Saxon calculations. An interesting side-effect of the spin-orbit force is that states fromdifferent major oscillators mix with each other. For example, in figure 1.3 the N = 4, 0g9/2 statebecomes part of the group of states made up from the N = 3, p and f states. Including thesestates in large-scale shell model calculations can be important in reproducing experiments [8,9, 10].As seen in figure 1.3, the spin-orbit term gives rise to the so-called nuclear “magic num-bers” [11, 12]. The nuclear magic numbers are conceptually similar to the atomic closed shellnumbers, in which elements having large first ionization potentials are non-reactive. These61.1. THE SHELL MODELææææààààààìììììì ììò òòò òòòò òòôô ôôôô ôô ôôôçç ç ççççç ççççáá á á áááá áááááíí í í í íííííííííóóóóóóóóóóóóóóóõõ õõ õõõõõõõõõõõõõææææææ ææææ ææææææææààààààààààààààààààààììììììììììììììì ì ìììììòòòòòòòòòòòòòòòòò òòòòò òôôôôôôôôôôôôôôôô ô ôôô ôôô ôççççççççççççççççç ççç ççç çáááááááááááááááááááá ááááííííííííííííííííííííí ííóóóóó óóóóóóóóóóóó óóóó óóõõõõõõõõõõõõõõõõõõ õõæææææææææææææææææ ææààààààààààààààà à ààììììììììììììììì ììòòòòòòò òòòòòòòòòôôôôôôôôôôôôôç ççççççççç çááááááááááí ííííííííóóóóóóó óõõõõõõææææ æàààìì31Ne32Na33Mg34Al35Si36P37S38Cl39Ar40K41Ca42Sc43Ti44V45Cr46Mn47Fe49Co50Ni54Cu 56Zn58Ga60Ge62As66Se69Br71Kr 73Rb75Sr78Y80Zr83Nb85MoN = ZN = 28010000200003000040000S2nHkeVL20 25 30 35 40 45Neutron Number, NFigure 1.5: Two neutron separation energies near N = 28. The region in red shows theWigner energy along N = Z, while the blue region shows the magic number atN = 28. Data from [14].elements are called the noble gases. Because of these similarities, this model of the nucleusis called the shell model. Experiments had found that certain nuclei with neutron N or pro-ton Z numbers of N,Z = 2,8,20,28,50,82 and N = 126 were significantly more tightly boundthan their neighbours. For example, nuclei with N = 50 or 82 exhibit higher natural chemicalabundances than could be explained by the existing models [13]. At Z = 50, the chemicalelement tin shows the most number of stable isotopes with a total of 11. Further, there are 6stable nuclei with N = 50 and 7 with N = 82. Oxygen-16, a “doubly” magic nucleus with 8protons and 8 neutrons, requires 15.6 MeV of energy to remove one neutron, while 17O, withone additional neutron, requires only 4.1 MeV of energy to remove one neutron.The enhanced binding near the magic numbers can be seen in systematic studies of thebinding energy BEBE(N,Z) = ZmHc2+Nmnc2−M(N,Z)c2 (1.5)where mH and mn are the masses of hydrogen and the neutron, respectively, M(N,Z) is theatomic mass of a nuclide, and c is the speed of light. The observed behaviour of the nu-clear binding energy changes at the magic numbers, hence differences—or derivatives—of the71.1. THE SHELL MODELDn HMeVL-2.502.55.07.510.012.5-64-2046deformationMagicNumbers820285082AtomicNumber,Z8 20 28 50 82 126Neutron Number, NFigure 1.6: ∆n surface. The neutron magic numbers are clearly seen as bright verticalbands. Areas of deformation can be seen near (N,Z) = (60,40) and (90,62). Datafrom [14].binding energy highlights these areas. Two commonly used differences are the two-neutronseparation energy S2nS2n(N,Z) = BE(N,Z)−BE(N−2,Z), (1.6)which is the energy required to remove two neutrons from a nucleus, and the empirical neutronshell gap ∆n [15]∆n = S2n(N,Z)−S2n(N+2,Z), (1.7)which is similar to a second derivative of the binding energies. In fact, regions of interestingunderlying nuclear structure, can be seen at the S2n or ∆n surfaces shown in figures 1.5 and 1.6,respectively. In the ∆n surface, some of the clearly seen features include:• the conventional neutron magic numbers, which appear as bright vertical bands due tosudden changes the the amount of binding when crossing a magic number,• the N = Z line, where increased binding occurs due to the Wigner energy [16],• the disappearance of the N = 20 and 28 magic numbers near the proton numbers Z = 1281.2. PRECISION POTENTIALS AND THE NEED FOR THREE-BODY FORCESand 16 (near the nuclides 32Mg and 40Mg as seen by the disapperance of the brightvertical bands,• regions of deformation near (N,Z) = (60,40) and (90,62), as evidenced by a negative(black) ∆n,• and the appearance of ‘new’ magic numbers at N = 16 and N = 32 near Z = 8 and Z = 20as evidenced by the appearance of bright bands,The independent particle model of the nucleus has been confirmed by many experiments.The results of electron induced knock-out reactions of protons from 206,208Pb confirms theexpected ordering, spacing, and occupancy of the orbitals [17, 18]. In the (d, p) transfer reac-tion on 132Sn [19] the single-particle nature of the states in 133Sn were confirmed. Both 208Pband 132Sn are doubly magic nuclei, thus, nuclei near doubly magic nuclei obey the indepen-dent particle model. However, as already seen in the ∆n surface, there are regions with largeneutron-to-proton ratios where the standard magic numbers seem to vanish. Thus, a majorfrontier of nuclear theory is to accurately describe the change in the nuclear interaction as onemoves away from stability.1.2 Precision potentials and the need for three-body forcesThe general quantum many-body problem for the nuclear Hamiltonian can only be exactlysolved in the lightest nuclear systems (A< 20), using what are called ab initio (Latin for “fromthe beginning”) calculations. The general nuclear Hamiltonian can be written asHˆ = T +V = T +∑i< jV 2Ni j + ∑i< j<kV 3Ni jk + . . . (1.8)where the potential is expanded in terms of the two-body V 2N, three-body V 3N, and higherorder terms. Many of the widely used nuclear potentials in ab initio calculations only includethe 2N interaction because it is expected that higher orders will have a small contribution inthe calculation [20]. The first nucleon-nucleon (NN) potential was the Yukawa model [21], inwhich the force between nucleons is mediated by pion exchange,V (r) ∝e−mpi rr(1.9)where mpi is the mass of the pion. Intuitively, this internucleon force can be thought of as theresidual interaction between colour-neutral nucleons, similar to Van der Waal forces between91.2. PRECISION POTENTIALS AND THE NEED FOR THREE-BODY FORCESArgonne V18Reid93repulsive core2ΠΡ, Ω, ΣΠ-1000100200300VCHMeVL0.0 0.5 1.0 1.5 2.0 2.5r HfmLFigure 1.7: Examples of the nuclear potential between two nucleons in the 1S0 channel.Note the resemblance to Van der Waals forces. The dashed lines show the regionsdominated by one and two pion exchange.charge-neutral atoms [20]. As seen in figure 1.7 three distinct regions of the NN interactioncan be identified: a strongly repulsive core at short ranges, an attractive well at mid-ranges,and a weak long-range attraction. Most models of this type include a central term, spin-spin,spin-orbit, and tensor interactions, with each of these terms included once without isospindependence and once with isospin dependence. (Isospin is discussed further in section 2.5).The models are constructed using the most general potential that obeys the symmetries of thenucleus: rotation, translation, isospin, etc. [22]. The exact form of the potentials depends bothon the method in which the potential operators were derived, and on the choice of the couplingstrength for those operators. While the derived potentials may differ in their functional form,they share in common that mesons are the force carriers. The most important meson in thenuclear potential is the pion pi (mpi± ≈ 138MeV/c2), with small contributions coming fromheavier mesons, such as the ρ (mρ ≈ 760MeV/c2), η (mη ≈ 549MeV/c2), etc. Examples ofoften used potentials are Argonne V18 [23], Reid93 [24] and Urbana14 [25].Up until the 1990’s, only the so-called NN potentials were considered, constructed to fitthe large body of nn and np elastic scattering data. However, systematic shifts in the bindingenergies for multi-nucleon systems could not be accounted for without the inclusion of NNNor 3N interactions. Ab initio calculations in light nuclei have demonstrated the need to include3N-forces in nuclear structure calculations [26]. Figure 1.8 shows the results of the binding101.2. PRECISION POTENTIALS AND THE NEED FOR THREE-BODY FORCESFigure 1.8: Binding energies calculated in Green’s Function Monte Carlo (GFMC) usingthe NN potential AV18 (blue) and the 3N potential Illinois-7 (red), as compared tothe experimental values (green). Figure reproduced with permission from [26].and excitation energies of several light nuclei, calculated both using the NN potential ArgonneV18 [23], and supplemented with the 3N interaction Illinois-7 [27]. Argonne V18 is the latestNN potential developed at Argonne National Laboratory, and its name is derived from the 18operator terms used in the model. The 3N Illinois potentials are divided into separate inter-actions, each with slightly different choices for the potential coefficients [27]. The Illinois-2and -7 interactions have been widely used in the nuclear community. When studying figure1.8, we notice that the NN AV18 results (blue), while shifted in energy from the experimentalvalues (green), largely follow the correct level ordering and spacing. However, this is not truein the ordering of the states in 10B, where the calculated ground state is a Jpi = 1+, while themeasured ground state is Jpi = 3+. The calculations including the 3N Illinois-7 (red+yellow),show agreement between theory and experiment. The systematic shift in the binding energiesis largely accounted for, and the ordering of the states in 10B is reproduced.While the addition of a 3N Hamiltonian to the precision NN potentials greatly increases theagreement between theory and experiment, there is no natural way to include their effects in111.2. PRECISION POTENTIALS AND THE NEED FOR THREE-BODY FORCESbeyond which we plot only the data locating on the coor-dinate axes and their nearest neighbors. As is clear fromFig. 2, the wave function is suppressed at short distanceand has a slight enhancement at medium distance, whichsuggests that the NN system has a repulsion (attraction) atshort (medium) distance.Figure 3 shows the central (effective central) NN poten-tial in the 1S0 (3S1) channel at t! t0 " 6. As for r2 inEq. (2), we take the discrete form of the Laplacian with thenearest-neighbor points. E is obtained from the Green’sfunction G# ~r;E$ which is a solution of the Helmholtzequation on the lattice [9]. By fitting the wave function!#~r$ at the points ~r " #10–16; 0; 0$ and #10–16; 1; 0$ byG#~r;E$, we obtain E#1S0$"!0:49#15$MeV and E#3S1$ "!0:67#18$ MeV. Namely, there is a slight attraction be-tween the two nucleons in a finite box. To make an inde-pendent check of the ground state saturation, we plot the tdependence of VC#r$ in the 1S0 channel at several distancesr " 0, 0.14, 0.19, 0.69, 1.37, and 2.19 fm in Fig. 4. Thesaturation indeed holds for t! t0 % 6 within errors.As anticipated from Fig. 2, VC#r$ and VeffC #r$ haverepulsive core at r & 0:5 fm with the height of about afew hundred MeV. Also, they have an attraction of about!#20–30$ MeV at the distance 0:5 & r & 1:0 fm. Thesolid lines in Fig. 3 show the one-pion exchange contribu-tion to the central potential calculated from V"C #r$ "g2"N4"# ~#1 & ~#2$# ~$1 & ~$2$3!m"2mN"2 e!m"rr; (5)where we have used m" ’ 0:53 GeV and mN ’ 1:34 GeVto be consistent with our data, while the physical value ofthe "N coupling constant is used, g2"N=#4"$ ’ 14:0. Evenin the quenched approximation, the one-pion exchange ispossible as the connected quark exchange between the twonucleons. In addition, there is in principle a quenchedartifact to the NN potential from the flavor-singlet hairpindiagram (the ghost exchange) between the nucleons [13].Its contribution to the central potential reads [14]: V%C #r$ "g2%N4"~$1& ~$23 # m"2mN$2#1r !m202m"$e!m"r. Here g%N and m0 are the%N coupling constant and a mass parameter of the ghost,respectively. The ghost potential has an exponential tailwhich dominates over the Yukawa potential at large dis-tances. Its significance can be estimated by comparing thesign and the magnitude of em"rVC#r$ and em"rVeffC #r$ atlarge distances, because V%C #r$ has an opposite sign be-tween 1S0 and 3S1. Our present data show no evidence ofthe ghost at large distances within errors, which mayindicate g%N ' g"N .Several comments are in order here. (1) The asymptoticwave function at low energy (E! 0) is approximated as!asy#r$" sin(kr)&0#k$*kr ! r)a0r , where &0#k$ (a0) is the s-wave0. 0.5 1.0 1.5 2.0NN wave function φ(r)r [fm]1S03S1-2-1 0 1 2 -2-10φ(x,y,z=0;1S0)x[fm]    y[fm]FIG. 2 (color online). The lattice QCD result of the radialdependence of the NN wave function at t! t0 " 6 in the 1S0and 3S1 channels. Inset shows the two-dimensional view in thex! y plane.  01002003004005006000.0 0.5 1.0 1.5 2.0V C(r) [MeV]r [fm]-50  0 501000.0 0.5 1.0 1.5 2.01S03S1OPEPFIG. 3 (color online). The lattice QCD result of the central(effective central) part of the NN potential VC#r$ [VeffC #r$] in the1S0 (3S1) channel for m"=m' " 0:595. The inset shows itsenlargement. The solid lines correspond to the one-pion ex-change potential (OPEP) given in Eq. (5).3004005006002 3 4 5 6 7 8V C(r) [MeV] r=0.00fm r=0.14fm r=0.19fm-40-20  0 202 3 4 5 6 7 8V C(r) [MeV]t-t0 [lattice unit] r=0.69fm r=1.37fm  r=2.19fm FIG. 4 (color online). t! t0 dependence of VC#r$ in the 1S0channel for several different values of the distance r.PRL 99, 022001 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending13 JULY 2007022001-3Figure 1.9: NN-potential in the singlet and triplet channels, calculated in lattice QCD.The long-range behaviour is similar to the one-pion exchange potential (OPEP)and the hard core is reproduced. Figure reproduced with permission from [30].these models. Specifically, the inter-nucleon force must derive from Quantum Chromodynam-ics (QCD), but the precision NN potentials have no strong theoretical foundation in QCD. Cur-rently, modern nuclear interactions are based on Chiral Effective Field Theory (χEFT). χEFTattempts to model the interactions between pions and nucleon fields (nucleons, ∆-resonances,etc.) by expanding the scattering amplitudes in small powers of the ratio of the pion mass tothe breakdown scale of the theory [28]. In this way, 2N, 3N, and higher order interactionsarise naturally in the theory, and there is a clear connection to the underlying QCD. Moderninteractions based on χEFT are discussed further in section 2.1.While ab initio calculations are powerful in predicting the structure of light nuclei, theycannot be directly applied to heavier nuclei due to large increases in the required model space.For example, in the calculation of the ground state of 12C, as shown in figure 1.8, the solu-tion to the Hamiltonian requires solving a system of 270 336 complex second-order coupledequations in 33 coordinates [29]. Such calculations become intractable for heavier systems.Instead, effective interactions are constructed to reduce the model space. How such effectiveinteractions are constructed, and the role of 3N forces is discussed in chapter 2.A question that often arises is how well do these potentials connect with the underlyingQCD? While these potentials do not intrinsically start from QCD, they do, however, reproduce121.3. MASS MEASUREMENT TECHNIQUESthe inter-nucleon potential [22]. One direction is to determine the NN potential directly fromQCD by calculating the NN potential in lattice QCD. Pioneering work by Ishii et al. [30, 31,32], has shown that the picture of an attractive potential with a hard-core at short distancesis correct. Figure 1.9 shows the lattice QCD result for the central part of the interaction inthe 1S0 and 3S1 channels. The calculations are compared at long range with the one-pionexchange potential, showing that the expected long-range behaviour is reproduced. Furtherwork [31] shows the hardness, or repulsiveness, of the core depends on the quark mass usedin the calculations – as more realistic quark masses are used, the harder the core becomes.As a consequence of the harder core, the medium range attraction is slightly enhanced. Thedifference between the precision potentials at short range may be related to differences in theunderlying quark mass, even though the quark mass is not a direct input.1.3 Mass measurement techniquesAs the atomic mass is an important component in determining nuclear structure, several differ-ent measurement techniques have been developed, broadly falling into two categories [33, 15]:indirect and direct methods. A common indirect method is a mass measurement through nu-clear reactions of the form A(b,c)D, where the beam particle b reacts with the target particleA, producing a beam-like ejectile c and a target-like recoil D. Determining the mass of any ofthese particles requires knowing at least three of the masses, the kinematics of the incomingchannel and the energy of one of the particles in the outgoing channel. Traditionally, reactionscan give accurate and precise mass values. Some of the highest precision mass values are from(n,γ), (p,γ), (n, p), and (n,α) reactions. For example, the separation energy of the deuteron isknown to 0.4 eV from the 1H(n,γ)2H reaction [34]. Neutron-capture reactions require stable,or very long-lived, targets, limiting the possible cases to nuclides close to stability. Slightlymore exotic nuclei can be investigated with such reactions by transferring several nucleonsfrom the beam to the target. The reaction mechanism is, however, much more complex, andwhen combined with low statistics from the small reaction cross-sections, can potentially leadto incorrect results.Another indirect method determines masses by measuring Q-values. The Q-value is thetotal amount of energy available in a decay or reaction, and is related to the difference betweenthe masses of mother and daughter asQ =∑KFinal−∑KInitial = (∑MInitial−∑MFinal)c2 (1.10)where K and M are the kinetic energy and atomic masses of the particles before and after the131.3. MASS MEASUREMENT TECHNIQUESdecay, respectively. Two common modes for nuclei to decay are α- and β -decay. Q-valuesfrom α-decays provide accurate and precise values owing to the simplicity of the decay—there are only two particles in the out-going channel—and the total energy of the decay canbe measured by implanting the parent in a suitable detector. Q-values from β -decays tendto be prone to under-estimating systematic errors because the energy of the decay is sharedbetween three particles: the daughter nucleus, the β -particle and the neutrino. The Q-valueis determined by measuring the β -energy endpoint. However, the response function of thedetector has to be well understood, otherwise the extracted end-point energy will be dominatedby systematic errors [15]. Further complicating matters, is the fact that in nuclei far from thestable nuclides, the decay Q-values are large, which opens up a significant number of decaychannels. If these are not properly accounted for, the extracted mass value can be systematicallyshifted.While both reactions and β -endpoint measurements can provide accurate mass values, theycan give incorrect results arising from complex systematics in the measurement device and co-produced contamination. An example of an incorrectly determined mass was 46V, where adirect Penning trap mass measurement differed by more than 3σ from a (p,γ) reaction [35].The mass value of 46V was subsequently confirmed by another direct Penning trap systemmeasurement [36, 37]. In β -endpoint measurements, it is generally seen that mass values fromβ -decay measurements underestimate the binding. For example, the Q-value for the decayof 85Nb was found to be 900 keV larger in a Penning trap measurement [38] than in the β -endpoint measurement [39]. Many of the known mass values used in r-process calculations(section 1.4.2) critically depend on the results of β -endpoint measurements; thus, independentverification of these masses is required.To overcome the limitations of these indirect methods, several direct mass measurementtechniques have been developed. The data collected in direct measurements are often sim-pler to interpret than the data from indirect methods, allowing for precise and accurate massmeasurements of nuclides not only with short half-lives (t1/2 < 100 ms), but also those pro-duced at very low rates. The main methods [33, 15] rely on measuring either the time-of-flight(TOF) [40] or the cyclotron frequency of an ion [41, 42] (section 3.3.4). Traditionally, time-of-flight techniques have used magnetic spectroscopy systems, such as SPEG at Ganil [43],where the time-of-flight is measured between two microchannel plate detectors. Time-of-flighttechniques have generally been relegated to facilities where fast (Ekin ≈ several MeV/nucleonto GeV/nucleon) production beams are used. One such device is the Experimental StorageRing [44] at the GSI facility in Darmstat, Germany. By measuring the revolution frequencyof an ion stored in the ring [40], it is possible to determine the mass when the flight path, and141.4. THE IMPORTANCE OF PENNING-TRAP MASS MEASUREMENTShence the total length is known. Resolving powers of up to m/δm ≈ 106 can be reached withstorages times of ≈ 1 s, and resolving powers of ≈ 105 can be reached after 50 µs [45]. Arecent development is the Multi-Reflection Time-of-Flight device, which captures low-energybeams (Ekin ≈ 2 keV) between a pair of electrostatic mirrors. In this way, the flight path of theion can be increased to several kilometres, allowing for resolving powers of ≈ 105 for flighttimes of several tens of milliseconds [46, 47, 48].For the highest precision, the tool of choice is a Penning trap. High-precision measurementsare achievable due to the unique storage environment: Single charged particles are held for longtimes in high vacuum, with well defined trapping potentials. Relative atomic mass precisionsof below 10−8 have been reached for unstable nuclei with half-lives below 100 ms [41], and10−11 for stable nuclei [42]. Examples of the performance of Penning traps are:• some of the most stringent tests of CPT symmetry, comparing the antiproton and protoncharge-to-mass ratios to 90 ppt [49]• the most precise atomic mass values for stable nuclei, 11 ppt for 16O, and 94 ppt for themass of the electron [50]• and precise Q-value measurements of superallowed β -emitters [51] to determine thequark mixing matrix element Vud of the Cabibbo-Kobayashi-Maskawa (CKM) matrix.TRIUMF’s Ion Trap for Atomic and Nuclear science (TITAN) [52] is one such Penning trapsystem. Due to the ion production source and the ion injection method into the trap, TITAN iscapable of performing precise and accurate measurements on short-lived isotopes that cannotbe measured by other Penning traps. The work described here was carried out using the TITANsystem, and is described in detail in section The importance of Penning-trap mass measurementsAtomic mass measurements play an important role in many aspects of nuclear physics, relevantfor questions from the smallest scale — is the quark mixing matrix unitary within the StandardModel of particle physics? — to the largest scales — how are elements heavier than ironmade in hot astrophysical environments? The atomic mass is important in studying new andemerging phenomena in nuclei, e.g. nuclear halos [53], in studying the evolution of “magicnumbers” [54, 55], and in studying the onset of deformation [56, 57]. Precise and accuratemass values provide stringent tests of nuclear theory [58, 59].The binding energy (Eq. 1.5), which is derived from the atomic mass, is important becauseit reflects the sum of all the interactions at play in the nucleus, making it sensitive to interactions151.4. THE IMPORTANCE OF PENNING-TRAP MASS MEASUREMENTSthat may only be observed far from stability. The following describes examples of regionswhere mass measurements are important for nuclear physics.1.4.1 Cabibbo-Kobayashi-Maskawa unitarityThe Cabibbo-Kobayashi-Maskawa (CKM) matrix is a unitary transformation matrix relatingthe quark mass eigenstates to the flavour eigenstates:dwswbw=Vud Vus VubVcd Vcs VcbVtd Vts Vtbdsssbs (1.11)where u, c, t, d, s, b are the up, charm, top, down, strange, and bottom quarks, and the subscripts and w denote the strong and weak eigenstates. Because the CKM matrix is defined to beunitary in the Standard Model, the sum of the squares of any row or column should be 1, withany deviation from this indicating that extra quark generations or other physics beyond theStandard Model may be required. Currently, the most stringent unitary test is done using thefirst row of the matrix|Vud|2+ |Vus|2+ |Vub|2 = 1. (1.12)The element |Vud|2 accounts for nearly 95% of the first row’s size. Vud is special among theCKM elements as it can be accessed by nuclear physics through superallowed 0+→ 0+ Fermiβ -decays, due to the simplistic nuclear structure. Occurring between isobaric analogue states(states that are related by isospin-lowering and isosping-raising operators), superallowed β -decays used for tests of the CKM matrix are almost independent of nuclear structure. The nu-clear matrix element of the decay differs from a Clebsch-Gordan coefficient at the few percentlevel, largely due to charge-dependent effects in the nuclear interaction. The other elements ofthe CKM matrix lie in the domain of particle physics, and are typically measured at colliderfacilities [60]. Electron capture and β -decay in nuclear systems transforms a proton (u,u,d)into a neutron (u,d,d), or vice-versa for β+-decay. From this one can access |Vud|. Assumingthe conserved vector current hypothesis [61], the β -decay f t-value can be written asf t =KGV |MF |2(1.13)where K is a constant, GV is the vector coupling constant, and MF is the matrix element con-necting the initial and final states. The f t-value is called the “comparative half-life” of thedecay, accounting for the available phase space f and the half-life t of the decay. The vector161.4. THE IMPORTANCE OF PENNING-TRAP MASS MEASUREMENTScoupling constant can be written as GV = VudGµ , where Gµ is the coupling constant for thepurely leptonic decay of the muon. The experimental f t-value depends on the three experi-mental values:1. the half-life of the superallowed decay,2. the branching ratio to the 0+ state,3. and the Q-value of the decay to the 0+ state.Because the f t-value depends on the Q-value to the fifth power Q5 [51], precise and accuratemass values are required.1.4.2 Nuclear astrophysicsExtremely neutron rich nuclei can be produced in hot astrophysical environments, such as core-collapse supernovae [62] or neutron star mergers [63], which are considered as sites for theso-called rapid-neutron capture process (r-process). Because the involved nuclei are extremelyneutron rich and difficult to produce in the laboratory, there is a distinct lack of experimentalknowledge on all required quantities: half-lives, β -delayed neutron decay probabilities, sep-aration energies, etc. This information is needed to develop a complete understanding of theprocess and the resulting chemical element abundances. Besides the nuclear physics properties,astrophysical sites need to be investigated and understood. Without experimental values for therequired nuclear properties, nuclear astrophysicists rely on nuclear models. Figure 1.10 showsthe range of calculated elemental abundances for four different mass models. These models areable to accurately predict masses where data exists, but they greatly diverge from each otherwhere the mass values are not known. This results in a wide range in the predicted chemicalabundances. Through precise and accurate mass measurements, more reliable and realistic de-scriptions of nucleosynthesis are possible. Therefore, sensitivity studies have been performedto determine the nuclei where masses have the largest influence on the final abundances [64].1.4.3 Nuclear halosIn 1985, a remarkable observation was made in a radioactive beam experiment at the LawrenceBerkeley National Laboratory: The matter radius of 11Li was found to be much larger than thatof adjacent nuclei [66]. Subsequent theoretical and experimental studies led to a coining of thename “halo nucleus”. Nuclear halos are characterized by a long tail in the matter distribution,related to the weak binding of the halo nucleons. Departing from the normal r ∝ A1/3 scaling171.4. THE IMPORTANCE OF PENNING-TRAP MASS MEASUREMENTSWe find that, when the wind termination shock occurs at high temperature (∼ 1 GK), ther-process takes place in (n,γ)-(γ,n) equilibrium, as in the classical r-process. In this case theneutron separation energies determine the abundances along an isotopic chain. On the otherhand, when the reverse shock is at low temperatures (< 0.5 GK) photo-dissociation becomesnegligible and there is a competition between neutron capture and beta decay. Consequentlythe relevant nuclear physics input will depend on the dynamical evolution of the outflow. Thistranslates into different r-process paths and differences in the final abundances, as shown inFig. 