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Decay spectroscopy of neutron-rich cadmium around the N = 82 shell closure Bernier, Nikita 2018

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Decay Spectroscopyof Neutron-Rich CadmiumAround the N = 82 Shell ClosurebyNikita BernierB.Sc., Universite´ Laval, 2011M.Sc., Universite´ Laval, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2018c© Nikita Bernier 2018The following individuals certify that they have read, and recommendto the Faculty of Graduate and Postdoctoral Studies for acceptance, thedissertation entitled:Decay Spectroscopy of Neutron-Rich CadmiumAround the N = 82 Shell Closuresubmitted by Nikita Bernier in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in Physics.Examining Committee:Dr Reiner Kru¨cken, PhysicsSupervisorDr Colin Gay, PhysicsSupervisory Committee MemberDr Janis McKenna, PhysicsUniversity ExaminerDr Chris Orvig, ChemistryUniversity ExaminerAdditional Supervisory Committee Members:Dr Sonia Bacca, PhysicsSupervisory Committee MemberDr Robert Kiefl, PhysicsSupervisory Committee MemberiiAbstractThe neutron-rich cadmium isotopes (Z = 49) near the well-known magicnumbers at Z = 50 and N = 82 are prime candidates to study the evolvingshell structure observed in exotic nuclei. Additionally, nuclei around thedoubly-magic 132Sn have been demonstrated to have direct implications forastrophysical models, leading to the r-process abundance peak at A ≈ 130and the corresponding waiting-point nuclei around N = 82. The β-decay ofthe N = 82 isotope 130Cd into 130In was investigated in 2002 [1], but theinformation for states of the lighter indium isotope 128In is still limited.Detailed β-γ-spectroscopy of 128,131,132Cd was accomplished using theGRIFFIN [2] facility at TRIUMF. In 128In, 32 new transitions and 11 newstates have been observed in addition to the four previously observed excitedstates [3]. The 128Cd half-life has also been remeasured via the time distribu-tion of the strongest γ-rays in the decay scheme with a higher precision [4].For the decay of 131,132Cd, results are compared with the recent EURICAdata [5, 6]. These new results are compared with recent shell model andIMSRG [7, 8, 9] calculations, which highlight the necessity to re-investigateeven “well-known” decay schemes for missing transitions.iiiLay SummaryThe discovery of radioactivity (1896) and the atomic nucleus (1911) are fairlyrecent in the history of mankind, but our understanding of the nucleus hasadvanced rapidly through numerous experiments. The Earth and its inhabi-tants are composed of various elements, such as gold and uranium, which arenot produced in our Solar system but in massive stars and are transferredinto the Solar System via the Interstellar Medium. Thus, every atom aroundus is made of previous stardust. Such radioactive nuclei could not be studieduntil we produced them with particle accelerators. These new experimentspush the limits of our theories on the nuclear structure: how neutrons andprotons work together to make up matter. Nuclear astrophysics works onexplaining how these elements are created in stars. This work highlights re-sults from experiments at TRIUMF with radioactive cadmium nuclei, whichbring important information on the structure of neutron-rich nuclei.ivPrefaceChapter 5 is based on work conducted at the TRIUMF laboratory under thesupervision of Professor Reiner Kru¨cken [TRIUMF/UBC] and Dr Iris Dill-mann [TRIUMF/University of Victoria, Canada]. I was responsible for theanalysis of the data sets for the β-decay of 128,131,132Cd collected in August2015. The data files were sorted using the analysis framework GRSISort[10], a code written in ROOT [11]. The figures of level schemes for this the-sis have been created using the SciDraw scientific figure preparation system[12].Chapter 6 is based on work conducted at the TRIUMF laboratory underthe supervision Professor Reiner Kru¨cken and Dr Jason Holt [TRIUMF].I was responsible for running the NuShellX@MSU [13] code provided byProfessor Alex Brown [Michigan State University/National Superconduct-ing Cyclotron Laboratory, USA] for 128,131In. I was also responsible forrunning the NuShellX@MSU code with the IMSRG interaction provided byDr Jason Holt for 127,128,129,130,131Sn, 127,128,129,131In, 125,126,127,128,129Cd and125,127Ag. Finally, I was responsible for running the effective single parti-cle energy (ESPE) code provided by Dr Jason Holt using the calculationspreviously mentioned.Section 5.1.5 presents the analysis of the half-life of 128Cd, which wasindependently extracted from the same data set by Ryan Dunlop [Universityof Guelph, Canada] and published in 2016 [4]. A manuscript describingthe current work on the nuclear structure of 128In is in preparation forsubmission to Physical Review C.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Symbols and Acronyms . . . . . . . . . . . . . . . . . . . xiiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Motivation and Theory . . . . . . . . . . . . . . . . . . . . . . 32.1 Nuclear Structure . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 Non-Interacting Shell Model . . . . . . . . . . . . . . 42.1.2 Interacting Shell Model . . . . . . . . . . . . . . . . . 82.1.3 Recent Developments . . . . . . . . . . . . . . . . . . 92.2 Nuclear Astrophysics . . . . . . . . . . . . . . . . . . . . . . 92.3 Nuclear Decay . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Decay Law . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Beta Decay . . . . . . . . . . . . . . . . . . . . . . . . 142.3.3 Gamma Decay . . . . . . . . . . . . . . . . . . . . . . 19vi3 Review of Literature . . . . . . . . . . . . . . . . . . . . . . . . 243.1 128Cd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 131Cd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 132Cd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.1 Beam Production . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . 415 Data Analysis and Results . . . . . . . . . . . . . . . . . . . . 475.1 128Cd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.1.1 β-Gated γ-Singles Measurements . . . . . . . . . . . . 475.1.2 β-Gated γ-γ Coincidence Measurements . . . . . . . 525.1.3 Decay Scheme . . . . . . . . . . . . . . . . . . . . . . 605.1.4 Spin Assignments . . . . . . . . . . . . . . . . . . . . 675.1.5 Half-Life . . . . . . . . . . . . . . . . . . . . . . . . . 705.1.6 248-keV Isomer . . . . . . . . . . . . . . . . . . . . . 725.2 131Cd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2.1 β-Gated γ-Singles Measurements . . . . . . . . . . . . 775.2.2 β-Gated γ-γ Coincidence Measurements . . . . . . . 825.2.3 Decay Scheme . . . . . . . . . . . . . . . . . . . . . . 855.3 132Cd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.3.1 β-Gated γ-Singles Measurements . . . . . . . . . . . . 945.3.2 β-Gated γ-γ Coincidence Measurements . . . . . . . 976 Shell Model Calculations . . . . . . . . . . . . . . . . . . . . . 1016.1 128In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.1.1 Level Energies . . . . . . . . . . . . . . . . . . . . . . 1026.1.2 Configurations . . . . . . . . . . . . . . . . . . . . . . 1046.1.3 Effective Single-Particle Energies . . . . . . . . . . . . 1106.2 131In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.2.1 Level Energies . . . . . . . . . . . . . . . . . . . . . . 1126.2.2 Configurations . . . . . . . . . . . . . . . . . . . . . . 1127 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . 114Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117viiAppendicesA Data Calibration and Processing . . . . . . . . . . . . . . . . 123B Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126viiiList of Tables2.1 Selection rules for β-decay angular momentum and parity . . 182.2 Selection rules for γ-decay angular momentum and parity . . 215.1 γ-ray energies in 128In, their intensities relative to Iγ(247.96)= 100 % and the initial energy levels are compared to previouswork [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2 Level energies in 128In, their β-feeding intensities per 100 de-cays and the log(ft) values . . . . . . . . . . . . . . . . . . . 685.3 γ-ray energies in 131In, their intensities relative to Iγ(988) =100 %, absolute intensities per 100 decays, and the initialenergy levels are compared to previous work Ref. [6]. . . . . 915.4 Level energies in 131In, their β-feeding intensities per 100 de-cays and the log(ft) values . . . . . . . . . . . . . . . . . . . 936.1 Single-Particle Energies for the jj45pn model space . . . . . . 1026.2 Comparison of proton-neutron coupling configurations in 128In 1056.3 Orbitals occupancy and configuration in 131In . . . . . . . . . 113ixList of Figures2.1 Nuclear shell structure with various potentials . . . . . . . . . 62.2 Proton (pi) and neutron (ν) valence orbitals for 128In (Z = 49,N = 79) and single-particle energies (SPE) [in MeV] . . . . . 72.3 Nuclide chart with one potential rapid neutron capture (r-)process path and r-process solar abundances . . . . . . . . . . 122.4 N = 82 region of the nuclide chart close to Z = 50 . . . . . . 132.5 Number of β-decays as a function of time for 128Cd and 128In 152.6 β-decay and β-delayed neutron decay processes . . . . . . . . 172.7 Examples of γ-γ angular correlations . . . . . . . . . . . . . . 223.1 Published decay schemes of 128Cd . . . . . . . . . . . . . . . . 263.2 Evolution of the ground state, first 1+ and isomeric state(s)in even-mass 122−130In . . . . . . . . . . . . . . . . . . . . . . 273.3 Published decay schemes of 131Cd . . . . . . . . . . . . . . . . 293.4 Evolution of the 1/2–9/2 states in odd-mass 123−131In . . . . 313.5 Single-particle orbitals in the 132Sn region [6] . . . . . . . . . 323.6 Published decay schemes of 132Cd. . . . . . . . . . . . . . . . 343.7 Tentative levels energies [in keV] for 132In . . . . . . . . . . . 354.1 TRIUMF ISAC experimental hall layout . . . . . . . . . . . . 374.2 Concept of the Ion Guide Laser Ion source (IG-LIS) . . . . . 384.3 124−130Cd yields at ISAC using the Ion Guide Laser Ion source 394.4 GRIFFIN γ-ray spectrometer . . . . . . . . . . . . . . . . . . 404.5 SCEPTAR scintillator array and moving-tape collector . . . . 404.6 Comparison of spectra observed for a 60Co source with andwithout crosstalk correction . . . . . . . . . . . . . . . . . . . 424.7 Comparison of clover addback [blue] and γ-singles [red] spec-tra observed for a 60Co source . . . . . . . . . . . . . . . . . . 444.8 Time difference between consecutive triggers as a function ofcrystal number for a 152Eu source . . . . . . . . . . . . . . . . 454.9 Absolute γ-ray detection efficiency for the GRIFFIN spec-trometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46x5.1 Difference between time stamps of β-particles and γ-rays . . . 485.2 Comparison of β-gated γ-singles [blue] and γ-singles [red]spectra observed for the decay of 128Cd . . . . . . . . . . . . 495.3 Comparison of β-gated γ-ray spectra observed for the decay of128Cd in addback mode with lasers on [blue] and laser blocked[red] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.4 Number of β-particles as a function of cycle time for the β-decay of 128Cd in (a) laser-on mode and (b) laser-blocked mode 535.5 Comparison of β-gated γ-ray spectra observed for the decayof 128Cd in addback mode as a function of cycle structure . . 545.6 Difference between the time stamp of a γ-ray coincident witha β-particle and the time stamp of a second γ-ray as a functionof the energy of the second γ-ray . . . . . . . . . . . . . . . . 555.7 Symmetrized β-gated γ-γ coincidence matrix for 128Cd data . 565.8 β-gated background-subtracted γ-gated spectra observed withthe GRIFFIN spectrometer for the β-decay of 128Cd: 68 keVand 173 keV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.9 β-gated background-subtracted γ-gated spectra observed withthe GRIFFIN spectrometer for the β-decay of 128Cd: 462 keVand 857 keV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.10 β-gated background-subtracted γ-gated spectra observed withthe GRIFFIN spectrometer for the β-decay of 128Cd: 408 keVand 336 keV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.11 β-gated background-subtracted γ-gated spectra observed withthe GRIFFIN spectrometer for the β-decay of 128Cd: 305 keVand 619 keV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.12 Energy levels [in keV] and γ-ray transitions in 128In followingthe β-decay of 128Cd . . . . . . . . . . . . . . . . . . . . . . . 635.13 Coefficients and mixing ratios of γ-γ angular correlations . . 715.14 Normalized γ-γ angular correlation data and fit for the 857-68keV cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.15 Fitted activity of selected γ-rays in 128In . . . . . . . . . . . . 735.16 Fitted activity of the sum of the 857- and 925-keV γ-rays in128In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.17 Effect of changing the fitting region on the extracted 128Cdhalf-life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.18 Difference between time stamps of β-particles and γ-rays(zoom in) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.19 Number of β-particles as a function of cycle time for the β-decay of 131Cd in (a) laser-on mode and (b) laser-blocked mode 78xi5.20 Comparison of β-gated γ-ray spectra observed for the decayof 131Cd in addback mode: lasers on [blue] and laser blocked[red] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.21 β-gated γ-ray spectra observed for the decay of 131Cd in ad-dback mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.22 β-gated γ-ray spectra around peaks with multiplet structuresin the decay of 131Cd . . . . . . . . . . . . . . . . . . . . . . . 815.23 β-gated γ-ray spectra around possible transitions in 130Infrom the βn-decay of 131Cd . . . . . . . . . . . . . . . . . . . 835.24 Partial decay scheme for the β-decay of 130Cd . . . . . . . . . 845.25 Symmetrized β-gated γ-γ coincidence matrix for 131Cd data . 865.26 Symmetrized β-gated γ-γ coincidence matrix for the 988-keVtransition 131Cd . . . . . . . . . . . . . . . . . . . . . . . . . . 875.27 β-gated background-subtracted γ-gated spectra observed withthe GRIFFIN spectrometer for the β-decay of 131Cd . . . . . 895.28 Energy levels [in keV] and γ-ray transitions in 131In followingthe β-decay of 131In . . . . . . . . . . . . . . . . . . . . . . . 905.29 β-gated γ-ray spectrum observed for the decay of 132Cd inaddback mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.30 Comparison of 131Cd and 132Cd data sets around 988 keV . . 965.31 Partial decay scheme for the β-decay of 132Sb . . . . . . . . . 975.32 Comparison of the activity of selected γ-rays in the 131Cd and132Cd data sets . . . . . . . . . . . . . . . . . . . . . . . . . . 985.33 β-γ-γ coincidence matrix for A = 132 data . . . . . . . . . . 1006.1 Comparison of excitation energies [in keV] in 128In . . . . . . 1036.2 Single-particle energies (SPEs) in the jj45pn model space, andeffective single-particle energies (ESPEs) [in MeV] for the foureven Z, even N neighbouring isotopes of 12849 In79 . . . . . . . . 1116.3 Comparison of excitation energies [in keV] in 121In . . . . . . 