Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

First mass measurements of highly charged, short-lived nuclides in a Penning trap and the mass of 74Rb Ettenauer, Stephan 2012

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata


24-ubc_2012_fall_ettenauer_stephan.pdf [ 21.06MB ]
JSON: 24-1.0072798.json
JSON-LD: 24-1.0072798-ld.json
RDF/XML (Pretty): 24-1.0072798-rdf.xml
RDF/JSON: 24-1.0072798-rdf.json
Turtle: 24-1.0072798-turtle.txt
N-Triples: 24-1.0072798-rdf-ntriples.txt
Original Record: 24-1.0072798-source.json
Full Text

Full Text

First Mass Measurements of Highly Charged, Short-lived Nuclides in a Penning Trap and the Mass of 74 Rb  by Stephan Ettenauer Dipl. Ing., Vienna University of Technology, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Physics)  The University Of British Columbia (Vancouver) May 2012 c Stephan Ettenauer, 2012  Abstract To date, Vud of the Cabibbo-Kobayashi-Maskawa quark mixing matrix is most precisely determined from superallowed 0+ → 0+ nuclear β-decays. In addition to half-life, Branching Ratio, and transition energy (called QEC -value) of a superallowed decay, theoretical corrections have to be considered to extract Vud . Among those, the isospin symmetry breaking corrections, δC , show discrepancies between different theoretical models, which are critical to be resolved. 74 Rb has the largest δC of all 13 superallowed β-emitters used to obtain Vud and would carry particular weight to discriminate between models were it not limited by the uncertainty in the QEC -value. However, 74 Rb’s half-life of 65 ms has previously posed a real challenge to the experimental precision in its QEC -value, which is best determined by direct mass measurements in Penning traps. In this work, Penning trap mass measurements of short-lived nuclides have been performed for the first time with highly-charged ions, using the TITAN facility. Compared to singly-charged ions, this provides an improvement in experimental precision that scales with the charge state q. Neutron-deficient Rb-isotopes have been prepared in an electron beam ion trap to q = 8 − 12+ prior to the mass measurements. In combination with a Ramsey scheme, this opens the door to unrivalled precision with gains of 1-2 orders of magnitude. The method is particularly suited for short-lived nuclides such as 74 Rb and its mass has been determined. In the realm of fundamental symmetries studied in low-energy nuclear systems such as in 74 Rb, the precision achieved by highly-charged ions is essential. For mass measurements motivated by nuclear structure or nuclear astrophysics, where the present experimental precision is already sufficient, this novel technique significantly reduces the measurement time and thus allows one to map the nuclear mass landscape more broadly. In the exploration towards the limits of nuclear existence where experimental efforts typically face shorter half-lives and lower production yields of radioisotopes, the same precision can be achieved by compensating for both challenges with the higher charge state. Finally, highly-charged ions provide opportunities for an unprecedented resolving power to identify low-lying nuclear isomers. This potential has firstly been demonstrated with 78,78m Rbq=8+ . ii  Preface As with almost every experimental work in contemporary nuclear physics, the studies presented in this thesis are the result of a collaboration between many people. The multi-ion-trap setup at TRIUMF’s Ion Trap for Atomic and Nuclear science (TITAN) has been operational since 2007. For the present work, several improvements and modifications of the setup, or in its operation, were necessary of which my most significant contributions are listed below. All people in the TITAN group at TRIUMF helped at all stages of the work. • From fall 2009 to spring 2011, I was in charge of the mass measurement program at TITAN. This involved the planning and preparation of measurements, leadership during the experiments, and the coordination with TRIUMF’s accelerator division before and during experimental beamtimes. • The preparation of the online beamtime of 44 K4+ was done together with Maxime Brodeur, Paul Finlay, and Alain Lapierre (see Section 3.4). This beamtime represented a proof-of-principle experiment and was crucial to identify challenges in the program of Penning trap mass measurements with Highly Charged Ions (HCI). • Performance tests of the Pulsed Drift Tube (PLT) after the Radio-Frequency Quadrupole (RFQ) were carried out initially together with Thomas Brunner and later with Ernesto Man´e (Section 3.3.1 and Section A.1). As a result, critical issues regarding the reliability of the PLT could be resolved. • In collaboration with Mel Good, I carried out the work regarding the baking of the Measurement Penning Trap (MPET), the hardware upgrade of its vacuum system as well as MPET setup modifications (Section 3.7.8 and Section A.5). Due to charge exchange with the residual gas in the MPET, an improved vacuum was essential for HCI.  iii  • The major part of the preparation for the neutron deficient Rb runs I conducted together with Martin Simon (among others Section 3.5 and Section A.3). This required the development of a suitable optimization procedure to yield fast and efficient charge breeding to higher charge states in the Electron Beam Ion Trap (EBIT). Based on the previous experience for 44 K4+ , the efficiency had been identified as one of the most challenging aspects of the present work. • The extension of the mass measurements to 78,78m Rb to demonstrate the improved resolving power for low-lying isomeric states was based on my idea. • For the first time at TITAN, I implemented and tested the Ramsey excitation scheme (Section 3.7.4). The Ramsey excitation technique led to an improvement in measurement precision by a factor of ≈ 2. • By utilizing Aaron Gallant’s work on a code which fits the theoretical line shape to a Time-Of-Flight (TOF)-resonance, I wrote an analysis code to perform the present analysis (Chapter 4). Aaron Gallant and I independently derived the covariant matrix to take into account correlations between extracted frequency ratios (Appendix B). Analysis progress was discussed weekly with Ankur Chaudhuri, Aaron Gallant, and Vanessa Simon. • I developed the idea for the new test of the Isospin Symmetry Breaking (ISB) corrections for superallowed β-decays (Section 5.1). A letter describing the main part of the work, i.e. the mass measurement of neutrondeficient Rb-isotopes, is published in S. Ettenauer et al., Phys. Rev. Lett. 107, 272501 (2011) First Use of High Charge States for Mass Measurements of Short-Lived Nuclides in a Penning Trap A second publication is planned to cover more details of the experimental setup, the analysis, as well as the implications of the result. The mass measurement of 44 K4+ (see Section 3.4) is part of two publications” A. Lapierre et al., Phys. Rev. C 85,024317 (2012) Penning-trap mass measurements of the neutron-rich K and Ca isotopes: Resurgence of the N = 28 shell strength and  iv  A. Lapierre et al., Nucl. Instr. and Meth. A 624, 54 (2010) The TITAN EBIT charge breeder for mass measurements on highly charged shortlived isotopes- First online operation The newly demonstrated technique to resolve low-lying isomeric states (Section 5.4) will be published. A manuscript is available at A. T. Gallant et al., arXiv:1112.0614 Highly charged ions in Penning traps, a new tool for resolving low lying isomeric states  v  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xi  Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xv  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . .  2  The Standard Model and Theory of Nuclear β-Decay . . . . . . . 2.1 The Standard Model of particle physics . . . . . . . . . . . . . 2.1.1 Local gauge invariance . . . . . . . . . . . . . . . . . . 2.1.2 Symmetries and particle content of the Standard Model . 2.1.3 Quantum Chromodynamics (QCD) and nuclear structure 2.2 Electroweak interaction in the Standard Model and Vud . . . . . 2.2.1 Masses of fermions . . . . . . . . . . . . . . . . . . . . 2.2.2 The Cabibbo-Kobayashi-Maskawa matrix . . . . . . . . 2.3 CKM-unitarity test via |Vud |2 + |Vus |2 + |Vub |2 = 1 . . . . . . . 2.3.1 Implications for physics beyond the Standard Model . . 2.4 Nuclear β-decays from first principles of the Standard Model . . 2.4.1 Neutron β-decay . . . . . . . . . . . . . . . . . . . . . 2.4.2 Nuclear β-decays . . . . . . . . . . . . . . . . . . . . . 2.5 Superallowed 0+ → 0+ β decays . . . . . . . . . . . . . . . . . vi  . . . . . . . . . . . . . .  1 2 4 5 6 8 9 15 16 17 19 22 24 27 32 38  2.6  2.7  2.8  2.9  3  Theoretical corrections to the f t-values . . . . . . . . . . . . . . 2.6.1 Corrections to the statistical rate function f . . . . . . . . 2.6.2 Radiative corrections . . . . . . . . . . . . . . . . . . . . 2.6.3 Corrected Ft-values . . . . . . . . . . . . . . . . . . . . Different models for isospin symmetry breaking corrections . . . 2.7.1 Nuclear shell model with Saxon-Woods radial wave-functions 2.7.2 Nuclear shell model with Hartree-Fock radial wave-functions 2.7.3 Isovector monopole resonance . . . . . . . . . . . . . . . 2.7.4 Self-consistent relativistic random-phase approximation . 2.7.5 Nuclear density functional theory . . . . . . . . . . . . . 2.7.6 Ab-initio calculation by the no-core shell model in 10 C . . 2.7.7 Exact formalism for δC . . . . . . . . . . . . . . . . . . . 2.7.8 Implications of different models for ISB corrections . . . . Experimental input to the debate around the isospin symmetry breaking corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Input parameters and independent benchmark quantities . 2.8.2 Improvement of f t-values . . . . . . . . . . . . . . . . . The case of 74 Rb and the need for highly charged ions in Penning trap mass measurements . . . . . . . . . . . . . . . . . . . . . .  Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction and overview of the TITAN facility . . . . . . . . . . 3.2 ISAC: production and delivery of radioactive beam . . . . . . . . 3.2.1 Production and delivery of neutron-deficient Rb-beams . . 3.3 TITAN’s Radio-Frequency Quadrupole (RFQ) cooler and buncher . 3.3.1 A PLT after the RFQ cooler and buncher . . . . . . . . . . 3.4 The electron beam ion trap for charge breeding of radioactive ions 3.4.1 Charge breeding time of Rb isotopes . . . . . . . . . . . . 3.5 Beam transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 A/q selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 The measurement Penning trap . . . . . . . . . . . . . . . . . . . 3.7.1 Motion of charged particles in a Penning trap . . . . . . . 3.7.2 The Fourier-transform ion-cyclotron-resonance method . . 3.7.3 Quadrupole excitations and the time-of-flight ion-cyclotronresonance method . . . . . . . . . . . . . . . . . . . . . . 3.7.4 The Ramsey excitation scheme . . . . . . . . . . . . . . . 3.7.5 Determination of atomic masses and QEC -values . . . . . 3.7.6 Dipole excitations . . . . . . . . . . . . . . . . . . . . . 3.7.7 TITAN’s MPET setup . . . . . . . . . . . . . . . . . . . . 3.7.8 MPET vacuum system . . . . . . . . . . . . . . . . . . . vii  40 40 41 43 46 46 49 50 53 54 55 58 58 62 62 63 66  71 71 75 78 81 86 87 90 91 95 97 97 103 104 113 117 119 120 124  4  5  6  Measurement and Analysis . . . . . . . . . . . . . . . . . . . . . . . 4.1 Measurement summary . . . . . . . . . . . . . . . . . . . . . . . 4.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Determination of the cyclotron frequency νc . . . . . . . . 4.2.2 Determination of the frequency ratio . . . . . . . . . . . . 4.2.3 Analysis software . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Analysis results . . . . . . . . . . . . . . . . . . . . . . . 4.3 Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Spatial magnetic field inhomogeneities . . . . . . . . . . 4.3.2 Harmonic distortion and misalignment of the magnetic field axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Non-harmonic imperfections of the trapping potential . . . 4.3.4 Relativistic effects . . . . . . . . . . . . . . . . . . . . . 4.3.5 Magnetic field stability . . . . . . . . . . . . . . . . . . . 4.3.6 Image charges . . . . . . . . . . . . . . . . . . . . . . . 4.3.7 Ambiguity in the TOF-range selection . . . . . . . . . . . 4.3.8 Remaining uncertainties and independent accuracy checks 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  129 130 134 134 136 137 138 147 148  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The QEC -and f t-value of 74 Rb . . . . . . . . . . . . . . . . . . . 5.2 The Atomic Mass Evaluation (AME) . . . . . . . . . . . . . . . . 5.3 General considerations for Penning trap mass measurements with radioactive HCI . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Resolving low-lying isomeric states with HCI . . . . . . . . . . .  158 158 161  148 148 149 149 150 150 152 154  162 163  Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 168  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Appendices A Further Details about the Experimental Setup . . . . . . A.1 Investigations regarding the PLT after the RFQ . . . . . A.2 More details on the beam transport . . . . . . . . . . . A.3 Optimization of EBIT injection, trapping, and extraction A.4 Measurement control system . . . . . . . . . . . . . . A.5 The MPET vacuum . . . . . . . . . . . . . . . . . . . A.5.1 Hardware upgrade for MPET vacuum . . . . .  viii  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  203 203 207 209 212 214 215  B Derivation of the Covariance Matrix due to Shared Reference Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219  ix  List of Tables Table 2.1 Table 2.2 Table 2.3 Table 2.4 Table 2.5  Quark generations in the Standard Model . . . . . . . . . . . . Fermions and scalar particles in the Standard Model . . . . . . Vud and Vus extracted from various decays. . . . . . . . . . . . Averaged Ft with different models for δC . . . . . . . . . . . Averaged Ft over the same 6 cases with different models for δC  Table 3.1 Table 3.2  Yields and contamination for neutron-rich Rb . . . . . . . Eigenfrequencies of an ion with mass A = 74 and charge q 8+ in the MPET . . . . . . . . . . . . . . . . . . . . . . . Increased uncertainty for 39 K4+ due to charge-exchange . .  Table 3.3  9 10 21 60 60  . . 80 = . . 124 . . 126  Table 4.9  Frequency ratios for 76 Rbq+ versus 85 Rb9+ . . . . . . . . . . . Frequency ratio between 75 Rb8+ and 85 Rb9+ . . . . . . . . . . Frequency ratios between 74 Rb8+ with 85 Rb9+ . . . . . . . . . Frequency ratio between74 Ga8+ with 85 Rb9+ . . . . . . . . . Differences in mass number A, m/q, and q/m for ion species studied in this work in comparison to the reference ion 86 Rb9+ Error budget for the frequency ratio of the measurements of 75−76 Rb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error budget for the frequency ratio of the measurements of 74 Ga and 74 Rb . . . . . . . . . . . . . . . . . . . . . . . . . . Mean frequency ratios between 76,75,74 Rb8+ and 74 Ga8+ and 85 Rb9+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Uncertainties for the Mass of 74 Rb in keV . . . . . . . .  Table 5.1 Table 5.2  Mass excess of 74 Rb and 74 Kr and QEC -value of 74 Rb . . . . . 158 Mass excess for measured nuclides. . . . . . . . . . . . . . . . 161  Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 4.7 Table 4.8  x  141 143 145 146 147 155 155 156 157  List of Figures Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7 Figure 2.8 Figure 2.9 Figure 2.10 Figure 2.11 Figure 2.12 Figure 2.13 Figure 2.14 Figure 2.15 Figure 2.16 Figure 2.17 Figure 2.18 Figure 2.19 Figure 2.20 Figure 2.21 Figure 2.22 Figure 2.23 Figure 2.24 Figure 2.25 Figure 2.26 Figure 2.27 Figure 2.28  Chiral expansion of nuclear forces . . . . . . . . . . . . . . . 12 Chiral expansion at leading order . . . . . . . . . . . . . . . 13 Repulsive core in nuclear potentials . . . . . . . . . . . . . . 14 Determination of CKM matrix elements . . . . . . . . . . . . 19 Example for a K 3 decay . . . . . . . . . . . . . . . . . . . . 20 K + and π + decay branches to extract |Vus | · fK /(|Vud | · fπ ) . 21 |Vus | · fK /(|Vud | · fπ ) determined from τ -decay . . . . . . . 22 Electroweak interaction between quarks and leptons involving an extra Z boson . . . . . . . . . . . . . . . . . . . . . . . . 24 u-quark decay . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Neutron decay . . . . . . . . . . . . . . . . . . . . . . . . . 28 Renormalization of gA due to the strong interaction . . . . . . 29 Neutron lifetime . . . . . . . . . . . . . . . . . . . . . . . . 32 Q-values of β − , β + -decay, and electron capture . . . . . . . . 36 Vud extracted from different decays . . . . . . . . . . . . . . 38 γW -box and ZW -box diagram . . . . . . . . . . . . . . . . 42 f t- and Ft-values for superallowed β-decays . . . . . . . . . 44 Error budget for Vud . . . . . . . . . . . . . . . . . . . . . . 44 δC -corrections with and without core orbitals . . . . . . . . . 48 δC with Saxon-Woods and Hartree-Fock radial wave-functions 49 δC for Saxon-Woods and recent Hartree-Fock radial wave-functions 51 δC in the IVMR- Model . . . . . . . . . . . . . . . . . . . . 52 δC with RHF + RPA and RH+RPA . . . . . . . . . . . . . . . . 53 δC in the Density Functional Theory (DFT)-calculation . . . . 55 δC in the No-Core Shell Model (NCSM) compared to other models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Ft and | Vud |2 in different models of δC . . . . . . . . . . . 59 Test of δC calculations . . . . . . . . . . . . . . . . . . . . . 64 A χ2 test for δC of different models . . . . . . . . . . . . . . 65 Experimental status of 74 Rb . . . . . . . . . . . . . . . . . . 66 xi  Figure 2.29 Half-life and Branching Ratio (BR) measurements of 74 Rb . . Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8 Figure 3.9 Figure 3.10 Figure 3.11 Figure 3.12 Figure 3.13 Figure 3.14 Figure 3.15 Figure 3.16 Figure 3.17 Figure 3.18 Figure 3.19 Figure 3.20 Figure 3.21 Figure 3.22 Figure 3.23 Figure 3.24 Figure 3.25 Figure 3.26 Figure 3.27 Figure 3.28 Figure 3.29 Figure 3.30 Figure 3.31 Figure 3.32 Figure 3.33  68  Nuclides with half-lives larger than 65 ms and 8 ms . . . . . . 72 The TITAN setup . . . . . . . . . . . . . . . . . . . . . . . . 73 Isotope Separator and ACcelerator (ISAC) . . . . . . . . . . . 76 ISAC experimental areas . . . . . . . . . . . . . . . . . . . . 77 Half-life and BR measurements at ISAC . . . . . . . . . . . . 78 Rb yields at ISAC for various targets . . . . . . . . . . . . . . 79 Yields of Ga, Rb, and Sr isotopes for three Nb-targets . . . . . 81 Longitudinal potential of the RFQ . . . . . . . . . . . . . . . 83 RFQ schematics . . . . . . . . . . . . . . . . . . . . . . . . . 84 Energy change of trapped ions in the buffer gas of the RFQ . . 84 RFQ DC potential during this work . . . . . . . . . . . . . . . 85 The PLT after the RFQ . . . . . . . . . . . . . . . . . . . . . 86 Schematic of an EBIT . . . . . . . . . . . . . . . . . . . . . 87 Charge breeding times in the TITAN EBIT . . . . . . . . . . . 88 Schematic of the EBIT and the injection, charge breeding, and extraction potentials . . . . . . . . . . . . . . . . . . . . . . 89 TOF -spectrum of charge-bred 44 K . . . . . . . . . . . . . . . 91 85 Charge breeding time of Rb . . . . . . . . . . . . . . . . . 92 TOF -spectrum of HCI of 75 Rb . . . . . . . . . . . . . . . . . 93 TITAN transport beamline . . . . . . . . . . . . . . . . . . . 94 Concept of a Bradbury Nielsen ion Gate (BNG) . . . . . . . . 96 New design of BNG based on chemically etched wires . . . . 96 Schematics of an ideal Penning trap . . . . . . . . . . . . . . 98 Motion of a positively charged particle in a Penning trap . . . 101 Radial energy of an ion after the Radio-Frequency (RF)-excitation106 Extraction of the ions from the MPET . . . . . . . . . . . . . 109 Example for a Time-Of-Flight Ion-Cyclotron-Resonance (TOF ICR ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 The Ramsey excitation scheme . . . . . . . . . . . . . . . . . 114 Comparison of the conventional and the Ramsey excitation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Fourier transform of the conventional and the Ramsey excitation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Linear interpolation of reference measurements . . . . . . . . 118 Application of the RF-fields . . . . . . . . . . . . . . . . . . 120 The injection, trap, and extraction setup of TITAN’s Measurement Penning Trap (MPET) . . . . . . . . . . . . . . . . . . . 121 Schematics and picture of TITAN’s MPET . . . . . . . . . . . 122 xii  Figure 3.34 Cross sectional view of the Lorentz steerer . . . . . . . . . . 123 Figure 3.35 Charge exchange for 39 K4+ . . . . . . . . . . . . . . . . . . 126 Figure 3.36 Baking of the MPET Setup . . . . . . . . . . . . . . . . . . . 128 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Figure 4.8 Figure 4.9 Figure 4.10 Figure 4.11 Figure 4.12 Figure 4.13 Figure 4.14 Figure 4.15 Figure 4.16 Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6  Figure A.1 Figure A.2 Figure A.3 Figure A.4 Figure A.5 Figure A.6 Figure A.7  TOF - ICR  of neutron-deficient Rb-isotopes . . . . . . . . . . . 131 histogram of 76 Rb8+ . . . . . . . . . . . . . . . . . . . . 132 TOF histogram 76 Rb8+ after being stored in MPET for 1s . . . 133 χ2 -histogram of fits of the theoretical line shape to the resonances137 Correlations between frequency ratios . . . . . . . . . . . . . 138 The time and durations of measurement runs for the mass determination of 76 Rb . . . . . . . . . . . . . . . . . . . . . . . 139 Measurements of 76 Rb8+ . . . . . . . . . . . . . . . . . . . . 140 Impact of correlations in R . . . . . . . . . . . . . . . . . . . 141 Measurement of 75 Rb8+ . . . . . . . . . . . . . . . . . . . . 142 Main TOF minimum for 74 Rb8+ . . . . . . . . . . . . . . . . 143 Measurements of 74 Rb8+ . . . . . . . . . . . . . . . . . . . . 145 Measurements of the frequency ratio R = νr /ν for 74 Ga8+ with 85 Rb9+ as the reference. . . . . . . . . . . . . . . . . . 146 Dependency of R on the TOF-range . . . . . . . . . . . . . . 151 Dependency of R on the TOF-range 2 . . . . . . . . . . . . . 152 Confirmation of the accuracy of the setup . . . . . . . . . . . 153 Atomic masses of 76,75,74 Rb and 74 Ga . . . . . . . . . . . . 157 TOF  Improvements in the Ft-value of 74 Rb . . . . . . . . . . . . . Test of the ISB correction term δC for the heavier superallowed β-decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resolving the low-lying isomeric state of 78 Rb . . . . . . . . Improved resolving power for low lying isomers with HCI. . . The required resolving power R to separate isomer and ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The required resolving power R to separate isomer and ground state 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HV switch for the PLT after the RFQ . . . . . . . Circuit diagram of the PLT . . . . . . . . . . . . Performance test of the PLT’s HV switch circuit . Beam gate in front of the RFQ . . . . . . . . . . Optimization of the injection into the EBIT . . . Experimental control system . . . . . . . . . . . Timings for beam transport and charge breeding.  xiii  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  159 160 164 165 166 167 204 205 206 208 210 212 213  Figure A.8 Diagram of the vacuum system for the MPET . . . . . . . . . 215 Figure A.9 New optic elements on the MPET extraction path . . . . . . . 216 Figure A.10 Baking of the detector cross . . . . . . . . . . . . . . . . . . 217 Figure B.1  Correlations in frequency ratios . . . . . . . . . . . . . . . . 220  xiv  Glossary AME  Atomic Mass Evaluation  ARIEL  Advanced Rare IsotopE Laboratory  BNG  Bradbury Nielsen ion Gate  BNL  Brookhaven National Laboratory  BR  Branching Ratio  χEFT  chiral Effective Field Theory  CKM  Cabibbo-Kobayashi-Maskawa  CPET  Cooler Penning Trap  CVC  Conserved Vector Current  DFT  Density Functional Theory  EBIT  Electron Beam Ion Trap  EC  Electron Capture  EFT  Effective Field Theory  EXO  Enriched Xenon Observatory  FAIR  Facility for Antiproton and Ion Research  FEBIAD  Forced Electron Beam Induced Arc Discharge  FRIB  Facility for Rare Isotope Beams  FSU  Florida State University  xv  FT- ICR  Fourier-Transform Ion-Cyclotron-Resonance  GUT  Grand Unified Theory  HCI  Highly Charged Ions  IMME  Isobaric Mass Multiplet Equation  ISAC  Isotope Separator and ACcelerator  ISB  Isospin Symmetry Breaking  ISOL  Isotope Separator On-Line  LEBIT  Low-Energy Beam and Ion Trap facility  MCP  Multi Channel Plate  MCS  Multi Channel Scaler  MOSFET  Metal-Oxide Semiconductor Field-Effect Transistors  MPET  Measurement Penning Trap  NCSM  No-Core Shell Model  NSCL  National Superconducting Cyclotron Laboratory  OLIS  Off-Line Ion Source  PDG  Particle Data Group  PLT  Pulsed Drift Tube  PMNS  Pontecorvo-Maki-Nakagawa-Sakata  PPB  Parts-Per-Billion  PPG  Programmable Pulse Generator  QCD  Quantum Chromodynamics  QED  Quantum Electrodynamics  RF  Radio-Frequency  RFQ  Radio-Frequency Quadrupole  xvi  RH  Relativistic Hartree  RHF  Relativistic Hartree-Fock  RIBF  RadioIsotope Beam Factory  RMS  Root-Mean-Square  RPA  Random-Phase Approximation  SCEPTAR SCI  Scintillating Electron Positron Tagging Array  Singly Charged Ions  SMILETRAP  Stockholm-Mainz-Ion-LEvitation-TRAP  SUSY  SUperSYmmetry  SWIFT  Stored Waveform Inverse Fourier Transform  TITAN  TRIUMF’s Ion Trap for Atomic and Nuclear science  TOF  Time-Of-Flight  TOF - ICR  Time-Of-Flight Ion-Cyclotron-Resonance  UCN  Ultra Cold Neutrons  xvii  Acknowledgements Naturally, the path to the first use of highly charged ions for Penning trap mass measurements of short-lived nuclides spans a much longer period than one PhD degree. I am indebted to very many people who have dedicated their time and effort to the TITAN setup which made the work for this thesis possible. First of all, I would like to thank my supervisor, Jens Dilling, for his continuous support, advice and -most importantly- sharing his infinite optimism. Thank you for your encouragement and trust which allowed me to take wider responsibilities and to broaden my horizon by engaging in other projects besides the thesis topic. I am especially grateful for the many opportunities to present results and interact with the community at conferences and workshops. I am very thankful to all members of the TITAN collaboration with whom I enjoyed working. I am particularly grateful to Maxime Brodeur, Thomas Brunner, Alain Lapierre, and Ryan Ringle who introduced me to the TITAN setup. None of the sucesses at TITAN are possible without Mel Good. I was very fortunate to learn from him about vacuum technology, design and assembly of parts, etc. I greatly appreciated the support of post-docs and other students to the present work: Ankur Chaudhuri, Usman Chowdhury, Aaron Gallant, Alexander Großheim, Ernesto Man´e, Martin Simon, Vanessa Simon, and Brad Schultz as well as Damien Robertson, a summer student who I had the pleasure of supervising on his project to improve the MPET vacuum system. I am especially thankful to Martin Simon, lab-colleague and close friend since undergraduate times, who ‘rushed to Vancouver when things looked grim’ to ‘concentrate forces’ and to get this measurement done. Special thanks to Aaron Gallant for his work on the fitting code of TOFresonances which greatly simplified the data analysis and to Matt Pearson who often helped in case I ran into an unexpected hardware problem. I would like to express my gratitude to all members of TRIUMF’s technical and scientific staff, especially the ISAC beam delivery and data acquisition group, who provided help or information during the course of my PhD.  xviii  I deeply enjoyed many discussions with Achim Schwenk, Sonia Bacca, Dave Lunney, Smarajit Triambak, Daryl Bishop, and many others from whom I learned a lot during the last years. I would like to express my gratitude to Georges Audi and Dave Lunney for their hospitality during my visit at Orsay for the mass evaluation of the present results. I would also like to thank Ian Towner for discussions and his calculations. Thank you to all people collaborating in the half-life measurement of 26m Al, especially to Gordon Ball, Paul Finlay, Hamish Leslie, and Carl Svensson who offered me the opportunity to get involved in the analysis. I am thankful to Barry Davids for inviting me to join the TRIUMF seminar committee which was a very enjoyable and interesting activity. Thank you to Aaron Gallant, Ania Kwiatkowski, Martin Simon, Vanessa Simon, and Brad Schultz for their reading, corrections, and comments to (parts of) the thesis. I would like to thank the members of my supervisory committee, Colin Gay, Jeremy Heyl, John Ng, and Achim Schwenk, for their guidance and the careful reading of the thesis. I am grateful to the Vanier Canada Graduate Scholarship and UBC Killam Doctoral Scholarship Programs for their financial and travel support. The time of my PhD would not have been the same without the great friends I was very fortunate to meet in Vancouver. A very special thanks to my family, especially my parents and sisters, and Chlo´e’s family for their continuous encouragement and support from overseas. Above all, I would like to thank Chlo´e for the wonderful time we have spent together in British Colombia, even when stress at work sometimes made it difficult to keep the focus on the important things. Thank you for your critical mind reflecting upon our lives, which prepares us well for our decisions in the future.  xix  test To my family  xx  Chapter 1  Introduction Since their introduction into the research of radioisotopes over twenty years ago [3, 4], Penning traps have made major contributions to the exploration of the nuclear mass surface. This is evidenced by the large number of existing and proposed facilities [5] as well as the wealth of experimental results [6]. Advances in experimental techniques now allow measurements for virtually all low energy, rare isotope beams as Penning traps have been able to access nuclides with half-lives below 10 ms [7] as well as superheavies (elements with proton number Z > 92) with production cross sections that correspond to yields of less than 1 particle per second [8, 9]. The widespread success of Penning traps is due to their precision following the expression δm m √ ∝ (1.1) m qBTrf Nion [10], where δm/m is the achievable relative precision in mass m, q is the ion’s charge state and B is the magnetic field strength. The measurement time Trf and the number of ions Nion are limited by a nuclide’s half-life and possibly by its production yield at radioactive beam facilities, but also by the efficiency of the spectrometer. Measurements are generally performed with Singly Charged Ions (SCI) or in special cases, where coupled to a gas stopper cell, with q = 2+. Penning trap mass studies utilizing Highly Charged Ions (HCI) have been successfully pioneered with stable nuclides [11]. Here the requirements of high efficiency and short measurement times are less relevant compared to the requirements when working with radioactive ions. In the realm of rare isotope science with Penning traps, HCI represent a thus far unexplored opportunity to improve the experimental precision further circumventing constraints imposed by short half-lives and lower yields when probing the limits of nuclear existence. The superallowed β emitter 74 Rb is a prime example where a short half-life of 1  only 65 ms poses a real challenge to experiment. Despite several Penning trap mass measurements [12–14], the total transition energy, QEC , still contributes significantly to the uncertainty of its corrected Ft-value, only surpassed by theoretical uncertainties of the isospin-symmetry breaking corrections δC [15]. The latter have recently been reduced by experimentally providing the 74 Rb Root-MeanSquare (RMS) charge radius as an input for the calculation of δC [16]. The QEC value and δc are now close to sharing the same weight to the total uncertainty of the Ft-value. Among all superallowed β emitters used to extract Vud of the CabibboKobayashi-Maskawa (CKM) matrix [15], 74 Rb has the highest atomic number, Z. Hence, it is of special importance in attempts to distinguish between conflicting nuclear models of δC since δC approximately scales as Z 2 [17, 18]. This work presents the first Penning trap mass measurements of short-lived HCI, performed with TRIUMF’s Ion Trap for Atomic and Nuclear science (TITAN) [19], including a successful mass determination of 74 Rb8+ . The high charge states were attained by breeding SCI delivered from the Isotope Separator and ACcelerator (ISAC) facility in an Electron Beam Ion Trap (EBIT).  1.1  Outline of the thesis  Theoretical background to the physics motivation of the present work, i.e. the measurement of the QEC -value of the superallowed β-emitter 74 Rb, is provided in Chapter 2. The theory of nuclear β-decays is derived from first principles of the Standard Model (Section 2.1 to Section 2.4) and is discussed in more detail for superallowed β-decays (Section 2.5). The determination of Vud from this type of decays requires theoretical corrections which are explained in Section 2.6. Among those corrections, the isospin-symmetry breaking corrections, δC , show discrepancies between different theoretical models which are reviewed in Section 2.7. Experimental data are critical to discriminate between different models (Section 2.8). Particularly, an improved QEC -value of 74 Rb (Section 2.9) could make tests of δC calculated in different models more stringent because 74 Rb has the largest δC among the 13 precise superallowed β-decays which are considered for the determination of Vud . Here, HCI for Penning trap mass spectrometry can provide the needed gain in experimental precision in the QEC of 74 Rb. The experimental setup utilized for the Penning trap mass measurements of radioactive HCI is introduced in Chapter 3. The radioactive beam of SCI produced at ISAC (Section 3.2) is cooled and bunched at TITAN ’s Radio-Frequency Quadrupole (RFQ) cooler and buncher (Section 3.3). Ion bunches are further transferred into an EBIT (Section 3.4) where the charge breeding to higher states takes place. Finally, the mass measurements themselves are carried out in TITAN’s Measurement Penning Trap (MPET) which is discussed in Section 3.7. 2  The mass measurements of 74−76,78 Rb and 74 Ga in charge states q = 8 − 12+, the analysis of the data, as well as systematic uncertainties are described in Chapter 4. The consequences of the present result are discussed for the QEC -value of 74 Rb and a new test for models of δC is introduced in Section 5.1. A general perspective of the opportunities offered by the introduction of HCI for Penning trap mass measurements of short-lived nuclides is given in Section 5.3 and its implications for resolving low-lying nuclear isomers (Section 5.4) is explained. The thesis is concluded by Chapter 6, which summarizes the experimental work and provides an outlook on the use of HCI for Penning trap mass measurements at radioactive beam facilities in general and for the QEC -value of 74 R in particular.  3  Chapter 2  The Standard Model and Theory of Nuclear β-Decay The current formulation of the Standard Model of particle physics was developed in the 1960s and 1970s and has since withstood experimental tests both in the low and high energy regimes. Neutrino oscillations which were established during the last decade [20] imply neutrinos are massive particles, in contradiction to the Standard Model. To account for neutrinos with non-vanishing masses, an extension of the theory is necessary for the first time. Some proposed models deviate quite substantially from the principles which the current Standard Model is built upon, and introduce new concepts, for instance, non-renormalizable interaction terms (compare to Section 2.1). An alternative solution is based on the assumption that the Standard Model is incomplete in its particle content. In contrast to quarks or the other leptons, the Standard Model does not include any right handed neutrinos (see Section 2.1). Adding right handed neutrinos while keeping the fundamental principles and symmetries intact generates a mass term for neutrinos. The new neutrinos would be sterile, i.e. they would not couple to any other Standard Model particles, explaining why they have not been observed so far. While such an extension might appear to be natural, it raises questions about the tiny coupling constants it would constitute to match with the experimental and cosmological upper limits on the neutrino masses. Further experimental data are required to constrain and to guide theory in the effort to correctly incorporate massive neutrinos. Particularly, the search for a neutrino-less double β-decay is expected to shed light onto the origin of neutrino masses. Apart from this manifested shortcoming, many other issues remain open and the emergence of a more general theory of particle physics is expected. However, the strong agreement of all experimental data, except for neutrino-oscillation, with the 4  Standard Model provides confidence that the current description must at least be a low energy approximation of a more complete theory. When interpreting the Standard Model as an effective theory, its fundamental particles and forces would resemble the degrees of freedom which are relevant for energies below several hundreds of GeV. Any search for new physics, especially on the low- energy, highprecision frontier, has to rely on the Standard Model as a starting point. This Chapter will introduce the basic concepts of the Standard Model as relevant to the experimental work and its theoretical description of (nuclear) β-decay. An attempt is made to bridge between today’s description of the Standard Model in particle physics and that of nuclear β-decays which are typically described within Fermi’s theoretical framework. As Vud is part of the former’s language but not necessarily of the latter, it is sought to accentuate their connection more explicitly in the following sections. To properly establish the link, some generally accepted basics of the Standard Model will be introduced. The first sections aim to emphasize the connection of the fundamentals of particle physics to one of its phenomena, the β-decay, and hence natural units ( = c = 1) will be used in the beginning. SI units will be introduced starting from Section 2.7.8, when theory and experiment of nuclear β- decays are more closely explained. The (implicitly) used metric follows the West Coast convention (see [21]) with gµν = diag[1, −1, −1, −1]. Although neutrinos are now known to be massive, their absolute masses and relative mass differences are small. β-decay experiments generally do not detect neutrinos directly. Consequently, neutrino oscillation or their non-vanishing masses will not affect the described β-decay experiments and neutrinos are assumed to be massless following the original formulation of the Standard Model.1  2.1  The Standard Model of particle physics  The Standard Model is a quantum field theory which is grounded on • a set of basic principles such as Lorentz invariance and the conservation of probability (i.e. unitarity of the Hamiltonian), • the experimentally observed particle content of fermions, • fundamental symmetries, • a scalar particle, i.e. the Higgs field, whose negative mass term is responsible for the spontaneous symmetry breaking, and 1  The discussion of the Standard Model and its connection to β-decays is, if not otherwise indicated, based on the following references: [21][22][23][24][25][26][27][28]  5  • the use of the most general, renormalizable Lagrangian given the previous constraints. Before introducing the specifics of the Standard Model, the relation between continuous symmetry groups and fundamental forces in our current description of particle physics will be discussed. By imposing invariance of a theory under local symmetry transformation, so-called gauge bosons are generated in the Lagrangian which are seen as the mediators of the force. A fundamental interaction is thus given by the underlying symmetry or gauge group and its representations in which different particle fields transform under the symmetry. Motivated by its success, theoretical work on physics beyond the Standard Model attempts to unify fundamental forces and utilizes a similar framework but employs different symmetry groups.  2.1.1  Local gauge invariance  In a Lie group any infinitesimal group element g can be expressed as an expansion around the identity with the expansion parameters αa following αa T a + O(α2 ).  g(α) = 1 + i a  (2.1)  T a are the generators of the group and are Hermitian operators, i.e. T a = T a+ . For notational clarity, sums are implicit over all doubled indices and the -symbol will be omitted. Any finite group member can be constructed by repeated application of the infinitesimal group element. G(α) = lim (1 + N →∞  i a a N a a α T ) = eiα T N  (2.2)  Since G depends on the parameters αa in a continuous way, a Lie group is a continuous group. Its generators follow the commutator relation [T a , T b ] = if abc T b ,  (2.3)  where the fully antisymmetric f abc are the group’s structure constants. Different Lie groups may have the same commutation relations. Up to this point, we are dealing with abstract objects. A mapping of all group generators onto n × n Hermitian matrices ta which fulfill the same commutator relations as in Equation 2.3 is called a representation of the group. Essential for our purposes is that different particles live in different representations of the same group. They act differently under the symmetry group and, hence, under the respective fundamental force. 6  Without loss of generality, we introduce an n-plet of Dirac particle fields ψ(x)T = ψ(x)1 , ..., ψ(x)n . The n × n representation ta of the group G transforms ψ(x) according to a a ψ(x) → eiα (x)·t ψ(x), (2.4) where α is also evaluated at the space-time position x. The α-dependence on x depicts this transformation as local, in contrast to a global transformation which would be independent of x. If we require the Lagrangian to be invariant under the continuous symmetry group G, a Dirac mass term mψψ is trivially left unmodified / involving a derivate ∂µ is by the transformation. However, the kinetic term ψi∂ψ not. In order to compensate for the additional terms arising through the local gauge transformation, the derivate ∂µ requires a generalization to the covariant derivative Dµ Dµ = ∂µ − igAaµ (x) · ta (2.5)  where the new vector fields Aaµ (x) are added and g is, at this point, an arbitrary constant extracted from the vector field. The transformation properties of the vector fields under the group G have to follow the expression Aaµ (x) → Aaµ (x) +  1 ∂α(x)a + f abc Abµ αc (x) g  (2.6)  to cancel the terms arising in the derivative of ψ(x) → 1 + iαa (x) · ta ψ(x). This can be shown by applying the commutator rules in Equation 2.3. Although the initial Lagrangian of the n-plet of free Dirac fields is now gauge invariant under the transformation of G, it is incomplete. Through the introduction of the new vector field Aaµ (x) and the intention to use the most general renormalizable Lagrangian, additional terms are accommodated. Indeed, Aaµ (x) is lacking a kinetic term. However, it can be added with the help of the definition of the field tensor a Fµν = ∂µ Aaν − ∂ν Aaµ + gf abc Abµ Acν (2.7) to gain a new, gauge invariant Lagrangian2 / − m)ψ − L = ψ iD  1 a F 4 µν  2  = ψ i∂/ − m)ψ −  1 a F 4 µν  2  / a · ta ψ (2.8) + gψ A  The first two terms are the free Dirac equations of the n Dirac fields followed by a 2 = F a · F a,µν . the kinetic and interaction terms among the vector fields, Fµν µν / a · ta ψ reflects the interaction between the Dirac fields and the vector Finally, gψ A 2  For simplicity, the subtlety of an additional, renormalizible, and gauge invariant term violating Parity and Time Reversal conservation is not discussed.  7  fields Aaµ (x) with g being the coupling constant. Since Aaµ (x) are consequences of the postulation of gauge invariance, they are called gauge bosons of the local gauge transformation G. Note that gauge invariance does not allow for a mass term for the gauge bosons. Since the generators of the group do not commute (see Equation 2.3), it is referred to as a non-Abelian gauge theory. The classical Lagrangian equations for Equation 2.8 lead to the Dirac equation for / a · ta ψ and to the equation of motions for the gauge the fermion (i∂/ − m)ψ = −g A boson, a c ∂µ Fµν + gf abc Ab,µ Fµ,ν = −gψγν ta ψ ≡ −gJνa , (2.9) where in analogy to electromagnetism a current of g- charged fermions was defined as Jνa = ψγν ta ψ. To summarize, an interaction is characterized by its Lie group; when a Lagrangian is required to be invariant under the action of this local symmetry, massless gauge bosons are created to establish the invariance. These manifest as the exchange particles of the interaction.  2.1.2  Symmetries and particle content of the Standard Model  Our most fundamental description of particle physics requires three symmetry groups, SU(3), SU(2), and U(1). SU(N) are the groups corresponding to all unitary transformations of N dimensional vectors with det(U ) = 1. The last requirement is relieved in U(N) symmetry groups. U(1) is associated with a simple phase rotation, exp(ia), and has only one generator. Trivially, this implies that all generators commute with each other and U(1) creates an Abelian gauge theory. Quantum Chromodynamics (QCD), the part of the Standard Model which deals with the strong interaction, only acts among quarks and is reflected by SU(3). We know of 6 types of quarks, called flavours, which are organized in 3 generations (see Table 2.1). The strong interaction, however, acts upon the colors of a quark, which motivates the often used notation SU(3)c . As there are three colors, the representation of SU(3)c is of dimension 3. The derivation of the Lagrangian essentially follows Equation 2.3 to Equation 2.8 where the generators of SU(3)c lead to the 8 gauge bosons called gluons. More details about QCD as it is relevant for nuclear physics will be discussed below. At the fundamental level, Quantum Electrodynamics (QED) and the weak interaction are unified in the electroweak force represented by SU(2)xU(1). QED also follows U(1), but instead of the electric charge U(1) refers to the so-called weak hypercharge Y . In comparison to the previous discussions, the force carriers of the weak force are experimentally known to be massive, although gauge invariance forbids massive gauge bosons. This problem can be overcome through the Higgs  8  Table 2.1: Quark generations in the Standard Model. All masses as listed in [29]. up-type 1st 2nd 3rd  name up charm top  u c t  mass 1.7-3.1 MeV 1.29+0.05 −0.11 GeV 172.9±0.6 ± 0.9 GeV  name down strange bottom  down-type mass d 4.1-5.7 MeV s 100+30 −20 MeV b 4.19+0.18 −0.06 GeV  mechanism which spontaneously breaks the SU(2)xU(1) symmetry. Here, a scalar particle is introduced whose ground state has a vacuum expectation value that does not follow the symmetry. This generates mass terms for 3 gauge bosons (W + , W − , and Z) and another massless gauge boson which is the photon γ, the force carrier of QED. With respect to SU(2), the left handed fermions are arranged in doublets. Right handed fermions are not affected by this local symmetry and are thus singlets. Without the Higgs mechanism, such a construction would not allow any Dirac mass term for fermions, because left and right handed fermion spinors are in different representation of the same group. Again, the spontaneous symmetry breaking restores the masses of fermions. There is no direct experimental evidence for right-handed neutrinos and they are omitted from the particle content of the Standard Model. Therefore, it is not possible to form a Dirac mass term mν ν L νR even after the spontaneous symmetry breaking. A more detailed discussion of the electroweak interaction can be found in Section 2.2. Table 2.2 summarizes all fermions and the scalar Higgs particle of the Standard Model (SM) as well as their transformation properties under the respective local symmetries. With the exception of the Higgs particle, all particles have been observed in experiments.  