1.3.2. Dependence on mass modelsHere we describe only the case where the evolution takes place at high temperatures and thereforein (n,γ)-(γ,n) equilibrium. Other cases and more details can be found in Ref. [5]. In Fig. 2 wepresent the final abundances obtained for the same trajectory but different mass models. We findthat the region before the third peak is remarkable different but also the abundances betweenpeaks vary considerably.Figure 2. Final abundancesobtained from different massmodels. All cases are cal-culated with the same tra-jectory which reaches (n,γ)-(γ,n) equilibrium, because thereverse shock is at 1 GK.In order to assess the impact of nuclear masses, we compare the abundances obtained fromFRDM to ETFSI-Q at freeze-out (Yn/Yseed = 1) and when all decays have occurred. In Fig. 3 weobserve remarkable odd-even effects following the behavior of the neutron separation energies.However, the final abundances in Fig. 4 are smoother similar to solar abundances.In the long-time evolution there is a competition between beta decay and neutron captureand we have found that neutron captures still play an important role when matter moves backto stability, even when neutron densities and neutron-to-seed ratios are low (Nn ≈ 1017cm−3and Yn/Yseed ≈ 10−5). Neutron captures can fill holes, move peaks to higher mass numberand reduce odd-even effects in the abundances. Moreover, the masses also enter in the neutroncapture cross sections, and this can explain the differences in Fig. 4 between the two massmodels.4. ConclusionsThe late-time evolution of the ejecta, also after freeze-out of the r-process, is very importantto determine details in the final abundances. Therefore, we performed nucleosynthesis studiesin neutrino-driven winds by means of long-time hydrodynamical simulations of core-collapsesupernova explosions. The conditions found in the simulations (low wind entropies and/or highelectron fraction) do not allow the formation of heavy elements. However, an artificial increaseof the entropy by a factor around two is enough to reach A=195 and allow us to explore thesensitivity of the wind termination shock and the nuclear physics input.3Figure 1.10: Model dependence of r-process abundances for three different mass models.Figure reproduced with permission from [65].in nuclei near stability, 11Li is comparable in size to 208Pb, even though they are ≈ 200 massunits apart.In 11Li two neutrons are loosely bound to a 9Li core, displaying a two-neutron separationenergy of only S2n = 369 keV [67]. From the extended matter distribution several key featurescan be extracted: the halo nucleons are in low angular momentum states, otherwise the cen-trifugal barrier would suppress the wave function. Moreover, the separation energy of the halonucleons needs to be small, otherwise the potential well would suppress the wave function atlong distances [68]. The extended matter distribution also manifests itself in a much largerreaction interaction cross-section than would normally be expected [68]. There are severaltypes of halo nuclei, classified by the number of nucleons that comprise the halo [68]: the oneneutron halos 11Be and 19C, the two-neutron halos 11Li and 6He, amongst others, and the four-neutron halos 8He and 14Be. Several proton halo nuclei are considered, such as the one-protonhalo 26P and the two-proton halo 17Ne, however there are fewer proton halo nuclei than neutronhalo nuclei [68]. Since halo nuclei are so weakly bound, the separation energy, and hence, themass, is an important component in determining their structure, and a key ingredient for testingtheoretical predictions.181.4. THE IMPORTANCE OF PENNING-TRAP MASS MEASUREMENTSæ ææææææææææææææææææææææææææ10 12 14 16 18 20 22Atomic Number Z0200040006000800010000DnHkeVLHaL16 18 20 22 24 26 28 30 32Atomic Number ZHbLFigure 1.11: Disappearance of (a) N = 20 and (b) N = 28 magic numbers, seen throughthe semi-empirical shell gap ∆n. Data from [69].1.4.4 Shell structure evolutionMost nuclear models have only been constrained with data near the stable nuclides, and onlyrecently have data from nuclei with extreme proton to neutron ratios become available. Isospinis an approximate symmetry of nucleons, and the nuclear interaction can be parametrized interms of isospin (isospin is discussed further in section 2.5). The important point for the presentdiscussion, is that the nuclear interaction is different in the T = 0 (proton-neutron) channel thanin the T = 1 channel (proton-proton or neutron-neutron) [70]. Near the stable nuclides the T =0 channel dominates because of the nearly equal number of protons and neutrons, but movingfar from these stable regions the T = 1 channel becomes increasingly more important, a changethat may affect the location of the standard magic numbers [70]. As an example, figure 1.11shows the empirical shell gap ∆n 1.7 along the N = 20 and 28 isotones. In progressivelyneutron-rich nuclei the shell gap drops below 2 MeV, an indication that the neutron number isno longer magic, as typical shell gaps are found to be around 4 MeV.In these regions of reduced shell gap strength, the ground state may be an “intruder”state [71]. An intuitive understanding of intruder states can be found through the following191.4. THE IMPORTANCE OF PENNING-TRAP MASS MEASUREMENTSexample. Consider a Hamiltonian with n degenerate states, all coupled with the same strength:Hˆ =−ε ∆ ∆∆ ε ∆ . . .∆ ∆ ε... . . .where ε is the energy of the degenerate states, and ∆ is the interaction strength between them.One state, the correlated state, shifts down by−(n−1)∆, while the other states shift up by ∆. Ifthe coupling between the states is large, this correlated state can drop below any lower energystates, thus becoming an “intruder” state. In atomic nuclei, an intruder state occurs whennucleons populate a state that would normally be higher in energy and, through a large gain incorrelation energy due to particle-hole excitations, causes the state to drop below the “normal”,or expected, state [71]. Regions of inversion have been experimentally found to lie wherethe magic numbers begin to vanish. There may still be a large shell gap in the single-particleenergies, which is an indication of a shell closure, but the large gain in correlation energycauses a re-ordering of the states [72]. A prime example is 32Mg, which is the start of the“island-of-inversion” (figure 1.11 (a)) [73, 74]. The expected ground state would be Jpi = 0+,made up of protons in the 0d5/2 and neutrons in the 0d3/2 states. Instead, the ground state isa deformed 0+ state, formed by neutrons predominantly in the 1p3/2 and 0 f7/2 orbitals [75].This is also confirmed by the observation that the ground state of 33Mg is largely a neutron inthe 1p3/2 on top of a 32Mg core [76].A similar region of inversion occurs along the N = 28 magic number. Large deformationis seen in the ground-state of 42Si [77, 78], and there are indications that this deformationmight continue into 40Mg [79, 80]. Early theoretical investigations show that the ground-statestructure of the N = 28 isotones rapidly changes from a spherical configuration in 48Ca througha vibrational configuration in 46Ar, to oblate (flattened spheroid shape) in 42Si, and prolate(elongated spheriod shape) in 40Mg [54]. More data is needed to clarify the structure evolutionin this region, and mass measurements will play a vital role in these investigations.Magic numbers cannot only disappear, they can also appear unexpectedly [81]. Oneexample that has received much attention is the appearance of a new magic number atN = 16 [82, 83, 84] in the oxygen isotopic chain. By now 24O is fully considered to be adoubly magic nucleus: systematic trends in the mass surface point to an increase in binding at24O, the ground state of 24O is s-wave, as predicted by the shell model, and a large excitationenergy for the first 2+ state. Other proposed new magic numbers are at N = 32,34 near Z = 20201.5. PENNING-TRAP MASS MEASUREMENTS TO TEST 3N FORCESæææææææ48CaH14C,11CL51Ca48CaH18O,15OL51CaTOFGSI51CaHaL-37-36-35-34MassExcessHMeVL0 2 4 6 8MeasurementææΒ-end pointTOF52CaHbL-36-35-34-33-320 1 2 3MeasurementFigure 1.12: Historical mass excesses for (a) 51Ca, and (b) 52Ca. The red band is thevalue adopted in AME03 [85]. Individual measurements are discussed further insections 4.1.1 and 4.1.2.[81].1.5 Penning-trap mass measurements to test 3N forcesAs we have seen, the atomic mass is an important component in many aspects of nuclearphysics, and provides a significant tool in testing nuclear models. Three-nucleon forces wereshown to be crucial in reproducing the global binding energies of light nuclei in ab initio cal-culations, however, their role in medium mass nuclei is unknown. In this thesis we will test therole of 3N forces in medium mass nuclei through precision mass measurements. Specifically,we will test established phenomenological interactions that only include two-body effects, andnew interactions, derived from chiral effective field theory (χEFT), a low-energy perturbationexpansion of QCD, that provides a framework for including 3N effects. How such models areconstructed is discussed in chapter 2.First, the masses of the neutron-rich nuclei 51,52Ca were measured to test the possibility ofnew shell closures near N = 32 and 34. The measurements are compared to predictions fromwell established interactions based on 2N forces, and modern interactions that include the effectof 3N forces. Further, the measurements that exist in this region are in strong disagreement, ascan be seen in figure 1.12. A precision mass measurement was needed to clarify the mass values211.5. PENNING-TRAP MASS MEASUREMENTS TO TEST 3N FORCESof these nuclides. Second, the masses of 20,21Mg were measured to test the role of 3N forcesin the isobaric multiplet mass equation (section 2.5). These two measurements probe differentaspects of the 3N force. As the calcium isotopes are at a proton magic number (Z = 20),the protons do not strongly interact with the active neutrons, providing insight into the 3Nneutron-neutron interaction. On the other hand, the proton-rich Mg measurements probe the3N interaction when both protons and neutrons are active in the calculation.22Chapter 2Nuclear theoryNN potentials currently lead to unsatisfactory results in comparison with experiment, but thiscan be overcome through the use of interactions based on chiral effective field theory (χEFT)which gives a systematic approach to including the effect of 3N forces. This chapter willintroduce how χEFT based interactions are used in modern nuclear physics calculations, howthese calculations compare to existing interactions based on phenomenology, and how massmeasurements can be used to tests these theories.2.1 Chiral effective field theoryChiral symmetry is only a true symmetry in the limit of massless quarks [61]. Since the massesof the u and d quarks are light compared to the mass of a nucleon, the chiral symmetry canbe treated as an approximate symmetry. As in any EFT, the degrees of freedom must be de-termined, and in the nucleus, the relevant degrees of freedom are the protons, neutrons andexchange pions. Generally, the chiral effective Lagrangian used for nuclear theory only cond-siders the u and d quarks, and from the spontaneous symmetry breaking of the chiral symmetry,three pseudo-Goldstone bosons [61] act as the force carriers. In modern χEFT interactions onlythe pion is considered, which is a natural choice due to the large mass gap of ≈ 600 MeV/c2to the ρ-meson. The breakdown energy Λ of the EFT is chosen be between the pion massand the nucleon mass, and, in practice, Λ is taken between 500-700 MeV. The interaction canthen be expanded in powers of Q/Λ, where Q is the “soft scale” of the EFT and is typicallyclose to the mass of the pion [86]. This chiral expansion of QCD in the nucleonic sector solvesmany of the problems with the precision NN interactions: χEFT allows for an expansion of thenuclear interaction order by order, allowing for theoretical uncertainties to be assigned, χEFTnaturally explains the observed hierarchy of the NN, 3N, etc. forces in a consistent framework.232.1. CHIRAL EFFECTIVE FIELD THEORY98 S.K. Bogner et al. / Progress in Particle and Nuclear Physics 65 (2010) 94–147Fig. 4. (a) Chiral EFT for nuclear forces. (b) Improvement in neutron–proton phase shifts shown by shaded bands from cutoff variation at NLO (dashed),N2LO (light), and N3LO (dark) compared to extractions from experiment (points) [31]. The dashed line is from the N3LO potential of Ref. [20].– 31– 32– 33– 34NLO NNLO Expt– 30– 35E [MeV]abFig. 5. (a) Differential cross section (inmb/sr) and vector analyzing power for elastic neutron–deuteron scattering at 10MeV (top) and 65MeV (bottom) atNLO (light) and N2LO (dark) from Ref. [36]. (b) Ground-state energy of 6Li at NLO and N2LOwith bands corresponding to the⇤ variation over 500–600MeVcompared to experiment (solid line, see Ref. [36] for details).is still considerable off-diagonal strength above k = 2 fm1, which remains problematic for nuclear structure calculations(and the coupled 3S1–3D1 channel is generally worse).3 One might think the solution is to simply fit with a smaller ⇤, butthen the fit worsens significantly as the truncation error grows with Q/⇤.3 Note that the cutoff associated with the potential in Fig. 6(a) is ⇤ = 500 MeV, which might lead one to expect no strength above k ⇡ 2.5 fm1.However, the regulator does not sharply cut off relative momenta.Figure 2.1: Order by order χEFT diagrams for NN, 3N and 4N forces. Orders are: lead-ing order (LO), next-to-leading order (NLO), next-to-next-to-leading order (N2LO),etc. Figure reproduced with permission from [87].Lastly, χEFT has a clear connection to QCD. Figure 2.1 shows the leading terms in the χEFTinteraction.An essential ingredient to any effective field theory are contact interactions that capture thephysics of the neglected degrees of freedom. These contact interactions are captured in contactterms that can be either calculated from existing theories or fit to experimental data. For thenuclear interactions we are concerned with, the two-body terms are generally fit to the pi-N andN-N scattering data, while the 3N terms are fit to reproduce observables in light many-bodysystems, such as 3H and 4He. Herein lies the power of the χEFT formulation: the couplingconstants are fit once to experiment, and the resulting interactions should then be applicable tothe whole nuclear chart.A problem with using the bare chiral potential is the strong coupling between low- andhigh-momentum states. T e coupling o high and low momentum components in these bare242.1. CHIRAL EFFECTIVE FIELD THEORY102 S.K. Bogner et al. / Progress in Particle and Nuclear Physics 65 (2010) 94–147Fig. 9. Schematic illustration of two types of RG evolution for NN potentials in momentum space: (a) Vlow k running in⇤ and (b) SRG running in . At each⇤i or i , the matrix elements outside of the corresponding lines are zero, so that high- and low-momentum states are decoupled.Fig. 10. Two types of RG evolution applied to one of the chiral N3LO NN potentials (550/600 MeV) of Ref. [44] in the 3S1 channel: (a) Vlow k running in⇤and (b) SRG running in  (see Fig. 27 for plots in k2, which show the diagonal width of order 2).‘‘At each scale, we have different degrees of freedom and different dynamics. Physics at a larger scale (largely)decouples from the physics at a smaller scale. . . . Thus, a theory at a larger scale remembers only finitely manyparameters from the theories at smaller scales, and throws the rest of the details away. More precisely, when wepass from a smaller scale to a larger scale, we average over irrelevant degrees of freedom. . . . The general aim of the RGmethod is to explain how this decoupling takes place and why exactly information is transmitted from scale to scalethrough finitely many parameters.’’The common features of RG for critical phenomena and high-energy scattering are discussed by StevenWeinberg in an essayin Ref. [64]. He summarizes:‘‘Themethod in itsmost general form can I think be understood as away to arrange in various theories that the degreesof freedom that you’re talking about are the relevant degrees of freedom for the problem at hand.’’This is the heart ofwhat is donewith low-momentum interaction approaches: arrange for the degrees of freedom for nuclearstructure to be the relevant ones. This does not mean that other degrees of freedom cannot be used, but to again quoteWeinberg [64]: ‘‘You can use any degrees of freedom you want, but if you use the wrong ones, you’ll be sorry.’’There are two major classes of RG transformations used to construct low-momentum interactions, which are illustratedschematically in Fig. 9. In the Vlow k approach, decoupling is achieved by lowering amomentum cutoff⇤ abovewhichmatrixelements go to zero. In the SRG approach, decoupling is achieved by lowering a cutoff  (in energy differences 2) using flowequations, whichmeans evolving toward the diagonal inmomentum space. The technology for carrying out these is outlinedin Section 3, but the effects can be readily seen in the series of contour plots in Fig. 10(a) and (b).With either approach, lowering the cutoff leaves low-energy observables unchanged by construction, but shiftscontributions between the interaction strengths and the sums over intermediate states in loop integrals. The evolutionof phenomenological or chiral EFT interactions to lower resolution is beneficial, because these shifts can weaken or largelyeliminate sources of non-perturbative behavior, and because lower cutoffs require smaller bases inmany-body calculations,leading to improved convergence for nuclei. The RG cutoff variation estimates theoretical uncertainties due to higher-order contributions, to neglected many-body interactions or to an incomplete many-body treatment. When initialized withFigure 2.2: Renormalization group (RG) evolved χEFT potentials at N3LO with Λ =500 and 600 MeV in the 1S0 channel. (a) Vlowk (b) SRG. Figure reproduced withpermission from [87].inter ctions equires extended model spaces to achi ve conv rg d results for nuclear physicscalculations, presenting a large computational challenge. The hard core also causes uncorre-lated two-body wave functions to diverge because the wave functions has a non-zero value atdistances less than the hard core radius. The high-momentum components can be removedby evolving the interaction to low momentum through renormalization group techniques, onesuch technique being the Vlowk approach. An interesting side-effect of these procedures is thatat low momentum, all potentials have the same form – there is a universal potential at l wmomentum – as can be seen in figure 2.2. This happens because the renormalization procedureintegrates out th high-momentum components. A side effect of this renormal zation is thatthe high-momentum components are “shuffled” to higher-body forces (e.g. high-momentumNN terms ar moved into the ffective 3N and higher terms). Th short range potential differsbetween the various interactions, however the long-range, low-momentum parts are the same,leading to comparable low momentum interactions.Several other techniques to soften the potential have been introduced in recent years. Oneclass, called the Similarity Renormalization Group (SRG) [88], uses a continuous sequence ofunitary transformations to drive th Hamiltonian to a band diagonal form (figure 2.2 (b)). Thisresults in an evolved form that differs from the global Vlowk form, although the low momentumparts of the two are quite similar. SRG evolved potentials have the advantage that high-energyphase shifts are preserved, unlike in Vlowk potentials. Extensions to the SRG, called the In-252.2. EFFECTIVE INTERACTIONSrVValenceCoreExcludedFigure 2.3: Schematic valence space for an effective interaction. Valence nucleons canonly interact with themselves. The core and higher lying states are inert and ex-cluded from the calculation.Medium SRG[89, 90] and multireference IM-SRG [91], have also been developed. Instead ofevolving the interaction in free space as done in SRG, IM-SRG evolves the potential in themedium of the many-body system being studied. This allows for the evolution of arbitrary or-der operators using only the machinery required for the two-body case. All of these techniquessuccessfully reproduce experiment in a wide range of nuclear systems [88, 89, 90, 91].2.2 Effective interactionsFor all but the lightest systems, full ab initio calculations are not possible due to the exponentialincrease in the required model space. Theory then turns to effective interactions, where effec-tive Hamiltonians are built in a valence space on top of a core nucleus, shown schematicallyin figure 2.3. This is done to reduce the model space of the calculation. To show this, we nowsketch the steps required to build an effective interaction [92]. First, we set up the Schro¨dingerequationHΨλ = EλΨλ , (2.1)where the Hamiltonian can be written asH = T +V = (T +U)+(V −U) = H0+H1 (2.2)where V is the internucleon interaction and U is a convenient potential, typically chosen to be262.2. EFFECTIVE INTERACTIONSthe harmonic oscillator,U =A∑i=112mω2r2i . (2.3)The wavefunction Ψλ is then expanded in the basis states φ0 of H0. To reduce the dimension-ality of the problem, the basis states are written in terms of the closed core |c〉. The core isusually a doubly magic nucleus, such as 16O or 40Ca, in which nucleon excitations from thecore into the valence space are prohibited. Next, two projection operators are defined, P andQ = 1−P, where P acts to project from the complete space into the valence space, while Qacts to project into the excluded space. The eigenvalue problem now reduces toPHe f f PΨλ = EλPΨλ , (2.4)and if we calculate the binding energies relative to the closed core,PH ′e f f PΨλ = (Eλ −Ec)PΨλ , (2.5)where He f f is the effective Hamiltonian in the valence space of interest, and H ′e f f is the shellmodel effective Hamiltonian. The effective Hamiltonian can then be decomposed into twopartsH ′e f f = H′0+ ve f f (2.6)where H ′0 is the one-body Hamiltonian, measuring the binding energy of single particles withrespect to the core, and ve f f is the effective interaction between all nucleons in the valencespace. In general, there are up to n-body interactions in the valence space, where n is thenumber of nucleons in the valence space, but typically only the two-body matrix elementswere considered. Using the two-body approximation, the general effective Hamiltonian isHe f f =∑εαa†αaα + 14∑〈αβ |V |δγ〉a†αa†βaδaγ (2.7)where εα is the single particle energy, Greek indices label states in the valence space, and Vis the effective interaction between two valence nucleons. The single particle energies can betaken empirically to be the difference in binding energy between the state α in a closed-shell +1 nucleon nucleus and the corresponding closed shell nucleus. Alternatively, the single particleenergies can be calculated self-consistently by calculating the one-body attached states in thenuclei of interest. Three issues need to be solved to use the above formalism: the valence spacehas to be chosen to contain the degrees of freedom for the specific physical quantity of interest,272.2. EFFECTIVE INTERACTIONSthe effective interaction ve f f needs to be determined from the original Hamiltonian H, and anumerical framework must be developed to diagonalize the resultant matrix.First, valence spaces were historically chosen to be the major oscillator shells of the har-monic oscillator (see figure 1.3):• the p-shell consisting of the 0p3/2,1/2 orbits,• the sd-shell consisting of the 0d5/2,3/2 and 1s1/2 orbits,• the p f -shell consisting of the 0 f7/2,5/2 and 1p3/2,1/2 orbits.Several widely used interactions are the the following: the Universial sd (USD) interactionsUSDA and USDB [93], which describe nuclei in the sd-shell, and the mass-dependent Kuo-Brown interaction (KB3G) [94] and the G-matrix, experimentally fit interaction (GXPF1A)[95], which describe nuclei in the p f -shell.Second, the effective interaction Hamiltonian needs to be determined. One method to pro-duce effective interactions, is to perform a perturbation calculation of an existing NN potential,including effects of the nuclear medium to the matrix elements [92]. Once the set of interactionmatrix elements is produced, minor adjustments are generally performed to known experimen-tal values, resulting in a interaction that not only reproduces experiment where data is available,but also offers some predictive power. Falling under this approach is the KB3G interaction, thelatest version of the Kuo-Brown interaction, which was one of the first attempts at a realisticinteraction. Kuo and Brown started from a precision NN potential of the time, employed aG-matrix renormalization [96], and calculated the matrix elements to second order. The KBinteraction showed spectacular agreement to the energy levels in 18O and 18F [97]. The KB3Ginteraction is the modern incarnation of the original KB interaction, having had the matrix el-ements and single particle gaps adjusted to provide better agreement with experiment. Otherapproaches consist of fitting the matrix elements to all of the existing data available. Fallingunder this approach are the USD [98] and GXPF1 interactions. The USD interaction was firstformulated in the 1980’s, and had 63 matrix elements fit to experimental data. Even this modestnumber of elements to be fitted required two years of computer time, which would only takean afternoon on a modern PC [99]. The GXPF1 interaction was also fit to experimental data,with 195 two-body elements and four single particle energies fit to 699 energy data.Lastly, codes need to be developed to diagonalize the large matrices generated in thevalence space. Many such codes exist, such as ANTOINE [100], NATHAN [100], andOXBASH [101]. Each code employs different coupling schemes to generate the basis states,and thus, the selection of code depends on the specific nucleus being calculated. For example,282.3. MODERN χEFT BASED CALCULATIONSthe number of elements in the Hamiltonian to be diagonalized for 56Ni in the full f p-shellis 1087455228 in the shell model code ANTOINE, and 15443684 in the code NATHAN.Such large matrices are diagonalized using the Lanczos algorithm [102] until convergence isreached.In order to provide better agreement with experiment, new two-body interaction terms havebeen explored. One possibility is the tensor force, which acts between S = 1 coupled protonsand neutrons [103]. When the proton and neutron have total angular momentums of j> = l+1/2 and j′<= l′−1/2, the tensor force is attractive. Alternatively, when the proton and neutronare both in j> or j< states the tensor force is repulsive. The tensor interaction reproduces theobserved magic and sub-magic shell closures, while being completely two-body in nature.Others argue that rather than introducing new NN terms or fitting the matrix elements to alarge body of experimental data, one should include 3N terms [104]. It may be that these newtwo-body terms or adjustments mimic three-body interactions. As shown earlier, 3N forceswere required to provide good agreement between theory and experiment in light systems, thusit may be that 3N forces are also required in these effective interactions.2.3 Modern χEFT based calculationsA commonality between all of the above interactions is they largely start from effective in-teractions derived from precision NN potentials and are then phenomenologically adjust thetwo-body matrix elements to reproduce experimental observables in the valence space of in-terest. This approach has lead to several iterations of existing models. For example, the USDinteraction was originally developed in the 1980’s [98], but in 2006 two updated USD interac-tions, USDA and USDB [93], were developed to account for both the increase in experimentalknowledge in the sd-shell, and the increases in theoretical tools to develop the effective inter-action. Both interactions start from the two-body matrix elements derived from a renormalizedG-matrix effective interaction. On one hand, the USDA was constrained to remain close tothe starting effective interaction, giving a reasonable fit to the data but still remaining close tothe initial derived effective interaction; on the other hand, this constraint was removed for theUSDB, resulting in an interaction that is the best fit to the data. Similar changes have beenmade to interactions in the p f -shell, such as the KB family: KB [105], KB3 [106], and KB3G[94]; and the GXPF family: GXPF1 [107] and GXPF1A [95]. Modern effective interactionsnow start with the χEFT internucleon interactions, and derive the effective interaction withoutresorting to phenomenology, i. e., the fitting of matrix elements to data in the region applica-ble to the model. Since these χEFT based interactions tend to have their coupling constants292.3. MODERN χEFT BASED CALCULATIONSdetermined in light nuclear systems, a truly predictive nuclear interaction is obtained.There are several approaches for solving the many-body Schro¨dinger equation in the va-lence space. One approach is to adapt ab initio type calculations for use in medium massnuclei. For example, calculations with the importance truncated no-core shell model have re-produced the binding energies of 16,24O and 40,48Ca [108], providing important benchmarksfor methods based on coupled-cluster theory [108]. In principle, these methods are exact, butthey are truncated to make the calculation computationally tractable. Coupled-cluster theoryuses a similarity transformed Hamiltonian H = eT Hˆe−T , where T is the cluster operator. Tacts to create n-particle n-hole states with respect to a reference state as,T = T1+T2+T3+T4+ . . .