113xiiList of Symbols andAcronymsA Total nucleon (mass) numberα Internal conversion coefficient (ICC)Bn Binding energy of an electron in the n-shellBRγ Branching ratio of a γ-ray transitionδ Mixing ratioE Energyγ Detection efficiency of a γ-ray transitionH HamiltonianH0 Non-interacting HamiltonianIγ Absolute/Relative intensity of a γ-ray transitionj Total angular momentum of a nucleonJ Nuclear spin (Total angular momentum of a nucleus)K Kinetic energyl Orbital angular momentumL Total orbital angular momentumλ Decay constantm Mass of a nucleonme Electron massmu Atomic mass unitM Mass of the Sunµ(r) Mean-field potentialn Radial quantum numberN Neutron numbern(A,Z) Abundance of element A,Znn Neutron densityν Neutronpˆ Momentum operatorpi ProtonΠ ParityxiiiΨ, ψ EigenstatesQβ Q-value of the β-decayr Position of a nucleons Spin of a nucleonS Total spin of a systemSn Neutron separation energyS2n Two-neutron separation energyS2p Two-proton separation energyT Temperaturet1/2 Half-lifeτ LifetimeV (r) Potential energyVso(r) Spin-orbit potentialWRES Residual interactionZ Proton (atomic) number3N Three-nucleon forcesβn β-delayed neutron emissionDAQ Data acquisition systemEFT Effective field theoryESPE Effective single particle energy (J)FWHM Full width at half maximumGRIFFIN Gamma-Ray Infrastructure For FundamentalInvestigations of NucleiHPGe High purity germaniumIG-LIS Ion Guide Laser Ion sourceIMSRG In-Medium Similarity Renormalization GroupISAC Isotope Separator and ACceleratorISOL Isotope separation on-lineMIDAS Maximum Integrated Data Acquisition SystemNN Two-nucleon forcespp chain Proton-proton chainQCD Quantum chromodynamicsr-process Rapid neutron capture processs-process Slow neutron capture processSCEPTAR SCintillating Electron-Positron Tagging ARraySPE Single particle energyTBMEs Two-body matrix elementsxivAcknowledgementsMy sincere thanks toReiner Kru¨cken, for his precise explanations and open-mindednessIris Dillmann, for her alternate perspective and advicePeter C. Bender, for his invaluable support and understandingShaun Georges, for his dedication and care of the gamma-ray groupGordon Ball, for his irreplaceable experience and discussionsJason Holt, for his patient teachings to an experimentalistJens Lassen, for his continued support since my first day at TRIUMFSonia Bacca, for her continued support since my first course at UBCTammy Zidar, for her life-changing points of viewAurelia Laxdal, for her timeless loveEvidently, the support ofxvPour l’exercice d’urgence nucle´aire qui a re´veille´ mes parentssix heures avant la naissance de leur premie`re fille.Il n’y a pas eu d’autre exercice depuis.For the nuclear emergency drill which woke my parents upsix hours before the birth of their first daughter.There has not been another drill since.xviChapter 1IntroductionThe discovery of neutrons in nuclei (1932, J. Chadwick [15]), following thediscovery of nuclei in atoms (1911, E. Rutherford [16]), prompted the ad-dition of a new fundamental force. The observation of a bound systemcomposed of only neutral and positively charged particles revealed a forcestronger than the well-known electromagnetic force, which was named thestrong nuclear force. As we know it today, this nuclear force is only a residualforce that is felt outside of the nucleons from the interaction of the quarksinside the nucleons, similarly to the van der Waals interaction between neu-tral molecules. This fundamental strong interaction is governed by quantumchromodynamics (QCD), which provides the basis of modern nuclear forcemodeling between two and three nucleons that are then used to describemore complex nuclei. To this day, the understanding of the nuclear forcefrom first principles is still limited and nuclear theory struggles to accuratelypredict properties of heavy nuclei.Nuclear astrophysicists have been theorizing about locations where theheavy elements could be created, with such candidates including core-collapse supernovae, neutron-star mergers and certain burning phases oflow-mass (1-3 solar masses, M) and massive (M > 8 M) stars. The Suncan fuse protons into heavier atoms up to 16O [17] (1967 Nobel Prize inPhysics, H.A. Bethe). Massive stars can ignite further advanced burningphases and create nuclei up to 56Ni (Z = N = 28) which decays to 56Fe.Yet, the Earth and its inhabitants are composed of various elements up to238U (Z = 92, N = 146). Three different processes in different astrophysicalscenarios were described in 1957 [18] (1983 Nobel Prize in Physics, S. Chan-drasekhar and W.A. Fowler): the rapid neutron capture (r-) process, theslow neutron capture (s-) process, and several production mechanisms forproton-rich (neutron-deficient) nuclei summarized as p-process. Each one ofthe first two processes accounts for ∼50% of the nucleosynthesis of elementsheavier than 56Fe, while the p-processes contribute ∼1%. Modern astrophys-ical simulations calculate the probable chains of nuclear reactions and theircorresponding isotopic abundances for the different production mechanisms,which are then compared to the observed abundances in the Solar System.1The neutron-rich region of the nuclide chart around A = 132 is of spe-cial interest to both nuclear structure and nuclear astrophysics. For nu-clear structure studies, the neighbours of the doubly-magic 132Sn (Z = 50,N = 82) are an ideal test ground for nuclear structure theories, such as thefamous nuclear shell model (1963 Nobel Prize in Physics, E.P. Wigner, M.G.Mayer and J.H.D. Jensen). From an astrophysics perspective, this region isconnected to the waiting-point nuclei around N = 82, at which the r-processmaterial within an isotopic chain accumulates and transfers the material tothe next isotopic chain, and the corresponding abundance peak at A ' 130.Together, the shell structure and half-lives far off stability provide criticalinformation on the position and the shape of the abundance peaks for ther-process.The decay of the N = 82 isotope 130Cd into 130In was investigated15 years ago [1], but puzzling questions remained open. The informationfor the decay of the lighter, less exotic Cadmium isotopes 128,129Cd wasalso very limited. For the β-decay of 128Cd (t1/2 = 246.2(21) ms [4]), onlyseven transitions were published [3] and the last known level in 128In at1173 keV is still 4146 keV away from the neutron separation energy (Sn) at5321(155) keV [19].For the N = 82 isotope 131In, the EURICA collaboration at RIKEN hasrecently published results for the proton (pi) hole states in 131In from theβ-decay of 131Cd [6] and the observation of the βn-decay of 132Cd [5].This thesis presents the detailed γ-ray spectroscopy of the β-decay of128,131,132Cd using laser ionization and moving tape cycle methods with theGRIFFIN spectrometer at TRIUMF. The relevant nuclear theory and mo-tivation are presented in Chapter 2, while Chapter 3 details the previouspublished works on the decay of 128,131,132Cd isotopes. The beam produc-tion techniques and the detectors of β- and γ-radiation used are described inChapter 4. Chapter 5 presents the data analysis based on coincidence anal-ysis and angular correlations. These new results are then compared withshell model calculations in Chapter 6.2Chapter 2Motivation and Theory2.1 Nuclear StructureSeveral pieces of experimental evidence observed in the early 20th centuryexhibit a shell structure for neutrons and protons inside nuclei. For exam-ple, the two-proton separation energy (S2p) and the two-neutron separationenergy (S2n), which are the energies required to remove two protons andneutrons from a nucleus, respectively, show sharp decreases just above spe-cific number of protons and neutrons. Moreover, the numbers at whichthe discontinuities occur are the same for protons and neutrons: 2, 20, 28,50, 82, and 126. An additional piece of evidence is seen with the nuclearcharge radius, for which a sharp increase is noticed at the same numbers.These three observations suggest an increased binding of the nucleus com-ponents for particular numbers that were rightly named “magic”, which arereminiscent of the structure of electron shells around the atomic nucleus. Anucleus with both its proton and neutron shells exactly full (closed) is calleddoubly-magic, such as 132Sn (Z = 50, N = 82).Quantum mechanically, the shell structure at the atomic and nuclearscales is described by the eigenstates Ψ which solve the Schro¨dinger equation:HΨ = EΨ. (2.1)The Hamiltonian H has the form:H = K + V =A∑i=1pˆ2i2mi+A∑i 6=kV (~ri,k) , (2.2)where K is the kinetic energy, A is the mass number, pˆ and m are the mo-mentum operator (pˆ = −i~∇) and the mass of a nucleon, respectively, andV (~ri,k) is the nucleon-nucleon interaction as a function of the coordinatesof the i-th and k-th particles. The nuclear potential V (~ri,k) describes howprotons and neutrons interact if we neglect three-body and higher orderforces. This potential is ultimately related to the interaction of the quarksinside the nucleons. The simple existence of a bound system of neutrons3and protons reveals an attractive component of the nuclear force strongerthan the repulsion of the electromagnetic force, at least at the nuclear scale.2.1.1 Non-Interacting Shell ModelBy adding and subtracting a mean-field potential v (~ri) (omitting spin andisospin degrees of freedom), the Hamiltonian of Equation (2.2) can be ex-pressed by the sum of a non-interacting part H0 and a residual interactionWRES :H =[A∑i=1pˆ2i2m+A∑iv (~ri)]+ A∑i 6=kV (~ri,k)−A∑iv (~ri)= H0 +WRES ,(2.3)whereWRES = 0 for the non-interacting shell model (or independent particlemodel).An infinite square well potential is a reasonable first-order approxima-tion of the nuclear potential, as shown in Figure 2.1. A specific numberof protons/neutrons can occupy each level according to the Pauli exclusionprinciple for fermions before filling up the next level sequentially. Followingthe electron nomenclature, the orbital angular momentum l of a level definesits type, which is labelled on the left of each level in Figure 2.1: s (l = 0),p (l = 1), d (l = 2), f (l = 3), g (l = 4), etc. The number in front ofthe orbital angular momentum label simply indicates the major shell. Thenumber of nucleons allowed per level, or degeneracy, is 2(2l+ 1). The factorof (2l+1) arises from the ml degeneracy and the factor 2 comes from the msdegeneracy [20]. Groups of levels form shells which are separated by largeenergy gaps for some total numbers of nucleons. These gaps are called shellclosures and the infinite square well reproduces only the first three magicnumbers observed experimentally.The second potential, which already explains a large number of phenom-ena in various fields, is the harmonic oscillator. The potential has the formV (ri) = mω20r2i /2 with solutions En,l = ~ω0(2n + l − 1/2), where ~ is thereduced Planck’s constant, ω0 is the classical angular frequency of the oscilla-tor, and n = 0, 1, 2, 3... is the radial quantum number and l = 0, 1, 2, ..., n−1is the angular momentum. Here again, only the first three magic numbersare reproduced.Both these approximations have the considerable flaw of requiring aninfinite amount of energy to remove a nucleon from the potential. A realisticnuclear potential would include the flat bottom of the interior well, the4parabolic raise of the harmonic oscillator, and also a finite saturation at thenuclear scale.The third potential shown in Figure 2.1 is the Woods-Saxon. It effec-tively creates the three desired properties from above, and is given by:V (ri) =−V01 + exp [(ri −R)/a] , (2.4)where V0 is the well depth, R is the mean radius following R = 1.25A1/3 witha = 0.524 fm, the skin thickness of the nucleus. The Woods-Saxon potentialin a one-body problem can only be solved numerically, whereas the harmonicoscillator can be solved analytically. Only the first three magic numbers arereproduced by the Woods-Saxon potential. However, the addition of a spin-orbit component causes the levels to be reordered and all magic numbers tobe reproduced. The spin-orbit term was introduced by M. Goeppert-Mayer[21] and H. Jensen [22], who shared the 1963 Nobel Prize in Physics. Whilethe atomic spin-orbit interaction arises from the electron’s magnetic momentinteracting with the magnetic field generated by the motion of the electrons,the nuclear spin-orbit results from a force between the nucleons themselves[20]. The spin-orbit term is written as Vso(ri) ~l ·~s, where s is the spin of thenucleon (s = 1/2).The total angular momentum of a level (labelled as a subscript on theright) is given by ~j = ~l + ~s, such that j = l ± 1/2. The degeneracy ofeach level is (2j + 1) and its parity is Π = (−1)l. This energy splittingdoesn’t affect the magic numbers 2, 8 and 20, however it brings the 1f7/2level low enough to create a shell closure at 28. The 1g is split into 1g11/2 (12nucleons) and 1g9/2 (10 nucleons), adding 10 nucleons to the previous magicnumber of 40 to form a new one at 50 nucleons as observed empirically.In the independent particle shell model, only the unpaired nucleons con-tribute to the ground state properties of the nucleus. For a nucleus with aneven number of neutrons N and an even number of protons Z, all nucleonsare paired and therefore the ground state (the configuration with the lowestenergy) has a spin-parity of 0+. For an even Z-odd N or odd Z-even Nnucleus, the properties of the ground state are defined by the total angularmomentum j and parity (−1)l of the level of the unpaired proton or neutron.For an odd-odd nucleus, the coupling of the unpaired proton and neutrondetermines the possible spin-parity combinations for the ground state.For example, 128In is made of 49 protons (pi) and 79 neutrons (ν). Asshown in Figure 2.2, 128In is one pi-hole and 3ν-holes from the double shellclosure at Z = 50 and N = 82. According to the independent particle shellmodel, the single proton in pi2p1/2 would couple to the neutron in ν1g7/2 to5(a) Square well1s1p1d2s1f2p1g2d1h3s2f1i3p1j2g2820345892138(b) Harmonic oscillator1s1p2s,1d2p,1f3s,2d,1f3p,2f,1h4s,3d,2g,1i28204070112168(c) Woods-Saxon1s1p1d2s1f2p1g2d3s1h2f3p1i2g3d4s2820405892112(d) WS+spin-orbit1 s1/2 21 p3/2 41 p1/2 21 d5/2 62 s1/2 21 d3/2 41 f7/2 82 p3/2 41 f5/2 62 p1/2 21 g9/2 101 g7/2 82 d5/2 62 d3/2 43 s1/2 21 h11/2 121 h9/2 102 f7/2 82 f5/2 63 p3/2 43 p1/2 21 i13/2 142 g9/2 103 d5/2 61 i11/2 122 g7/2 84 s1/2 23 d3/2 41 j15/2 162820285082126184Figure 2.1: Nuclear shell structure with (a) infinite square well potential, (b)harmonic oscillator potential, (c) Woods-Saxon potential, and (d) Woods-Saxon potential with spin-orbit. (Adapted from [20].)6π1 f5/2 -0.71662p3/2 1.11842p1/2 1.12621g9/2 0.178528501g7/2 5.74022d5/2 2.44222d3/2 2.5148ν3s1/2 2.17381h11/2 2.67955082Figure 2.2: Proton (pi) and neutron (ν) valence orbitals for 128In (Z =49, N = 79) and single-particle energies (SPE) [in MeV] for the jj45pnainteraction in NuShellX.give a ground state with j = 7/2±1/2 = 3 or 4, and Π = (−1)1 ·(−1)4 = −1.In this case, the simple model prediction fails to reproduce the measured 3+ground state.While the relatively simple shell model described here works well to ex-plain magic numbers in stable nuclei and ground state properties observedin nuclei close to magic shell closures, contemporary experiments with ra-dioactive nuclei have produced new results which do not directly agree withthese simple nuclear theories. Since the effective potential resulting fromthe interaction between the nucleons (proton-proton, neutron-neutron andproton-neutron) is responsible for the energy of the levels, the number ofnucleons of the same type and the number of nucleons of the other typeboth have a critical impact on the shell evolution (as seen in Figure 2.1).Therefore, exotic nuclei with N  Z (proton rich) and N  Z (neutronrich) especially test our understanding of the nucleon-nucleon forces.This shell evolution is already seen in Figures 2.1 and 2.2, where theorbitals within the shells of interest are displayed in a different order. These7differences arise from the fact that large asymmetries in Z and N , such asin the region around Z = 50 and N = 82, produce a nuclear potential withlevels which have slightly different energies than stable nuclei. Figures 2.2shows the orbitals in order of their single-particle energies (SPE) as definedin the shell model calculation code NuShellX [13], which will be discussedin Chapter 6.2.1.2 Interacting Shell ModelThe interacting shell model considers the residual interaction (WRES 6= 0).Calculations for the interacting shell model divide the proton and neutronorbitals in three spaces: a non-interacting core, a valence space and anexternal space. For calculations in the model space between 28 < Z ≤ 50and 50 < N ≤ 82, the four pi- and five ν-orbitals orbitals shown in Figure 2.2are included in the valence space and contribute to the mean-field potential.While the lower closed shells form an inert 78Ni core, the empty orbitalsabove Z > 50 and N > 82 form the external space, which is always empty.The single-particle energies (SPEs) of the valence orbitals are a result of amean field calculation. They can be derived from many-body perturbationtheory or by phenomenologically fitting matrix elements to experimentaldata from nuclei in the region [23]. Including these orbitals as part of thevalence space means that the Hamiltonian takes into account the interactionsof the nucleons in these orbitals with all other nucleons in the valence spaceorbitals via the respective two-body matrix elements (TBMEs). The shellmodel code diagonalizes the Hamiltonian matrix that is set up by the SPEsand TBMEs.When the effective Hamiltonian is applied, the orbital SPEs shift toeffective single-particle energies J (ESPE). The ESPE of an