2.1.3  QCD  and nuclear structure  Given that protons and neutrons, each composites of quarks, can be bound to atomic nuclei by the strong force, QCD does in principle play a decisive role in studies of weak decays in nuclear systems. As explained later in Section 2.6, theoretical, nuclear structure dependent corrections dominate the uncertainty on the extraction of Vud from superallowed 0+ → 0+ β-decays. Owing to its much smaller coupling constant, the weak force is not of relevance for the structure of nuclei, which are governed by the strong and the electromagnetic force only. For practical purposes, however, peculiar features of QCD such as its non-perturbative character  9  Table 2.2: Fermions and scalar particles in the Standard Model. U-type quarks u are up, charm, and top quark, while down-type quarks d stand for down, strange, and bottom quark. denotes the massive leptons e, µ, and τ . ν are their respective neutrinos (νe , νµ , and ντ ). The last two columns show the dimension of the representation in which the particles transform under SU(3)C and SU(2). Particle(s) right handed up-type quarks  Notation uR  SU(3)C 3  SU(2) 1  right handed down-type quarks  dR  3  1  left handed quarks  u d  3  2  1  1  1  2  1  2  right handed  L  R  ν  left handed leptons  L  φ+ φ0  Higgs scalar  at lower energies make direct calculations of large quark compounds unpractical. In the past, nuclear structure research has thus treated nucleons as its elementary particles and developed its phenomenological nucleon-nucleon potentials which are to one degree or another inspired by, but not ultimately based on QCD. The nuclear force, i.e. the force between nucleons, is a residual interaction of QCD comparable to the van der Waals force in QED: Just as electrically charged electrons and atomic nuclei bond to form electrically neutral and energetically favourable atoms or molecules, quarks form color-neutral or color-less hadrons. Gluons acting only on color-charged objects will not be directly exchanged between nucleons. Modern phenomenological potentials are constructed based on meson-exchange between nucleons (e.g. [30][31][32][33][34]). For instance, the long range part of the potential is described by a one-pion exchange model proposed by H. Yukawa. While very successful in many applications for nuclear structure, these potentials struggle with a consistent formulation of many-nucleon forces and lack an explicit link between QCD and nuclear forces. These problems are overcome in nuclear potentials derived from effective field theory. Noting that the light quarks, u and d, which neutrons and protons are made of, have small masses compared to their typical  10  momenta, they can be treated ultra-relativistically as (almost) massless particles. Their QCD-Lagrangian, / − L = q iD)q  1 a 2 F , 4 µν  (2.10)  decouples the left- and the right-handed quarks. It is thus invariant under any transform among left (or right) handed quark fields, i.e. u d  i a σ a PL/R → αL/R 2  u d  .  (2.11)  σ a is the 2x2 representation of the three generators of U(2), which act upon (u, d)T , thus in flavour space. PL = 21 · (1 − γ5 ) and PR = 21 · (1 + γ5 ) are the leftand right-handed projection operators acting in the 4 dimensional spinor space. U(2)L xU(2)R is not anomaly-free and the correct symmetry is in fact SU(2)L x SU(2)R xU(1)B . The latter rotates all quarks by a phase and is an exact symmetry not only of QCD but of the entire Standard Model. Since each continuous symmetry implies (according to Noether’s theorem) a conserved quantity, U(1)B corresponds to the quark (or baryon-) number conservation, B = 1/3 · (nq − nq ) = const. SU(2)L xSU(2)R is called the chiral symmetry of massless (or light) QCD. This symmetry is explicitly broken by the mass term, however, since the u and d quark masses are small, it is considered to be an approximate symmetry. Additionally, a subgroup of the chiral symmetry is considered to be broken spontaneously, meaning that the ground state of QCD is, despite the symmetry, an asymmetric state. This is a necessary construction to reproduce the experimentally observed spectrum of compound quark particles. According to the Goldstone theorem (see [23]), a spontaneously broken symmetry induces a massless Goldstone boson. In the case of chiral symmetry, pions are identified as these Goldstone bosons. Their non-vanishing mass is due to the approximate character of the symmetry. More details on the general mechanism of spontaneous symmetry breaking can be found in Section 2.2. Although it remains a challenge to derive nuclear forces based on QCD, it is possible to construct an Effective Field Theory (EFT) which bridges QCD and the internucleon interaction. An EFT is a low energy approximation of a more complete theory taking only those degrees of freedom into account which are relevant at the energy scale to be described by the EFT. In modelling the nuclear forces, chiral Effective Field Theory (χEFT)[35, 36] considers neutrons and protons, as well as pions. Pions have to be included explicitly because their mass corresponds to typical nucleon momenta Q in atomic nuclei. The link to QCD is established by the most general Lagrangian involving nucleons and pions which is consistent with the 11  Figure 2.1: Chiral expansion of nuclear forces. Figure from [37].  chiral symmetries discussed above, thus the name chiral Effective Field Theory. The Lagrangian can be systematically expanded order-by-order with higher orders contributing less than lower orders. Figure 2.1 shows the expansion up to next-tonext-to-next-to leading order or N3 LO. Individual terms consist of (multiple) pion exchange reflecting the long range part of the force (see Figure 2.2a) or contact terms (Figure 2.2b) which cover short range physics not resolved at this energy scale. As the short range details will become visible at higher energies the theory naturally has a hard cut-off scale ΛB above which it is expected to fail. For chiral Effective Field Theory, this scale is typically ΛB ≈ 0.5 GeV and the perturbative expansion of the Lagrangian follows in orders of Q/ΛB . Short range physics just above ΛB also manifests itself in the non-renormalizible character of the theory. As a consequence new terms will appear order-by-order in contrast to a renormaliz12  (a)  (b)  π  N  N  N  N  Figure 2.2: Chiral expansion at leading order with one-pion exchange between two nucleons N in (a) and a contact term (b).  ible theory which only produces counter-terms to already existing terms. However, since the cut-off corresponds to a true, physical separation of scales these terms are suppressed by (Q/ΛB )n with n being the order of the expansion. Many-body forces naturally appear within the frame work of chiral Effective Field Theory. Because they contribute only at higher orders, their impact is smaller than the twobody force. This hierarchy is consistent with phenomenological models, but puts this observation on solid theoretical ground. The absence of (Q/ΛB ) terms in Figure 2.1 is related to parity conservation. As long as the low energy coupling constants cannot be derived from first principles, they have to be matched with experiment. Typically, nucleon(N)-nucleon(N) and pion-nucleon scattering phase shifts are used for the NN-force while 3-body (3N) forces are derived by fitting the couplings to binding energies or other observables of light nuclei (e.g. [36]). The chiral expansion can be connected to potentials which can be employed in the Schr¨odinger equation. The solution of nuclear many-body problems is found there with the help of modern few- and many-body techniques. A major problem already present in phenomenological potentials is the strong repulsive part of the nuclear potentials for small distances between nucleons (see Figure 2.3a). It is also referred to as the hard repulsive core of the potential. As illustrated in Figure 2.3b, it couples the low and the high momentum modes. Such off-diagonal couplings in momentum space complicate practical calculations severely because a larger model space is required or they even become non-perturbartive. Modern potentials, both phenomenological and χEFT based, can be further transformed to lower energies by utilizing renormalization group techniques [37]. While observables remain unchanged, potentials and basis are evolved by integrating out the  13  S.K. Bogner et al. / Progress in Particle and Nuclear Physics 65 (2010) 94–147  96  Fig. 1. NN phase shifts for the Argonne v18 [18] (solid), CD-Bonn [19] (dashed), and one of the chiral N3 LO [20] (dotted) potentials in selected channels (using non-relativistic kinematics). All agree with experiment up to about 300 MeV.  Fig. 2. (a) Several phenomenological NN potentials in the 1 S0 channel from Ref. [21]. (b) Momentum–space matrix elements of the Argonne v18 (AV18) 1 S0 potential after Fourier (Bessel) transformation (see footnote 1).  Figure 2.3: (a) Illustration of the repulsive core for three phenomenological 1 S channel. The repulNN-potentials (Bonn, Reid93, and heavy meson exchange (ρ , ω, ‘‘σ ’’). The short-range part of the AV18) potentialsin inthe Fig. 2(a) 0is a repulsive core (often called a ‘‘hard core’’). sive core leads to coupling of high and low momentum modes as shown Nuclear structure calculations are complicated due to the coupling of low to high momenta by these potentials. This is 1 S channel. Figure inFourier (b) for the Argonne the wave), made clear by the transform (that is, thevBessel transform inagain a givenin partial 18 potential, 0 as shown in Fig. 2(b). We feature the Argonne v18 potential [18] because it is used in the most successful high precision (� 1% accuracy) nuclear structure from [37]. calculations of nuclei with mass number A � 12 [22–24]. For our purposes, the equivalent contour plot in Fig. 3 is a clearer  representation and we use such plots throughout this review.1 The elastic regime for NN scattering corresponds to relative momenta k � 2 fm−1 . The strong low- to high-momentum coupling driven by the short-range repulsion is manifested in Fig. 3(a) by the large regions of non-zero off-diagonal matrix elements. A consequence is a suppression of probability in the relative wave function (‘‘short-range correlations’’), as seen for the deuteron in Fig. 3(b). short-range correlations abovelocal; a cut-off Λ.eachThis softens the core ofseparation the po- r alone. The potentials in Fig. 2(a) are partial-wave that is, in partial wave they arehard functions of the 2 This condition, which simplifies certain types of numerical calculations, constrains the radial dependence to be tentials and consequently decouples low and high momenta, which improves thesimilar to Fig. 2(a) if the potential is to reproduce elastic phase shifts, and in particular necessitates a strong short-range repulsion of nuclear calculations dramatically. It is from important to stress in theconvergence S-waves. The similarity of all structure such potentials, perhaps combined with experience the Coulomb potential, has led tothat the (often implicit) misconception the and nuclear must haveparameter, this form. Thisare prejudice has been reinforced the hard-cut off ΛB of that χEFT Λ,potential the evolution not the same. recently by QCD lattice calculations that apparently validate a repulsive core [25–28]. Simply reducing Λparticles, energies would have led distances, to a larger truncation error would B to lower For finite-mass composite locality is a feature we expect at long but non-local interactions be more natural at short distances. In fact, the potential short range nucleon is far removed from an observable, and locality is when neglecting more and more physicsat at higher momenta. On the other imposed on potentials for convenience, not because of physical necessity. Recall that we are free to apply a short-range hand, in the evolution of Λ to (and lower momenta the renormalization keeps the short unitary transformation U to the Hamiltonian to other operators at the same time),  � χ EFT � � � range parts of the expansion in ΛB�/Q, but integrates them out by shifting �n |� En = �Ψn |H |Ψn � = �Ψn |U Ď UHU Ď U |Ψn � = �Ψ H |Ψn �, (1) strength between different coupling constants. Different initial potentials converge to similar ones after the renormalization which is referred to as the universality 1 In units where h = c = m = 1 (with nucleon mass m), the momentum–space potential is given in fm. In addition, we typically express momenta in ¯ − of the evolved low-momentum potentials. Aside from the renormalization group, fm (the conversion to MeV is using h¯ c ≈ 197 MeV fm). 2 For example, in current implementations of Green’s Function Monte Carlo (GFMC) calculations [22], the potential must be (almost) diagonal in other methods have been successfully developed to tackle the problem of short coordinate space, such as the Argonne v potential. range correlations due to the hard core as, for example, the Unitary Correlation Operator Method [38]. χEFT has become the bond between QCD and nuclear structure. In combination with modern techniques to soften nuclear potentials, the research of atomic nuclei can now be approached in a consistent framework over the whole nuclear chart [39]. Different methods to solve the nuclear many-body problem will remain to be more suitable to their specific mass regions. While different in their techniques, 1  18  14  their underlying physics input, i.e. the interaction potentials between nucleons, can now be solidly linked to QCD. Although χEFT based potentials have so far not been used in the context of superallowed β decays, two recent developments shall be noted: nuclear matrix elements for neutrinoless-double β decays and thus related to weak interaction studies with atomic nuclei have been calculated for the first time with χEFT [40]. Secondly, theoretical isospin symmetry corrections important for superallowed nuclear β decays have been derived microscopically within density functional theory [41]. Hence, it would be interesting to calculate these corrections also based on χEFT. Before concluding the topic of QCD with light quarks, let’s rearrange the symmetry transformation due to PL/R = 21 (1 ± γ5 ) in Equation 2.11 to a vector u d  i → αa σ a 2  u d  i → α a σ a γ5 2  u d  (2.12)  and an axial-vector transformation u d  .  (2.13)  While the latter symmetry is broken by any mass term in the Lagrangian, the vector transformation in SU(2) is a pure rotation in flavour space and remains an exact symmetry as long as the masses of u and d quark are identical, which is assumed here. Its associated quantum numbers are called isospin T . We define its z-projection for the u-quark as Tz = −1/2 and for d-quark as Tz = +1/2, such that in the baryon doublet with T = 1/2 of neutron (udd) and proton (uud), Tz = +1/2 corresponds to the neutron. The isospin symmetry is in fact broken, too. This gives rise to the small mass difference between neutron and proton.  2.2  Electroweak interaction in the Standard Model and Vud  The existence of massive gauge bosons in the electroweak interaction necessitates a modification of the protocol for gauge theories outlined in Section 2.1. In the Higgs mechanism, spontaneous symmetry breaking appears through a scalar doublet φT = (φ1 , φ2 ) which transforms under SU(2)xU(1) according to φ → eiα  a σa  β  ei 2  15  φ1 φ2  .  (2.14)  σ a are the Pauli matrices. Following Equation 2.5, its covariant derivate is expressed as Dµ φ = ∂µ − ig1 Aaµ (x) · σ a − ig2 Bµ (x) φ. (2.15) The condition of the Standard Model to use the most general, renormalizible Lagrangian requires the addition of ∆L = +µ2 φ+ φ − λ(φ+ φ)2 .  (2.16)  Although µ2 φ has the form of a scalar mass term, we are allowed to choose µ > 0 as long as λ > 0. Then, VHiggs = −µ2 φ+ φ + λ(φ+ φ)2 remains bound from below. When expanding around the minimum of the potential, φ(x) = √ (0, µ/ 2λ + H(x)), the kinetic term of the scalar field, (Dµ φ)(Dµ φ) contains a term independent of H(x). It resembles mass terms for three gauge bosons when the four initial gauge fields Aaµ and Bµ are redefined to W± = Z =  1 √ A1µ ∓ iA2µ 2 1 g1 A3µ − g2 Bµ . 2 2 g1 + g2  (2.17)  These are identified as the exchange particles for the weak interaction. The remaining field, 1 Aµ = g2 A3µ + g1 Bµ , (2.18) 2 g1 + g22 is the massless photon as the force carrier of QED. With respect to U(1) of QED, the Z- boson is neutral, i.e. it does not couple to the photon alone. W + and W − are charged with +e and −e, respectively, when the electromagnetic coupling constant is defined as e = g1 · g2 / g12 + g22 .  2.2.1  Masses of fermions  The weak interaction only operates on left-handed fermions. Left- and righthanded fermions act in two different representations of SU(2). Latter ones are unchanged under SU(2) and are singlets. Left-handed fermions are arranged in doublets, each generation within one doublet. Hence, νe e−  , L  νµ µ−  16  , L  ντ τ−  (2.19) L  for the leptons and u d  , L  c s  , L  t b  (2.20) L  for quarks. Consequently, no direct mass term can be part of the initial Lagrangian, because it would not be gauge invariant as left- and right-handed fermions transform differently. Fermionic masses are created due to the so-called Yukawa coupling which ties left-handed doublets and right-handed singlets to the scalar field. For instance, in the quark sector it becomes 0 djR µ/ 2λ + H(x) √ −µ/ 2λ − H(x) ujR + h.c., 0 √  ∆L = − Λij · (u, d)iL · ˜ ij · (u, d)i · − Λ L  (2.21)  where i and j sum over the 3 generations with ui=1,2,3 = (u, c, t)and di=1,2,3 = (d, s, b). In the second part of the sum the scalar doublet is included as −iσ 2 φ∗ . All other combinations of uiR , diR , (u, d)iL , and φ would either violate SU(2) or U(1). The components without H(x) are seen as quark mass terms. i  Lmq = −Mdij dL djR − Muij uiL ujR + h.c.  (2.22)  Mdij and Muij are not constrained by any symmetry and are thus not necessarily diagonal. This implies that the quark eigenbasis of the weak interaction is not identical to its mass eigenbasis. With the unitarity transformations, uLi = ULij ujL ,  ij j dLi = DL dL  uRi = URij ujL ,  ij j dR dRi = DL  (2.23)  we rotate the weak eigenbasis into the mass eigenbasis, where the mass matrices are diagonal.  2.2.2  The Cabibbo-Kobayashi-Maskawa matrix  Most terms in the Standard Model Lagrangian contain both uR/L and uR/L (or dR/L and dR/L ). Hence, in these cases there is no difference in the Lagrangian between bases connected by the unitarity transformations of Equation 2.23. The only exceptions are the contractions of quarks with W -bosons which arise from  17  the covariant derivatives of (left-handed) quarks. g i L = √ uiL γ µ diL · Wµ+ + dL γ µ uiL · Wµ− 2  (2.24)  in the weak eigenbasis turns into g i +,jk ij +,jk L = √ ULij DL · uLi γ µ dLk · Wµ+ + DL UL · dL γ µ uLk · Wµ− 2  (2.25)  when transforming into the mass eigenbasis. Since these are the only terms where the transformations of Equation 2.23 can explicitly appear in the Lagrangian of the Standard Model, we set for convenience ij = ULij = δ ij , URij = DR  (2.26)  + which transforms the where δ ij is the Kronecker Delta and define V = UL · DL left-handed, down-type quarks from the mass into the weak eigenbasis d = V · d or       d Vud Vus Vub d  s  =  Vcd Vcs Vcb  ·  s  . (2.27) b L Vtd Vts Vtb b L  This unitary matrix is called the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. The physical quarks are the mass eigenstates. However, the weak interaction acts within a different basis which results in mixing of quark flavours as seen by the contraction of one up-type quark with all three flavours of the downtype quarks in Equation 2.25. The unitarity of the CKM matrix requires that its Hermitian conjugate is its inverse, i.e. V + · V = 1 or Vji∗ · Vjk = δik . From these, 9 linearly independent equations reduce the 18 real parameters of any 3 x 3 complex matrix to 9 real parameters (3 real amplitudes or rotation angles and 6 phases) of a unitary matrix. Additionally, not all of the 9 real parameters of the CKM matrix can actually appear in the Lagrangian of the SM, analogously to the rotational matrices UR and DR in Equation 2.23. Hence, they cannot be observed in any physical process. The physically significant parameters of the CKM matrix further reduce to three real numbers and one complex phase. As previously mentioned, these four parameters are not given by the Standard Model itself. But the requirements of unitarity and the consistency with only four parameters in the CKM matrix allows stringent experimental tests of the electro-weak quark sector of the Standard Model. While in mathematical sense V · V + = 1 is equivalent to V + · V = 1, we ∗ = δ . These are can experimentally examine both Vji∗ · Vjk = δik and Vij · Vkj ik each 3, hence totally 6, vanishing equations which one can picture as triangles in  18  !"#$!%&'()*#+,-./&  234&  ",-!&&  •! 5,6*.71&8.+7&-.679/&  ",/!&&  •! :&7;-&<&-.679/&=:>?@A&B7B71A&B.**.C&  ",8!&&  •! DE71(*.//&B&-.679/&=B7B71A&B.**.C&  ;-&234&  "6-!A!"6/!&&  •! G&-.679/&=D>@?H6C&  "68!&&  •! B&-.679/&=B7B71A&B.**.C&  &234&  "+-!A!"+/!&&&  Figure 2.4: Feynman diagrams corresponding to possible experimental deter•! B-&A&B/&J#K#;L&=DG0A&GMA&B7B71A&B.**.C& minations of the CKM matrix elements. Figure from [42].  "+8!&&  •! I&-.679/&=DG0A&GMC&&  .8&QHRSA&TMMQ&  the complex plane. Among these, the so-called unitarity triangle  D:J&PE9/#6/&H&2U&-.&F7;L13&=O505H>50C&  Q&  ∗ + Vcd Vcb∗ + Vtd Vtb∗ = 0 Vud Vub  (2.28)  is commonly tested. The triangle in the complex plane is constructed by dividing the equation by Vcd Vcb∗ , identifying the absolute value of each summand as a side length of the triangle. The area of the triangle is related to the violation of CP, i.e. combined charge conjugation and parity symmetry, in the CKM sector. Additionally, the non-vanishing parts of V + · V = V · V + = 1 constitute 6 real equations among which the first row |Vud |2 + |Vus |2 + |Vub |2 = 1  (2.29)  sets the most stringent limits on physics beyond the Standard Model [26]. Figure 2.4 provides an overview of how individual entries of the CKM matrix can be accessed experimentally.  2.3  CKM -unitarity  test via |Vud |2 + |Vus |2 + |Vub |2 = 1  In order to probe the unitarity of the CKM quark mixing matrix within the first row, precise and accurate experimental data in addition to theory are required to extract  19  u  K+  u  π0  f + (0)  s Vus  W  u e+  +  νe  time Figure 2.5: Feynman diagram of an example for a K π 0 e+ νe can be used to measure Vus .  3  decay: K + →  Vud and Vus . At the current level of precision of Vud and Vus , the value of Vub is too small, |Vub | = (3.93 ± 0.36) · 10−3 [26], and only marginally affects the unitarity test. This particular test had for a long time fallen short of unity by more than 2 standard deviations. Less than a decade ago, this deviation had been resolved by new measurements of Vus , which is most precisely known from kaon decays. In semi-leptonic decays called K 3 , a kaon disintegrates into a pion and a pair of leptons following K → π ν. An example for a K 3 decay is shown in Figure 2.5. Following the E865 experiment [43] at Brookhaven National Laboratory (BNL) several collaborations [44, 45][46][47][48][49] have confirmed a new value of Vus which re-establishes the unitarity condition in the first row of the CKM matrix. Previous, conflicting measurements were rejected by the Particle Data Group (PDG) due to inadequate treatment of radiative decay [29]. Results of the recent measurements provide |Vus | · f + (0), where f + (0) is the form factor of the respective K 3 decay at momentum transfer q 2 = 0. Form factors take into account that we are not dealing with free quarks, but quarks bound to form hadrons. A kaon’s quark content also involves an s- quark, the third light quark. Although heavier than the uand d- quark, it is sufficiently light that the SU(2) of light QCD in Section 2.1.3 can be extended to SU(3) when adding the s-quark. In this limit and assuming the later discussed Conserved Vector Current hypothesis, f + (0) = 1 holds. As this SU(3) is broken, QCD calculations are needed as small symmetry breaking corrections. Recent lattice QCD calculations yield f + (0) = 0.960+5 −6 [50, 51] and 0.9560(84)[52], respectively, when correcting for isopin-symmetry breaking, SU(2), separately. Information about Vus can also be gained through pure leptonic kaon decays, K 2 , from K → ν. Their form factors (in this case also often called decay constants) 20  (a)  K+  s fK  u (b)  π+  d fπ  u  W+ Vus W+ Vud  µ+ νµ µ+ νµ  Figure 2.6: Feynman diagrams of K + (a) and π + (b) decay branches to extract |Vus | · fK /(|Vud | · fπ ). Table 2.3: Vud and Vus extracted from various decays. Quantity |Vud | |Vus | |Vus |/|Vud | |Vus |/|Vud |  Type of Decay(s) superallowed 0+ → 0+ β-decays K3 K2 τ -decay  Value 0.97425(22) 0.2254(13) 0.2312(13) 0.2314(24)  Reference [15] FlaviaNet [54]a FlaviaNet [54]b [55]c  a  Values are taken from [54] because other recent evaluations in [56] and the PDG [29] do not include a new value for f + (0) in [51]. b Values are taken from [54] because it takes all calculations of f /f into K π account. c [55] uses 1.189(7) from [53] for f /f instead of the averaged 1.193(6) in [54]. K π are calculable, but at lower precision. Thus, K 2 cannot directly compete with K 3 . However, the ratio of the kaon’s form factor fK to the pion’s form factor fπ is known from lattice QCD with similar relative precision [53]. Hence, decay rate measurements of K + → µνµ and π + → µνµ provide |Vus | · fK /(|Vud | · fπ ) (see Figure 2.6), from which |Vus |/|Vud | is inferred. An interesting cross check of the previous determination of |Vus |/|Vud | is found in the hadronic decay of the τ - lepton which is heavy enough to decay into both kaon (τ − → K − ντ ) and pion (τ − → π − ντ ) (see Figure 2.7). Analogously to K 2 decays, |Vus | · fK /(|Vud | · fπ ) can be accessed in experiment with the same fK /fπ as above. The measurement of Vud is the main topic of this work and will be discussed in 21  (a)  Vus τ−  W−  u fK  K-  s ντ (b)  Vud τ−  W−  u fπ  π−  d ντ Figure 2.7: Feynman diagrams of τ − decay branches to K − (a) and π − (b) which allow the determination of |Vus | · fK /(|Vud | · fπ ).  more detail in the next chapter. Nuclear β- decays, the neutron decay and the pion’s β-decay branch are studied to yield Vud . Table 2.3 summarizes measurements of Vus and the most precise value for Vud from superallowed 0+ → 0+ nuclear β-decays. Fitting Vud and Vus from this data neglecting the less precise τ decay leaves |Vud | = 0.97425(22) unchanged, but yields |Vus | = 0.2253(9) [54]. The correlation between the two is negligible. Hence, the unitarity test of the first row of the CKM matrix results in |Vud |2 + |Vus |2 + |Vub |2 = 0.9999(4)(4),  (2.30)  in agreement with unity. Respective uncertainties are due to Vud and Vus . Even though Vud is known much more precisely than Vus , its larger value causes an equal share of Vud and Vus to the uncertainty of the unitarity test.  2.3.1  Implications for physics beyond the Standard Model  While Equation 2.30 does not highlight any deviations, it puts stringent limits on models beyond the Standard Model. • 4th quark generation: Missing decay strength could indicate a fourth generations of quarks usually denoted t’ and b’. Equation 2.30 sets constraints on Vub according to the given uncertainty. However, at the present level of precision, sensitivity to a fourth generation is only possible if it does not follow the hierarchy |Vud | > |Vus | > |Vub | > |Vub | as observed for the first three 22  generations. As mentioned before, to date not even |Vub | has a significant impact on this unitarity test. A priori, the SM does not demand this hierarchy and on the (massive) neutrino sector the CKM equivalent, the PontecorvoMaki-Nakagawa-Sakata (PMNS) matrix, is not as hierarchically structured. This constraint is complementary to direct searches at colliders which put limits on the masses of new quarks [29]. As the number of quark and lepton generations have to be equal in order to cancel the gauge anomaly in the weak interaction, limits on a fourth generation also arise from respective heavy lepton searches [29]. • Coupling universality: In the Standard Model the left-handed lepton doublets couple to the W ± -bosons analogously to the quark doublets (see Equation 2.24). The quark coupling only differs by the additional factor due to the CKM-element Vij . In the low-energy effective Lagrangian the coupling constant is called the Fermi constant GF (see section Section 2.4). In the Standard Model |Vud |2 + |Vus |2 + |Vub |2 = GτF = GbF GµF = GCKM F  (2.31)  holds, where the respective Fermi constants are extracted from muon decay, CKM-studies, leptonic tau-decay, and from the fine structure constant α in combination with weak boson properties. Although in agreement with the other estimates, GτF ’s precision is too low for stringent Standard Model tests. Assuming CKM unitarity, deviations from GµF = GCKM F could be caused by exotic muon-decays.3 In fact, some possibilities for non-standard muon decays are orders of magnitude more constrained by direct searches. But others such as µ+ → e+ ν e νµ , where both neutrinos are the respective antineutrinos of the regular µ- decay ( µ+ → e+ νe ν µ ), receive the most stringent [56, 57]. limit by GµF = GCKM F GbF is known about 4 times less precise than GCKM (again assuming unitarity F in the CKM matrix). The experimental agreement between GµF , GCKM F , and b ∗± GF poses limits on new heavy SU(2)L doublets or excited W bosons in extra dimensions with mW ∗ > 2 − 3 TeV [56, 57]. • New boson Zχ in SO(10): A candidate for a Grand Unified Theory (GUT), the unification of all forces in the Standard Model, could be SO(10) which is the group of all orthogonal transformations in 10 dimensions with det = 1, i.e. all rotations. SO(10) can be broken down to SU(3) x SU(2) x U(1) x CKM Note that Gµ |Vud |2 + |Vus |2 + |Vub |2 and |Vud |2 + |Vus |2 + |Vub | = 1 are not F = GF two independent tests of the Standard Model. The CKM- matrix elements are calculated assuming CKM Gµ F = GF (see Section 2.6.3). 3  23  the V„j, beyond already been acin Eq. (5) to  +00021+~.  he bounds  mass effects. (3) The summation in Eq. (10) is over the Z; Effects of mixing' are included mass eigenstates. Q' in that expression. (4) Because the through the C; and diagrams in Fig. 1 are dominated by high-frequency loop corrections to the hamomenta, QCD strong-interaction dronic P-decay amplitudes are calculable. Following the analysis in Refs. 10 and 16, we find those corrections e  Ve  can contribute to e can, therefore, ch bosons via the evoted to discuss-  a simple exo an effective ry with N addimz. Their couetrized by  ng  (9a)  I  I  I  I  W  , I I  I  Z  /  W I  e  Ve  I  I  Z;  I I  d  e  Ve  W  /  Ve  /  ~~Zj /  I I  d  U  U  e  U  d  U  Figure 2.8: Feynman diagrams in the electroweak interaction between quarks einvolving e "e an extraeZ boson. Ve e Ve and leptons Figure from [58]. I I I  I  Zj  W  I I  /  I I  IW  W  I I  I  ~  Z  'll  ~  Zj)  /  W  I  I  / U(1) where the first three groups mirror the Standard Model and the last V~ V~ V~ I V~ U(1) corresponds to a new, neutral gauge boson Zχ . Zχ enters in Feynman diagrams used to measure CKM-matrix elements (Figure 2.8), but also to the muon [58]. Currently, the lower to mass limit on Zχ muon based onPthe FICJ. deI. decay Box-diagram corrections and quark unitarity test is m > 460 GeV which is less stringent than the tightest — χ For the case of V„, and V„b, d +s, or cays involving Z; Zbosons. bound from collider searches (mZχ > 822 GeV) [26]. in b the above diagrams. I  I  Moreover, the unitarity test constrains right-handed currents, in addition to the Standard Model’s left-handed current in the weak interaction [26] or aspects of SUperSYmmetry (SUSY) [59]. Many of these probes of the Standard Model are at or close to the TeV scale and impose tight constraints on models for new physics. Due to the equal contributions of Vud and Vus to the uncertainty in the unitarity test, progress on both sides is essential for a further error reduction. Additionally, all doubts about the accuracy of theory and experiment in respect to Vud and Vus have to be carefully investigated. Otherwise the precision is not worth as much as is claimed. The importance of later considerations has been demonstrated in the already discussed shift of Vus which had incorrectly created a tension of the unitarity in the CKM matrix. We will next turn to the determination of Vud . Here, concern was raised about theoretical isospin-symmetry corrections (see Section 2.7) and discrepancies in the measurement of β-decay transition energies were found and resolved through Penning traps (Section 2.9). Searches for physics beyond the Standard Model on the level of the effective Lagrangian of nuclear β-decays such as scalar or tensor interactions are also discussed in [15, 60].  2.4  Nuclear β-decays from first principles of the Standard Model  Before we develop the formalism for nuclear β-decay, we describe the decay at the quark-level only. Hence, we neglect the hadronic structure in which the quark is 24  embedded in and treat the quark as a free particle. In the rest frame of the decaying particle, a decay rate Γ in quantum field theory is given as dΓ =  1 2mi  f  d3 pf 1 (2π)3 2Ef  · |M (i → f )|2 · (2π)4 δ (4) (pi −  pf ),  (2.32)  f  where mi is the mass of the particle, the product and sum are over all particles in the final state, and Ef is the energy of a particle after the decay. The four Dirac deltas, notated as δ (4) , enforce energy and momentum conservation and M (i → f ) is the transition matrix element. The interaction Lagrangian responsible for the weak decay of a u- to a d-quark and vice versa follows according to Equation 2.25 g LI = √ 2  ∗ Vud · uL γ µ dL · Wµ+ + Vud · dL γ µ uL · Wµ−  + eL γ µ νe,L · Wµ− + ν e,L γ µ eL · Wµ+ .  (2.33)  Here we have dropped the u because we will from now on always operate in the physical mass basis. The (charged) weak interaction Lagrangian for electrons e and electron neutrinos νe is identical to the quarks with the exception of the additional factor Vud . Second and third generation leptons are not considered, because the transition energy in nuclear β-decays is less than the muon or tau mass. It is common to drop the L-subscript in favour of the left-handed projection operator PL = 1/2 · (1 − γ 5 ), e.g. eL γ µ νe,L = 1/2 · eγ µ (1 − γ 5 )νe . The right handed part of the fermion fields will thus vanish. The transition matrix element of the decay u → de+ νe is constructed by utilizing respective Feynman rules. Although these are typically derived in standard textbooks, we will recapitulate some critical steps which will later be essential to expand the formalism to hadrons and to derive the effective low-energy Lagrangian. The transition matrix element M is related to the S-matrix of the process via (pe+ , se+ ), (pνe , sνe ), (pd , sd ) | S | (pu , su ) .  (2.34)  Here, each fermion is annotated with its 3-momentum p and its z-projection in spin, i.e. s = ±1/2. With S = 1 + iT , it can be evaluated to4 (pe+ , se+ ), (pνe , sνe ), (pd , sd ) | iT | (pu , su ) = (pe+ , se+ ), (pνe , sνe ), (pd , sd ) | T exp i 4  d4 xLI (x) | (pu , su )  (2.35) f.c.d.  f.c.d. stands for fully connected diagrams and T in front of the exponential means time-ordered.  25  Non-trivial terms will first appear at the second order of the exponential’s expansion. Since (pe+ , se+ ) | e and (pe+ , se+ ), (pνe , sνe ), (pd , sd ) | u cannot contract, the only remaining terms of Equation 2.33 at second order expansion are 2i2 g 2 2! · 2  (pe+ , se+ ), (pνe , sνe ), (pd , sd ) | 1 − γ5 u(x) · Wµ− (x) × 2 1 − γ5 d4 yν e (y)γ ν e(y) · Wν+ (y) | (pu , su ) . 2 ∗ d4 xVud · d(x)γ µ  T  (2.36)  Putting the numerical factor and the integrals aside, the bracket can be separated by Wick’s theorem (e.g. in [23]) to 1 − γ5 e(y) | 0 × 2 0 | TWν+ (y)Wµ− (x) | 0 ×  (pe+ , se+ ), (pνe , snue ) | ν e (y)γ ν ∗ · d(x)γ µ (pd , sd ) | Vud  1 − γ5 u(x) | (pu , su ) . 2  (2.37)  0 | TWν+ (y)Wµ− (x) | 0 is the W-boson’s propagator which can be expressed as 0 | TWν+ (y)Wµ− (x) | 0 =  d4 q −igµν iq·(x−y) e . (2π)4 q 2 − m2W  (2.38)  The contraction of the brackets with the fields and the integrations over the fourvectors x and y create δ (4) -functions which establish 4-momentum conservation at each vertex, hence, q µ = pµu − pµd . Considering the available transition energies of a few MeV in nuclear β-decays the momentum transfer q is negligible compared to the mass of the W-bosons (mW ∼ 80 GeV). The propagator (in momentum space) can be approximated by igµν /m2W . The resulting transition matrix element ∗ g2 Vud s d d (pd )γ µ (1−γ 5 )usu (pu )·ν seν (pν )γµ (1−γ 5 )ese+ (pe+ ) 8m2W (2.39) could have also be obtained at first order expansion of Equation 2.36 with an effective Lagrangian  i·M (u → de+ νe ) =  LI (x) =  g2 V ∗ d(x)γ µ (1 − γ 5 )u(x) · ν e (x)(1 − γ 5 )e(x). 8m2W ud  26  (2.40)  (a)  W+ g  u  νe  gVud (b)  u  e+  d  Vud GF  e+ νe d  Figure 2.9: Lowest order Feynman diagram for the ‘decay‘ of a u-quark within the Standard Model (a) and its low energy approximation in means of a contact term (b).  All fermions are now evaluated in the expansion at the same space-time point x. As expected, in the effective Lagrangian the short-range physics (i.e. the exchange of the W-boson) is not resolved at the lower energy. Instead, it is moved into a contact term (see Figure√2.9). The coupling constant relates to the definition of the Fermi constant as GF / 2 = g 2 /(8m2W ). In correspondence to Equation 2.9, ν e (1−γ 5 )e and dγ µ (1 − γ 5 )u are called the leptonic and quark current, respectively. Equation 2.32 can now be evaluated utilizing the derived matrix element. However, since quarks only exist in colourless hadrons, we will first discuss the hadronic complications to the matrix elements.  2.4.1  Neutron β-decay  The computation of the transition matrix element of the neutron decay follows analogously the derivation on the previous pages. Within the neutron (udd) a dquark decays into a u-quark resulting in a proton (uud) (Figure 2.10). Pending an exact description of the neutron by means of QCD, we can not evaluate the hadronic part (compare to Equation 2.37 for the u-quark decay) from first principles. Instead, one arrives at Vud GF √ p | uγ µ (1 − γ 5 )d | n · e, ν e | eγµ (1 − γ 5 )νe | 0 . 2  (2.41)  The leptonic current can be dealt with as before. The hardronic part is usually split into the vector (or Fermi) p | uγ µ d | n and an axial-vector (or Gamow-Teller) part p | uγ µ γ 5 d | n . Lacking knowledge about the QCD details, we parameterize 27  W  −  Figure 2.10: Feynman diagram of the neutron decay  each part and introduce form factors to cover the hadronic structure. In momentum space, the most general vector based on all available momenta and spins allows the Fermi component to be written as sp  p | uγ µ d | n = ψ p (pp ) f1 pµ + f2 q µ + f3 γ µ + f4 iσ µν pν + f5 iσ µν qν ψnsn (pn ) (2.42) with pµ = pµn +pµp and q µ = pµn −pµp . ψp and ψn are the Dirac spinors of the neutron and proton respectively which are themselves spin-1/2 fermions. Applying the Gordon identity (see e.g. in [23]) for free Dirac particles, the previous expression can be simplified to fM (q 2 ) µν fS (q 2 ) µ sn iσ qν + q ψn (pn ). 2M 2M (2.43) M is an arbitrarily chosen mass to keep the form factors dimensionless. gV and fS are called the vector and the induced scalar form factors. In analogy to QED, where the corresponding form factor gives rise to the anomalous magnetic moment, fM is known as the weak magnetism form factor. A second analogy to QED will lead to another reduction in the number of required form factors: For an electromagnetic current J µ , charge conservation is manifested in ∂µ J µ = 0. With a Fourier transform, J(q = p1 − p2 ) = d4 x · exp(iqx)J(x), the continuity equation is qµ J µ = 0 in momentum space. Since the structure of the weak vector current in Equation 2.43 is identical to the electromagnetic current, it is expected that qµ p | uγ µ d | n = 0 holds. This is one aspect of the Conserved sp  p | uγ µ d | n = ψ p (pp ) gV (q 2 )γ µ +  28  W−  W−  Figure 2.11: Two examples of Feynman diagrams which modify the weak axial-vector coupling constant gA due to the strong interaction.  Vector Current (CVC) hypothesis [61]. Then, the weak magnetic component vanishes naturally due to the antisymmetry of the Dirac matrices, σ µν qµ qν = 0. In the limit of exact isospin-symmetry the masses of protons and neutrons are equal. Hence, ψ p (pp )qµ γ µ ψn (pn ) = ψ p (pp )(p /n − p /p )ψn (pn ) = (mn − mp )ψ p (pp )ψn (pn ) = 0 (2.44) when the free Dirac equation is employed. In contrast, the induced scalar contribution does not vanish; therefore, CVC, qµ J µ = 0, requires fS to vanish. Similar steps can be undertaken for the axial part of the hadronic weak current. Due to the negligible momentum transfer in neutron and nuclear β-decays, we continue by working in the limit q 2 → 0 and neglect all terms in q. sp  p | uγ µ (1 − γ 5 )d | n = ψ p (pp ) gV (0)γ µ − gA (0)γ µ γ 5 ψnsn (pn )  (2.45)  gA is renormalized in the presence of the strong force as illustrated by meson exchange diagrams in Figure 2.11. However, gV is protected by the CVC hypothesis. The strong interaction does not renormalize the electric charge as evident by the equality of the absolute charge value of electrons and protons. In the same manner, the CVC hypothesis assumes gV is not affected by the strong force and should for q 2 → 0 be equal to the free quark limit, gV (0) = 1. The validity of CVC itself is subject to intense experimental scrutiny, particularly through studies of superallowed nuclear β-decays. In the standard or Dirac representation of the γ matrices (see e.g. in [22]) the (4  29  dimensional) spinor of a free fermion (which we will associate with a nucleon) ψ s (p) =  χs  σi ·pi s E+m χ  −−−−→ p m  χs 0  (2.46)  is in the non-relativistic limit p m represented by the 2-dimensional χs only. For the neutron at rest this is exact. For the proton it is a good approximation due to the small momentum transfer, q 2 m2p , i.e. the recoil of the proton is neglected. When applying Equation 2.46 to Equation 2.45 and using again the standard representation, one obtains µ0 µi p | uγ µ (1 − γ 5 )d | n = gV (0)χ+ − gA (0)χ+ p χn δ p σ i χn δ .  (2.47)  The vector or Fermi component consequently cannot flip the nucleon’s spin (∆S = 0), while the axial or Gamow-Teller part may, due to the presence of σi , change the spin (∆S = 1). This implies that in the Fermi decay contribution the anti-neutrino and electron couple to S = 0, and for Gamow-Teller to S = 1. Since the neutron decay is spin 1/2 to spin 1/2, it involves both components. Combining the results (Equation 2.47, Equation 2.41) we end up with Vud GF 5 sν s µ0 s µi se √ (gV χs,+ − gA χs,+ p χn δ p σi χn δ ) · e (pe )γµ (1 − γ ))ν (pν ) 2 (2.48) for the transition matrix element. We have dropped the explicit notation of q 2 → 0 for gV and gA . The leptonic current was obtained analogously to the u-quark decay. Equation 2.32 yields the neutron’s lifetime τ = Γ−1 , but also momentum correlations between the outgoing fermions. Correlation measurements in neutron and nuclear β-decay are themselves a wide field of study to probe the Standard Model because they could be sensitive to new physics in the form of scalar or tensor interactions or deviations from the maximal parity violation in the V − A theory [60]. The measurements discussed in this thesis are neither with spin-polarized neutrons or nuclei nor are they sensitive to the spin of the outgoing particles. When averaging over all initial spin states and summing over all final spins the total decay rate of the neutron becomes iM =  Γ=  G2F | Vud |2 2 2 (gV + 3gA ) 2π 3  Q+me me  dEe (Ee − Q − me )2 Ee  Ee2 − m2e (2.49)  where the phase space integral is over the electrons’ total energy and Q is the mass difference between initial and final (massive) particles, Q = mn − mp − me . The literature commonly defines λ = gV /gA and normalizes the integral to the  30  electron’s rest energy, me . Vud is then related to the neutron’s mean lifetime τ and λ according to 2π 2 1 | Vud |2 = 2 2 5 · (2.50) GF gV me τ (1 + 3λ2 )f with the statistical rate function f W0  f= 1  dW (W − W0 )2 W  W 2 − 1.  (2.51)  Hence a measurement of the neutron’s half-life as well as λ allows for an experimental determination of Vud , when taking GF from measurements of the muon decay. The mass difference between neutron and proton is well known. Although a more exact treatment has to take radiative corrections as well as a modification of f into account, the neutron decay is a theoretically clean way to determine Vud because there are no nuclear corrections to consider. This feature is also shared by the β-decay of the pion, but this pion decay branch has a tiny Branching Ratio (BR) in the order of only 10−8 . Thus, the neutron β-decay is often considered to be the most attractive option to determine Vud . However, the uncertainty is dominated by experiment, particularly, since the neutron lifetime remains intensely debated with considerable shifts in its mean-value over the decades (Figure 2.12). More recently, a measurement by Serebrov et al. in 2005 [62] deviated from the previous world average by 6.5 σ and from the previously most precise single measurement [63] by 5.6 σ. The debate about these seriously conflicting data was intensified in 2010 by Pichlmaier et al. [64] who reported a neutron lifetime closer to [62] (see Figure 2.12b). Furthermore, Serebrov and Fomin [65] have challenged the analysis of systematic errors in previous experiments of trapped Ultra Cold Neutrons (UCN), including the one in [63]. Following a recent review [66], these discrepancies can only be resolved by detailed re-analysis of existing measurements or through improved next generation experiments. In the latter context, consistency between results from trapped UCN and currently less precise neutron beam experiments (see Figure 2.12b) would be of particular importance as both are dealing with very different sources of systematic errors. In addition to the unsatisfactory situation regarding the neutron lifetime, the determination of Vud by studying the neutron β-decay requires knowledge about λ = gV /gA (see Equation 2.50). Experimental results for λ show a larger spread than statistically expected, too, and the PDG had to inflate the uncertainty on the weighted average accordingly [29]. Hence, all these difficulties make Vud from neutron decay less competitive in comparison to superallowed nuclear β-decays.  31  Introduction  (b) Neutron lifetime [s]  (a)  ! 895  Trap experiments Beam experiments 890  885  880  875  1990  1995  2000  2005  2010  !!!!!  Figure 2.12: Neutron lifetime τ : (a) historical evolution of τ measurements over the last half-century. Figure from [29]. (b) Experimental results currently considered by the Particle Data Group (PDG)[29]. The horizontal band represents the PDG average which includes a scale factor to take into account the inconsistencies between the data. See text for details. !!  !  1  2.4.2  Nuclear β-decays  !!  !!  