+TA (2.8)where the Ti’s are the i-particle i-hole cluster operators. This formulation is exact, provided thatT is allowed to create A-body excitations. In practice, T is truncated to only include one- andtwo-body excitations, an approximation termed the Coupled-Cluster Singles-Doubles (CCSD).The inclusion of 3N forces poses a problem for CC-based calculations, as the computationalcost increases by orders of magnitude. To overcome this, the 3N force is reduced to an effectivetwo-body force by integrating the chiral 3N force over the Fermi sea in symmetric nuclearmatter [109]. Other methods involve constructing an effective interaction from the bare χEFTpotentials and then solving the Schro¨dinger equation using many-body perturbation theory [8].These effective interactions can then be used with existing shell model codes.Another approach uses Many-Body Perturbation Theory (MBPT) in a traditional shellmodel framework. The effective interaction is built by evolving the χEFT interaction withΛ = 500 MeV to low momentum, and is called Vlowk. The NN interaction is included at thenext-to–next-to–next-to leading order (N3LO) level, while the 3N interaction is included at theN2LO level (see figure 2.1). Three-body effects are included by including the normal orderedone- and two-body 3N interaction, corresponding to interactions between valence nucleons andcore nucleons. The residual 3N interaction between the valence nucleons is not included, asCC calculations have shown that these interactions are small compared to the normal ordered3N interaction [110]. These effective 3N interactions provide important repulsion between thevalence nucleons, increasing the spin-orbit splitting of the single particle energies [111].The CC and shell model methods have had great success in reproducing experiment. More-over, these two different approaches provide quite similar results, provided both calculationsstart with consistent single-particle energies. Figure 2.4 compares CC and MBPT ground statecalculations in the calcium chain, where each calculation is based on the same NN potential.302.3. MODERN χEFT BASED CALCULATIONSHOLT, MEN ´ENDEZ, SIMONIS, AND SCHWENK PHYSICAL REVIEW C 90, 024312 (2014)2 4 6 8 10 12 14 16 18Nh_ ω-12-10-8-6-4-2Single-Particle Energy (MeV)2 4 6 8 10 12 14 16 18Nh_ ω-4-2024 notorP)b(nortueN)a(p3/2f7/2p1/2f5/2 f5/2p1/2f7/2p3/2FIG. 2. (Color online) Convergence of (a) neutron and (b) proton SPEs as a function of increasing intermediate-state excitation N~ω.Calculations are based on NN forces in 13 major harmonic-oscillator shells.order converging faster than second order. For all orders, theground-state energies of both 42,48Ca are well converged by∼12~ω. Similarly, Fig. 2 shows the convergence of neutronand proton SPEs in the pf -shell as a function of N~ω. Whileconvergence is slower compared to the ground-state energies,all SPEs are converged by 14~ω, with neutron and protonSPEs following a very similar convergence pattern. Finally,all calculations with 3N forces seem to be converged whenincluded in five major shells. In two-body matrix elements andSPEs, the change from four to five major shells is ∼10 keVand ∼50 keV, respectively. Work to extend 3N forces beyondfive major shells is currently in progress.D. Benchmark with coupled-cluster theoryWe can also benchmark the MBPT energies with abinitio methods by using identical starting interactions andworking in the same single-particle basis. Here, we performCC calculations for the ground-state energies of the calciumisotopes by using the same Vlow k interaction in a single-particle basis of 13 major harmonic-oscillator shells with~ω = 12 MeV. The results are shown in Fig. 3 relative to theground-state energy of 40Ca. The closed j -subshell systems40,48,52,54,60Ca are calculated at the "-CCSD(T) level [42,55].The A± 1 systems 47,49,51,53,55,59Ca are obtained with the CCparticle-attached-or-removed equations of motion method atthe singles and doubles level (PA-PR-EOM-CCSD) [42,55].To compare with CC results, we perform the MBPTcalculations in the pf shell, where the SPEs are taken as thePA-EOM-CCSD (f7/2, p3/2, p1/2, f5/2) energies in 41Ca. Theparticle-attached g9/2 is not of single-particle character, sothe MBPT pf -shell comparison provides the cleanest bench-mark. This comparison probes the two-body part of thevalence-space Hamiltonian, assessing the reliability of theconvergence trend illustrated in Fig. 1.In Fig. 3, we find that the MBPT ground-state energiesare within 5% (in most cases much better) of those of CCtheory. This shows that MBPT can be comparable to CCtheory for Vlow k interactions, provided that consistent SPEsare employed.While the CC ground-state energies agree well with MBPTto 55Ca, this agreement deteriorates for heavier isotopes.The reason is that the CC calculations begin to fill the g9/2orbit, which is lower in energy than the calculated f5/2. Thismakes a comparison of the CC and pf -shell valence-spacecalculations unreliable for 59,60Ca. Moreover, a benchmark inthe pfg9/2 space is not possible because, as mentioned, the CCone-particle-attached g9/2 state in 41Ca is not of single-particlecharacter.E. Valence-space calculationsFor neutron-rich oxygen and calcium isotopes, we haveshown that it is necessary in MBPT calculations of valence-space Hamiltonians to expand the valence space beyondthe standard one-major harmonic-oscillator shell [28,30–32].40 44 48 52 56 60Mass Number A-300-250-200-150-100-500Energy (MeV)CCMBPTFIG. 3. (Color online) Comparison of MBPT and CC ground-state energies of calcium isotopes relative to 40Ca based on the sameNN interaction (for details see text). The MBPT results use the SPEsobtained in CC theory.024312-4Figure 2.4: Comparison in calcium ground state energies calculated in coupled-cluster(CC) and valence space many-body perturbation theory (MBPT). Figure repro-duced with permission from [111].The calculated ground state energies are quite close to each other, only deviating in the heavycalciums. This is due to differences in the filling of the orbitals in these nuclides [111]. An-other example is the long standing problem of the oxygen drip-line anomaly. The neutrondrip-line in the C, N and O isotopic chains ends at N = 16, while, with the addition of oneproton, the drip-line in the F isotopic chain extends to at least 30F. This is unexpected, asnaively one would expect 28O to be bound, as it is doubly magic when considering the conven-tional magic numbers. With χEFT it was shown that the 3N part of the interaction providesthe necessary repulsive force, leading to the observed drip-line. As seen in figure 2.5, the NNonly interaction [10] over binds in the n-rich nuclei, leading to a bound 28O. By introducing the3N-interaction, a repulsive force arises, pushing the drip-line back to N = 16. The phenomeno-logical effective interaction USDB [93] is in excellent agreement with the experimental datafrom the Atomic Mass Evaluation 2012 (AME12) [69], and it correctly predicts the drip-lineto be at N = 16. The Coupled-Cluster approach [109] also predicts the correct position of thedrip-line, although the heaviest nuclei are severely under-bound.312.4. TESTING χEFT FORCES: THE N = 32,34 SUB-SHELL CLOSURESææææææææAME12USDBNNNN+3NNN+3N HCCL-80-60-40-200BindingEnergyHMeVL8 10 12 14 16 18 20Neutron Number NFigure 2.5: Oxygen binding energies relative to 16O calculated with USDB, NN + 3Nshell model and CC interactions. NN and 3N + NN calculations from [10], and CCcalculations from [109]. Experimental data is taken from AME12 [69].2.4 Testing χEFT forces: The N = 32,34 sub-shell closuresMuch experimental and theoretical efforts have been spent trying to understand the propertiesnear N = 32 and 34. It has been predicted that new magic numbers may appear here, thusleading to two new doubly magic nuclei: 52,54Ca.There are several signatures of magicity that can be studied. One simple measure is theexcitation energy of the first excited 2+ state. A high energy first excited 2+ state indicates amagic nucleus because of the cost in energy associated with constructing this state in magicnuclei. Moreover, this cost is especially high in doubly-magic nuclei. Due to the pairingmechanism in doubly-magic nuclei all pairs of nucleons are coupled to Jpi = 0+ states fromwhich it is impossible to construct Jpi = 2+ states. The only way to construct these statesis to break the nucleon pair, costing ≈ 1 MeV in energy, and promoting one of these nucleonsacross the shell gap. The energy associated both with breaking the nucleon pair, and promotinga nucleon across the shell gap leads to large energies for the first 2+ state.A second measure of magicity is the reduced transition probability B(E2), a quantity thatmeasures the transition probability between the ground state and first excited 2+ state. The322.4. TESTING χEFT FORCES: THE N = 32,34 SUB-SHELL CLOSURESæææææCrBMFGXPF1AKB3GHaL0.æææææTiHbLæææææCrHcL200400600800100012001400BHE2LHe2fm4L26 28 30 32 34Neutron Number NæææææTiHdL26 28 30 32 34Neutron Number NFigure 2.6: E(2+) and B(E2) values for Cr and Ti isotopic chains near N = 32,34. E(2+)for the even Cr (a) and Ti (b) isotopes, and B(E2) values for Cr (c) and Ti (d)isotopes. Beyond-mean-field (BMF) calculations from [112]. Cr data are from[113] and [114], Ti data are from [115] and [114].B(E2) is proportional to the electric quadrupole moment,B(E2) ∝∣∣〈2+∣∣ Qˆ ∣∣0+〉∣∣2 , (2.9)where Qˆ is the electric quadrupole operator. A small B(E2) is interpreted as being a nearspherical nucleus, while a large B(E2) corresponds to a deformed nucleus. Thus, large B(E2)values should be found in collective nuclei that are mid-shell, while small B(E2) values will befound at magic nuclei [116].We can now study the existing spectroscopic information in the nuclei around 52Ca. In fig-ure 2.6 we show the experimental E(2+) and B(E2) values, along with theoretical calculationsusing GXPF1A, KBG3, and a beyond-mean field approach [112]. The GXPF1A and KB3G332.4. TESTING χEFT FORCES: THE N = 32,34 SUB-SHELL CLOSURESpredictions were calculated in the full p f -space on top of a 40Ca core, using the shell modelcode ANTOINE [117, 100, 99]. In figure 2.6 (a) and (b), there is a clear increase in the E(2+)energy at both N = 28 and 32, as compared to the surrounding nuclei. The N = 32 gap isreduced in the Cr isotopes as compared to the Ti chain, as evidenced by the decrease in theE(2+). There is evidence that N = 34 is magic in 56Ti, however, in 58Cr the value returns tothe non-magic value. There is a corresponding decrease in the experimental B(E2) values atN = 28 and 32 in both chain, but no such dip is seen at N = 34. From this we can concludethat N = 32 is a good magic number, while N = 34 may be magic in Ti.The beyond-mean-field approach reproduces the trend in the E(2+) energies, however, thevalues are systematically too high. The GXPF1 and KB3G calculations do a much better job inpredicting the absolute values of the excitation energies, the GXPF1 calculation reproduces theincrease at N = 32, while the KB3G calculation does not. For the B(E2) values, the GXPF1Aand KB3G both give very similar results. In the Cr chain, the experimental B(E2)’s are quitewell reproduced, however, the drop at N = 32 is not. In the Ti chain, none of the staggering inthe B(E2)’s is reproduced. This lack of staggering is due to the choice of the effective chargeof the proton and neutron [70]. The effective charge is not the bare charge of the nucleon sincethe effect of the core nucleons has been absorbed by the valence nucleons during the processof defining the effective interaction. If the effective charge is changed, a staggering becomesapparent, however, it still does not completely reproduce experiment.All three theoretical models are mostly able to reproduce the measured results in the Crand Ti chains. As seen from both data and theory, it is evident that there is a sub-shell closureat N = 32, given the above introduced signatures, and a weak sub-shell at N = 34 may existin Ti. A true test would be to examine the trends in the calcium chain. Verification of thesepredictions thus far have not been possible due to the difficulty in performing experiments inthis region, resulting in an absence of data.In figure 2.7 we plot the E(2+) values for the Ca isotopic chain. Again, all theories agreequite well with experiment, however, they start to deviate from each other at N = 34. Infigure 2.7 (b) we also show the results based on χEFT interactions [120, 119]. The NN-onlyinteraction fails at reproducing the data, even to the point of missing the N = 28 shell closure in48Ca. The calculations including 3N-forces achieve much better agreement with experiment,not only reproducing the N = 28 magic number, but also in predicting the excitation valueat N = 34. The E(2+) was recently measured at the RIKEN facility [118], confirming that asub-shell exists at N = 34.Another method to determine if N = 32 and 34 are closed shells, or sub-shells, is throughmass measurements, specifically, by examining the S2n values. Figure 2.8 presents the mea-342.4. TESTING χEFT FORCES: THE N = 32,34 SUB-SHELL CLOSURESææææøBMFGXPF1AKB3GHaL012345EH2+LHMeVL26 28 30 32 34Neutron Number NææææøNNNN+3N HMBPTLNN+3N HCCLHbL26 28 30 32 34Neutron Number NFigure 2.7: E(2+) values in the Ca isotopic chain. Both the GXPF1A and KB3G calcu-lations agree where data is known, but they disagree in their predictions at N = 34.The star (F) is the recently measured value by Steppenbeck et al. [118], and isconsidered to be a confirmation of a sub-shell closure at N = 34, the other exper-imental data are from [114]. the BMF calculations from [112], the NN and NN +3N calculations are from [119], and the CC calculations are from [120].æææææææAME03GXPF1AKB3GNN+3N HMBPTLNN+3N HCCL500075001000012500150001750020000S2nHkeVL28 30 32 34Neutron Number NFigure 2.8: Experimental S2n’s, taken from AME11 [121], for the calcium isotopes, ascompared to theory. Values for MBPT and CC calculations are taken from [59].352.5. TESTING χEFT FORCES: THE ISOBARIC MULTIPLET MASS EQUATIONsured values from the Atomic Mass Evaluation 2011 (AME11) [121]. The GXPF1A andMBPT [119] results agree quite well with the data to N = 31. The CC [120] result also agreesquite well with the GXPF1A and MBPT calculations, however, there is a large dip in the S2nvalue at N = 31. The KB3G calculation agrees with the both data and the other calculationsuntil N = 30, where the KB3G values become systematically lower than the other calculations.2.5 Testing χEFT forces: The isobaric multiplet massequationThe proton and neutron are both spin-1/2 particles and are nearly degenerate in mass. Thereis, however, a striking difference between the two particles, which is their charge. Due to thesesimilarities, it is possible to consider the proton and neutron as members of a doublet in theabstract isospin T space, where the proton has a z-projection of Tz = −1/2 and the neutronhas a z-projection of Tz = 1/2. This concept was originally proposed by Heisenberg [122].Isospin is a good quantum number, thus for any nucleus the z-projection of the isospin of theground-state is given byTz =N−Z2. (2.10)For a given collection of protons and neutrons, there can be isospin configurations between∣∣∣∣N−Z2∣∣∣∣≤ T ≤ N+Z2 . (2.11)Thus, states with the same isospin in different isobaric nuclides form an isospin multiplet.If the nuclear force is isospin independent, then the binding energy of states in an isospinmultiplet should be degenerate. Furthermore, the excited states of such nuclei should be similar.As an example, figure 2.9 shows the A = 9 multiplets. Note the similarity of energy levels in9Be and 9B, and between 9Li and 9C. A special case of this are the T = 1/2 and 1 “mirrornuclei” (see [124, 125, 126] for examples), where two nuclides have the same mass numberbut the number of protons and neutron are swapped. These nuclides sit on either side of theN = Z line. Because the ground states of these nuclei have the same T , but opposite Tz, theyshould have very similar structure in their excited states, as can be seen in figure 2.10 in theground state rotational band in 50Fe and 50Cr. The similarity of the two spectra demonstratesthat isospin is a symmetry of the nuclear interaction; however, the small differences point to anisospin non-conserving interaction.Isospin is an approximate symmetry, manifesting itself in the mass difference between the362.5. TESTING χEFT FORCES: THE ISOBARIC MULTIPLET MASS EQUATION25410nD.R.Tilleyetal./NuclearPhysicsA745(2004)155–362Fig. 10. Isobar diagram, A= 9. The diagrams for individual isobars have been shifted vertically to eliminate the neutron–proton mass difference and the Coulomb energy,taken as EC = 0.60Z(Z − 1)/A1/3. Energies in square brackets represent the (approximate) nuclear energy, EN =M(Z, A) − ZM(H) − NM(n) − EC, minus thecorresponding quantity for 9Be: here M represents the atomic mass excess in MeV. Levels which are presumed to be isospin multiplets are connected by dashed lines.Figure 2.9: Measured energy levels for the A= 9 systems. The ground-state energy levelshave been shifted so that the isobaric multiplets lie at approximately the same en-ergy. Isobaric multiplets are connected with dashed lines. The J = 5/2+, T = 3/2ground-state multiplet has been highlighted in red. Figure reproduced with permis-sion from [123].proton and neutron. The isospin symmetry is also broken by the isospin-dependent part ofthe nuclear Hamiltonian and the Coulomb interaction. These interactions break the symmetryand lift the degeneracy of the isospin multiplet, but the largest contribution comes from theCoulomb interaction. The Coulomb interaction in isospin space isVcoul =∑i< jQiQ j∣∣ri− r j∣∣ = e2∑i< j(12− tz(i))(12− tz( j))1∣∣~ri−~r j∣∣ , (2.12)where Q is the charge operator, e is the electron charge, and tz is the isospin operator. This canthen be expanded as a sum of isoscalar, isovector, and isotensor operatorsV (0)coul = e2∑i< j(14+13~t(i) ·~t( j))1∣∣~ri−~r j∣∣ ,372.5. TESTING χEFT FORCES: THE ISOBARIC MULTIPLET MASS EQUATION0+0.2+7834+18816+31638+474410+63390+02+7654+18526+31598+478610+63677831098128215811595765108713081627158150Cr50FeFigure 2.10: The level structure of the ground-state rotational bands in the T = 1 nuclei50Fe and 50Cr. The arrows indicate a transition between the connected states, withthe transition energy listed in keV. The left labels are the Jpi of the state, the rightlabel are the excitation energy in keV of the state. Data from [124].V (1)coul =−e22 ∑i< j(tz(i)+ tz( j))1∣∣~ri−~r j∣∣ ,andV (2)coul = e2∑i< j(tz(i)tz( j)− 13~t(i) ·~t( j))1∣∣~ri−~r j∣∣ ,where~t is the isospin operator. The operator V (0)coul does not depend on the operator Tz, whileV (1)coul and V(2)coul depends on the operators Tz and T2z , respectively.If we treat the Coulomb interaction as a perturbation, the first-order energy shift is given bythe expectation value of the Coulomb interactionEcoul = 〈α,T,Tz|Vcoul |α,T,Tz〉 (2.13)where α represents all of the good quantum numbers that do not depend on the isospin. Ap-plying the Wigner-Eckart theorem [127], it is possible to extract the isospin dependence of the382.5. TESTING χEFT FORCES: THE ISOBARIC MULTIPLET MASS EQUATIONCoulomb energy shift,Ecoul = 〈α,T,Tz| ∑q=0,1,2V (q)coul |α,T,Tz〉 (2.14)=∑q(−1)T−Tz(T q T−Tz 0 Tz)〈α,T,Tz| |V (q)coul| |α,T,Tz〉 (2.15)= E(0)coul(α,T )+E(1)coul(α,T )Tz+E(2)coul(α,T )(3T2z −T (T +1)). (2.16)where the Coulomb energy shifts areE(0)coul =1√2T +1〈α,T | |V (0)coul| |α,T 〉E(1)coul =1√T (2T +1)(T +1)〈α,T | |V (1)coul| |α,T 〉E(2)coul =1√T (2T +3)(2T +1)(T +1)(2T −1) 〈α,T | |V(2)coul| |α,T 〉The double-bar elements are reduced matrix elements, indicating that they are independent ofTz. The Tz dependence can be factored out, leading to a quadratic relationship [128]ME(A,Tz) = a(α,T )+b(α,T )Tz+ c(α,T )T 2z . (2.17)This equation is called the Isobaric Multiplet Mass Equation (IMME), first introduced by E.Wigner. The a term is the mass excess of the Tz = 0 for integer T multiplet. In the cases ofhalf-integer T , the a term is related to the difference E(0)coul−T (T +1)E(2)coul. The b term dependson the expectation value of V (1)coul and gives the largest contribution to the IMME. The c termdepends on the expectation value of V (2)coul and describes the interaction between states that differby two units of isospin.An intuitive understanding of the b and c terms can be gained by considering the energy ofa uniformly charged sphere with radius R = r0A1/3Ecoul =3e25RZ(Z−1) (2.18)=3e25r0A1/3(A4(A−2)+(1−A)Tz+T 2z)(2.19)392.5. TESTING χEFT FORCES: THE ISOBARIC MULTIPLET MASS EQUATIONY.H. Lam et al. / Atomic Data and Nuclear Data Tables ( ) – 3Fig. 1. (Color online) The b coefficients of the quadratic IMME (8) as a function of A2/3 for all T = 1/2, 1, 3/2, and 2multiplets. A weighted fit to b coefficients, b = 690.98(±89)A2/3+1473.02(±93) (keV) is displayed by the solid line. The dashed line shows the unweighted fit to b coefficients, b = 726.64A2/3+1952.7 (keV). The dash–dottedline is b =  3e2(A1)5r0A1/3 . The double-dot–dashed line is b = 3e25r0A2/3.b(↵, T ) = nH  E(1)coul(↵, T ) and,c(↵, T ) = 3E(2)coul(↵, T ). (9)The neutron–hydrogen mass (or mass excess) difference isnH = Mn MH = 782.34664 keV.Obviously, charge-dependent forces of nuclear origin of a two-body type can be treated in the samemanner, so the form of Eq. (8)stays valid.2. Compilation of IMMEOver recent years, more experimental data of higher precisionon nuclear mass excess and level schemes have been accumu-lated for most of the N ⇡ Z nuclei, in particular, for nuclei withmass number ranging from A = 41 to 71. Incorporating all re-cent mass measurements from the evaluation [6] and experimen-tal level schemes [7], we have revised and extended the databaseof IMME coefficients compiled previously by Britz et al. [8]. The to-tal number of multiplets presented in our work incorporates 382doublets, 132 triplets, 25 quartets, and 7 quintets. In particular, itincludes recent experimental data on pf -shell nuclei. This new setof IMME a, b, c (and d, e, defined below in Section 4) coefficients islisted in Tables 1–4. The b, c , and d coefficients (sometimes for thelowest-lyingmultiplets only) are also shown in Figs. 1–4, Figs. 5–7,and Fig. 8, respectively.As seen from Eq. (8), a and b coefficients for doublets1 and a, b,and c coefficients for triplets can be determined in a unique wayfrom the exact solution of a system of two and three linear equa-tions for two or threemultipletmembers, respectively. For triplets,the a coefficient is equal to the mass excess of the Tz = 0 nucleus,so it is not mentioned separately in Table 2.In principle, knowledge of three mass excesses from a givenquartet or quintet is also sufficient to determine the a, b, and ccoefficients of the quadratic IMME. However, we do not consider1 There are no c coefficients for doublets.such incomplete multiplets. In the present compilation an isobaricmultiplet is taken into consideration only if the mass excesses ofall multiplet members are known experimentally. To this end, thea, b, and c coefficients of the quadratic IMME, Eq. (8), are obtainedby a least-square fit to four or five mass excesses for a quartet orfor a quintet, respectively.3. IMME b and c coefficients3.1. Uniformly charged sphere estimatesBefore we discuss the trends of the experimental b and ccoefficients, let us recall a simple model prediction. If we assumethat the Coulomb interaction is the only contribution shiftingisobaric-analogue states, forming an isobaric multiplet, and treata nucleus as a uniformly charged sphere of radius R = r0A1/3, thetotal Coulomb energy of a nucleus is given byEcoul = 3e25RZ(Z  1)= 3e25r0A13A4(A 2)+ (1 A)Tz + T 2z. (10)Putting this expression into the IMME form, one can get the follow-ing estimates of the IMME b and c coefficients [9–11]:b = 3e25r0(A 1)A13, c = 3e25r01A13, (11)where e2 = 1.44 MeV fm.Using the approximation Z(Z  1) ⇡ Z2, one can get instead ofthe first equation in Eq. (11) an even simpler, often used expressionfor the b-coefficient [12]:b = 3e25r0A2/3. (12)These crude estimates of the Coulomb contribution to the b and ccoefficients are shown in Fig. 1 and Figs. 