occupied orbitis calculated by taking the average of the one-nucleon separation energiesweighted by the probability to reach the corresponding A± 1 eigenstates byadding/removing a nucleon to/from a single-particle state ΨJ :J =∑iS+i E+i +∑kS−k E−k , (2.5)where S± are the spectroscopic probabilities of the single-particle state ofenergy E± for the A±1 neighbouring isotopes [24]. These calculated ESPEsevolve as a function of the spectroscopic factors, which characterize theoccupation of the levels.82.1.3 Recent DevelopmentsThe most recent development for the derivation of realistic interactions is theconnection of nuclear forces to the fundamental theory of quantum chromo-dynamics (QCD) using effective field theories (EFT). The latter preserves allthe symmetries of the underlying fundamental strong interaction, but useseffective degrees of freedom such as neutrons, protons and pions. Thesetheories, such as the chiral EFT [25, 26, 27], focus on the characterizationof two-nucleon (NN) and three-nucleon (3N) forces with an expansion toseveral leading orders.Ideally, calculations would take root in physics first principles withoutapproximations. Those forces have been used in ab initio calculations ofnuclear properties for light nuclei and near closed shells. Also, they havelead to the development of interactions that can be used in large-scale shellmodel calculations, such as the ones used in this work. The advanced com-putational techniques which are required to find solutions to these quan-tum problems represent one of the main limitations of the theory and alsoprogress quickly with modern technologies [28].Experiments on nuclei at and around shell closures provide empiricalparameters which are indispensable to test and benchmark nuclear structuretheories. Data on the odd-odd 128In and 132In, and the odd-even 131In canprovide information on the proton-hole structure along the Z = 49 isotopicchain. The configurations of key states and the size of shell gaps are examplesof fundamental information required to develop and understand two-bodymatrix elements of the effective interaction.2.2 Nuclear AstrophysicsThe Big Bang nucleosynthesis predominantly produced hydrogen, helium,deuterium, and small amounts of lithium. Beryllium and boron can be pro-duced by galactic cosmic-ray spallation, which is the bombardment of pre-existing matter by high-energy cosmic-ray particles. Two hydrogen fusioncycles in stars can fuse protons into 4He: the proton-proton chain reactions(pp chains), and the carbon-nitrogen-oxygen (CNO) cycle. In the followinghelium burning phase, two 4He nuclei are fused in a first step to unstable8Be, and then further to 12C due to the high density of 4He and the ex-istence of a resonant state (“Hoyle state”), and then partially by anotherα-particle capture to 16O. Stars more than eight times the mass of the Sun gothrough further advanced nuclear burning phases, such as oxygen, carbon,neon and silicon burning, of which the ashes can reach up to 56Fe (Z = 26,9N = 30). Finally, heavy elements beyond iron cannot be created via fusionreactions but only via neutron-capture reactions: the rapid neutron capture(r-) process and the slow neutron capture (s-) process [18].About half of the nuclei heavier than iron are produced by the r-process[29, 30]. The path of the r-process across the nuclide chart is formed by“waiting-point” nuclei, which are defined in two different cases. Withinan isotopic chain away from a neutron shell closure, the maximum of theabundance distribution occurs at one neutron number, which is considered awaiting-point nucleus. This local maximum is produced by the equilibriumbetween radiative neutron capture reactions (n,γ), where the nucleus absorbsa neutron and emits a high energy photon (γ-ray), and photodisintegrations(γ,n), where the nucleus absorbs a γ-ray and emits a neutron. Once β-decay occurs and the r-process material is transferred to the Z + 1 isotopicchain, multiple neutron captures and photodisintegrations happen until anew equilibrium distribution is established. Another β-decay to the nextisotopic chain then takes place, most likely from the waiting point withinthat isotopic chain.The main astrophysics parameters determining the reaction equilibriumfor each element are the neutron separation energy Sn, the environmenttemperature T and the density of neutrons nn. Therefore, there can beseveral waiting-point nuclei within an isotopic chain, for given T and nn, ifthe Sn-values are similar. The r-process abundances of element A,Z andelement A + 1, Z at nuclear statistical equilibrium are determined by thenuclear Saha equation for neutron capture [29]:n (A+ 1, Z)n (A,Z)= nn ·(A+ 1A· 2pi~2kBT ·mu)3/2· exp(−Sn (A+ 1, Z)kBT), (2.6)where kB is Boltzmann constant, and mu is the atomic mass unit. This pro-duction mechanism requires very neutron-rich environments. In explosivescenarios, typical neutron densities are higher than 1020 neutrons/cm3 andtemperatures are higher than 1 GK.The second, “classic” waiting-point nuclei are caused by neutron shellclosures and is the most relevant for the cadmium isotopes of interest in thiswork. At neutron shell closures, the energy required to capture an extraneutron above the neutron magic number rises and the neutron capturecross section drops. This leads to an accumulation of r-process material atthe neutron shell closures, and provides the required waiting time for theprobability of β-decay to increase and to overcome the waiting point. Oncethe β-decay and another neutron capture happen, the material is still at10the neutron shell closure and thus another classic waiting point is reached.The β-decays followed by neutron captures are repeated until the neutroncapture cross section rises due to lower Sn-values, and therefore the r-processfollows the neutron shell closure for several nuclei until it is able to breakout to more neutron-rich nuclei.Figure 2.3 shows a possible path of the r-process identified with the re-spective waiting-point nuclei (red boxes) across the nuclide chart (with Z onthe y-axis and N on the x-axis). Nuclei which were identified experimentallybut for which no physical properties like half-lives or masses are known areshown in grey; isotopes for which the half-life was measured are shown inblue. The r-process path follows the waiting-point nuclei, which accumulater-process material before β-decaying to stability (freeze-out). Since severalsuch nuclei are located at the neutron shell closures at N = 82 and 126, theshell closures directly translate to peaks at masses A ' 130 and 195 in thesolar r-abundance curve. The solar r-abundance (Nr) curve is deduced bysubtracting the well-known calculated s-process abundances (Ns) from theobserved solar abundances (N): Nr = N −Ns. For example, the N = 82isotope 130Cd is responsible for the peak maximum at A ' 130 and providescritical information on the second abundance peak of the r-process.Sensitivity studies of the r-process guide nuclear experiments by deter-mining the nuclear properties and the isotopes which have the most impacton the calculations [32, 33]. Experimental information for isotopes betweenreaction path and stability affect calculations of the element abundancesand astrophysical processes. First, the masses and the half-lives define theposition and shape of r-process reaction path. Second, the nuclear struc-ture far off stability defines the position of the abundance peaks. So far, noevidence for any deviation from the known shell structure like shell quench-ing has been observed for N = 50, 82, 126, and the robust location of thepeaks of the r-abundance curve also do not hint to any new phenomena.For waiting-point nuclei, the important nuclear-structure properties are thedecay half-life, the Qβ-value, the Sn-value, the excitation energies of thekey levels and their comparative half-lives (log(ft) values) [1]. Unique levelstructures, such as long-lived isomeric states, are characteristic to the decayof the isotopes toward stability. Finally, neutrons from β-delayed neutronemission (βn) or (γ,n) have a smoothing effect on the abundance distribu-tion.The region of the nuclide chart around Z = 50 and N = 82 has beeninvestigated extensively in the past decades. However, the detailed explo-ration of the region “south” and “south-east” of 132Sn (Z < 50, N > 82)requires the new generation of radioactive beam facilities and powerful γ-ray1115020025010−210−1100101Solar r abundancesN=184N=126N=82N=5028 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58606264666870727476788082848688 9092 94969810010210410610811011211411611812012212412612813013213413613814014214414614815015215415615816016216416616817017217417617818018218426283032343638404244464850525456586062646668707274767880828486889092949698100r−process waiting pointIdentifiedKnown half−lifeFigure 2.3: Nuclide chart with one potential rapid neutron capture (r-)process path and corresponding r-process solar abundances [31].spectrometers. Figure 2.4 highlights this area of the nuclear chart.2.3 Nuclear Decay2.3.1 Decay LawNuclear decay is a random process for which the decay probability withina time interval dt is constant. Therefore, the probability of observing nevents in a given dt follows a Poisson distribution with a standard deviationσ =√n.The number of radioactive nuclei dN to decay within a time interval dtis equal to the decay constant λ times the number of nuclei in the sampleN at time t:dN = −λN(t)dt. (2.7)The decay constant λ is simply ln (2)/t1/2, where t1/2 is the half-life ofthe isotope: the characteristic time for half the nuclei in a sample to decay.12Figure 2.4: N = 82 region of the nuclide chart close to Z = 50. (Adapted from ENSDF withupdated half-lives from [34])Alternatively, the lifetime of an excited state τ is defined as 1/λ, i.e. thetime for the number of nuclei in a sample to decay by a factor 1/e.The exponential law of radioactive decay is given by integrating Equa-tion (2.7):N(t) = N0e−λt = N(0)e− ln (2)·t/t1/2 . (2.8)When new nuclei N1 are being produced and added to the sample at arate R1, with the initial condition N1(0) = 0, the size of the sample growsas:dN1 = R1dt− λ1N1dt (2.9a)N1(t) =R1λ1(1− e−λ1t)(2.9b)A1(t) = λ1N1(t) = R1(1− e−λ1t). (2.9c)The activity of the sample as a function of time A1(t) is given by mul-tiplying N1(t) by the decay constant λ. For a production time t  t1/2,secular equilibrium (saturation) is reached, the exponential goes to zero and13A1(t) ∼= R1. When the beam of produced radioactive nuclei is turned offafter reaching saturation, the isotope decays according to:N1(t) =R1λ1e−λ1t (2.10a)A1(t) = λ1N1(t) = R1(t)e−λ1t. (2.10b)Exotic nuclei typically decay to daughter nuclei which have longer half-lives. For t1/2,daughter  t1/2,parent, the number of the first daughter nucleiN2(t) with decay constant λ2 and activity A2(t) = λ2N2(t) follows, whilethe beam is on:N2(t) =R1λ2(1− e−λ2t)+R1λ2 − λ1(e−λ2t − e−λ1t). (2.11)While the beam is off, the daughter isotope decays following:N2(t) =R1λ2 − λ1(e−λ1t − e−λ2t)+N2(0)e−λ2t. (2.12)Figure 2.5 illustrates the decay curves for 128Cd and its first daughter128In (Equations (2.9) to (2.12)).2.3.2 Beta DecayBeta (β-) decay is a type of nuclear decay involving three particles. Decayprocesses are energetically possible for positive Q-values, which are definedas the difference between the atomic masses of the parent and the daughteratoms.Neutron-deficient nuclei β+-decay by changing a proton p into a neutronn and emitting a positron e+ and an electron neutrino νe :p→ n+ e+ + νe, (2.13a)AZXN →AZ−1 X∗N+1 + e+ + νe, (2.13b)from which the Q+β -value follows:Q+β =[m(AZXN)−m (AZ−1X∗N+1)− 2me] c2, (2.13c)where m(AZXN)and m(AZ−1X∗N+1)are the neutral atomic masses of parentand daughter nuclei, respectively, me is the mass of the positron and c is the14Figure 2.5: Number of β-decays as a function of time for 128Cd (t1/2,parent =0.246 s) and 128In (t1/2,daughter = 0.840 s) for a beam intensity R1 =1000 pps, and N1(t = 0) = N2(t = 0) = 0. The beam is turned on fromt = 0 to 10 s.speed of light. The last equation highlights that β+ decay is only possibleif the atomic mass energy difference is greater than 2mec2 = 1.022 MeV.If E < 1.022 MeV, the predominant decay mechanism of neutron-deficient nuclei is via capture of an electron by the nucleus. The atomis left in an excited state (X ′) which will then emit characteristic X-rays.The electron-capture process also changes by Z − 1 and N + 1:p+ e− → n+ νe, (2.14a)AZXN + e− →AZ−1 X ′N+1 + νe, (2.14b)andQ =[m(AZXN)−m (AZ−1X ′N+1)] c2 −Bn, (2.14c)where Bn is the binding energy of the captured n-shell (n = K, L, ...)electron.15Neutron-rich nuclei β−-decay by changing a neutron into a proton andemitting an electron and an electron antineutrino:n→ p+ e− + ν¯e, (2.15a)AZXN →AZ+1 X∗N−1 + e− + ν¯e, (2.15b)andQ−β =[m(AZXN)−m (AZ+1X∗N−1)] c2. (2.15c)Energy Release in β-decayThe maximum energy Ee that the electron can carry away is called the Qβ-value, which is the difference between the atomic masses of the parent andthe daughter atoms. In Figure 2.6, the Qβ-value represents the differencebetween the ground state energies of the parent and the daughter nuclei.When the electron is released Ee < Qβ, the β-decay typically also populatesexcited states in the daughter nucleus. These excited states, which aremarked by ∗ in Equations (2.13) and (2.15), then emit the extra energy asgamma-radiation in order to reach the ground state of the daughter nucleusor a β-decaying isomeric state. This process is called gamma (γ-) decay andis discussed in the next section.If the Qβ-value is larger than the Sn-value of the daughter, the β-decaycan populate states above the Sn-value which will decay either by emission ofa neutron or γ-decay. Therefore, the final nucleus from a β-delayed neutronemission (βn) has Z + 1, N − 2 and A − 1. The probability of a βn-decaydepends on the difference between the Qβ-value of the parent and the Sn ofthe daughter.Selection rulesThe orbital angular momentum l of the emitted electron defines the typeof β− decay transition, or the degree of forbiddenness, which can either besuperallowed or allowed (l = 0), first forbidden (l = 1), second forbidden(l = 2), etc. The parity of the transition follows (−1)l. Forbidden transitionsare suppressed in decay rate compared to allowed transitions [36].The change in angular momentum between the parent nucleus (Ji) andthe state populated in the daughter (Jf ) is defined by ∆J = |Jf−Ji| = L+S,where S is the vector sum of the electron and neutrino intrinsic spins (s =16Figure 2.6: β-decay and β-delayed neutron decay processes [35].1/2 for both). For parallel spins (S = 1), the decay is called Gamow-Teller(GT). For antiparallel spins (S = 0), the decay is called Fermi (F).These selection rules are summarized in Table 2.1. Parentheses indicate a∆J which is not possible for some exceptions, such as 0→ 0 and 1/2→ 1/2.Superallowed 0+ → 0+ are pure Fermi type, because Gamow-Teller transi-tions must carry one unit of angular momentum. It follows that transitionsfor which ∆J = L + 1 are only possible via Gamow-Teller transitions andare called “unique”.In addition, Fermi decays populate isobaric analog states, which meansthat a neutron from a neutron orbital decays to a proton in a proton orbitalwith the same l and J . Since most very exotic nuclei do not have neutronand proton valence orbitals in the same shells, Fermi decays are not possibleand only Gamow-Teller decays are expected.For example, considering the 0+ ground state of the even-even 128Cdparent nucleus, it follows from the selection rules that its β-decay only feedslow-spin states in the daughter nucleus 128In. Isobaric analog states donot exist in this very exotic isotope (48 protons and 80 neutrons) and onlyGamow-Teller decays are expected. Therefore, allowed decays populate 1+states, and first forbidden decays populate 0−, 1− or 2− states.17Table 2.1: Selection rules for β-decay angular momentum and parity. Note:Parentheses indicate transitions which are not possible if either Ji or Jf is0.Fermi Gamow-TellerTransition Type l ∆Π ∆J ∆J log(ft)Superallowed 0 No 0 – ∼3.5Allowed 0 No 0 (0),1 ∼4.0–7.5First Forbidden 1 Yes (0),1 0,1,2 ∼6.0–9.0Second Forbidden 2 No (1),2 2,3 ∼10–13Third Forbidden 3 Yes (2),3 3,4 ∼14–20Fourth Forbidden 4 No (3),4 4,5 ∼23Comparative Half-LivesThe explicit link between the experimental data and the quantum mechan-ical description of the decays resides in the comparative half-lives (log(ft)).That is, the transition matrix elements between initial and final states ψi andψf and the transition operators τ , στ can be extracted from measurementsof the Qβ-value and the half-life t1/2 of the parent nucleus, and the intensityof the beta-feeding to a state Ef in the daughter as shown in Equation (2.16).B(F) and B(GT) are the strengths of the Fermi and Gamow-Teller transi-tions; gV is the vector coupling constant and gA is the axial-vector couplingconstant.ft =f(Qβ − Ef , Z)t1/2Iβ(Ef )=2 ln 2pi3~7m5ec41g2VB(F) + g2AB(GT)(2.16a)whereB(F) = |〈ψ∗f |τ |ψi〉|2, (2.16b)B(GT) = |〈ψ∗f |στ |ψi〉|2. (2.16c)The log(ft) values allow an identification of possible types of transitionand a comparison of β-decay probabilities in different nuclei. Typical rangesof log(ft) values for known decays are listed in Table 2.1.182.3.3 Gamma DecaySimilar to X-rays being emitted as electrons move to lower-energy orbitals,nuclear excited states (Ei) transition to lower-energy excited states (Ef )by releasing energy as γ-radiation, such that Eγ = Ei − Ef . Successiveγ-ray emissions happen as a cascade until the ground state or a β-decayingisomeric state is reached. The γ-rays carry crucial information on the changein energy, angular momentum and parity between the states.