In the previous section the formalism of the neutron’s β-decay has been introduced. The proton β-decay is energetically forbidden (mn > mp ), but its decay formalism could be developed along the same lines. Within an atomic nucleus this energy might be available and in the nuclear environment the decay of a neutron into a proton and vice versa are observed. According to the electric charge of the lepton, they are known as β − and β + . Related to each other by the crossing symmetry, the β + is in competition with the nuclear Electron Capture (EC). In this process, an electron from an atomic shell is captured into the nucleus. With the introduction of form factors, the formalism of the previous section is operating on nucleons instead of the fundamental quarks. Nucleons are also the entities nuclear theory is working with. In this sense, the framework is already well suited for the correct resolution scale. However, it needs to be further adapted to properly integrate the nuclear surrounding in the β-decay. Following Equation 2.41 we will need to evaluate i · Mβ − =  Vud GF √ ψD | γ µ (1 − γ 5 ) | ψP · ese (pe )γµ (1 − γ 5 ))ν sν (pν ) (2.52) 2  32  Figure 2: A historical perspective of values of a few particle properties tabulated in this Review as a function of date of publication of the Review. A full error bar indicates the quoted error; a thick-lined portion indicates the same but without the “scale factor.”  for β − and i · Mβ + =  ∗G Vud F √ ψD | γ µ (1 − γ 5 ) | ψP · ν seν (pν )γµ (1 − γ 5 )ese+ (pe+ ) (2.53) 2  for β + decays. ψP and ψD denote the parent and daughter nucleus of the decay. But in contrast to Equation 2.42 and later equations, we cannot describe a nucleus as a Dirac spinor. Nuclei are generally not spin 1/2 objects and are compound objects with a spectrum of excited states built upon the ground-state. It is our aim to translate ψD | γ µ (1−γ 5 ) | ψP into a language of nuclear theory such that nuclear wavefunctions can be employed. To do so, let’s reconsider the neutron decay. According to Equation 2.47 and Equation 2.48, the non-vanishing components in the approximation of pp mp for the proton are Vud GF √ d4 xeix(pp +pν +pe −pn ) × 2 s,+ χp (pp ) gV δ 0µ − gA σi δ µi χsn (pn ) · µ ,  p, e, ν | iT | n =  (2.54)  where µ is the leptonic current in momentum space. Neglecting the recoil of the proton (pp mp ) yields a decay rate dΓ which is independent of pp except for the Dirac delta in three momentum δ (3) (pp + pν + pe ). Hence, the integration over the proton’s three-momentum in Equation 2.32 will not have any impact on the decay rate and we drop d3 pp in Equation 2.32. In other words, neglecting pp does not mean pν = −pe from the Dirac delta, but the proton absorbs any momentum which the electron and anti-neutrino might have. Consequently, the integration of x in Equation 2.54 which generates the δ (4) is only needed to enforce energy conservation. The transition matrix element is then redefined as p, e, ν | iT | n = 2πδ(En − Ep − Eν − Ee ) · iM  (2.55)  with iM =  Vud GF √ d3 xeixpp χs,+ p (pp ) × 2 gV δ 0µ − gA σi δ µi e−ixpn χsn (pn ) ·  ix(pe +pν ) . µe  (2.56)  In a plane wave ansatz χsp (x) = e−ixpp χsp (pp ) and χsn (x) = e−ixpn χsn (pn ) this is further simplified to iM =  Vud GF √ 2  0µ d3 xχs,+ − gA σi δ µi χsn (x) · p (x) gV δ  33  ix(pe +pν ) . µe  (2.57)  Next, we re-call the isospin symmetry of Section 2.1.3 and associate neutron and proton as isospin Tz = 1/2 and Tz = −1/2 projections of the same doublet respectively. Isospin is in many ways analogous to regular spin or angular momentum and we can describe the neutron-decay matrix element with the isospin lowering operator tˆ− as iM =  Vud GF √ d3 xχ+ (x, s, −1/2) × 2 gV δ 0µ − gA σi δ µi tˆ− χ(x, s , +1/2) ·  ix(pe +pν ) . µe  (2.58)  Because the isospin-symmetry is broken, we will at a later point need to introduce theoretical corrections to this approximation. In this formulation we are dealing with nucleon wave-functions in position space including their isospin projections. In this picture, the β-decay on nucleon level is a change in the z-projection of the isospin and is taken care of by the isospin-lowering operator. This result can now be generalized to nuclear wave-functions where the nuclear isospin lowering and raising operators are defined as the sum over all A nucleons in a nucleus A  A  Tˆ− =  Tˆ+ =  t− (i) ,  t+ (i) ,  (2.59)  i=1  i=1  t± (i)  acts upon nucleon i. Suppose a parent nucleus P with A nucleons of where which Z are protons; it is described by a nuclear wave-function ψP (α, Tz ) where α denotes all quantum numbers including its total angular momentum J, but not its isospin projection Tz = (N − Z)/2. The wave-function is the Slater determinant over the individual nucleons, ψP (α, Tz ) = A[χp1 (x1 ) · χp2 (x2 ) · ... · χ(xZ )pZ · χ(xZ+1 )n1 · ... · χ(xA )nA−Z ], (2.60) with A yielding the full anti-symmetrization. If we assume non-relativistic nucleons (pi m) in the atomic nucleus, the same approximations as in the neutrondecay can be followed. In the β − decay of a nucleus ψP (α, Tz ) to the daughter nucleus ψD (α , Tz − 1), Equation 2.58 is then generalized by summing over all nucleons. This combines to the nuclear isospin lowering operator as defined above (Equation 2.59). Instead of integrating over just one nucleon’s spatial coordinates, the integration over all nucleons’ positions is necessary. Thus, the transition matrix  34  element follows Vud GF √ d3 x1 ...d3 xA ψD (x, α , Tz − 1) × 2 gV δ 0µ − gA σi δ µi Tˆ− ψP (x, α, Tz ) · µ .  iM =  (2.61)  In the previous step, the exponential with the lepton momenta was expanded, exp ix(pe + pν ) ≈ 1+ix·(pe +pν ) and only the first constant term was kept. This is possible due to the small momentum transfer in nuclear β-decays in comparison to the size of the nucleus. The expansion corresponds in the phenomenology of β-decays to allowed (first term) and forbidden transitions (all remaining terms). As the expansion term of the 1st forbidden transition, ix · (pe + pν ), is linear in x, the parity of parent and daughter wave-function must be opposite parity states, i.e. πP = πD · (−1), otherwise the spatial integration would vanish. Conversely, for allowed transition the parity of the two wave-functions must be the same. In allowed transitions, the total angular momentum of the leptons couples to L = 0 and, hence, LP = LD . The derivation will continue in the allowed approximation only. In the last equation, the only part which remains dependent on the lepton momenta is the leptonic current, which is formally independent of any spatial coordinate. This was correct for the neutron decay, however, in the nuclear decay one cannot assume a free outgoing (anti-) electron as the other nucleons will distort the β-particle. This is usually accounted for by adding a term F (Z, Ee ), which is calculated by solving the Dirac equation for continuum states in the presence of a point-like nucleus with charge Ze and compare the solution to the free Dirac equation. F (Z, Ee ) is known as the Fermi function. In this formulation, the leptonic current µ remains identical to the neutron decay and independent of the spatial integrations. The hadronic and leptonic part are thus separated. When summing over all initial and averaging over all final spin projections, we obtain analogously to the neutron a decay rate of Γ=  G2F | Vud |2 m5e 2 f · gV2 | MF |2 +gA | MGT |2 2π 3  (2.62)  with a statistical rate function of W0  f= 1  dW (W − W0 )2 W  W 2 − 1 · F (Z, W ).  (2.63)  Following its definition in Equation 2.51, the maximal (normalized) energy of the outgoing electron is W0 = (Q + 1)/me . MF and MGT are known as the nu-  35  β − − decay  (a)  (b)  (c)  Qβ − = m(Z, N) − m(Z + 1, N − 1)  Z x e-  Z x e-  Z N  Z+1 N-1  β + − decay  +  e-  +  νe  Qβ + = m(Z, N) − m(Z − 1, N + 1) − 2me  Z x e-  Z x e-  Z N  Z-1 N+1  Electron Capture  +  e+  +  νe  QEC = m(Z, N) − m(Z − 1, N + 1)  Z x e-  (Z-1) x e-  Z N  Z-1 N+1  +  νe  Figure 2.13: Schematics to illustrate the definition of Q-values with atomic masses m(Z,N) for β − , β + -decay, and nuclear electron capture.  clear matrix elements for the Fermi (vector) or Gamow-Teller (axial-vector) parts. According to Equation 2.61, they are expressed as MF  ≡  MGF,i ≡  f | Tˆ− | i ≡  d3 x1 ...d3 xA ψD (x, α , Tz − 1)Tˆ− ψP (x, α, Tz ) (2.64)  f | σi · Tˆ− | i ≡  d3 x1 ...d3 xA ψD (x, α , Tz − 1)σi · Tˆ− ψP (x, α, Tz )  For β + -decays, the same equations hold except that Tˆ− is replaced by Tˆ+ and the transition undergoes from Tz to Tz + 1. Additionally, the Q-values for β − and β + -decays are defined differently, see Figure 2.13. When energetically possible, i.e. Q > 0, the decay can proceed to the ground state or to excited states in the daughter nucleus. As discussed above, according to the selection rules for the relation between the initial and final states, the leptons couple to S = 0 for Fermi-transitions and to S = 1 for Gamow-Teller. Allowed 36  transitions further require ∆L = 0 and according to πf = πi · (−1)∆L also ∆π = 0. Generally, transitions involve both Fermi and Gamow-Teller parts. However, when the total angular momentum J = S + L changes by ∆J = 1 the transition is a pure Gamow-Teller decay. Pure Fermi decays are transitions between nuclear states with total angular momentum Ji = Jf = 0. The half-life T1/2 of a nuclide is related to the individual transitions by T1/2 =  ln(2) , f Γf  (2.65)  where the sum is over the transition rates to all populated states in the daughter nucleus. For β + -decays this includes the electron-capture branches as well. The Branching Ratio (BR), R, of a specific transition is defined as R = Γk /( f Γf ). Individual transitions are usually characterized by their f t-values. f is the statistical rate function with the transition energy or Q-value to this specific state and t is the partial half-life t = T1/2 /R = ln(2)/Γ. In β + -decays with competing electron-capture it is further corrected for the probability of an electron capture to this state, T1/2 t= (1 + PEC ), (2.66) R when we define the BR as the transition strength to a particular nuclear level, independent of whether it was populated by EC or β + -decay. PEC is the electron capture fraction, which is discussed in [67]. Its calculation depends, among others, also on the Q-value. In other words, the f t-value describes how ‘fast’ a particular transition is, correcting for the available phase space. To determine an f t-value experimentally, the half-life and the BR are required for the partial half-life t and the transition energy has to be measured which enters into the statistical rate function f . For allowed transitions it is equal to (f t)−1 =  ln(2)G2F | Vud |2 m5e 2 · gV2 | MF |2 +gA | MGT |2 . 2π 3  (2.67)  In order to extract Vud from nuclear β-decays, the f t-value, the nuclear matrix elements MF and MGT , as well as its gA have to be known. The nuclear matrix elements are generally not calculable precisely enough to obtain a competitive estimate of Vud . Isospin T = 1/2 mirror and superallowed 0+ → 0+ β-decays are more favourable because they proceed between isobaric analog states whose wavefunctions only differ by the isospin projection. Consequently, the nuclear matrix elements are simpler to calculate. The value of Vud from T = 1/2 mirror decays has been evaluated recently for the first time [68]. Since these decays are Fermi37  Rep. Prog. Phys. 73 (2010) 046301  I S Towner and J C Hardy  0.9800  Vud 0.9750  0.9700  nuclear 0+ 0+  neutron  nuclear mirrors  pion  Uncertainty  0.003  0.002  0.001  Experiment  Radiative correction  Nuclear correction  Figure 8. The five valuesFigure for Vud derived in the Vtext are shown in the top panel, the grey band being the average value. The four panels at the 2.14: Top: ud extracted from different types of decays. For the neubottom of the figure show the error budgets for the four results shown in the top panel with points and error bars. The three contributors to tron decay, the larger errorcorrection—are bar reflects the large, identified. non-statistical spread the uncertainties—experiment, radiative correction and nuclear separately  of neutron lifetime measurements, while the smaller one only takes the neutron-lifetime of [62] and asymmetry measurement [69] intoof 0.02%, which is the Finally, the Particle Data Group [111] combines thethe derived in section 6.1.4 has in a precision account. The lowerofpanels display partial contribution of experiinclusive (8.3) and exclusive (8.6) determinations |Vub | by mostthe precise result so far obtained for this matrix element and ment theoretical, Figure fromthan [26]. taking a weighted average of the twoand to obtain by more an order of magnitude, the most precisely determined value for any element in the CKM matrix. The CKM matrix element Vus can be obtained from K 3 (see section 7.1) and K 2 (section 7.2) decays. The SU (3)a result that is dominated by the inclusive measurements. Gamow-Teller admixtures and involve gA symmetry-breaking , which is renormalized in theis presence correction too uncertain to include the of the strong force, additional experimentalresults input from in thehyperon form ofdecay a correlation mea-while values from τ (section 7.3) 9. CKM unitaritysurement and its significance is necessary. Superallowed 0+ decay → 0+(section β-decays are less cumbersome; 7.4) are at the present time lacking sufficient because of Jf = Ji = 0 they are pure Fermi decays. They are currently the most experimental precision. For K 3 decay, we have (7.4) In section 6 we described four different classes of precise way to gain V (see Figure 2.14). β-decay measurement used to determine Vud ud experimentally: (9.2) |Vus | = 0.2246 ± 0.0012 + + superallowed 0 → 0 nuclear transitions, neutron decay, mirror nuclear transitions and pion decay. Each of the four has from K 2 decay (7.10) 2.5 Superallowed 0+ → 0+ β and decays produced a value—two values in the case of neutron decay— |Vus |for transitions befor Vud . The result for nuclear superallowed appears Quantum mechanical transitions addition of angular momentum allows = 0.2319 ± 0.0015. (9.3) in (6.8); those for neutron arestates in (6.17) (6.18); the momentum Ji = |V tween decay nuclear withand total angular Jud f | = 0 to have both mirror-transition value appears (6.24) and thatcontributions. for pion decay However, for Ji = Jf = 0 and ∆L = 0 in Fermi and in Gamow-Teller Thus, we now have three pieces of data—|Vud | from is in (6.29). All five results are plotted in the top panel of nuclear decays, (9.1), |Vus | from K 3 decays, (9.2), and the ratio figure 8. Obviously, they are consistent with one another but, |V |/|V ud | from K 2 decays, (9.3)—from which to determine because the nuclear superallowed value has an uncertainty at 38 us least a factor of six less than all other results, it dominates any two parameters, |Vud | and |Vus |. We perform a non-linear least average. Furthermore, the more precise of the two neutron squares fit to obtain the result results can hardly be considered definitive since it involves |Vud | = 0.974 25(22), |Vus | = 0.225 21(94). (9.4) only two selected measurements. Consequently we use the nuclear superallowed result, (6.8), as the appropriate value for Note that the value of |Vud | obtained from this fitting Vud to use in testing CKM unitarity. procedure is left unchanged compared with (9.1); while |Vus | is To date, the most demanding test of CKM unitarity comes increased compared with (9.2), but still well within the quoted from the sum of squares of the top-row elements (2.13), |Vub | = (3.93 ± 0.36) × 10−3 ,  (8.7)  the allowed approximation, any change in spin S is forbidden, thus ∆S = 0, too, and we deal with a pure Fermi decay and MGT = 0. As a consequence of the (approximate) isospin-symmetry in the nuclear force, there are states in isobars, i.e. nuclides with the same mass number A, which are (almost) identical except for their Tz quantum number. These states are called isobaric analog states or members of an isobaric multiplet. In accordance with angular momentum rules, a multiplet with total isospin T will have 2T + 1 isobaric analog states with Tz ranging from −T to +T . Since the isospin-symmetry is broken, isobaric analog states are not degenerate. Their relative position in energy is described by the Isobaric Mass Multiplet Equation (IMME) [70]. Assuming that (i) all tz dependent contributions (including Coulomb) to the nuclear force can be expressed like the Coulomb force as a sum of isoscalar, isovector, and isotensor parts (i.e. spherical tensors with rank ≤ 2 in isospin space), the masses of isobaric analog states follow a parabola in Tz . M (A, T, Tz ) = a(A, T ) + b(A, T ) · Tz + c(A, T ) · Tz2 .  (2.68)  The IMME can be derived in first order perturbation theory by utilizing the WignerEckart theorem. It is essential for the description of the β-decay branch between isobaric analog states that the nuclear wave-functions of parent and daughter are identical requiring ∆L = ∆J = ∆S = ∆T = 0. To first order, they only differ by their isospin projection Tz . All nuclear structure details of the wave-functions are consequently irrelevant when the nuclear matrix element MF is evaluated in Equation 2.64. Referring once more to the analogy of isospin to angular momentum, the action of the isospin raising or lowering operator Tˆ± is well understood and we obtain MF = α, T, Tz ± 1 | Tˆ± | α, T, Tz =  (T ∓ Tz ) · (T ± Tz + 1).  (2.69)  Due to the maximal overlap of the nuclear wave-function, β-decays between isobaric analog states are called superallowed decays. A superallowed 0+ → 0+ nuclear β-decay occurs between isobaric analog states which have both a total angular momentum of J = 0. It combines the advantages of a pure Fermi decay and simple nuclear matrix elements. Due to the Coulomb repulsion, isobaric analog states with larger Z are generally less bound and all superallowed decays discussed here are β + -decays. T = 1 cases are studied most extensively although f t-values of T = 2 superallowed β emitters have been measured (e.g. [71]). The T = 1 cases are divided into two groups of decays, Tz = −1 →√ 0 and Tz = 0 → 1. In both cases, the nuclear matrix element yields MF = 2 (see 39  Equation 2.69). The f t-value for superallowed 0+ → 0+ , T = 1 nuclear β-decays is simply π3 K ft = = 2 2 , (2.70) 2 2 ln(2)gV GF | Vud |2 m5e 2gV GF | Vud |2  where the constant K = 2π 3 ln 2/m5e has been introduced. Note that this result for the f t-value is independent of any specifics of the transition and should be the same for all superallowed 0+ → 0+ , T = 1 nuclear β-decays. This will allow one to test the validity of the CVC-hypothesis experimentally, which requires that gV = 1 for all β-decays. Before doing so, the f t-values need to be adjusted by several theoretical corrections to account for critical approximations in the previous derivations.  2.6  Theoretical corrections to the f t-values  Theoretical corrections to the f t-values are small and only a few percent. But at the present experimental precision they dominate the uncertainty in the so-called corrected Ft-values. The origin of these corrections are • isospin symmetry breaking, • radiative correction, and • shape correction and atomic overlap correction in the statistical rate function. In the evaluation of the nuclear matrix element, isospin-symmetry has been assumed. Although an approximate symmetry of chiral QCD, it is broken even in QCD itself but most notably by the Coulomb interaction as neutrons are electrically neutral while protons are not. Respective inaccuracies are repaired by the isospinsymmetry breaking corrections δC . They are dependent of the nuclear structure and are thus specific to each decay. More details are elaborated in Section 2.7. In this section, we will only consider the to-date most reliable calculations of δC based on the nuclear shell model with Saxon-Woods radial wave functions [72, 73].  2.6.1  Corrections to the statistical rate function f  In Equation 2.63, the statistical rate function was defined as the normalized phase space integral over the β-particle’s energy. It was already corrected with the Fermi function F (Z, Ee ), which took into account that the outgoing β particle is not a free Dirac spinor but is influenced by the charge of the daughter nucleus. The framework of F (Z, Ee ) assumed the nucleus as a point-like particle. However, at the level of experimental precision this approximation is insufficient. The finite 40  size of the nucleus cannot be ignored and a more accurate model for the nuclear charge distribution is appropriate. Additionally, the atomic electrons are shielding the charge of the nucleus. Both effects are accommodated by adding the shapecorrection function S(Z, W ) to the statistical rate function [74]. S(Z, W ) incorporates a mild dependence on the nuclear structure of the involved nucleus. The theoretical uncertainty in f (0.01% for light nuclei up to 0.1% for A = 74) due to the different shell model calculations is currently small compared to the uncertainty due to the Q-value and is not added to the error budget of f [75]. Since the β-decay changes the charge Z of the nucleus, the atomic electron wavefunction of the parent and daughter nucleus are not identical. The mismatch in the atomic wave-functions hinders the β-decay slightly. This is taken into account by the atomic overlap correction r(Z, W ) [15]. That such a small effect is considered necessary to be dealt with at all, accentuates the level of experimental precision in regards to the Q-value. With these two new modifications, the statistical rate function is corrected to W0  f= 1  dW (W − W0 )2 W  W 2 − 1 · F (Z, W ) · S(Z, W ) · r(Z, W ). (2.71)  In this treatment, care has to be taken that parts of the electromagnetic interaction between the β-particle and the daughter nucleus incorporated in F (Z, W ) are not accounted for in the radiative corrections which are discussed in the next subsection. Neglecting the shape correction function S(Z, W ) would change the statistical rate function by 0.2% at A = 10 up to 5.7% at A = 74 [74]. When considering r(Z, W ), f is modified by maximally 0.02% [15].  2.6.2  Radiative corrections  Radiative corrections consider the emission of bremsstrahlung photons, which are usually not detected in experiment, as well as loop effects of exchange of photons and Z-bosons [76][77, 78][79][80]. For superallowed β-decays, the β-decay rate Γ0 is correct to Γ = Γ0 · (1 + RC) with RC =  α mZ mZ g(Em ) + δ2 + δ3 + 3 ln + ln + Ag + 2C . 2π mp mp  (2.72)  For details, the reader is referred to the literature. Here, only the origin of the corrections is discussed. The first four terms are due to loop corrections and bremsstrahlung which involve the electromagnetic and weak vector interactions. A comprehensive description of g(Em ) is found in [76]. It is a universal func41  A. SirIin: Current a  590  e+  y',  P  Z  (a)  Figure 2.15: γW -box and ZW -box diagram. Figure from [77].  (b)  of p an the exchange 6.whichBox diagrams FIG. tion dependsdi on the energy of the involving (anti-)electron with a maximal energy and is averaged over the available β-decay spectrum. δ and δ are higher and leptons. tween Eorderhadrons expansion terms in Zα and Z α . The remaining terms are related to the m  2  2 3  2  3  weak axial-current, which, as derived in the previous sections, does not enter to first order (or tree-level) for pure Fermi-decays. Loop corrections at higher orders are nevertheless sensitive to the axial-vector interaction, where gA is not protected by CVC. A dominant contribution to these terms is the loop in the γW - box diagram [80] (see Figure 2.15). For instance, the weak axial-vector might change the spin, but the electromagnetic interaction between the nucleus and the outgoing β-particle could reverse it, which is allowed even for Ji = Jf = 0. This particular contribution is part of the correction term C. In [72], C is divided into the two components. In one part, the same nucleon is spin-flipped twice, while in the second component two different nucleons are involved. Consequently, the latter one depends on nuclear structure. Furthermore, the coupling of spin-flip processes is quenched in the nuclear medium. Hence, they are nuclear structure dependent, too, but different from free nucleons [72]. As a consequence, the nuclear structure dependent contributions in C are separated from the universal part according to C = C(free) + CNS . It is conventional to separate these terms into transition independent (∆VR ) and transition dependent parts. Latter ones are further divided into those which depend on the specifics of the transition, i.e. the charge Z of the daughter nucleus and the Q-value, but are independent of nuclear structure (δR ) and those which depend on  B. Corrections to the vertex  Qh'h  T o simplify the analysis we assume in this s that @', has been defined to have exactly zero v pectation value, so that tadpole contributions Figs. 5(i), (j) can be disregarded (Taylor, 197 We first discuss the counterterms of order g ated with the vertex @h'h. Referring back to E we note that in this equation g and wa e b are ties. Imagining that these constants are written and mw, performing 42 the shifts g'=g- 5g, rn' —5m~, remembering Eqs. (6.4) and (6.5), and the field renormalization of (p, we see that up of order g' the interaction of (p with the quarks  the structure of the nucleus δN S . ∆VR = δR = δN S  =  mZ α mZ + ln + Ag + 2C(free) 3 ln 2π mp mp α g(Em ) + δ2 + δ3 ] 2π α CNS . 2π  (2.73) (2.74) (2.75)  Recent improvements in the radiative corrections [80] have led to an alteration of these expressions, but the same conventions for ∆VR , δR , and δN S remain [73]. For the transition independent radiative correction one obtains ∆VR = (2.361±0.038)% [73]. δR are ≈ 1.5% and the nuclear structure dependent corrections δN S range from 0.005(20)% in 26m Al to −0.345(35)% in 10 C [73].  2.6.3  Corrected Ft-values  Collecting all contributions together, we end up with the corrected Ft-value, which is related to Vud as follows: Ft = f t(1 + δR )(1 + δN S − δC ) =  2gV2 G2F  K = const, (2.76) | Vud |2 (1 + ∆VR )  where gV = 1 according to CVC. This equation summarizes the program of superallowed 0+ → 0+ nuclear T = 1 β-decays. In principle, a single superallowed βdecay would allow one to derive Ft. However, in order to minimize errors it is important to measure as many superallowed decays as possible with high precision to establish their f t-values experimentally. Figure 2.16(a) illustrates the experimental status of the 13 most precisely measured cases whose Ft have uncertainties of less than 0.4 %. Appyling the transition dependent theoretical corrections, δR , δN S , and δC yields the corrected Ft-value, which should, according to the CVC-hypothesis, be identical for all T = 1 superallowed decays. Hence, the agreement of all Ftvalues represents a stringent test of CVC. When considering the isospin-symmetry breaking corrections from shell model with Saxon-Woods radial wave functions (see Figure 2.16(b)), the CVC hypothesis is indeed confirmed at the 1.3 · 10−4 level [15]. This consistency between Ft-values also highlights another advantage of the Vud extraction from superallowed β-decays over neutron and pion decay: experimental or theoretical errors are minimized because the same quantity is extracted from and cross checked between a group (currently 13) of individual decays. Any deviations should show up in Figure 2.16(b) before the calculation of Vud . In combination with ∆VR and the Fermi constant GF from muon decay, the weighted average of currently Ft = 3072.08(79) s [15] finally yields Vud . Figure 2.17 dis43  (b)  3095 10C  3090  3090  3080  3085  3070  3080  3060  34Ar  46V  38mK  34Cl  42Sc  15  20  62Ga  74Rb  50Mn 56Co  3075  3050  3070  3040  3065  3030  22Mg  14O 26mAl  Ft [s] Ft [s]  f tft [s] [s]  (a) 3100  0  5  10  15  20  25  30  35  3060  40  Z of daughter  0  5  10  25  30  35  40  Z of daughter  Figure 2.16: f t-values (a) and corrected Ft-values (b) for the 13 most precisely measured superallowed 0+ → 0+ β decays. The grey band in (b) represents the weighted average. The plots are based on data from [15].  total uncertainty  total uncertainty total uncertainty  ∆VR  Figure 2.17: Development of partial uncertainties of | Vud |2 since 2005. The estimated error due to transition dependent corrections is dominated by the Isospin Symmetry Breaking (ISB) corrections δC . The reduction from 2005 [74] to 2006 is due to the improved calculation for ∆VR [80]. The development of a new Hartree-Fock protocol while rejecting the old one led to smaller systematic discrepancies between models of δC and a subsequently reduced partial uncertainty in 2009 [15].  44  plays the partial uncertainties to Vud from experiment, nuclear structure dependent corrections (δC and δN S ), and the radiative corrections δR and ∆VR . The dominating contribution has its source in ∆VR . A future error reduction over today’s value in [80] might be feasible [81]. The second largest source of uncertainty is found in the isospin-symmetry breaking corrections. Previous to [15], their contribution to the total uncertainty in Vud was almost identical to ∆VR . In earlier surveys of superallowed β-decays, the uncertainty of δC had to be inflated because their determination in the shell model with radial wave-function from Hartree-Fock calculations with Skyrme-type interactions [82, 83] showed systematic discrepancies to the already mentioned shell model with Saxon-Woods radial wave-functions. In their latest survey [15], Towner and Hardy have performed their own HartreeFock calculations. This was motivated by the fact that the older Hartree-Fock corrections used a model space which was shown to be too small [73]. Towner and Hardy’s Hartree-Fock calculations of δC are in much better agreement with their own Saxon-Woods approach and the inflation of the theoretical uncertainty in δC was reduced accordingly. However, in [15], the protocol of the HartreeFock method has been altered in comparison to the original calculations [82, 83]. As expressed by Towner and Hardy themselves [15], it remains desirable to compare the current values for δC to independent calculations. As pointed out in [84], the isospin-symmetry breaking corrections δC were subject to significant revisions over the last 10 years. From 2002 [72] to 2008 [73], the uncertainties of δC for individual superallowed β emitters actually increased (see for instance Table II in [73]). The reduction on its associated error for Vud is due to the smaller systematic difference between the Saxon-Woods and Hartree-Fock calculation only. This situation has triggered a lot of effort both on the experimental and theoretical side. New models of δC have produced their first results. These will be discussed in the next subsection. Although the experimental uncertainty is smaller than the error contributions from theory, measurements of superallowed β-decays have by no means lost their importance. Efforts continue to add new precision cases to the current 13 most studied decays. Experimental data provide input for the isospin symmetry breaking corrections such as spectroscopic factors or branching ratios to non-analog 0+ states. Very recently, laser spectroscopy determined a more precise charge radius of 74 Rb [16] which is used in the Saxon-Woods calculation of δC . New, more precise measurements of the f t-values of the 13 most-studied cases allow more stringent comparisons of conflicting theoretical models. This could challenge perceived consistencies between a set of δC -calculations, experimental results, and the conserved vector current hypothesis. These experimental ‘tests’ of the ISB corrections are described in Section 2.8.  45  2.7  Different models for isospin symmetry breaking corrections  The result of the Fermi matrix element for the superallowed decay branch, Equation 2.69, assumed isospin to reflect an exact symmetry. Already in Section 2.1.3 it was stated, that even within QCD the symmetry is broken by the quark mass difference. In the full Lagrangian of the Standard Model (SM) the difference in the electric charge of u- and d- quarks (or neutron and proton) in QED is also inconsistent with an exact isospin symmetry. For the present purpose, the latter is in fact the dominating effect [85]. As a consequence, the simplicity of Equation 2.69 only holds as an approximation and the ISB corrections add to the Fermi matrix element. | MF |2 =| M0 |2 (1 − δC )  (2.77)  where for T = 1 decays | M0 |2 = 2 as before. These corrections are of the order of ∼ 1%, but have to be known to about 10% of their value to fully take advantage of the experimental precision. During the last years, these corrections have received a new focus in theoretical and experimental research on weak interaction studies with nuclear systems. Several new approaches have been explored recently, mostly in disagreement with the most developed δC by Towner and Hardy. This section provides an overview over models and ideas in this ongoing discussion.  2.7.1  Nuclear shell model with Saxon-Woods radial wave-functions  Due to its many improvements over the last decades, the nuclear shell model approach with Saxon-Woods radial wave-functions is, to date, the most advanced calculation of δC which all new models are compared against. The surveys of superallowed β-decays [15, 74] are presented around this set of ISB corrections though the comparison to the shell model with Hartree-Fock radial wave-functions has always served to highlight potential shortcomings. In order to calcualte δC , the nuclear matrix element of the Fermi decay MF = f | Tˆ+ | i =  d3 x1 ...d3 xA ψD (x, α , Tz − 1)Tˆ+ ψP (x, α, Tz ) (2.78)  + with Tˆ+ = A i=1 t(i) is expressed in second quantization by utilizing creation and annihilation operators for neutron and proton quantum states. This means that the initial and final nuclear wave-functions are embodied by (combinations of) creation operators a+ acting upon the vacuum state | 0 in Fock space. More importantly for our purposes, operators are also expressed as annihilation and creation operators. Hence, for the total isospin raising operator Tˆ+ , the sum is taken over all states of  46  a single-particle basis and not over the nucleons. Towner and Hardy [73] write the Fermi matrix element as MF = f | Tˆ+ | i =  α,β  f | a+ ˆ+ | β , α aβ | i α | τ  (2.79)  where aβ annihilates a proton in state β and a+ α creates a neutron in state α instead. α | τˆ+ | β is the single-particle matrix element between the neutron state α and the proton state β. It is intended to take the difference in the radial wave-function Rn (r) and Rp (r) of neutron and proton in the single-particle basis into account. α | τˆ+ | β = δβα  ∞ 0  drr2 Rn (r)Rp (r) ≡ δβα rα .  (2.80)  Finally, as both f | a+ α and aβ | i are wave-functions of (A−1) nucleons, Towner and Hardy [73] insert a full basis of (A − 1) nucleon states | π , MF = π,α  f | a+ α | π π | aα | i rα .  (2.81)  ∗ Noting that without ISB, rα would be unity and f | a+ α | π = π | aα | i , this result allows an important separation in δC which is used by some other models as well. Assuming that both contributions are small, the leading order correction for ISB would be a sum δC = δC1 + δC2 . The corrections due to the deviations from rα = 1 are reflected in δC2 , where it is assumed that f | a+ α | π and π | aα | i are Hermitian conjugates. Conversely, for δC1 , rα = 1 holds and the corrections ∗ are due to f | a+ α | π = π | aα | i . Details of these calculations are found in [72, 73]. As pointed out in [73] the consideration of shell model orbitals from closed-shells can be essential for δC2 . However, to open up all shells is computationally intractable. Spectroscopic factors from pick-up reactions are used as an experimental criterion which shells are expected to contribute. The difference in the radial wave functions between Rn (r) and Rp (r) is calculated with a Saxon-Woods potential. Here the potential’s well 1/2 radius was matched to reproduce the empirical RMS charge radius r2 ch . Furthermore, the asymptotic form of Rn (r) and Rp (r) was fixed to the respective neutron and proton separation energies Sn and Sp . Towner and Hardy use sets of different parameterizations of the Saxon-Woods potential as well as different shell model interactions. The spread of the result is taken as the uncertainty on δC2 in addition 1/2 to the error which is due to the uncertainty in r2 ch . Their numerical values increase from δC2 = 0.165(15)% for 10 C to δC2 = 1.50(30)% for 74 Rb [73]. The determination of δC1 is also constrained by experimental input. Differences  47  0.4  1.8  0.35  1.6  Towner & Hardy 2002 Towner & Hardy 2008  1.4  0.3  1.2  bC2 [%]  bC1 [%]  0.25 0.2  1 0.8  0.15 0.6 0.1  0.4  0.05 0 0  0.2  10  20  30  0 0  40  Z of daughter  10  20  30  40  Z of daughter  Figure 2.18: Comparison of shell-model ISB corrections without [72](2002) and with [73](2008) the inclusion of core orbitals. The radial wavefunctions for δC2 were calculated based on Saxon-Woods potentials.  in single-particle states for neutrons and protons were modified to match the experimental single-particle shifts in energies. Charge dependent interactions were adjusted to reproduce the linear and quadratic coefficients b and c of the IMME (see Equation 2.68), which are in most cases known from experiment [86]. δC1 can be linked to the Fermi matrix elements to other 0+ states which are not isobaric analog states to the parent nucleus. If isospin was an exact symmetry, these matrix elements would vanish. In a sense, isospin mixing due to the ISB moves decay strength from the superallowed decay branch into branches to other 0+ states. In perturbation theory the matrix elements are then related to each other by the difference in energy between analog and the non-analog 0+ state through 1/∆E 2 . The comparison between the experimental ∆E and the one calculated in the shell model are used to scale δC1 accordingly. The latest determination lists δC1 = 0.010(10)% for 10 C which increased to maximally δC1 = 0.350(40)% for 70 Br. Hence δC1 is, in comparison to δC2 , a factor of 5-10 smaller. All numerical values for the ISB corrections are shown in Figure 2.18. It also compares the most recent results from 2008 with those of 2002. The latter did not consider closed shells in the shell model calculations.  48  2 1.8  Saxon−Woods (2002) Hartree−Fock (1995) Hartree−Fock (1995) [range]  1.6 1.4  δC [%]  1.2 1 0.8 0.6 0.4 0.2 0 0  5  10  15  20  25  30  35  40  Z of daughter  Figure 2.19: Comparison of shell-model ISB corrections with radial wavefunctions from Saxon-Woods [72] and Hartree-Fock potentials [83]. Only those cases are displayed where results are available in both procedures. For the Saxon-Woods radial wave-functions, the calculations from 2002 are shown, because both use the same shell model space. For the heavier superallowed emitters 62 Ga, 66 As, 70 Br, and 74 Rb, reference [83] does not provide uncertainties, but two different calculations. Here, the respective data points refer to the mean and spread of these two calculations.  2.7.2  Nuclear shell model with Hartree-Fock radial wave-functions  corrections in which the radial wave-functions were based on Skyrme-HartreeFock calculations were initially proposed and evaluated by Ormand and Brown [82, 83, 87, 88]. This approach was motivated by the fact that in a Hartree-Fock calculation the mean field is inherently proportional to the proton and neutron densities. The effect of the Coulomb repulsion, which pushes the protons further out and leads to the extended proton radial wave-functions, is, in the Hartree-Fock procedure, reduced by an isovector potential. Induced by extended proton densities, this isovector potential counteracts the Coulomb repulsion and as a consequence results in smaller δC2 compared to the Saxon-Woods potential. This expectation is generally confirmed as shown in Figure 2.19. In the past, both sets of δC led to corrected Ft-values which were each consistent with the CVC hypotheses, but resulted in different averaged Ft-values. The difference between the two models was added as a systematic uncertainty to the extraction of Vud . The total partial unISB  49  certainty on Vud from δC was similar to the error contribution due to the transition independent radiative corrections ∆VR [80] which, to date, dominate the uncertainty in Vud . The latest assessment of the Hartree-Fock procedure by Ormand and Brown in 1995 does not cover all of today’s 13 superallowed precision cases. Furthermore, a direct comparison between [83] and [73] is no longer appropriate because of different shell model spaces. In [15], Towner and Hardy introduced their own HartreeFock calculations for δC2 utilizing the same model space of [73], however, with a substantial deviation from the method of Ormand and Brown. In their approach, they note that the direct term of the Hartree-Fock equation for the parent nucleus leads to an asymptotic Coulomb potential limr→∞ VCdirect = (Z + 1)e2 /r, with Z being the number of protons in the daughter nucleus. This appears as an unphysical charge. It is too large by one unit in e, because we are interested in the mean field for the last proton in the field of Z protons and A − Z − 1 neutrons. In the complete Hartree-Fock formalism, the exchange term would account for this but only at the price of a non-local integral in the potential which is notoriously hard to solve and is often approximated. In Towner and Hardy’s view, such an approximation would not yield satisfactory results for asymptotic quantities such as the required asymptotic differences in neutron and proton radial wave-functions. So, instead of two Hartree-Fock calculations for the parent nucleus (Z + 1, N ) and the daughter (Z, N + 1), they perform only one with the nucleus (Z, N ). The resulting mean field is used to derive the single particle wave-functions for the last proton and neutron in the parent or daughter nucleus respectively. Although some discrepancies between the Saxon-Woods and the Hartree-Fock approaches remain, their overall agreements are improved (see Figure 2.20). Consequently, the uncertainties in Vud associated with δC are reduced in the latest survey of superallowed β-decays [15]. Figure 2.17 shows how the partial uncertainties to Vud have evolved over the last years. Despite the success of the recent calculations, a better understanding of the discrepancies to the Hartree-Fock method designed by Ormand and Brown remains desirable. Assuming CVC, an experimental test of isospin breaking corrections for the T = 2 superallowed β-decay 32 Ar [71] obtained δC = 2.1 ± 0.8% which is in agreement with Brown’s calculations of δC = 2.0 ± 0.4% in the same publication. Generally, T = 2 cases are considered to provide a good testing ground to discriminate between different models of δC as they are expected to be larger for T = 2 superallowed decays [89].  2.7.3  Isovector monopole resonance  In the debate on the ISB corrections, Auerbach [85] has developed a qualitative model in which charge-dependent components of the complete nuclear Hamilto50  1.8 1.6  Saxon−Woods (2008) Hartree−Fock (2009)  1.4  δC2 [%]  1.2 1 0.8 0.6 0.4 0.2 0 0  5  10  15  20  25  30  35  40  Z of daughter  Figure 2.20: Comparison of the most recent shell-model ISB corrections with radial wave-functions from Saxon-Woods [73] and Hartree-Fock potentials [15], both performed by Towner and Hardy.  nian H = H0 + VC are treated perturbatively on top of a charge independent Hamiltonian H0 [85]. For β + -decays between eigenstates of H0 the nuclear transition matrix element would √ follow the derived relations due to the isospin raising operator T + , hence, MF = 2 for a decay within a T = 1 multiplet. Auerbauch considers the Coulomb part as the dominant part of the charge dependent part VC , which is approximated by a uniformly charged sphere inside a radius R. Ze2 VC (r) = − 3 R  A i=1  ri2 3R2 − 2 2  1 − tz (i) 2  r≤R  (2.82)  of which the isovector part is linked to the giant isovector monopole state. This relationship is used to calculate the matrix element and a simple equation for the ISB correction is obtained. V1 2 δC = 8 (2.83) 41ξA2/3 1 The symmetry potential strength V1 and the model dependent ξ are chosen by Auerbach to be V1 = 100 MeV and ξ = 3. For the isospin admixture 21 several models  51  2  Saxon−Woods (2008) Hydrodynamical (IVMS) NEWSR (IVMS) EWSR (IVMS) Microscopic (IVMS)  1.8 1.6 1.4  δC [%]  1.2 1 0.8 0.6 0.4 0.2 0 0  5  10  15  20  25  30  35  40  Z of daughter  Figure 2.21: ISB corrections by Auerbach [85] based on the model employing the isovector monopole resonance state in comparison to shellmodel ISB corrections with radial wave-functions from Saxon-Woods potentials [73].  are being discussed which result in the following expressions for δC . δC  = 6.0 · 10−7 A2 −7  = 0.67 · 10  −7  = 5.7 · 10  A  −7  = 18.0 · 10  A  (hydrodynamical) 7/3  2  A  (NEWSR) (EWSR)  5/3  (‘microscopic‘) (2.84)  Although this approach lacks the structural details of the considered nuclei and omits charge dependent interactions other than the Coulomb force, it has some critical features. While previous calculations do not include collectivity, it is considered here since the giant isovector monopole state itself is a collective excitation. Furthermore, it does not require the division of δC into δC1 and δC2 , which is model dependent [85, 90]. The difference to all previous calculations is apparent (see Figure 2.21). Even if Auerbach assumes a 50 % uncertainty his results are still significantly smaller than the ISB corrections by Towner and Hardy.  52  (b)  (a)  1.4 1.2  0.1  1  bC [%]  PKO1/2/3 (RHF+RPA) DDïME1/2 (RH+RPA) NL3 / TM1 (RH+RPA)  0.15  6=bC,PKO1* ï bC,x [%]  1.6  SaxonïWoods (2008) PKO1 (RHF+RPA) PKO1 (RHF+RPA) PKO3 (RHF+RPA) DDïME1 (RH+RPA) DDïME2 (RH+RPA) NL3 (RH+RPA) TM1 (RH+RPA)  0.8 0.6  0.05  0  0.4 ï0.05  0.2 0 0  10  20  30  0  40  Z of daughter  5  10  15  20  25  30  35  40  Z of daughter  Figure 2.22: (a) Results for δC of Relativistic Hartree-Fock (RHF) + RandomPhase Approximation (RPA) and Relativistic Hartree (RH)+RPA[91] compared to Saxon-Woods shell model calculations [73]. Only those cases are displayed where both models have published results. (b) In PKO1* the exchange term of the Coulomb force is turned off. The results are contrasted to the full RHF + RPA calculation as well as to RH+RPA with the density dependent meson-nucleon Lagrangian in DD-ME1 and DD-ME2 and the Lagrangian with nonlinear meson coupling (NL3 and, TM1). Error bars reflect the spread of the respective results as shown in (a).  2.7.4  Self-consistent relativistic random-phase approximation  In [91], the ISB corrections are approached in a self-consistent, relativistic RandomPhase Approximation (RPA). The starting point is a relativistic Hartree-Fock theory in which a Lagrangian density is built upon Dirac spinors describing the nucleons which interact through exchange-mesons and photons. In order to investigate the importance of an exact treatment of the Coulomb interaction, RPA calculations were performed on the basis of both RHF and RH mean field. As an approximation, latter ones neglect the exchange or Fock term in the potential. Additionally, for RHF + RPA and RH + RPA different effective interactions have been used to probe for the model dependence of the interaction. The results are shown in Figure 2.22(a), of which all are consistently lower than the values from the shell model. The full calculations (RHF+RPA) show very little dependence on the interaction and yield 53  distinctively smaller ISB corrections δC than the approach of RH+RPA. Within the latter the effective interaction DD-ME1 and DD-ME2 are, with regard to δC , almost identical. Both are based on a Lagrangian density with density dependent nucleonmeson couplings. However, they are still different to the RH+RPA approach using a Lagrangian density with nonlinear couplings (TM1 and NL3). Utilizing the effective interactions TM1 and NL3, very similar results are generally obtained, although for δC of the decays in 30 S, 34 Ar, 38 Ca, as well as 38m K the deviations between the two interactions are rather large. Unfortunately, neither the differences between DD-ME1/2 on the one hand and TM1 / NL3 on the other nor the large differences for some cases among TM1 and NL3 are highlighted or discussed in [91]. However, the distinction between results of RH+RPA and RHF+RPA is well explained: Utilizing the interaction PKO1 another RHF+RPA is being performed but this time the exchange term regarding the Coulomb interaction is turned off (PKO1*). This calculation recovers most of the differences to RH+RPA (compare Figure 2.22(b)) in which all exchange terms are neglected. Hence, the observed deviations between RH+RPA and RHF+RPA are not due to the respectively used interactions, but caused by the incomplete treatment of the Coulomb interaction in RH. This reflects Towner and Hardy’s concern regarding the correct implementation of the Fock term in Hartree-Fock calculations [15]. RHF+RPA in [91] represents a full treatment of the exchange term, but it appears that by including the exchange term, δC is reduced further in respect to the shell model calculations. Liang and collaborators speculate in [91] why their result could be different and envision more work on the implementation of the correct neutron-proton mass difference, isoscalar and isovector pairing, and deformation. The proper mass difference between neutron and protons has been investigated [92], but although this tends to shift δC up, the effect is rather small.  