6–7, respectively.Figure 2.11: b-coefficients of the IMME (eq. 2.17) as a function of A2/3 for theT = 1/2,1,3/2,2 multiplets. The solid line is a weighted fit with b =−690.98(±89)A2/3 + 1473.02(±93) (keV), the dashed line is an unweighted fitwith b =−726.64A2/3+1952.7 (keV), the dash-dotted line is b =−3e2(A−1)5r0A1/3, andthe double-dot-dashed line is b = −3e25r0 A2/3. Figure reproduced with permissionfrom [129].where we have used Z = A/2−Tz. The b and c coefficients are thenb =−3e35r0(A−1)A1/3, c =3e25r01A1/3. (2.20)From these simple estimates, th b term is by far the leading contribution to the IMME, asgenerally A >> Tz or T 2z . The general scaling of b y A2/3 and c by A−1/3 can be seen in thetrend of the fitted b and c terms [129]; however this simple picture of the Coulomb energyshift does not reproduce all of the observed features. Figure 2.11 shows the behavior of theb-co ffici nts with respect to A2/3. The imple b term derived in equation 2.20 reproduces theoverall slope seen in the experimentally determined b values, however, there seems to be anoverall offset of ≈ 1500 keV.402.5. TESTING χEFT FORCES: THE ISOBARIC MULTIPLET MASS EQUATIONY.H. Lam et al. / Atomic Data and Nuclear Data Tables ( ) – 7Fig. 7. (Color online) The c coefficients of the lowest, first excited and second excited triplets as a function of A1/3. The solid (red) line connects the lowest-lying triplets’c coefficients. The (blue) double-dot–dashed line links (blue) dots which display the first higher-lying triplets’ c coefficients. The dotted line links triangles which representthe second higher-lying triplets’ c coefficients. The (black) dashed line is c = 3e25r0 A1/3.Fig. 8. (Color online) The experimental d coefficients as a function of A for all quartets and quintets. These d coefficients are defined by the cubic IMME (Eq. (13) withe = 0). The (blue) dots and (red) squares are d coefficients of the lowest-lying quartets and quintets, respectively; whereas (black) triangles are d coefficients of higher-lyingquartets.rately the experimental c coefficients for the lowest-lying triplets(upper panel) and quartets and quintets (lower panel) as a functionofA1/3. Dashed straight lines indicated in the figures represent theestimate deduced from a classical homogeneously charged sphere,Eq. (11). For the lowest-lying triplets’ and quintets’ c coefficients,indicated as red triangles and red squares, respectively, this classi-cal assumption is roughly valid. However, no clear dependence canbe seen for quartets (Fig. 6, lower panel), or for the known higher-lying quartets’ c coefficients (not shown in the figures).Figs. 5–6 give evidence that the c coefficients of triplets hav-ing A = 4n and A = 4n + 2 form two distinct families. A regu-lar staggering effect is clearly visible. Quintets in the sd-shell spacealso show a small staggering behavior when they are plotted as afunction of A1/3 in Fig. 6. No oscillatory behavior can be noticedin quartets’ c coefficients.The staggering also takes place in higher lying triplets, as shownin Fig. 7. In this figure, we connect by solid, dash–dotted and dottedlines for the lowest, first, and second excited triplets, respectively.These lines stop as soon as there are breaks in experimental datapoints (e.g., absence of data for A = 44, 52, 56 triplets). In addition,the ground state multiplet (J⇡ = 2+) of the A = 8 triplet is notconsidered in our work due to well confirmed large isospin mixingbetween two neighboring 2+ states in 8Be at 16.626 MeV and at16.922 MeV excitation energy. Two higher lying triplets, J⇡ = 1+Figure 2.12: Experimental d coefficients from the cubic form of the IMME, for all quar-tets and quintets. The blue dots and red squares are the d coefficients of the lowest-lying quartets and quintets, while the black triangles are the d coefficients of higherlying quartets. Figure reproduced with permission from [129].This offset of≈ 1500 keV can be corrected for by including the difference in mass betweenthe proton and neutron. Nuclei with higher Tz should be heavier than nuclei with lower Tz. Theb-term now becomesb = ∆nH− 3e35r0(A−1)A1/3. (2.21)The mass difference ∆nH = 782 keV, which is the mass difference between the neutron andhydrogen atom, corresponds to half of the difference needed to correct the shift in bindingenergies.It may be that the quadratic form of the IMME can not explain the measured mass excesses,and quartic d and quintic e terms may be required. For example, large deviations from theIMME have been observed in the A= 9 Jpi = 3/2+ and A= 33 and 35 Jpi = 3/2+ quartets [130,131, 129] and in the A= 8 and 32 quintets [132, 133, 134]. These higher order terms could arisefrom isospin mixing between nearby states, second-order Coulomb effects [129], or missing3N interacitons. It is precisely these effects that provide a stringent test of theory.412.5. TESTING χEFT FORCES: THE ISOBARIC MULTIPLET MASS EQUATIONIn the extended IMME, where d and e terms are considered, the d and e terms may bedirectly determined for IMME quartets and quintets, respectively. In an isospin quartet, the dterm is given byd =16(−ME(Tz =−3/2)+3ME(Tz =−1/2) (2.22)−3ME(Tz = 1/2)+ME(Tz = 3/2))and the d and e coefficients in an isospin quintet are given byd =112(−ME(Tz =−2)+2ME(Tz =−1) (2.23)−2ME(Tz = 1)+ME(Tz = 2)),e =124(ME(Tz =−2)−4ME(Tz =−1) (2.24)+6ME(Tz = 0)−4ME(Tz = 1)+ME(Tz = 2)).The uncertainty of these terms can be found by a simple propagation of errors. In general,the experimental d terms are consistent with zero, except in a few cases as mentioned above.Figure 2.12 shows the experimental d terms for all known isospin quartets and quintets. It isremarkable that, except in a handful of cases, the experimental d terms are all close to zero.2.5.1 Two-level mixing and the d termThe primary cause of the d term is from two-level mixing of nearby states with the samespin but different isospin. This causes the perturbed wave functions to have a mixed isospincharacter; thus the state no longer belong to the isobaric multiplet. As an example, we taketwo nearly degenerate states with a matrix element V connecting them [135]. The good wavefunctions can then be found by diagonalizing the matrix(E VV E +∆)(2.25)where E is the energy of one state and ∆ is the difference in energy. The eigenvalues areλ = E +∆2± 12√∆2+4V 2. (2.26)The resulting energy shift is quite complicated because of interference between the Coulomb,422.5. TESTING χEFT FORCES: THE ISOBARIC MULTIPLET MASS EQUATIONisovector, and isotensor parts of the isospin non-conserving interactions. Such a mechanismhas been employed in the A = 9 isospin quartet [130], and it was shown to be the main driverfor the observed d-term.2.5.2 Testing the IMMEHistorically, the quadratic behaviour of the IMME has been confirmed in a number of experi-ments. However, much of the data – ground state and excitation energies – tend to have quitelarge uncertainties, limiting the precision of the investigation of isospin-symmetry-breakingeffects in nuclei. For example, understanding these isospin-symmetry-breaking effects is im-portant for calculations of the isospin-symmetry-breaking correction δC in super-allowed Fermibeta decays [136]. It has only been in recent years, with the advent of Penning trap mass spec-trometers, that some of the IMME multiplets have been found to deviate from the quadraticform of the IMME.In order to test the predictions of both effective interactions and χEFT interactions we havemeasured the masses of 20,21Mg, which are the most proton-rich members of the A= 20, T = 2isospin quintet and the A= 21, T = 3/2 isospin quartet. The test of the χEFT based interactionis quite interesting as this is the first time this interaction will be tested with both active protonsand neutrons in the valence space.43Chapter 3Experimental setupTRIUMF’s Ion Trap for Atomic and Nuclear science (TITAN) is located in the Isotope Sepa-rator Accelerator (ISAC) [137] facility at TRIUMF in Vancouver, British Columbia. TITANcurrently consists of three ion traps: (1) a Radio-Frequency Quadrupole (RFQ) cooler andbuncher, used to prepare the beam from ISAC, (2) an Electron Beam Ion Trap (EBIT), used tocharge breed the beam to increase the achievable precision of a mass measurement, and (3) aMeasurement Penning Trap (MPET), used to perform high precision mass measurements onshort-lived (t1/2 / 100 ms) nuclides. A schematic outline of the TITAN system is shown infigure 3.1. From its first operation in 2007, TITAN has focussed on measuring the masses ofhalo nuclei. For example, TITAN has measured the masses of the halo nuclei 6,8He [138],11Li [67] and 11Be [139]. Beryllium-12 [140] is an interesting case as the halo state is not theground state, but instead is an excited state [141]. Since then, TITAN has conducted severalmeasurement campaigns of medium mass nuclei to investigate such phenomena as the Island ofInversion [142, 143], the presence of deformation in potential r-process nuclei in the neutron-rich Rb and Sr isotopic chains [144, 145], measuring the 71Ge-71Ga Q-value to calibrate theSAGE and GALLEX neutrino detectors [146], and measuring the mass of 74Rb [147] to testthe unitarity of the CKM matrix.Two properties distinguish TITAN from other on-line Penning Trap Mass Spectrometry(PTMS) systems: The ability to charge breed exotic beams leads to increased precision andresolving power. The unique combination of production source and MPET injection opticspermit measurements of the shortest-lived nuclides at TITAN. These qualities are exemplifiedby the mass measurement of 11Li, whose half-life of 8.8 ms, is the shortest lived nuclide tohave its mass measured in a Penning trap.The precision of a PTMS measurement is inversely proportional to the charge state of the442.1. Beam production and separation at ISACoff-line ion sourceSCISCIa) SCISCIb) HCIFigure 2.2: The TITAN experimental setup which includes a RFQ, a high-precision Penning trap, an EBIT and an o-line ion source. a) Shown inred is the path of the beam when mass measurement on singly charge ions(SCI) is performed. b) In blue is the path for highly charged ions (HCI)mass measurement.gas-filled linear radio-frequency quadrupolar (RFQ) trap ([Smi06, Smi08a]).The subsequent step depends on whether a mass measurement using singlycharged ions (SCI), or highly charged ions is performed. The ions can eitherbe transferred to an electron-beam ion trap (EBIT) [Fro06], where chargebreeding takes place (blue path in figure 2.2) or directly sent to the Penningtrap (MPET) where the mass of the ion of interest is determined (red pathin Figure 2.2).In this chapter, we present how an ion beam is produced and deliveredby the ISAC facility, how the beam preparation devices (i.e. the RFQ, theEBIT and the transport optics) are employed, and how the mass of an ionis determined by the TITAN Penning trap. The Penning trap is presentedin more detail as this device is at the core of this thesis.2.1 Beam production and separation at ISACThe Isotope Separation and ACceleration (ISAC) facility produces radioac-tive ion beams by the Isotope Separator On-Line (ISOL) method [Dom02].In this well-established method, unstable ions are produced by bombardinga thick target, such as the one shown in figure 2.3, with a 500 MeV con-36Figure 3.2: Schematic of the experimental setup of TRIUMF’s Ion Trap forAtomic and Nuclear science (TITAN). The different paths for SCI (a)and HCI (b) are marked.lenges of highly charged, radioactive nuclides to explore the advantages of HCI.A schematic overview of the TITAN facility is shown in Figure 3.2. The radioac-tive beam of SCI from ISAC is injected in a RFQ cooler and buncher [134, 135]which is floated just below (V ⇡ 5  20 V) the ISAC beam energy to deceler-ate and to trap the beam. Buffer gas cooling takes place through collisions withHe or H2 gas, which thermalize the ions to room temperature while an oscillat-ing Radio-Frequency (RF) field provides radial confinement. The net result is anoverall cooling of the ‘hot’ ISAC beam. In addition, a longitudinal electrostaticpotential leads to a confinement which allows the continuous beam from ISAC tobe accumulated in the RFQ. Ions are extracted from the RFQ by opening the con-73Figure 3.1: Rendering of TITAN. Beam is delivered from ISAC or the TITAN ion sourceto the RFQ. Si gly charged ions (SCI) are sent either to EBIT, for charge breedingand decay spectroscopy, or to MPET. Highly charged ions (HCI) can be extractedfrom EBIT and sent to MPET for precision mass measurements.ion [148]δmm∝mqBTRF√N(3.1)where q is the charge state o e i n, B is the m gnetic field strength f the trap, TRF is theexcitation time of the ion, and N is the number of detected ions. Several factors conspire tolimit the achievable precision in on-line mass measurements: (1) the magnetic field strengthB is limited—large homogeneous magnetic fields represent a technological challenge, (2) theexcitation time TRF is limited by the half-life of the nuclide, and (3) N is fixed both by the yieldof the ion of interest and the limited access to online beam time at rare isotope beam facilities.These limitations can be overcome by charge breeding, the process of removing electrons fromthe trapped beam through impact ionization with the electron beam, in the EBIT. An increasein the charge state q leads to an increase in the achievable precision, and also greatly shortensthe m asurements time (≈ TRFN) to reach a given precision. For example, in 22 hours theTITAN measurement of 74Rb in a charge state q = 8+ [147] achieved a precision comparableto that of the ISOLTRAP system [149], which needed more than 150 hours of data collection453.1. BEAM PRODUCTIONusing singly charged ions [150].3.1 Beam productionCurrently there are two primary methods for producing exotic beams: the fragmentation ofheavy ion beams on a thin target, called in-flight fragmentation [151, 152], and the spallationand fragmentation of a thick, high-temperature target by a light beam, called Isotope Separa-tion on-line (ISOL) [152]. In-flight fragmentation has the ability to produce any beam, as it isessentially free of chemistry effects, because the high-energy secondary beam cannot chemi-cally react with the target material. While wide in its reach, in-flight fragmentation can sufferfrom low yields of the nuclide of interest, especially for nuclei far from the valley of stability.On the other hand, ISOL facilities can have very high yields, even for beams far from stability.Nevertheless, these high yields quite often suffer losses due to in-target chemistry effects/re-actions (such as binding to the lattice of the target material), because the reaction products areproduced nearly at rest with respect to the target, and must diffuse to the target surface to beionized and extracted. Several other niche production methods are in use, notably the CARIBUfacility [153] at Argonne National Laboratory, where the spontaneous fission products from thedecay of 252Cf are caught in a gas cell, and the ion guide isotope separator on-line (IGISOL)facility [154] at JFYL in Jyva¨skyla¨, Finland, in which fission products from the reaction of pro-tons on U or Th targets are caught in a gas cell. At both CARIBU and IGISOL, the producedbeam does not need to diffuse out of a thick ISOL target, greatly reducing in-target losses dueto chemistry.Stopped beam experiments at fragmentation facilities can also be affected by chemistry.The high-energy beams are stopped in a gas cell, usually filled with a He buffer gas to slow thebeam through collisions. The stopping beam creates a harsh environment, with large amountsof space charge, allowing the beam and buffer gas impurities to form exotic molecules. Manytimes these molecules are close in mass to the beam of interest, which can be a problem forPenning-trap-based experiments due to contamination effects.3.2 ISACAt TRIUMF beams are produced by bombarding a thick production target with a high-current,high-energy (up to 100 µA at 480 MeV) proton beam. A schematic of the ISAC target andextraction front end are shown in figure 3.2. The target material is composed of many foils,stacked along the beam axis, helping to speed the diffusion of the fragments to the surface ofthe target material. To further speed diffusion of the fragments, the tube housing the target463.2. ISAC2.1. Beam production and separation at ISACtinuous proton beam coming from the TRIUMF cyclotron. The current ontarget of that beam can go as high as 100 µA. Once produced, di↵erent nu-PROTONBEAMEXITING PROTONBEAMTARGETIONISATIONTUBEIONBEAM4ARGET ANDION SOURCEION BEAMTO 4)4!. OTHER EXPPROTONBEAM)3!# BEAM PRODUCTION ANDSEPARATION ROOM4ARGET ANDION SOURCE(IGHRESOLUTIONMAGNET SEPARATOR0RESEPARATORFigure 2.3: The ISAC production and separation room. This room includestwo target stations, target and ion sources, a pre-separator and a high-resolution (m/m = 3000) magnet separator. Also shown is a rendering ofthe target and a surface ion source.clei di↵use out of the target and are then ionized by an ion source [Dom02].Subsequently, the ionized isotopes are extracted and formed into a beamwhich is electrostatically accelerated to an energy of 12 to 60 keV. It is laterguided to a two-stage dipole magnet separator that include a pre-separatorand a high-resolution magnet separator (figure 2.3). This separates and se-lects the ions of interest according to their mass-to-charge ratio (m/q) at aresolving power of typically m/m = 3000. Finally, the separated beam isdelivered to the ISAC hall where various experiments are located.The two species of interest to this thesis, 6He and 8He, are producedusing a SiC target and ionized by the so-called Forced Electron Beam IonArc Discharge (FEBIAD) source [Bri08]. Using this technique, ionization isdone via a plasma generated by injecting atomic gas into a chamber where37Figure 3.2: Overview of the ISAC target and dipole separation ag et. Figurefrom [155].is ohmically heated to ≈ 2000◦C. It is during this diffusion that chemistry can occur: thesynthesized radio-nuclides may chemically bind to the target material and will not be released,or the ionization potential of the desired element is not suitable to the ion source, resultingin no ionized beam. The produced rare isotopes are released from the target’s surface andtravel in a random walk to the target exit. H re a he ted tube that is coated with high-work-function material, typically rhenium, surface ionizes the beam, allowing species with ionizationpotentials below approximately 6 eV to be ionized [156]. Atoms are surface ionized by beingdesorbed from a hot surface, and in the p ocess are spontan ou ly ionized. Many species, suchas refractory elements and gases, can not be readily ionized in such a scheme. For example,phosphorous is a very reactive element and will readily react with the target material, and bebound to the target. Gases, such s the halogens and noble gases, have very high ionizationpotentials so a special ionizer, for example a forced electron beam ion arc discharge [157],must be used to ionize the beam. Elements, such as the alkaline earth metals and the transitionmetals, with ionization energies between 6-9 eV are not efficiently surface ionized, instead,they can be laser ionized, as discussed in section 3.2.1.Once the beam is ionized and extracted from the target, it is electrostatically accelerated toenergies between 10−60 keV. The desired isotope is then selected by passing the beam through473.2. ISAC80693.01 cm-'3p3d'D°j tested TiSa excitation schemes 285.3 nm Aji=4.91 10' Av (A,-Aj)=0.05 cm' / f_ Ocm-i 2p«3s='S„ proton ban FIGURE 1: Laser excitation schemes used for Mg RIS. The most efficient scheme uses an auto-ionizing state (AI). The transition into the AI is saturated with merely 5mW UV laser light inside the 3mm in diameter ionization region of the TRIUMF target ion source transfer tube. The cross section of the TRIUMF target ion source module indicates to scale, the target container and the ionization region for RILIS. The ions are extracted from a source potential at 20kV to 60kV. TRIUMF RESONANT IONIZATION LASER ION SOURCE At TRIUMF the laser laboratory is located about 20m from the target-ion source. Initially the lasers were at 15m, however, at proton beam intensities above 70|j,A occupancy of the area had to be restricted due to high neutron fluences. Therefore the laser laboratory location was relocated further away in the ISAC experimental hall. Laser beam transport is from the laser table down trough the Bl level to the B2 mass separator level as shown in figure 2. In order to maintain pressure zoning, anti-reflection coated windows are used in the 6 individual laser beam transport paths. On the mass separator level the beams are focused over a distance of 10m into the ionization region, and overlapped in line of sight of the target ion source with dielectric mirrors. Additional shielding separates the mass separator from the target ion source and pre-separator magnet with the laser view-port so that personnel access to the optics in the mass separator area is possible then the proton beam is disabled. In line of sight to the target a coUimated beam of residual activity (mainly y-radiation of about 2mSv/h) remains when the port shielding is removed and the irradiated target is in place. This setup is depicted in figure 2. Cleanliness requirements for RILIS laser beam transport are also important for low maintenance operation and optics lifetime. An IS06 or similar class clean room environment for the laser laboratory is a good environment for sensitive, high power dielectric optics. To extend this the laser beam transport path can be either enclosed in high vacuum type beam tubes, areas with laminar flow-fan filter unit supplied air, and sealed surfaces. Concrete surfaces must be sealed everywhere en route, in order to avoid optics contamination - especially in hard to access, high radiation areas, which in general have vast amounts of concrete shielding. 11  This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: to  IP: On: Fri, 14 Nov 2014 23:07:12Figure 3.3: Laser ionization schemes for magnesium (Z = 12). Only the states relevantto the ionization scheme are shown. The rightmost scheme is the most efficient.Figure reproduced with permission fr m [158].a dipole magnet, which separates the beam based on m/q. The dipole separator has a resolvingpower of m/∆m≈ 3000, which is sometimes sufficient to select the element of interest from thecontaminants; however, often this resolving power is insufficient for providing a pure beam.These background contaminants can render many experiments impossible due to extremelybad signal-to-contaminant ratios. A new ion source technique, discussed in section 3.2.1, hasbeen developed to confront these issues.3.2.1 TRIUMF resonance ionization laser ion source and the ion-guidelaser ion sourceFor certain elements that may not be efficiently surface ionized, it is possible to use a Reso-nance Ionization Laser Ion Source (RILIS), by inducing transitions to auto-ionizing states in theelement of interest, provided that a suitable ionization scheme is known. By using step-wiseexcitation to the auto-ionizing state, el ment specific ionization is achieved. Since the laser483.2. ISACFigure 3.4: Schematic of the ISAC target with IG-LIS. Surface ionized species arestopped by the repeller, while neutral atoms can drift into the RFQ volume. Res-onant laser ionization ionizes only the element of interest, allowing dramatic in-creases in beam purity. Figure from [162].ionization is element specific, while the mass separator selects on the mass number A, theircombination represents a powerful approach to producing isotopically pure beams. As an ex-ample, the demonstrated ionization schemes for Mg are shown in figure 3.3. At ISAC the TRI-UMF’s Resonance Ionization Laser Ion Source (TRILIS) source uses three frequency tunabletitanium:sapphire (Ti:Sa) lasers that are pumped by a frequency doubled Nd:YAG laser [158].The Ti:Sa laser can be frequency doubled, tripled and quadrupled to nearly cover the wave-length range 200− 1000 nm [158, 156]. Once the most efficient ionization scheme has beenfound, the total efficiency of the laser ionization system most strongly depends on the availablelaser power for a given wavelength. At TRILIS for the ionization of Mg, the first ionizationstep (285.3 nm) is nearly saturated, while the second ionization step (880.9 nm) is fully satu-rated [159]. Increasing the laser power in either of these transitions would have a small effecton the overall ionization rate. However, the final ionization step (291.6 nm) is not saturated,and can benefit from any increase in laser power. The RILIS technique has been successivelyused at many ISOL facilities, such as ISOLDE [160], IGISOL [161], and ISAC [158].In some cases the number of surface-ionized contaminants still overwhelms the RILIS-produced beam. To overcome this, a new ion source has been developed at TRIUMF, the IonGuide Laser Ion Source (IG-LIS) [159], which is a variation of the originally proposed laserion source trap (LIST)[163, 164]. In both cases, a repeller electrode is located at the exit of thetarget, preventing surface-ionized species from leaving the target volume, allowing only neu-tral species to drift into an ion-guide volume. The neutral atoms are then exposed to laser light493.2. ISACææææææææææçççççççææææææ æææççççççççNaMg10-11011031051071091011YieldHparticlessL20 22 24 26 28Mass HuLFigure 3.5: IG-LIS yields measured during the 20,21Mg experiment. Open circles areyields measured with a previous surface ion-source target, filled circles are theyields measured with IG-LIS. The Na yields are reduced by a factor of ≈ 106,while the Mg yields are reduced by ≈ 10. Data from [159].which provides the element specific ionization, suppressing the background contaminants bymany orders of magnitude. The LIST source uses an RFQ buncher to bunch the laser-ionizedbeam coming from the target, creating an ion bunch with well defined beam properties. Beambunching also results in more brilliant beams for reaction experiments. By synchronizing thedata acquisition to the extraction pulse, these reaction experiments can reduce backgroundscoming from any “leaky” beam escaping the trap. The IG-LIS source does not trap the beamlongitudinally, instead the RFQ is used as an ion-guide to radially confine and guide the beamto the extraction electrode. A schematic of IG-LIS is shown in figure 3.4. During the present20Mg experiment, IG-LIS improved the signal-to-contaminant ratio by more than a factor of104. Figure 3.5 summarizes the IG-LIS yield measurements for the laser-ionized species mag-nesium, and the surface-ionized species sodium. These species provided an excellent test ofIG-LIS’s ion suppression capabilities. On the proton-rich side of stability, the alkali metalsodium is closer to stability than magnesium, meaning the sodium will be produced in muchlarger quantities than magnesium. As seen in figure 3.5, the sodium yields could be suppressedby up to 6 orders of magnitude. IG-LIS also reduces the magnesium yield by approximatelyone order of magnitude due to both shorter ion-laser interaction times in the short IG-LIS vol-ume and the suppression of any surface ionization of the beam of interest; however, for many503.3. TITAN2.2. Beam preparation: the radio-frequency quadrupole (RFQ) cooler and buncherVDCR CVACVz01 3 2422-6.6 V5 V13.4 V-26.6 V++++++++continuous beambunched beamextraction potentialcapture potentialrodcooling and accumulationFigure 2.4: Top: Schematic sideview of TITAN’s RFQ which is composedof four 24-segmented rods that create a longitudinal trapping potential. Awell