γ-decay typically happens on a much shorter time scale than β-decay:on the order 10−9−10−15 s. However, particular states called isomeric existwith half-lives longer than 10−8 s. Isomeric transitions can have a big impacton nuclear astrophysics simulations as they halt the r-process flow back tostability because of their much longer half-lives. In addition, the suppressedtransition probability of isomeric states provides important information onthe nuclear configuration of these states.Electromagnetic RadiationFrom classical electrodynamics, the energy emitted per unit time by a dipoleradiation field expends to higher multipole orders with:P (σL) =2(L+ 1)c0L [(2L+ 1)!!]2(ωc)2L+1[m(σL)]2 , (2.17)where σ = E or M for electric or magnetic radiation, the multipole orderis defined as 2L (L = 1 for dipole, L = 2 for quadrupole, etc.), ω is theangular frequency, and m(σL) is the amplitude of the time-varying electricor magnetic multipole moment.In quantum mechanics, the probability of photon emission per unit timeis obtained by integrating the average radiated power over the volume of thenucleus:λ(EL) ' 8pi(L+ 1)L [(2L+ 1)!!]2e24pi0~c(E~c)2L+1( 3L+ 3)2cR2L (2.18a)for electric transitions, andλ(ML) ' 8pi(L+ 1)L [(2L+ 1)!!]2(µp − 1L+ 1)2( ~mpc)2( e24pi0~c)×(E~c)2L+1( 3L+ 2)2cR2L−2(2.18b)19for magnetic transitions, where E is the transition energy, R is the nuclearradius, and µp and mp are the proton magnetic dipole moment and mass,respectively. The (e2/4pi0~c) dimensionless factor is the fine structure con-stant ('1/137).The Weisskopf estimates of the transition probability for the lower mul-tipoles are obtained by replacing R = R0A1/3 and [µp − 1/(L+ 1)]2 = 10:λ(E1) ' 1.0× 1014A2/3E3λ(E2) ' 7.3× 107A4/3E5λ(E3) ' 34A2E7λ(E4) ' 1.1× 10−5A8/3E9(2.19a)λ(M1) ' 5.6× 1013E3λ(M2) ' 3.5× 107A2/3E5λ(M3) ' 16A4/3E7λ(M4) ' 4.5× 10−6A2E9,(2.19b)where λ is in s−1 and E is in MeV [20].It is obvious from the Weisskopf estimates that the lower multipolaritiesare dominant by orders of magnitude. In a decay experiment, for example,E1, M1 and E2 transitions are the most likely to be detected, as well asisomeric transitions with multipolarities M2 and E3. Also, in medium andheavy nuclei, electric radiation is more likely than magnetic radiation for agiven multipole order.The unobserved transitions contribute to the Pandemonium effect whichaffects the uncertainty of the measured transitions intensities [37, 38]. There-fore, experimental knowledge of higher energy excited states, which de-exciteto lower states with both low and high multipole orders, can reduce the im-pact of the Pandemonium effect.Selection rulesTransitions between excited states Ji and Jf are characterized by the changein total angular momentum |Ji−Jf | 6 L 6 Ji+Jf , where L gives the order ofthe multipole. Electric and magnetic multipoles of same order have oppositeparity, such that Π(EL) = (−1)L and Π(ML) = (−1)(L+1). These selectionrules are summarized in Table 2.2. Since a photon carries a minimum of20L = 1 and ~Ji = ~L+ ~Jf , it is important to note that a 0→ 0 transition cannotdecay through γ-decay, only via emission of internal conversion electrons.Table 2.2: Selection rules for γ-decay angular momentum and parity. Note:A 0 → 0 transition cannot decay by γ-decay, only via emission of internalconversion electrons.L 0 1 2 3 4 5ΠiΠf = −1 E1 E1 M2 E3 M4 E5ΠiΠf = +1 M1 M1 E2 M3 E4 M5Angular correlationsA state of given angular momentum l is also characterized by its magneticsubstate m, which runs from -l to l. For a cascade of two γ-ray transitionsdetected in coincidence by two different detectors, an uneven population ofm-substates is observed for the distribution of one γ-ray with respect to thedirection of the other γ-ray [20]. The probability of observing one γ-ray asa function of the angle to the direction of the other γ-ray, is given by:W (θ) = A0 + a2P2(cos θ) + a4P4(cos θ)= A0 [1 + a22P2(cos θ) + a44P4(cos θ)] ,(2.20a)whereP2(cos θ) =12(3 cos2 θ − 1) , (2.20b)P4(cos θ) =18(35 cos4 θ − 30 cos2 θ + 3) (2.20c)are the Legendre polynomials [39, 40].The a22 and a44 coefficients are characteristic to the transition charactersand their mixing ratio δ:δ =〈ψf |E(L+ 1)|ψi〉〈ψf |M |ψi〉 , (2.21)which compares the relative matrix elements for electric and magnetic tran-sitions. The electric transition is in the numerator since the mixing of higherorder electric transitions with lower order magnetic transitions is possible.This mixing is possible since the transition rates of higher order electrictransitions and lower order magnetic transitions can be comparable (from21Figure 2.7: Examples of γ-γ angular correlations for cascades (a) 4→ 2→ 0,(b) 1 → 2 → 0 with mixing ratio δ = −0.5, (c) 2 → 1 → 0 with δ = 1, and(d) 2→ 1→ 0 with δ = −1.Equations (2.19a) and (2.19b)). The mixing of higher order magnetic tran-sitions with lower order electric transitions is not common, but still possible.Information on the spin and parities of different levels can be extractedby obtaining experimental coefficients and comparing them to theoreticala22 and a44 coefficients for given γ-ray cascades, given that enough statisticsare available. For example, Figure 2.7 highlights a variety of γ-γ angularcorrelations curves [41]. A 4→ 2→ 0 cascade with coefficients a22 = 0.1020and a44 = 0.0091 is shown. The angular distributions are also shown fora 1 → 2 → 0 cascade with a22 = 0.2829 and a44 = −0.1524 (mixing ratioδ = −0.5), for a 2 → 1 → 0 cascade with a22 = −0.1854 and a44 = 0(δ = 1.0), and for a 2 → 1 → 0 cascade with a22 = 0.4854 and a44 = 0(δ = −1.0).Internal ConversionIt is possible for the excited nucleus to interact electromagnetically withan atomic electron, which is then emitted. This internal conversion pro-22cess competes with the γ-decay and the total decay probability of a levelλt is then the sum of the γ-emission probability λγ and the internal con-version probability λe. Therefore, a level decays more rapidly than it wouldfrom γ-decay only. Internal conversion includes E0 transitions (0→ 0) (seeTable 2.2).The internal conversion coefficient (ICC) α is defined as λe/λγ and isestimated by:α(EL) ' Z3n3(LL+ 1)(e24pi0~c)4(2mec2E)L+5/2(2.22a)α(ML) ' Z3n3(e24pi0~c)4(2mec2E)L+3/2. (2.22b)where n is the principal quantum number of the atomic shell and me arethe electron mass.Firstly, the Z3 dependence of α means that the internal conversion pro-cess is more important for heavy nuclei. Secondly, α decreases rapidly forhigher atomic shells (1/n3) and with the increasing energy of the transition.Therefore, internal conversion is a dominant decay mode at low energies.Thirdly, these estimates show that α increases with increasing multipoleorder. The ICCs can be calculated very precisely with the BrIcc code [42].23Chapter 3Review of LiteratureNuclear structure studies on the neighbours of the doubly-magic 132Sn(Z = 50, N = 82) contribute to our knowledge of the shell evolution andof the extreme limits of the nuclear force, as well as the application of theinformation to calculations of the astrophysical r-process, which is thoughtto be the main source for the synthesis of the heavy elements in the cosmos.The precise theoretical prediction of the relevant nuclear properties is chal-lenging since the structure of nuclei in the N = 82 region, the furthest offstability experimentally accessible today, is still not well known.The doubly magic character of 132Sn (Z = 50, N = 82) was estab-lished through (d,p) neutron transfer reaction with a radioactive ion beamof 132Sn in Ref. [43]. The spectroscopic factors calculated were consistentwith S = 1, which supports 132Sn as being a closed core. The results val-idated that the N = 82 shell closure is very robust in this region. Veryrecently, the results of a Coulomb-excitation experiment of 132Sn at CERN-ISOLDE were considered to be the first direct verification of the sphericityand double magicity of 132Sn [44]. Their new experimental values for thereduced transition strengths B(E2) agreed well with the large-scale shell-model and Monte Carlo shell-model calculations presented.The first decay spectroscopy study of the N = 82 r-process waitingpoint 130Cd was published in 2003 from ISOLDE data [1]. It claimed thatonly mass models including the phenomenon of N = 82 shell quenching(where the shell gap is reduced) below 132Sn agreed with the high Qβ-valueof 8.34 MeV. In 2016, Ref. [45] used high statistics γ-γ coincidences from theEURICA array at RIKEN to revise the decay scheme of 130In. These newresults were found to be in good agreement with shell model calculationsand do not show shell quenching.3.1 128CdThe half-life of the ground state of 128Cd was first measured in 1986 (t1/2 =340(30) ms [46]). The EURICA collaboration recently measured a shorterhalf-life of 245(5) ms [34]. A consistent result (t1/2 = 246.2(21) ms) was24extracted from the data set presented in this work and published in 2016[4].The Qβ-value of128Cd is reported to be 6.904(153) MeV [19], while theSn-value in128In is at 5.321(155) MeV [19]. No neutron-branching ratio(Pn) has been published.The amount of spectroscopic information for the decay of the lightestisotope studied in this work, 128Cd, is very limited. The relevant publicationsare based on an experiment at the OSIRIS ISOL-facility in Sweden. Asshown in Figure 3.1, seven transitions between five states in 128In werepublished [3]. The large majority of the β-decay of the 0+ ground state of128Cd feeds the 1+ excited state in 128In at 1173 keV. It is worth noting thatthis level is the highest known level and it is still 4148 keV below the Sn-value. The multipolarities of the 248 (M2, E3) and 68-keV (M1) transitionswere deduced by conversion electron measurements. It is important to notethat the decay scheme presented comes from a work which was not publishedand therefore only few details are known about the experiment and theanalysis. The decay scheme reported in 1988 was re-evaluated in 2015 byENSDF (Evaluated Nuclear Structure Data File) evaluators [14] and thetentative spin and parity JΠ assignment of the 316 keV level was revisedfrom a 2− to a 1− [14].The neutron-rich even In isotopes (odd N , odd Z) have common proper-ties: a low-lying 1+ excited state and isomeric state(s). The evolution of theimportant states across the neutron-rich even-mass In isotopes from A = 122to 130 are shown in Figure 3.2. The energy of the first excited 1+ state isimportant for r-process simulations because it is the main β-decay branch(GT feeding) [1]. Its predominant configuration is noted as pig−19/2 ⊗ νg−17/2,indicating the coupling of a proton hole in the pig9/2 orbital with a neutronhole in the νg7/2 orbital.In 2004, the half-life of the 248-keV transition was measured to be23(2) µs in Ref. [47]. This 1− isomeric state was reported to result pre-dominantly from the coupling pig−19/2⊗νh−111/2. The main configuration of theground state is pig−19/2 ⊗ νd−13/2, with a resulting spin-parity of 3+. Therefore,the 248-keV transition is a neutron moving from νh11/2 to νd3/2.Finally, there is a second β-decaying isomer in 128In (8−), which is notobserved in this experiment since it is not populated by the β-decay of theparent 0+ ground state. Its energy was first reported at 340(60) keV [48]and was recently remeasured at 262(13) keV [49].25(a) (b)Figure 3.1: Published decay schemes of 128In: (a) 1988 work by B. Ekstrom quoted in Ref. [3], and(b) 2015 Nuclear Data Sheets evaluation [14]. The main difference is the revised spin assignmentsof the excited states at 316 keV and 711 keV.2649122 In731+ 05+ 40(8-) 290049124 In75(1)+ 0(8-) 50(1)+ 24249126 In773(+) 0(8-) 90(1+) 68849128 In79(3)+ 0(1-) 248(8-) 2621+ 117349130 In811(-) 0(10-) 50(5+) 3591+ 2120Figure 3.2: Evolution of the ground state, first 1+ and isomeric state(s) ineven-mass 122−130In. Level energies [in keV] shown with left and right wingsare not to scale. [Data taken from ENSDF]273.2 131CdThe large difference between the Qβ-value of131Cd (12806(103) keV [19])and the Sn of131In (6213(38) keV [19]) leaves a large window of 6593 keV forthe βn-decay to 130In. However, the half-life and neutron-branching ratiomeasured at ISOLDE showed surprising results [50]. While Quasi-ParticleRandom Phase Approximation (QRPA) calculations deduced a half-life of286 ms and a neutron-branching ratio (Pn) of 39.8% [51], experimental valuesfound only t1/2 = 68(3) ms and Pn = 3.5(10)%. In 2015, Lorusso et al.reported a value of t1/2 = 98(2) ms [34] from the EURICA experiment, whichrepresents a difference of 30% compared to the 2010 ISOLDE measurement.The neutron-branching ratio was recently remeasured at BRIKEN (Pn ∼10%) [31].In addition, one high-spin isomer (t1/2 = 630(60) ns) with a decay energyof 3782 keV and a tentative spin assignment of 17/2+ was identified in a GSIexperiment with the RISING setup [52] (E4 transition to the 9/2+ ground-state). The excitation of the 1/2− isomer at 365(8) keV was deduced via atrap measurement published in Ref. [53].The β-decays of 131−132Cd have received a lot of attention in the lasttwo decades. The first spectroscopic information was reported in 2000 ina PhD thesis with transitions at 995, 1587, 1737, 1910, 360, and 3870 keV[54]. Information on γ-transitions from an experiment at CERN-ISOLDE isreported in [55] and in a PhD thesis of the same group [56]. However, bothreferences reported inconsistent information which was not very detailed.Together they identified eight transitions (844, 988, 2428, 2640, 3556, 3866,4403, and 6039 keV), but the identification of some of these lines is question-able due to the low statistics. Figure 3.3a shows a level scheme placing threeof the transitions in a single-particle picture: the 988-keV γ-ray correspondsto the transition between the pip3/2 and the pip1/2, the 2428-keV betweenthe pi5/2 and the pip1/2, and tentatively the 844-keV transition between thepif5/2 and the pip3/2.The EURICA collaboration at RIKEN has recently published results forthe pi-hole states (with respect to the 132Sn core) in 131In from the β-decayof 131Cd and the βn-decay of 132Cd, which confirmed the main transition at988 keV [5]. This transition had also been seen in the ISOLDE data [55], butwas not mentioned in the following PhD thesis [56]. A more detailed analysisof the EURICA data identified more than 20 transitions [6]. A tentativelevel scheme for each data set is shown in Figure 3.3. While the transitionof 988 keV was observed again between the pip3/2 and pip1/2 orbitals, the(5/2)− levels were not assigned. While the pig−19/2, pip−11/2 and pip−13/2 states28(a) (b)Figure 3.3: Published decay schemes of 131In from (a) ISOLDE (2009) [55], and (b) RIKEN (2016)[6]. Energies are displayed in keV.29were observed in both experiments, the location of the 5/2−pi-hole state isstill uncertain. The evolution of the 1/2–9/2 states across the neutron-richodd-mass In isotopes is shown in Figure 3.4.The ground state of 131Cd with configuration ν1f7/2 [6] is in very goodapproximation the ground state of 130Cd coupled to a 1f7/2 neutron acrossthe N = 82 shell gap. A single-particle decay should convert the 1f7/2neutron into a proton, which can only occupy a single-particle state