2.7.5  Nuclear density functional theory  The most recent model of δC by Satula et al. employs the self-consistent nuclear density theory [41]. As the Density Functional Theory (DFT) breaks rotational and isospin symmetry spontaneously, both have to be restored through projection onto a good angular momentum and isospin basis while retaining the true physical isospin breaking of the Coulomb interaction. Other charge-dependent interactions in the nuclear interaction were not considered. A technical difficulty regarding the application of DFT for odd-odd N = Z nuclei limits the approach to use the specific Skyrme V energy density functional. Skyrme V is of low spectroscopic quality which according to Satula et al. [41] particularly impacts δC for lighter nuclei in a negative manner. Additionally, ISB corrections of δC = 10 % were obtained for the superallowed β emitter 38m K, which appears much too large in 54  2  Saxon−Woods (2008) DFT  δC [%]  1.5  1  0.5  0 0  5  10  15  20  25  30  35  40  Z of daughter  Figure 2.23: δC obtained from nuclear Density Functional Theory (DFT) [41] in contrast to the shell-model with Saxon-Woods radial wave-functions [73]. Only those cases are displayed where both models have published results.  comparison with any other model or the CVC hypothesis. This deviation is considered to be a consequence of an incorrectly predicted shell structure in Skyrme V. 38m K was consequently disregarded in [41]. Despite these difficulties, one should note that the nuclear DFT approach has no adjustable parameters and represents a fully microscopic model. Indeed, for heavier, open shell nuclides it is presently the only microscopic description available. We note that the DFT is the only existing model which tends to predict larger δC than the shell model with Saxon-Woods radial wave-functions. Figure 2.23 shows the DFT results for δC together with the shell model calculations. Compared with other previous models, the agreement is reasonable, apart from the exception in 62 Ga. As mentioned before, caution is advised for lighter nuclei.  2.7.6  Ab-initio calculation by the no-core shell model in 10 C  Ab-initio methods play a particularly important role in contemporary nuclear physics. They attempt to describe atomic nuclei from first principles. This means that they work with a full A-body Hamiltonian in contrast to cluster models, the shell model  55  0.7  DFT  RH + RPA  RHF + RPA  PT + IVMR  0.3  Hartree-Fock O&B (1995)  0.4  Hartree-Fock T&H (2009)  10  bC [%] for 12C  0.5  Saxon-Woods (2008)  0.6  NCSM+PT  0.2  NCSM (8h 1) 0.1  0  Figure 2.24: Direct comparison of δC for 10 C from the No-Core Shell Model (NCSM) with the largest model space, but not converged (NCSM 8 Ω) and the perturbation theory based on the NCSM results (NCSM+PT) to all other presented models.  (assuming closed shells and effective interactions between valence nucleons), DFT, etc. Ab-initio methods are in this sense exact, provided that the employed nuclear potentials are accurate. Due to numerical reasons (note A!), these methods are limited to light nuclei with typically A ≤ 16. In light of modern high precision, nuclear potentials such as those from χEFT (compare Section 2.1.3), this mass range naturally bridges the underlying physics of quarks with the rest of the nuclear chart. As such, nuclear potentials and ab-initio methods can be benchmarked in the light mass sector providing confidence in the nuclear potential especially in their many-body forces, which can then be used in many-body methods for all heavier nucleon systems. With respect to superallowed β-decays, 10 C and 14 O are within the currently accepted range accessible by ab-initio methods. A determination of the ISB corrections have been attempted for 10 C within the No-Core Shell Model (NCSM) [93]. Although the calculation did not converge, perturbation theory was employed to gain δC from the non-converged result. The NCSM and the regular shell model both use a harmonic oscillator basis and have a similar second quantization framework, but the NCSM is distinct in treating all nucleons as ‘active’ and not in an inert core, hence the name. The calculation reported in [93] only included two-body forces in 56  the nuclear potential, more specifically for the superallowed decay matrix element it used the CD-Bonn 2000 NN potential [94] which includes the isospin symmetry breaking. As a cross check, the NCSM calculation of 10 C also determined the coefficients of the IMME (Equation 2.68) in the A = 10, T = 1 multiplet. The agreement with experiment is within ≈ 2% of the experimental value for b , but is less accurate for c (0.535 MeV in the NCSM compared to experimental 0.362 MeV) although the trend with increasing model space reduced the differences. In the largest tractable model space δC = 0.12% is obtained, however, the calculation is not converged and a larger model space would be required. Utilizing this result, an estimate based on perturbation theory yields δC ≈ 0.19%. It is stated in [93] that the perturbative result might be overestimated. As δC increases with increasing model space, we compare in Figure 2.24 both results to the other models of ISB corrections assuming that the converged δC in the NCSM would lie in-between, likely closer to δC = 0.19%. The shell model calculations are all in agreement with the NCSM, though for the Hartree-Fock approach by Ormand and Brown the large uncertainty covers a wide range of possibilities. Towner and Hardy’s Hartree-Fock method barely touches the NCSM results from above. The calculations utilizing the giant isovector monopole state and DFT are both in strong disagreement with the NCSM, one far below, one far above. However, the lighter nuclides are due to Skyrme V more problematic for the DFT and their descriptions are thus expected to be less reliable in DFT. Finally, assuming that the perturbative result based on the NCSM is indeed overestimated, the NCSM agrees well with RPA+RH. However, on a conceptual basis, RPA+RHF should obtain the more accurate value as it also incorporates the Coulomb exchange term. This result is, however, below the nonconverged δC in the NCSM. So, if one considered the NCSM due to its ab-initio character as the most reliable result, a few conclusions about the reliability of other calculations could be drawn. Auerbach’s perturbative method would be rejected and so would the DFT. RPA results would be somewhat problematic because the RHF would perform less accurately than the approximate RH. No discrimination regarding the shell model variants could be made. Making a broader statement based on such a (dis-)agreement with NCSM on the general performance of a method would arguably be inappropriate as the full set of superallowed β-decays spans a large mass range with different shell effects involved. This is particularly true for the implementation of the DFT, where better validity for heavier systems is expected (compare discussion in previous subsection). Generally, all of the conclusions in the previous paragraph for which the NCSM is considered as the benchmark result, have to be dealt with cautiously especially as the NCSM calculation is not converged. A better comparison could be made with converged results for 10 C and 14 O, possibly also including 3N-forces. 57  2.7.7  Exact formalism for δC  Miller and Schwenk have reviewed the formalism of Towner and Hardy in terms of the ISB corrections [90, 95] though without numerical results of their own. They observe that the implementation of the isospin-raising operator for the calculation of the Fermi matrix element in Equation 2.79 and Equation 2.80 is in fact not following the exact definition of isospin in the Standard Model. Instead Towner and Hardy’s operator is the analog spin or W -spin raising operator. In simplified words, instead of a pure transformation of a proton into a neutron, it transforms a proton from a certain proton state into a neutron in the neutron state with identical quantum numbers as the initial proton state. This is apparent in Equation 2.80 by the Kronecker delta between the states of the neutron and proton single particlebasis, α | τˆ+ | β = δβα  ∞  0  drr2 Rn (r)Rp (r) ≡ δβα rα .  (2.85)  This implies that on the level of single-particle states, the only difference is due to the differences in radial wave-function between neutron and proton. On the contrary, in [90, 95] it is argued that the correct Tˆ+ operator can result in changes in the radial quantum numbers, hence leading to radial excitations, too. Moreover, the use of the W -spin raising operator does not follow the usual commutation rules which one expects to hold for the isospin raising operator. It can be + , though part of the shown that Towner and Hardy’s isospin raising operator TˆTH Standard Model operator Tˆ+ , is incomplete. It is argued that the component in the isospin raising operator which is missing in the description of Towner and Hardy might even cancel δC2 and could lead to reduced ISB corrections.  2.7.8  Implications of different models for ISB corrections  After the introduction of different models of ISB corrections in superallowed 0+ → 0+ , T = 1 nuclear β-decays in the previous subsections, we will discuss aspects of what the discrepancies between different models imply for the CVC hypothesis, for Vud , and the CKM unitarity test. Table 2.4 lists the weighted averages for the corrected Ft-values when considering different theoretical models of the ISB corrections. Some authors have not published δC for all 13 well-measured superallowed β-decays for their respective model. Ft of Table 2.4 includes all superallowed emitters for each model for which a δC has been calculated. From the Hartree-Fock approach by Ormand and Brown, which generally provides theoretical uncertainties for δC , the heavier cases 62 Ga, 66 As, 70 Br, 74 Rb were omitted because no uncertainties are given. This follows 58  3090 0.95 3085 0.949  1 ï |Vus|2  0.948  |Vud|2  Ft [s] Ft [s]  3080  3075  0.947 0.946 0.945  3070 0.944 0.943  SM SM -H -W SM F ( S -H T& F H) ( PT O& B ) + R IV H M F + S R RP H A + R PA D FT  FT D  SM SM -H -W SM F ( S -H T& H F ) ( PT O& B ) + R IV H F MS + R RP H A + R PA  3065  Figure 2.25: Comparison of averaged Ft and | Vud |2 with the ISB corrections δC taken from different models. Experimental data from [15]. For PT+IVMS, RHF+RPA, and RH+RPA the error in δC is due to the spread of different models or effective interactions only. In the plot on the right, | Vud |2 is compared to 1− | Vus |2 , with | Vus | from K 3 -decays only.  the approach of the superallowed survey from 2005 [74]. As the RHF and RH plus RPA calculations do not estimate an uncertainty of δC , we take in each the spread of results caused by different effective interaction as an estimate of the error. Similarly, in the case of Auerbach’s δC with perturbation theory and isovector monopole state, the uncertainty was chosen to be the spread in δC due to different models. The Ft-values of Table 2.4 are plotted in Figure 2.25 highlighting the already discussed discrepancies between different models. Since the averaged Ft only has a physical meaning if the CVC hypothesis holds, the reduced χ2 for a constant fit through the individual Ft is also found in Table 2.4. Its interpretation in terms of a goodness of fit in comparison to the χ2 distribution has to be regarded with caution because of the non-statistical character of the theoretical uncertainties of δC , δN S , and δR . Furthermore, the compilation of the experimental data in [15] proceeds very conservatively and often inflates experimental uncertainties if a possibility of inconsistent measurements occurs. Hence, the total uncertainty on a superallowed β emitter’s Ft-value is largely of non-statistical origin. Assuming CVC as valid, a 59  Table 2.4: Averaged Ft with different models for δC . Experimental data, δN S , and δR were taken from [15]. For each model all available δC were taken out of the 13 well-measured superallowed β emitters. Model SM-WS SM-HF T&H SM-HF O&B PT + IVMS RHF+RPA RH+RPA DFT  ref. [73] [15] [83] [85] [91] [91] [41]  # cases 13 13 9 13 9 9 12  Ft [s] 3072.1(0.8) 3071.5(0.9) 3077.0(1.2) 3087.8(0.7) 3081.3(0.7) 3079.0(0.8) 3070.1(1.0)  χ2 /d.o.f 0.3 0.9 0.9 7.9 2.3 1.4 2.6  Table 2.5: Averaged Ft over the same 6 cases with different models for δC . Experimental data, δN S , and δR were taken from [15]. The superallowed β-emitters 10 C, 14 O, 26m Al, 34 Cl, 42 Sc, and 54 Co were considered for which all listed models have published ISB corrections. Model SM-WS SM-HF T&H SM-HF O&B PT + IVMS RHF+RPA RH+RPA DFT  Ft [s] 3072.0(0.9) 3070.7(1.1) 3076.2(1.5) 3085.2(0.9) 3080.5(0.8) 3078.7(0.9) 3069.0(1.2)  χ2 /d.o.f 0.4 0.8 0.9 4.9 2.4 1.8 0.9  smaller χ2 than usually expected is likely. By far the worst value for the reduced χ2 /ν = 7.9 is found in the perturbative approach with isovector monopole state. This is not too surprising considering that details of nuclear structure are not included in this calculation. The model rather attempts to estimate the typical size of δC for different mass ranges. Assuming that the right physics is taken into account when linking the isovector monopole state to δC , it is concluded that the Ft should be larger than generally assumed in the established calculations. The relatively large value χ2 /ν = 2.6 for the DFT is dominated by 62 Ga. With δC from RHF and RH plus RPA a χ2 /ν of 2.3 and 1.4 are obtained, respectively,  60  which appears too large given the non-statistical contributions to the uncertainty.5 However, the present evaluation might underestimate these models’ uncertainties. All shell-model calculations result in reasonable χ2 /ν although the difference in Ft between Hardy and Towner on one hand and Ormand and Brown on the other is significant [15]. One might reject the older Hartree-Fock model by Ormand and Brown based on doubts regarding the correct implementation of the Fock term or because of its smaller model space (as it is done in the latest survey on superallowed decays [15]), but certainly not because of unexpected violation of the CVC hypothesis. To treat all methods on an equal footing, we have performed a second evaluation; we only take those 6 superallowed β-decays into account for which values of δC are published in all models. The results in Table 2.5 generally confirm the previous observations. Since the problematic case of 62 Ga is not part of this group, the DFT model is also characterized by a χ2 /ν ≈ 1. As none of the models can claim a breakdown of CVC, we proceed with a calculation of | Vud |2 according to Equation 2.76 with Ft from Table 2.4, K/( c)6 = (8120.2787 ± 0.0011) × 10−10 GeV−4 s, gV = 1 (CVC), and GF /( c)3 = 1.16637(1) × 10−5 GeV−2 from muon decay [29]. The results are shown in Figure 2.25 in comparison to 1− | Vus |2 . A deviation from unitarity in the CKM matrix would be a consequence of some of the recent models. Notably, even Ormand and Brown’s shell model calculation would lead to a deviation by falling short of unitarity. To summarize this section, there are many indications that the most advanced calculations of Towner and Hardy yield correct δC : • agreement with the macroscopic calculations of NCSM for 10 C and generally with DFT, • consistency in δC2 between their own Saxon-Woods and Hartree-Fock radial wave-functions, • considerations of core orbitals, and • agreement of constant Ft-values with the expectation from CVC. Other models show systematic differences to Towner and Hardy’s results and all of these sets obtain smaller δC . This is also to be expected from the exact formalism by Miller and Schwenk. However, all of these calculations are either in an exploratory state and require more refinement or, for the case of Ormand and Liang et al. have used a non-standard format of a χ2 - test [96] which yields their small χ2 /ν in [91]. 5  61  Brown’s δC , should extend their model space. A discussion about the correct implementation of the Hartree-Fock procedure would also be crucial. In light of the importance of Vud for the Standard Model the issue of discrepancies in δC needs to be resolved. Otherwise an inflation of the uncertainties in δC might be required with a subsequent increase in the partial uncertainty for transition dependent corrections (see Figure 2.17). The next section will explain how experiments can impact this debate.  2.8  Experimental input to the debate around the isospin symmetry breaking corrections  Generally, there are two areas where experimental results can influence the debate on ISB corrections and help to discriminate between different models of δC . First, some models (usually the shell model based approaches) rely on experimental quantities such as the nucleon separation energies, IMME coefficients, etc. as input to the calculation. Improved input quantities or measurements for those cases, where only extrapolated values exists, will help the respective model. Conversely, an observable which is not needed to fine-tune a model, could serve as an independent benchmark of the model. Secondly, more precise f t-values, either of the 13 most studied cases, or of new ones, can be used for experimental tests of the ISB corrections. Usually, in these tests the validity of the CVC hypothesis is assumed to some level. Since CVC is an hypothesis and needs to be confirmed experimentally itself in superallowed β-decays, it is more accurate to formulate that these measurements can highlight discrepancies between a set of δC , the CVC hypothesis, and experimental results.  2.8.1  Input parameters and independent benchmark quantities  The shell-model methods typically use neutron and proton separation energies, charge radii, spectroscopic factors, coefficients of the IMME, and excitation energy of other 0+ states as input for their calculations. Particularly for the heavier superallowed decays this information is only sparsely available. As a consequence the uncertainties in δC are larger than for the lighter masses leaving lots of room for improvements. For instance, the Coulomb part of the Saxon-Woods potential is adjusted to reflect the experimental RMS charge radii [72, 73]. These have not been measured for all superallowed emitters and are then extrapolated from stable isotopes. Their uncertainties are consequently large and if sudden unexpected structural changes occur towards more exotic nuclides, they might be incorrect altogether. For this purpose, a campaign has been started at TRIUMF to measure iso-  62  tope shifts [97] which are sensitive to differences in RMS charge radii. A successful measurement was performed recently involving TITAN’s cooler and buncher which confirmed the extrapolated RMS charge radius of 74 Rb with a tenfold error reduction compared to the extrapolation. As a result the theoretical uncertainty on δC2 for 74 Rb in the shell model with Saxon-Woods radial wave-functions could be reduced by about 20 % [16]. An analogous measurement is planned for 62 Ga. In the discussion of the calculation of 10 C with the NCSM it was mentioned that the IMME parameters have also been calculated and compared to experiment. This is an example when independent benchmark parameters have been used to test the reliability of a calculation. The shell-model calculations have also been tested by a comparison of measured and theoretical transition strength to non-analog 0+ states [98].  2.8.2  Improvement of f t-values  Different strategies are employed to test for inconsistencies between a set of δC , experimental results, and the conserved vector current hypothesis. For instance, in [15] Towner and Hardy assume the validity of CVC and compare the individual f t-values with the f t-values extracted from ft =  Ft (1 + δR )(1 − δC + δN S )  (2.86)  where Ft is the weighted average of all cases (see Figure 2.26). This test is circular and hence insensitive to the correct absolute value of Ft, but it can help to identify inconsistencies. Similarly (and with analog deficiencies), ‘experimental’ nuclear structure dependent corrections (δC − δN S )exp are calculated (δC − δN S )exp = 1 −  Ft (1 + δR ) · f t  (2.87)  and compared to the individual theoretical (δC − δN S ) [99]. Avoiding the circularity of using the same δC to extract Ft and for the test, the 13 precise superallowed β-decays can be grouped and the Ft-value of each group is compared against each other. After a high precision measurement of the halflife of 26m Al [84] its Ft = 3073.0(12) s rivals the precision of the other 12 cases combined, Ft = 3072.0(10), both with the Saxon-Woods ISB corrections. This is a particularly strong test because δC for 26m Al is among the smallest absolute values of δC and its theoretical uncertainty is the smallest. Considering the other 12 cases, the Ft of 3072.3(10) s with Hardy and Towner’s Hartree-Fock 63  PHYSICAL REVIEW C 79, 055502 (2009)  WED 0+ → 0+ NUCLEAR . . . V. CONCLUSIONS  ft (s)  74 Rb evious survey [5], only four years ago, we the excellent agreement among the derived Ft 3080 62 Ga nted that the results of the unitarity test were us, and predicted that the already well-measured Calculated the “traditional nine” superallowed decays were 34 22 Ar Mg improved dramatically in the near future. Much 3060 since then, not all of it expected. Today, we can 38m K 46V excellent Ft value consistency remains—or, to 54 34 rate, it has been restored after Penning-trap QEC Cl 42Sc 50MnCo ements, nonexistant at the time of the last survey, 3040 14 O make important improvements (and changes) in 10 26m C Al t values, which in turn prompted improvements ) in the calculated isospin-symmetry-breaking At the same time, the calculation of the nucleus3020 radiative correction, VR , was improved, leading 10 40 0 20 30 cise result for Vud , and the kaon-decay community Z of daughter oncerted effort, which led to a new and reliable FIG. 9. Experimental f t values plotted as a function of the CKM s . With these new results, and others, Figure 2.26:charge Comparison of experimental f t-values to one which is extracted on the daughter nucleus, Z. Both bands represent the quantity now been tested to unprecedented precision . . . the transition dependent correcfrom the weighted Ft/[(1 + δR )(1 − δaverage The and two separate bands distinguish C + δNS )]. Ft sed the test with flying colors. = −1 (darker those β emitters whose parent nuclei have isospin T z tions. See text for details. Figure from [15]. re, we have demonstrated in Sec. IV how shading) from those with Tz = 0 (lighter shading). se improved results can be in setting limits on beyond the standard model, whether that new scalar interaction, right-hand currents, or extra Z ave seen that tiny uncertainties on approach the f t values the quantity Ft/[(1 + δvalue δC + δNS )] shown as a Saxon-Woods band, R )(1 − calculated is in agreement with the with the radial ngredients of a demanding test of CKM unitarity, the width of26m which represents the assigned theory error. The wave-functions. Butcorresponds for Altoformer corrections lead to 3069.0(19) s which is ads to tight limits on new physics. The challenge band the calculated corrections normalized in agreement withviathe other cases Hartree-Fock er those uncertainties can be reducedneither still further? to the data the12 measured averagewith Ft value, Ft, takencorrections nor on is as strong as ever: to identifywith the need for Ft-value from Table IX. Thus, this comparison not test its own derived fromalthough Saxon-Woods based does δC . This difference is due —or to limit the possible candidate theories even the absolute values of the correction terms, it does test the to a much larger δC2 = 0.410(50)% for the new Hartee-Fock calculation comvely. collective ability of all three calculated correction terms to pared to δC2 = reproduce 0.280(15)% in Saxon-Woods [73] and Brown’s older aken pains throughout this work to pay careful the significant variations in f t and from Ormand one transition l uncertainties, both theoretical and experimental. Hartree-Fock with δC2 = 0.29(9)%. is important to resolve this discrepancy independent of Z when to another. In fact, since δR isItalmost A we detail the various contributions to the that, Z> 10, this only test really probes directly the effectiveness of the considering to date, the Towner and Hardy’s Hartree Fock calculations n |Vud |2 . Of these, by far the largest is still from calculated values of (δC − δNS ). are employed as benchmarks of their Saxon-Woods approach. In fact the new halfugh its uncertainty has recently been improved It can be seen that there is remarkable agreement between measurement leads an increase the δC the model associated uncertainty. [6]. To improve it more must remainlife an important theory andtoexperiment. In in assessing significance of this oal. agreement, it is important to recognize that the calculations Recently,2 Towner and Hardy have introduced a comparative test between models argest contributor to the error budget for |Vud | is of δCFollowing and δNS for Z 26 are based on well-established shellof ISB corrections. ructure-dependent corrections (δC − δNS ). Their model wave functions that were further tuned to reproduce arise both from the input parameters used in their measured binding energies, charge radii, and coefficients of two-body matrix elements in the shell-model the isobaricδ multiplet mass equation Ft [7,179]. The origins (2.88) C = 1 + δN S − experimental uncertainties in charge-radii, etc. of the calculated correction terms forδRall (1 + ) ·cases f t are comfrom possible systematic differences between pletely independent of the superallowed decay data. Thus, the t methods used for calculating radial wave the figure between the measured superallowed Ft is treated asagreement a single in free parameter to minimize the difference in all available e Sec. III C). From a theoretical point of view, data points and the theoretical band is already a powerful cases between δvalidation whose uncertainty is dominated C and 1+δ N S iously be desirable to have a third completely of the −Ft/[(1+δ calculated corrections in determining R )·f t],used 2 wasconvincing ulation, to reinforce the assessment from of systematic thatAs band. The validation becomes evenχmore experiment. a figure of merit the reduced employed. Figure 2.27 However, in the absence of such a calculation, when we consider that it would require a pathological fault y on experiment to test the accuracy of these indeed in the theory to allow the observed nucleus-to-nucleus rrections. This has become, and should remain, a variations in δC and δNS to be reproduced in such detail or experiment. while failing to obtain the 64 absolute values to comparable od, which is best described with reference to precision. As satisfactory as the agreement in Fig. 9 is, though, ased on the validity of the CVC hypothesis new experiments can still improve the test, making it even ected Ft values for the superallowed 0+ → 0+ more demanding, and can ultimately serve to reduce the und be constant. In the figure we compare the certainty in the nuclear-structure-dependent corrections even measured f t values (points and error bars) with further.  055502-19  (%)  PHYSICAL REV  C  C  CC  (%)  %)) ((%  COMPARATIVE TESTS OF ISOSPIN-SYMMETRY-COMPARATIVE ... PHYSICAL. REVIEW C 82, 065501 (2010) TESTS OF ISOSPIN-SYMMETRY..  PHYSICAL REVIEW C 82, 065501 (2010)  (%) (%) C  C C  C C  %)) ((%  (%)  C  C C  C C  (%)  ((% %))  ISOSPIN-SYMMETRY- . . .  Z of daughter  of daughter daughter ZZ of  Z of daughter  C  (%)  FIG. 1. plotted Isospin-symmetry-breaking correction δZC ,ofinthe percent, plotted as a function FIG. 1. Isospin-symmetry-breaking correction δC , in percent, as a function of atomic daughter nucleus. The solidof atomic number Z of the d Figure 2.27: Results for circular δC from of points aEq. χ2(5), minimization with anumber free parameter Ft. obtained from of Eq. experimental ft values and the val with errorthe bars are the values of δC and with experimental ft values the values δR(5), andwith δNS the (and their circular points with error bars are the values of δC obtained all taken from Table I. In effect, wejoined treat as the “experimental” δC values. The X’s joined by li The I.solid lines the ISB corrections while the these points with The X’s by lines represent the δC values uncertainties) all taken from Table In effect, we are treatuncertainties) thesetheoretical as the “experimental” δC values. by results. the various models described in the textFigures and thebeen top left of each graph. The value of Ft i Ftidentified in Eq. (5)inhas adjusted calculated by the various models described in theminimization text calculated and identified in the top left of text each graph. The value of errors are the See for details. from in each case by least-squares fittingδCtovalues optimize thecorresponding experimental δC values and the calculat andthe theagreement calculatedbetween ones. The in each case by least-squares fitting to optimize the agreement between the experimental 2 [18]. values next-to-last row of Table I. of χ /nd are listed in the next-to-last row of Table I. values of χ 2 /nd are listed in the  Z of daughter  Z of daughter confidence levels well below 0.5%. Because the two other  course, does not prove that the SM-  levels as well below of 0.5%. Because theof two other nucleus. course, does y-breaking correction δCconfidence , in percent, plotted a function atomic number Z the daughter The solidnot prove that the SM-SW model is correct in analyses included nonstatistical uncertainties on the theoretical way; however, it does demonstr analyses included nonstatistical uncertainties on the theoretical every way; however, it does demonstrate that the every other models from Eq. (5), with the experimental ft values and the values of δ and δ (and their are the values of δC obtained shows details of the test results. Overall shell model with Saxon-Woods radial R the NS correction terms in addition to the statistical experimental ones, in their correction terms in addition toδCthe statistical experimental ones, in the their for form cannot be used values. The X’s by lines represent δC present values form cannot be used to extract a numberpresent Table I. In effect, we treat these as the “experimental” 2 wave-functions yields thejoined best internal agreement. 2 their are substantially lower, but test CKM unitarity. As we dand ud and values ofin χthe /ntop but relative to test CKM unitarity. Asthe werelative note in Sec.VII, if thetoFt V/n Ftvalues in Eq. of (5)χhas been adjusted dels described in the texttheir and identified leftsubstantially of each graph.lower, The value ofthe d are ud ranking the The sixvalues models approximately preserved: in allthen valuesaverage are not consistent with one an alternative, viewpoint [17] tries toconsistent emphasize similarities, andtheir ranking of the An six is approximately inofones. all areisnot with one another, values and preserved: the calculated corresponding fitting to optimize the agreement between themodels experimental δCsemi-empirical cases the that SM-SWhas is by farsignificance the best. It is remarkable that has no defined significance since eithe the next-to-last row of Table casesI.the SM-SW is by far thebetween best. It is remarkable no defined since either the symmetry-breaking notmodel differences, different models.model Especially, the shell model calcula-  thetheoretical model which model becomes secondorbest model is wrong or CVC itself has fai the model which becomes second best when the is wrong CVCwhen itselfthe hastheoretical failed. tions of δC show similar relative the development of arguably δC overthe Z,most though uncertainties included isisthe earliestmodel, and There is a second model, SM-HF, uncertainties are included is the earliest and arguably thepatterns most are in There a second SM-HF, which has many promisprimitive one. Its success evidently stems from its treatment ing features. low 0.5%. Because primitive the two other course, does not prove that SM-SW model is features. correct inAs can absolute values In view of Auerbach’s [85]anand one. their Its success evidently stemsmight fromthe itsdiffer. treatment ing bepublication appreciated from examination of As can be appreciated 2 of the radial mismatch between the parent and daughter states, Fig. 1, its relatively large χ 2 is due stical uncertainties on of thethe theoretical every way; however, it does demonstrate that the other models radial mismatch between(see the parent and daughter states,for N Fig.∼1,Zitsthe relatively large dependent χ is due to correcits failure to match other works references in [17]) transition which accounts rather well for the sharp increase in δ between the experimental δC values for the n to the statistical experimental ones, rather in their present formincrease cannot in beδused to extractthe a number C which accounts well for the sharp cases with Z 30. If experimental δC values for the 2 . for C between tions are expected to scale approximately as Z Hence, the individual models Z = 12 and Z = 16 and between Z = 26 and Z = 30. It is we were to restrict ourselves only t substantially lower, but the relative and to test CKM unitarity. As we note in Sec. II, if the Ft V Z = 12 and Z = 16udand between Z = 26 and Z = 30. It is we were to restrict ourselves only to the lighter cases, then equally striking that the most recentwell IVMR model the model at would agree well with C is approximately preserved: in all are values arethe notmost consistent with perhaps one another, their average onlythat considered for IVMR their representation ofmodel nuclear structure effects butfails not fordifference perhaps equally striking recent model fails then the would agree with CVC. This to either the trend of theZdata or any of its the the highest Z values between the SM 2 behaviour. by far the best. It is remarkable thatthe hasgeneral no significance since the symmetry-breaking to reproduce trend ofdefined the data or any of its characteristic theInstead highest between SM-SW andsuSM-HF model their (approximate) Zreproduce ofvalues correcting allcharacteristic individual calculations s second best when the theoretical model is wrong or CVC itself features. has failed. features. calculations has been known for 15 years, having first beenhas been known for 15 to the transition independent one only corrects by thepointed out by Ormand and Brown [6 is the earliest and arguably the most perallowed There iscases a second model, SM-HF, which has pointed many Ft-values, promisout by Ormand and Brown [6] even before the decays of the emitters, 62 Ga an 62 to the charge74 evidently stems from its treatment shell-structure ing features. As can beinappreciated from an examination of effects δC and extrapolates the resultant f t-values of the highest-Z emitters, Ga and Rb, had yet highest-Z been 2 precisely measured. Prompted by th ween the parent and daughter states, independent Fig. 1, its relatively large χ is due to its failure to match measured. Promptedeffects by the are results reported here, limit where isospin-symmetryprecisely breaking and Coulomb negVI. CONCLUSIONS VI.experimental CONCLUSIONS we are currently examining whether t for the sharp increase in δC between Z 30. If the δC values for the cases with we are currently examining whether feature of the SM-HF ligible. This is not the case only for an extrapolation towards Z = 0, but atthisZ = 0.5 model (as described in Sec. IV C) is s Evidently, the model shell model with Saxon-Woods wave between Z = 26 and Z Evidently, = 30. It isthe we were restrict ourselves towave the lighter cases, (asthen described in Sec. IV C)radial is sensitive to the particular shell modeltowith Saxon-Woods radial used [18]. We ha functions, SM-SW, is the only model tested that yields isospinat the most recent IVMR model fails the model would agree wellyields withwithin CVC. Thisisospin difference at where the total mass splitting an multiplet vanishes [17]. Finally, Skyrme interaction used [18]. We have, by now,Skyrme sampledinteraction 12 functions, SM-SW, is the only model tested that isospindifferent interactions and have also symmetry-breaking corrections which, and when combined witha pairing the data or any of itssymmetry-breaking characteristic these thecorrections highest Z values between the SM-SW and SM-HF model different interactions have also added term to which, combined extrapolated Ftwhen values can bewith compared between different models of δC andthe interaction, turning the calculati the experimental ft values, produce Ft values that agree with calculations has been known foragree 15 years, first been the interaction, turning the calculation into a Hartree-Fockthe experimental values, produce Ft values that with having a Vftud can be obtained. Bogolyubov one. However, under no the CVC hypothesis over the full range of Z values. This, of pointed out by Ormand and Brown [6] even before the decays Bogolyubov one. However, under no circumstances have we the CVC hypothesis over the full range of Z values. This, 74 of 62 of the highest-Z Ga benefit and Rb, hadnew yet precision been In summary, all ofemitters, these tests from cases or improved f tprecisely measured. Prompted by the results reported here, 065501-7 ONCLUSIONS 065501-7 we are currently examining whether this feature of the SM-HF model (as described in Sec. IV C) is sensitive to the particular del with Saxon-Woods radial wave 65 Skyrme interaction used [18]. We have, by now, sampled 12 nly model tested that yields isospindifferent interactions and have also added a pairing term to ctions which, when combined with the interaction, turning the calculation into a Hartree-Fockproduce Ft values that agree with Bogolyubov one. However, under no circumstances have we the full range of Z values. This, of 065501-7  1233  200  *  *  2 ϩ c 3 ϩ c 3 ϩ  1  ϩ 2 ϩ 1 ϩ c 1  a  1160 1180 1200 1220 1240 1260 1280  b c  4244  d  Energy [keV]  G. 2. Spectrum of ␥ rays from the 74Rb decay. Arrows and ndicate contaminating radiation from the 74Ga and 74Br de38 m 46 22 respectively.  Mg  K  V  26 m 34 74 50 62 Al constant Aras a function Mn peaks were of time, Ga as expected Rb if come from 34 a very short-lived source. Finally, in an in54 Cl 42Sc ͓13͔, it wasCo found that the 456ndent ␤-␥ experiment 1198-keV lines decay with the characteristic 74Rb halfThis ␤␥ experiment also confirmed the intensities of two transitions as presented below. he intensities of the observed transitions per 74Rb decay iven in Table I. The values are corrected for summing ts related to positron emission, which were determined the experimental ␥-ray spectrum to be ϳ2%. Beyond the intensity of each ␥ ray is corrected for summing other, coincident, ␥ rays. The ␥-␥ summing probabiliwere determined from standard calibration sources and ϩ ϩ ϩ between 5% and 10%. For the 0 ϩ 2 →0 1 and 2 1 →0 1 tions, 30 and con10 theoretical K/total 20intensity ratios ͓14͔ on coefficients ͓12͔ were used to determine the total tion intensities the measured K-converted radiaZ offrom daughter  (a)  40  20  0  18  Ne  60  0 62  Ga  74  TZ = 0  60  (b) 0 18 3022 Mg 4026Si Ne Q-value SMg 3426 Ar Si  22  66  As  40  70  74  Br  Parent nucleus 40  TZ = -1  20  38 30  Branching ratio  20  60  R  C  Rb0 40  18  Ne  -  NS  22  Mg  26  !20 %  Si  30  Ga  66  As  70 62  Br Ga  66 74  As Rb  70  38  TiAr  Ca  Ti  TZ = -1  TZ = 0  TZ = 0  34  S  Ar  38  Ca  42  Ti  Q-value  Half-life  THalf-life Z = -1  Branching ratio  ratio TBranching Z = 0  d’  dR’  R  dC - dNS  Br  42  TZ = -1  Q-value  FIG. 5. Summary histogram of the fractional uncertainties at60 20 20 tributable to each experimental and theoretical input factor that contributes to the final Ft values for the 11 other superallowed transitions. Where the error is shown as exceeding 60 parts in 104 , no 40 useful experimental measurement has been0made. 0  he decay scheme for Rb is shown in Fig. 3. With the rected experimental f t values of ption of the 1198- and 4244-keV ␥ rays, allas of athefunction obd transitions can be attributed to the decay of known hter nucleus (top panel) and the corresponding 62 s in 74Kr ͓8,9,15͔. We have tentatively assigned the ϩ ray latter as a (1differ →0 ϩ transition. -keV ␥the nel); theThe f t1204-keV values by the 1 ) from tion from the decay of the 2 ϩ was not directly FIG. 3. Partial decay scheme of 74Rb. Intensity of transitions and2 δlevel horizontal gray tion terms δR , δNSin, energy C . The 74 ved, since it coincides with a stronger 1204are given in units of 10Ϫ5 per 74Rb decay.  42 34  Ca S  Half-life  4  4120 4160 4200 4240 4280 4320 4360  from the intensity of the 748-keV ␥ ray and the intensity ratio of the 1204- and 747-keV ␥ rays taken from Ref. ͓15͔. ϩ The 1742-keV, (2 ϩ 3 )→0 1 transition, predicted by the shell20 model calculation to have 60 an intensity of one third of the 1286-keV transition, could not be observed because of the presence of the 1745-keV ␥ ray from the 74Ga decay. From the energy spectrum of the positrons detected in the thick plastic scintillator in coincidence with the observed  4 Parts in 10 Parts in 10  Rb  Parts in 104  74  Parts in 104  0  1233͑1͒ 1286͑1͒ 4244͑1͒  *  100  14 12 10 8 6 4 2 0  Fig. 4 as well as Ga and Rb in Fig. 5 – the theoretical uncertainties are greater tha 29͑4͒ 6 (2 ) →0 experimental ones. In these cases, the nuclear-structure-dependent correction, δC − δN S , c 9͑5͒ 10 (2 ) →2 26 12͑2͒ 4 (1 →0 ) 60all F t values between 60 of 3-7 parts Alm and 54 Co but jumps up to 20-30 parts i 20 in 10 to Prediction corresponds to column 3 in Table II. because of nuclear-model ambiguities. For its part, the nucleus-dependent radiative correctio Prediction corresponds to sum of 32͑7͒ϩ48͑5͒. Tentative assignments ͑see text͒. that starts very small but grows smoothly with Z 2 . This is because the contribution to δR f Intensity determined indirectly ͑see text͒. 0 40 been estimated from2240 its leading logarithm [176] and the magnitude of this estimate has been 18 26 34 Ne Mgwas inferred Si 30S Ar 38Ca 42Ti keV ␥ ray from the 74Ga decay. Its intensity  1286  74  Rb  dC - dNS  Q-value Half-life  element is 20 Branching ratio Parent nucleus Parent nucleus Figure 2.28: (a) Decay scheme of Rb. The superallowed decay branch is † 2 d’ anel gives one standard deviation around the | f |aα |π | . (5) M0 = 051305-3 a ground-state to ground-state transition. Figure from [100]. (b) Pard - and d experimental FIG. 5: Summary FIG.histogram 5: Summary of the histogram fractionalofα,π uncertainties the fractional uncertainties to attributable each experimental to each theoretical 0 attributable tial uncertainties for the Ft-value of 62 Ga and62 74 Rb. 66The uncertainty 70 74 contributes to the contributes final Ft to values the final for the Ft eleven values other for the superallowed eleven other transitions. superallowed Where transitions. the error Where is shown the as erre BrRMS Rb 74 Rbisisnot If isospin exact |i Ga and |fAsareofnot to an about 20symmetry, % due to then the extrapolation its 4 analogs of each 4in δC for nd |f are exactinisospin 10 , no useful in experimental 10 , no useful measurement experimental has measurement been made. has been made. isospin analogs and measured a correction to M0 needs toParent be evaluated. f |aα† |π ∗ , and the symmetry-limit matrixcharge radius. This was recently at TRIUMF [16] nucleus and led to This is the isospin-symmetry-breaking correction, δC , we seek a reduction in the associated uncertainty. Adaptation of a figure from is defined by FIG.to5:determine. SummaryIthistogram of the fractional uncertainties attributable to each experimental and t [15]. contributes to the final Ft2 values2 for the eleven other superallowed transitions. Where the error is s MF = M Q-value in 104 , no useful experimental measurement has been made.(6) 0 (1 − δC ). R R  C  Half-life  Branching ratio  Alm 34Cl  38  Km  42  Sc 46V  Parent nucleus  -  Ideally, to obtain δC one would compute Eq. (4) using the shell and introduce Coulomb and other The charge-dependent values for the model 13 well-studied superallowed decays. heavier superallowed terms into importance the shell-model Hamiltonian. However, because the emitters are of particular as they have larger δC corrections. Figure 2.26 74 Coulomb force is long range, the shell-model space would and Figure 2.27 highlight that an improvement in the f t-value for Rb would lead to be huge to include all the potential states with which to much more have stringent tests and discrimination between different models of ISB the Coulomb interaction might connect. Currently, this is not corrections. Inathe next section the present experimental situation regarding 74 Rb practical proposition. will be reviewed. To proceed with a manageable calculation, we have developed a model approach [7,178,179] in which δC is divided into two parts:74 50 54 C  NS  Mn 2.9 Co The  case of Rb and the need for highly charged ions = δC1 + δC2 . (7) in Penning trap massδCmeasurements  histogram of the fractional uncertainties atwe compute For δ0C1 +, → The superallowed 0+ , T = 1 β-decay of 74 Rb proceeds to the isobaric erimental and theoretical input factor that conanalog ground state of 74 Kr.f¯Its scheme is displayed1/2 in, Figure 2.28(a). As values for the “traditional nine” superallowed |aα†decay |π π |b (8) α |ı = M0 (1 − δC1 )  74 the superallowed decay in α,π Rb connects two ground states its only detectable signature is the outgoing positron or the neutrino of which the latter is usually not detected.055502-12 The kinematics of a decay into three particles (daughter, positron, and  66  NS  neutrino) results in a spectrum of positron energies which overlays with the spectra of transitions to non-analog states. A direct measurement of the superallowed branching ratio is thus not easily possible. Instead the sum of all non-analog decays is determined to obtain the superallowed BR. In those decay branches, the β-decay populates excited states in 74 Kr which decay via emission of γ-rays towards the ground-state. Except for longer lived isomers, the life-times of these states are very short and the γ-decay occurs instantaneously. γ − γ coincidences further allow the reconstruction of the level-scheme and, normalized to the counted β- particles, a BR can be obtained. Because of the relatively large Q-value in 74 Rb many high lying 1+ excited states in 74 Kr are populated by Gamow-Teller β-decay transitions. Their individual population is small and below the experimental sensitivity level. However, when their BR are added, the value is not negligible for the extraction of the superallowed branching ratio. The remedy to this problem is found in lower lying levels in 74 Kr, which act as collector states of the weak Gamow-Teller transition. This means that the many high lying states will not decay directly into the ground state, but feed to a few low lying states. The γ- decays from these levels can be detected. However, to achieve complete accounting for the total non-analog decays, assistance from theory is required as some direct feeding of the ground state from high lying levels will remain unobserved. With the benchmark that the theoretical description properly describes the relative feeding of the observed transitions, the remaining component of Gamow-Teller strength can be calculated by theory. Such a measurement and analysis has been performed in [100] with shellmodel calculations by Towner and Hardy. The total non-analog BR was determined to be 0.5(1)% resulting in a superallowed BR of 99.5(1)%. The most recent survey of superallowed β-decays lists two precision measurements of the half-life of 74 Rb. Their uncertainties are almost overlapping (see Figure 2.29(a)), but following the conservative procedures of the survey a scaling factor is introduced to account for a potential non-statistical difference in the two measurements. The combined half-life is 64.776(43) ms, which makes 74 Rb the shortest-lived nuclide among all 13 well measured superallowed decays. An approximation of Equation 2.63 shows that the statistical rate function roughly scales with the fifth power in the Q-value. Hence, the Q-value has to be measured 5 times more precisely than half-life and BR for the determination of the f t-value. In the past, masses and Q-values for β-decays have been measured by nuclear reactions or β-decay end-point measurements. In order to obtain the required precision for superallowed β-decay studies, these have been restricted to the following: (3 He,t) reactions, (p,n) threshold measurements, or (p,γ) and (n,γ) on the same target to reach parent and daughter of the superallowed decay. The emergence of Penning traps in the realm of rare isotope research led to unprecedented precision in atomic masses and, thus, also for Q-values. The impact of Penning trap 67  (a)  (b) 65  mass excess ï 51 905 [keV]  20  TRIUMF  64.9 64.85 64.8 64.75  ISOLDE  halfïlife [ms]  64.95  64.7  15  2000  10 5  2003  0 ï5  2002  ï10 ï15 ï20  Figure 2.29: (a) Precision half-life measurements of 74 Rb at ISOLDE [101] and TRIUMF [102]. (b) Direct mass measurement of 74 Rb with ISOLTRAP [103] from [12, 13], and summarized in [14].  