allows for beam accumulation and subsequent bunching. A square-wave RF is applied to the opposite segments to provide radial confinement.Bottom: Schematic potential distribution for accumulation (solid line) andbunch extraction (dashed line).beam with ✏99% = 50 ⇡ mm mrad transverse emittance at 60 keV energy[Smi08a]. The transverse emittance of the beam leaving the RFQ is ap-proximately ✏99% ⇡ 10 ⇡ mm mrad at 1 keV. The measured full width halfmaximum (FWHM) of the beam energy spread at this energy is typicallyaround 6 eV [Cha09].The typical buer gas used to cool the beam is helium, due to its inertnature and light mass allowing favorable momentum transfers for ecientenergy spread dissipation. However, for the 6,8He mass measurements thebeam was cooled using hydrogen to avoid resonant charge exchange reac-tions. Figure 2.4 shows that the TITANs RFQ is composed of a four rodstructures on which a radio-frequency quadrupolar field is applied to create39Figure 3.8: Top: Schematic of the longitudinal segmentation of TITAN’s RFQint 24 sections. The segment number is indicated at the top. Bottom:Schematic of the typically applied potential. The DC field drags thebuffer gas cooled ions to the minimum of the trapping potential (solidline). The beam is extracted in ion bunches by switching the potentialsof the electrodes 22 and 24 (dashed-line). Figure from [146]..tor which da ps the RF to protect the DC power supplies.An effective net buffer gas cooling in an RFQ takes place when the mass of theion is larger than the coolant gas particles [165, 168]. In the presence of an RF-field for trapping, a drastic change in energy would disturb the ion’s micr -motionsignificantly and bring the motion out of phase with the RF-field. Hence, if themass of the coolant is larger than the ion’s mass the energy of the ion is on averageincreased, an effect which is referred to as RF-heating [169]. In a buffer gas oflighter masses, little momentum is transferred per collision, and the disturbance ofthe micro-motion is less relevant. As a consequence the harmonic, macro-motioncan be damped while keeping the micro-motion coherent to the external RF-field.The average, relative energy change per hard-sphere collision (Figure 3.10) follows[168]< ✏ >= 1 (1 + )2(3.1)83Figure 3.6: (top) Axial segmentation all ws for a drag and trapping field to be created.Segments 22 and 24 ar witched for ejection. (bottom) Axial field in the RFQduring trapping and ejection. Figure reproduced with p rmission from [155].applications this is mor than compensated by the large background suppression. There is alimit to the achievable suppression, as neutral contaminant ions can also drift in to the IG-LISvolume, and become ionized due to the hot electrode surfaces, resulting in background rates of100−1000 ions/second. This is generally only a problem when the contaminant beam yield ismany orders of magnitud more than th beam of interest.3.3 TITAN3.3.1 TITAN RFQ cooling and bunchingAt TITAN, the ISAC beam is first delivered to the radio-freque cy quadrupole (RFQ) linearPaul trap cooler and buncher [165]. The RFQ is biased several volts below the beam energyso that the beam enters with little energy. Through collisions with the helium buffer gas, theoverall bea emittance is reduced, a requirement for precision mass measurements. Further-mor , th beam delivered from ISAC is continuous, so the RFQ bunches th beam permittingefficient injection into either EBIT or MPET.513.3. TITANTITAN’s RFQ is segmented into 24 axial electrodes, each of which can be individuallybiased to create an axial drag field to pull ions into a potential well. The potential well atsegment 23 provides axial confinement, while transverse confinement is provided through theapplication of a quadrupole RF field on the RFQ electrodes. A schematic of the electrodestructure and the applied potentials during injection and extraction is shown in figure 3.6. Theradio frequency is driven by a square wave with frequencies up to 1.2 MHz and peak-to-peakamplitudes up to 400 Vpp. The inner radius of the radio frequency rods is r0 = 11 mm, and thetotal length of the RFQ is 700 mm. To cool the beam, a He buffer gas is introduced to the RFQvolume at a pressure of ≈ 0.01 mbar. Helium is chosen for two reasons: First, the ionizationenergy of He is 24 eV which reduces the probability of charge exchange reactions, and second,He is much lighter than most isotopes measured using TITAN, which is beneficial for cooling.If the buffer gas is heavier than the injected beam, the energy of the injected beam increases ina process called RF-heating [166, 167]. For a model using so-called hard-sphere collisions ina Paul trap, the average energy transfer to the beam ion can be calculated as [167]〈εRF〉= κ κ−1(1+κ)2(3.2)where κ = M/m is the ratio of the buffer gas with mass M and the beam particle with mass m.For 0< κ < 1 the beam will be cooled, but for κ > 1 the beam will gain energy. For beams withA < 12, a buffer gas of H2 is used instead of He, increasing the extraction efficiency by nearlya factor of 2 [165]. Once the ions are thermalized with the buffer gas, a process taking severalmilliseconds, they are ejected from the RFQ by quickly changing the voltages on segments 22and 24. The beam is then accelerated to 1− 2 keV to the pulsed drift tube, where the beamis then pulsed to ground. The RFQ has an overall transfer efficiency of between 7− 15%,depending on the beam used. Alkali metals typically have the highest efficiencies becausethey do not react with impurities in the buffer gas, while a beam of noble gases does reactwith impurities in the buffer gas, resulting in greatly decreased efficiencies. The probabilityfor charge exchange can be reduced by decreasing the overall cooling time, however, this bothreduces the total cooling time, potentially affecting the beam quality and it also reduces thetotal accumulation time, leading to a reduction in the total efficiency.The potential felt by the ions in the well formed at electrode 23 isΦ(x,y,z; t) =ψ(t)r20(x2− y2)+Uendgz20(2z2− x2− y2) (3.3)where ψ(t) is a time-varying RF-signal, Uend is the axial trap depth, r0 is the distance from the523.3. TITANaxis to the outside of a rod, z0 is the length of the trapping electrode and g is a geometric factor.The first term provides radial confinement, while the second term provides axial confinement.In essence, this is a combined linear mass filter and Paul trap, and it is given the name of linearPaul trap [168, 169]. This differs from a Paul trap in that the trapping potential Uend is heldconstant in a linear Paul trap, while in a Paul trap it is a function of time. A general choice forψ(t) isψ(t) =Udc−URFS(Ωt) (3.4)where UDC is a potential offset applied between adjacent rods, URF is the amplitude of the timevarying field S(Ωt), andΩ is the angular frequency of the field. We introduce the dimensionlesstime parameter ξ = Ωt/2 to simplify the following derivations. This leads to the followingequations of motion∂ 2x∂ξ 2+(ax+aend−2qxS(2ξ ))x = 0 (3.5)∂ 2y∂ξ 2+(ay+aend−2qyS(2ξ ))y = 0 (3.6)∂ 2z∂ξ 2−2aendz = 0 (3.7)withau = ax =−ay = 8qeUdcmr20Ω2(3.8)qu = qx =−qy = 4qeURFmr20Ω2(3.9)aend = −8qeUendgmz20Ω2. (3.10)An important note is that for positive ions aend is always negative [170]. If Uend is negative, theaxial potential becomes a hill, causing ions to be lost axially. If we now introduce the effectiveterm a′u = au + aend, we can write the radial equations of motion in the well known Mathieuform∂ 2u∂ξ 2+(a′u−2quS(2ξ ))u = 0 (3.11)where u corresponds to either the x or y solution. This is slightly different from the normalMathieu equation due to the extra aend term. These are the same equations for a linear massfilter; however, ax and ay are shifted up by −aend.Several methods exist to solve the Mathieu equations. As the time-varying signal is peri-533.3. TITANodic, the solution lends itself to matrix methods. The transition matrix of the RF field movesthe initial position and velocity of the particle to the final position and velocity(xn+1vn+1)= M ·(xnvn)= Mn ·(x0v0)(3.12)where x0 and v0 are the initial position and velocity of the ion, and xn and vn are the positionand velocity after n applications of the periodic waveform. We can rewrite this using theeigenvectors ~mi and eigenvalues λi of M(xnvn)= Mn ·(x0v0)=C1λ n1~m1+C2λn2~m2. (3.13)where the Ci’s describe the ion’s initial conditions in terms of the eigenvectors of M. If anion’s motion is to be stable, the position and velocities must remain finite as n→ ∞, requiring∣∣λ1,2∣∣≤ 1. The eigenvalues of M areλ1,2 =Tr{M}2± i√|M|−(Tr{M}2)2. (3.14)From Liouville’s theorem (valid without buffer gas as the forces are conservative), the totalphase space area of the ion bunch in the RFQ must be conserved, so the determinant of M mustbe 1. Substituting s = Tr{M}/2 the eigenvalues becomeλ1,2 = s± i√1− s2. (3.15)There are several interesting properties of the eigenvalues: λ ∗1 = λ2 and λ1λ2 = 1. From theearlier stability requirement that∣∣λ1,2∣∣ ≤ 1 it follows that s ≤ 1, otherwise the eigenvaluesbecome real and greater than 1. The stability requirement is then simplyTr{M} ≤ 2 (3.16)for any given transition matrix.To solve for M we can divide the waveform into time regions of constant voltage, solve theequations of motion in each time section, and take the product of the resultant set of matrices.543.3. TITANFigure 3.7: Stable regions (shaded) in a 50% duty-cycle square-wave driven linear Paultrap for different values of aend. As the trap becomes deeper, the smaller the stableregion becomes. For comparison, a sine-wave filter aend = 0 is plotted as a dashedregion.Solving the equations of motion for a constant S(t), yields the transition matrix [171]M(τ, f ) =(cos(τ√f)sin(τ√f)/√f−√ f sin(τ√ f ) cos(τ√ f ))(3.17)where f = a′−2q, and τ is the length of time that the waveform is constant. It is then possibleto build-up any given waveform through the application of M(τ, f ). The two most commonwaveforms are sinusoidal and square-wave, however, nearly all RFQ’s in use at RIB facilitiesare sinusoidal.Axially, the ions are confined for any choice of aend, however, while providing axial con-finement, the axial potential also adds a repulsive radial force. If the axial trap is too deep, theions will collide with the RFQ rods and be lost. Figure 3.7 shows the effect of increasing theaxial trap depth with regions inside the curves being stable, while the regions outside beingunstable. In this mode of operation, it is possible to operate the linear Paul trap as a mass filter.By increasing the trap depth, only species with the correct m/q will be confined, the otherswill be lost radially. At TITAN, the RFQ is operated with a trap well of Uend = −1 or −2 V,corresponding to an aend of ≈ −0.01. The DC offset, ax,y, is kept at zero, while q is typically553.3. TITANFigure 2.7: Illustration of an EBIT. From left to right: the electron gun as-sembly, the magnet coils and drift tube assembly, and the collector as-sembly. Typical trapping potentials are shown. (figure from Ref. [38]cMaxime Brodeur. Reproduced with permission).Figure 2.8: Schematic of the TITAN EBIT. From left to right: the electron gunassembly, the magnet chamber and drift tube assembly, the injectionoptics, and the collector assembly (Credit: Image Courtesy of TITAN).14Figure 3.8: Schematic of the ion trap and electron beam in the lectron beam ion trap.The central electrodes create a potential well, confining the ions axially, while themagnetic field and space charge from the electron beam provides radial confine-ment.chosen to be close to Electron beam ion trapA unique feature of TITAN, in the context of rare isotope science, is the ability to charge breedradioactive nuclides in an Electron Beam Ion Trap (EBIT) [172], creating what are known asHighly Charged Ions (HCI). In an EBIT, axial ion confinement is provided by an electrostaticpotential well, while radial confinement is provided by a strong magnetic field and the spacecharge of the electron beam. Currently at TITAN, the EBIT typically use electron beam ener-gies of up to ≈ 5 keV and currents of up to 400 mA. High electron beam energies are requiredto reach the highest charge states of heavy nuclides. As an example, the ionization energy ofhydrogen-like U91+ is ≈ 130 keV. The magnetic field compresses the electron beam near thetrap centre, where the field is the strongest, creating a high current density, leading to fastercharge breeding, provided good overlap of the electron beam and the ion cloud. Figure 3.8shows a schematic drawing of the EBIT.Charge-bred ions from EBIT are primarily used to increase the precision of mass measure-ments, as can be seen in equation 3.1. Several mass measurements that have benefited fromthe use of HCI’s include: 74Rb [147], the mass of which is important for tests of the CKM ma-trix, the 71Ge-Ga [146] and 51Cr-V [173] Q-values which are important for neutrino sources563.3. TITAN - 5831495.0 (Hz)RFi-10 -5 0 5 10 15 20s)µTime of Flight (222426283032s)µTime of Flight (Figure 3.9: Resolving the 100 keV isomer in 78Rb using ions charge bred to q = 8+,and an excitation time of 197 ms. An equivalent separation with SCI would needexcitation times of ≈ 1.6 s. Figure from [174].that are used to calibrate neutrino detectors, and measurements of neutron-rich Rb and Sr iso-topes [144, 145] that provide important input for astrophysical r-process calculations. HCI’scan also increase the achievable resolving power R of a Penning trap mass spectrometer, asthe resolving power goes as [148]R ≈ ωcTRF = qBTRFm . (3.18)This was demonstrated in 78Rb [174], where the ground state and 100 keV isomer could beresolved with an excitation time of 197 ms, the separation is clearly seen in figure 3.9. ForSingly Charged Ions (SCI), an equivalent resolving power would have required excitation timesof > 1 s.3.3.3 Cooler Penning trapCharge breeding increases the energy spread of the ion beam, which has a detrimental effect onmass measurements in MPET. To reduce the energy spread a cooler Penning trap (CPET) [175]has been constructed, and is being commissioned off-line. By using either electrons or protons,CPET will cool the charge breed beam sympathetically through collisions. A He buffer gas isnot used, because excessive ion losses will result from charge exchange reactions between theHCI’s and the He gas. Electrons are an ideal candidate because they quickly self cool through573.3. TITANK. Blaum / Physics Reports 425 (2006) 1 –78 9zρ0z0UdcB(b)zρ0z0(a)~5 cmzBUdc(c)Udc+Vrf ρρρFig. 5. Electrode configurations of a Paul (a) and Penning trap (b, c), consisting of two end electrodes and a ring electrode with hyperboloidal (a, b)or cylindrical shape (c). For charged particle storage a trap voltage with proper polarity is applied between the ring electrode and the end electrodes.Fig. 6. Left: Radiofrequency quadrupole mass filter electrodes having hyperbolic cross-section. Right: Equipotential lines for a quadrupole fieldgenerated with the electrode structure shown left.3.2. Radiofrequency quadrupole and Paul trapsPaul and Steinwedel first described the linear radiofrequency quadrupole mass spectrometer (QMS), also named theradiofrequency quadrupole mass filter (QMF) or ion guide, in 1953 [95,97]. This device provides two-dimensional ionconfinement and mass separation by oscillating electric fields. It was continuously improved and extended to threedimensions [98,122] in the now-called Paul trap. Both are widely used in various branches of science. The principlesand applications of a quadrupole mass spectrometer are summarized in the textbook by Dawson [20].An ideal quadrupolar geometry (see Fig. 6) is formed by four hyperbolic electrodes of infinite length with twoperpendicular zero-potential planes that lie between the electrodes and intersect along the center-line z-axis. For massanalysis both a static electric (dc) potential and an alternating (ac) potential in the rf range are applied to the electrodesof the linear Paul mass filter which is used, for example, in rest gas analysis or analytical chemistry [123]. The relativeFigure 3.10: Schematic of a Penning trap. Figure reproduced with permission from [41].synchrotron radiation in a strong 7 T magnetic field, while protons do not. A detriment of usingelectrons is that they can be captured by the HCI, causing a loss of the ion of interest; however,simulations have shown survival rates for U92+ of more than 90% for a cooling time of 500 ms[176].CPET is planned to be installed in the TITAN beam line in early 2016.3.3.4 Measurement Penning trapThe measurement Penning trap (MPET) is the principle trap of TITAN, dedicated to performingaccurate and precise mass measurements. The mass is determined by measuring the cyclotronfrequency ωc = qB/m of an ion in a homogeneous magnetic field. The magnetic field only pro-vides radial confinement, while axial confinement is provided by a three dimensional electricquadrupole field. A natural choice for the electric field is a harmonic potential,V = ax2+by2+ cz2 (3.19)where a, b and c are undetermined coefficients. By solving the Laplace equation, we find thatthe sum of the coefficients must be zero. The natural choice is to preserve cylindrical symmetry583.3. TITANTable 3.1: Characteristic trap dimensions for MPET.Parameter Length (mm)r0 15z0 11.785d0 11.21by setting a = b, constraining a =−c/2, leading to the potentialV (z,r) =c2(2z2− r2). (3.20)c is determined by taking the difference between the two equipotentials, as shown in fig-ure 3.10, where the top and bottom sheets are called the “end-cap” electrodes and the middlesheet is called the “ring” electrode. This leads to the potentialV0 =V (z0,0)−V (0,r0) = c(2z20− r20) (3.21)c =V02d20(3.22)where r0 is the distance from the trap centre to the closest approach of the ring electrode,z0 is the distance from the trap centre to the closest approach of the end-cap electrodes, andd20 = (2z20+r20)/4 is called the characteristic trap distance. These trap measurements for MPETare summarized in table 3.1. The quadrupole potential is thenV (z,r) =V02d20(2z2− r2) . (3.23) Ion motion in a Penning trapIn a Penning trap the ions are affected by both the electric field and the magnetic field. Asuperconducting solenoid magnet provides a strong and homogeneous magnetic field ~B = B0zˆin the trapping volume. The combination of these fields yields the equations of motion [177]x¨−ωcy˙− ω2z2x = 0 (3.24)y¨+ωcx˙− ω2z2y = 0 (3.25)z¨+ω2z z = 0 (3.26)593.3. TITANTable 3.2: Eigenfrequencies for 39K+ in MPET.Motion Frequencyνc 1457822.6 Hzν+ 1451683.4 Hzνz 133508.2 Hzν− 6139.2 Hzwhere we have defined ω2z = qV0/md20 . To solve for the radial motions, we introduce thecomplex coordinate u = x+ iy, transforming the radial equation of motion tou¨+ iωcu− ω2z2u = 0. (3.27)The radial motion should be periodic, so we try a solution of the form u ∝ exp(−iωt+φ).This yields two eigenfrequenciesω± =12(ωc±√ω2c −2ω2z), (3.28)where ω± are called the reduced cyclotron and magnetron frequencies. For the eigenfrequen-cies to be real, the condition ωc >√2ωz, or in terms of the applied fields qB20/m > 4V0/d20 ,must be fulfilled. For typical choices of trapping voltages, this leads to the hierarchy ωc >ω+  ωz  ω−. Typical values for the eigenfrequencies of 39K+ in MPET are shown intable 3.2. The solution to the radial equation of motion is thenx(t) = r+ cos(ω+t+φ+)+ r− cos(ω−t+φ−) (3.29)y(t) = r+ sin(ω+t+φ+)− r− sin(ω−t+φ−) , (3.30)where φ± are the initial phases of the ion motion in the reduced cyclotron and magnetronmodes, respectively.From equation 3.28 the eigenfrequencies can be combined into several useful relationships[177]:ωc = ω++ω− (3.31)ω2z = 2ω+ω− (3.32)ω2c = ω2++ω2z +ω2−. (3.33)603.3. TITANFrom the above, we see that the cyclotron frequency is not an eigenfrequency of the ion’s mo-tion, it is instead a combination of the radial eigenfrequencies. By measuring the eigenfrequen-cies directly, or by measuring a “side-band” frequency, a frequency that is a linear combinationof the eigenfrequencies, the cyclotron frequency can be determined. Two methods for mea-suring the eigenfrequencies directly are Fourier-Transform Ion Cyclotron Resonance (FT-ICR)[178] and Phase-Imaging Ion Cyclotron Resonance (PI-ICR) [179]. FT-ICR measures the cur-rent on a trapping electrode induced by an ion’s motion. This method produces the most precisemass values; however, it involves ion observation times of several tens of seconds, a problemfor the short-lived nuclides measured with TITAN. Another draw back is detecting the inducedcurrent requires a high-quality LC circuit tuned to the desired eigenfrequency, limiting the abil-ity to quickly change isotopes, as such a resonant circuit would also have to be changed. Inon-line measurements, several different isotopes are usually measured in a single beam time,limiting the FT-ICR technique to stable and very long-lived nuclides.The PI-ICR technique also measures the eigenfrequencies, not by measuring induced cur-rents, but instead projects the phase of the ion’s motion onto a position-sensitive detector.This technique has shown great promise in measuring stable isotopes, reaching precisions of0.2 ppb [180].The PI-ICR technique can in principle reach the same precision as the FT-ICRtechnique, provided the same care is taken in preparing the ion, but with the benefit of notneeding a tuned LC circuit. Sideband quadrupole excitationOne way to access the cyclotron frequency is to excite the frequency sideband ωc = ω++ω−,and measure the amount of conversion from a state of pure magnetron motion to a state of purereduced cyclotron motion. An ion is excited with a weak quadrupole fieldVRF =Vq2a2cos(ωRFt+φRF)(x2− y2) (3.34)where Vq is the excitation amplitude at a distance a from the trap centre, with frequency ωRFand phase φRF . This field is applied radially by split electrodes, as schematically shown infigure 3.11. Usually the excitation voltage is applied on a split ring electrode, but at TITANthe excitation is applied on the split correction guard electrodes. To produce a quadrupolarexcitation field, adjacent electrodes receive signals that are 180◦ out of phase. By breaking thecylindrical symmetry of the trapping field with the excitation, the two radial eigenmotions canbe coupled, leading to an interconversion of modes. Similarly, quadrupole excitations can beapplied in the xz or yz planes, allowing the axial and radial motions to be coupled [181].613.3. TITANquadrupole dipoleV cos (!rf · t)V cos (!rf · t)V cos (!rf · t)V cos (!rf · t)Figure 3.31: Schematic of a cross sectional view of a segmented ring elec-trode which can be used to apply an azimuthal quadrupole or dipolefield on top of the DC trapping potential.into the setup in 2012. It will be able to separate the ions of interest from isobariccontaminations in the ISAC beam such as the unwanted 74Ga ions next to 74Rb inthe present work.3.7.7 TITAN’s MPET setupThe Penning trap structure dedicated to the high precision mass measurements atTITAN is mounted in the bore of a superconducting magnet. The field strength of3.7 T is comparable to other Penning trap facilities at radioactive beam facilitiesalthough most of those employ larger field strengths. But the unique feature of HCIat TITAN can boost the precision according to Equation 3.45 to a level above theone attainable even with the current strongest magnet field of 9.4 T at the Low-Energy Beam and Ion Trap facility (LEBIT) [110] at the National SuperconductingCyclotron Laboratory (NSCL) .The trap setup as well as neighbouring beam optics for extraction and injectionare shown in Figure 3.32 all of which are mounted on the same support structure.It is installed in a vacuum tube in the inside of the magnet’s bore. To minimizemagnetic field inhomogeneities, all material used for the trap support structure andeven for the vacuum tube is chosen to be non-magnetic. The main components ofthe injection path in Figure 3.32 are a PLT and a Lorentz steerer. The purpose ofthe PLT is to remove the majority of the kinetic energy from the ions. It is biasedbelow the transport energy of the ions. When the ions are in the centre of the PLTat time tPLT, it is switched to a bias voltage below the Penning trap. The ions canenter the trap through a hole in the end cap electrode of the trap (see Figure 3.33).120Figure 3.11: RF application for quadrupole and dipole excitation. The annular segmentrepresents the guard electrodes of MPET as shown in figure 3.22, which are seg-mented into four parts.The equation of motion can be solved classically [182], however, the solution is much morereadily obtained in the quantum domain [183]. First, we write the Hamiltonian asH =12m(~p−q~A)2+qV (z,r) (3.35)where ~A is the vector potential of the magnetic field, chosen to be ~A = (B/2)(−yxˆ+ xyˆ) forconvenience. It is possible to write the canonical coordinates as [184, 183]q+ =−√mω1(y˙+ω−x) p+ =√mω1(x˙−ω−y) (3.36)q− =√mω1(y˙+ω+x) p− =√mω1(x˙−ω+y) (3.37)q3 =√mωzz p3 =√mωzz˙ (3.38)which leads to the HamiltonianH =ω+2(q2++ p2+)− ω−2(q2−+ p2−)+ωz2(q2z + p2z). (3.39)This is the Hamiltonian for two normal simple harmonic oscillators, in + and z, and an invertedoscillator in −. The quantum problem can now be formulated by constructing the annihilation623.3. TITANand creation operatorsa± =1√2h¯(q±+ ip±) , a†± =1√2h¯(q±− ip±) , (3.40)which follow the standard commutation relations. The quadrupole excitation field can now bewritten in terms of the creation and annihilation operators:Vr f =Vq2a2(e−i(ωr f t+φr f )(a†2+ +a2−+2a†+a−)+ ei(ωr f t+φr f )(a2++a†2− +2a†−a+)). (3.41)The first term describes the process of absorbing a photon from the exciting field, and creatingtwo quanta of reduced cyclotron motion with energy 2h¯ω+. The second term describes theprocess of absorbing a photon from the exciting field, and annihilating two quanta of magnetronmotion with energy 2h¯ω−. The third term describes the process of absorbing a photon withenergy h¯ωc and converting a quanta of magnetron motion into a quanta of reduced cyclotronmotion. The last three terms describe the inverse process. Only the interconversion of modesis of interest, thus we arrive at the HamiltonianH(t) = h¯g(e−i(ωr f t+φr f )a†+(t)a−(t)+ ei(ωr f t+φr f )a†−(t)a+(t))(3.42)where g = qVq/(2m√ω2c −2ω2z ) is the coupling constant between the magnetron and reducedcyclotron modes. Ignoring the axial motion, the complete Hamiltonian for the radial motionduring an excitation is [183]H(t) = h¯ω+(a†+a++12)− h¯ω−(a†−a−+12)+ h¯g(e−i(ωr f t+φr f )a†+(t)a−(t)+ ei(ωr f t+φr f )a†−(t)a+(t)).Rather than solving the Schro¨dinger equation for the Hamiltonian above, we find theamount of conversion from one mode to the other by considering a quantum two-level systemexcited by a time-varying potential. We start with the time-dependent Schro¨dinger equationih¯∂Ψ∂ t=(Hˆ0+Vˆ)Ψ (3.43)where Hˆ0 is the time independent Hamiltonian and Vˆ is the sinusoidal excitation. Expanding633.3. TITANHaL0.Ω=ΩcLæææææææææææææææææææææææææææææææææææææææææææææHbL5860626466687072Time-of-FlightHΜsL0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4RF Amplitude HVLFigure 3.12: Conversion between magnetron (slower) and reduced cyclotron (faster) asa function the excitation amplitude for an excitation time of 0.1 s. (a) Transitionprobability (b) Observed time-of-flight as a function of the RF amplitude. Theblue line is a fit to the data. One full conversion occurs near Vr f ≈ 0.24 V.