that isnot occupied in the ground state of 131Cd. For protons, the ground stateof 131Cd is equal to the ground state configuration of 130Cd. Figure 3.5summarizes the possible allowed (GT) and first forbidden (ff) decays asdiscussed in Ref. [6].Firstly, the allowed Gamow-Teller decay (L = 0, ∆I = 1, ∆Π = 0) ofa 0g7/2 neutron into the 0g9/2 orbital is expected to dominate the decay of131Cd. This decay, which has been observed in the decay of many othercadmium and indium isotopes, populates states of the pi−19/2 ⊗ ν(1f7/20g−17/2)multiplet and should be high in energy. The transition of a 1f7/2 neutroninto a 0f5/2 proton is not expected to be directly populated since the twoorbitals are separated by two harmonic oscillator shells [6].Secondly, several first forbidden decays (L = 1, ∆I = 0, 1, 2, ∆Π = 1) arepossible. A 1f7/2 neutron can decay to a 0g9/2 proton, which populates the9/2+ ground state of 131In. Also, a 1f7/2 neutron can decay to a 0g7/2 protonacross the Z = 50 shell. This decay results in a 7/2+ cross-shell excited stateat very high energy due to the cross-shell excitation (5-6 MeV).A ν0h11/2 → pi0g9/2 transition populates a multiplet of states that resultsfrom the coupling of an unpaired 0g9/2 proton with a 1f7/2 neutron and a0h11/2 neutron hole, i.e. pi0g−19/2⊗ν(1f7/20h−111/2). According to Ref. [6], thesestates are expected around 4 MeV.Finally, a 1d3/2 neutron can decay to a proton in one of the negative-parity orbitals: 1p1/2, 1p3/2 or 0f5/2. These decays lead to multiplets ofstates pi(1p, 0f−15/2)⊗ν(1f7/21d−13/2) from coupling an unpaired 1f7/2 neutron,a 1d3/2 neutron hole and a 1p1/2,3/2 or 0f5/2 proton hole. However, thesedecays are possible only to the fractions of the wave function which havethe appropriate 1p−21/2, 1p−23/2 or 0f−25/2 holes in the130Cd ground state. Thesestates are also expected at ∼4 MeV. However, in comparison to the allowedtransition, these first forbidden transitions would be even further suppressedand one would expect only larger log(ft) values on the order of∼6 and higher.3049123 In74(9/2)+ 0(1/2)- 327(3/2-) 698(5/2+) 1052(5/2-) 1137(9/2+) 1512(7/2+) 156649125 In769/2+ 01/2(-) 360(3/2-) 797(5/2+) 1099(5/2-) 1220(9/2+) 1564(7/2+) 157849127 In78(9/2+) 0(1/2-) 408(3/2-) 932(5/2+) 1202(5/2-) 1588(9/2+) 168649129 In80(9/2+) 0(1/2-) 459(3/2-) (983)(5/2+) 1423(5/2-)(1525)(9/2+) 158649131 In82(9/2+) 0(1/2-) 365(3/2-) 1353(5/2-)(2134)(5/2-)(2730)Figure 3.4: Evolution of the 1/2–9/2 states in odd-mass 123−131In. Levelenergies [in keV] shown with left and right wings are not to scale [Datataken from ENSDF and Ref. [55]].31Figure 3.5: Single-particle orbitals in the 132Sn region [6]. The ground stateconfiguration of 131Cd is shown with open (proton holes) and filled (neutronparticle) dots. The allowed (GT) and first forbidden (ff) single-particledecays are indicated by solid and dashed arrows, respectively.323.3 132CdThe large difference between the Qβ-value of132Cd (12 150(510) keV [19])and the Sn of132In (2450(60) keV [19]) predicts an important β-delayedneutron (βn) branch. A neutron-branching ratio of Pn = 60(15)% and ahalf-life for the ground state of 132Cd of t1/2 = 97(10) ms were reportedfrom a previous ISOLDE experiment [50]. The half-life was remeasured atRIKEN to t1/2 = 82(4) ms [34].A QRPA calculation published in [50] and shown in Figure 3.6a indicatesa 1+ state at ∼1200 keV which would be fed by an allowed Gamow-Tellerdecay. It is very likely that the transitions from the 1+ state to the ten-tatively assigned (7)− ground-state or to the first 1− state (E1 transition)are very weak and not visible due to the low statistics, or that the 1+ stateis a β-decaying isomer. These two cases could be distinguished and indi-rectly measured. If there is a β-decaying isomeric state in 132In, transitionsfrom excited states in 132Sn would be seen. Otherwise, the appearance ofthe βn-daughter 131In would dominate the spectra due to the high neutron-branching ratio.The latter case is what the EURICA collaboration observed in 2014 [5](see Figure 3.6b). Only the 988 keV transition in 131In was seen from the βn-decay branch and therefore they were not able to extract information aboutany excited states in 132In. The non-observation of any transition in 132Incould hint to a Pn-value closer to 100% than the previously measured 60%[50]. The Pn-value of132Cd has been remeasured in 2017 with the BRIKENsetup at RIKEN, and a preliminary value of Pn ∼ 88(5)% was presentedrecently by A. Estrade at Nuclear Structure 2018 [57], which would confirmthe EURICA value. The question remaining is where the remaining ∼ 12%of feeding go. The P2n channel can be excluded experimentally from theBRIKEN data (P2n ∼ 0%, Qβ2n=3480(200) keV).Different insight into missing pi-hole states in 131In is gained from the132Cd βn-decay compared to the 131Cd β-decay. 132Cd is an even-evennucleus with a 0+ ground-state and so the βn-decay populates only low-spinstates in 131In, like 3/2− and 1/2−, via emission of l = 0 or l = 1 β-delayed neutrons (see Figure 3.6b). Since 131Cd has a 7/2− ground state, itpopulates only high-spin states by its β-decay. Additionally, the probabilityof β-decay to a specific state depends on the energy transition to the fifthpower (∝ E5). β-delayed neutron emission tends to populate states at lowexcitation energy in the grand-daughter nucleus [58].The only transitions seen so far in 132In were observed following the β-delayed neutron emission of 133Cd with energies of 50, 86, 103, 227, 357 and33(a) 2010 QRPA calculations [50] (b) 2016 EURICA data with the observed 988 keV transition [6]Figure 3.6: Published decay schemes of 132Cd.602 keV [58]. Again due to the E5 dependence of the transition probability,it was assumed that the βn-decay of 133Cd populated states at low energiesin 132In and therefore the six γ-rays are believed to form a cascade of M1transitions between the 1− multiplet (pig−19/2⊗νf−17/2) and the 7− ground state(see Figure 3.7).34Figure 3.7: Tentative levels energies [in keV] for 132In [Figure from XUNDL].35Chapter 4ExperimentIn August 2015, the five neutron-rich isotopes of Cadmium were deliveredto the GRIFFIN spectrometer in the low-energy experimental area of theISAC-I facility at TRIUMF over seven days of beam time (13 shifts). Thischapter covers the production of the beam, the detector hardware as well asthe data processing.4.1 Beam ProductionA 480-MeV 9.8-µA proton beam from the TRIUMF main cyclotron wasdirected to the East production target in the ISAC1 facility [59]. The ISACexperimental hall layout is shown in Figure 4.1. An Isotope SeparationOn-Line (ISOL) production target of uranium carbide (UCx) was used toproduce the neutron-rich cadmium beams studied in this work via protonspallation and fission reactions.The ion guide laser ion source (IG-LIS) [60] was used to suppress surface-ionized ions as described in Figure 4.2. The production target operating attemperatures around 1500◦C releases strong surface-ionized isobaric con-taminants of the same mass such as cesium, barium and indium, along withthe cadmium atoms of interest. First, an electrostatic potential barrier re-pels these contaminants, allowing only neutral atoms to enter the cold regionof the resonance ionization laser ion source (RILIS), which is protected bya heat shield. Second, a three-step laser excitation, which depends on theproton number Z, enables to selectively ionize the cadmium atoms [61]. Theexcitation scheme for cadmium (Z = 48) consists three successive transitionsat wavelengths 228.9 nm, 466.3 nm and 1064 nm, respectively. Finally, alinear radio frequency quadrupole (RFQ) guides the ions toward the highvoltage exit electrode.Next, the isotope of interest was isolated with the ISAC high resolutionmass separator based on the A/q ratio, where A is the mass number andq is the charge of the ions, with a resolving power in the order of 1/2000.1Isotope Separator and ACcelerator36Figure 4.1: TRIUMF ISAC experimental hall layout [59]. The 8pi γ-rayspectrometer was replaced by the GRIFFIN array, which was commissionedin 2014.The beam was then delivered to the experimental hall with an energy of 20-30 keV in cyclic mode. An electrostatic potential barrier (kicker) enabledto control the length of time for which the beam was received.These steps allow the suppression of surface-ionized indium and cesiumto the extent that the spectroscopy of laser-ionized cadmium becomes pos-sible, as the measured background is suppressed by factors 105 to 106. Fig-ure 4.3 shows the yields of neutron-rich cadmium beams as measured atISAC’s yield station after the high-resolution mass-separator isolated theisotope of interest. When the repeller is off, the yield of laser-ionized cad-mium (blue) is drowned in background (pink). When the repeller is on, thecadmium yield is comparable to the remaining contamination (red), makingthe study of these cadmium beams possible. Because of the low intensity ofthe 131−132Cd beams, their yields were measured for the first time directly37Figure 4.2: Concept of the Ion Guide Laser Ion source (IG-LIS) [60].at the GRIFFIN2 γ-ray spectrometer [2, 62], which has a higher efficiencythan the ISAC yield station.4.2 DetectorsThe delivered beam was implanted on an in-vacuum moving tape collec-tor (MTC) system at the center of both the GRIFFIN spectrometer and theSCEPTAR3 plastic scintillator array [64], shown in Figures 4.4 and 4.5. Thealuminum coated Mylar tape removes long-lived activity from the centre ofthe array to behind a lead shielding wall, reducing the background radiationfrom such long-lived nuclei. The moving tape is operated in cycles whichtypically consists of the tape move, a background measurement, a collec-tion time (beam on) and a decay time (beam off). The transition betweenthe beam on and off parts of the cycle are controlled by the kicker. Thecyclic motion of the tape station enables crucial analysis techniques whichwill be discussed in Chapter 5, such as the measurement of half-lives anddiscrimination of background γ-rays.SCEPTAR is an array of 20 scintillating paddles enabling β-particletagging. While SCEPTAR was positioned inside the vacuum chamber, alow-Z Delrin plastic sphere (10 or 20 mm thick) was installed on the outsideof the chamber in order to absorb the high energy electrons and protectthe high purity germanium (HPGe) crystals. Details on the geometry ofSCEPTAR are given in Ref. [66].2Gamma-Ray Infrastructure For Fundamental Investigations of Nuclei3SCintillating Electron-Positron Tagging ARray38Figure 4.3: 124−130Cd yields at ISAC using the Ion Guide Laser Ion source(IG-LIS) [63]. For 125Cd, the decays of the isomeric (m) and ground (g)states are identified separately.GRIFFIN is an array of up to 16 large-volume HPGe clover-type de-tectors, for a total of 64 coaxial semiconductor crystals, dedicated to decayspectroscopy of the low-energy radioactive ion beams at TRIUMF. Detailson the geometry of the GRIFFIN array are given in Ref. [66].γ-rays interact with matter through three mechanisms: photoelectricabsorption, Compton scattering and pair production. With photoelectricabsorption, an atom completely absorbs an incoming γ-ray and emits anelectron with energy Ee = hf −Bn, where f is the frequency of the photonand Bn is the binding energy of the electron. The high energy electron theninteracts with the charge carriers in the germanium crystal. For an n-typesemiconductor, electrons are the majority carriers and holes are the minoritycarriers. The electron/holes pairs migrate in opposite directions with thehigh bias voltage applied between the p- and n-contacts, where the totalcharge deposited is collected. The photoelectric process is the dominantmode of interaction for γ-rays of relatively low energy [67]. With Comptonscattering, an incoming γ-ray transfers part of its energy to an electron in the39Figure 4.4: The west hemisphere (8 of 16 High Purity Germanium (HPGe)clover detectors) of the GRIFFIN γ-ray spectrometer is shown, with theSCEPTAR photomultiplier tubes [top] and the vacuum chamber [top right].Kathia Bernier visited TRIUMF in March 2015.Figure 4.5: Downstream hemisphere of the SCEPTAR scintillator arrayand moving-tape collector inside the vacuum chamber at the centre of theGRIFFIN array. [65]40detector and is deflected. When the γ-ray is deflected outside of the crystal,there is incomplete charge collection, which creates background in the γ-ray energy spectrum. This interaction is discussed further in 4.3. Finally,an incoming γ-ray with Eγ > 1.022 MeV can produce an electron and apositron, both with masses of 0.511 MeV/c2. The positron then annihilateswith another electron and two annihilation photons of 511 keV are created.If one of the annihilation γ-ray escapes the crystal, it can be detected by aneighbouring detector and form a peak at 511 keV.The GRIFFIN signals are processed by four digitizer modules with 16channels (GRIF-16). A fifth GRIF-16 collects the SCEPTAR signals. Col-lector modules (GRIF-C Slave) concentrate the 16 outputs of a GRIF-16into a single output, which is linked to the GRIF-C Master collector. Themaster clock of the GRIF-C Master collector provides the reference clockto the GRIF-Clk Slave modules, which give a time stamp to each signal.Finally, the GRIF-C Master collector filters all the data according to thepreset filter and sends the accepted signals to the MIDAS [68] frontend tobe written to disk. The current data were collected in “singles” mode, whichmean that every γ-ray and every β-particle detected was written. The fullydigital data acquisition (DAQ) system is described in [69].The MIDAS data files were sorted using an in-house analysis frameworknamed GRSISort [10], based on the code ROOT [11] from CERN. GRSISortfirst sorts the raw data into fragment trees. Then, analysis trees are writtenby ordering the hits by time stamps and grouping them in events defined bya coarse coincidence time window typically of 2 µs. The code also includesinteractive analysis tools which enable rapid visualization and analysis ofthe data. From this point, the user writes scripts to display the full data setor parts of it by applying one or several constraints as required.4.3 Data ProcessingCrosstalk CorrectionAn important effect of clover-type HPGe detectors is the crosstalk betweenthe four crystals. A γ-ray detected in a crystal can induce signals in theneighbouring crystals and hence decreases the resolution of the detectors.This energy-dependant effect is corrected by applying a clover-by-clover cor-rection matrix, which is determined by studying the amplitude of the in-duced signal between each pair of crystals as a function of energy [70, 66].The effect of the crosstalk correction is presented in Figure 4.6: the energyresolution is recovered.411100 1150 1200 1250 1300 1350 1400Energy [keV]210310410510Counts per 1 keV Clover Addback Gamma-ray singles(a) Without crosstalk correction1100 1150 1200 1250 1300 1350 1400Energy [keV]210310410510Counts per 1 keV Clover Addback Gamma-ray singles(b) With crosstalk correctionFigure 4.6: Comparison of clover addback [blue] and γ-singles [red] spectraobserved for a 60Co source (a) without and (b) with crosstalk correction.42Addback MethodCompton scattering of γ-rays between the four crystals or even betweenneighbouring detectors is detected as background. Any γ-ray entering acrystal can deposit a fraction of its energy before scattering in one or moreother crystals. Therefore partial energies are collected and build a contin-uous background underneath the full-energy photopeaks. The full energyof the scattered γ-ray can be recovered by adding back the partial ener-gies which were detected in neighbouring crystals. Therefore, the addbackmethod enables us to increase the detection efficiency [66]. Obviously, par-tial energies for γ-rays which were scattered to the outside of the arraycannot be recovered. This undesirable effect can be reduced with Comptonsuppression shields, which were installed on GRIFFIN in 2018, after theneutron-rich Cd data sets were collected.The addback mode is defined with both a time condition and a spacecondition. The γ-rays added-back have to be detected within 300 ns of eachother, and within the same clover detector. Figure 4.7 (which shows the fullrange of the data in Figure 4.6) presents the effect of addback: the Comptonbackground is reduced and the signal-to-noise ratio is improved.Summing CorrectionThe summing of γ-rays is another effect