mass measurements was indeed critical not only from a precision point of view, but it also brought deficiencies in accuracy of older measurements to light, e.g. for 46 V [104]. In the meantime, all Q-values of superallowed β-decays have been measured in Penning traps, except for 14 O [15, 105]. Penning traps are able to perform mass measurements even of short-lived nuclides with relative precisions of δm/m ≈ 10−9 . By taking the mass differences the Q-value of the β-decay is obtained (see Figure 2.13). Traditionally, Towner and Hardy list the QEC -value of the electron capture, which is simply the difference in atomic mass between the parent and the daughter. In those cases where the superallowed decay populates an excited state in the daughter nucleus, precise knowledge of the excitation energy of the state is also required. This can be achieved via γ-ray spectroscopy. The atomic mass of 74 Rb’s daughter, 74 Kr, has been measured with a precision of 2.1 keV at the Penning trap facility ISOLTRAP at CERN [106]. The mass measurement of 74 Rb in a Penning trap is, due to its short half-life of 65 ms, more challenging. Before TITAN’s measurement of 11 Li (T1/2 = 8.8 ms)[7], 74 Rb was the shortest-lived nuclide whose mass has been measured in a Penning trap. In order to achieve this, ISOLTRAP carried out three campaigns on 74 Rb [12–14] (see Figure 2.29). Other measurements of the mass of 74 Rb published in [107, 108] are in agreement with the ISOLTRAP result, but because of their poorer precision, they do not carry any weight. Since the nuclear binding energies with typically ≈ 8 MeV per nucleon are small compared to the mass of a nucleon (≈ 1 GeV) it 68  is common to express the atomic mass of a nuclide with mass number A in terms of the mass excess m.e. = [M − A · M (12 C)/12] · c2 . For 74 Rb, the three mass measurements at ISOLTRAP lead to a mass excess of 51914.7(3.9) keV and combined with the mass of 74 Kr to QEC = 10417.3(44) keV. The partial uncertainties to the corrected Ft of 74 Rb are shown in Figure 2.28b, when considering δC from the shell model with Saxon-Woods radial wave-functions. As mentioned before, its associated uncertainty contains the RMS charge radius of 74 Rb, which was previously not known experimentally. With the recent laser spectroscopy measurement, the RMS charge radius was determined and reduced the uncertainty on δC2 = 1.50(30)% to 0.25% while confirming the mean value [16]. As a consequence, the QEC value contributes to the uncertainty of the Ft at a similar amount as the nuclear structure dependent corrections. This implies two strong motivations to improve the QEC -value. Firstly, an error reduction on QEC would have a direct impact on the corrected Ft whose uncertainty could be improved by 20%. Secondly, an improved f t-value would allow more stringent tests regarding the discrepancies between different models of the ISB corrections (Section 2.8). As mentioned before, 74 Rb, with its largest δC among all superallowed β emitters, would carry particular weight were it not limited by the current precision in its QEC -value. Considering the importance of the CKM matrix and the tension of some ISB models with unitarity, an error reduction in the f t-value of 74 Rb is crucial. For all of these reasons, a new measurement of the superllowed BR in 74 Rb has recently been performed at TRIUMF and its analysis is underway [109]. But since the uncertainty of the f t-value is dominated by the QEC it is most important to improve the precision of involved masses. Another dedicated measurement of 74 Kr in a Penning trap would likely lead to a more precise atomic mass. However, 74 Rb poses, due to its half-life of 65 ms, a real challenge to experiment. Its mass is known by a factor 2 less precisely than that of 74 Kr, although it has already been determined 3 times in a Penning trap. This indicates that an improved precision is unlikely to be reached due to limits of the technique. To understand its limitation better, it is important to consider the achievable precision of a mass measurement in a Penning trap which follows δm m √ ∝ m qBTrf Nion  (2.89)  [10], where δm/m is the achievable relative precision in mass m. q is the ion’s charge state utilized in the measurements, which are generally performed with SCI . B is the magnetic field strength of typically a few Tesla. The highest field strength used at Penning trap facilities to measure masses of radioactive nuclides is 9.4 T [110]. The requirement of very homogeneous fields makes even larger field 69  strengths very difficult to explore. The number of ions Nion is limited by the available experimental time at radioactive beam facilities where radioactive nuclides are being produced (see Section 3.2). It is further constrained by the production yield at radioactive beam facilities and efficiency of the spectrometer. Finally, the precision is dependent on the measurement time Trf on each individual ion. The range of possible Trf is obviously restricted by a nuclide’s half-life, which is indeed the limiting factor for the precision in the mass of 74 Rb. Hence, an option for a more precise mass of 74 Rb is to develop experimental techniques which enable Penning trap mass measurements in higher charge states. Another approach would be a novel octupolar excitation scheme in Penning trap spectrometry [111, 112] which promises gain in experimental precision. However, this excitation is more sensitive to the initial conditions of the measurement and its theoretical foundation has been laid out only very recently [113]. On the other hand, Penning trap mass studies utilizing HCI have already been successfully pioneered with stable nuclides [11, 114]. In the realm of rare isotope science with Penning traps, HCI represent a thus far unexplored opportunity to improve the experimental precision further circumventing constraints imposed by short half-lives and lower yields when probing the limits of nuclear existence. However, in contrast to measurements with stable nuclides, the requirements of high efficiency and short measurement times are critical when working with radioactive ions. The experimental part of this work is focused on the very first mass measurement of HCI of radioactive, short-lived nuclides in a Penning trap including a successful mass measurement of 74 Rb+8 .  70  Chapter 3  Experimental Setup 3.1  Introduction and overview of the TITAN facility  The mass measurements described in this thesis are performed with TRIUMF’s Ion Trap for Atomic and Nuclear science (TITAN), which is coupled to the radioactive beam facility ISAC at TRIUMF. Operational for online measurements since 2007, TITAN’s measurement program had initially focused on mass measurements of so-called halo nuclei [115, 116]. The masses of 8 He [117, 118], 6 He [118], 11 Li [7], 11 Be [119], and 12 Be [120] have successfully been measured. The ground state of 12 Be is not a halo nucleus itself, but in its first excited 0+ -state a neutron halo-like structure might be formed [121]. The ability to carry out precision mass measurements, even for very-short lived nuclides with T1/2 < 10 ms distinguishes TITAN from other facilities and is due to TITAN ’s fast measurement preparation which does not require a preparation trap and uses a Lorentz steerer for beam injection into its Measurement Penning Trap (MPET)(see Section 3.7.7). In fact, the demonstration of the mass measurement of 11 Li with its half-life of 8.8 ms makes TITAN the fastest online Penning trap system worldwide. This is a critical feature for further exploration towards the limits of nuclear existence. Considering that the half-lives tend to get shorter further away from the valley of stability, this ability becomes increasingly important. Figure 3.1 illustrates the increased accessibility of short-lived nuclides at TITAN for masses up to Z ≤ 50 and N ≤ 50. In the field of metrology, TITAN has helped to resolve a discrepancy in the mass of the stable nuclide 6 Li [124]. This measurement, with δm/m = 4.4 PartsPer-Billion (PPB), has highlighted the achievable accuracy which is competitive with facilities dedicated to high precision mass measurements of stable nuclides. More recently, TITAN has performed measurements on 30,31 Na motivated by the socalled island of inversion [125, 126] and of 47−50 K, 49−50 Ca [1] and 51 K, 51,52 Ca 71  (a)  (b)  50  50  T1/2 > 65 ms  T1/2 > 8 ms  45  40  40  35  35  30  30  Protons  Protons  45  25  20  %  20  !"#$&  15  !"#$  15  10  25  10  '"#$&  5  5  ()*+,-.,/0 0  0 0  5  10  15  20  25  30  35  40  45  50  0  Neutrons  5  10  15  20  25  30  35  40  45  50  Neutrons  Figure 3.1: Lower section of the chart of nuclides with nuclides with halflives larger than 65 ms (a) and 8 ms (b). Before TITAN’s mass measurement of 11 Li, 74 Rb (T1/2 = 65 ms) used to be the shortest lived nuclide whose mass has been determined in a Penning trap [13]. The color-code reflects the relative mass uncertainty δm/m of a nuclide. The mass data were taken from the 2011 preview of the atomic mass evaluation [122] and the half-lives are from [123].  [127]. The latter mass range is interesting due to predictions about shell closures at N = 32 and N = 34 [128, 129] to which the two-neutron separation energies S2n = −M (A, Z) + M (A − 2, Z) + 2mn are sensitive. Theoretical models with only NN-forces fail to reproduce the experimental masses even when soft, momentum-evolved potentials are used (compare with Section 2.1.3). But when 3-body forces are added, the qualitative trend follows experiment [127, 129]. Up to now, TITAN measurements have been performed with Singly Charged Ions (SCI) as it is done at all other Penning trap setups [110][130][131][132][133] coupled to radioactive beam facilities. As the precision scales inversely with the charge state of the ion (Equation 2.89), the use of Highly Charged Ions (HCI) offers great gains for Penning trap mass measurements. Indeed, the Stockholm-Mainz-IonLEvitation-TRAP (SMILETRAP) has pioneered this approach for stable nuclides [11, 114]. In the realm of rare-isotope science, short half-lives and the need for high efficiency complicates the use of HCI in Penning trap mass spectrometry, and this path has so far not been exploited. In consequence, HCI represent an opportunity for new limits in the precision of Penning trap mass measurements. A second cornerstone of TITAN’s scientific program is hence to set out to master the chal72  2.1. Beam production and separation at ISAC  a) SCI  SCI  b) HCI  SCI SCI  off-line ion source Figure 3.2: Schematic of the experimental setup of TRIUMF’s Ion Trap for Atomic and Nuclear science (TITAN). The different paths for SCI (a) and HCI (b) are marked.  Figure 2.2: The TITAN experimental setup which includes a RFQ, recision Penning trap, an EBIT and an off-line ion source. a) Sh of highlybeam charged, radioactive to explore the advantages of HCI. on singly char ed is the path lenges of the whennuclides mass measurement A schematic overview of the TITAN facility is shown in Figure 3.2. The radioactive beam of b) SCI from ISAC is injected inthe a RFQ cooler and buncher [134, 135] charged ions SCI) is performed. In blue is path for highly which is floated just below (∆V ≈ 5 − 20 V) the ISAC beam energy to decelerate and to trap the beam. Buffer gas cooling takes place through collisions with mass measurement. He or H2 gas, which thermalize the ions to room temperature while an oscillating Radio-Frequency (RF) field provides radial confinement. The net result is an overall cooling of the ‘hot’ ISAC beam. In addition, a longitudinal electrostatic potential leads to a confinement which allows the continuous beam from ISAC to be accumulated in the RFQ. Ions are extracted from the RFQ by opening the con-  as-filled linear radio-frequency quadrupolar (RFQ) trap ([Smi06, Sm 73 The subsequent step depends on whether a mass measurement using harged ions (SCI), or highly charged ions is performed. The ions can e transferred to an electron-beam ion trap (EBIT) [Fro06], where reeding takes place (blue path in figure 2.2) or directly sent to the P  fining potential, and ion bunches are released. Beam extraction from the RFQ can proceed in the reverse or forward direction [97]. In the reverse mode, the beam is sent back into the ISAC beam line (as indicated in Figure 3.2 with ‘beam to next experiment’) where other experimental setups can take advantage of the cooled and bunched beam. Recent laser spectroscopy measurements of isotope shifts have benefited from this technique [16]. When the bunched beam from the RFQ is extracted in the forward direction, it can follow two paths within the TITAN setup. First, it can be transferred directly into the precision MPET. In this case, the mass measurement is carried out with SCI (see a in Figure 3.2). Alternatively (Figure 3.2b), the ion bunches can be sent into the EBIT charge breeder [2, 136]. In an EBIT , higher charge states are reached through electron-impact ionization by an intense electron beam. The design of the EBIT reaches electron beam energies up to 70 keV. As the electron binding energies increase with the number of removed electrons, such an electron beam energy in the EBIT allows removal of even deeply bound electrons of the ion. After the charge breeding, the HCI are extracted as pulses and are delivered to the MPET. An unavoidable effect of the charge breeding in the EBIT is an increased energy spread which can negatively impact the capture efficiency at the MPET or the mass measurement itself (see Section 3.7.3). A Cooler Penning Trap (CPET) [137] [138] [139] is currently being built at TITAN to prepare the HCI for injection into the MPET by providing a cooling mechanism for highly charged ions. Because of the high charge state and the rapid charge exchange (compare with Equation 3.57 to be discussed later), buffer gas cooling with neutral atoms is not an option. Recently, sympathetic cooling of HCI in a laser-cooled plasma of singly charged ions (e.g. 24 Mg+ ) has been suggested as a fast cooling mechanism [140] [141] [142]. However, at TITAN’s CPET, electron or proton cooling will be utilized because both techniques can be developed and studied based on the same trap setup. Moreover, electron cooling of protons has already been demonstrated in [143]. The selfcooling of electrons in a strong magnetic field due to synchrotron radiation offers simple availability of a cold electron gas. Simulations suggest [137] that during the time until the ions are cooled to ≈ 1eV/q only a few percent of the ions undergo recombination. Losses could then be minimized by separating the HCI from the electrons before recombination sets in. In proton cooling, recombination of HCI with the coolant is avoided. Due to short half-lives well below one second, the hot ion bunches from the EBIT will need to be trapped, cooled, and transferred to MPET quickly. Access to large amounts of cold protons, Np ∼ 108 , on this time scale is more difficult because the larger proton mass hinders any self-cooling in the B-field. Once tested in an off-line setup with its own source of SCI [139], the CPET will be inserted into the TITAN beamline just in front of the MPET . Apart from the time limitations imposed by the respective nuclide’s half-life there 74  are no fundamental differences between stable and radioactive ions in terms of applying these techniques. Thus, a surface ion source is installed at TITAN which allows test and optimization of all individual components. Generally, it provides different beams of singly charged alkali metals. A recent test of a Ca source has also been successful. Hence, measurements can be prepared using ions from TITAN’s ion source previous to ISAC beam delivery of radioactive beams. As an additional advantage systematic effects can be studied well in advance of an experiment by measuring well known masses of stable nuclides. As a result, major contributions to the systematic uncertainty can be investigated independently of the actual measurement. This chapter will introduce the different stages of the beam formation and preparation necessary for the mass measurement.  3.2  ISAC :  production and delivery of radioactive beam  Various techniques are established to produce beams of short-lived radioactive nuclides. Most common are fragmentation of a fast (typically 100s of MeV/u), heavy ion beam on a target of light nuclei (called in-flight fragmentation) [144] and the spallation, fission, and fragmentation of a heavy target by a driver beam of light nuclides. The second technique is known as Isotope Separator On-Line (ISOL). The ISAC facility [145] at TRIUMF falls into this category. It is driven by a proton beam from TRIUMF’s cyclotron, which accelerates H− to 500 MeV. When passing through stripper foils inside the cyclotron a fraction of a H− bunch is converted to H+ and extracted. The proton beam is delivered to ISAC where it impinges on a socalled production target. The ISAC facility receives up to 100 µA of proton beam which is currently the highest beam intensity on-target for any ISOL facility. Hence, the availability of beam powers with up to 50 kW in combination with thick, high power targets represents unique prerequisites to achieve the highest ISOL production yields. The products from the bombardment of the proton beam have to diffuse out of the target. Thus, target temperature is a critical parameter which is controlled by the energy loss of the proton beam and possibly by heating the target resistively. For high beam powers, targets require cooling to withstand deposited heat during the continuous proton bombardment. Once the products diffuse out of the target several ion sources [147] are available at ISAC to ionize the neutral atoms including a surface, laser [148], and Forced Electron Beam Induced Arc Discharge (FEBIAD) ion source. Surface ionization is accomplished by heating an ionizer tube and target to about 2000 o C. Following the Langmuir-Saha surface ionization theory (see its application for surface ion sources for instance in [149, 150]), elements with an ionization potential Ip lower than the work function φ of the surface, e.g. tungsten with φ = 4.6 eV, can efficiently be ionized. Hence, a surface ion source is well 75  2.1. Beam production and separation at ISAC tinuous proton beam coming from the TRIUMF cyclotron. The current on target of that beam can go as high as 100 µA. Once produced, different nu-  (a)  )3!# BEAM PRODUCTION AND SEPARATION ROOM  (IGH
SEPARATOR  PROTON BEAM  Figure ISAC production and separation room. room includes Figure 3.3:2.3: (a) The Schematics of production, ionization, andThis separation of ratwo target stations, and inset ion sources, a pre-separator highdioactive beams at target ISAC . The (b) shows the target andand ion asource resolution (m/∆m = 3000) magnet separator. Also shown is a rendering of setup. Figure from [146]. the target and a surface ion source.  clei alkali diffuseelements out of the target andstudied are then by(Ian = ion4.2 source suited for such as the Rbionized isotopes keV),[Dom02]. but noble p Subsequently, the ionized isotopes are extracted and formed into a beam gases are not accessible owing to the highest ionization potentials of all elements. which is electrostatically accelerated to an energy of 12 to 60 keV. It is later The ions are electrostatically accelerated to 60 keV energy. They pass a guided to a two-stage dipole magnetup separator thatbeam include a pre-separator two-stage magnet mass separator unit (figure (pre- and main and dipole a high-resolution magnet separator 2.3). Thisseparator) separateswhich and se-selects ions on their mass-to-charge ratio. with different massat numlectsbased the ions of interest according to theirNuclides mass-to-charge ratio (m/q) a bers are easily separated due to them/∆m large mass differences. Butseparated differentbeam elements resolving power of typically = 3000. Finally, the is delivered to thenumber ISAC hall various areclose located. with the same mass (i.e.where isobars) canexperiments often be too in mass to be 8 The species of interest to this thesis, 6 He and are delivered produced to resolved by thetwo mass separator. Isobaric contamination in theHe, beam using a SiC target and ionized by the so-called Forced Electron Beam the experiment has to be expected if a typical resolving power of m/δm ≈ Ion 3000 Arc Discharge (FEBIAD) source [Bri08]. Using this technique, ionization is is surpassed. This could represent a problem for experiments such as the mass done via a plasma generated by injecting atomic gas into a chamber where measurements at TITAN which require contamination-free beams of the nuclides of interest. Contaminants could affect the measurement or reduce the sensitivity. 37 The schematics of production, ionization, and separation of the radioactive beam at ISAC are shown in Figure 3.3. The ISAC experimental halls (see Figure 3.4) are divided into three areas with distinct energy regimes. All measurements which are relevant for superallowed β-decays require low energy beams. Indeed, TRIUMF is in the unique position to measure all quantities to characterize a superallowed β-  76  ISAC II S RES TIG  A  EMM  S  E ACL HER  ISAC I  RFQ  AN TIT  Figure 3.4: Schematics of the ISAC facility and the experimental facilities. ISAC experimental areas are distinguished by their respective beam energy. First, the low energy section with up to 60 keV is dedicated to ground state and decay properties investigated by trapping experiments, β-decay studies, or collinear laser spectroscopy work. An RFQ and a drift tube linac accelerate the radioactive ion beam up to 1.5 MeV per nucleon in the so-called mid-energy regime. There, the beam is used for reaction measurements important for nuclear astrophysics. Moving into the third energy regime, the ISAC-II superconducting linear accelerator brings the beam to 5-11 MeV/u for nuclear reactions at higher energy.  77  (a)  (b)  Figure 3.5: (a) Photo of the 8π facility. (b) Schematic of the gas proportional counter with tape station. Both from [153].  decay. BR are determined at the 8π γ-ray spectrometer [151] which consists of 20 Compton-suppressed HPGe detectors. It works in conjunction with a tape-station setup and β-particle counter such as the Scintillating Electron Positron Tagging Array (SCEPTAR) [152], see Figure 3.5. Half-lives of β-decays are measured at a tape station which transports the implanted sample into a gas proportional counter [154]. Finally, the QEC -value can be deduced from direct mass measurements at TITAN . If the superallowed decay branch populates an excited state in the daughter nucleus, its excitation energy can be precisely measured through the emitted γ-rays at the 8π facility.  3.2.1  Production and delivery of neutron-deficient Rb-beams  Different target materials produce different radioactive nuclides. Figure 3.6 compares various targets in their yields of Rb isotopes. This study aimed towards the proton dripline, i.e. the line where members of an isotopic chain become unbound when removing another neutron. These neutron-deficient Rb isotopes can be delivered in high quantities from ZrC and Nb targets. For the yields shown in Figure 3.6, ZrC targets were typically operated with a 35-75 µA proton beam, while Nb targets were bombarded by 98 µA of protons. The sudden drop for 74 Rb in the use of a ZrC target is likely caused by the short half-life of 65 ms (compared to 19 s for 75 Rb) and the lower proton beam intensity of 35 µA compared to the Nb-target 78  12  10  ZrC Ta UCx Nb  10  measured ISAC yield [1/s]  10  8  10  6  10  74Rb 4  10  2  10  0  10  70  75  80  85  90  95  100  105  mass number A  Figure 3.6: Measured yields of ground state Rb isotopes at ISAC for various targets. These are measured absolute numbers which are not normalized to the respective beam currents for different targets. Due to the regulator licence restrictions, the ISAC UCx target was only irradiated with a 2 µA proton beam which is a factor of 10-50 less than all other targets. Uranium targets owing to their large excess of neutrons are particularly suited for beams of neutron-rich nuclides but less favourable for the present studies. The gap at A = 85 − 87 is due to the stable 85,87 Rb as well as the long-lived 86 Rb with a half-life of 18.6 days. Data are taken from ISAC yield database [155].  [156][157][158]. This has led to the choice of a Nb-target for the present measurements. As Rb is an alkali metal the studies presented in this thesis required the surface ion source. Unavoidably, this also ionized other elements such as Ga (Ip = 6.0 eV) or Sr (Ip = 5.7 eV), although at a lower ionization efficiency. Most other elements such as Kr (Ip = 14.0 eV) were not surface ionized due to their higher ionization potentials. The resolving powers R = m/δm needed to separate isobaric Ga and Sr from the Rb beam were beyond the capabilities of the ISAC mass separator (see Table 3.1). 74 Rb could be separated with R ≈ 4300 from the isomeric and ground states of 74 Ga which did not much exceed the performance of the mass separator. It was possible to fine tune the magnets of the mass separator to a more favourable ratio between contaminant and beam of interest at the cost of a lower intensity of  79  Table 3.1: Contamination, mass excess m.e., required resolving power R, and results of yield measurements [159] previous to the TITAN beamtime Nov. 20-24, 2010. A  element  m.e. [keV]  R to A Rb Nov. 7  78  76  75  74  Ga Rb m Rb Sr Ga Rb Sr Ga Rb Sr Ga m Ga Rb Sr  -63706.0(1.9) -66936.2(7.5) -66825.1(7.5) -63173.9(7.5) -66296.6(2.0) -60478.1(1.2) -54248(35) -68464.6(2.4) -57218.7(1.6) -46619(220) -68049.6(3.7) -67989.6(3.7) -51917.0(3.7) mass unknown  Yield [1/s] Nov 12  Nov 20  22492 653445 19312 -12167 11363 -6212 6591 -4273 -4289  74 Rb  2.1 · 109 4.4 · 109  1.6 · 108 2.2 · 109  4.4 · 107  2.3 · 107  4.6 · 105 2.1 · 106  2.0 · 105 1.1 · 106  3.1 · 105 1.2 · 103 1.7 · 104  4.5 · 105 2.4 · 103 1.2 · 104  6.4 · 103  (see Section 4.1). Table 3.1 also lists the measured yields previous to this measurement. Ga was the dominant nuclide at A = 74, and for A = 75 a beam contamination of ∼ 15 % was found. The yields of heavier Ga isotopes were not measured this time. But beam intensities for Ga (see Figure 3.7) typically become smaller with every added neutron for masses larger than ≈ 70. The Ga isotopes found were on the neutron-rich side in respect to stable Ga, and higher masses represent isotopes further away from stability. Hence, 76 Ga (or 78 Ga) was not expected to be a significant contamination for 76 Rb (or 78 Rb). Sr yields were not determined directly. Simulations of in-target production [159] indicated that on the neutron-deficient side Sr was about one order of magnitude less abundant than Rb and was dropping down to more than 2 orders of magnitude for A = 74. This trend was consistent with older yield measurements for 79,81 Sr (Figure 3.7). Hence, contamination was an important factor to consider for the mass measurements of 74,75 Rb. Since Ga was a dominant component in the A = 74 beam, fine tuning of the mass separator and isobaric cleaning with ion-trap techniques at TITAN was required (see Section 3.7.6). It is interesting to note that the beam intensities from this target decreased over time (see Table 3.1) indicating deterio80  9  10  Sr Rb Ga  normalized ISAC yield [1/s/µA]  8  10  7  10  6  10  5  10  4  10  3  10  2  10  1  10  60  65  70  75  80  85  90  95  mass number A  Figure 3.7: Measured yields of Ga, Rb, and Sr isotopes for three Nb-targets at ISAC. The yield is normalized to 1 µA proton beam. Data taken from [155].  ration due to irradiation. When 74 Rb was first delivered to TITAN its intensity had dropped by almost a factor of ≈ 2.5 in comparison to the target performance when it had been first irradiated by the proton beam.  3.3  TITAN ’s  Radio-Frequency Quadrupole (RFQ) cooler and buncher  For the mass measurement, the ISAC beam with an energy up to 60 keV needs to be brought to quasi-rest and have a typical remaining kinetic energy of a few eV in the MPET. The formation of the radioactive beam introduces an energy spread of a few 10’s of eV in the extraction process from the source as opposed to the measurement requirement of low energy spread. Moreover, ISAC delivers a continuous beam. Trap experiments on the contrary load the trap in pulsed beams, and usually no new particles are added during the measurement itself. To efficiently use the ISAC beam, it has to be accumulated and bunched. All of these tasks are accomplished by a buffer gas filled RFQ cooler and buncher. Such devices are now widely used at radioactive beam facilities [5, 160, 161], 81  mostly for mass measurements of rare isotopes but increasingly also for collinear laser spectroscopy [16, 162, 163]. In order to stop the radioactive beam TITAN’s RFQ [134, 135] is biased a few eV below beam energy. Care has to be taken during injection to match the acceptance of the RFQ to the emittance of the incoming beam. This is achieved by deceleration optics forming an electrostatic azimuthal quadrupole potential [134, 164]. This decelerates the ions in axial direction while maintaining the transverse emittance as the ions oscillate radially with a harmonic motion. TITAN’s RFQ is designed for beam energies between 12 - 60 keV with a transverse emittance of up to 50 πmm mrad in x − θx -space. The ions are cooled through collisions with a room-temperature buffer gas to thermal energies. Without additional forces, the ions could not be confined in an accumulation region during the collisional cooling process. Thus, the buffer gas cooling takes place in a linear Paul trap. A linear Paul trap is a 2D trap, hence without a longitudinal confinement as in a 3D hyperboloidal Paul trap [5]. In a linear Paul trap, the ions are trapped in longitudinal direction by an electrostatic potential (see Figure 3.8). It is applied by biasing 24 longitudinal segments at well defined DC voltages. The gradient is chosen to drag ions into the minimum of the potential. Since the Laplace equation does not have solutions with minima in all three dimensions for a source free volume, additional confinement is necessary. For this reason, the radial confinement is accomplished by a quadrupolar RF-field generated by four, pairwise connected longitudinally segmented rods. A cross-sectional view is given in Figure 3.9a and the respective electric potential in Figure 3.9b. The saddle of the quadrupole field results in a force on the ions towards the radial centre of the trap in one direction and outwards in the other direction. By switching the polarity of the applied voltage the ion can be confined radially when the combination of applied RF frequency, amplitude, and the ions’ mass-to-charge ratio leads to stable trajectories in the trap. Radially, the ions follow along a harmonic macro-motion disturbed by a small micro-motion. The harmonic motion can be described by a so-called pseudo-potential [165] in comparison to the trap depth in a true harmonic potential. Traditionally, sinusoidal RF-fields are used in these devices. However, it can be shown [134] that a square wave driven RF field can increase the pseudo-potential by a factor of about 1.5. The square-wave drive is also referred to as digitally driven [166]. TITAN’s RFQ operates with a square-wave RF with peak-to-peak amplitudes up to Vpp = 400 V in a frequency range of 0.2 to 1.4 MHz. Such a drive is a technical challenge for large capacitive loads as in the case of the TITAN RFQ with about 1500 pF. It is achieved by stacking fast switching, Metal-Oxide Semiconductor Field-Effect Transistors (MOSFET) in a push-pull configuration [134, 167]. The use of several MOSFETs reduces the power dissipation on each individual transistor. The RF is capacitively coupled to the individual electrodes (see Figure 3.9c) while the DC bias to create the longitudinal field gradient is delivered over a resis82  2.2. Beam preparation: the radio-frequency quadrupole (RFQ) cooler and buncher  VDC R 1  3  22  C  24 VAC rod  V  continuous beam + +  0  + +  13.4 V 5V  +  capture potential cooling and accumulation bunched beam -6.6 V  + + +  extraction potential -26.6 V z  Figure 2.4: Top: Schematic of TITAN’s RFQ which is composed Figure 3.8: Top: Schematic of thesideview longitudinal segmentation of TITAN ’s RFQ of four 24-segmented rods that create a longitudinal trapping potential. into 24 sections. The segment number is indicated at the top. Bottom: A well allows of for the beam accumulation subsequent bunching. A squareSchematic typically applied and potential. The DC field drags the wave RF is applied to the opposite segments to provide radial confinement. buffer gas cooled ions to the minimum of the trapping potential (solid Bottom: Schematic potential distribution for accumulation (solid line) and line). The beam is extracted in ion bunches by switching the potentials bunch extraction (dashed line). of the electrodes 22 and 24 (dashed-line). Figure from [146]. . beam with �99% = 50 π mm mrad transverse emittance at 60 keV energy [Smi08a]. The transverse emittance of the beam leaving the RFQ is aptor which damps the �RF to protect the DC power supplies. proximately 99% ≈ 10 π mm mrad at 1 keV. The measured full width half An effective net buffer gas cooling in anenergy RFQ takes place when the mass of the maximum (FWHM) of the beam spread at this energy is typically ion is larger than the[Cha09]. coolant gas particles [165, 168]. In the presence of an RFaround 6 eV The typical buffer gas used to cool the beam is the helium, to its inert field for trapping, a drastic change in energy would disturb ion’sdue micro-motion nature and light mass allowing favorable momentum transfers for significantly and bring the motion out of phase with the RF-field. Hence,efficient if the 6,8 He mass measurements the spread dissipation. for the theenergy mass of energy the coolant is larger than theHowever, ion’s mass of the ion is on average beam was cooled using hydrogen to avoid resonant charge exchange reacincreased, an effect which is referred to as RF-heating [169]. In a buffer gas of tions. Figure 2.4 shows that the TITANs RFQ is composed of a four rod lighter masses, little momentum is transferred per collision, and the disturbance of structures on which a radio-frequency quadrupolar field is applied to create  the micro-motion is less relevant. As a consequence the harmonic, macro-motion can be damped while keeping the micro-motion coherent to the external RF-field. 39 The average, relative energy change per hard-sphere collision (Figure 3.10) follows [168] 1−κ < >= κ (3.1) (1 + κ)2 83  2.2. Beam preparation: the radio-frequency quadrupole (RFQ) cooler and buncher  (a)  (b)  (c) VDC R  1  3  22  C  24 VAC rod  V  continuous beam  13.4 V 5V  1: A schematic of an ideal hyperbolic electrostatic quadrupole (thick line) and the + + + + + Figure 3.9: Schematic crossFigure sectional view of electrode structure (a), which capture potential circular electrodes used to closely approximate the hyperbolic shape are shown on the top cooling and accumulation generates a radial quadrupole field (b). The polarity of the applied 0 left. The saddle-shaped harmonic potential it creates pois illustrated at the bottom left. The Figure 1: A schematic of an ideal hyperbolic electrostatic quadrupole (thick the geometry ofline) anand formed from rods and pair-wise applied alternating tentials in (a) is switched following aRFQ square-wave RFhalf-cylindrical driver, circular electrodes used to closely approximate the hyperbolic shape are shown on the top bunchedwhich beam conleft. The saddle-shaped harmonic potential it creates is potential illustrated at the (V bottom + is on and the right. rf )left. + the ions radially. coupling ofTheshown the DC RF+ components to the geometryfines RFQ formed half-cylindrical rodsThe and pair-wise applied alternating Figure 1: of A an schematic of anfrom ideal hyperbolic electrostatic quadrupole (thick line) and the-6.6 V potential (V ) is shown on the right. circular electrodes used to closely approximate the hyperbolic shape are shown on the top individual electrodes of a longitudinal segment is shown in (c). Figure left. The saddle-shaped harmonic potential it creates is illustrated at the bottom left. The extraction potential geometry of an RFQ formed from half-cylindrical rods and pair-wise applied alternating (b) are from [134] and (c) from [146]. potential(a) (V )and is shown on the right. rf  rf  -26.6 V z  average energy transfer per collision  4  Figure 2.4: Top: Schematic sideview of TITAN’s RFQ which is composed of four 24-segmented rods that create a longitudinal trapping potential. A 4 0.15allows for beam well accumulation and subsequent bunching.4 A squarewave RF is applied to the opposite segments to provide radial confinement. Bottom: Schematic potential distribution for accumulation (solid line) and 0.1 bunch extraction (dashed line). 0.05  beam 0 with �99% = 50 π mm mrad transverse emittance at 60 keV energy [Smi08a]. The transverse emittance of the beam leaving the RFQ is approximately �99% ≈ 10 π mm mrad at 1 keV. The measured full width half −0.05 maximum (FWHM) of the beam energy spread at this energy is typically around 6 eV [Cha09]. −0.1 The typical buffer gas used to cool the beam is helium, due to its inert nature and light mass allowing favorable momentum transfers for efficient −0.15 spread dissipation. However, for the 6,8 He mass measurements the energy beam was cooled using hydrogen to avoid resonant charge exchange reac−0.2 Figure 2.4 shows that the TITANs RFQ is composed of a four rod tions. 0 0.25 0.5 0.75 1 1.25 1.5 structures on which a radio-frequency field is applied to create mcoolant / mquadrupolar ion  39 Figure 3.10: Average, relative energy change per hard-sphere collision between an ion trapped in an RFQ and a particle from the buffer gas [168].  84  0 -6.6 V  V  -22 V  z Figure 3.11: Schematic of RFQ DC potential during the neutron deficient Rb-mass measurements.  when κ = mcoolant /mion is the mass ratio between coolant gas particle and ion. At TITAN , a He buffer gas is used at a pressure of order ∼ 1 · 10−2 mbar. The choice of the buffer gas is also influenced by the charge exchange cross section as neutralized ions-of-interest are lost in this process by becoming neutral atoms. With the largest ionization energy, He is in this respect the optimal choice. However, due to resonant charge exchange reactions during radioactive He-isotope measurements, TITAN ’s RFQ can also be operated with H2 as a buffer gas. Figure 3.10 suggests that a buffer gas of N2 or Ne would have been even more favourable for 74−78 Rb from a cooling kinematics point of view. He was used nevertheless as simulations of the TITAN RFQ [134, 135] showed that thermalization of the even heavier 133 Cs can be accomplished faster than typical accumulation times and suggested that the necessary condition for effective and efficient transport into the next unit of the TITAN system can be achieved. Once the ions are thermalized in the RFQ, they can be extracted in an ion bunch from the RFQ by switching the segments neighbouring the minimum in potential (see Figure 3.8). In all segments for which the DC bias needs to be switched, the DC and RF-fields cannot be coupled to the electrode following Figure 3.9c. Such a setup would lead to a too long rise time when switching the DC voltage. Instead dedicated switch-boxes have been built at TRIUMF allowing the coupling of the RF-field as well as fast switching of the DC component. For mass measurements of singly charged ions the kicking strength of the extraction is typically ∆V = 20 V around segment 23, the minimum of the trapping potential (Figure 3.8). A softer  85  Figure 3.12: Extraction optics from the RFQ into a PLT which is floated to Figure 13: Extraction optics of the TITAN RFQ. An ion pulse accelerated to 1 keV lonU kinetic − energy ∆V . before Whenentering the ion bunch is in the centre of the PLT it is gitudinalRFQ a pulsed drift tube. The potential on the tube to ground without affecting The SCI is thenswitched switched using a push-pull switch such thatthe theion ionsbunch. leave the tube at leave ground.the The ions subsequently so that pass throughfrom a 5 mm differential pumping PLTarewith a kineticfocused energy of ethey · ∆V . Figure [134]. aperture using an Einzel lens placed half way between the RFQ and the aperture.  delivered vertically to either the measurement Penning trap [53] or electron trapin [25] (see Figure 5). The twenty RFQ electrodes, including kick beam wouldion result a smaller energy spread, butfour an increased pulse-width in time. injection and extraction optics, are illustrated in Figure 14. This figure also Both effects are relevant for the trapping and mass measurement at MPET. For opthe drag applied in forward extraction (top schematic). timaldisplays performance forpotential SCI , the pulse-width is artificially shortened to 300 - 1000 ns. This achieved by onlyof accepting a certain fraction of the phase space and 3.4. is Reverse Extraction Cooled Bunches eliminating in the low energy tailofofelectrodes the ion bunch. consequence, this reTheions symmetric arrangement inside As the aTITAN RFQ (see ducesFigure the overall measurement efficiency. For measurement of HCI , energy spread 10) allows one to apply a mirror drag potential to the RFQ electrodes. and pulse width influenced mainly by the charge breeding the EBIT. In this case,are ions are cooled in collisions with the buffer process gas but in instead of work, being dragged through the RFQ direction, they are collectedpulse In this the kicking strength was in ∆Vforward = 100 V to achieve a smaller the potential at the entrance of bunch the RFQ. Figure (bottom widthatwhich allowed minimum efficient trapping of the ion in the EBIT14 . An additional schematic) displays the applied drag potential for reverse extraction and as pomodification of the RFQ DC settings was done by lowering the minimum of the a comparison, also the field applied during forward extraction. This cooled tential (see Figure 3.11). This was motivated firstly by simulations of TITAN’s RFQ ion bunch can then be extracted towards the ISAC beam line. Following [170]extraction, which showed good for thisthe potential forofheavier masses the ions areperformance accelerated towards ground shape potential the beam 133 such line as since Cs. no Secondly, it allowed storage ions at the with the pulsed drift tube is larger installed at theofentrance sidesegment of the RFQ. Passing two 45◦ Though electrostatic thea difference bunched and ion pulse carried is potential minimum. it didbenders, not make forcooled measurements 74 then sent towards the laser spectroscopy beam line. So far, several stable out with lower beam intensities such as Rb, more ions per bunch simplified the 6,7 23 74,78,85,87 85 74 Rb measurement radioactive isotopes (e.g.which Li,was Na, been extracted setupand of the done with Rb) Rbhave from TITAN ’s own ion from the RFQ in reverse direction. A detailed description of the on-line laser source. Particularly to find a tuneable signal for injection, charge breeding, and extraction from the EBIT, large beam intensities were helpful. 21  3.3.1  A Pulsed Drift Tube (PLT) after the RFQ cooler and buncher  The kinetic energy of bunched beams along the TITAN beam lines connecting RFQ, EBIT , and MPET ranges between 1-2 keV. At these energies, electrostatic beam optics with moderate requirements on power supplies can be used (a few 100 V for 86  segmented central drift tube  Figure 3.13: Right: Schematic of an EBIT with a magnetic field formed by Helmholtz coils. The ions are trapped in axial direction by an electrostatic field. Left: A cross sectional view of the central trapping electrode. It is segmented which provides direct visual access to the trapped HCI and allows monitoring of the charge breeding process. Moreover, the segmentation can be used for RF- cleaning (see Section 3.7.6).  steering and <5 keV for focusing). The beam can be brought to rest by floating a trap potential just below the beam energy. Ions in the RFQ are trapped, but at an electrostatic potential URFQ of a few 10 kV which is adjusted to stop the incoming ISAC beam. A Pulsed Drift Tube (PLT) is used as an ‘elevator’ to lower the ion bunch to ground potential [161]. As shown in Figure 3.12, the ion bunch is accelerated towards the PLT after extraction from the RFQ. The PLT is initially biased to URFQ − ∆V . When the ions reach the centre of the PLT it is switched quickly to ground potential. Due to the length and the injection hole of the PLT, the shape of the potential within the PLT itself is not affected by the fast switching, hence, avoiding any extra accelerating force on the ions. The ion bunch leaves the PLT while on ground potential with a kinetic energy of e · ∆V . In the presented measurements, ∆V was set to about 2 kV. Recent investigations of the setup of the PLT are summarized in Section A.1.  3.4  The electron beam ion trap for charge breeding of radioactive ions  The main function of the EBIT is to convert singly charged ions into multiple or Highly Charged Ions (HCI). The short half-lives and limited yield of radioactive 87  3.14:breeding Simulation times inbased the TITAN EBIT to reachmodeling a cer- using FigureFigure 2.5: Charge timesofofbreeding high Z elements on EBIT theoretical the SUK program by Becker [5]. The charge state of various and elements plotted as a taindeveloped charge state for various elements. Simulation figureis from function of the [171]. breeding time in the TITAN EBIT with a current density of 25,000 A/cm2 .  