Ψ in terms of the unperturbed wave functions Ψ0Ψ=∑kbkΨ(0)k ,we are lead to the following set of differential equations [185, 186]ih¯∂bm∂ t=∑kbkVmkeiωmkt (3.44)where Vmk is the matrix element connecting states m and k, and ωmk = (Em−Ek)/h¯. We now643.3. TITANassume a sinusoidal perturbation with frequency ωVˆ = Fˆ cos(ωt) (3.45)=Fˆ2(eiωt + e−iωt)(3.46)where Fˆ is a general operator. Substituting this into equation 3.44 leads toih¯∂bm∂ t=∑kbkFmk(ei(ωmk+ω)t + ei(ωmk−ω)t). (3.47)We note that the right hand side of the above is identical in form to equation 3.42. If ω isclose to ωmn, then only these states will contribute to the solution. In the other states, thefrequency terms are large and will be averaged out over the time that Vˆ is applied. By makingthis “rotating wall” approximation we only need to examine the slowly varying term. Thisleads to the two coupled equationsih¯∂bm∂ t= bnFmneiεt (3.48)ih¯∂bn∂ t= bmFnme−iεt (3.49)where ε = ωmn−ω is the frequency detuning. Here we care about converting magnetronmotion into reduced cyclotron motion, so the frequency that will create maximal conversion isω+− = (E+−E−)/h¯ = ω++ω− = ωc. Solving these equations leads tob−(t) = Aeiεt/2(cosΩt− iε2ΩsinΩt)−Beiεt/2 igΩsinΩt (3.50)b+(t) =−Ae−iεt/2 igΩ sinΩt+Be−iεt/2(cosΩt− iε2ΩsinΩt)(3.51)where g= Fmn/h¯, Ω=√ε2/4+g2, and A and B are determined from the initial normalizationof the wave function. This solution is much easier to work with when it is expressed as a matrix(b−(t)b+(t))=W (ε, t)M(ε,g, t)(b−(0)b+(0))=(eiεt/2 00 e−iεt/2)(cosΩt+ iε2Ω sinΩt − igΩ sinΩt− igΩ sinΩt cosΩt− iε2Ω sinΩt)(b−(0)b+(0))(3.52)where W (ε, t) is the phase evolution of the state vector, and M(ε,g, t) is the propagation matrix.653.3. TITANg = А2g = 3А20. -2 -1 0 1 2 3Ε HHzLFigure 3.13: Quadrupole excitation line shape for g = pi/2, 3pi/2 for an excitation timeof 1 second.If we start with a state of pure magnetron motion, then the probability F1(ε,g, t) for an ion tobe converted to a state of pure reduced cyclotron is the (1,2) component of the propagationmatrix [183]F1(ε,g, t) =∣∣M1,2(ε,g, t)∣∣= g2Ω2 sin2Ωt. (3.53)Maximal conversion for ε = 0 occurs when Ωt = gt = (2n+1)pi/2 for integer n. Conversely,minimal conversion occurs with gt = npi . This is identical in form to Rabbi flopping, meaningthe ion motion will change between magnetron and reduced cyclotron motion as a function ofg or t. This behaviour is demonstrated in figure 3.12, where the excitation amplitude, whichis proportional to g, was varied for the ion 39K+ with a fixed excitation time of 100 ms. Fig-ure 3.13 demonstrates the probability to be converted from a state of pure magnetron motionto pure reduced cyclotron motion as a function of the frequency detuning. The conversion lineshape is narrowest when g = pi/2, a feature making it the most interesting for experiment.We must now connect the quantum solution with the classical ion motion. In MPET, an iontypically has a few electron volts of energy, corresponding to quantum numbers of ≈ 109. Byconstructing coherent states of the magnetron and reduced cyclotron oscillators the classicalmotions [182] can be recovered [183]. Figure 3.14 shows the radial evolution of an ion subjectto a quadrupole excitation. During the excitation, the radius of the reduced cyclotron motionslowly grows, while the radius of the magnetron motion slowly decreases.663.3. TITANHaL-1.5-1.0- -1.0 -0.5 0.0 0.5 1.0 1.5x HmmLHbL-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5x HmmLFigure 3.14: Ion motion under a quadrupolar excitation. (a) t ∈ [0,τ/2], (b) t ∈ [τ/2,τ],where τ is the time required for one complete conversion. The ion is started on amagnetron radius of 1 mm, as denoted by the red circle.HaL-1.0- 0.2 0.4 0.6 0.8 1.0Time Harb.LFigure 3.15: Excitation pulses for (a) quadrupole and (b) Ramsey excitations. The blacklines show when the signal is applied, and the red line shows the sinusoidal ex-citation. The red dashed line shows that the excitation is phase coherent betweenthe first and second excitation pulses of the Ramsey scheme.673.3. TITAN0. -2 -1 0 1 2 3Ε HHzLFigure 3.16: Ramsey excitation line shape (black), compared with a quadrupole exci-tation (red). The Ramsey line shape uses an excitation scheme t1 − t0 − t2 of0.1−0.8−0.1 s, while the quadrupole excitation uses an excitation time of 1 s. Ramsey excitationInstead of applying the excitation field in one pulse, the excitation can be applied at two differ-ent times, allowing for a phase to accumulate between the RF-field and the ion motion. Thistime-separated oscillatory field technique [187, 183, 188] was first pioneered by Ramsey formolecular beams, for which he received the Nobel prize, and as such are called “Ramsey” ex-citations. The Ramsey method leads to an interference-type line shape, narrowing the centrallobe, thereby increasing the precision of the measurement. The line shape can be found by suc-cessive applications of M(ε,g, t), being careful to track the phase difference between the RFand ion motion. The excitation is split into two pulses, one of length t1, and the other of lengtht2, separated by a waiting period of length t0. For full conversion g(t1 + t2) = pi/2, which isthe same condition for single pulse quadrupole excitation length of t1+ t2. Figure 3.15 showsthe difference in the time structure between a single pulse and a Ramsey pulse. An impor-tant part of the Ramsey excitation is that the phase between the first and second RF pulses bephase coherent; otherwise, the maximum conversion will be shifted in frequency [183, 189].At TITAN, phase coherence is accomplished by using an RF switch to turn on and off the RFthat is applied to the trap, while keeping the RF function generator continually running. The683.3. TITANtransition probability isF2(ε,g, t2, t0, t1) = |M(ε,g, t2)M(ε,0, t0)M(ε,g, t1)|21,2 (3.54)F2(ε,g, t2, t0, t1) =g2Ω2(sin2(εt02)sin2 (Ω(t1− t2))+(cos(εt02)sin(Ω(t1+ t2))+ε2Ωsin(εt02)(cos(Ω(t1+ t2))− cos(Ω(t1− t2))))2), (3.55)which is found through successive applications of the propagation matrix M(ε,g, t). The lineshape is narrowest when t1 = t2, reducing toF2(ε,g, t2, t0, t1) =g2Ω2(cos(εt02)sin(2Ωt)+ε2Ωsin(εt02)(cos(2Ωt)−1))2. (3.56)If t0 = 0, the normal line shape (equation 3.53) is recovered. A Ramsey conversion line shapeis plotted in figure 3.16. The Ramsey technique increases the overall precision by a factorof 2-3, depending on the choice of excitation and waiting times, allowing for more precisemeasurements, or for the same precision as with the one pulse excitations but in a shorteramount of time. Dipole cleaningContaminant ions may be delivered simultaneously with the ion of interest. These contami-nants can potentially shift the measured cyclotron frequency in the trap through their mutualinteractions [190, 191]. By applying a dipole field to the trap at one of the eigenfrequencies it ispossible to remove unwanted ions from the trapping region [181]. The dipole field is created byapplying 180◦ out of phase signals to opposite electrodes, as shown in figure 3.11. The growthof the eigenmotion depends on the relative phase difference between the RF excitation and theion motion, but given a large enough excitation amplitude, and a long enough excitation pulse,the motion will grow linearly [192] until the ion is lost on the trap electrodes, either radiallyin the case of radial excitations, or ejected from the trap by axial excitations. At TITAN, thereduced cyclotron motion of unwanted ions is excited, causing the ions orbit to increase inradius. The magnetron motion could also be excited; however, the magnetron motion is nearlyconstant with mass, limiting the resolving power.693.3. TITAN3.3.4.5 Time-of-flight ion cyclotron resonanceIn order to measure the conversion line shape, the Time-of-Flight Ion Cyclotron Resonance(TOF-ICR) technique [182] is used at TITAN. By converting the magnetron motion to reducedcyclotron motion, a large gain in radial energy occurs (because ω+ >> ω−), and this gain inenergy can be measured in the TOF of an ion from the trap to a detector. The ion acts as acurrent-carrying loop, having a magnetic dipole moment depending on the radial energy of theion µ(ω) = E(ω)/B0, where ω is the frequency of the applied excitation. Extracting an ionthrough a magnetic field gradient creates a force on the magnetic dipole, changing the radialenergy into axial energy, and causing the time-of-flight to the detector to change depending onthe radial energy of the ion in the trap. If the electric and magnetic fields are known along theflight path to the detector, the TOF can be calculated asT =∫ z1z0(m2(E0−qV (z)−µ(ω)B(z)))1/2dz (3.57)where E0 is the initial total energy of the ion, V (z) and B(z) are the electric potential andmagnetic field strength along the z-axis from the trap at z0 to the detector at z1. The radialenergy of the ion after an excitation is dominated by the kinetic energy of the reduced cyclotronmotionE(ω)≈ 12mω2+F1(ωc−ω,g, t)ρ2−(0) (3.58)where ρ−(0) is the initial radius of the magnetron motion. The cyclotron frequency is the foundby scanning the excitation frequency ωr f and finding the minimum in the TOF distribution, asseen in figure 3.17 for 23Na+. Measuring the axial frequency of the Penning trapUsing TOF-ICR methods it is possible to directly determine the reduced cyclotron frequencyω+, and to infer the magnetron frequency using the relationship ω−=ωc−ω+. These methodsare not directly applicable to determining the axial frequency, since there is no convenient wayto couple the axial and radial modes. Moreover, if one could couple the axial and radial modes,the resonance would be “washed out” due to the axial motion of the ion in the trap. Instead,we can measure the phase evolution of the ions by varying the trapping time. By intentionallyclosing the trap at the incorrect time, we can create ions which have sizeable axial oscillations.Usually, the correct time to close the trap is the time when they are near the trap centre, suchthat they have the minimum possible axial energy. As the switch timings are well controlled,we can start the ions on the same axial phase on each injection cycle. The TOF to the detector703.3. TITANæææææææææææææææææææææææææææææææææææææææææ-30 -20 -10 0 10 20 304042444648505254ΝRF-2.4707904´106 HHzLTime-of-flightHΜsLFigure 3.17: TOF-ICR resonance of 23Na+ for an excitation time of 97 ms. The blue lineis a fit of the theoretical line shape [182].zUHzLFigure 3.18: Ion trajectories when being ejected. The ion on the left is already travellingto the right, so no additional turn-around time is required. The ion on the right istravelling to the left, and must turn-around before leaving the trap.713.3. TITANis thenT =∫ z1z0(m2(E0−qU(z)))1/2. (3.59)There is the added complication that the TOF depends on the axial phase of the motion – thevelocity vector of the ion matters. This is because the ion can be moving away from the exitwhen the trap is opened. To calculate the correct TOF, we must account for the “turn-around”time of the ion bunch. Because the potential in the z-direction is harmonic, we know thevelocity of the ion at the moment the trap is opened,E =12mA2ω2z cos2(ωzt+φ0) (3.60)where A is the amplitude of the axial oscillation, and φ0 is the initial phase of the ion. Tosimplify calculating the turn-around time, we assume that the trap electrodes switch instan-taneously, preserving the ion’s kinetic energy and spatial position, changing only the poten-tial energy. The position where the ion turns around is found by solving U(z) = E, with theturn-around time being twice the length of time for the ion to go from the starting positionAcos(ωzt + φ0) to the turn-around position. A drawing of extracting an ion from the trap isshown in figure 3.18. In the figure two ions are shown: an ion on the right of the trap movingtowards the left, and an ion on the left of the trap moving to the right. For the ion moving left, itcontinues travelling to the left until it turns around and can exit the trap, while the ion movingto the right can immediately leave the trap. The rest of the TOF to the detector is calculatedas normal. A measurement of the axial frequency is shown in figure 3.19. Each data point isthe average of 205 injection-ejection cycles, with the total trapping time varied between 1 and100 ms. The “kink” in the TOF, at ≈ 2 µs, occurs just before the ions turn around and begintravelling towards the exit. The longest TOF occurs just after turning around, when the ionsare heading away from the detector, while the fastest time of flight occurs when the ions arenear the centre of the trap, and are travelling towards the detector. This is expected, as the ionson the exit side of the trap have their potential energy reduced when the trap is opened. Ions onthe entrance side of the trap have more potential energy, leading to slightly higher velocities,and a shorter TOF. For 39K+, the axial frequency was measured to be 133508.24(18)Hz, witha fitted oscillation amplitude of 1.8(1) mm.This technique can also be used to minimize the axial oscillation amplitude of the ionbunch. Because of higher order components in the trapping potential, the axial and radialmodes in a Penning trap can couple, leading to a potential shift in the measured cyclotron fre-quency. By carefully eliminating axial oscillations, any frequency shifts related to the axial723.3. TITANæææææææææææææææHaL6869707172737475Time-of-flightHΜsL0.000 0.008 0.016æææææææææææææææHbL50.000 50.008 50.016Evolution Time HmsLæææææææææææææææHcL100.000 100.008 100.016Figure 3.19: Axial frequency measurement of 39K by evolving the axial phase of the ionbunch. The evolution times are (a) 0.001−0.015 ms, (b) 50.001−50.015 ms, (c)100.001−100.015 ms. The axial frequency was found to be 133508.24(18)Hz.motion can be eliminated. Further, large axial oscillations can “wash-out” the measured reso-nance, affecting the achievable statistical precision of the measurement. By varying both thetime when the trap closes and the incoming energy of the ion bunch, it is possible to eliminatenearly all axial oscillations [155].3.3.5 Technical setupThe field strength of the superconducting magnet is 3.7 T, which is nearly half of the averagefield strength of other on-line Penning trap spectrometers, most having field strengths rangingbetween 6–9.4 T [149, 193, 194]. To compensate for the low magnetic field strength, highlycharged ions can be used.To prepare the initial magnetron motion, a Lorentz Steerer (LS) is used [195]. This isdifferent from the usual method of dipole excitation at the magnetron frequency. Not only isthis dipole excitation time consuming, requiring excitation times of > 10 ms, but the RF phaseof the magnetron pulse must be locked to the ion capture time [196]. Long excitation times areneeded because the magnetron frequency is very low and the driving voltage is limited by theRF amplifier. The RF phase must be locked because ions cannot be injected with zero initialmagnetron motion and the magnetron radius after excitation depends on the phase difference733.3. TITAN-7.5-5.0- -5.0 -2.5 0.0 2.5 5.0 7.5x HmmLFigure 3.20: Lorentz Steerer schematic, with equipotentials [195] (black), and E-fielddirection lines (blue). Near the centre of the Lorentz steerer, the electric field isalmost entirely along the y-axis. The top electrode is at 1 V, the bottom electrodeis at −1 V, and the side electrodes are at 0 V.between the magnetron motion and RF pulse. The LS is located near the trap, and it is whollycontained in the strong magnetic field of the superconducting solenoid. The LS eliminatesthis preparation stage by starting the ion on an initial magnetron motion during injection, adevelopment allowing access to nuclides with half-lives below 50 ms. Among others, the LShas allowed TITAN to measure the mass of 11Li [67], having a half-life of only 8.8 ms, whichis the shortest lived nuclide measured in a Penning trap. When an ion passes through the LS itexperiences an ~E×~B field, causing the ion to drift off axis. Figure 3.20 illustrates the electricpotential and field direction for an ion inside the LS. Typically the LS voltages are set suchthat the ions are injected into MPET with magnetron radii of ≈ 1 mm. This is much smallerthan the 7.6 mm inner radius of the LS, meaning the electric field the ion experiences is nearlyunidirectional. The LS can also correct for off axis injection caused by ion optics upstream ofthe trap. By manipulating the voltages on the four electrodes the ion’s initial position can beaccurately set. For example, the phase of the magnetron motion can be controlled to within afew degrees [195].After passing through the LS, the ions must be pulsed down in energy to be captured in thetrap, as the transport energy of the ions is typically ≈ 2 keV, while the trap is held at ground743.3. TITANLS PLTV2 keV>100 eV>100 eV>0 eVFigure 3.21: Schematic of an ion’s energy during injection into MPET. The pulsed drifttube (PLT) is pulsed down when ions are passing through, removing nearly allof the transport energy. The ions then climb into the trap, and are captured bychanging the potential on the trap end-cap.potential. A schematic of this is shown in figure 3.21. A long pulsed drift-tube (PLT) acts asan ion energy elevator, removing enough kinetic energy so that the ion when trapped has, atmost, a few electron volts of energy. This is an important step, as excess axial energy may“wash-out” the resonance due to an increase in the TOF spread of the extracted ion beam.Optimal injection parameters are found by scanning both the capture time and the lower levelof the PLT. A general procedure for optimizing injection is discussed in [155]. However, if onewishes to measure the axial frequency, a slight change of the capture time from the optimalsetting induces axial oscillations. Once optimal settings are found, the timings for nuclidesnear-by in mass can be calculated by a simple scaling of the timings with the mass-to-chargeratio. This is particularly useful when the nuclide of interest is produced at very low rates, astoo much time would be needed to optimize the injection.For the electrode configurations, MPET uses hyperboloids of revolution for the end capand ring electrodes of the trap. To correct for higher order terms arising both from truncatingthese electrodes and from holes to allow for injection and ejection, so-called guard electrodescorrect for the electrode truncation while correction tubes correct for the holes in the end caps.The characteristic distances are r0 = 15 mm, z0 = 11.785 mm, and d0 = 11.21 mm. Figure 3.22shows a schematic of the real trap electrodes. Between the ring and end caps a potential differ-ence of 35.75 V is used, with the end caps set to 20 V and the ring set to−15.75 V. Determiningthe correct settings for the correction electrodes is a time consuming process, however, to findthe correct settings one can follow the method presented in [198, 155]. Following this proce-753.3. TITANM. BRODEUR et al. PHYSICAL REVIEW C 80, 044318 (2009)MCPMCPto EBITOff-line ion sourceFIG. 1. (Color online) The TITAN experimental setup thatincludes a RFQ, a high-precision Penning trap, an EBIT, an off-lineion source, and MCPs for beam diagnostics.A. The off-line ion source and radiofrequency quadrupoleA surface ionization source is located below the RFQ, asshown in Fig. 1. This ion source produces both 6Li and 7Lias well as smaller quantities of the alkali metals 23Na, 39K,41K, 85Rb, and 87Rb. The ion source is biased at the sameelectrostatic potential as the RFQ, which is 5 to 40 kV abovethe beam line ground potential. The ions are extracted fromthe source and then accelerated and focused toward the RFQ.The purpose of the RFQ is to accumulate, cool, and bunchthe continuous beam coming from either the TITAN off-lineion source or ISAC. It is similar to other RFQ’s at radioactivebeam facilities as described, for example, in Refs. [21] and[22].B. The mass measurement Penning trapPenning traps [23] are devices used to perform high-precision mass measurements on stable and exotic nuclei(see review [24]). TITAN’s Penning trap is located insidethe bore of the superconducting magnet, shown in Fig. 1.Prior to injection into the trap, the ions are moved off-axisusing a Lorentz steerer [25], similar to the one currently usedat the LEBIT facility [26]. The Lorentz steerer reduces thepreparation time in the Penning trap by inducing an initialradial displacement of the ion bunch prior to its capture, asopposed to using RF fields inside the trap, which is typicallythe case for on-line Penning trap experiments [27].Figure 2 shows a model of the hyperbolical electrodes of theTITAN’s high-precision Penning trap. It is composed of twohyperboloids of revolution forming one ring [label (1) in Fig. 2]and two end cap electrodes (2). The ions are axially trappedby a harmonic quadrupole electrostatic potential produced bya potential difference !Udc = 36 V between the ring and theBOut to MCP3214Corr. tube el.End cap el.Ring el.Corr. guard el.VRFIons in from RFQ or EBITFIG. 2. (Color online) Illustration of the TITAN Penning trapformed from the hyperbolic ring (1) and end cap electrodes (2) thatproduces the harmonic potential, tube (3), and guard (4) correctionelectrodes that produce the harmonic potential. The RF is appliedon (4).end cap electrodes, as shown in Fig. 2. Some anharmonicitiesin the trapping potential are introduced by the holes in theend cap electrodes and by the finite size of the hyperbolicelectrodes. Two sets of correction electrodes [labeled (3) and(4) in Fig. 2], are used to compensate for higher-order electricfield components. The radial confinement is provided by ahomogenous 3.7-T magnetic field produced by a persistent,actively shielded superconducting magnet. The linear decayof the magnetic field due to flux creep [28] depends on thepressure in the liquid helium vessel and during the 6Li-7Limeasurement it was measured to be (1/B)× (δB/δt) <0.25 ppb/h.The ion motion in a Penning trap is well understood [23,29]and is composed of three different eigenmotions: an axial mo-tion with frequency νz and two radial motions with frequenciesν±. In an ideal trap, the sum of the frequencies of the two radialmotions is equal to the true cyclotron frequency of the ion[30]:ν+ + ν− = νc = 12πq × Bmion, (1)where q/mion is the charge-to-mass ratio of the trapped ionand B is the magnetic field at the trap center. The two radialmotions can be coupled by applying an azimuthal quadrupolarRF signal on the sliced correction electrode [(4) in of the trapFig. 2]. The cyclotron frequency is determined using the time-of-flight (TOF) resonance detection technique [29,31,32].For a proper choice of the RF amplitude VRF, at a given RFexcitation time TRF, and when νRF = νc a full conversion ofthe initial magnetron motion into reduced cyclotron motionwill occurs. This leads to an increase in the kinetic energyof the trapped ions. After the excitation has been applied, theions are released from the trap and their TOF is recorded on anMCP located outside of the high magnetic field region. Due totheir larger kinetic energy gained during their excitation phasethe TOF of the ions with νRF = νc will be shorter. A cyclotronresonance curve is obtained by scanning the RF frequency in044318-21 CAD$coin Figure 3.33: Schematics and picture of TITAN’s MPET electrodes. The Cana-dian dollar coin (with a diameter f 26.5 mm) is given for scale (bot-tom). Figures from [146].PLT before it is pulsed down. In this configuration the ions lose most of their kineticenergy before entering the Lorentz steerer (see Figure 3.32). The Lorentz steereris composed of a segmented, cylindrical tube which creates a dipole field when bi-ased following Figure 3.34. The radial displacement is proportional to the Lorentzsteering strength VLS and the time it takes the ions to pass through the Lorentzsteerer (Equation 3.55). The latter time depends onm/Q since every ion species isaccelerated by the same electric potential difference after the thermalization in theRFQ or the EBIT. Since, t / pm/Q ion species with smaller m/Q ratios need astro ger Lorentz steering strength VLS to be positioned to the same initial mag-netron radius. Since for a full conversion the initial magnetron radius and the finalcyclotron motion are identical in size (see Equation 3.25) a larger ⇢0 will result ina larger gain in energy during the RF-excitation in the MPET. The maximal ⇢0 is122Figure 3.22: Schematic of MPET’s trap e ec odes. The guard electrodes are coloured tomatch the colour sc me in figure 3.11. The grey circles are sapphire balls thatare used to elec rically separate he trap electrodes. Figu e from [197].dure, the guard electrodes are set to 0.189 V, while the correction tubes are set to 28.17 V. Thisprocedure only needs to be completed once, as all parameters, including the trap geometry, arekept constant.After trapping and excitation, the ion is ej cted from the trap towards a detector with single-ion sensitivity. MPET uses two detectors, a Micro-Channel Plate (MCP) mounted in-line withthe trap and a Daly detector [199] mounted perpendicular to the optical axis [197]. MCP’s are≈ 40% efficient when detecting singly charged ions, while a Daly detector can have efficienciesof > 90%. When an ion impinges on an MCP, it releases electrons from low work functionmaterial in one channel of th MCP. These released electrons start a cascade, amplifying theinitial signal up to 106 times. If an ion hits in between channels, the probability for a cascade,and detection, is reduced. A Daly detector first impinges the ion beam on a material witha low work function, releasing several el ctrons for ach incoming ion. At TITAN, a plateof naturally anodized aluminium is used, releasing ≈ 3 electrons for incident ion en rg es of5 keV [200]. These released electrons are then accelerated t wards another MCP where theyare detected. Increasing the initial number of charged particles increases the likelihood that atleast one will be detected, leading to a large overall increase in the detection efficiency. TheDaly detector was used during the 20,21Mg experiment, and was found to be two-fold moreefficient than the on-axis MCP.763.3. TITAN3.3.6 Determining the massOnce an ion’s cyclotron frequency has been measured, the mass can be determined providedthe magnetic field is well known. This is not possible because the field strength is not constant.It varies with pressure and temperature, and it is slowly decaying due to residual resistance inthe superconducting coils. By taking the frequency ratio between two different species thesefluctuations can be largely eliminated. In the following, we call one ion the “ion of interest”and the other the “calibrant” or “reference” ion. The frequency ratio R is defined to beR =νc,refνc=qrefqmmref. (3.61)where m is the mass of the ion. To obtain the atomic mass, we must correct for the missingelectrons and their binding energyM =qqrefR(Mref−qrefme+Be,ref)+qme−Be (3.62)where M is the atomic mass of the species, me is the mass of the electron, and Be is the totalbinding energy of the missing electrons. The electron binding energies for singly chargedions are usually quite small, having values between ≈ 5− 10 eV. In most measurements thestatistical uncertainty (≥ 100 eV) dominates, so the binding energy can be ignored. Quiteoften, atomic masses are reported in short-hand notation as mass excesses ME = M−A · u,where u is the atomic mass unit, defined such that the mass of 12C is exactly 12 u. The massexcess removes the bulk of the mass that comes from the constituent protons and neutrons,enabling a clearer picture of the differences in binding energy between isobaric nuclides.While the simple frequency ratio above can largely calibrate the magnetic field, there re-mains the issue that the two measurements are not performed at the same time. This can becorrected for by performing two reference