seen in arrays with large angularcoverage. Since the isotopes are implanted and stopped in the tape, theirdecays happen isotropically. The isotropic angular distribution is a resultof the random orientation of nuclear spins and thus a lack of quantizationaxis. Therefore, there is a non-zero probability that two γ-rays are emittedin the same direction and detected in the same crystal. γ-rays from differentnuclei on the tape can also deposit their energies in a single crystal, whichwill only detect the total energy deposited. However, the probability oftwo γ-rays being emitted at 0◦ is the same as being emitted at 180◦. Thesumming effect can be corrected by measuring the total number of events inthe γ-gated coincidence spectrum between two opposite detectors separatedby 180◦.Energy Resolution and Detection EfficiencyThe average energy resolution at full width at half maximum (FWHM) at122.0 keV and 1332.5 keV for all 64 crystals is 1.12 (6) keV and 1.89 (6) keV,respectively [62]. As a comparison, the theoretical highest resolution of a430 200 400 600 800 1000 1200 1400Energy [keV]210310410510Counts per 1 keV Clover Addback Gamma-ray singlesFigure 4.7: Comparison of clover addback [blue] and γ-singles [red] spectraobserved for a 60Co source.germanium detector, which has a band gap of ∼0.7 eV, is given by the sta-tistical deviation of the number of electrons emitted divided by the numberof electrons. For a 1332-keV γ-ray, the number of electrons emitted Ne =1332 keV/(0.7 eV/electron) = 1902857 electrons and the theoretical bestresolution is√Ne/Ne ∗ 1332 = 0.97 keV.The absolute γ-ray detection efficiency γ is defined as:γ =Number of γ-rays detectedNumber of γ-rays emitted by source=Nγ,detectedIγ ·A · t , (4.1)where Iγ is the absolute decay intensity of the γ-ray, A is the activity of thecalibration source at the time of the data collection, and t is the live time.The live time is the difference between the run time and the dead time, whichis the minimum amount of time for which two γ-rays can be distinguishedas separate signals [67]. The detection efficiency γ as a function of energyE is determined by fitting the efficiency of all the individual γ-rays γ to:γ(E) = 10p0+p1 log(E)+p2 log2(E)+p3/E2 . (4.2)44Figure 4.8: Time difference between consecutive triggers as a function ofcrystal number for a 152Eu source. The dead time, i.e. the minimum timebetween triggers, is found to be 7.5 µs per trigger.The dead time per trigger is determined by looking at the time differencebetween consecutive triggers as shown in Figure 4.8. The minimum time be-tween triggers is found to be 7.5 µs per trigger per crystal. The average deadtime per crystal, i.e. 7.5 µs/(trigger·crystal) the number of triggers/crystal,gives the total dead time for the full array. The few triggers seen with a timedifference less than 7.5 µs are called pile-up events and are rejected in theanalysis. Newer versions of the DAQ are able to distinguish these pile-upevents in to two or more separate events [69].The absolute γ-ray detection efficiency curves shown in Figure 4.9 forthe energy range of 53–3450 keV were determined with calibration sources of133Ba, 152Eu and 56Co and include both the addback and summing correc-tions. For the 131,132Cd data sets, the 20 mm Delrin plastic sphere was usedwith 62 operational crystals. Step-by-step methods for energy and efficiencycalibrations are described in Appendix A.45(a) 64 crystals, with SCEPTAR and 10 mm Delrin plastic sphere(b) 62 crystals, with SCEPTAR and 20 mm Delrin plastic sphereFigure 4.9: Absolute γ-ray detection efficiency for the GRIFFIN spectrom-eter in addback mode with summing correction: (a) with SCEPTAR and10 mm Delrin plastic sphere as used for the 128Cd data set, (b) with SCEP-TAR and 20 mm Delrin plastic sphere as used for the 131−132Cd data sets.46Chapter 5Data Analysis and Results5.1 128Cd5.1.1 β-Gated γ-Singles MeasurementsApproximately seven hours of 128Cd data were collected with a beam in-tensity of ∼1000 pps. With these statistics, a detailed coincidence analysisof weaker lines became possible. Also enabled is the determination of spinand parities with γ-γ angular correlation measurements, where only a fewthousand β-γ-γ coincidences are required.The data set was collected in “singles” mode, which means each γ-rayand each β-particle detected was written to disk. Therefore, looking atγ-singles data includes room background radiation in addition to the γ-rays and X-rays emitted by the isotopes of the beam composition. TheSCEPTAR scintillator array is the first tool used to isolate the γ-rays ofinterest, by only looking at the ones which were detected in GRIFFIN withina set coincidence time window relative to the detection of a β-particle inSCEPTAR.Figure 5.1 shows the energy of γ-ray plotted as a function of the differ-ence between the time stamp of a β-particle and the time stamp of a γ-ray.The majority of the β-γ time differences are centered around zero and seenwithin a 200 ns wide time window. This window is called the prompt coin-cidence window and points to a time correlation between the β-particle andthe γ-ray, meaning they are statistically more likely to come from the samedecay event. At lower energies, a positive tail, which is a symptom of theslower charge collection in the germanium crystals for smaller amounts ofelectrons, can be noticed. γ-rays which are seen consistently with larger timedifferences, appearing as horizontal lines on the matrix, are most likely un-correlated, e.g. coming from different isotopes, and are called time-randomcoincidences. Therefore the gate (or cut) selects the correlated β-γ eventsand rejects a majority of uncorrelated events. Step-by-step methods for timeand energy gates, along further data analysis techniques, can be found inAppendix B.47Figure 5.1: Difference between time stamps of β-particles and γ-rays (γ−β)as a function of the energy of the γ-ray. The 2-dimensional time cut isoverlayed on the matrix.The effect of this β-tagging is shown in Figure 5.2, which shows γ-singlesspectra with and without β-tagging. The β-tagged data is defined by thecoincidence time gate drawn on the time difference matrix in Figure 5.1.One can nicely see that the peaks with a black asterisk correspond to γ-rays from the β-decay of 128Cd published in Ref. [3]. The first advantageis the significant reduction of the background at lower energy. Environ-mental background γ-rays include decay products from the natural decayof 235,238U and 232Th (U/Th-series), 40K from concrete, and a continuousbackground from cosmic-rays mesons [67]; all of which can be detected byGRIFFIN while collecting the data of interest. However, most of the back-ground β-particles will not be detected by SCEPTAR, which is shieldedfrom the experimental hall by the GRIFFIN clovers. Therefore a coinci-dence spectrum suppresses the room background while also increasing thephotopeak-to-total ratio for the beam species.Another feature of the β-tagging is the rejection of isomeric state decays,which are seen with larger time differences between their β-particle emissionand their γ-ray(s). The 248-keV isomer seen in Ref. [3] with a half-life of23 µs is confirmed here. In Figure 5.2, this transition disappears in the480 200 400 600 800 1000 1200Energy [keV]02004006008001000120014001600310×Counts per 1 keV*68*247*462e-/e+*857*925*1172 Beta-gated singles Gamma-ray singlesFigure 5.2: Comparison of β-gated γ-singles [blue] and γ-singles [red] spectraobserved for the decay of 128Cd. No normalization is applied. The peakswith a black asterisk correspond to γ-rays published in Ref. [3]. The 248-keV line is an isomeric transition, which mostly decays outside of the β-γcoincidence window of 200 ns.β-gated spectrum since most of the decays only happen outside of the β-γcoincidence window of 200 ns. Hence, few γ-rays were observed to be incoincidence with the 248-keV isomeric transition.Finally, this comparison also displays the efficiency of the SCEPTARarray. By taking the ratio of the area of a photopeak in β-gated γ-raysingles (Nβγ) and in γ-ray singles (Nγ):NβγNγ=β · γ · Iγγ · Iγ = β, (5.1)we find a β-tagging efficiency of 65-70 % between 462 and 925 keV.Next, laser ionization has been proven to be a powerful tool for further49discrimination of isobaric background [55]. When the lasers are on, thespectrum is composed of γ-rays from surface- and laser-ionized species (ce-sium, indium, cadmium), whereas only γ-rays from surface-ionized species(cesium, indium) are visible when one of the laser transitions is blocked.Hence, new γ-transitions can be unambiguously identified by comparingsubsequent spectra taken with lasers on and with lasers blocked.The comparison of the γ-spectra for 128Cd with the lasers on and blockedcan be seen in Figure 5.3. Approximately 20% of the nine hours of 128Cdbeam time was used to collect background data while one of the three laserswas blocked. The peaks with a black asterisk correspond to γ-rays fromthe β-decay of 128Cd published in Ref. [3], whereas those with an orangeasterisk are newly observed lines.In both the lasers-on and laser-blocked data sets, there is a transitionof 831 keV which is fed by the 8− isomer (0.72(10) s) in 128In only and notthe 3+ ground state of 128In. However, when scaled to the 1460 keV roombackground line (from 40K) in γ-ray singles, the 831 keV transition is 5.5times more intense with the lasers on than with the first laser blocked. First,the 8− isomer cannot be populated from the β-decay of the 0+ ground stateof 128Cd. There is a 10+ isomer (3.56(6) µs) at 2711.5(11) keV in 128Cd[14], but its β-decay to the 8− would be a third forbidden, which is highlyunlikely. Since the isomer cannot be populated by the β-decay of 128Cd, the128In isomer has to be part of the beam composition. The first excitationstep of cadmium with the 228.9 nm laser has enough energy to excite anindium electron into a Rydberg state, from which any of the two otherlasers can non-resonantly ionize the indium. Some neutral indium can passthrough the repeller electrode or, depending on the operation mode, intothe ionization volume inside the ion guide, where it will then be ionized andextracted. Based on the intensity of the 831-keV gamma-ray, 45(2) pps forthe 8− isomer were delivered with the lasers on, while only 8(1) pps weredelivered with the first laser blocked.In the case of a heavily contaminated spectrum, i.e. with beam contam-inants or decay chains of short-lived daughters, peaks of interest can alsobe identified by looking at counts as a function of the time within the tapecycle. The cycle, which is shown in Figure 5.4, starts with the tape move,removing the longer-lived nuclei from the previous implantation and expos-ing a clean piece of tape. It is seen in Figure 5.4a that the activity dropsrapidly has the used tape is moved behind the shielding bricks. With thelaser blocked (Figure 5.4b), the lower activity highlights the noise causedby the moving tape collector system. Then, the background is measured for1.5 s before the beam is implanted on the tape (10 s). During the subsequent500 200 400 600 800 1000 12000100200300400500310×Counts per 1 keV*68*173*221*315*408*462e-/e+*619*857*925*1089 + 1089 (128Sn)*1172Lasers on    0 200 400 600 800 1000 1200Energy [keV]0100200300400500600700800900Counts per 1 keV75 (128Sb)121 (128Sn)168313 (128Te)442 (128Xe)482 (128Sb)e-/e+589743 (128Te)753 (128Te)788 (128Te)8081168 (128Sn)Laser blockedFigure 5.3: Comparison of β-gated γ-ray spectra observed for the decay of 128Cd in addbackmode with lasers on [blue] and laser blocked [red]. The laser-on spectrum was collected incycle mode for ∼6.5 hours and the laser-blocked spectrum, for ∼2.5 hours. The peaks witha black asterisk correspond to γ-rays published in Ref. [3], whereas those with an orangeasterisk are newly observed lines.51time window (2 s), data is collected with the beam off. One can see thatdue to the difference in half-lives, the first seconds of the beam-on windoware dominated by the Cd activity (t1/2= 246.2(21) ms [4]), whereas the Inactivity (t1/2= 810(30) ms [34]) dominates most of the beam-off window,enabling identification of new lines based on time structure. Additionally,since a single decay can be registered by more than one SCEPTAR paddle,multiple beta-particle counts with the same time stamp are only countedonce.The plotted curves show the number of decays for the cadmium isotopes(red line), the first daughter indium (yellow line), the indium present inthe beam (4% of the beam intensity) (violet line), the constant background(green line), and the sum of the four curves (light blue line). In a way similarto how the laser on/blocked comparison highlighted the cadmium lines, thespectra for the beam-on (Cd + In) and beam-off (In) time windows alsopoint to peaks of interest likely to be related to the Cd decay, as shown inFigure 5.5.The discrepancy between the fit and the data for the decay part of thecycle comes from a small fraction (∼5%) of the cycles which do not includea decay part. The transitions intensities were calculated from the beam-onpart of the cycle only, which is not affected by the missing decay parts.5.1.2 β-Gated γ-γ Coincidence MeasurementsSimilar to the β-γ coincidence gate in Figure 5.1, a β-γ-γ coincidence gatecan be defined by looking only at β-particles which were detected in coin-cidence with at least two γ-rays, as shown in Figure 5.6. This further cutsan order of magnitude of statistics and enables us to extract information oncascades of γ-rays which are emitted from successive γ-decays.In Figure 5.7, a β-gated γ-γ coincidence matrix is built by plotting theenergy of the first γ-ray on the x-axis and the energy of the second γ-rayon the y-axis. To ensure the symmetry of the matrix, the energy of thesecond γ-ray is then plotted on the x-axis and the energy of the first γ-ray,on the y-axis. Several features are observed regarding this matrix, includinghorizontal, vertical and diagonal lines. Continuous horizontal and verticalstraight lines consist of γ-rays coincident with every γ-ray energy and sug-gest random coincidences, as seen before in Figure 5.1. Diagonal lines showincomplete charge deposition when γ-rays scatter between crystals. Someof the scattered γ-rays were already corrected with the addback mode, how-ever, some scatter events cannot be recovered and appear on the coincidencematrix as diagonal lines.52(a)(b)Figure 5.4: Number of β-particles as a function of cycle time for the β-decay of 128Cd in (a) laser-on mode and (b) laser-blocked mode. The cycleconsists of the tape move, background measurement (1.5 s), beam on (10 s)and beam off (2 s).530 200 400 600 800 1000 1200050100150200250300350400450310×Counts per 1 keV*68*173*221*315*408*462e-/e+*619*857*925*1089*1172Beam on    0 200 400 600 800 1000 1200Energy [keV]05000100001500020000250003000035000Counts per 1 keV *68121 (128Sn)257 (128Sn)*462e-/e+538 (128Sn)831 (128Sn)*857*925 935 (128Sn)*1089 + 1089 (128Sn)1168 (128Sn)Beam off   Figure 5.5: Comparison of β-gated γ-ray spectra observed for the decay of 128Cd in addbackmode as a function of cycle structure. The beam-on spectrum was collected for 10 s percycle and the beam-off spectrum, for 2 s. The peaks with a black asterisk correspond to γ-rays from the β-decay of 128Cd published in Ref. [3], whereas those with an orange asteriskare identified as newly observed lines.54Figure 5.6: Difference between the time stamp of a γ-ray coincident with aβ-particle and the time stamp of a second γ-ray (γ1 − γ2) as a function ofthe energy of the second γ-ray. The 2-dimensional time cut is overlayed onthe matrix.When zooming in on the coincidence matrix (Figures 5.7b and 5.7c),the most important feature becomes obvious: spots of increased statistics.The region of the matrix shown in Figure 5.7b around 68 keV, which isthe second most intense transition in 128In, shows two spots: at 173 keVand 221 keV. In Figure 5.7c, there is a spot at the intersection of 316 and857 keV, with the latter being the third most intense transition in 128In.The increased number of counts suggest a higher probability of observingγ-rays of these two energies at the same time when compared to γ-rays ofneighbouring energies, and hence coincidence relationships are identified.Background subtracted γ-gated spectra are shown in Figures 5.8 to 5.11,where newly observed coincidence lines are identified by orange asterisks.These spectra are produced