nuclides requires the charge breeding process to be fast and efficient. Moreover, a charge state breeding process is desired which maintains a clean environment. Charge states of q ≈ 10 − 20+ should be attainable in less than half of the nuclide’s half-life while maintaining an overall breeding efficiency of a few percent. In an EBIT (see Figure 3.13), ions are trapped axially by an electrostatic potential applied to an EBIT’s drift tube. The confinement in the radial direction is achieved by an intense electron beam and a strong magnetic field. Higher charge states are produced by electron impact ionization of the ions with the electron beam. The TITAN EBIT [2, 136, 171] was designed and built in collaboration with the 27 Max-Planck-Institut for Nuclear Physics (MPI-K) in Heidelberg and brought to TRIUMF in 2006. It is designed to operate with electron beam energies and currents of up to ∼ 70 keV with 500 mA (and later 5 A), of which 400 mA at ∼7 keV and 25 keV at 200 mA have been demonstrated experimentally [2]. In a magnetic field strength of 6 T the 70 keV, 500 mA electron beam is compressed to a density of 30,000A/cm2 . Simulations of the charge breeding in the TITAN EBIT [171]  88  !)$*+,-."2$/3"4-."5  $)$*+,-. 6$/3  (a)  3/;.$+<* C<$)8  & */+7-8$ $)$*+,-. *-))$*+-,  (b)  +,/0"$)$*+,-8$  : */0+9,$ &+,/0  AB  & &*/+  1 !""""""""""#$"%&"""""'"&""""("" $)$*+,-. +,/0 */+  = *7/,;$"6,$$8<.;  ? $>+,/*+<-.  <-."*)-98  A@B  &+,/0  1 !""""#@"& <-. +,/0  Figure 3.15: Schematic of the EBIT (a) and the injection, charge breeding, and extraction potentials (b).  confirm the rapid charge breeding process (Figure 3.14). In particular, high charge states of q = 20+ and more can be reached within tens of milliseconds which is hence compatible even for short-lived nuclides such as 74 Rb. At TITAN, the ion bunch of SCI from the RFQ is injected into the EBIT where it is decelerated by floating the central drift tube to a potential Utrap slightly below the transfer beam energy (compare with Figure 3.15). The ion bunch is dynamically captured by switching the neighbouring drift tube to a higher potential which establishes the axial trapping potential. The electron beam energy is defined by the difference between Utrap and the bias voltage of the cathode of the electron gun Ucat , E = e(Utrap − Ucat ). The charge breeding time is set by the time the ions are kept in the EBIT. Typical breeding times used so far range from 20 ms to 200 ms depending on half-life of the nuclide and desired charge state. Once this charge state is maximized in the abundance distribution, the beam is extracted from the same side of the EBIT as it had been injected (see Figure 3.15). Due to the higher charge state the total kinetic energy of the HCI is increased to Eion = q · Utrap after 89  extraction. In our operation, the electron beam is continuously on, even during injection and extraction of the ion bunches. The first charge breeding of radioactive beam for a mass measurement was achieved in fall 2009. 44 K1+ beam from ISAC was enhanced to a charge state q = 4+ in the EBIT , and its mass was determined in the MPET [1, 2]. This was done with a small electron beam current of less than 1 mA by only warming up the cathode but not applying any bias. The electron beam energy was ≈ 2 keV. Due to this low electron beam current 200 ms of breeding time were necessary to reach a sufficiently large q = 4+ abundance to perform a mass measurement. A Time-Of-Flight (TOF) spectrum of ion bunches extracted from the EBIT is displayed in Figure 3.16. Peaks at the charge states q = 2−5+ were readily distinguishable from charge bred residual gas. However, residual gas is always present in the EBIT despite the good vacuum conditions achieved by operating the trap’s drift tube at 4 Kelvin. The long halflife of 22 min and the high ISAC yields of more than 107 ions / s for 44 K were favourable for this proof-of-principle measurement. In an attempt to perform a mass measurement with the ground state of the superallowed β emitter 38m K during the same beamtime with ISAC yields of ∼ 105 ions per second, no signs of charge bred 38 K extracted from the EBIT could be observed. Hence, despite the success it was concluded that significant improvements in breeding time, electron beam current, and efficiency would have to be accomplished in preparation for the first physics-motivated measurement, which should also be carried out in a higher charge state. Hence, for the radioactive beam measurement of neutron-deficient Rb isotopes in November 2010 the EBIT was operated with a beam current of 10 mA. The optimization of the charge breeding time is discussed in the next section.  3.4.1  Charge breeding time of Rb isotopes  The optimal charge breeding time for the Rb measurements was determined with stable 85 Rb, however, taking into account that the total breeding time should not exceed about a half of the half-life of 74 Rb (65 ms). For the purpose of charge breeding, isotopic differences were negligible, and it was considered to be identical to the radioactive Rb regarding the cross sections of electron impact ionization. The relative abundances of the charge states were measured via TOF to a Multi Channel Plate (MCP), and the results are displayed in Figure 3.17. Two trends were apparent. First, with longer breeding time higher charge states were favoured over lower. Second, the higher the charge state the smaller was the maximum in abundance. The latter was due to the fact that more charge states were reached, and the total number of ions was spread over a wider range of charge states. In Figure 3.17, a plateau for q = 8+ was reached at ≈ 20 ms. For the measurement of 74 Rb in the same charge state, the ions were hence kept for 23 ms in the EBIT 90  rom the RFQCT is 2 keV, while ctor extraction electrode is at À 2 kV). As a result, when during charge-breeding tests, pierre et al. / Nuclear Instruments and Methods in Physics Research A 624 (2010) 54–64 IT had to be aligned with the  ractionthe electrode which can igned, electron gun MCP is he beams injected into and se to the drift-tube assembly. ue at of aits potential on the sed high voltage,iswarming ion beam energy. Typically, weak electron beam current rom thewithout RFQCT isapplying 2 keV, while even any ctor extraction electrode is anode electrodes. This beam at À 2 kV). As aions result, when ultiply charged from the duringThe charge-breeding ams. presence of tests, such IT had to be aligned with and the f the extracted beam  EBIT  61  MCP  TOF path 2+ T beamline and switch yard He gned, the electron gun MCPthe is ect is that upon injection, se an to energy the drift-tube assembly. to of around 100 eV, ed at a high voltage, between their initialwarming kinetic eak electron beam current potential (multiplied by the eventhe without applying with any ture, ions interact electrodes. This beam s,anode and consequently, some of ultiply charged ions no from the ction energy is then longer N+ ams. The presence such rgy but is equal to theofcentral + O+ N+ f 1:95 thekeV extracted beam and 2 /CO Á q. Spectra of stable T beamline and charged switch yard from the RFQCT, bred ect is that upon injection, the extracted onto the MPET-MCP to an energy of around 100 eV, eir large A/q ratios, such lowbetween their initial kinetic asily be resolved from ion 39 44 46 Fig.3.16: 12. TOF of multiply( TOF ionized K (stable), K, and K isotopes potential (multiplied the Figure ted ions charge bred tobyhighA spectra Time-Of-Flight ) spectrum of ions extracted from the extracted from the EBIT. Small quantities of multiply charged Ar, which had been 44 + ure, the at ions extracted theinteract momentwith and EBITa. few Forweeks the shaded distribution no Kin the ions were injected injected earlier as a gas, are observed background spectra.into The the , and consequently, some of lanned for the next year. spectra taken with slightly different experimental conditions EBITwere . These TOF peaks correspond to charge-bred residualsuch gasasfrom ction energy is then no longer injected beam current,on beam tuning, and acquisition the EBIT . Figure the bottom modified fromtime. [2]. The cathode was unbiased (warmed up only, see text). The electron beam, injected-ion beam, and rgy but is equal to the central extracted-ion beam energies were 3.95, 2 keV, and $ 1:95 keV Á q, respectively. Ca 1:95 keV q. Spectra of stable ments forÁ high-precision isobaric contamination exists in the 44K and 46K spectra. The gray-shaded region from the RFQCT, charged bred represents a background measurement with no K injected. extracted onto the MPET-MCP for charge breeding. In the cases of 75,76 Rb the breeding time was 35 ms for which eir fast large A/q ratios, lowve decay times, such the use qof= 9+ was the most populated charge state. A TOF spectrum of charge bred 75 Rb asily be resolved ion their masses is onlyfrom advantais presented in Figure 3.18. Hints of q = 12+ ions can44be seen in this spectrum. Fig. 12. TOF spectra of multiply ionized 39K (stable), À8 K, and 46K isotopes ted ions charge bred to highmeasurements of higha mass precision ðdm=m t10 time (time of production) is charge Þ on isotopes of This state was measurement of 76 Rb. extracted from theused EBIT.for Small quantities of multiply charged Ar, which had been extracted the moment and stages thatatare high enough to very short ðTas 100 In particular, precision mass 1=2at injected a fewhalf-lives gas, aremsÞ. in the background spectra. The charge In future studies, itweeks willearlier be interesting toobserved compare measurements of the for the next year. .anned For instance, if the time to measurements are needed to experimentally test the theoretical spectra were taken with slightly different experimental conditions such as breeding time beam with current, simulations such as and the acquisition one in Figure 3.14. For the current injected tuning, The cathode was more than 7 times the isotope corrections to the ftbeam values of superallowed btime. emitters employed measurement this was not possible as the breeding conditions (e.g. electron unbiased (warmedthe up only, see text). matrix The electron beam, injected-ion beam, and beam lving power does to determine dominant elements of the Cabibbo– pffiffiffiffi not then extracted-ion beam energies were 3.95, 2 keV, and $ 1:95 keV Á q, respectively. radius or overlap of the electron(CKM) beam with themixing trapped ions) were, dueCKM toCa the time ments for high-precision ics scaling as N. Hence, for Kobayashi–Maskawa quark matrix. The isobaric contamination exists in the 44K and 46K spectra. The gray-shaded region constraint of the online beamtime, not fully determined. isotopes, the charge breeding matrix should be unitary, based with on fundamental represents a background measurement no K injected. concepts. Should be shorter or approximately a deviation from this unitary be measured, this would strongly e fast decay times, the useisofa interests. Charge breeding imply physics beyond the Standard Model. 74Rb is one of the main 3.5 Beam transport theirinto masses is only advantaake account changes in envisaged candidates at TITAN. Other candidates are listed in À8 measurements of with high precision time (time production) is m=m Þ on sections for ofdifferent charge Table 2, together half-lives andt10 charge breeding times After cooling and bunching in their the RFQ andðdcharge breeding in isotopes the EBIT ,ofthe ions stages that are high enough to very shorta half-lives ðT1=2 t 100 msÞ. In particular, precision mass ed by electronic shell closures to reach He-like charge state. The present mass determination are delivered to the MPET . An illustration of the TITAN beamline is shown in Figinstance, if the time to measurements are experimentally test theby theoretical ndFor recombination of ions with uncertainty of 74 Rbneeded is dm $to keV, which is limited the short ure 3.19 which provides details about19beam optics and diagnostic elements. Howmore thanresidual 7 times background the isotope corrections to the ft values superallowed performed b emitters on employed nge with half-life of the isotope. Massofmeasurements He-like ever, minor components are omitted for clarity of the figure. The entire set of beam 74 35 þ ving power does not then to determine the dominant matrix elements of the Cabibbo– ng times and p ion Rb would allow us to reduce this uncertainty by a factor of ffiffiffiffi abundance ics scaling as rate N. differential Hence, for Kobayashi–Maskawa (CKM) quark tomixing matrix. CKM lving coupled approximately 35, that is, down dm $ 500 eV. AsThe seen in charge breeding matrix be use unitary, on fundamental Should y,isotopes, mainly,the between electronTable 2,should with the of anbased electron-beam energy concepts. and high current 91 measured, shorter or approximately aofdeviation from be this would 25 keV and 500this mA,unitary respectively, the breeding time ofstrongly He-like onberates [7,25]. 74 74 interests. breeding a imply beyond the Standard one of the main he TITANCharge EBIT such as is the Rb isphysics approximately 32 ms, whichModel. is bellowRb itsishalf-life of 65 ms. ake changes by in envisaged candidates atisTITAN. Other candidates are Rb listed A 25-keV electron beam sufficient to produce He-like ions in in neticinto fieldaccount were prescribed ections densities for different charge Table 2, together their half-liveselectron-beam and charge breeding current in the trap large amount. Thewith 70-keV maximum energy times of the by electronic to reach a He-like charge Thethe present mass TITAN EBIT will allow us state. to reach vicinity of determination the maximum gd times shorter shell than closures 100 ms. 74 nd recombination of ions mass with uncertainty Rb is dm $ 19 keV, which is limited by the short desirable for accurate value of the of electron-impact ionization cross-section distribution nge with residual background half-life of the isotope. Mass measurements performed on He-like 74 35 þ ng times and ion abundance Rb would allow us to reduce this uncertainty by a factor of  charge breeding time of  85  Rb  3000 7+ 8+ 9+ 10+  2500  # of counts  2000  1500  1000  500  0 10  15  20  25  30  35  40  45  charge breeding time [ms] Figure 3.17: Measured number of ions of 85 Rb in various charge states for different charge breeding times in the EBIT with a 2.5 keV and 10 mA electron beam. The plotted number of counts for each breeding time is the sum of 200 extractions from the EBIT.  optics at TITAN is electrostatic. Here only those elements and devices are discussed which were relevant for the measurement. After the ions were extracted from the RFQ, the PLT (Section 3.3.1) allowed for a adjustment of the kinetic energy to 2 keV. The ion bunch was transferred through a series of benders into the EBIT, where the charge breeding took place. A detailed description of the beam optimization for injection, trapping, and extraction in the EBIT is presented in Section A.3. A series of retractable MCP detectors were installed at critical positions along the beamline for beam diagnostics and were used for beam transport optimization. Three of them (MCP1, MCP3, and MCP4) were further equipped with phosphor screens which provided a visualization of the beam spot impinging upon the MCP. A more comprehensive explanation of the beam transport system and optics can be found in Section A.2. After charge breeding in the EBIT, a bunch of the HCI was extracted from the EBIT and was bent by 90o into the MPET beamline. There, MCP0 was utilized for beam  92  600  75  Rb1+ injected into EBIT no ions injected into EBIT  75  Rb9+  500  Rb8+  ion counts  75  400  75  10+  Rb  75  Rb7+  300 200 14 2+ 75  100 0 24  12+  N  16 2+  O  Rb  75  Rb6+  26  28  30  32  34  TOF [µs]  Figure 3.18: TOF spectrum of 500 ion bunches of radioactive 75 Rb extracted from the EBIT with an 800 ns extraction pulse after 35 ms of charge breeding with a 10 mA, 2.5 keV electron beam. 16 O2+ and 14 N2+ were due to ionized and further charge-bred residual gas in the EBIT.  optimization. MCP0 was operated in a single-ion counting mode by coupling the MCP ’s anode signal via a preamplifier to a Multi Channel Scaler ( MCS ). TOF spectra analogous to Figure 3.16 and Figure 3.18 were recorded. When the count-rate of ions in the TOF peak of interest was too high and led to signal pile-up, the beam intensity of ions sent into the RFQ was reduced. Through the single-ion-counting mode, different beam transport and EBIT settings could be compared quantitatively. An optimization software was developed to scan individual beamline parameters and to record the number of counts for specific peaks in the TOF spectrum in each setting. By maximizing the counts in the TOF peak associated with the desired charge state, this greatly improved the overall beam transfer efficiency and EBIT performance. The resolving power of the TOF depended on the energy spread and the spatial extent of the ions in the EBIT. The distance between the two end caps in the EBIT is approximately 77 mm. If the trap were fully emptied, then the TOF peaks would have been smeared out entirely in the TOF spectrum. As the resolving of different m/q was essential for the mass measurement (see Section 3.6), the EBIT was only opened with an extraction pulse of a few hundred nanoseconds. The remaining HCI were dumped onto the beampipe a few milliseconds after the initial ion ex-  93  Schn  links  oben e-gun MCP4  45° D  rehun  middle tube of einzel lens is segmented  g  u  ntebeamline horizontal Abb EBIT n ildun g 4.2 recht zwei Ele 4 s fehler mente. M : (links) N fast-switchable n in ach e it unt e in er nochm in Sikler lens e Sikler Lens al du er Ebene schiedliche m schrägMCP3 en Scwith einer r a r Spa einzel lens + steering h Einze ch und (o bgelenkt n phosphor-screen nitt d n u n b g ll w das d en re inse a urch erden beauf ie ch e u . s sogen Richtungs f diese W ts), so en Führt ma chlagt kö ine hohlzy nnen fast-switchable linder tstehe eise ( annte korrek n ◦ 9 0 Ixoand un -förm n 45ro benders verd t sorgt S nenyssteerer t r dafür teerer-Len ur und Fok ten recht 2 xvie E e h le t daz platesrahlen m ige Elektr ment s), so , dass ussier s ode links, u ode e it ge ( ( d r u e u o recht ie ein n s). zelne Sikler-Lins ng von Ion tsteht ein nten Mitte ben Mitte ringsten A entstehen n Elem ) enstra MCP6 e. De ) MCP2 bbildbeamline e . d x Teilt envertical trem rE un hle ente wfast-switchable gle ie zwe inbau mit n◦ (allg. ge kompakte man das m ichen Sc gsMCP0 h x-steereri plates s elek 45 V laden it nitt Ablen t lere S pairs e Teil t 45o bender erdrofe Wie a k-Pla egme chens rostatisch hung t t u quadrupoles n e s n e p t zu Ab s ra aare a pair tof to nicht ngest den Schn hlen) erm Elemequadrupoles nt, in der bildung 4.MPET ö e it u g tlagen ert w li 18 ers Die E Ablen erd (rech cht: die ich n k 45o bender 45o bender en könn en (o ts unten) h erfo dispe ergie-disp ebene des tlicBradburyben, u lg e Nielsen gate rsive I nten, used to transfer Eben rsive Ebe onenstrah t die Sepbeamline sigb a o  deflection angle  # counts  }  ne are e d ls, SCIra from MCP1 with t RFQ to MPET Komp Magnetfe es Wien- des elektr sondern n ion der Io phosphor-screen 45 bender FiltTOF-gate o n o ld de s onent e r tatisc n nac m al d ers w capabilities: r sup en in Komp h h a e e z L n Ab rden ral o x lenke u. Das hat adungszu RFQ-PLT tische nente weg - und z -R eitenden S damit e r stand s z wei G u n e n n ichtu pulen tkopp n vˆ = d die r ü n n vorlie Ablenkun e g d x G ˆ d l t. (ii) er FL e: (i) (Wie e in er ge g Das n schwindig te. Au nde Orien zu einer u ster Ordn n-Filter K ASH-EBI k e i c its h ne un T ha ti t o f t in d vernachl die A der Streck erung des rwünschte g ohne A ordinaten ä i uswir ). Di Wien bweic e zwi n ger e z -K esem Ber s- RFQ ku -F sc in hu dann eich ompo erst l ng von d hen dem e ilters erla gfügigen L ng) führt ne te u inear e n l wirku (gate mit d r geradlin ektrostati bt eine be adungszus ach der el nbeam xngen s s e i t e c g sere K and-S r Ent k(fast-switchable des F heTOF en Fl trosta alten n A [µs] f e e L o u b p r m steerer plates) A gb le n sollte a der W SH-EBIT- ung vom A ahn in y - nker und pensation ration. Die dem W Richt Magn dieser from ISAC blenk ien-F u et er (si ilter d ehe A ng zunäch ien-Filter Effekaher feldes auf n s st qu mögli adrat kaliert chst n die Ionent hang, Seit i e s rajTITAN Figure 3.19: Layout transport beamline the RFQ cooler ah am of the c 14 h ufrom nd Ablen ektorien m 6). Um d ie and buncher to the EBIT . Only the elements ker p and finally ö to the MPET ositio glichst ge Ausnierare r i which are mentioned in the text displayed while omitting other n g t wer zu den. Nachin red are controlled einzel lenses and steerers. Elements displayed by fast HV switches. Devices with blue double arrows are retractable. The insert at the top right shows a schematic of73a so-called Sikler lens. Figure of the Sikler lens taken from [172].  94  itt 2  traction. Although this implied a significant reduction in efficiency, accumulation of short-lived nuclides in the EBIT was limited by the half-life. More importantly, ionized and further charge bred residual gas could have filled the trap hindering the injection and storage of new SCI. The trap was hence left open after emptying the EBIT of remaining HCI until the arrival of the new ion bunch.  3.6  A/q selection  The mass measurement in a Penning trap by the technique applied at TITAN (see Section 3.7.3) is ideally performed with ions of only one ion species in the trap at a time. Other contaminant ions reduce the measurement signal and might lead to a disturbance of the measurement result. Ions with different ratios of mass number A to charge state q differ in their TOF (see Figure 3.16). They can be rejected by a gate which only lets ions with a certain A/q pass. In the past, this TOF gate was implemented by switching the applied voltage on one element of a steerer plate assembly which is located right after the switchyard (see Figure 3.19). The steerer assembly deflected the beam into the beam pipe except at a specified time, when the beam was directed along the beamline. The minimum gate width of the setup was ∼ 1.2 µs, which was in most cases sufficient to separate different mass numbers for singly charged ions. For example, this method was used in the mass measurement of 6 Li [124] to separate the two stable Li-isotopes with A = 6 and 7, which both were simultaneously injected into the RFQ from TITAN’s surface ion source. With the capability of HCI, the spacing in TOF over a similar TOF distance was much tighter. Additionally, the EBIT produced background in the form of charge bred residual gas. For instance, 44 K3+ , N+ , and O+ in Figure 3.16 were all within a window of 3 µs in TOF. Selecting a clean ion bunch of 44 K3+ would be impossible with a gate width of 1.2 µs. Therefore, a better TOF gate was necessary. An investigation of the limiting factors of this TOF gate highlighted three shortcomings, an overly long cable between HV switch and steerer plate, strong voltage oscillation after the fast HV switch itself, and a large capacitance of ∼ 70 pF. The first two issues were addressed by a reduced cable length and a new HV switch designed and built at TRIUMF. To reduce the capacitance for a TOF gate, a Bradbury Nielsen ion Gate (BNG) [173] was designed and tested [174] in a collaboration with the Enriched Xenon Observatory (EXO) [175] group at Stanford and TITAN. Instead of deflector plates, a BNG consists of small, parallel wires arranged in a plane which is perpendicular to the propagation of the ions (see Figure 3.20). Every second wire is biased at the same voltage. If both sets of wires are at the same potential (normally on the beamline ground) the ion bunch can pass through the gate. However, any potential difference leads to a steering force and deflects the 95  G Model  G Model  ARTICLE IN PRESS MASPEC-14578; No. of Pages 7  MASPEC-14578; No. of Pages 7 2  ARTICLE IN PRESS T. Brunner et al. / International Journal of Mass Spectrometry xxx (2011) xxx–xxx  T. Brunner et al. / International Journal of Mass Spectrometry xxx (2011) xxx–xxx  G Model MASPEC-14578;  No. of Pages 7  2  ARTICLE IN PRESS T. Brunner et al. / International Journal of Mass Spectrometry xxx (2011) xxx–xxx  Fig. 1. Schematic of the working principle of a BN gate showing idealized ion trajectories. If both wire sets are at the same potential, most of the ions cross the gate and reach the detector (left). If opposite voltages are applied to the wires, the beam is deflected, effectively turning off the gate (right). Also shown are equi-potential lines of the wires.  the ions are deflected in such a way that they cannot reach the detector (a state referred to as gate closed in this paper). Deflected  The wire grids described here were photo-chemically etched from a 50.8 ␮m thick stainless steel sheet by Newcut Inc. [12].  are deposited somewhere along the beam line. For a detailed Thus, the resulting wires exhibit the approximate cross section of a Fig. 1. Schematic of theions working principle of a BN gate showing idealized ion trajectories. If both wire sets are at the same potential, most of the ions cross the gate and reach discussion on the deflection angle ˛ see [7]. diamond with similar dimensions of 50 ␮m in both thickness and An important property of BN gatesapplied is that they only influence width. While this leads a field configuration that is different turning off the BNG voltages are to the the wires, the beam istodeflected, effectively gate (right). Also shown are equi-potential lines of the wires. the detector (left). If opposite motion of the beam at distances comparable to the wire spacing d. from that of ideal, circular wires, such an effect is only important  Figure 3.20: Concept of a Bradbury Nielsen ion Gate ( ). The ion beam is passing through a set of wires perpendicular to the direction of the ion ARTICLE IN PRESS the ions are deflected in suchpropagation. a way that they cannot reach the The wire grids described here photo-chemically etched The wires are arranged in pairs such that to every were second detector (a state referred to as gate closed in this paper). Deflected from a 50.8 ␮m thick stainless steel sheet by Newcut Inc. [12]. potential is applied.Thus, When both sets of exhibit wires the areapproximate at the ions are deposited somewherewire alongthe the same beam line. For a detailed the resulting wires cross section of a same potential (normally on beamline ground) the ions will pass discussion on the deflection angle ˛ see [7]. diamond with similar dimensions of 50 the ␮m in both thickness and 2 While this leads to a field configuration that is different An important property of BNgate gates(figure is that they width. ononly the influence left). Athe potential difference between the two wire motion of the beam at distances comparable to the wire spacing d. from that of ideal, circular wires, such an effect is only important sets, however, deflects the beam (figure on the right). Figures This results in a compact design and fast switching, providing a near the surfaces and we do not expectfrom it to alter the general behavclose to ideal definition of a gate in contrast to deflection (kicker) ior of the gate, described by Eq. (1). This is because of the large [174]. 2  This results in a compact design and fast switching, providing near the surfaces and we do not expect it to alter the general behavG Model a close to ideal definition of a gate in contrast to deflection (kicker) ior of the gate, described by Eq. (1). This is because of the large No. Pages 7 to the wire diameter. Mounting holes were plates. The geometry of BN gates generally also resultsMASPEC-14578; in smaller spacing asof compared capacitance, simplifying the drive circuitry and further aiding the located on each side of the wire grids. The grid-to-grid alignment T. Brunner / International Journal of Mass Spectrometry xxx (2011) xxx–xxx holes. In et oural. design the goal of fast switching. More importantly, the spatial dimensions of is provided by the position of the mounting the gate along the beam axis and thus the disturbance of the ion’s grid is centered by the mounting screws. If more precise tolerances flight path is greatly reduced for BN gates compared to kicker plates. are required, the positioning of the wire grids could be realized by Fig. 3. Exploded view of one half BN gate showing stainless steel mounting frame BN gates of large (cm) size are typically built by stretching wires alignment pins. In order to be able to use identical mounting frames structure, wire grid and mounting ceramics (top). A section view of the fully assemon frames in various arrangements [8–11]. This results in substanfor the two grids, the mounting holes are offset with respect to the bled BN gate consisting of two identical frame structures rotated by 180◦ to each tial complexity and delicate devices that are difficult to assemble symmetry axis of the grid. other corresponding to the two polarities (bottom). and not reliable. This is particularly true when only clean and Each grid is mounted onto a pair of macor isolators that is in ultra-high vacuum components can be used. In order to overcome turn installed onto a stainless steel frame, as shown in Fig. 3. The these drawbacks, a new type of BN gate design has been develtension of the grid is set by the screws holding the macor blocks oped based on chemically etched wires. The new design simplifies onto the frame and is easily adjusted at the time of assembly. A the TITAN setup prior to injecting them into a Penning trap [21]. the construction while at the same time providing far more robust dedicated tensioning screw pushing the macor blocks apart was Furthermore, the BN gate is used to isolate ions with certain chargeand reliable assemblies. Additionally, the size of the gate is easily found unnecessary for the sizes tested. A rounded edge of the macor to-mass ratios q/m after charge breeding in an EBIT [19]. For highly scalable, covering large active areas. Currently, an active area of blocks allows each grid to have one end bent out of the way, so that 900 mm2 has been realized where the wire spacing can easily be when the two frames are mounted against each other, the wires charged ions fast switching times are of particular importance to changed simply by replacing the grids. are interleaved and the two grids are not shorted with each other. separate different ion species with very similar q/m. A time-of-flight For simplicity of fabrication, grids, macor blocks and frames are all spectrum of radioactive 76 Rb in different charge states is shown in identical and assembled in a mirrored configuration. A picture of a 2. Gate design fully assembled and mounted gate is presented in Fig. 4. Fig. 6 along with residual background contamination. large extension of the gate body in beam direction proThe The critical part in assembling a BN gate is the precise posiFor the characterization presented here, a continuous beam of tects the grids and acts as the beam dump if the gate is closed. tioning of the two sets of wires isolated from each other. In the singly charged 39,41 K and 85,87 Rb ions was extracted from a test While stainless steel was used to fabricate the grids described here, design described here, these issues are overcome by using photoother metals can be used as well. Because of the intrinsically clean ion source through a radio-frequency quadrupole cooler trap, the etched grids, as shown in Fig. 2. The basic idea is to handle two construction materials (macor and stainless steel) a vacuum of wire grids instead of individual wires. In addition, uniform wire TITAN RFQ [14], at a beam energy of 3 keV. This beam was then 2 × 10−10 mbar was achieved during all measurements. tensioning is automatically achieved and the handling of the grids  Fig. 5. Schematic of the TITAN setup. For the characterization of th RFQ was delivering a continuous beam of K+ /Rb+ ions at an energy of 3 by the test ion source (TIS) [14]. MCP detectors and beam was delivered (FC) can be moved into the beam line while the BN gate is installed The position of the MCP detector used in the presented studies is indi arrow. (For interpretation of the references to color in this figure lege is referred to the web version of the article.)  sent towards a micro-channel plate (MCP) ion detecto BN gate was installed between the RFQ and the MCP d distance of ∼1.35 m from the MCP detector as illustrate For this work, the gate was generally closed and only s transparent state (gate open) for a short, well-defined ti voltage was switched by two solid state devices develo UMF. These switches were set up to output either the tw Vpos and Vneg to block the beam, or 0 V to let ions pass t gate. The rise times at the output of the switches were a voltage set of ±160 V. A TTL control signal was prov switches by a Tektronix pulse generator (model AFG 3 a rise time of 18 ns. A synchronized TTL signal was th trigger the data acquisition, a multi-channel scaler (MC Research SR 430). For the characterization of the gate, th of flight distribution was recorded varying either the ap ages Vpos and Vneg or the opening time T. Typically, the  plates. The geometry of BN gates generally also results in smaller spacing as compared to the wire diameter. Mounting holes were capacitance, simplifying the drive circuitry and further aiding the located on each side of the wire grids. The grid-to-grid alignment goal of fast switching. More importantly, the spatial dimensions of is provided by the position of the mounting holes. In our design the is substantially easier than that of individual wires. 3. Characterization the gate along the beam axis and thus the disturbance grid is centered by the mounting screws. If more precise tolerances of the ion’s The presented large-area BN gate was installed at are TRIUMF’srequired, Ion flight path is greatly reduced for BN gates compared to kicker plates. the positioning of the wire grids could be realized by Trap setup for Atomic and Nuclear science (TITAN) [15]. The TITAN setup consists of several ion traps and is dedicated to high precision BN gates of large (cm) size are typically built by alignment stretching wires pins. In order to be able to use identical mounting frames measurements of radioactive, short lived nuclei (see, e.g., [16–19]). A schematic of the TITAN facility is presented in Fig. 5 indicating the location of the gate. Radioactive isotopes arefor produced by inBNsubstanthe two grids, the mounting holes are offset with respect to the on frames in various arrangements [8–11]. This results bombarding an ISOL type [20] target at the Isotope Separation and ACceleration (ISAC) facility [3] with 500 MeV protons. During typtial complexity and delicate devices that are difficult to assemble symmetry axis of the grid. ical operation, the BN gate is used to separate different isotopes by their time-of-flight when ejected from the first ion trap within and not reliable. This is particularly true when only clean and Each grid is mounted onto a pair of macor isolators that is in ultra-high vacuum components can be used. In order to overcome turn installed onto a stainless steel frame, as shown in Fig. 3. The Fig. 5. Schematic the TITAN For the characterization BN gate, the these drawbacks, a new type of BN gate design has been develtension of the grid is setof by thesetup. screws holdingof the the macor blocks Fig. 6. Time-of-flight distribution of charge-bred radioactive Rb (r The ion RFQ was delivering a continuous beam of K /Rb ions at an energy of 3 keV. The bunch was extracted from was delivered by the test ion source (TIS) [14]. detectors and Faraday cups oped based on chemically etched wires. The new design simplifies onto the framebeam and is easily adjusted atMCP the time ofionassembly. Athe EBIT and detected with the same used for the characterization presented in this work. The background c (FC) can be moved into the beam line while the BN gate is installed permanently. originating from the EBIT is also displayed (green spectrum). (For int far more robust tensioning pushing the macor blocks the construction while at the same time providing dedicated the MCP detector used in the presented studies is indicated by a redapart was The position ofscrew the references to color in this figure legend, the reader is referred to th Fig. 4. A fully assembled BN gate with mounting bracket and electrical connections. Fig. 3. Exploded view of one half BN gate showing stainless steel mounting frame reader arrow. (For interpretation of the references to color in this figure legend,ofthe the article.) This particular gate had d = 1.1 mm wire spacing and a total aperture of 900 cm . wire grid and mounting ceramics (top). A section view of the fully assemand reliable assemblies. Additionally, the structure, found unnecessary size of the gate is easily for the to the websizes version oftested. the article.) A rounded edge of the macor is referred bled BN gate consisting of two identical frame structures rotated by 180 to each Figure 3.21: New design of BNG based on chemically etched wires. The other corresponding to the two polarities (bottom). scalable, covering large active areas. Currently, an active area of blocks allows each grid to have one end bent out of the way, so that Pleasesent cite this article in as: T. Brunner, et al.,(MCP) Int. J. Mass doi:10.1016/j.ijms.2011.09.004 towards a press micro-channel plate ionSpectrom. detector(2011), [22]. The etched (a), the ofwhen the the BNG and the fully assem900 mm2 has been realized where the wires wire spacing can schematic easily be two(b), frames mounted other, BN gate wasare installed between theagainst RFQ and theeach MCP detector at athe wires the TITAN setup prior to injecting them into a Penning trap [21]. distance of ∼1.35 m from the MCP detector as illustrated in Fig. 5. changed simply by replacing the grids. interleaved and the two grids are not shorted with each other. arecertain BN gate is used to isolate ions from with chargebled BNGFurthermore, (c) aretheshown. Figures [174]. For this work, the gate was generally closed and only switched to to-mass ratios q/m after charge breeding in an EBIT [19]. For highly transparent state (gategrids, open) formacor a short, well-defined time frames T. The For simplicity of fabrication, blocks and are all charged ions fast switching times are of particular importance to voltage was switched by two solid state devices developed at TRIseparate different ion species with very similar q/m. A time-of-flight identical and assembled in a mirrored configuration. A picture of a UMF. These switches were set up to output either the two voltages 2. Gate design spectrum of radioactive 76 Rb in different charge states is shown in and mounted Vneg to block the beam,is or 0 V to let ions pass the fully assembledVpos and gate presented inthrough Fig. 4. Fig. 6 along with residual background contamination. gate. The rise times at the output of the switches were ∼24 ns for For the characterization presented here, a continuous of extension of the gate body in beam direction prolarge The beam a voltage set of ±160 V. A TTL control signal was provided to the The critical part in assembling a BN gate is the39,41precise singly charged K and 85,87posiRb ions was extracted from a test bymain a Tektronix pulse generator (model 3022B) with switches beam. Hence, such a setup can be used as a TOF gate. Theacts advantage of ifaAFGthe tects the grids and as the beam dump gate is closed. ion source through a radio-frequency quadrupole cooler trap, the a rise time of 18 ns. A synchronized TTL signal was then used to tioning of the two sets of wires isolated TITAN from each other. In the RFQ [14], at a beam energy of 3 keV. This beam was then trigger the data acquisition, a multi-channel scaler (MCS Stanford While stainless steel was used to fabricate the grids described here, BNG over deflection platesbyisusing its photoreduced capacitance and hence faster switching design described here, these issues are overcome Research SR 430). For the characterization of the gate, the ion’s time other metals can be used as well. Because of the intrinsically clean of flight distribution was recorded varying either the applied voltetched grids, as shown in Fig. 2. The basic idea is to handle two time. Narrower TOF gates can be realized. Additionally, when the potentials on thesteel) ages Vpos and V(macor T. Typically, the ion’satime neg or the opening construction materials andtime stainless vacuum of wire grids instead of individual wires. In addition, uniform wire −10 wire pairs are equal but of opposite polarity, the field alterations due to the BNG 2 × 10 mbar was achieved during all measurements. tensioning is automatically achieved and the handling of the grids  (a)  (b)  Fig. 2. A photo-etched grid for a BN gate. Note that multiple wire spacings (three in this case) can be obtained on the same etched foil.  (c)  2 The tolerances of metal sheet thickness and wire diameter are 10% and 13 ␮m, respectively (Private communication with P. Engel, Newcut.) The wire diameter cannot be smaller than the thickness of the metal sheet.  +  76  +  Please cite this article in press as: T. Brunner, et al., Int. J. Mass Spectrom. (2011), doi:10.1016/j.ijms.2011.09.004  2  ◦  arethan minimal only extend is substantially easier that of and individual wires.  spatially over lengths comparable to the wire spacing. Characterization So, the ions which pass the BNG first after3.the gate is opened are less influenced than in the case of deflector plates. As a drawback the transmission of an open The presented large-area BN gate was installed at TRIUMF’s Ion BNG is not unity because of the physical barrier the wires themselves represent. for Atomic and Nuclear science (TITAN) [15]. The TITAN Trap setup consists of several and is dedicated to high precision The newly designed BNG [174] represents setup an advancement in ion thetraps manufacturing Fig. 6. Time-of-flight distribution of charge-bred radioactive 76 Rb (red spectrum). The ion bunch was extracted from the EBIT and detected with the same MCP detector  the characterization presented this work. nuclei The background contamination used measurements originating of for radioactive, shortinlived (see, e.g., [16–19]). from the EBIT is also displayed (green spectrum). (For interpretation of Fig. 4. A fully assembled BN gate with mounting bracket and electrical connections. of the references to color in this figure legend, the reader is referredin to theFig. web version schematic the TITAN facility is presented 5 indicating A the article.) This particular gate had d = 1.1 mm wire spacing and a total aperture of 900 cm . 96 the location of ofthe BN gate. Radioactive isotopes are produced by bombarding an ISOL type [20] target at the Isotope Separation and Please cite this article in press as: T. Brunner, et al., Int. J. Mass Spectrom. (2011), doi:10.1016/j.ijms.2011.09.004 ACceleration (ISAC) facility [3] with 500 MeV protons. During typical operation, the BN gate is used to separate different isotopes by their time-of-flight when ejected from the first ion trap within 2  Fig. 2. A photo-etched grid for a BN gate. Note that multiple wire spacings (three in this case) can be obtained on the same etched foil.  2 The tolerances of metal sheet thickness and wire diameter are 10% and 13 ␮m, respectively (Private communication with P. Engel, Newcut.) The wire diameter cannot be smaller than the thickness of the metal sheet.  of such a gate. In previous designs the wires were stretched over a frame which is difficult to assemble and requires significant effort when changing to a different wire spacing. These complications are overcome by chemically etched wires to be used in the new BNG [174] as shown in Figure 3.21. A wire set is pictured in (a), two of which are individually mounted on its frame (b) to form the BNG wire structure. The fully assembled BNG in (c) was inserted into the TITAN beamline downstream of the steerer plate assembly which had been used at the TOF gate in the past (see Figure 3.19). Its capacitance was ∼ 20 pF, and the bias voltages of the full setup could be switched such that a 50 ns pulse width could be achieved. Typical widths of TOF peaks were much longer than 50 ns. Thus, the BNG performance was well suited for the purpose of a TOF gate. During the presented measurements, the TOF gate was opened between 300-500 ns to let the Rb charge state of choice pass the gate. For the wire sets in place, wires with a diameter of 51 µm were spaced 2.2 mm apart. This resulted in an approximate beam transmission of 95%. In this configuration, the advantages offered by the BNG outweighed the losses in efficiency.  3.7 3.7.1  The measurement Penning trap Motion of charged particles in a Penning trap  In a homogeneous magnetic field B, the Lorentz force F = Q · v × B directs a charged particle of charge Q = q · e in a circular motion.1 The angular frequency of this motion, called the cyclotron frequency, is ωc =  QB . m  (3.2)  It is linear inversely proportional to the mass, the quantity of interest, but independent of the radius of the circular motion ρ. Considering an ion of charge state q, mass number A, and of radial energy Eρ = 1 eV, the radius ρ is according to Eρ = mωc2 · ρ2 /2 √ A −4 ρ ≈ 1.4 · 10 m. (3.3) q · B[T]  Hence, even the heaviest atomic ions with Aq+ > 2501+ could be confined with a magnetic field strength of 2 T to ρ ≈ 1 mm. However, such a magnetic field does not restrict the ion motion along the field direction. If the axial motion contributes 1 If not otherwise stated, the description of the ion motion in Penning traps follows the references [5, 165, 176–178]  97  K. Blaum / Physics Reports 425 (2006) 1 – 78  9  ~5 cm B  B  z  z  z0 Udc+Vrf  z  z0 ρ0  Udc  ρ  ring electrode ρ  ρ0  Udc  ρ  end cap electrode (a)  (b)  (c)  Figure 3.22: Schematics of a Penning trap. The hyperbolic shape of the elecFig. 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 particletrodes storage a trap voltage with proper is applied between the ring electrode and the end electrodes. generates thepolarity electrostatic quadrupole potential of Equation 3.4. Figure modified from [5]. √ to the energy with 1 eV, the axial velocity of va ≈ 1.4/ A · 104 m/s would move the ion very rapidly out of a measurement region. To trap the ion along the magnetic field axis, a weak electrostatic field is superimposed to form a Penning trap. For simplicity, this electrostatic potential is usually chosen to follow a quadrupole form, U 1 V (ρ, z) = 2 z 2 − ρ2 . (3.4) 2 2d0 In the cylindrical symmetry, ρ is the radial coordinate, ρ2 = x2 + y 2 . The homogeneous magnetic field shall be taken along the z-direction. The potential of a quadrupole field has been illustrated already in Figure 3.9. An accurate forma3.2. Radiofrequency quadrupole and Paul traps tion of such a potential can be realized for instance by shaping electrodes along equipotential Thesequadrupole are, according to Equation 3.4, two Paul and Steinwedel first described thesurfaces. linear radiofrequency mass spectrometer (QMS), also named the hyperboloids of radiofrequency quadrupole mass filter (QMF) or ion guide, in 1953 [95,97]. This device provides two-dimensional ion revolutions (see Figure 3.22). The potential difference between the one-sheeted hyconfinement and mass separation by oscillating electric fields. It was continuously improved and extended to three dimensions [98,122]perboloid in the now-called Paul trap. Both are widely used in various branches of science. The principles (the ring electrode) and the two-sheeted one (the two end caps) is given and applications of a quadrupole mass spectrometer are summarized in the textbook by Dawson [20]. as An ideal quadrupolar geometry (see Fig. 6) is formed by four hyperbolic electrodes of infinite length with two Fig. 6. Left: Radiofrequency quadrupole mass filter electrodes having hyperbolic cross-section. Right: Equipotential lines for a quadrupole field generated with the electrode structure shown left.  perpendicular zero-potential planes that lie between the electrodes and intersect along the center-line z-axis. For mass 1 2 analysis both a static electric (dc) potential and an alternating (ac) potential in the rf range are applied to U the electrodes 2 ∆V is=used, V (ρ = 0, z in=restz0gas )− V (ρor= ρ0 , z chemistry = 0) =[123]. The 0 + ρ0 ). of the linear Paul mass filter which for example, analysis analytical 2 (zrelative  2d0  2  (3.5)  The characteristic trap dimension d0 is defined by 1 1 d20 = (z02 + ρ20 ) 2 2  98  (3.6)  such that the potential difference between the end caps and the ring electrode equals U . The classical equations of motion of a charged particle in the trap are 1 x ¨ = ωc y˙ + ωz2 x 2 1 y¨ = −ωc x˙ + ωz2 y 2 z¨ = −ωz2 z  (3.7)  of which the axial motion is completely decoupled from the radial motion and follows a harmonic motion with a frequency Q·U . md20  ωz =  (3.8)  Thus, Q · U > 0 must hold to form a confining force in axial direction. The radial equations are most easily solved by combining the two coordinates x and y to the complex u = x + iy, which transforms the two coupled real differential equations into one complex, 1 u ¨ = −iωc u˙ + ωz2 u. (3.9) 2 Its general solution u = A+ exp(−iω+ t) + A− exp(−iω− t) is governed by the two angular eigenfrequencies2 ω± =  ωc ± 2  ωc2 ωz2 − . 