measurements, one before and one after the mea-surement of the ion of interest. The reference frequency can be linearly interpolated to the timeof the measurement of the ion of interest, removing magnetic field instabilities that are linear intime. Not only does the frequency ratio eliminate magnetic field fluctuations, it also eliminatesmany other systematic effects.3.3.7 Systematic shiftsDue to many differing effects, the measured cyclotron frequency may be shifted from the truecyclotron frequency. Here we will examine these potential shifts, and assign upper limits ontheir size. In nearly all cases we will find that the systematic effects are much smaller than773.3. TITANthe desired measurement precision. We parametrize the shift in the frequency ratio ∆R/R bycalculating the difference between the measured ratio and the ideal ratio∆RR=Rmeasured−RidealRideal. (3.63)One can see that systematic shift will cancel to high order through this procedure. Assume thatthe ion of interest and the reference ions have different systematic shifts δνc and δνc,ref. Themeasured frequency ratio would beRmeasured =νc,ref+δνc,refνc+δνc=νc,refνc· 1+δνc,ref/νc,ref1+δνc/νc≈ Rideal(1+δνc,refνc,ref− δνcνc)(3.64)If both ions have the same m/q, then the systematic shifts will be close in size, causing thesystematics to largely cancel. Thus, we can expect the systematic shifts to be small. To simplifycalculations, the shifts are expressed in terms of ∆(m/q), where m is expressed in atomic massunits, and q is the charge state, or number of removed electrons. Suitable references can alwaysbe found, so ∆(m/q) of 2 or 3 is typical, however, the difference can be as large as 10 or more. Relativistic effectAn ion’s velocity in the reduced cyclotron mode is v+ = ω+ρ+, a value depending on themagnetic field strength and radius of the motion. For example in MPET, for a 39K+ ion on a1 mm orbit, the velocity is v/c ≈ 3 · 10−5, resulting in a relativistic correction factor γ − 1 of4.6 ·10−10. The relativistic cyclotron frequency is given byωc =qBγm, (3.65)resulting in a measured frequency shift upwards of 0.4 ppb. For heavy SCI, like 39K+, therelativistic effect is evidently quite small, and can be neglected. For light SCI or HCI, therelativistic shift can be quite large (several ppb), and must must be corrected for [201].In the measurements presented here, the relativistic shift can be neglected because the pre-cision of all measurements were > 80 ppb. Spatial magnetic field inhomogeneitiesCare was taken during the construction of the magnet to ensure the magnetic field was homo-geneous in the trapping region, but some inhomogeneities still exist due to the finite size of thesolenoid, and by inhomogeneities caused by the material used to construct the trap, vacuum783.3. TITANvessel, etc. The frequency shift is given by [202]∆νc = β2((z2−ρ2+)− ν−νc(ρ2++ρ2−))(3.66)where β2 is the quadrupole coefficient of the magnetic field inhomogeneity, and z is the am-plitude of the axial motion. For TITAN the upper limit on the shift in the frequency ratiois [198]∆RR< 4.3 ·10−10∆(m/q). (3.67) Non-harmonic imperfections of the trapping potentialThe electric potential of the trap is not a pure quadrupole field, because higher order termsarise both from truncating the trap electrode surfaces, and from the holes in the end caps forinjecting and ejecting ions. These higher order terms are corrected for by adding electrodesto compensate these finite size effects. “Guard” electrodes are added between the end cap andring electrodes to correct for the electrode truncation, while “tube” electrodes are added nearthe end caps to correct for the injection and ejection holes. These electrodes are shown infigure 3.22. A general procedure to minimize these non-harmonic potentials was developed in[155] to determine the optimal trap settings. The frequency shift is less than∆RR< 3.6 ·10−10∆(m/q). (3.68) Harmonic distortion and magnetic field misalignmentPrecision machining and setting of the trap electrodes is a difficult procedure. Any distortionof the ring electrode from cylindrical symmetry can lead to a frequency shift. This distortion isparametrized by an ellipticity factor η . Further, the trap axis may be at an angle θ with respectto the magnetic field axis. Both of these misalignments lead to a frequency shift of [203]∆νc =(94θ 2− 12η2)ν−. (3.69)The shift in the frequency ratio is then given by [203]∆RR=(94θ 2− 12η2)(∆AAref)(ν−ν+,ref)(3.70)793.3. TITANwhere A is the mass number. The shift can be estimated by considering what the maximummachining tolerances in the components holding the trap together are. The largest potentialshift arises from the maximal tolerances in the sapphire balls that separate the trap electrodes(see figure 3.22). The maximum angle is then estimated to be θ ≈ 4.2 · 10−3 rad [198]. Theshift could be of the order∆RR< 4.3 ·10−9∆(m/q). (3.71)This is certainly a very conservative estimate, since it is unlikely that sapphire balls withopposite tolerances would be placed to give the maximal deviation. The alignment and dis-tortion parameters can be measured through a specific combination of the eigenfrequencies[203] (94θ 2− 12η2)≈ 2ω−ω+ω2z−1. (3.72)Using the measured eigenfrequencies in table 3.2, and using a conservative error of 0.2 Hz forthe magnetron frequency, the frequency ratio shift becomes∆RR=−0.6(17) ·10−9∆(m/q). (3.73)Again, shifts on the order of 1 ppb are much smaller than the precision usually measured in anon-line experiment. Ion-ion interactionsIdeally only a single ion would be trapped at a time, but quite often multiple ions, either ofthe same species or of a contaminant species, will be trapped simultaneously. These additionalcharges not only modify the potential inside the trap, but they also interact with each otherthrough their mutual Coulomb interactions. Through these interactions, the observed eigen-frequencies may be shifted, leading to a systematic shift in the measured cyclotron frequency.These shifts have been observed, and they vary linearly with the number density, assumingthe same charge state [190]. Other shifts arise due to the simultaneous trapping of differentspecies [191].To correct for these shifts, one can determine the cyclotron frequency as a function of thenumber of detected ions and extrapolating to the detector efficiency, correcting for both theion-ion interaction and the efficiency of the detector. Such an analysis is called a “count-class” analysis [204], and it can be applied when a total of greater than ≈ 1000 ions havebeen collected in a resonance spectrum. Trap extractions are divided into classes based on thenumber of ions detected. A typical count-class analysis for 23Na with 4 count-classes is shown803.3. TITANææææ-0.30-0.25-0.20-0.15-0.10-Ν-2470790.5HHzL0 1 2 3 4 5Count ClassFigure 3.23: Count class analysis for 23Na with four count classes. The dashed lines arethe ±1σ line fits, while the filled area show the error band when the count classis extrapolated to the detector efficiency. A typical detector efficiency of 0.6 figure 3.23. The classes are divided in such a way that each combined class has as close toan equal number of ions as possible. In figure 3.23, the first, second, and third classes containextraction events where 1, 2, or 3 ions were detected, while the fourth class contains all eventswith 4 or more detected ions. The position of the class on the x-axis is taken to be the centreof gravity of the class. In the fourth class of figure 3.23, it is close to 4 because most events inthat class have 4 detected ions. A linear fit is done to the count-class data, and is extrapolatedto the detector efficiency. In this way the cyclotron frequency when one ion is in the trap canbe extracted. In figure 3.23, the solid blue line is a linear fit to the count-class data, while thedotted blue lines show the±1σ error bands. In cases where statistics are too low, the differencebetween the analysis with only one detected ion to that of an analysis with all detected ions canbe used. The difference is taken to be the systematic error.In the present measurements, a count-class analysis is done for each measured isotope. Non-linear magnetic field fluctuationsAs mentioned at the beginning of this section, the cyclotron frequency of both the ion of interestand the calibrant ion are measured. However with TITAN, it is not possible to measure the813.3. TITANfrequency of both ions simultaneously. In the time between measurements, the magnetic fieldmay decay, or otherwise fluctuate, in a non-linear manner, the result of which would be asystematic shift of the measured mass. In order to minimize these potential shifts, the timebetween reference measurements is usually kept below one hour. The effect of changing thetime between the reference measurements was determined to be δv/v = 0.04(11)ppb/h [201].This is below the sensitivity of the present measurements, and is not included in the analysisherein.82Chapter 4Results and discussionIn this chapter we discuss the results of the mass measurements of 51,52Ca, 51K, and 20,21Mg.The Ca and K mass values, along with the recent measurements by ISOLTRAP [59], will becompared to existing phenomenological interactions and interactions based on χEFT, with theaim of elucidating the ground state structure near the N = 34 shell closure. The Mg mass valueswill be used to test the IMME in the A = 20 and 21 isotopic chains. This will be comparedto the IMME calculated with the USDA/B interactions, and χEFT based calculations. TheχEFT calculations are particularly interesting, as they are the first χEFT based calculations toincluded both active neutrons and protons in the valence space.The measurements were completed in three separate experiments. First, beams of 51,52Cawere produced by bombarding a Ta target with 75µA of 480 MeV protons, and ionized withTRILIS. Second, the 51K beam was made with a UCx target with 1.4µA of protons, with asurface ion source. Third, beams of 20,21Mg were produced by bombarding a SiC target with40µA of protons, and were ionized using IG-LIS. Yields are presented in table 4.1.Table 4.1: Ion yields [205] for 51,52Ca, 51K and 20,21Mg.Species T1/2 Yield (ions/s)51Ca 10.0 s 1.4 ·10452Ca 4.6 s 1.3 ·10351K 365 ms ≈ 7520Mg 90.8 ms 5021Mg 122 ms 2.7 ·103834.1. EXISTING DATA4.1 Existing data4.1.1 51CaCreating beams of neutron rich Ca isotopes has been a challenge for rare-beam facilities. Be-cause of this, mass measurements in this region have generally relied on multi-nucleon trans-fer reactions. The mass of 51Ca, as of the Atomic Mass Evaluation 2003 (AME03) [206],is derived from three-neutron-transfer reactions, using beams of 14C or 18O on a 48Ca target[207, 208, 209, 210]. Additionally, two TOF mass measurements of 51Ca agree with each other[211, 212], but disagree at the ≈ 1σ level with the reaction based experiments. A more recentmeasurement at GSI using the fragment separator and experimental storage ring (FRS-ESR)[213], agrees with the TOF measurements, but is in strong disagreement with the reactionvalues. Figure 4.1 summarizes these results, with the TITAN value for comparison.First, we will examine the reaction based measurements. Three-nucleon transfer reactionsare quite complicated, due to the possibility of multiple steps during the transfer, leading to lowcross-sections, and resulting in low statistics. Further, due to the possibility of contaminantsin the target material, it is possible that observed states in the outgoing reaction channel maybe misidentified. As all four reaction experiments used enriched 48Ca targets, the residual40Ca contamination resulted in large backgrounds. Other target contaminants include 16O and14C. In reactions on these contaminants, the outgoing projectile-like particle (15O or 11C)have energies that are close to the ground state energy of the reaction on 48Ca, however, thespectrometers used in these reactions were capable of separating these reactions. Becauseof this, it is unlikely that a peak was misidentified. Another potential source of error in thesereactions is false calibration of the reaction spectrometer. In order to calibrate the spectrometer,reactions on well known targets are performed. This further eliminates the possibility that acontaminant reaction peak was identified as belonging to the reaction of interest.Beyond the general disagreement of the reaction experiments on the ground state mass of51Ca, none of the experiments agree on the energies of the excited states. Two recent measure-ments also corroborate the conclusion that these early multi-nucleon transfer reactions iden-tified the wrong state as the ground state. One measurement used deep inelastic collisions of238U on a target of 48Ca [214], while the other used the β -decay of 51K and the β −n decay of52K [215]. These measurements agree with each other, but disagree with the values extractedfrom the above reaction-based experiments. For example, in [210] the lowest excited state wasfound to be 1.01(11)MeV, while in the two recent measurements the lowest excited state wasfound to be 1.72 MeV. Further, the energy levels from the recent experiments agree well with844.1. EXISTING DATAææææææææ48CaH14C,11CL51Ca48CaH18O,15OL51CaTOFGSI TITAN51Ca-37-36-35-34MEHMeVL1 2 3 4 5 6 7 8MeasurementFigure 4.1: All 51Ca mass measurements, as compared to the TITAN value. Only tworeaction based measurements agree with each other, while the other two are in greatdisagreement. The TOF measurements agree with each other, and are in slightdisagreement with the reaction measurements. The red band shows the AME03 [85]value. The TITAN value is shown for comparison.calculations done with both KB3G and GXPF1A.4.1.2 52CaIn the case of 52Ca, only two prior mass measurements exist and are included in the AtomicMass Evaluation 2003 [206]. The first measurement comes from a β -end point measure-ment [216], while the second comes from a TOF measurement [211]. The measurementsstrongly disagree with each other. The mass excess from the β -decay measurement is−35.75(32)MeV, while the TOF measurement is−32.5(5)MeV. In the AME2003, the evalua-tors chose to disregard the β -end point measurement, taking the value of the TOF measurement,and slightly inflating the error bar. The end-point measurement also determined the Q-value ofthe 52Sc→52 Ti, allowing for a determination of the mass excess of 52Ca. However, if any ofthe intermediate measurements were wrong, the value for 52Ca would suffer large systematicshifts. Figure 4.2 shows the previous measurements together with our TITAN value.854.1. EXISTING DATAæææΒ-end pointTOFTITAN52Ca-36-35-34-33-32MEHMeVL1 2 3MeasurementFigure 4.2: All 52Ca mass measurements. All measurements disagree. The red bandshows the AME03 [85] value.4.1.3 53,54CaThe masses of 53,54Ca were measured for the first time using the multi-reflection time-of-flightdevice at ISOLTRAP [59]. ISOLTRAP is a multi-trap experiment at the ISOLDE/CERN facil-ity, which employs a Penning trap and a recently added MR-TOF system for isobar separationand mass measurement. The measured mass excesses are −229387.8(43.3) keV for 53Ca and−225161.0(48.6) keV for 54Ca. ISOLTRAP also measured the masses of 51,52Ca in a Penningtrap and the mass of 52Ca with the multi-reflection device. In each case, the masses agree wellwith the values measured with TITAN. This lends credence to the accuracy of the mass valuesfor 53,54Ca.4.1.4 51KThe mass of 51K has not been measured prior to the present measurements. The mass valuetabulated in the AME03 is based on observed trends in the mass surface [206]. In general,the mass surface varies slowly and regularly as a function of N and Z. Rapid changes in thisregularity signals changes in structure, structure such as new sub-shell closures or deformation.Quite often, these predictions are accurate, agreeing well with new experimental data [15].864.2. DISCUSSION AND MEASUREMENTS FROM THIS STUDYThe TITAN mass excess of −22516(13) keV agrees at the 1σ level with the AME03 value of−22000(500) keV.4.1.5 20,21MgPrior to our TITAN mass measurements, the mass of 20Mg was measured using the24Mg(4He,8 He)20Mg reaction [217, 218], while the mass of 21Mg was measured usingthe 24Mg(3He,6 He)21Mg reaction [219, 220]. For 20Mg the measured Q-values were -60900(210) and -60677(27) keV, and for 21Mg the measured Q-values were -27488(40) and-27512(18) keV. In each case, the measurements are in good agreement with each other. Isospin multiplet energy levelsDetermining the energy level of an isospin multiplet member relies on knowing both theground-state and excited state energies accurately. Except in the cases discussed below, the ex-citation energies will be taken from the National Nuclear Data Center [114], while the ground-state masses will be taken from the AME2012 [69]. The energy of the Jpi = 0, T = 2 state in20Na depends on knowing the proton separation energy. Recent measurements of the ground-states of 20Na and 19Ne, led to an improved proton separation energy of 2190.1(11) keV. Com-bining this with a new excitation energy measurement with the value compiled in [221], leadsto an average value of 6524.0(98) keV, a value that is shifted by 10 keV as compared to the tab-ulated value [114]. In 21Mg, a new measurement of the Jpi = 1/2+ state was completed [222]which, when averaged with the NNDC [114] value, yields 200.5(28) keV.4.2 Discussion and measurements from this studyTo determine the atomic masses presented here, we follow the procedure described in sec-tion 3.3.6. For 51Ca, 51V was used as a reference, for 52Ca, 58Ni and 52Cr were used asreferences in two separate measurements, while for 20,21Mg, 23Na was used as a reference.In all cases, the mass of the reference is much better known than the precision achieved inthe experiment. Because high precision was not required in these measurements, the standardone-pulse quadrupole excitation was used. Further, the measurements were completed withsingly charged ions, as the gain in precision from charge breeding were not needed. The Caand K measurements will be compared to the values from AME03, as previous measurementsin [223], and the measurements presented here, dominate the data in both AME11 and AME12.In order to correct for the ion-ion interaction arising from potential contaminants simulta-neously trapped with the ion of interest, a “count-class” analysis [204] (section was874.2. DISCUSSION AND MEASUREMENTS FROM THIS STUDYconducted. In the case of the 51,52Ca and 51K measurements, large amounts of contamina-tion from V and Cr were observed. This contamination was removed from the trap via dipolecleaning.For the 20,21Mg measurements a count-class analysis was not strictly needed, because theIG-LIS blocked nearly all contaminants from being ionized. However, to be conservative acount-class analysis was done, and the corresponding errors were folded into the total un-certainty. Enough statistics were collected so that the count-class error was included in thestatistical analysis for 21Mg, while for 20Mg the difference between the analyses with one de-tected ion and with all detected ions yielded a systematic error of 38 ppb. Because these resultsare of a higher precision than the Ca and K measurements we also include the conservativeharmonic distortion and magnetic field misalignment shift of 4.3∆(m/q) ppb, resulting in sys-tematic errors of 9.6 ppb for 21Mg and 12.9 ppb for 20Mg.4.2.1 Calcium and Potassium at N = 32Typical resonances for 51,52Ca are shown in figure 4.3. For 51Ca, the measured mass excessof −36339(23) keV is in disagreement with the AME03 value of −35863(94) keV, differingby 476(97) keV. As shown in figure 4.1, this result is in agreement with the TOF based mea-surements, but is in strong disagreement with the 4 reaction based measurements. Recently,ISOLTRAP also measured the mass of 51Ca [59], obtaining a value of −36332.07(58) keV,which agrees with our measurement.For 52Ca, the measured mass excess of −34245(61) keV disagrees with the AME03 valueof−32509(699) keV. The TITAN mass value is 1.74 MeV more bound than the AME03 value.This is comparable to the deuteron’s binding energy of 2.22 MeV. ISOLTRAP has also mea-sured 52Ca, obtaining a value of −34266.02(71) keV [59], which also agrees will with theTITAN measurement.These measurements, combined with the measurement of 51K, create much more bindingleading up to the sub-shell closure at N = 32. This is quite significant, and is in line with theobserved high excitation energy of the E(2+) state in 52Ca [118]. Figure 4.4 shows the S2nvalues tabulated in AME03, the TITAN values and the recent ISOLTRAP values. The valuesfor 50Ca and 48−50K are from a previous TITAN measurement campaign [223]. The TITANand ISOLTRAP mass values clearly show a sub-shell closure at N = 32, using the previousexplained signature for the behaviour.Next, we examine if theory is able to reproduce these results. Figure 4.5 shows the S2n en-ergies for the calcium chain, while figure 4.6 shows the difference between the experimentallymeasured and theoretical values. The NN+3N(MBPT) was calculated in the extended p f g9/2884.2. DISCUSSION AND MEASUREMENTS FROM THIS STUDYæææææææææææææææææææææææææææææææææææææææææ51CaΝcent = 1114621HaL58606264666870æææææææææææææææææææææ52CaΝcent = 1093124HbL656871747780Time-of-flightHΜsLæææææææææææææææææææææææææææææææææææææææææ51KΝcent = 1114298HcL727680848892-20 -10 0 10 20Ν - Νcent HHzLFigure 4.3: Typical TOF-ICR resonances (as in figure 3.17) of 51,52Ca and 51K. The blueline is a fit of the theoretical line shape [182].894.2. DISCUSSION AND MEASUREMENTS FROM THIS STUDYæææææææàààààààìììììììææææàà47K48Ca49ScAME03TITANISOLTRAP2500500075001000012500150001750020000S2nHkeVL26 28 30 32 34Neutron Number NFigure 4.4: S2n for the Ca, K and Sc isotopic chains near N = 32. Points in black arethose tabulated in AME03, the red points are the TITAN measurements, and theblue points show the recent ISOLTRAP measurements [59]. The values for 50Caand 48−50K are from a previous TITAN measurement campaign [223].æææææææææææAME03TITANISOLTRAPGXPF1AKB3GNN+3N HMBPTLNN+3N HCCL500075001000012500150001750020000S2nHkeVL28 30 32 34Neutron Number NFigure 4.5: S2n energies for Ca comparing theory to the TITAN and ISOLTRAP values.Calculations with the GXPF1A and NN+3N(MBPT) agree well with the experi-mental data.904.2. DISCUSSION AND MEASUREMENTS FROM THIS STUDYNN+3NHMBPTLNN+3NHCCLGXPF1AKB3G-2000-1000010002000S2nHTheoryL-S2nHExptLHkeVL30 31 32 33 34Neutron Number NFigure 4.6: Difference between calculated and experimental S2n energies. TheNN+3N(MBPT) calculation agrees well with the experimental values.valence space, on top of a closed 40Ca core. The NN forces are included at next-to-next-to-next-to leading order (N3LO), while the 3N forces are included at N2LO. For the 3N interaction,the short-range coupling constants were fit to the binding energy of 3H and the charge radiusof 4He [224]. The dominant component of the 3N interaction amongst the valence neutronsis due to the long range, two-pion exchange component of the 3N force [58, 119]. In the CCcalculation, the chiral NN interaction was included at N2LO, while a schematic 3N interac-tion was included by integrating one nucleon in the leading order 3N force over the Fermimomentum in symmetric nuclear matter [120]. The short range couplings were adjusted toreproduce the binding energies of 48,52Ca. The calculation is done using the CCSD approxima-tion, and includes 3-particle-3-hole excitations perturbatively within the Λ-CCSD(T) approach[225]. The NN+3N(MBPT) calculation reproduces the experimental values quite well, whilethe GXPF1A calculation is in fair agreement. The CC and KB3G calculations have much largerdeviations, with both the CC and KB3G calculations consistently underbinding, as comparedto experiment. As already shown in figure 2.7, both the NN+3N(MBPT) and CC calculationsare able to reproduce the measured E(2+) in 54Ca, while the phenomenological interactionsGXPF1A and KB3G do not. Mass measurements thus provide an alternative way of differen-tiating between the models. Here, the calculations using NN+3N(MBPT) seems to provide abetter description than the CC calculations or the phenomenological interactions.914.2.DISCUSSIONANDMEASUREMENTSFROMTHISSTUDYTable 4.2: Measured mass values for 51,52Ca, 51K, and 20,21Mg compared with the atomic mass evaluation [85, 69]. The Caand K values are compared to AME03, while the Mg values are compared to AME12.Nuclide Reference T1/2 TRF (ms) Frequency Ratio r ME (keV) MEAME (keV) ∆ME (keV)51Ca 58Ni 10.0 (8) s 77 0.87961718 (42) −36339. (23) −35863. (94) 476. (97)52Ca 58Ni 4.6 (3) s 77 0.89691649 (187) −34260. (101) −32509. (699) 1750. (700)52Ca 52Cr 4.6 (3) s 77 1.00043782 (158) −34236. (76) −32509. (699) 1730. (700)52Ca average −34245. (61) −32509. (699) 1740. (700)51K 51V 365 (5) ms 77 1.00062561 (28) −22516. (13) −22002. (503) 510. (500)20Mg 23Mg 90 (6) ms 97 0.870765248 (87) 17477.7 (18) 17559. (27) 81. (27)21Mg 23Mg 122 (2) ms 97 0.913956913 (35) 10903.85 (74) 10914. (16) 10. (16)924.2. DISCUSSION AND MEASUREMENTS FROM THIS STUDYæææææææææææææææææææææææææææææææææææææææææ21MgΝcent = 2703399HaL3638404244464850Time-of-flightHΜsL-20 -10 0 10 20Ν - Νcent HHzLæ æ æææ ææææææææææææææææ20MgΝcent = 2837494HbL-30 -20 -10 0 10 20 30Ν - Νcent HHzLFigure 4.7: Typical TOF-ICR resonances (as in figure 3.17) for 20,21Mg. The blue line isa fit of the theoretical line shape [182].4.2.2 A = 20, 21 isobaric multiplet mass equationResonances for 20,21Mg are shown in figure 4.7. The measured mass excess of10903.85(74) keV for 21Mg agrees with the tabulated AME12 value of 10914(16) keV, how-ever, the measured uncertainty has been improved by over an order of magnitude. The mea-sured mass excess of 17477.7(18) keV for 20Mg disagrees with the tabulated AME12 value of17559(27) keV at the 3σ level, with the uncertainty being improved by an order of magnitude.The measured values are summarized in table 4.2.Table 4.3 summarizes the fit results of the quadratic and quartic forms of the IMME for theA = 20 and 21 multiplets. For each multiplet the χ2 of the fit increased as compared to thevalues tabulated in [129]. For the A = 20 multiplet, nearly all of the uncertainty now resides inthe excitation energy of the T = 2 state in 20Na. The χ2 of the quadratic fit increased from 1.1to 10.2, an increase of nearly an order of magnitude. The best fit is obtained with the cubic fit,Table 4.3: Extracted IMME parameters for the A = 20 and 21 multiplets. Mass excessesare taken from [69] and excitation energies Ex from [114] and [226], except wherenoted. Also shown are the d and e coefficients for cubic and quartic fits and theχ2 values of the fit. Shell model calculation results using the USDA/B plus INCinteractions are presented.934.2. DISCUSSION AND MEASUREMENTS FROM THIS STUDYTable 4.3: Continued from previous page.Nuclide Tz ME(g.s.) (keV) Ex (keV)A = 20, Jpi = 0+, T = 220O +2 3796.17 (89) 0.020F +1 -17.45 (3) 6519.0 (30)20Ne 0 -7041.9306 (16) 16732.9 (27)20Na -1 6850.6 (11) 6524.0 (97) a20Mg -2 17477.7 (18) b 0.0Ref. a (keV) b (keV) c (keV) χ2This Work 9689.79 (22) -3420.57 (50) 236.83 (61) 10.2Ref. [129] 9693 (2) -3438 (4) 245 (2) 1.1Fit d (keV) e (keV) χ2Cubic 2.8 (11) - 3.7Quartic Only - 0.89 (12) 9.9Quartic 5.4 (17) −3.5 (18) -USDA −0.1 -USDA - −1.7USDB −0.1 -A = 21, Jpi = 5/2+, T = 3/221F +3/2 -47.6 (18) 0.021Ne +1/2 -5731.78 (4) 8859.2 (14)21Na -1/2 -2184.6 (3) 8976.0 (20)21Mg -3/2 10903.85 (74) b 0.0Ref. a (keV) b (keV) c (keV) χ2This Work 4898.4 (13) -3651.36 (63) 235.00 (77) 28.0Ref. [129] 4894 (1) -3662 (2) 243 (2) 3.0Fit d (keV) χ2Cubic 6.7 (13) -USDA −0.3USDB 0.3A = 21, Jpi = 1/2+, T = 3/221F +3/2 -47.6 (18) 279.93 (6)21Ne +1/2 -5731.78 (4) 9148.9 (16)21Na -1/2 -2184.6 (3) 9217.0 (20)21Mg -3/2 10903.85 (74) b 200.5 (28) cRef. a (keV) b (keV) c (keV) χ2This Work 5170.4 (14) -3633.6 (10) 220.9 (10) 9.7Ref. [129] 5171 (10) -3617 (2) 217 (2) 3.5Fit d (keV) χ2Cubic −4.4 (14) -USDA −1.2USDB 1.9aAverage of Refs. [227, 221]bPresent workcAverage of Refs. [226, 222]944.2. DISCUSSION AND MEASUREMENTS FROM THIS STUDYæææææAMEHaL-1001020304050FitResidualHkeVL-2 -1 0 1 2TzæææææTITANHbL-2 -1 0 1 2TzFigure 4.8: A = 20 Jpi = 0+ T = 2 quadratic residuals for (a) the AME2012 [69] massvalues, and (b) using the TITAN mass value for 20Mg. The large error bar for 20Na(Tz =−1) is due to the uncertainty in the excitation energy. Excitated state energiesare listed in table 4.3resulting in d = 2.8(11) keV and a χ2 of 3.7. A quadratic only fit results in e = 0.89(12) keV,and a χ2 of 9.9.For A = 21 there are two T = 3/2 isobaric multiplets, a ground state multiplet with Jpi =5/2+ and an excited state multiplet with Jpi = 1/2+. For the Jpi = 5/2+ multiplet, the χ2 ofthe quadratic fit increased from 3 to 28.0, an increase of nearly an order of magnitude. Forthe Jpi = 1/2+ multiplet, the χ2 of the quadratic fit increased from 3.5 to 9.7. According tothis, the IMME has failed in both instances. Large cubic terms are required for both multiplets,taking the values d = 6.7(13) kev for Jpi = 5/2+ and d =−4.4(14) keV for Jpi = 1/2+.To test the role of 3N forces in these nuclei, the values for the IMME were calculatedusing both the phenomenological interactions USDA/B supplemented with an isospin non-conserving (INC) Hamiltonian of reference [228] and the NN+3N χEFT valence space inter-action. The results for the USDA/B d and e coefficients are presented in table 4.3. For A = 20in the USDA, the e term comes from mixing of states of similar energy but different isospin in20F, 20Ne, and 20Na. The largest mixing comes from a pair of close by T = 0, 2 states in 20Ne.The largest mixing for a single level stems from a Jpi = 0+, T = 0 state in 20Ne that is 641 keVabove the T = 2 state. The INC mixing matrix element of 49 keV pushes the T = 2 state downby 3.8 keV, resulting in a quintic coefficient of e =−1.7 keV. With the USDB, these states arenearly degenerate, resulting in an uncertainty that is too large to give a meaningful result. With954.2. DISCUSSION AND MEASUREMENTS FROM THIS STUDYæææ æAMEHaL-50-40-30-20-10010FitResidualHkeVL-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5TzææææTITANHbL-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5TzFigure 4.9: Ground state A = 21 Jpi = 1/2+ T = 3/2 quadratic residuals for (a) theAME2012 [69] mass values, and (b) using the TITAN mass value for 21Mg. Exci-tated state energies are listed in table 4.3æææ æAMEHaL01020304050FitResidualHkeVL-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5TzææææTITANHbL-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5TzFigure 4.10: Excited state A = 21, Jpi = 1/2+ T = 3/2 quadratic residuals for (a) theAME2012 [69] mass values, and (b) using the TITAN mass value for 21Mg. Ex-citated state energies are listed in table 4.3.964.2. DISCUSSION AND MEASUREMENTS FROM THIS STUDYan INC mixing matrix element of 49 keV, a T = 0 level 350 keV above the T = 2 state wouldreproduce the experimental e value. There is a state 657(15) keV above the T = 2 state withunknown spin that could reproduce the experimental e value if the INC matrix element was≈ 70 keV.The calculated d term for A = 20 comes from mixing in 20F and 20Na, however, the T = 2states are well separated from nearby T = 1 states, resulting in a small shift and a too smalld term. The d-term for both the USDA and USDB are −0.1 keV. Experimentally, there aremany T = 1 states with unknown spin near the Jpi = 0+, T = 2 state. If one of these states wasan intruder (p- to sd-shell excitation) 0+ state there could be enough isospin mixing to give alarge d value.The results for the A = 21 d-coefficients for the J = 5/2+ multiplet are −0.3 keV withthe USDA and 0.3 keV with the USDB, while for the J = 1/2+ multiplet are 1.2 keV withthe USDA and 1.9 keV with the USDB. These do not agree with experiment. These non-zerovalues come from mixing with nearby T = 1/2 states in 21Ne and 21Na that can be interpretedin terms of a two-level repulsion due to the INC Hamiltonian, as outlined in section 2.5.1.For the J = 5/2+ state the largest two-level shift is due to a T = 1/2, J = 5/2+ state in 21Ne,which in the USDA is 372 keV below the T = 3/2 isobaric analogue state. The INC mixingmatrix element is 25 keV, pushing the T = 3/2 level up by 1.6 keV, contributing +0.8 keV tothe d-coefficient. In order for this single state to give +6.7 keV for d it would have to lie about50 keV below the J = 5/2+, T = 3/2 isobaric analogue state. There are experimental levelsthat lie 10, 58 and 77 keV below the isobaric analogue state with unknown spins [114], whichmay contribute to the observed d-coefficient. Further experimental investigation is required todetermine the spins of these states, which would shed light on the large measured d-term.For the J = 1/2+ state the largest two-level shift is due to a T = 1/2, J = 1/2+ state in21Ne that in the USDA is 246 keV above the T = 3/2, J = 1/2+ isobaric analogue state. TheINC mixing matrix element is 27 keV, pushing the J = 1/2+, T = 3/2 level down by 3.0 keV,and giving a contribution of 1.5 keV to the d-coefficient. In order for this single level to gived = −4.4 keV it would have to lie about 100 keV below the J = 1/2+, T = 3/2 IAS. Thereis an experimental level 71 keV below with an unknown spin [114] that may contribute to thelarge measured d-term. Again, further experimental investigation is required to determine thespins of these states, which would shed light on the large measured d-term.For the A = 21 multiplets, it is possible that the experimental results can be explained byINC mixing with nearby T = 1/2 states, however, a full understanding from theory, and itsrelationship to experiment, must be explored in more detail. In the two-level discussion above,we only give the results for the most important state, but there are other states, including those974.2. DISCUSSION AND MEASUREMENTS FROM THIS STUDYTable 4.4: Experimental and calculated ground-state energies (in MeV) of 20,21Mg withrespect to 16O.Nuclide Exp. USDA USDB NN + 3N Exp. - NN + 3N20Mg −6.94 −6.71 −6.83 −6.89 −0.0521Mg −21.59 −21.79 −21.81 −23.18 1.59in 21Na, that contribute to the total.The A = 20, 21 IMME’s were also calculated using χEFT interactions. These calculationsrepresent the first time that active protons and neutrons in the valence space have been usedwith χEFT interactions in the shell model framework [162]. The calculated ground state en-ergies for both 20,21Mg are listed in table 4.4, along with the results from the USDA/B. TheUSDA and USDB both reproduce the ground state energies, while the χEFT calculation onlyreproduces 20Mg. However, the ground state of 21Mg is overbound by 1.6 MeV. Because ofthese large deviations in the cases where protons and neutrons are active in the valence space,the calculated d and e terms have uncertainties that are too large to make a quantitative judge-ment on their accuracy. For example, the A = 20 d term was calculated to be −18 keV, whichis vastly different from the experimental value of 2.8(11) keV. An interesting feature of theseχEFT calculations is the overbinding decreases as the Tz of the nuclei increases. While 21Mgis 1.6 MeV overbound, 21F is only 0.8 MeV overbound. This results in a large cubic term ofd = −38 keV for the A = 21 multiplet. The χEFT calculations are currently being improved,and recent developments [229] may result in closer agreement with experiment for these mul-tiplets. While these calculations cannot reproduce the experimental values, they do representan important first step in developing interactions based on χEFT.98Chapter 5SummaryThe abundance of data in experimental nuclear physics is only possible due to the increase inthe power and range of exotic beam facilities. The addition of the proposed facilities, such asthe Radioisotope Beam Factory (RIBF) in Japan, the Facility for Rare Isotope Beams (FRIB)and the CARIBU facility in the USA, and the Advanced Rare Isotope Laboratory (ARIEL) inCanada will greatly increase the reach of experiments to access the approximately 7000 boundnuclei that are predicted to exist [1]. Many of these nuclei are neutron-rich, and touch on manyimportant aspects of nuclear physics, as their masses are important inputs for astrophysical r-process calculations, and in determining the evolution of nuclear structure towards the neutrondripline.In the past decade, three-nucleon forces have been shown to be crucial in determining thestructure of neutron rich nuclei, as three-nucleon forces become increasingly important far fromstability. These three-nucleon forces have been derived in the framework of effective field theo-ries based on quantum chromodynamics. χEFT based interactions offer predictive power to thewhole nuclear chart, as only a few coupling constants need to be fit to existing data. Currently,these χEFT calculations are able to account for the two-nucleon interactions at next-to-next-to-next-to-leading order (N3LO), while the three-nucleon interactions are included at N2LO.Three-nucleon interactions are required, because when bare two-nucleon interactions are used,experiment is not reproduced. This is particularly seen in the magic number N = 28 in the cal-cium isotopic chain, as it is only reproduced with the inclusion of the three-nucleon interaction.This is in contrast to the established calculations performed with phenomenological models,which are fit to a large amount of experimental data in the region applicable to the model.Further, these phenomenological models only include the effect of two-nucleon interactions.While these phenomenological models can reproduce the magic number at N = 28, it may bethat fitting the matrix elements in these models may mimic the effects of three-nucleon forces.99In this work, the first Penning trap mass measurements of the radioactive nuclei 51,52Ca,51K, and 20,21Mg were performed at TRIUMF’s Ion Trap for Atomic and Nuclear science(TITAN). The measurements of 20,21Mg required the first use of the Ion Guide Laser IonSource (IG-LIS), which suppressed the sodium contaminants by up to a factor of 106.The measured Ca and K nuclides were used to test the predictions of χEFT based calcula-tions in the vicinity of the predicted neutron magic numbers N = 32 and 34. The mass mea-surements showed a significant flattening of the two-neutron separation energies leading up toN = 32, with large deviations from the values tabulated in the 2003 Atomic Mass Evaluation.In fact, the mass of 52Ca was found to deviate from the tabulated value by 1700(700) keV. TheTITAN mass values for 51,52Ca were confirmed by a later mass measurement by the ISOLTRAPPenning trap mass spectrometer. The masses of 53,54Ca were measured by ISOLTRAP’s multi-reflection time-of-flight mass spectrometer. The combination of the TITAN and ISOLTRAPmass measurements showed excellent agreement with the χEFT based calculation and theGFPX1A phenomenological interaction. A measurement of the first E(2+) in 54Ca at theRIKEN facility agrees with the prediction of the χEFT based interaction.The masses of 20,21Mg were used to test the predicted quadratic behaviour of the isobaricmultiplet mass equation (IMME) using both the phenomenological interactions USDA andUSDB, and the χEFT three-nucleon interaction. This is the first time that χEFT calculationsbased in the shell model were used in open shell nuclei, representing an important step in theinvestigation of χEFT based calculations. It was found that large cubic terms in the IMMEwere required to reproduce the experimental data. Neither the USDA/B nor the χEFT calcula-tions were able to reproduce the experimental cubic terms. The USDA/B calculations typicallyproduced cubic terms near 0 keV, in disagreement with the 3-7 keV values found experimen-tally. The χEFT based calculations produced very large cubic terms of between −20 to −40keV with quite large errors, preventing any definitive statements as to their origin.In summary, the influence of three-nucleon forces in χEFT have:• Reproduced the E(2+) in 48Ca, showing the need for 3N forces,• Correctly predicted the E(2+) in 54Ca, as confirmed by a measurement at RIKEN [118]• Correctly predicted the behaviour of the S2n’s in the calcium isotopes, as confirmed by themeasurements presented in this thesis, and by subsequent measurements by ISOLTRAP[59].Further, we have performed the first mass measurement of 51K. The A = 20 and 21 isobaricmultiplet mass equations were aslo tested with TITAN:100• The phenomenological interactions USDA and USDB could not reproduce the large cu-bic d and quartic e terms observed• Interactions using 3N forces showed large deviations and errors, indicating that furtherwork is required for cases where both protons and neutrons are active in the calculation.In each case, theory must further refine their calculations to understand the origin of the largeobserved d and e terms.101Bibliography[1] J. Erler, N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, et al.. The limits of thenuclear landscape. Nature, 486, 509–512 (2012). doi:10.1038/nature11188. → pages1, 2, 99[2] S. Borsanyi, S. Durr, Z. Fodor, C. Hoelbling, S. D. Katz, et al.. Ab initio calculation ofthe neutron-proton mass difference. Science, 347, 1452–1455 (2015).doi:10.1126/science.1257050. → pages 2[3] A. Frank, J. Jolie, and P. Van Isacker. Symmetries in atomic nuclei: from isospin tosupersymmetry. 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Phase-modulated storedwaveform inverse Fourier transform excitation for trapped ion mass spectrometry.Analytical chemistry, 59, 449–454 (1987). → pages 123121Appendix AContributions to TITANA.1 Axial Frequency MeasurementsPenning trap experiments that use the TOF-ICR or PI-ICR detection methods do not rely ondetecting the pick-up signals of an ion’s motion on the trap electrodes. Instead, the radialfrequencies are determined either by extracting the ion through the magnetic field gradientof the superconducting solenoid magnet, and measuring the change in time-of-flight (section3.3.4), or by projecting the motion of the ion on to a position sensitive detector and measuringthe phase accumulation of the ion [179]. Both of these methods only allow for the detection ofthe radial motion of an ion.In section I describe a new method to measure the axial frequency of an ion in aPenning trap using destructive detection methods.A.2 Arbitrary function generator programmingTo perform a cyclotron frequency measurement a list of desired frequencies must be given toan Arbitrary/Function Generator (AFG). Previously at TITAN this was accomplished throughthe use of two AFGs, the first called the “frequency generator”, which supplied the radio-frequency for the ion excitation, and the second called the “ladder”. The “ladder” generatorwas programmed with a staircase waveform, with the number of steps corresponding to thenumber of frequencies to be applied when generating a resonance. The “frequency” generatorwas programmed to be frequency modulated about a user supplied centre frequency νcent and amodulation depth of±νmod. The output of the “ladder” was sent to the modulation input of the“frequency” generator, thus generating a series of frequencies from νcent−νmod to νcent+νmod.Several drawbacks of this technique are:122A.2. ARBITRARY FUNCTION GENERATOR PROGRAMMING• Output voltage noise from the “ladder” generator will cause jitter of the “frequency”generator’s output frequency• To ease the understanding the total system cycle, each step in the “ladder” waveform wasgenerally set to a convenient length (20 ms, 50 ms, 100 ms, etc.), limiting the range ofdipole and quadrupole excitation times• Drifts in the calibration constants of the “ladder” output digital-to-analog converter couldcause systematic shifts in the absolute output frequency of the “frequency” generatorTo overcome these problems, a model 33521A AFG from Agilent (now Keysight) waspurchased to apply the quadrupole field, while a model 33500B AFG with expanded memoryfrom Agilent was purchased for dipole cleaning and stored waveform inverse Fourier transform(SWIFT) cleaning [230]. Both AFG models have a “list” mode, where a list of frequencies canbe programmed to the AFG. The AFG can then be externally triggered to step through the list,not only eliminating the need for the “ladder” generator, but also eliminating potential jitter andoffset issues arising from the digital-to-analog converter of the “ladder” generator. A potentialsystematic from the use of the “ladder” generator was discovered during a measurement of the51Cr Q-value [173] (affected data was not included in [173]), where it was found that the outputof the “ladder” generator caused the frequency modulation range to be smaller than desired.This did not affect resonance data where the central portion of the resonance lineshape wasnear the centre of the scan range, instead, only affecting data near the edges of the scan range.By using the “list” mode of the 33521A AFG, systematics from the frequency modulationrange are eliminated. The 33521A AFG also expands the capabilities of the TITAN system,as there is no need for a “ladder” generator, meaning that an arbitrary series of frequenciescan be programmed into the AFG, and can be applied for any length of time, eliminating thecycle-length dependence on the “ladder” waveform.One potential application of the new frequency generation system is the measurement ofco-trapped ions. As an example, suppose that two ion species are delivered to MPET simul-taneously, each with cyclotron frequencies of ν1,2. The dipole AFG can be programmed suchthat for the first half of the measurement ion2 is dipole cleaned, while the quadrupole AFGsteps through the frequencies required to generate a resonance of ion1. During the secondhalf of the measurement, the dipole AFG will then dipole clean ion1, while the quadrupoleAFG steps through the frequencies required to generate a resonance of ion2. In this way, theresonance of each ion are generated at the same time. This is useful if one of the ions has awell known mass and can serve as a reference to calibrate the magnetic field. Any systematicsarising from magnetic field drifts are removed since measurements are built in such a way that123A.3. SORTEVAeach ion will see the same drifts, assuming the magnetic field drifts are slow (i.e. if magneticfield fluctuations occur on time scales longer than several minutes or longer).The code to program the AFGs is hosted at SortEVASortEVA is a program written to streamline the analysis of large numbers of resonance files,particularly when optimizing the trapping voltages in MPET. The general method used at TI-TAN to optimize the trapping electrodes can be found in [155]. The optimizing proceduregenerates a large quantity of data, all of which must be fit individually in order to ensure thequality of the data, and due to limitations of the existing fitting program. To overcome theselimitations, SortEVA (figure A.1) was written to organize the files generated from the data ac-quisition system, and to perform fits to each of the files. The resulting fits are then stored inmemory, allowing the user to examine the quality of each fit, ensuring both that the data in thefile is “good”, and that the fit is correct.An earlier version of this program played a key role in determining the systematic errorsduring the first uses of on-line, highly-charged ions in the following publications [147, 174,144].A.4 Correlations between adjacent frequency ratiosTo eliminate systematics effects, as described in section 3.3.7, the ratio of frequencies is takenbetween the ion of interest and a well known calibrant, or reference, ion (section 3.3.6). Gen-erally, the sequence of measurements is as follows: reference, ion of interest, reference, ion ofinterest, reference, etc. This pattern of measurements can be seen in figure A.2. The referencefrequency is interpolated to the time of the measurement of the ion of interest, thus leadingto correlations between the frequency ratios Ri and Ri+1, as both ratios depend on a sharedreference measurement. It may also be the case that two or more measurements of the ionof interest are between the same set of reference measurements. Here we will derive the co-variance between frequency measurements that share reference measurements. This work wasderived independently by Stephan Ettenauer [197], and was published in [174].The frequency between any two references can be interpolated asν(T ) =ν j+1−ν jt j+1− t j(T − t j)+ν j (A.1)where T is a time between the reference measurements at times ti and ti+1. The covariance124A.4. CORRELATIONS BETWEEN ADJACENT FREQUENCY RATIOSFigure A.1: Screen capture of SortEVA analysing a data set. Clockwise from upper left:Z-class histogram (number of detected ions after extracting from the trap), plot offitted frequencies against the measurement number along with a running list of thefit results (filename, frequency, frequency uncertainty, reduced χ2), resonance andfit along with TOF histogram, and the main fit dialog for monitoring the program’sprogress.between two frequency measurements that share one reference measurement (for example, thecorrelation between the frequency ν(Ti+2) and ν(Ti+3) in figure A.2) iscov(ν(Ti),ν(Ti+1) =(∂ν(Ti)∂νi+1)(∂ν(Ti)∂νi+1)cov(νi+1,νi+1)=(Ti− t jt j+1− t j)(t j+2−Ti+1t j+2− t j+1)σ2j+1(A.2)where σ j+1 is the uncertainty of the ( j+ 1)th reference measurement. If we define the fre-quency ratio to be R = νre f /ν , the covariance between the frequency ratios is thencov(Ri,Ri+1) =1νiνi+1cov(ν(Ti),ν(Ti+1)) . (A.3)where νi is the ith frequency measurement of the ion of interest.125A.4. CORRELATIONS BETWEEN ADJACENT FREQUENCY RATIOSA. T. GALLANT et al. PHYSICAL REVIEW C 85, 044311 (2012)Time (s)1000 2000 3000 4000 (Hz)6 - 10cνRef. Frequency, 00.51jtj+1tj+2tj+3tiTi+1T i+2Ti+3TFIG. 4. Illustration of the correlation introduced between adjacentfrequency ratio measurements from shared references. The solidcircles (tj ’s) represent reference measurements of νc,ref and the opencircles (Ti’s) show the interpolation of νc,ref to the center time of ameasurement of an ion of interest. From the figure it is clear that Ti iscorrelated with Ti+1, Ti and Ti+1 with Ti+2, and Ti+2 with Ti+3 (detailsfollow in the text).measurements of the ion of interest will often share a referencemeasurement introducing correlations between the frequencyratios. With the use of highly charged ions and the high levelof precision that can be reached it is important to include thesecorrelations when determining the final averaged frequencyratio. The relative statistical uncertainty of the cyclotronfrequency in a measurement is related to the resolving poweras δma/ma ∝ R−1 ×√N−1 [48] where N is the number ofdetected ions. Here we present two cases shown in Fig. 4. First,the most likely case where two measurements share a referencemeasurement, and second, the case where several measure-ments occur between two reference measurements. In practicethe second case does not occur because these data are generallysummed. In the second case, the analysis with the timecorrelations and with the summed data will yield nearly thesame result because the summed data implicitly include timecorrelations between the frequency measurements. For the firstcase, the covariance relation between frequency ratios iscovar(Ri, Ri+ 1)= 1νc,iνc,i+ 1(Ti − tjtj + 1− tj)(tj + 2− Ti+ 1tj + 2 − tj + 1)σ 2j + 1, (5)where the i and i + 1 refer to the ith and ith+1 measurementsof the ion of interest and the j ’s refer to the referencemeasurements. For the second case, the covariance betweenfrequency ratios that share both references iscovar(Ri, Ri+1) = 1νc,iνc,i+1[(tj+1 − Titj+1 − tj)(tj+1 − Ti+1tj+1 − tj)σ 2j+(Ti − tjtj+1 − tj)(Ti+1 − tjtj+1 − tj)σ 2j+1]. (6)Figure 4 illustrates the relationship between the variablesgiven in the above equations. In both cases the covariance isproportional to the variance of the reference measurements. Itis desirable to measure the reference ion much more preciselythan the ion of interest to reduce correlation effects, however,a trade-off must be made to maximize the statistics collected,and hence, the precision, for the ion of interest.B. Systematic errors and uncertaintiesSeveral systematics must be taken into account. Systematicsrelating to misalignment between magnetic and trap axes,electric field miscompensation, relativistic effects, etc., areminimized by choosing a reference ion which is close in m/qto the ion of interest as these effects scale with the differencein the charge-to-mass ratio $(m/q) [34]. To determine anypotential shifts from different m/q effects between the ion ofinterest and the reference ion, a series of mass measurementson 85Rb10,8+ and 87Rb9+ using 85Rb9+ as the reference werecompleted. The extracted masses all agree within 1σ of theliterature value. Although no shifts were observed to beconservative we take, as an upper limit on any systematiceffects, a systematic uncertainty of 42 parts per billion (ppb)in the frequency ratio.A second systematic effect stems from the ambiguity inselecting the upper and lower time cuts on the time-of-flight spectrum. The ambiguity arises from charge exchangeprocesses in the trap. If an ion undergoes charge exchangewith residual gas in the vacuum, these ions will manifestthemselves as a long tail in the time-of-flight spectrum. Inthe present analysis the lower and upper levels were setat 12 and 40 µs, respectively. The lower level was setto 12 µs to maximize the number of on resonance ionswhile minimizing background counts from the nearby H+2peak resulting from charge exchange in the trapping region.Figure 5 shows a typical time-of-flight spectrum for 78Rb8+which was trapped for 197 ms. The dashed-blue lines showthe lower and upper time cuts whereas the solid-red lines showwhen, on average, 78Rb ions with different charge states wouldarrive. To determine the systematic effect ¯R was determinedfor upper level time cuts of 30, 35, 40, 45, and 55 µs forboth the ground and isomeric states. If the average frequencyratio determined at 40 µs for either case was an extremumthe systematic effect was assigned to be the full range of theextracted ¯R’s, otherwise half of the range was assigned.s)µTime of Flight (0 10 20 30 40 50 60 70Counts051015202530354045sµ12 sµ40 q=2+q=3++2H 8+Rb78 = 197 msRFTFIG. 5. (Color online) Time-of-flight spectrum of ions extractedfrom the MPET. See text for details.044311-4Figure A.2: Illustration of the correlation introduced between adjacent frequency ratiomeasurements from sha d refe ces. The solid circles (t j’s) represent referencemeasurements of νc,re f and the open circles (Ti’s) show the interpolation of νc,re fto the centre time of a measurement of an ion of interest. From the figure it is cle rthat Ti is correlated with Ti+1, Ti and Ti+1 with Ti+2, and Ti+2 with Ti+3. Figurereproduced from [174].The covariance between measurements that share both reference measurements (for exam-ple points Ti and Ti+1 in figure A.2) can be found in a similar manner. The covariance iscov(Ri,Ri+1) =1νiνi+1[(t j+1−Tit j+1− t j)(t j+1−Ti+1t j+1− t j)σ2j +(Ti− t jt j+1− t j)(Ti+1− t jt j+1− t j)σ2j+1].(A.4)The weighted average and uncertainty can then be calculated fromR¯ =∑i, j(V−1)i j R j∑i, j (V−1)i j(A.5)with an accompanying uncertainty ofσ2R¯ =1∑i, j (V−1)i j. (A.6)126


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