by defining 3 gates: one on a photopeak in theprojection, which should include the coincidence spots, and two backgroundgates on either sides of the photopeak in the projection spectrum. Thebackground and scatter included in the second and third cuts can then besubtracted from the first cut of interest. However, since the diagonal scatterlines are seen at different energies in the projection, they create dips in the55(a)(b) (c)Figure 5.7: Symmetrized β-gated γ-γ coincidence matrix for 128Cd data: (a)the full matrix, and zoomed in on (b) 68 keV and the region around 200 keVand (c) 857 keV and the region around 280 keV. All are displayed in 1 keVper bin.56background-subtracted projection.Firstly, the top and middle panels in Figure 5.8 show the gate on thethird most intense transition in 128In: 68-keV between the 316 and 248-keVlevels. The high intensity of the transition causes a lot of uncorrelated co-incidences from different nuclei decays on the tape. Also, the projectionincludes a high level of low energy background, which includes X-rays andseveral Compton scatter lines from all transitions. The background sub-traction is sensitive to these different sources of background, which can beover-subtracted (dips) or under-subtracted (peak). Therefore, it is useful togate on higher energy transitions which show cleaner coincidences to extractconvincing coincidence. The 2782, 3707 and 4432-keV transitions are placeddirectly feeding the 316-keV level.A gate on the 173-keV transition is shown in the last panel of Figure 5.8.Coincidence peak are seen 68, 221, 462 and 1097 keV. Therefore, this transi-tion was placed on top of the 68-keV transition and results in a new level at489 keV, which feeds the level at 316 keV and is directly fed by the 221-keVtransition, which is fed by the top 462-keV transition. The 1097-peak isseen in this gate only, which results in a level at 1585 keV and directly feedsthe 173-keV transition and 489-keV level. The subtraction of scatter linescreates dips around the peaks at 295 keV, 780 keV and 990 keV.Secondly, the gate on the 462-keV doublet is shown in the top and middlepanels of Figure 5.9. The peak at 711 keV corresponds to the transitionsfrom the level at 711 keV to the ground state. By comparing with the857-keV gate (bottom panel), transitions can be placed either feeding thetop 462-keV transition (1173 → 711 keV) or the bottom 462-keV transition(711 → 248 keV). The 1552, 2387 and 3313-keV transitions, which are notseen in the 857-keV gate, are placed directly feeding the 711-keV level. The1089, 1924 and 2486-keV transitions, which are seen in both the 462 and857-keV gates, are placed on top of the 1173-keV level. The 1089 and 1552-kev transitions both depopulate a level at 2263 keV. The 1924 and 2387-kev transitions both depopulate a level at 3097 keV. Finally, the 3313-keVtransition, which is not seen in coincidence with the 857-keV, places a levelat 4024 keV, and the 2486-keV transition, which is common to both gates,places at level at 3659 keV.Thirdly, Figure 5.10 shows gates on the 408-keV transition (top andmiddle panels) and on the 336-keV transition (bottom panel). The 408-keV transition is seen in coincidence with peaks at 211, 336, 516, 1608 and2441 keV, and are placed directly feeding the 408-keV transition. The totalenergy of the cascade formed by the 408 and 516-keV transitions is ∼924keV,which corresponds to the energy difference between the 248 and 1173-keV570 200 400 600 800 1000 120005001000150020002500Counts per 1 keV75 (Xray)*173*221Scatter*462Scatter*857*108968 keV gate (low)2600 2800 3000 3200 3400 3600 3800 4000 4200 440010−01020304050Counts per 1 keV *2782ScatterScatter*3707Scatter*4432Scatter68 keV gate (high)0 200 400 600 800 1000Energy [keV]0500100015002000Counts per 1 keV*68*221Scatter *462ScatterScatter*1097173 keV gateFigure 5.8: β-gated background-subtracted γ-gated spectra observed with the GRIFFINspectrometer for the β-decay of 128Cd. The top and middle panels show the coincidenceswith the 68-keV transition (316→ 248 keV) in 128In, and the bottom panel shows the coin-cident γ-rays with the 173-keV transition (489→ 316 keV). The black asterisks correspondto transitions published in Ref. [3], whereas the orange asterisks mark newly observed lines.580 200 400 600 800 1000 120005001000150020002500Counts per 1 keV*68*173*221Scatter *462*711*1089462 keV gate (low)1600 1800 2000 2200 2400 2600 2800 3000 3200 340010−0102030405060Counts per 1 keV*155216421797*1924 *2387*2486Scatter*3313462 keV gate (high)1000 1200 1400 1600 1800 2000 2200 2400 2600Energy [keV]20−020406080100120140Counts per 1 keV*68*315*1089*1924*2486857 keV gateFigure 5.9: β-gated background-subtracted γ-gated spectra observed with the GRIFFINspectrometer for the β-decay of 128Cd. The top and middle panels show the coincidenceswith the 462-keV doublet (1173 → 711 keV, and 711 → 248 keV) in 128In, and the bottompanel shows the coincident γ-rays with the 857-keV transition (1173→ 316 keV). The blackasterisks correspond to transitions published in Ref. [3], whereas the orange asterisks marknewly observed lines.59levels. Therefore, the 408-keV transition is placed on top of the 248-kevlevel, parallel to the 68-keV cascade, and results in a new level at 656 keV.It follows that the 211-keV transition places a level at 866 keV, and the336-keV transition places a level at 991 keV. Also, this places the 1608-keVtransition depopulating the level previously placed at 2263 keV, and the2441-keV transition depopulating the level previously placed at 3097 keV.Since the 1395-keV transition is also seen in coincidence with the 211-keVtransition (not shown here), it is placed on top of the 211-keV transition,between the previously determined levels at 2263 and 866 keV. The 336-keVgate (992 → 656 keV) sees a transition at 1270 keV, which again supportsthe position of the 2263-keV level.Fourthly, we look at gates on the 305-keV and 619-keV transitions (seeFigure 5.11). The total energy of the cascade formed by the 305, 211 and408-keV transitions is ∼924keV, which again corresponds to the energy dif-ference between the 248 and 1173-keV levels. The low-energy range of the305-keV gate (top panel) highlights the difference in intensity between the211-keV and 619-keV transitions, which are both directly fed by the 305-keV transition. This projection is also characterized by several scatter peaks.The middle panel shows coincidences between the 305 and 1089-keV tran-sitions, and the 305 and the 1924-keV transitions, which were both alreadyplaced as feeding the 1173-keV level. Finally, the 619-keV gate (bottompanel) shows coincidences with the 305-keV transition. The sum of their en-ergies is ∼924keV, which fits between the 248 and 1173-keV levels. There-fore, the 619-keV transition is placed underneath the 305-keV transition,which is also linked the parallel cascade with the 408 and 211-keV transi-tions. The 619-keV transition is also fed by transitions at 1395 keV (fromthe 2263-keV level) and 2230 keV (from the 3097-keV level).5.1.3 Decay SchemeThe identified transitions were placed in the level scheme shown in Fig-ure 5.12 and listed in Table 5.1. The four excited levels published in Ref. [3]are confirmed. This updated work using the GRIFFIN spectrometer shows32 new transitions and 11 new levels in 128In. The highest level was foundat 4747 keV, which is just 574 keV lower than the Sn at 5321(155) keV [19].The γ-ray intensities relative to the intensity of the 248-keV transitiongiven in Table 5.1 were measured by diving the efficiency-corrected area ofa peak in β-gated γ-singles by the area of the 248-keV peak:Iγ,rel. =Areaγ/γArea248/248. (5.2)600 200 400 600 800 1000 1200 1400 16000200400600800Counts per 1 keV*211*336Scatter*516Scatter*1395*1608408 keV gate (low)2400 2600 2800 3000 3200 3400 3600 3800 4000 420010−010203040Counts per 1 keV*2441 *2871Scatter31873376ScatterScatter*4092408 keV gate (high)0 200 400 600 800 1000 1200Energy [keV]200−100−0100200300400500Counts per 1 keVScatterScatter *408ScatterScatterScatter Scatter*1270336 keV gateFigure 5.10: β-gated background-subtracted γ-gated spectra observed with the GRIFFINspectrometer for the β-decay of 128Cd. The top and middle panels show the coincidenceswith the 408-keV transition (656→ 248 keV) in 128In, and the bottom panel shows the coin-cident γ-rays with the 336-keV transition (992→ 656 keV). The black asterisks correspondto transitions published in Ref. [3], whereas the orange asterisks mark newly observed lines.61200 300 400 500 600 700200−0200400600800Counts per 1 keVScatter*211Scatter*619305 keV gate (low)1000 1200 1400 1600 1800 2000 2200 2400 260020−10−01020304050Counts per 1 keV *1089Scatter*1924Scatter 305 keV gate (high)0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400Energy [keV]0100200300400500600700800Counts per 1 keV Scatter*305ScatterScatter*1395*2230619 keV gateFigure 5.11: β-gated background-subtracted γ-gated spectra observed with the GRIFFINspectrometer for the β-decay of 128Cd. The top and middle panels show the coincidenceswith the 305-keV transition (1173 → 868 keV) in 128In, and the bottom panel shows thecoincident γ-rays with the 619-keV transition (868 → 248 keV). The black asterisks corre-spond to transitions published in Ref. [3], whereas the orange asterisks mark newly observedlines.6249128 In79(3+) 0 810(30) ms(1-) 248.0 23(2) μs(1-,2-) 316.1(0-,1-,2-) 489.5(0-,1-,2-) 656.3(1-,2-) 710.9(0-,1-,2-) 867.8(0-,1-,2-) 992.41+ 1172.9(1+) 1587.2(1+) 2263.2(1+) 3097.2(1+) 3527.5(1+) 3659.0(1+) 4023.8(1+) 4748.5Sn = 5321(155)128 Cd0+ Qβ- = 6904(153)247.9668.16316.14173.36408.31221.97462.52710.5211.8619.50336.6743.8305.6461.8516.5856.92924.991173.01097.71090.21270.81395.51552.51607.52015.51923.372230.12386.52441.72781.43097.42871.23526.42486.63312.73708.34023.34091.84432.8Figure 5.12: Energy levels [in keV] and γ-ray transitions in 128In following the β-decay of 128Cd.Previously published levels from Ref. [3] are shown in bold black. The colour of the transitionsrepresents the intensity of the γ-ray relative to the 248-keV transition: Iγ > 10% in red, Iγ < 10%in blue, and Iγ < 2% in black. The 1+ assignment of the 1172-keV state was previously publishedin Ref. [3].63It is important to note that this calculation does not depend on the SCEP-TAR efficiency when both areas are obtained in β-gated γ-singles. Since the248-keV isomeric transition is suppressed in β-gated γ-singles, its intensityrelative to the strong 462-keV peak was fitted in γ-singles and then scaledwith the intensity of the 462-keV peak in β-gated γ-singles. The uncertain-ties on the fitted areas and centroids are determined by GRSISort, whilethe uncertainties on the detection efficiency is obtained from the fit shownin Figure 4.9a. The uncertainties are propagated through the calculationsby addition in quadrature. Also, the energy of the levels are determined bytaking the average of the transitions and cascades depopulating the givenlevels. The standard deviation of the cascades energy give the uncertaintieson the level energies.The areas for 17 of the 39 transitions were calculated from γ-ray branch-ing ratios because the statistics in β-gated γ-singles were too low to be seenor a close background peak prevented a clean fit. These ratios represent thefraction of decays which happen through a particular γ-transition when aparticular level is fed and they are obtained in β-tagged background sub-tracted γ-gated spectra. The γ-ray branching ratio BR for a transition γbdepopulating a level is found by gating on a strong transition γa feeding thesame level (i.e. gating from above):BRγb =Iγb(γa)k∑i=1Iγi(γa)=Nγb(γa)/γbk∑i=1Nγi(γa)/γi, (5.3)where Nγb(γa) is the area of the γb peak in the γa gate, and γb and γi arethe detection efficiency at the energies of γb and γi, respectively.When no strong transition is available to gate on from above, the inten-sity of a transition γa feeding a level is calculated by gating on a transitionγb depopulating the same level (i.e. gating from below):Iγa =Nγa(γb)γa · γb ·BRγb. (5.4)The intensity of the bottom 462-keV transition (710 → 248) was calcu-lated from its branching ratio in the gate on the 1552-keV transition feedingthe 710-keV level. The intensity of the top 462-keV transition (1173→ 710)was calculated by subtracting the intensity of the bottom 462-keV transitionfrom the total intensity of the 462-keV doublet in β-gated γ-singles.The absolute intensity of a transition represents the number of timesit occurs per 100 decays and is calculated by dividing the intensity of the64transition by the number of 128Cd decays, which is calculated by fitting thenumber of β-particles from 128Cd decays in Figure 5.4a. In addition, thenumber of β-particles is corrected by subtracting the number of β-particleswhich are observed during the same time in laser-blocked mode (see Fig-ure 5.4b). For these calculations, the intensities of the 68- and 248-keVtransitions have to be corrected for internal conversion. The multipolaritiesand conversion coefficients were used as reported in Ref. [3] (see Table 5.1).The absolute correction factor in the footnote a was determined by dividingthe conversion-corrected absolute intensities by the relative intensities. Theuncertainty on the correction factor was propagated from the calculationsof the two most intense transitions in β-gated γ-singles (68 and 857 keV).The right side of Table 5.1 compares the current results to those fromRef. [3]. There are no significant discrepancies for the level energies, andthe transitions energies and their intensities, except for the 462-keV doublet.Fogelberg et al. measured both transitions at 462.7(3) keV, while the bottomtransition was measured at 462.52(1) keV in the 1552-keV gate and thedoublet was measured at 462.20(2) keV in β-gated γ-singles. In addition,the intensities for both 462-keV transitions are not in agreement as the 462-keV doublet intensities were divided on the basis of the β-decay branchingratio to the 710-keV level by evaluators [14]. However, looking back atFigure 3.1a, intensities of 8.7% are given for both 462-keV transitions byRef. [3], which is very close to our new value of 8(3)%.The β-decay branching ratios, which represent the fraction of the parentdecays feeding a particular level in the daughter, are listed in Table 5.2.These ratios are calculated from the difference between the sum of the ab-solute γ-ray intensities which feed and depopulate a given level. The uncer-tainties are propagated through the calculations by addition in quadrature.In addition, since the β-decay of the 0+ ground state to the 3+ groundstate of 128In would be a second forbidden, direct β-feeding to the 3+ groundstate is assumed to be zero. The sum of the absolute γ-ray intensities ofthe seven transitions decaying to the ground state is 86(3)%, which gives anestimate of ∼14% for all unobserved ground state transitions. The sum ofthe β-decay branching ratios to all states observed is 86(12)%.Also given are the log(ft) values, which represent an important piece ofinformation for assigning spins and parities to the levels. They are calculatedusing the parent ground state energy, the parent half-life, the Qbeta-value, thedaughter level energy, the transition intensity and the transition uniqueness.In order to conservatively account for unseen low-intensity transitions andthe Pandemonium effect, limits on branching ratios and log(ft) values arereported for levels below 1173 keV. Decays to 0− or 1− states below 1173 keV65Table 5.1: γ-ray energies in 128In, their intensities relative to Iγ(247.96) = 100 % and the initialenergy levels are compared to previous work [14].this work Ref. [14]Ei(level) [keV] Eγ [keV] Iγ [%]a Ei(level) [keV] Eγ [keV] Iγ [%]b c Mult. α248.0(1) 247.96(1) 100(6) 247.87 247.92(10) 100 (M2,E3) 0.25(4)316.1(1) 68.16(1) 38(2) 315.86 68.02(10) 38(4) (M1) 1.55316.14(2) 3.6(2)489.5(1) 173.36(5) 2.4(1)656.3(1) 408.31(3) 2.6(1)710.9(4) 221.97(4) 1.7(1) 710.37462.52(1) 8(3) 462.7(3)d 4.8(10)710.5(1) 0.8(1)867.8(4) 211.8(1) 0.09(1)619.50(3) 2.6(1)992.4(7) 336.6(1) 1.2(1)743.8(1) 1.3(1)1172.9(3) 305.6(1) 1.1(1) 1172.88462.20(2) 8(3) 462.7(3)d 3.9(13)516.5(4) 1.1(1)856.92(1) 92(5) 857.05(10) 95(10)924.9(1) 11(1) 925.0(3) 12.4(12)1173.0(1) 10(1) 1172.4(3) 10.8(11)1587.2(2) 1097.7(2) 0.021(3)2263.2(3) 1090.2(1) 0.3(1)1270.8(2) 0.11(2)1395.5(1) 0.84(5)1552.5(1) 0.10(2)1607.5(2) 0.20(2)2015.5(1) 0.07(1)3097.2(6) 1923.37(4) 1.3(1)2230.1(3) 