4 2  (3.10)  The angular frequencies are tied to each other by the following relationships. ωc = ω+ + ω−  2 2 ωc2 = ω+ + ω− + ωz2  2ω+ ω− = ωz2 .  (3.11)  √ For ωc < 2 · ωz , ω± becomes complex which implies a real exponent in the exponential in u. Hence, the ion would not be confined radially. This condition for 2 Angular (ωx ) and ordinary frequencies νx = ωx /(2π) are often interchanged in the following text. While formal treatments are more conveniently expressed by ωx , the ordinary frequencies νx are preferred from an experimental viewpoint.  99  ion trapping translates to |U [V]| B[T]  102 ·  |q| · (d0 [mm])2 . A  (3.12)  A confinement even within 1 mm3 can be accomplished when employing a strong magnetic field in combination with a weak electrostatic potential. At the limit of the trapping condition, the two radial eigenfrequencies are identical, ω+ = ω− . √ Penning traps are usually operated well below this limit ( ωc 2·ωz ) to establish a hierarchy in all three eigenfrequencies ω+  ωz  ω− .  (3.13)  A Taylor expansion of Equation 3.10 under this condition allows the radial eigenfrequencies to be expressed as ω+ ≈ ωc − ω− ≈  ωz2 2ωc  ωz2 . 2ωc  (3.14)  Because of these relationships, ω+ is known as the reduced cyclotron frequency. Due to the influence of the superimposed electrostatic quadrupole field, the cyclotron motion in the magnetic field is slightly reduced in its angular frequency from ωc in a pure magnetic field to ω+ in the Penning trap. The radial eigenmotion at ω− is best understood by considering a drift velocity vd = E × B/B 2 . Because of the cross product with B, it is a radial vector but depends on the radial position ρ. For a charged particle at velocity vd in the Penning trap, the force in radial direction would vanish because F = Q · (E + vd × B) = QEz zˆ. Here, zˆ is the unit vector along the z-axis. Analogously to the principle of a Wien velocity filter, the charged particle is unaffected by the electric and magnetic fields and the drift velocity vd is unchanged. When we look more closely at vd vd =  E×B ωz2 = ρ × zˆ ≈ ω− ρ × zˆ, B2 2ωc  (3.15)  vd can be identified as the velocity vector of the radial motion characterized by ω− . Hence, its associated motion can be seen as an E × B drift motion. Again in analogy to the velocity filter, which works irrespective of the charged particle’s  100  008  gniR  0 0  pacdnE  ρ  2.5. The mass measurement Penning trap  006  With the ansatz u ∝ e−iω± t+α± , one finds the two radial eigenfrequencies � ωc ωc 2ω 2 ω± = ± 1 − 2z (2.18) 2 2 ωc  004  z 0U  002 0  parametric equations for x(t) and y(t) -tsisnoc part gninneP a fo noitarugfinoc edortcele :tfeand L .the 4 .gfollowing iF ))bb ( -repyh htiw edortcele gnir a dna sedortcele pacdne owt fo gni u(t) = r+ e−iω+ t + r− e−iω− t (2.19) htiw part gninneP depahs yllacirdnilyca :thgiR .epahs ladiolob x(t) = r+ cos (ω+ t) + r− cos (ω− t) (2.20) dnuora detaerc si laitnetop cinomrah A .sedortcele noitcerroc y(t) = −r+ sin (ω+ t) − r− sin (ω− t) (2.21) -netop dednetxe eht saerehw edortcele gnir eht fo retnec eht noi detcejni eht fo erutpac tneicffie eht rof desu swhere i llewwelaset itthe phases α+ = α− = 0 to simplify the expression. Equation (2.20) (a) .]24[ hcn(b) ub  6 4 2 0 24-  1.0  magnetron motion -w  6-  0.5  004  006  ρ−  y 0.0  ))cc (  +  002  0  )sµ( t  r −r ρ +  ρ−  06 05 04  0.5  axial w motion  reduced cyclotron +w motion  03 02 01  1.0  )Ve( ygrene latoT  y mm  r + rρ  z  )mm( r  B  )mm( z  noitasnepmoC  pacdnE  0 noi na fo snoitomnegie tnednepedni yllaedi eerht ehT .5 .giF 1.0 0.5 0.0 0.5 1.0 noitcerid laixa eht ni noitallicso cinomrah a :part gninneP a ni 0001 008 006 004 002 0 x x mm a si taht noitom laidar a dna ,) zω ycneuqerf htiw noitom laixa( )mm( Z ycneuqerf htiw noitom nortolcyc defiidom eht fo noitisoprepus Figure 2.11: Example curve of the epitrochoid motion of an ion in a Penning ω ycneuSchematics qerf htiw noitoof m nthe ortenmotion gam eht dof na aωpositively Figure. −3.23: sessecro−rp= n1charged omm, italru+m=uc0.2 cparticle amm, dnaω−gn=il1oin ca eand ht ωfo+ = no20 italumiS .3 .giF trap, with+ parameters: so−1 s−1 . See text forPdetails. ARTharmonic LOSI fo parteigenmotions, elopurdauq ycneuqerfoidar raenil eht ni Penning trap. The motion consists of three laixa eht )a( :rabm 2−01 fo erusserp muileh a ta snoi +sC rof dna edortceone le gnirinnethe ewteaxial b deilppdirection a egatlov ehwith t si and ereangular hw 0Uan a sa frequency oitiparametrization sop laidaω r zehofand t an)bepitrochoid. ( two ;emit in foAnnexample oitcnuf a sa noitisop (2.21) correspond tonthe .spacdne eht ot tcepser htiw ygrene laixa latot eht )c( ;emit fo noitcnuf the radial directions with frequencies ω . While (a) is an illustration ± -om nortolcyc eht sa nwonk era snoitom laidar owt ehT 50t( noitiso p laixa .]72[ )dettolp osla si laitnetop laixa eh  -serp eht otof euthe d( ycthree neuqerdimensional f nortolcyc defiimotion, dom htiw(b) noitis its projection onto the radial -gam eht dna +ω )dlefi cirtcele elopurdauq eht fo ecne plane.: See text for details. Figures are modified from [131] and [146]. spacdne eht neewteb deilppa 0U fo ecnereffid egatlov A −ω ycneuqerfnegie htiw noitom norten laitnetop ralopurdauq a secudorp edortcele gnir eht dna 2ω 2ω cω = ±ω )7( ,z − c ± 0U )4( . 2ρ − 2 z2 2 = ) z ,ρ( U 2 4 2 d4 mass or charge, ω− is (approximately) independent of m/q, h t i w p a r t g n i n n e P a r o f d n o i s n e m id part citsiretcarahc ehT q )8( . B = cω s a d e n fi e d s i s e dortcele lacilobrepyh htiw 2 m  ωz  U  noitom nortengam eht sretemarap gniω pp−ar≈ t lacipyt r= oF 2 .)5( , ) 20ρ + 20z2((3.16) = 2d4 c ols ads0iB ehT .dlefi B × E eht yb desuac noitom tfir2ω d w part gninneP a ni denfinoc noi na fo snoitomnegie eerht -solc eht si 0z2 dna suidar gnir renni eht setoned 0ρ erehw 4 .gfi ees( sρpasince cdne ow ht aneewteb ecnatsid tse eht yebo sspeaking, eicneuqerf laall idaof r ow t ehT .gfi ntrue i nwoat hs one era specific.)position Strictly this is .5only vdt eis noitaler noitarugfinoc dlefi-citengam dna -cirtcele laedi na nI function of ρ. Hence, there has to be a non-vanishing delputime oced sderivative i m ssam hon tiw vndoiinehthe t fo noitom laixa eht )9( . −ω + +ω = cω c i n o m r a h n a s m r o f r e p n oi eht dnaofdleafi citengam eht morf radial direction once the particle leaves the specific ρ, where the statement eht si eno siht ,snoitaler tnairavni rehto era ereht hguohT ycneuqerf eht htiw noitom yrotallicso vanishing PARTLOSI radial eht fo txforce etnoc is ehtexact. ni noitaThe uqe treason natropmwhy i tsomit is a good approximation is due to siht fact fo nothat itaniω m− retis edsmall, tcerid and a ecnits is associated ,retemortcepradial s ssam eigenmotion known as0Umagnetron q the ,2 = zω fo noi eht fo ssam eht enimreted ot swolla ycneuqerf mus )6( d m motion is slow.  The general motion in a Penning trap can be described as the combination of the three eignmotions. An example of a trajectory of a positively charged particle is shown in Figure 3.23. The total energy of the charged particle in the trap is 1 2 E = Ekin + Epot = mzmax + m(ω+ − ω− ) · (ρ2+ ω+ − ρ2− ω− ) 2  101  (3.17)  where zmax and ρ± are the amplitudes and radii of each eigenmotion. The radial contributions to the total energy are 1 1 2 2 mvρ2 = m ρ2+ ω+ + ρ2− ω− + 2ρ+ ρ− cos (ω+ − ω− )t + φ+ − φ− 2 2 1 = QV = − mωz2 ρ2+ + ρ2− + 2ρ+ ρ− cos (ω+ − ω− )t + φ+ − φ− . 4 (3.18)  Eρ,kin = Eρ,pot  Owing to the hierarchy of the involved frequencies (ω+ ωz ω− ) the kinetic term associated with the reduced cyclotron motion dominates over the potential part. For the magnetron motion the opposite is true. The radial potential of the quadrupole field is negative. Hence, the larger the magnetron radius ρ− the smaller its energy. The magnetron motion is strictly speaking unstable. While any dissipative energy loss damping the axial or reduced cyclotron motion forces the ions closer to the trap centre, a lower energy for the magnetron motion pushes the ions to larger and larger radii until the ions are lost from the trap, i.e. hit an electrode. The damping times in UHV are much longer than the ions are trapped for these experiments and, thus, the instability of the magnetron motion is irrelevant for any of the mass measurements at TITAN. Of key importance for the mass measurement technique employed at TITAN is that the energy of a cyclotron motion with radius ρ+ = ρ0 is significantly larger than a magnetron motion with the same radius ρ− = ρ0 . Quantum mechanical description The motion of a charged particle in a Penning trap can be quantized [176, 178] by formulating the Hamiltonian in canonical coordinates and momenta and replacing the classical Poisson brackets by the quantum-mechanical commutator relations. The Hamiltonian can be expressed by means of annihilation and creation operators or equivalently by the number operator n ˆ = a+ a for the three eigenmotions. 1 1 1 H = ωz (ˆ nz + ) + ω+ (ˆ n+ + ) − ω− (ˆ n− + ) 2 2 2  (3.19)  Among these three harmonic oscillators, the magnetron motion is inverted, i.e. the larger the quantum number n− the smaller the energy. This is reflected in the previous discussion for the classical magnetron motion of lower energy for larger magnetron radii. TITAN’s MPET is at room temperature. Assuming that the ions were in thermal equilibrium with their surrounding, their energy would be E = kB T ≈ 25 meV. Since ω+ is the largest of the three eigenfrequencies, we 102  expect the reduced cyclotron motion to be in quantum states n+ ≈  E E A > ≈ · 6 · 104 . ω+ ωc q · B[T]  (3.20)  The typical energy spread of ions extracted from the RFQ is in the range of a few eV. Hence, the classical description of the ion motion in the Penning trap will be sufficient for our purposes. However, the quantum mechanical treatment of creation and annihilation operators can provide an intuitive explanation of ion excitation by means of RF fields [113, 178], which will be discussed later.  3.7.2  The Fourier-transform ion-cyclotron-resonance method  To determine the mass of an ion in a Penning trap, the cyclotron frequency ωc is to be measured. However, ωc is not an eigenfrequency of any eigen motion and cannot be measured directly. Penning traps for stable nuclides often take advan2 + ω 2 + ω 2 (Equation 3.11), a relation known as the Browntage of ωc2 = ω+ − z Gabrielse invariance theorem [176]. In the technique called Fourier-Transform Ion-Cyclotron-Resonance (FT- ICR) [179], the image currents induced by the motion of the ions in electrodes of the Penning trap are detected. In a Fourier analysis of these signals the three eigenfrequencies can be determined and by utilizing the Brown-Gabrielse invariance theorem ωc , is obtained. Variations of this approach can achieve very high precision. For instance TITAN’s mass measurement of 6 Li [124] has been confirmed and improved by such a measurement at Florida State University (FSU) [180] by over one order of magnitude to ≈ 0.2 PPB. Moreover, mass comparisons of singly charged molecules and atoms with relative accuracies below 10−11 have been reported [181, 182]. Despite its appeal by means of the achievable precision, the FT- ICR method is generally not applicable for radioactive nuclides because of its extended measurement time which is in conflict with short half-lives. The exception to this statement is found for mass measurements of super-heavy elements or other special cases (e.g. the double magic nuclide 100 Sn), where half-lives in the order of seconds are combined with very low production yields (∼1 ion per minute or less). To take full advantage of few low numbers of ions available, the non-destructive character of the FT- ICR method might be advantageous over the destructive Time-Of-Flight Ion-Cyclotron-Resonance (TOF - ICR) technique discussed next. FT- ICR Penning traps for longer-lived, but ultra low yield nuclides are currently being built (e.g. [183]). To date, however, all Penning trap mass spectrometers at radioactive ion beam facilities utilize the TOF - ICR method.  103  3.7.3  Quadrupole excitations and the time-of-flight ion-cyclotron-resonance method  In the TOF - ICR [184, 185] the cyclotron frequency ωc is determined by coupling the two radial frequencies ω+ + ω− = ωc through the application of an external azimuthal quadrupole RF- field, Vrf = −  V0,rf cos (ωrf t + φrf )x · y. 2a2  (3.21)  V0,rf is the amplitude of the RF-field at a radius a. The angular frequency and phase of the RF-field are ωrf and φrf , respectively. Under the influence of such an RF-field magnetron motion is converted into reduced cyclotron motion and vice versa. Details of the general, classical ion motion and the technique are derived in [185]. In our measurement the ions are prepared on a pure magnetron orbit at the beginning of the measurement, ρ− (t = 0) = ρ0 and ρ+ (t = 0) = 0. The axial motion is minimized. Since it is irrelevant for the measurement principle it is ignored for the moment. Assuming these initial conditions, the amplitudes of the radial motions evolve owing to the RF-field according to ρ− (t) = ρ0 cos(ωB t) + ρ+ (t) = −  1 1 i(ωrf − ωc ) sin (ωB t) · ei 2 (ωrf −ωc )t 2 ωB  1 1 ρ0 k0 e−i∆φ sin (ωB t) · ei 2 (ωrf −ωc )t 2 ωB  (3.22)  with ωB = k0 =  1 (ωrf − ωc )2 + k02 2 V0,rf Q 1 . 2 2a m ω+ − ω−  (3.23) (3.24)  ∆φ is the difference between the phases of the RF, the magnetron and the reduced cyclotron motion. For simplicity, we set all phases to zero. It is important to note that phase independent results are only obtained for the initial conditions of ρ− (t = 0) = ρ0 and ρ+ (t = 0) = 0 [178]. In the case when ωrf = ωc , the amplitudes turn into k0 t 2 k0 t ρ+ (t) = ρ0 sin . 2 ρ− (t) = ρ0 cos  104  (3.25)  The quadrupole RF-field results in a beating between the two radial eigenmotions. For those times when there is a pure reduced cyclotron or pure magnetron motion, the ion revolves along the same circular radius, ρ+,max = ρ−,max = ρ0 . Considering the absolute values | ρ± |, the period of one full conversion from and back to pure magnetron motion is T =  2π πa2 m πa2 B = · (ω+ − ω− ) = k0 V0,rf Q V0,rf  1−  2ωz2 . ωc2  (3.26)  Assuming the hierarchy in the angular frequencies, the last expression can be expanded to ω2 mU πa2 B πa2 B 1 − z2 = 1− . (3.27) T ≈ V0,rf ωc V0,rf QB 2 d20 Thus, the period is mass and charge independent except for the small term A · U [V] mU = 10−4 . 2 2 q(B[T] · d0 [cm])2 QB d0  (3.28)  In order to convert an initially pure magnetron motion into pure cyclotron motion, the product of RF-amplitude and the time Trf , which the RF-field is applied for, has to be a certain constant: k0 =  π ⇒ V0,rf · Trf = const. Trf  (3.29)  Since ω+ ω− , the radial (kinetic) energy is dominated by the reduced cyclotron 2 . A conversion between the two radial energies leads motion, Eρ ≈ m/2 · ρ2+ ω+ to a drastic change in kinetic energy, i.e. a gain from starting in pure magnetron motion. For the product V0,rf · Trf at which the initial magnetron motion is fully converted into reduced cyclotron motion, the radial energy is maximal. Hence, the radial energy is an indication for the degree of conversion between the radial motions. In general, the radial energy is, according to Equation 3.25, proportional to sin2 (ωB Trf ) Eρ ∝ . (3.30) 2 ωB For k0 · Trf = π (one full conversion when ωrf = ωc ) one obtains Eρ = E0  sin2 (π/2 · 1 + (2∆νrf Trf )2 ) 1 + (2∆νrf Trf )2  (3.31)  when introducing the detuning frequency ∆νrf = (ωrf − ωc )/(2π). Figure 3.24a 105  (a)  (b)  1 full conversion b=ï0.3 b=0.3  0.9  0.8  0.7  0.7  0.6  0.6  El / E0  El / E0  0.8  1 0.9  0.5  0.5  0.4  0.4  0.3  0.3  0.2  0.2  0.1  0.1  0 ï4  ï2  0  6i u Trf  2  0 ï4  4  full conversion Fourier transform  ï2  0  6i u Trf  2  4  Figure 3.24: Calculated radial energy of an ion after the application of a quadrupole RF- field for a time Trf = π/k0 at an angular frequency ωrf = ωc + 2π∆ν. (a) The full conversion is compared to an underconverted (δ = −0.3) and an over-converted (δ = 0.3) case. (b) The shape of the Eρ - function is compared to the Fourier transform of a pulse of a scalar wave of the same duration.  displays the radial energy after the RF-field is applied as a function of the detuning function and Trf . A pronounced maximum is found for ∆νrf = 0, hence ωrf = ωc , because the initial magnetron motion is fully converted into reduced cyclotron motion. For all other ∆νrf some magnetron motion remains. Hence, for ∆νrf = 0 the reduced cyclotron motion has not reached its full amplitude ρ+ = ρ0 , and the radial energy is smaller. Consequently, the frequency of the RF-quadrupole field causes a resonance at the true cyclotron frequency ωc , which can be detected by the change in energy of the ion in the Penning trap. Figure 3.24a motivates the following measurement principle. Ions from an ion species whose mass is to be determined are injected into the Penning trap and prepared with initial magnetron motion. Then, a quadrupole RF-field is applied at a certain frequency ωrf for a time Trf . At the end of the RF-field the energy of the ion is determined. Repeating these steps for several ωrf , a resonance curve as a function of ωrf is obtain from which ωc of the ion, and thus its mass, can be calculated, when B and q are known. Before discussing how Er is measured experimentally, a few more details about the ion excitation with an azimuthal quadrupole RF-field are mentioned. The excitation time Trf enters into the shape of the energy distribution twice (Equation 3.30),  106  firstly through ∆ν · Trf and secondly via k0 in ωB because the product of k0 · Trf dictates the amount of conversion when ωrf = ωc . Experimentally, this is tuned by an optimal value of V0,rf · Trf (Equation 3.29). According to Equation 3.28, this condition is only approximately mass and charge independent. If we introduce a deviation δ from the exact condition k0 · Trf = π · (1 + δ)1/2 , then, the radial energy of Equation 3.31 is modified to Eρ = E0 (1 + δ) ·  sin2 (π/2 · 1 + δ + (2∆νrf Trf )2 ) . 1 + δ + (2∆νrf Trf )2  (3.32)  The radial energies with δ = ±0.3 are also shown in Figure 3.24a. As the ion motion is either under- or overconverted, the amplitude of the cyclotron motion is not maximized for ∆νrf = 0. Instead, some magnetron motion is present. In consequence, the radial energy at ∆νrf = 0 is reduced. However, the detuning frequency ∆νrf enters into Eρ in quadrature and the maximum at ∆νrf = 0 persists. Hence, slight deviations of the condition for full conversion k0 · Trf = π · (1 + δ)1/2 might cause a reduction in the measurement signal, but do not affect the position of the resonance at ωrf = ωc . The m/Q dependence at the ∼ 10−4 level in k0 · Trf is orders of magnitude larger than the aimed precision for the mass measurement, but does not matter for a measurement of ωc . This is a general feature of this measurement technique. Not everything in the setup has to be controlled or measured at the accuracy level of the mass measurement itself. Critical for the mass measurement is the accuracy in the determination of ωc only. The cases of δ = ±0.3 in Figure 3.24 are extreme cases to illustrate the behaviour. In reality, δ is much closer to zero, and the reduction in Eρ for ωrf = ωc is negligible for the measurement sensitivity. The over-conversion with δ = 0.3 cannot only be seen in a smaller radial energy at ∆ν = 0, but also in the maxima on either side of the main maximum (Figure 3.24). In the scenario of an over-conversion their heights are increased compared to the full conversion. For under-conversion the opposite holds. The reason for these additional maxima or so-called side-bands is due to the square-wave pattern of the amplitude of the quadrupole RF- field, V0,rf = VA,rf · Θ(t) · Θ(Trf − t). Θ(t) is the Heaviside step function in time t. A Fourier transform of a scalar wave s = exp(−iω0 t + φ) · Θ(t) · Θ(Trf − t) for example results in an intensity profile in frequency space of sin2 (π/2∆ν Trf ) 2 ˜ I(ω) =| S(ω)| = (∆νTrf )2  107  (3.33)  with ∆ν = (ω − ω0 )/(2π), whose analogy to Equation 3.31 is apparent. It is plotted normalized to the radial energy profile in Figure 3.24b. The width of the energy profile is in fact narrower than the Fourier transform of the scalar wave. When expanding Equation 3.31 in a Taylor series, an approximation of the line width of 0.8 ∆ν(FWHM) ≈ (3.34) Trf is found. Hence, the longer the observation time the more narrow the line profile and the more precise the measurement. Quantum mechanical description In [178]3 , the RF-quadrupole field Equation 3.21 is quantized by the previously introduced creation and annihilation operators of the radial motions. + 2 Q · Vrf ∝ e−i(ωrf t+φrf ) · a+2 + + a− + 2a+ a− + h.c.  (3.35)  The first term is responsible for the creation of two quanta of reduced cyclotron motion when a photon with an energy ωrf ≈ 2 ω+ from the RF- field is absorbed. Analogously, the second term could remove two quanta of the magnetron motion. Since the magnetron motion is an inverted, harmonic motion this requires additional energy which is due to an absorbed photon of E = ωrf ≈ 2 ω− . Finally, the last term is responsible for the conversion of magnetron motion into reduced cyclotron motion as one of the latter quanta is annihilated while a former quantum is created. A photon of the RF- field of energy E = ωrf ≈ ωc is absorbed. This is the part of the quadrupole excitation which is used for the mass measurement. Thus, the formalism provides an intuitive understanding of the quadrupole RF- field and how it leads to a conversion between the two radial eigenmotions. Furthermore, energy conservation through the creation and annihilation of respective quanta from the undisturbed Hamiltonian Equation 3.19 and the relation ωc = ω+ +ω− explains why the resonance of the quadrupole excitation occurs at ωrf ≈ ωc . The measurement of Eρ via TOF The previous sections outlined how an azimuthal quadrupole RF-field induces a resonant net gain in energy for ωrf ≈ ωc , which allows the determination of ωc = QB/m. The measurement of the radial energy to map out the resonance has yet to Note that the x − y coordinate system of [178] is rotated by 45o compared to [185] and Equation 3.21. 3  108  2.5. The mass measurement Penning trap  Figure 3.25: Radial energy (black) and magnetic field (blue) along the ion extraction path. The top shows a schematic of the ion optic compoFigure 2.14: Axial Bz magnetic field strength and kinetic energy related nents. After the extraction of the ion bunch from TITAN’s MPET, radial to the radialenergy motion Er as a in function of the distance theenergy. trap centre. is converted the magnetic field gradient from into axial Shown on top is atheschematic of the trapofand Thus, TOF from MPET to a ion MCPoptics detectorbetween provides athe measure the MCP detector. radial energy of the ions in MPET. Figure from [146].  Inbeaexplained. typical cyclotron measurement, the ions of aretheexcited This is donefrequency by a Time-Of-Flight (TOF) measurement excited at a extracted from trap to anand MCPthen detector. Along the extraction path The fixedion frequency νRF the forPenning a given time released from the trap. (Figure 3.25), are from transported and homogeneous time-of-flight of the theions ions the from trapthe tostrong the MCP detector magnetic is recorded. field at the trap to the (quasi-) field-free region where the MCP detector is located. This procedure is repeated while varying νRF within the vicinity of the While in the homogeneous magnetic field, the circulation of an ion around the expected νc and the TOF spectrum is obtained (figure 2.15). The cyclotron magnetic field axis generates a magnetic moment µ. Hence, the radial energy Eρ frequency determined by fitting themagnetic expected to the spectrum. can be is expressed as an interaction of the fieldline withshape the magnetic moment. The following analytical expression for the TOF [Kon95]: Eρ = U = µ · B (3.36) �1/2 � z1 � M T (νRF ) = the magnetic moment as dz (2.45) which defines 2 · [E0 − q · V (z) − µ(νRF ) · B(z)] z0 Eρ (ωrf ) is used to describe the line shape. µ= · zˆ. B  2.5.5  (3.37)  The mass determination  The mass m of the trapped ion is determined from the measurement of its 109 cyclotron frequency νc : 1 qB M= . (2.46) 2π νc 56  It was established above that the radial energy, and thus the magnetic moment as well, depends on the angular frequency of the external quadrupole RF- field. Once in the magnetic field gradient, an axial force given by F (ωrf , z) = −∇ µ(ωrf ) · B = −  Eρ (ωrf ) ∂z Bz zˆ B  (3.38)  transforms the radial energy of the ion into axial energy (Figure 3.25), which is measured by the TOF of the ion from the trap to the detection MCP. The analytic expression for the TOF to the MCP detector is [185] z1  T (ωrf ) = z0  m dz 2 | E0 − Q · V (z) − µ(ωrf ) · B(z) |  1 2  (3.39)  where E0 is the initial (axial) energy. V (z) and B(z) are the electric potential and the magnetic field along the TOF path. The sequence for a measurement, then, is as follows. Each ion bunch is injected into the Penning trap and prepared with pure magnetron motion. A quadruple RFfield is applied at νrf near νc . After the RF-excitation, the gain in radial energy is measured by the TOF to a detector outside of the magnetic field. When plotted as a function of the applied RF- frequency, a TOF resonance is obtained like in Figure 3.26. The maximized radial energy for νrf = νc is seen as a minimum in TOF. The axial energy and its spread of the ions when injected into MPET as well as background counts at the MCP can reduce the signal-to-baseline ratio, i.e. the difference between the observed minimum and the maximum in the TOF resonance in comparison to the uncertainty in the TOF. This ratio can be improved by collecting data in the form of multiple ion bunches at each νrf . In practice, one chooses a frequency scanning range νs ≤ νc ≤ νe before a measurement run. A frequency scan over this frequency range consists of n ion bunches (typically 41 or 21 at TITAN). For each ion bunch, a different νrf = νs + i · (νe − νs )/(n − 1) with i = 0, 1, ..., (n − 1) is applied, and the corresponding TOF is measured. A measurement run is composed of several frequency scans (refer also to Figure A.7). Based on these data, the averaged TOF of ions from the MPET to the MCP detector is calculated as a function of the applied νrf . Damping of the ion motion Interaction of the charged particles with the residual gas in the Penning trap can lead to a damping of the ion motion [185]. The damping arises from the polarization of the residual gas atoms in the presence of the ion(s). The averaged damping  110  30 29 28  <TOF> [µs]  27 26 25 24 23 22 21 20 −15  85  Rb9+ −10  Trf= 97 ms −5  0  5  10  15  νrf − 6 020 883 [Hz]  Figure 3.26: Example for a measurement of a resonance curve for the TOF ICR method. Ion bunches of stable 85 Rb+9 are injected in the TITAN MPET , where for each ion bunch a fixed quadrupole RF - frequency νrf is applied for a time Trf = 97 ms. The conversion of initial magnetron into reduced cyclotron motion is measured via TOF when an ion bunch is extracted from the MPET after the application of the RF- field. The minimum in TOF corresponds to the true cyclotron frequency νc = 1/(2π) · mB/Q. The fit (solid red line) is based on the theoretical line-shape Equation 3.39 as derived in [185].  force on an ion motion can be described by F = −δ · m · v.  (3.40)  The damping parameter δ is expressed as δ=  Q 1 p/pN m Mion T /TN  (3.41)  in relation to a normal pressure pN and temperature TN . The damping is proportional to Q/m and will thus play a larger role for HCI. The reduced ion mobility Mion is a coefficient which depends on ion species and the residual gas [186]. Mion  111  can be determined from the drift velocity of ions through a certain gas (at pN and TN ). The drift velocity is a result of the balance in forces between an applied electric field E and the damping and hence vd = Mion · E.  (3.42)  As mentioned earlier, a damping force will reduce the amplitude of the cyclotron motion, but increase the magnetron motion. The effect of the damping is folded into the theoretical resonance line shape in [185]. The maximal radial energy is reduced by the damping which results in a less pronounced TOF minimum. Additionally, the width of the resonance is broadened. The expression of the damping parameter in Equation 3.41 indicates which experimental conditions have to be addressed in order to minimize damping. An improved base pressure in the vacuum system not only makes the influence of damping less severe, but also reduces charge exchange between HCI and the neutral atoms from the residual gas. A good vacuum system is consequently a prerequisite for mass measurements of HCI to minimize damping and charge exchange (see Section 3.7.8). The measurements (mostly with q = 8, 9+) of this work did not exhibit signs which could be unambiguously linked to damping. But cyclotron resonances during a later experiment with neutron-rich Rb and Sr- isotopes with q = 15+ [187] were broader than the expected width based on Equation 3.34. Although it is not entirely resolved yet whether this is caused by damping, the inclusion of the damping into the fitting function can compensate for the wider resonance. Hence, a fitting function including damping was always employed for the present analysis (Section 4.2). Achievable measurement precision Following Equation 3.34, the width in the resonance is inversely proportional to the time Trf of the RF-application. The (mass) resolving power R can be formulated to be ∆m ∆ν 1 R= = ∝ . (3.43) m νc νc · Trf  Inserting the expression for the cyclotron frequency νc = 1/(2π) · QB/m, one obtains m (3.44) R∝ Q · B · Trf  112  for the resolving power. Finally, the achievable precision δm/m is improved by statistics, hence the number of ions in the measurement Nions . R m δm √ ∝√ ∝ . m Nions Q · B · Trf Nions  (3.45)  This is Equation 2.89 whose implications have been discussed in Section 2.9. The√proportionality constant between the achievable precision and m/(Q · B · Trf Nions ) is a quality factor of the resonance which is trap-dependent. The individual parameters depend on various aspects. In the area of radioactive ion research, the number of ions, owing to the radioactive decay, is also a function of the excitation time and the nuclide’s half-life, Nions = N0 · exp [−Trf ln 2/T1/2 ]. The depth of the trap U (U = 35.75 V in the present work) is for SCI in most cases low enough that ions would leave the trap due to the recoil following a βdecay. For HCI, the energy required for an ion to be expelled from the trap axially can be much larger because of the charge state dependence of the effective trap potential, E = Q · U . In this case, the daughter may remain trapped although with a much higher axial energy. Assuming that the number of surviving ions after the RF-excitation period in a measurement run is large enough to unambiguously record a TOF resonance, the best precision is achieved with an excitation time of Trf = 2 · T1/2 / ln 2 ≈ 2.9 · T1/2 .  3.7.4  The Ramsey excitation scheme  A variation of the quadrupole excitation is the Ramsey method of separated oscillatory fields. The original idea is from N.F. Ramsey [188] [189] [190] who proposed the technique for the molecular-beam magnetic resonance method. It was first introduced to Penning trap mass spectrometry in 1992 [191]. Penning trap measurements based on the same concept have been performed [114, 192], but for the TOF - ICR technique it took until 2007 when a sound theoretical foundation was established [178, 193]. Other studies of the same approach are published in [194–196]. [197] discusses the effect of damping on the Ramsey technique. The basic idea of the Ramsey method of separated oscillatory fields is to apply the RFfield in well defined pulses instead of continuously. Figure 3.27 illustrates different excitations schemes that are typically applied in Ramsey TOF - ICR methods. As described in Section 3.7.3, during a normal quadrupole excitation the ions are driven by a continuous RF-field for the whole excitation time Trf = τ , Vrf ∝ A1 cos(ωrf · t + φrf ) · Θ(t) · Θ(τ − t).  113  (3.46)  114  S. George et al. / International Journal of Mass Spectrometry 264 (2007) 110–121  waiting intervals of duration τ0 in betwee cycle time is τtot = nτ1 + (n − 1)τ0 . Note t pulses of the rf-quadrupole field must be co Assuming that the ion is initially in a state motion, the probability for the conversion o into quanta of the cyclotron motion by a twowith detuning δ = ωd − ωc has been calcula F2 (δ; τ0 , τ1 , g) =  4g2 cos ωR2  δτ0 2  sin(ωR τ  δ sin ωR  δτ0 2  [cos(ωR  + Fig. 4. Excitation schemes: (a) standard excitation, (b) excitation with two  This result and analogous ones for Ramse  Figure 3.27: Amplitudes of the quadrupole excitation in a conventional withex3–5 pulses can be found in [11]. If 100 ms Ramsey pulses, (c) excitation with three 60 ms Ramsey pulses, and quadrupole field equals the cyclotron citation scheme discussed in Section ampli(d) excitation withas four 40 ms Ramsey pulses. Here3.7.3 is τ =(a) 2 · τcompared 1 + τ0 . In the the the following, τ1 is theexcitation duration of one excitation period and τ0 is the duration of the amplitude of the field is chosen such tudes in Ramsey schemes. (b) represents a two-pulse mode one waiting period. The excitation amplitude is chosen in a way that at νc one tion intervals exactly add up to the convers in which the RF - pure fieldmagnetron is applied two motion pulsesis obtained. of duration full conversion from to purein cyclotron Thus, τ1 which the coupling constant g satisfies the relation sum of the by greyacolored areasperiod is identical all four are the separated waiting ofintime τ0schemes. . (c) and (d) are three- and then the profile function (11) reaches the va four- pulse excitations. Figure from [194]. F2 (δ = 0; τ0 , τc /2, g) = 1.  An elementary derivation of the two-pu F2 in Eq. (11) is possible by applying Eqs. successively to the time intervals 0 ≤ t ≤ τ1 For a two-pulse in two pulses 4g2 scheme, and τeach 2 Ramsey 2 ωR τthe RF 2 -field is only applied 2 1 + τ0 ≤ t ≤ 2τ1 + τ0 = τtot , with th R+ (τ) = 2 sin · R− (0) = F1 (δ; τ, g) · R− (0). of duration τ1 . The pulses are separated by a waiting period τ0 with aαsummed 2 ωR + (0) = 0 and arbitrary α− (0), in order to co of the time development of the radius of the duration of Trf = 2 · τ1 + τ0 (see Figure 3.27b). (10) R+ (τtot ). The first calculation yields the initi respect to time the shape of the − excitation pulse is rectanthe waiting period, the second calculation w Vrf,2−pulses ∝With A2 cos(ω ·t+φ )· Θ(t)·Θ(τ t)+Θ(t − T + τ )·Θ(T 1 1 rf rf rf rf − t) . gular. Thus, in frequency space it is expected that the excitation phase change during the waiting period and (3.47) resembles the intensity (i.e., the modulus squared) of the Fourier ues α± (τ1 + τ0 ) for the second excitation pe 2 To understand why theof Ramsey technique is an appealing extension the nortransform a rectangular profile, namely (4g2 /δ2 ) sin (δτ/2). tofinally results in α± (2τ1 + τ0 ) = α± (τtot ) an mal quadrupole excitation it is helpful to consider the Fourier transform of RFradius The actual lineshape, however, differs in two important respects thethe final of the cyclotron motion R+ (τ from 3.7.3, this Fourier transform: at resonance (δ = 0) it is between fields. In Section we have already(a)seen the close relationship Eqs. (5)the and (6) has to be replaced in the se a periodic function τ, describing the ion periodic conversion the corresponding phases of the quadrupo Fourier transform and the exactofdescription of the motion under the by influence and reconversion of the magnetron and modified cyclotron and ωd (τ0 + τ1 ) + χd , respectively. The fina of the RF-field. Figure 3.28 compares 2Equation 3.46 and Equation 3.47 by means modes, F1 (δ = 0; τ, g) = sin (πτ/τc ), whereas the intensity of of the amplitudes, the actual RF-field, and the respective Fourier transform. From the Fourier transform increases proportionally to τ 2 . (b) The R2+ (τtot ) = F2 (δ; τ0 , τ1 , g) · R2− (0). the perspective of precision experiments the most important advantage between central peak is actually narrower than for the Fourier transform the two excitation schemesThis is the the line width which of a rectangle. canreduction be deducedinfrom the position δ0 of facilitates The time adevelopment of the ion orbit during the in zero that separatesprecision. the centralTo peak first satellite significant gain experimental seefrom thisthe narrower line width better, periods is shown in Fig. 3(a and b), during t 2 2 as compared to (δ τ)2 = 4π 2 for the Fourier peak, (δ τ ) = 3π 0 c 0 ionexcitafollows a rosette shaped orbit. the extreme case of τ1 = Trf /1000 is contrasted to the normal quadrupole the rectangle, = τc . Plotting tion in Figuretransform 3.29. In of order to obtainassuming the sameτ strength in the intensity profile the the profile function (11) with a fix ing time τ0 as a function of the detuning δ one 2.2.2. The two-pulse excitation scheme and more general lineshape of the two-pulse Ramsey cycle. It b schemes to the Fourier transform of a signal consisting Although it is not obvious on first sight, a close formal analpulses, but as for a single pulse there are impo ogy exists between nuclear magnetic istic differences. Lineshapes for higher order 114 resonance on the one hand and the interconversion of the magnetron and cyclotron motional calculated in an analogous fashion using th modes of an ion in a Penning trap due to quadrupole excitation Fig. 5, generic results are displayed for n = on the other hand. This was shown in [11] using the concept of a total cycle time τtot = 300 ms and τ1 = τ a Bloch vector. It is therefore reasonable to expect that the use tion schemes. Note that with the n-pulse e of Ramsey’s method of separated oscillatory fields will lead to spectral distribution is obtained in which the increased precision in mass spectrometry too. first major sideband and the central peak con A symmetric n-pulse Ramsey cycle of a total duration τtot peaks, while the distance between the first m consists of n excitation intervals of duration τ1 with (n − 1) the central peak increases with n. and computing the time development of the radius of modified cyclotron motion:  normal quad.  Ramsey  0.8 0.6 0.4 0.2 0.0  3 A rf  A rf  (a) 1.0  2 1 0  0.0 0.2 0.4 0.6 0.8 1.0 time  time  V rf  V rf  (b) 1.0 0.5 0.0 �0.5 �1.0  0.0 0.2 0.4 0.6 0.8 1.0 time  �I�Ω��  �I�Ω��  4 2 0 �2 �4  0.0 0.2 0.4 0.6 0.8 1.0 time  (c) 0.20 0.15 0.10 0.05 0.00 �3 �2 �1 0 1 2 3  0.0 0.2 0.4 0.6 0.8 1.0  0.20 0.15 0.10 0.05 0.00 �3 �2 �1 0 1 2 3 �Ν  �Ν  Figure 3.28: Comparison of a one-pulse excitation scheme (‘normal quad.’) compared to a two-pulse Ramsey excitation (‘Ramsey’). In (a), the amplitudes of the respective excitation are displayed over time. The actual applied RF-fields following Equation 3.46 and Equation 3.47 are shown in (b). Finally, the Fourier transforms of each case are plotted in (c) as a function of ∆ν = (ω − ωrf )/(2π). The magnitude of the amplitude for the Ramsey excitation is adjusted such that the intensities in frequency space | I(ω = ωrf ) | are the same for both excitation modes.  115  0.20 IΩ  0.15 0.10 0.05 0.00 3  2  1 0  1  2  3  Ν Figure 3.29: Direct comparison of a Fourier transform of a regular one-pulse excitation with a duration of Trf to a two-pulse Ramsey excitation with a pulse width of each RF-pulse of τ1 = Trf /1000, but with the same total Trf = 2 · τ1 + τ0 .  amplitude A2 of the Ramsey excitation has to be scaled to A1 according to A2 = A1 ·  Trf . 2 · τ1  (3.48)  Hence, the shaded areas in Figure 3.27 have to be equal for each scheme. Strongly enhanced side maxima are a distinct feature of the Ramsey method. Indeed, in Figure 3.29 the side maxima are indistinguishable from the main maximum at ∆ν = 0. In practise, this implies that the main maximum is typically first identified with the continous quadrupole excitation, and only then the measurement technique is adapted to the Ramsey method. Studies of the application of the Ramsey technique for TOF - ICR have shown that the two pulse excitation scheme offers the largest gain in precision [194] and it was used throughout the present measurement. The smaller τ1 , the more narrow is the line width although the scaling law of the amplitude Equation 3.48 limits the minimal practical duration of τ1 . The exact model of the line shape in TOF - ICR is derived in [178] and was used for the fit of all recorded resonance spectra. In addition, our fitting function has been modified to consider damping as explained in [197].  116  3.7.5  Determination of atomic masses and QEC -values  In order to extract the mass from the measurement of the cyclotron frequency, one has to consider that the magnetic field strength is not known precisely enough, undergoes fluctuations, and decays with the residual resistance of the superconducting magnet. Hence, a simple inversion to mass from Equation 3.2 is not possible. Instead the magnetic field is calibrated by measuring νc of an ion species with a well known mass. From there, a frequency ratio R=  νc,r qr mion = · νc q mr,ion  (3.49)  between ion of interest and the reference ion (labeled with r in the subscript) is obtained. The atomic mass of the ion of interest is calculated by m=  q · R · (mr − qr · me + Be,r ) + q · me − Be . qr  (3.50)  Here, mr is the atomic mass of the reference ion, me is the mass of the electron, and the Be,x are the total binding energies of the qx removed electrons. Following the convention of the Atomic Mass Evaluation (AME) [198, 199], the atomic masses of nuclides are expressed as the mass excess m.e. = m − A · m(12 C)/12 and keV is used as the unit of mass. When ions of the parent (p) and daughter (d) nuclide involved in a β- decay are available during an experimental campaign, the QEC -value can be determined to QEC =  νc,d qp · −1 ·(md −qd ·me +Be,d )+me ·(qp −qd )−Be,p +Be,d . (3.51) νc,p qd  Since νc,d /νc,p · qp /qd ≈ 1, the uncertainty on the QEC -value due to the atomic mass of the daughter nuclide is suppressed and does not need to be known in the literature to the same precision as for their direct mass measurements. Measurements of νc for the ion of interest are performed between determinations of νc,r of the reference ion. As indicated in Figure 3.30, the cyclotron frequency for the reference ion is linearly interpolated to the mid-time of the measurement run for the ion of interest and a frequency ratio R = ν˜c,r /νc is used for the evaluation of Equation 3.50 or Equation 3.51. The difference in time between two consecutive reference measurements has to be short enough to minimize non-linear field fluctuations. At TITAN, this time is typically ≈ 1 h at a field decay of ≈ 0.25 PPB/h [124]. The access to ions with different charge states of the same nuclide for mass measurements at TITAN would in principle allow an absolute atomic mass determina-  117  ν˜c,r  νc,r  %&$'(  νc,r  νc  &$%$&$*-$+  ")*+,)%,"*!$&$+!  !"#$ Figure 3.30: Schematic of the linear interpolation of reference measurements for the determination of the frequency ratio R = ν˜c,r /νc .  tion according to m=  q1 q2  · R(−q2 · me + Be,2 ) + q1 · me − Be,1 1−  q1 q2  ·R  (3.52)  without the need for a reference ion species. The frequency ratio R = νc,2 /νc,1 taken is here between charge states q2 and q1 of the same nuclide. In practice, this approach is not feasible because the denominator is close to 0 and inflates the partial uncertainty on the mass dramatically σm,R =  m/R − q1 · me + Be,1 m σR · σR ≈ · q1 q1 1 − q2 · R 1 − q2 · R R  (3.53)  in comparison to the uncertainty in the measured frequency ratio σR . Hence, in contrast to Equation 3.50, the relative precision of the frequency ratio does not translate to the relative precision in the mass. Determination of electron binding energies In the evaluation of Equation 3.50, electron binding energies are taken from the literature (e.g. [200] for HCI of Rb). For conventional Penning trap mass measurements of radioactive nuclides with SCI, these are only a few eV. Be is in this case negligible compared to the measurement uncertainty. In the context of HCI,  118  the total electron binding energies can be much more influential and can be in the range of a few keV or more. Often, their uncertainties can account for several tens of eV and/or are based on theoretical calculations only. Although this is not the case for the present studies, these uncertainties could be a limiting factor for future mass measurements at higher charge states q of higher precision. In such a scenario, differences between binding energies for different charge state of the same element could be measured directly. Since σ∆Be ≈ mion  σR σR ≈A · 109 eV R R  (3.54)  uncertainties of ∼ 10 eV and below would be in reach provided that systematic errors can be kept under control. The contribution to the uncertainties due to the atomic mass itself is again suppressed by a term (1 − q1 /q2 · R), which is close to zero. This is particularly true as these measurements could be done with stable isotopes whose atomic masses are generally much better known.  3.7.6  Dipole excitations  Each individual eigenmotion can be excited by an electric dipole RF-field with a frequency equal to the respective eigenfrequency. For the excitation of the axial motion the dipole field is applied between the two end cap electrodes. Since the axial motion is mostly irrelevant for the TOF - ICR technique, an axial RF-field is rarely employed. For excitations of the radial eigenmotions the ring electrode can be segmented. As this electrode is often split in (at least) 4 radial segments for the quadrupole excitation, the dipole field is applied to two opposing segments. Figure 3.31 compares the application of quadrupole and dipole RF-fields. With the mass measurement program at TITAN, a dipole excitation is routinely used to push unwanted, contaminating ion species to a larger radius or out of the trap altogether. Here, one takes advantage of the mass and charge dependency of the reduced cyclotron frequency ω+ . Exact knowledge of ω+ is not necessary because the more power is driven into the system at the approximate ω+ the more frequencies in the Fourier transform carry sufficient intensity to drive an ion out of the trap. This cleaning method of contamination can be very effective, but each contaminating species has to be identified such that a dipole excitation on its individual ω+ can be carried out. When working with overwhelming, unidentified contamination other separation techniques such as the sideband cooling technique in a buffer-gas filled purification Penning trap [131, 201] or the Stored Waveform Inverse Fourier Transform (SWIFT) ion excitation [202] for advanced dipole cleaning are more appropriate. Both cannot currently be applied at TITAN. For contamination suppression a multi-reflection time-of-flight apparatus [203] is planned to be inserted 119  quadrupole  dipole  V cos (ωrf · t)  V cos (ωrf · t)  −V cos (ωrf · t)  −V cos (ωrf · t)  Figure 3.31: Schematic of a cross sectional view of a segmented ring electrode which can be used to apply an azimuthal quadrupole or dipole field on top of the DC trapping potential.  into the setup in 2012. It will be able to separate the ions of interest from isobaric contaminations in the ISAC beam such as the unwanted 74 Ga ions next to 74 Rb in the present work.  3.7.7  TITAN ’s MPET  setup  The Penning trap structure dedicated to the high precision mass measurements at TITAN is mounted in the bore of a superconducting magnet. The field strength of 3.7 T is comparable to other Penning trap facilities at radioactive beam facilities although most of those employ larger field strengths. But the unique feature of HCI at TITAN can boost the precision according to Equation 3.45 to a level above the one attainable even with the current strongest magnet field of 9.4 T at the LowEnergy Beam and Ion Trap facility (LEBIT) [110] at the National Superconducting Cyclotron Laboratory (NSCL) . The trap setup as well as neighbouring beam optics for extraction and injection are 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 minimize magnetic field inhomogeneities, all material used for the trap support structure and even for the vacuum tube is chosen to be non-magnetic. The main components of the injection path in Figure 3.32 are a PLT and a Lorentz steerer. The purpose of the PLT is to remove the majority of the kinetic energy from the ions. It is biased below the transport energy of the ions. When the ions are in the centre of the PLT at time tPLT , it is switched to a bias voltage below the Penning trap. The ions can enter the trap through a hole in the end cap electrode of the trap (see Figure 3.33). 120  %  10 /6  O  /8  2C 51  6  <  @7 @6 1: 1/ 7@ 3@ 2<6 27 16 3@ 71/ 80 <6 3@ /7 7B2 27 /F27 .0 .; 05 86 85 586 HG6 @. G6 1@ 45 F .0 /< 712 1: <9 1: 0 6 25 0/ 10 :21:@89 <9 /46 5052 AI53 1/ 2<6 73 29 @9 3583: .0 7G6 85 5< 0; 8@ CB6 16 @.5 2< 16 32C/ 69 627 39 6 G6 74 4/ 5= 6@ <:/4 01 61:C/1 4< 5< B@ C5 A@ .0 06 46 ;6 56 15 <6 = 0C 71 0/ 5< 9<5 7 @B6 <@ 06 @1 /7 /7 @ 1@ <5 :5 / =E H9 12 0; 86 8G6 16A5610 45G6 86 0F = <9 4/ @ 66 F02 627 6 29 5= 27 2<56 =E P 0B 0 == @06 11 6F 82<3CB6 349B@0C 7 /3 <. P LE) 56 53 /4 621576 :@ 4@ 4/A@<2D5 /1 B272 P LE# 2B25 2@ <6 .5 456 <5 46 <: LE 8 85 6050C 6@ 8G60/1@6/7 7 L& P 82C <2H7 LE .0 2<<2 06273@.2 0; 8 )Q 58 # <6 80 5< @7 6 58 G /F 57 6@ ./ G D0 27 C2 7 1/ B6 01 30 3: 6C @C 12D5 1: G 53 D0 51 244 M5 <E 5 50 05 8  *  3  93 '%  *  &%  BC0D  C/??  B  (  3  8  +  '  , !"#  $%  &'  !(  )*!%  &  +  extraction  trap  !  Injection # $ $ 0 % / !%> &'" . !%> ?,0B ()* !%> ?,0D , !%> ?,0U $( B !%> ?,00 0? %R 5): KF FG !%> ?,0W 0? ? (I GMKL HI %&' ?" ) 0? 0 %H TKIF) ST)( )#J (+# W. / 4S JKQ)! :FGH STI KLMN) ,-'!% $. FITKX TVIM I)# ):FG 3NNI "!* "G )&LI)' ):FG KLST JKLM HI)#HOPQ ) " H $. JJZ SY I)# ):FGH N)3NN JKL JKL I)3 IH MN)3 $. )&LKG LFI)O MN NNI OPQ NNI )&LK TL\ GPP)[ HO GT H)+ U HO H L\H PQ PQ )1 GNZI H \K F  ,  -  Lorentz Steerer (LS)  Pulsed Drift Tube (PLT)  previous position of Multi-Channel Plate (MCP)  .  /  (  !"#  0  !100 eV  +  %  PLT  '  LS  *  V TCAP  +  !100 eV  &  TPLT  !0 eV  + MPET  3 :3 44)5 :# "&%13'&" '): %' )3 3" ")3 '%)& &%1 ' # !1<%)1#)6%)7 )9 &)( 8* %&3 3 )( !4" 44%9 4%3 )= 1 9 '3)#7&) %9)6 +!1 # %: <)! 1)&8 #'% %> %)9 )3 ?, ' 3 " " % 0, @2 +!1 $64 <2) ; "% %  #  + ):' 3$  %)3  ""%  $6  4; )9  '3  +!1  <  +  2 keV  ,  -  Figure 3.32: Top: Technical drawing of the injection, trap, and extraction setup of TITAN’s Measurement Penning Trap (MPET) structure. The whole structure is located inside a titanium vacuum tube placed in the bore of the superconducting magnet. The figure at the bottom explains the electrostatic potentials on axis (not to scale) and timings for the dynamic capture of ions in the MPET. /  0  1#  &%"  2  !  The bias voltage of this end cap is lowered during injection. The energy difference between the PLT and the trap centre is adjusted such that the ions are at rest when the ions reach the trap. At this time, tcap , the potential at the end cap electrode is raised and the ions are dynamically captured. Potentials and timings of the capture process are illustrated in the bottom of Figure 3.32. The Lorentz steerer [204] is an electrostatic steerer setup in the presence of the magnetic field. The E × B drift motion allows the ion bunch to be steered off-axis. This prepares the ions in the initial magnetron motion. Due to the Lorentz steerer, the radial displacement of the ions is equal to [204] ρ=  E · t = vdrift · t. B  (3.55)  Hence, the axial velocity of the ions while passing through the Lorentz steerer should be small and its potential along the axis is the same as the potential of the 121  )  0 0/ 0/ //-  to new MCP  0  0  0  0  ./ &  )  PHYSICAL REVIEW C 80, 044318 (2009)  Out to MCP  B  Corr. tube el. 3 End cap el. 2  MCP  Ring el.  1  VRF  to E  BI  T  Corr. guard el. 4 Ions in from RFQ or EBIT FIG. 2. (Color online) Illustration of the TITAN Penning trap formed from the hyperbolic ring (1) and end cap electrodes (2) that 1 CAD$ produces the harmonic potential, tube (3), and guard (4) correction coin electrodes that produce the harmonic potential. The RF is applied on (4).  ne ion source  end cap electrodes, as shown in Fig. 2. Some anharmonicities  he TITAN experimental setup that in the trapping potential are introduced by the holes in the on Penning trap, an EBIT, an off-line Figure 3.33: Schematics and and picture TITAN ’s MPET The Canaend cap electrodes by ofthe finite size ofelectrodes. the hyperbolic m diagnostics. dian dollar coin (with a diameter of 26.5 mm) is given for scale (bot-  electrodes. Two sets of correction electrodes [labeled (3) and (4) inFigures Fig. 2],from are used tom). [146].to compensate for higher-order electric and radiofrequency quadrupole field components. The radial confinement is provided by a rce is located below the RFQ, as homogenous 3.7-T magnetic field produced by a persistent, ource produces both 6 Li and 7 Li actively shielded superconducting magnet. The linear decay 39 before it is pulsed down. In this configuration the ions lose most of their kinetic es of the alkali metals 23 Na,PLT K, of the magnetic field due to flux creep [28] depends on the entering steerer (see Figure 6 ion source is biased at the energy same before pressure in the the Lorentz liquid helium vessel and 3.32). duringThe the Lorentz Li-7 Listeerer composed of a segmented, tube which a dipole field when RFQ, which is 5 to 40 kV is above measurement it cylindrical was measured to becreates (1/B) × (δB/δt) < biased following Figure 3.34. The radial displacement is proportional to the Lorentz ntial. The ions are extracted from 0.25 ppb/h. steering strength ∆V and the time it takes the ions to pass through the Lorentz ated and focused toward the RFQ. The ionLSmotion in a Penning trap is well understood [23,29] steerer (Equation The latter time dependseigenmotions: on m/Q since an every ionmospecies is is to accumulate, cool, and bunch and is 3.55). composed of three different axial accelerated by the same electric potential difference after the thermalization g from either the TITAN off-line tion with frequency νz and two radial motions with frequencies in the RFQ or the νEBIT . Since, t ∝ m/Q ion species with smaller m/Q ratios need a milar to other RFQ’s at radioactive ± . In an ideal trap, the sum of the frequencies of the two radial stronger steering strength ∆Vtrue be positioned to the same LS to d, for example, in Refs. [21] and Lorentz motions is equal to the cyclotron frequency of theinitial ion magnetron radius. Since for a full conversion the initial magnetron radius and the final [30]: cyclotron motion are identical in size (see Equation 3.25) a larger ρ0 will result in q× 1 in a larger gain in energy during theν RF= -excitation theBMPET asurement Penning trap ν+ + νc = , . The maximal (1) ρ0 is − 2π mion devices used to perform high122 ratio of the trapped ion ents on stable and exotic nuclei where q/m is the charge-to-mass  s Penning trap is located inside ucting magnet, shown in Fig. 1. trap, the ions are moved off-axis , similar to the one currently used The Lorentz steerer reduces the nning trap by inducing an initial ion bunch prior to its capture, as  ion  and B is the magnetic field at the trap center. The two radial motions can be coupled by applying an azimuthal quadrupolar RF signal on the sliced correction electrode [(4) in of the trap Fig. 2]. The cyclotron frequency is determined using the timeof-flight (TOF) resonance detection technique [29,31,32]. For a proper choice of the RF amplitude VRF , at a given RF excitation time T , and when ν = ν a full conversion of  2.3. Beam preparation: beam transport to the Penning trap  Lorentz steerer configuration LSYP =V ∆VLS LSYP = PLT 1936 ++∆V LS  Β d = s ∆VLS Ε LSXN = 1936 V  LSXP = 1936 V d  LSXN = VPLT  LSXP = VPLT  LSYN = 1936 - ∆VLS LSYN = VPLT − ∆VLS  FigureCross 2.6: Schematic the Lorentz steerer comprise a quarted Figure 3.34: sectionalofschematices of thewhich Lorentz steerer. Figurecylinmoddrical electrode axially aligned with a magnetic field B. We show the poified from [146]. tential needed to steer the beam and the corresponding electric quadrupolar field produced. The displacement, d, of the beam is proportional to the offset potential, ∆VLS .  constrained by the radius of the injection hole in the end cap of MPET and similar sized holes in the PLT. The functionality of the Lorentz steerer described so far assumes injection of the ions into the magnetic field along its axis. The Lorentz steerer can also be used to correct for off-axis injection following the procedure in [204]. Most other Penning trap facilities operate without a Lorentz steerer and the initial magnetron motion is induced by a dipole excitation on the (almost) massor charge-independent magnetron frequency ν− . The duration of this excitation is on the order of a few 10’s of ms and is consequently a limiting factor for measure43 ments of short-lived nuclides. In contrast, the preparation by the Lorentz steerer takes place during the injection itself and comes without any loss in time. The measurement Penning trap The mass measurement itself takes place in a precision Penning trap. Schematics and a picture of this MPET are displayed in Figure 3.33. Its characteristic trap dimension is d0 ≈ 11.21 mm. The trap electrodes are made of ultra pure oxygenfree copper. They are silver and gold plated to impede localized oxidation which could result in field distortions and to optimize conductivity. The hole for injection and extraction in each end cap electrode as well as the finite size of the electrode cause higher order anharmonicities and deviations from the harmonic potential of an ideal Penning trap. To correct for these imperfections, in particular for the field 123  Table 3.2: Nominal frequencies of an ion with mass A = 74 and charge q = 8+ in TITAN’s MPET with B = 3.7 T and the trapping potential U = 35.75 V. νc ν+ νz ν−  6.142 MHz 6.136 MHz 274.1 kHz 6.123 kHz  in the trapping volume at the trap centre, correction-tube electrodes are placed next to the entrance and exit hole in the end caps. Correction guard (or octopole) electrodes are inserted into the space between the ring and the end cap electrodes. The trap compensation procedure to obtain a potential closest to the ideal case is discussed in detail in [205]. For the present measurements, a trapping potential between the end cap and the ring electrodes of U = 35.75 V has been used. Nominal frequencies of an ion with Aq+ = 748+ are listed in Table 3.2. The azimuthal (dipole and quadrupole) RFfields (Figure 3.31) are most commonly applied to segments of the ring electrode. In the TITAN setup the guard electrodes are segmented instead of the ring electrode to avoid distortions in the harmonic field due to the splitting of the ring electrode. This improvement, however, requires larger RF-amplitudes to reach equal field strengths at the position of the ions and hence the need for RF-amplifiers. At the end of the RF-excitation phase, the extraction end cap is lowered and ions are extracted from the MPET. During the extraction path the radial energy is converted into longitudinal energy as discussed earlier (Figure 3.25) and the TOF from the ion extraction to the arrival of the ions at the MCP detector is measured. Timings of the measurement process are either controlled by the MPET Programmable Pulse Generator (PPG) (e.g. PLT, RF, etc. ) or by a slow control unit (essentially all DC bias potentials for the Penning trap) which are operated by the MIDAS control and data acquisition system [206]. The TOF measurement itself is integrated into the MIDAS system by a time-to-digital converter based on the timing of the MCP detection signal.  3.7.8  MPET  vacuum system  The quality of the vacuum in the Penning trap is important due to increased damping and charge exchange for higher pressures. Requirements on the residual gas density are more severe when dealing with HCI compared to SCI. The MPET was designed as a room temperature trap system. Considering the success of Penning 124  trap mass measurements of stable HCI at SMILETRAP, which also operates at roomtemperature [114], it was concluded that cryogenic temperatures were unnecessary at TITAN’s MPET. The SMILETRAP group reports a pressure estimate in their Penning trap of less than 6.4(3.6) · 10−12 mbar. This estimate is based on a measured charge exchange rate in comparison to charge exchange cross sections from the literature (in this case from [207]). When keeping 76 Ge22+ for 3.7 s in the trap only 9 % of the initial ions underwent charge exchange. An empirical scaling law [208] of the cross section for charge exchange of ions with energies below 25 keV/A in a neutral gas gives σ QX [cm2 ] = 1.43 · 10−12 q 1.17 · Ip−2.76 [eV],  (3.56)  where Ip is the ionization potential of the neutral atom or molecule. Hence, the probability for an ion to pick up an electron from the residual gas in an infinitesimal time interval dt is P ∝  σ QX pvion q 1.17 pvion · dt ∝ · dt. kB T kB T  (3.57)  Charge exchange during the TOF - ICR measurement is consequently intensified for HCI due to two reasons. Firstly, the charge exchange cross section is proportional to the charge state q and, secondly, the reduced cyclotron frequency is increased for HCI leading to a larger ion velocity vion with νc = νrf . The pressure in the MPET section before working with charge-bred ions (earlier than summer 2009) was determined to be ≈ 2 · 10−9 mbar4 . Previous to the mass measurement of 44 K4+ [1] an ion pump in the MPET vacuum section was baked at ≈ 200o C which led to a vacuum pressure of ≈ 8 · 10−10 mbar. However, as illustrated in Figure 3.35 for data with 39 K4+ , strong signs of charge exchange were observed. A peak of H+ 2 appeared in the TOF spectra when storing the ions longer in the MPET. H2 was the most abundant residual gas as determined by a residual gas analyzer. Additionally, the tail of the TOF peak of 39 K4+ extended to longer TOF which is indicative of 39 K in lower charge states. As soon as an ion changes its initial charge, it has a different cyclotron frequency and is subsequently insensitive to the applied RF-field. Thus, the charge exchange affects the quality of the TOF resonance severely as seen in Figure 3.35 and Table 3.3. In order to successfully work with HCI the quality of the vacuum had to be improved further. This was achieved by multiple efforts including baking the whole 4  Please note that this is not necessarily the pressure in the trap itself. The pressure was measured with an ion gauge which is in the same cross as a turbo molecular pump. All pressures in the MPET vacuum section were, if not otherwise stated, measured at the same position and are at least a sign of relative improvements.  125  2  8 ms 0.5  1  0  150  1  197 ms 0.5  0 0  counts/scan  100  [!s]  50  offset  counts/scan  0 0  50  100  150  <TOF> ï T  counts/scan  1  0 0  ï2  ï3  1  0.5  ï1  497 ms  H+2  8 ms 197 ms 497  ï4  50  100  ï5  150  ï2  TOF [!s]  ï1  0  (irfïic)*Trf  1  2  Figure 3.35: TOF spectra of ion bunches of 39 K4+ measured with Trf = 8 ms, 197 ms, and 497 ms irrespective of the applied νrf (left part of the figure). The counts are normalized to the number of frequency scans for each measurement run. The ions were extracted from the MPET right after the application of the RF-field. The strong peak of 39 K4+ in the TOF spectrum for Trf = 8 ms is reduced for longer excitation times. Instead a rise in counts of H+ 2 and in the tail of the initial TOF peak is observed, which corresponds to 39 K3+,2+,1+ . The charge exchange reduces the quality of the resonance as seen in the plot on the right.  Table 3.3: Uncertainty in the cyclotron frequency νc of 39 K4+ for different excitation times Trf . The last column lists the expected uncertainty based on ∆ν for Trf = 8 ms and scaled to the larger Trf using Equation 3.45. Figure 3.35 shows the TOF - ICR of these measurement runs. Trf [ms] 8 197 497  frequency scans 100 200 199  ∆ν [Hz] 2.607 0.096 0.094  126  expected ∆ν [Hz] 0.074 0.030  setup at temperatures above 200o C and a hardware upgrade to the vacuum system. An example for the bake-out setup and for residual gas measurements of the MPET vacuum tube is shown in Figure 3.36. A more detailed description of the baking procedures and the modifications of the vacuum system are summarized in Section A.5. In the final setup, the pressure during the mass measurement of the neutron deficient Rb was ≈ 6 · 10−11 mbar. When the ion pump was tested and fully operational at a later time, the pressure was further reduced and eventually went below the limit of the ion gauge (≈ 2 · 10−11 mbar). During the course of the modifications of the MPET vacuum system prior to the Rbmass measurements, the MCP detector for the TOF measurement was moved further away from the MPET and a Daly detector [209, 210] was added. A detector in addition to the primary MCP detector provided an immediate alternative detection system for the TOF measurement in case of a detector failure. Details can be found in Section A.5. Finally, the MPET support structure and wiring system were modified for better reliability of the electrical connections and simplified maintenance (see documentation in [211]).  127  RGA TC TC  TC  TC  TC  IG  IG  heating tape  heating tape  heating rings  heating pads  turbo pump TV 551  ï3  10  Change to IG on tube end  ï4  10  ï5  pressure [mbar]  10  turn off heating elements  ï6  10  turn off heating elements  ï7  10  ï8  10  leak closed  ï9  10  ï10  10  heating elements turned on  leak opened leak closed  leak opened  ï11  10  6  8  10  12  14  16  18  20  March [days]  Figure 3.36: Top: Pictures of the baking setup for the vacuum tube with the trap structure, injection and extraction optics. The baking was monitored by two ion gauges (IG), a set of thermocouples (TC), and a residual gas analyzer (RGA). Several heating elements were mounted on the tube and the cross with the turbo molecular pump. The development of the pressure during and after the baking is shown in the plot at the bottom. A vacuum leak opened twice during a cool-down phase and was closed by tightening the bolts on the CF flange.  128  22  Chapter 4  Measurement and Analysis Neutron-deficient Rb nuclides represent a very appealing isotopic chain for the first Penning trap mass measurements of HCI at a radioactive beam facility. The halflives of 75−78 Rb range from tens of seconds to minutes. Radioactive decay does not need to be considered for those cases because the typical measurement times with a single ion bunch at TITAN are much shorter than the respective half-lives. In combination with the high yields (Table 3.1) for nuclides closer to the valley of stability, this allows for optimal experimental conditions while approaching the measurement of 74 Rb. The relatively well known masses of 75,76 Rb with uncertainties of 1.6 and 1.2 keV [14], respectively, provide immediate benchmarks for the accuracy of the new measurement approach. Both nuclides have been measured at the ISOLTRAP facility and are considered to be very reliable. The strong physics motivation and the necessity for a new method with improved precision (Section 2.9) make 74 Rb itself most interesting among neutron deficient Rb isotopes. Its short half-life of 65 ms further gives insight about the feasibility of the technique with respect to the lowest lifetimes measurable with Penning trap mass spectrometers. Moderate yields of a few thousand ions per second highlight the relationship between loss in efficiency due to the additional step of charge breeding in contrast to the gain in precision due to the higher charge state. Finally, a long lived isomeric state (with a half-life of 5.7 min) is known for 78 Rb, which lies only 111.19(22) keV above the ground state [212]. This corresponds to a resolving power of R ≈ 6.5 · 105 . Although isomer and ground state have been resolved before in a Penning trap [213], the case can illustrate the potential of HCI for isomer research in Penning traps. This approach is particularly well suited for low lying isomers with half-lives of 100’s of ms. This chapter will provide details about the mass measurement and its analysis of highly charged 74−76,78m,78 Rb as well as 74 Ga. The latter was a contaminant in the  129  A = 74 beam from ISAC. Systematic uncertainties will be discussed which also include measurements involving stable nuclides with well known masses as a cross check.  4.1  Measurement summary  The experimental facility was prepared previous to beam delivery from ISAC with a beam of stable 85,87 Rb from TITAN’s own surface ion source (see Chapter 3). A study to benchmark systematic uncertainties was carried out during this time. The ion-optic tune from ISAC to TITAN was established with a 20 keV ISAC beam of stable 69 Ga. Its intensity of ≈ 160 pA made the beam detectable on Faraday cups and allowed for better tuning of the beam into the RFQ and adjustments of RFQ parameters. During this process the RFQ was operated in a DC mode, i.e. ions were passing through the RFQ without being trapped longitudinally. A DC transfer efficiency of ≈ 75 % was reached which compared well with the RFQ’s typical performance [134]. Once optimized, the only parameters to be adjusted for different masses were mass dependent parameters such as beam transport timings and in some cases RF-frequency and amplitude or buffer gas settings in the RFQ . The rest of the TITAN beamline remained unchanged because the transfer beam energy was controlled by the PLT after the RFQ. Hence, modifications were not required in comparison to the preparation tune. Complementary to the ISAC beam, stable beam of 85 Rb or 87 Rb could be delivered from TRIUMF’s Off-Line Ion Source (OLIS) [214] at the same beam energy as the ISAC beam. Access to an ion species with well known mass was necessary to calibrate the magnetic field in MPET to the required level of precision (see Section 3.7.5). Determining the QEC - value of 74 Rb directly as outlined by Equation 3.51 was not possible because 74 Rb and its daughter nuclide 74 Kr could not be delivered due to different chemical properties and would have required different target- ion source combinations. The stable Rb beam from OLIS was preferable over TITAN’s own offline source because OLIS could provide dipole-magnet based mass separated and consequently contamination free 85 Rb or 87 Rb beam. In contrast, the beam from TITAN’s offline source was a combination of various alkali metals. Although the RFQ provided a mass selectivity, its resolving power was not high enough to separate the two stable isotopes of Rb. 85 Rb and 87 Rb could only be distinguished in TOF after extraction from the RFQ or EBIT. In the aim for the most identical experimental conditions for reference ions and radioactive ion of interest, OLIS was the better alternative even for stable Rb beam. An initial plan to deliver stable 74 Ge from OLIS could not be realized this time. Providing a reference ion of the same mass number as 74 Rb would have reduced systematic uncertainties (see Section 4.3) and 74 Ge’s atomic mass with a precision of 0.2 PPB [215] is very well known. 130  30  76  Rb12+  (a)  32  75  Rb8+  (b)  <TOF> [!s]  30 25  28 26  20  24 15 10  22 ï12  ï6  0  6  20  12  ï12  irf ï 8 977 118 [Hz] 32  74  (c)  8+  Rb  <TOF> [!s]  30  0  6  12  74  8+  (d)  Rb  40 35  28  30  26 24  25  22  20  ï50  ï6  irf ï 6 064 157 [Hz]  ï25  0  25  50  i ï 6 145 705 [Hz]  ï10  ï5  0  5  10  i ï 6 145 705 [Hz]  rf  rf  Figure 4.1: Time-of-flight ion-cyclotron resonances for Rb-isotopes in charge state q = 8+ and q = 12+. During (a) and (c) the RF- field was continuously applied for Trf = 97 ms and 30 ms, respectively, and a Ramsey excitation scheme with 6-85-6 ms was utilized in (b) and (d). The solid (red) lines represent the fit to the theoretical line shapes [178, 185].  The measurements of radioactive Rb isotopes were performed with charge states q = 8+ and q = 12+. The magnetic field was calibrated by measurements of 85 Rb9+ . The charge state q = 9+ was chosen as the reference to minimize the difference in m/q between the radioactive q = 8+ ions and the reference ions because a number of systematic effects are proportional to this difference (see Section 4.3). Typical resonances of radioactive Rb isotopes are displayed in Figure 4.1. Both, conventional quadrupole and Ramsey excitations were applied in the course of the measurement. 74 Ga was the dominat contribution to the A = 74 beam and its mass was also determined in a set of measurements. The required resolving power be131  160  q=8+  q=3+  q=2+  140  120  13 µs  # counts  100  50 µs  80  60  H+2 40 76  Rb+8 Trf= 97 ms  20  0 0  10  20  30  40  50  60  70  TOF [µs]  Figure 4.2: A TOF histogram of detected ions after they were released from the MPET. In addition to 76 Rb8+ , a peak corresponding to H+ 2 was observed, which was due to charge exchange in the MPET between the stored HCI and the residual gas. See text for details.  tween 74 Rb and 74 Ga was slightly above the performance of the resolving power of the ISAC mass separator. Nevertheless, it could be fine tuned to a more favourable ratio between the two components of the beam at the expense of a lower absolute 74 Rb yield. For the purpose of this fine adjustment, singly charged A = 74 beam was brought directly into MPET by bypassing the EBIT entirely (compare with Figure 3.19). In consecutive measurement runs the dipole cleaning (Section 3.7.6) was applied at the mass selective reduced cyclotron frequency ν+ for 74 Rb+ and 74 Ga+ . The number of ions lost because of the dipole RF -field at the respective ν + was interpreted as an indicator for the number of ions in the beam for Rb or Ga. The ISAC mass separator magnet and slits were adjusted until 74 Rb represented the majority in the beam. Then, the mass measurement continued with HCI. Figure 4.2 is a TOF histogram of all detected ions during a measurement run of 76 Rb8+ irrespective of the applied RF - frequency ν . The main peak in the TOF hisrf togram corresponded to 76 Rb8+ ions which were initially injected into the MPET.  132  70  34  50 !s  60  33  50  32  <TOF> [!s]  # counts  13 !s  40 30  31 30  20  29  10  28  0 0  10  20  30  40  50  60  27 ï2  70  ï1  0  1  2  irf ï 5 984 572 [Hz]  TOF [!s]  Figure 4.3: TOF histogram (left) and TOF resonance (right) for 76 Rb8+ with an excitation time of Trf = 997 ms. In comparison to Figure 4.2 the peak corresponding to H+ 2 (at ≈ 5 µs) was more prominent.  An additional peak at a TOF of ≈ 5 µs was due to charge exchange of the stored HCI with residual gas in the MPET . Previous work established that it represented + H+ 2 ions as the application of a dipole excitation at ν+ of H2 removed this peak. For longer storage times of the HCI in the MPET larger peaks associated with charge exchange appeared. This indicated that despite the improvements in the vacuum quality of the MPET (Section 3.7.8) charge exchange could not be avoided entirely. However, for an excitation time of Trf = 97 ms the fraction of ions which underwent charge exchange remained at an acceptable level (Figure 4.2); the number of counts in the H+ 2 peak was only ≈ 5% of all detected signals. Hence, the majority of the online mass measurement was carried out with the excitation time Trf = 97 ms. This corresponded to a 10 Hz repetition rate, i.e. the rate at which ions were extracted from the RFQ. In fact, the HCI could be held much longer in the MPET and a measurement with a quadrupole time of Trf = 997 ms was performed as well (hence at a 1 Hz repetition rate). Although the charge exchange was a more severe problem in this case, a TOF- resonance of 76 Rb8+ could be recorded (see Figure 4.3). As these measurements are carried out with only a few ions (typically 1-5) in the trap at a time to minimize frequency shifts by ion-ion interactions (compare Section 4.2.1), the ISAC beam of 76 Rb had to be attenuated to reduce the number of ions in MPET to the desired level. A TRIUMF site-wide power-outage at the end of the 74 Ga mass determination trig-  133  gered a turbo pump failure in the EBIT, which required the reconditioning of the electron beam and a re-tuning of the beam transfer from the EBIT to the MPET previous to the 74 Rb mass measurement. As a consequence, another set of measurements to cross check the accuracy of the setup had to be performed between 74 Rb measurement runs, since the comprehensive systematic tests performed before the power-outage could not reliably be applied to the data obtained after the reconditioning. Finally, the ground and isomeric state of 78 Rb were studied in the MPET. As the energy difference between the two states is only 111.2 keV it represented an interesting case to demonstrate the improved resolving power for low lying isomeric states when working with HCI. In order to show the relationship between high charge state and excitation time in the resolving power, the measurements were performed with Trf = 97 ms and Trf = 197 ms.  4.2  Analysis  The analysis of the data was separated into two main parts. First, the cyclotron frequency νc was determined for each measurement run by fitting the resonance data with Equation 3.39 and the theoretical line shapes [178, 185]. In the second part, individual frequency ratios (Equation 3.49) were summarized to weighted averages from which the atomic mass was determined (Equation 3.50).  4.2.1  Determination of the cyclotron frequency νc  Following Section 3.7.3, the raw data recorded during a measurement of a cyclotron resonance consisted of TOF measurements of ions which were released from the MPET at a certain RF- frequency νrf . Hence, for each ion bunch the number of counts at the MCP detector, their respective TOF, and the applied νrf was stored. As discussed in Section 3.7.3 the amount of damping was a free parameter in the fitting function. To obtain νc the analysis procedure described below was followed. Data selection H+ 2 originating from charge exchange did not respond to the RF -excitation due to its different (mass dependent) cyclotron frequency which was outside the νrf scanning range. Hence, the counts of ions at the MPET MCP detector associated with H+ 2 (compare to the TOF histograms of Figure 4.2 and Figure 4.3) would have reduced the quality of the actual resonance of a Rb- isotope. The peak in the + TOF spectrum corresponding to H2 was excluded from the analysis. Analogously, 134  which had picked up an electron from the residual gas while stored in the MPET were subsequently insensitive to the RF- excitation. Consequently, the TOF range which was considered in the analysis was fixed by the TOF- distribution for very short excitation times (Trf ≈ 10 ms) for which charge exchange was negligible. Hence, only counts in the TOF-range of 13-50 µs were selected for the analysis. In the measurement of 76 Rb12+ it was extended to 10-50 µs. The considered range is indicated in TOF histograms of Figure 4.2 and Figure 4.3 by labeled vertical lines. As shown in Figure 4.2 other charge states resulting from charge exchange could have fallen into the same TOF range. Ambiguities related to this choice of the considered TOF- range and potential systematic effects are described in Section 4.3.7. The cut on the data also excluded a large fraction of background counts which were recorded without a fixed time relation to the release of ions from the MPET. At the mostly used repetition rate of 10 Hz, the number of these background counts after 50 µs in the recorded TOF window (which spanned over 0-200 µs) was a few 10’s per hour of measurement. An additional selection criterion was applied to each ion bunch as a whole. To minimize potential shifts in νc due to the interactions between ions stored in the MPET at the same time, only ion bunches with 1-5 detected ions per bunch were processed in the analysis. These shifts were observed experimentally (e.g. in [213]) when ions of different ion species were simultaneously kept in the trap. HCI  TOF  uncertainty  In order to obtain the measured TOF-resonance (such as in Figure 4.1) the raw data had to be sorted according to the RF-frequency applied to each trapped ion bunch. After the cut on the TOF range was made the mean TOF, T k , was calculated for ion bunches with the same applied νrf,k . In previous work, the variance on T k was obtained by considering the width of the TOF distribution irrespective of νrf,k σ 2 (T ) =  N i=1 (Ti  − T )2 , N −1  (4.1)  where N was the number of ions in the selected TOF- range, Ti was the TOF of each ion, and T = ( N i=1 Ti )/N . The variance on T k was assigned according to σ 2 (T k ) =  σ 2 (T ) Nk  (4.2)  with Nk being the total number of counted ions at νrf,k . This procedure is called the sum-statistics method. Recently, it has been pointed out [216] that from a statistical 135  point of view the approach was not exact because it assumed Ti to be independent of νrf,k . This is opposed to the measurement principle of νc itself. In fact, for measurement runs with a sufficient number of ion counts per νrf , the sum-statistics is too conservative and overestimates the uncertainty. Although the correct statistical treatment was tested, the present analysis continued to use the sum-statistics method. This was motivated by two reasons. First, for measurement runs with very few counts for a given RF-frequency νrf,k the sum statistics approach was considered to be more appropriate [216]. The second reason is related to possible line shape distortions when working with HCI, e.g. due to charge exchange (compare for instance with the discussion on damping in Section 3.7.3). When the correct statistical treatment was utilized, the typical χ2 was larger than what was expected from the degrees of freedom in the fit. This situation would require the inflation of the uncertainties on the fit parameters in order to account for potential line shape distortions. However, when the sum-statistics method was applied, the distribution of χ2 in the line-shape fits followed the statistical expectation (see Figure 4.4). Hence, in order to take full advantage of the correct statistical treatment of the data in future work, the measured line shapes have to be investigated. Count class analysis To address previously mentioned potential shifts in νc due to ion-ion interaction, a so-called count-class [217] analysis was performed. In this method, data from a measurement run were grouped according to the number of detected ions per ion bunch, usually into three groups. The resonance of each group was fitted separately to determine a cyclotron frequency. Trends in the cyclotron frequency as a function of detected ions per ion bunch could be investigated and νc was extrapolated to the case of a single ion stored in the Penning trap. Considering the detector efficiency of the MCP, this corresponds to = 0.6 ± 0.2 detected ions [118]. Due to charge exchange of HCI, ions in different charge states could be present in the trap even when a contaminant free beam had been captured in the MPET. Hence, a countclass analysis was performed for all online measurements provided that a run had sufficient number of counts (typically a few thousand) to separate the data into three classes.  4.2.2  Determination of the frequency ratio  The frequency ratio R = νc,r /νc between the ion of interest (νc ) and the reference ion (νc,r ) was calculated by linearly interpolating the νc,r to the mid- time of the measurement run to determine νc as discussed in Section 3.7.5 (compare 136  8  r2 of fits to resonances r2ïdist. with 37 d.o.f.  # measurement runs  7 6 5 4 3 2 1 0 20  25  30  35  r2  40  45  50  Figure 4.4: A histogram of χ2 from fits of the theoretical line shape to experimental resonances is compared to the expectation of the statistical χ2 -distribution. Each fit was based on 41 data points in a resonance and 4 free parameters (νc , the initial ρ− , the TOF-offset, and the damping parameter), hence 37 degrees of freedom. For the resonance data, the sum-statistics method was used for the determination of the uncertainty of the TOF-measurements. The experimental data is a measurement set of 87 Rb9+ versus 85 Rb9+ which was performed previous to the online measurement.  with Figure 3.30). As a measure to maximize the use of available radioactive beam reference measurements were shared between consecutive measurement runs of the ion of interest (see Figure 4.5). Consequently, these frequency ratios were not statistically independent. These correlations due to shared reference measurements have so far not been considered in the literature. However, owing to the potential of HCI for the precision of mass measurement, correlations between frequency ratios were accounted for in the present work. The derivation of the complete covariance matrix to reflect these correlations is done in Appendix B.  4.2.3  Analysis software  For the present analysis, a new software package was developed which automated most of the analysis and allowed efficient investigations of the results as a function of various fitting settings. It consisted of a resonance fitting program [218], a sorting code to coordinate the fitting of individual measurement runs and to extract (averaged) frequency ratios and masses, and several plotting routines. As a cross 137  ν r,i  νr,i+1  ν r,i+1 νr,i+2  &$%$&$*-$+  %&$'(  νr,i  νj νj+1  ")*+,)%,"*!$&$+!  !"#$ Figure 4.5: Origin of correlations between measured frequency ratios. The reference ion’s cyclotron frequency is linearly interpolated to the midtime of a measurement run of the ion of interest. When the same reference measurement (e.g. νr,i+1 in this plot) is used for the interpolation of νr to two measurement runs (νj and νj+1 ), these two frequency ratios are statistically correlated.  check, a large part of the analysis was also performed with previously established software packages such as EValuation and Analysis (EVA) [219] for fitting the theoretical resonance line shapes to the TOF data and Simple Online Mass Analysis (SOMA) [220] for the calculation of frequency ratios and masses.  4.2.4  Analysis results  76 Rb  The data for 76 Rb were separated into four groups. Three sets of measurements were carried out in the conventional quadrupole excitation. This set consisted of three groups: firstly, measurements of 76 Rb8+ with an excitation time of Trf = 97 ms, secondly, two measurement runs with Rb-ions in a higher charge state (76 Rb12+ with Trf = 97 ms), and finally 76 Rb8+ resonances with an extended excitation time of Trf = 997 ms. All reference measurements to calibrate the magnetic field were based on ions of 85 Rb9+ with an excitation time of Trf = 97 ms. The last set consisted of measurements with the Ramsey excitation mode. Two RF pulses were applied for τ1 = 6 ms interrupted by a waiting period τ0 = 85 ms  138  0.2  0.1  0.1  ï0.4 ï0.5 ï0.6 0  0.5  1  0.1  ï0.2  0  ï0.3  ï0.1  ï0.4  ï0.2  ï0.5  ï0.3  ï0.6  ï0.4  0 0.5  0 ï0.1 ï0.2 ï0.3 ï0.4 ï0.5  reference  76Rb8+  76Rb8+  76Rb12+  with Trf = 97 ms  with Trf = 997 ms with Trf = 97 ms  ï0.6 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5  5  time since first reference. run [h]  ï0.5  0.5 1  1.50  time since firstrun reference. run [h] time since first reference. [h] Ramsey excitation 1 1.5  ï0.6  iref ï 6 020 883.36 [Hz]  ï0.3  ï0.1  ï 6 020 883.32 [Hz]  ï0.2  0.2  ref  ï0.1  0  i  0  conventional quadrupole excitation  0.1  iref ï 6 020 883.32 [Hz]  0.2  iref ï 6 020 883.32 [Hz]  iref ï 6 020 883.32 [Hz]  0.2  0.5  0.5  1.5 2  2 1  2 2.5  1.5  2.5 3  2.5 2  3 3.5  2.5  3  3.5 4  3  4 4.5  3.5 3.5  4.5 5  5  4.5 4  4  5  4.5  5  time since first reference. run run [h][h] time since first reference.  0.4 0.3 0.2 0.1 0  ï0.1 ï0.2 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8  time since first reference run [h]  Figure 4.6: The time and durations of measurements for reference ions 85 Rb9+ and for radioactive 76 Rb ions in charge state q = 8+ and 12+ are indicated by rectangles. For the reference ions the measured cyclotron frequency is shown on the vertical axis.  (which is labeled 6-85-6 ms in Figure 4.1) adding again to a total measurement time of Trf = 97 ms. The relative time and durations of individual measurement runs for 76 Rb and its references are shown in Figure 4.6. The scatter of the reference measurements are also displayed. Measurements with Trf = 97 ms of 76 Rb8+ were always alternated with calibrations with 85 Rb9+ as these were the main data of the 76 Rb mass determination. The supplementary data sets of Trf = 997 ms excitations or in charge state q = 12+ were each performed between two reference measurements. In Figure 4.7a, the error contributions to the frequency ratios for individual 76 Rb8+ measurements due the cyclotron frequency of 76 Rb8+ and of the (interpolated) reference are shown. The individual frequency ratios themselves are plotted in Figure 4.7b together with the weighted averages for each set. The latter’s uncertainty also includes the correlations due to shared reference measurements. The effect of taking correlations into account was studied in Figure 4.8. In (a), 139  70  10  (a)  interpolated ir i ionïofïinterest  50 40 30 20  0  Trf = 6-45-6 ms  ï5  Trf = 97 ms ï10  ï15  10 0  (b)  5  ( R ï1.0060674 )u108  Error contributions to R [ppb]  60  Trf = 997 ms 1  2  3  4  5  6  7  8  9  10  11  ï20  12  measurement number  1  2  3  4  5  6  7  8  9  10  11  12  measurement number  Figure 4.7: Measurements of 76 Rb8+ . All reference measurements of 85 Rb9+ were performed with a conventional excitation of T rf = 97 ms or with a Ramsey scheme of 6-85-6 ms, while 76 Rb8+ was split into sets of measurements with Trf = 97 ms and Trf = 997 ms in the conventional excitation and of a Ramsey scheme with 6-45-6 ms. (a) contributions to the frequency ratio R due the uncertainty in the cyclotron frequency of the reference ion and the ion of interest. (b) Individual R. The shaded areas represent the weighted average with uncertainties including correlations between the individual R.  all measurements of the ion of interest were separated by reference data. Due to correlations, the central value shifted by ≈ 8 % of the statistical uncertainty, which itself was increased by about a tenth of its value. Figure 4.8b was the special case that all three measurements shared both references, i.e. the three runs were taken in between two references (compare also to Figure B.1). Here, the correlations substantially inflated the uncertainty. As these three measurement runs were very close in time (see Figure 4.6), an alternative approach was to add their data together and to fit the resonance. This should implicitly take all correlations between the individual runs into account. As seen in Figure 4.8b, this result was indeed almost identical to the analysis of the individual runs demonstrating the importance to consider the correlations for this case. Table 4.1 summarizes all measurements of 76 Rb. 75 Rb  All data for 75 Rb were taken in charge state q = 8+ and with the Ramsey excitation method. As mentioned in Section 3.7.4, the side-minima in Ramsey schemes 140  0  (a)  ï8  data added  ï6  76Rb  ï0.5  ï4  with correlations  0  ( R ï1.0060674 )*108  0.5  with correlations  1  ï10  ï1 ï1.5  (b)  ï2  without correlations  ( R ï1.0060674 )*108  1.5  without correlations  2  1  ï12  2  1  2  3  Figure 4.8: Implications of correlations due to shared references studied with 76 Rb8+ measurement runs. (a) The weighted average for R of the data with a conventional Trf = 97 ms excitation is displayed with and without taking correlations into account. With correlations the central value is shifted by ≈ 8 % of the uncertainty σ. σ itself is increased by ≈ σ/10. (b) Same as (a) but for the Trf = 997 ms data. According to Figure 4.6 this is a special case, because all three runs share both references. When the data of the three runs are added together and analysed (3), its central value and uncertainty is essentially identical to (2), where correlations are taken into account between the three individual measurement runs.  Table 4.1: Averaged frequency ratios (with correlations) of 76 Rbq+ with respect to 85 Rb9+ ions. The table lists the charge state q, the excitation mode, the excitation time, the number of measurements per set, and the resulting frequency ratio R. Except for q = 12+, R are based on a countclass analysis and the given errors are statistical uncertainties. The count rate for 76 Rb12+ was too low to perform a count-class analysis with the recorded data. q of 76 Rb 8+ 12+ 8+ 8+  excitation mode conventional conventional conventional Ramsey  Trf [ms] 97 97 997 6-85-6  141  # measurements 5 2 3 4  R 1.006067401(15) 0.670692259(23) 1.006067338(55) 1.006067422(12)  30  4  (a)  3  (b)  29  ( R ï0.992864 )*108  <TOF> [!s]  28 27 26 25 24 23  21  75  ï15  Rb8+ ï10  ï5  0  5  10  1 0 ï1 ï2  Ramsey excitation 40ï17ï40 ms  22  2  ï3  15  irf ï 6 064 157 [Hz]  ï4  1  2  3  4  5  measurement number  Figure 4.9: Measurement of 75 Rb8+ . The position of the main minimum corresponding to the cyclotron frequency of 75 Rb8+ was determined with a Ramsey scheme of 40-17-40 ms such as (a) which had less pronounced side minima. In (a), these side minima are close to the edges of the frequency scanning region. Individual frequency ratios for measurements of 75 Rb8+ with respect to 85 Rb9+ with a Ramsey excitation of 6-85-6 ms are shown in (b). The error band corresponds to the weighted average with correlations considered.  are indistinguishable from the main minima when the application time τ1 is much shorter than the waiting period τ0 . In order to determine the position of the main minimum which corresponded to νc of 75 Rb, a Ramsey scheme of 40-17-40 ms was employed in which the side maxima were much less pronounced [178]. An example of such a measurement run is shown in Figure 4.9a. Once the main minimum was unambiguously identified, the data taking was continued with the more precise 6-85-6 ms excitation scheme. A count-class analysis of νc of 75 Rb8+ did not reveal frequency shifts due to potential contamination of 75 Ga8+ (see Table 3.1). Assuming a linear dependence of νc on the number of detected ions, the weighted average of the slope was k = 0.01 ± 0.03 Hz/(# detected ions), consistent with zero. The individual frequency ratios of the75 Rb data set are displayed in Figure 4.9b. Table 4.2 contrasts the averaged frequency ratio with and without correlations due to shared reference measurements. The central value was relatively insensitive to the correlations, but the uncertainty was increased by ≈ 12%.  142  Table 4.2: W