0.14(1)2386.5(2) 0.08(1)2441.7(3) 0.020(3)2781.4(2) 0.10(1)3097.4(5) 0.02(1)3527.5(2) 2871.2(2) 0.06(1)3526.4(4) 0.10(2)3659.0(2) 2486.6(2) 0.29(2)4023.8(8) 3312.7(3) 0.10(2)3708.3(4) 0.04(1)4023.3(5) 0.005(3)4748.5(6) 4091.8(7) 0.010(3)4432.8(2) 0.004(3)a For absolute intensity per 100 decays, multiply by 0.77(4).b For absolute intensity per 100 decays, multiply by 0.76(3).c Uncertainties are assumed as 10% by evaluators [14].d Divided on the basis of Iβ(to 710 level) by evaluators [14].66are first forbidden, while decays to 2− states are first forbidden unique.Unique transitions have larger log(ft) values, which are consistent with thelower limits listed in Table 5.2.Table 5.2 also compares the current results to those from Ref. [3]. Theprevious discrepancy on the intensities of the 462-keV doublet propagates tothe β-decay branching ratios for the 710- and 1173-keV levels. Elekes andTimar re-evaluated the branching ratio to 93(9)% for the 1173-keV level[14], while the 1988 work reported ∼100% [3] and this work sees 75(3)%.This discrepancy does not affect the agreement of the log(ft) value or the1+ assignment for the level. For the 710-keV level, our values increase thelimit to log(ft) > 6.4 from the published log(ft) > 5.8. The fact of onlycomparing limits, along with the significant uncertainties on the several newlow-intensity transitions feeding the level, make it more difficult to come toclear conclusions.5.1.4 Spin AssignmentsThe tentative spin assignments of levels presented in Table 5.2 are based onγ-ray spectroscopy and rely on several pieces of experimental information,including log(ft) values and angular correlations, together with the selectionrules of β-decay and γ-decay.First, considering the 0+ ground state of the even-even parent nucleus128Cd, its β-decay only feeds low-spin states in the daughter. Isobaric analogstates do not exist below the neutron separation energy in this very exoticisotope (48 protons and 84 neutrons) and only Gamow-Teller decays areexpected. Therefore, log(ft) values point to allowed decays, which wouldcorrespond to a spin of 1+, or at least first forbidden decays, which wouldcorrespond to spins of 0−, 1− or 2− [72]. For the 0− and 1− assignments,the β-decays are first forbidden and for the 2−, the decay is first forbiddenunique and the log(ft) values are larger.According to shell model calculations, which will be discussed further inChapter 6, only one positive-parity excited state is expected at low energiesand this 1+ state would be fed by an allowed β-decay, which agrees withthe previous assignment of 1+ for the 1173-keV state. Table 5.2 showsa log(ft) value of 4.06(6) for this state. This value is smaller than whatwould be expected for any forbidden transition and thus we conclude it isan allowed decay. Higher log(ft) values could also be allowed decays butsince they are in the same range than first forbidden transitions, they cannotbe distinguished.For the states above the strongly populated 1+ state at 1173 keV, first67Table 5.2: Level energies in 128In, their β-feeding intensities per 100 decays and the log(ft) valuesare compared to previous work Ref. [14], which include the multipolarity and conversion coeffcientα for the 68 and 248-keV transitions.this work Ref. [14]Elevel [keV] Iβ− [%] log(ft)a Jpi Elevel [keV] Iβ− [%] log(ft) Jpi248.0(1) <5.8 >6.4 (1−) b 247.87 <17 >5.1 (1−)316.1(1) <5.8 >6.2 (1−, 2−) b 315.86 <12 >5.2 (2−)489.5(1) <0.6 >6.56 (0−, 1−, 2−)656.3(1) <0.15 >7.6 (0−, 1−, 2−)710.9(4) <4.2 >6.4 (1−, 2−) b 710.37 <2.0 >5.8 (1−)867.8(4) <0.4 >6.56 (0−, 1−, 2−)992.4(7) <1.5 >5.88 (0−, 1−, 2−)1172.9(3) 75(3) 4.06(6) 1+ 1172.88 93(9) 4.03(9) 1+1587.2(2) 0.013(2) 7.68(9) (1+)2263.2(3) 1.01(4) 5.53(7) (1+)3097.2(6) 1.03(4) 5.15(8) (1+)3527.5(2) 0.10(1) 5.9(1) (1+)3659.0(2) 0.18(1) 5.61(9) (1+)4023.8(8) 0.09(1) 5.7(1) (1+)4748.5(6) 0.006(3) 6.3(3) (1+)a Calculated with the Logft web application [71] using the parent ground state energy, the parent half-life(246.2(21) ms [4]), the Q-value (6.904(153) MeV [19]), the daughter level energy, the transition intensity and thetransition uniqueness.b J = 0 was ruled out by γ-γ angular correlation analysis. (See Section 5.1.4.)forbidden decays would be too weak to be observed and therefore, onlyallowed transitions leading to a spin-parity of 1+ are considered.Assignments of J = 0 were ruled out for the levels at 248, 316 and710 keV based on γ-γ angular correlation analysis, which is described inthe next section. In addition, a tentative multipolarity of (M2, E3) wasobtained from conversion electron measurements for the 248-keV transitionin Ref. [3] (see Table 5.1), which suggests ∆J = 2 and a change of parityfrom the 3+ ground state. This constraint rules out Jpi = 2− and leaves atentative assignment of Jpi = 1− for the 248-keV level.68857-68 keV CascadeThe cascade of the 857-keV (1173 → 316) and 68-keV (316 → 248) tran-sitions has the potential of restricting the spin of the intermediate state at316 keV. The spin assignments of the 1173-kev level (1+) and of the 248-keVlevel (1−) are established, which leaves only the middle state at 316 keV withtwo possible spin-parities (1−, 2−). For the 1+ to (1−/2−) to 1− cascade,the first transition is predominantly E1 and the second transition is predom-inantly M1 (as reported in Ref. [3]). Our log(ft) values agree with both a1 → 2 → 1 cascade, as originally suggested in Ref. [3], and a 1 → 1 → 1cascade, as revised in Ref. [14]. We are trying to answer this question withγ-γ angular correlations.Since the three states involved in these two cascades have non-zero spins,each state has more than one m-substate and each transition has an indepen-dent mixing ratio. These multiple degrees of freedom increase the complexityof this analysis. All combinations of the two mixing ratios are summarizedin Figure 5.13. By looking at a fixed mixing ratio, one can see the range ofpossible values for the a22 and a44 coefficients and hopefully find a range ofvalues for which they do not overlap for different spins. Here, the obviousdifference is that a44 = 0 for a 1→ 1→ 1 cascade with any combination ofmixing ratios. A coefficient a44 = 0 does not rule out a 1→ 2→ 1 cascade,however a44 6= 0 can rule out a 1→ 1→ 1 cascade. Regions of non-overlapfor the a22 coefficient would be a22 < −0.92 and a22 > 0.82.The γ-γ angular correlation data for the 857-68 keV cascade is shownin Figure 5.14. Details on the normalization of the γ-γ angular correlationdata can be found in Appendix B. Fitting to Equation (2.20) extracts exper-imental coefficients of a44 = −0.012± 0.007, which is consistent with zerowithin two standard deviations σ, and a22 = 0.168± 0.005, which is far fromthe regions of non-overlap. While this cascade of the most intense transi-tions provided sufficient statistics (∼14000 counts) to perform an angularcorrelation analysis of the levels involved, the characters and coefficients ofthe possible spins for the 316-keV level are very similar and cannot be dis-tinguished within uncertainty. Therefore γ-γ angular correlations are notwell suited in this case to discriminate between 1 → 1 → 1 and 1 → 2 → 1cascades.It is important to note that the theoretical values shown in Figure 5.13do not take into account some experimental systematic factors, such as theopening angle of the detectors. In measuring angular correlations, the finitesize of the detectors attenuates the observed correlations with coefficientsQ22 and Q44. The a22 and a44 coefficients become Q22a22 and Q44a44. These69attenuation coefficients have been determined experimentally for GRIFFINby measuring the angular correlations for well known cascades [73, 74].The tentative spin assignments of 1− or 2− still hold for the 316-keV leveland the five others between the 1173-keV 1+ state and the 248-keV isomer,to which Ref. [3] assigned a spin of 1−. On the other hand, a level of spin 0can only have an m-substate equal to zero, which translates to an isotropicγ-γ distribution in space. Therefore, an anisotropic angular distributionbetween two coincident γ-rays, such as seen in Figure 5.14, rules out a 0−assignment for the 316-keV level between the 857- and 68-keV transitions.5.1.5 Half-LifeThe half-life of the ground state of 128Cd is measured by fitting the timedistribution of the β-gated γ-ray transitions in the daughter 128In during thebeam-off part of the tape cycle. Figure 5.15 shows the data for the 68-, 857-,462- and 925-keV γ-rays in 128In. The activity was fitted to an exponentialdecay (see Equation (2.10)) and a constant background. In Figure 5.16, thesummed activity of the 857- and 925-keV γ-rays is fitted to:Atotal(t) = Ae− ln 2·t/t1/2,parent +Be− ln 2·t/t1/2,daughter + C (5.5)A previous analysis of the 128Cd half-life from this data set was carriedindependently and published in 2016 [4]. The current analysis finds a half-lifeof 245(3) ms for the 68-keV transition, 247(2) ms for the 857-keV transition,246(5) ms for the 462-keV doublet, and 244(8) ms for the 925-keV transition.The two first values have the most statistics and show a strong agreement,while the later fits have at least a factor 5 less statistics and show higheruncertainties.The different parameters of the fit equation were studied with thesummed activity of the most intense γ-rays (see Figure 5.16). Assumingthat the delivered beam included 4% of 128In with a fixed half-life of 810 ms,the summed fit resulted in a half-life of 245.4(23) ms (fit uncertainty only).This value agrees with the 246.2(21) ms obtained by the sum of the 857-and 925-keV transitions published in [4]. Figure 5.16b shows that letting thedaughter parameters free does not affect the extracted half-life significantly.Changing the binning of the data from 10 ms per bin to 20 and 40 ms perbin did not significantly change the fitted half-life.Finally, systematic uncertainties in the fit are investigated in Figure 5.17.The results for the half-life are plotted as a function of the first and last timebins included in the fitting range. For the fit of the sum of the 857- and925-keV transitions, the range of bins does not have an obvious effect on70(a) a22 for a 1→ 1→ 1 cascade (b) a44 for a 1→ 1→ 1 cascade(c) a22 for a 1→ 2→ 1 cascade (d) a44 for a 1→ 2→ 1 cascadeFigure 5.13: Coefficients and mixing ratios of γ-γ angular correlations for a 1 → 1 → 1 cascade(a-b) and a 1 → 2 → 1 cascade (c-d). The mixing ratio δ1 of the top transition is plotted on thex-axis and the mixing ratio δ2 of the bottom transition, on the y-axis. The a22 and a44 coefficients,on the left and right plots respectively, are displayed on the z-axis [41].71Figure 5.14: Normalized γ-γ angular correlation data and fit for the 857-68keV cascade. The data point at 180◦ is not included in the fit.the result. Therefore, fitting range for Figures 5.15 and 5.16 was set to11.5 to 13.5 seconds, which is the entirety of the beam-off part of the cycle.The standard deviation of the values obtained within the range gives thestatistical uncertainty on the final value: 245.4 (23)(stat.) (0.7)(syst.) ms =245.4(23) ms.5.1.6 248-keV IsomerIn order to identify long-lived states and isomers, one needs to look againat the time difference between β-particles and γ-rays as previously seen inFigures 5.1 and zoomed-in in Figure 5.18. The 248 keV isomer was observedin Figure 5.2 as the most intense transition in 128In and reported with a half-life of 23(2) µs by Ref. [47]. It is seen in the matrix as an asymmetric linearound t = 0 which extends to longer times, with orders of magnitude morestatistics on the positive side. This happens when the β-particle is detectedbefore the γ-ray, i.e. the γ-ray time stamp is higher than the β-particle timestamp. Transitions which are correlated in time are seen as spots centeredaround t = 0, whereas γ-rays which are seen consistently with larger time72(a) 68 keV (b) 857 keV(c) 462 keV (d) 925 keVFigure 5.15: Fitted activity of selected γ-rays in 128In collected during the beam-off part of thetape cycle. The plotted curves show the activity of the parent nuclei 128Cd (red line), a constantbackground (green line) and the sum of the two curves (light blue line). The fitting parametersand reduced χ2 are shown in the respective insets. Half-lives are in seconds.73(a) Fixed 128In half-life and rate (4%) (b) Unfixed 128In half-life and rateFigure 5.16: Fitted activity of the sum of the 857- and 925-keV γ-rays in 128In collected duringthe beam-off part of the tape cycle. The fitting function (light blue line) considers the activitiesof parent nuclei (red line), the indium present in the beam (with t1/2 = 0.810(30) s [34]) (violetline), and a constant background (green line). The fit parameters and reduced χ2 are shown in therespective insets.(a) Number of the first bin included (b) Number of the last bin includedFigure 5.17: Effect of changing the fitting region on the extracted 128Cd half-life.74differences are uncorrelated, such as coming from different decaying nuclides,and are called time-random coincidences.Several long-lived transitions are visible in Figure 5.18. Such lines extendon the positive side at γ-ray energies of 121, 207, 248, 321, 426, 626, 1053,1061 and 1279 keV. Five of the nine lines (121, 207, 321, 426 and 1061 keV)are known transitions in 128Sn which are observed only from the β-decayof the 8− isomer in 128In (0.72(10) s). As discussed in Section 5.1, this8− isomer cannot be populated by the β-decay of 128Cd, but it can benon-resonantly ionized by the first of the three lasers. None of these ninelines are seen when the first laser is blocked, which is consistent with thelaser excitation of the isomer. In addition, the fact that only the 248-keVtransition is more intense during the beam-on part of the cycle relative tothe beam-off part suggests that the eight other lines are most likely notpopulated by the β-decay of 128Cd.75Figure 5.18: Difference between time stamps of β-particles and γ-rays (γ − β) as a function of theenergy of the γ-ray (zoom in). Long-lived transitions are seen at 121, 207, 247, 321, 426, 626, 1053,1061 and 1279 keV.765.2 131Cd5.2.1 β-Gated γ-Singles MeasurementsData from the decay of 131Cd were collected for 32 hours in August 2015with a beam intensity of 0.6-0.8 pps.For the 131Cd data set, the same β-γ coincidence window of 200 ns hasbeen used. The cycle structure shown in Figure 5.19 consists of the tapemove and background measurement (1 s), beam on (10 s), and beam off(1 s). The noise caused by the moving tape collector system during thetape move is clearly visible. The peak before the beam is turned on is noisefrom the tape move. The plotted curves show the number of decays forthe 131Cd (t1/2 = 98(2) ms [34]) isotopes (red line), the first daughter131In(t1/2 = 261(3) ms [34]) (yellow line), the βn-decaying131Cd isotopes into130In (Pn = 3.5(10)% [50]) (violet line), the constant background (greenline), and the sum of the four curves (light blue line).The comparison of β-gated γ-ray spectra observed with lasers on andlaser blocked, which enables the identification of transitions in 131In, isshown in Figure 5.20. Several published transitions from ISOLDE [55] andRIKEN [6] were confirmed, including the transition of 988 keV. 21 of 23 linesseen at RIKEN are confirmed. The current data set does not see the 3177-keV line, which was observed at RIKEN with 1.3(4)% absolute intensity,and thus no level is seen at 4531 keV.Five of the seven transitions seen at ISOLDE [55] are confirmed here,including the four transitions common to both Ref. [55] and Ref. [6] (988,3555/3556, 3866/3869, and 6039 keV). The strong 2434-keV line in 131Sn(Iγ,abs=90%) shows a tail which might hide a 2427-keV line seen in Ref. [55]which is not listed here. However, we did not observe the 844 and 4403-keVlines observed at ISOLDE. It is suspected that the 844 keV line was fromthe decay of the 131Te isomeric state (t1/2=33.25 h).The full range of the β-gated γ-ray spectra is displayed in Figure 5.21.Unidentified lines are observed at 331, 344, 590, 1923, 3042, 4086, and4194 keV. The 331-keV line could possibly be the one in 131Sn, while the344 and 1923-keV lines could be from 131I. The 590-keV line was observed inthe 128Cd data set with the laser blocked as well, but it remained unidenti-fied. It might originate from a long-lived beam contaminant which was stillpresent in the vacuum chamber. Finally, the 3042 and 4086-keV lines areseen in the 131Cd laser-blocked spectra. Figure 5.22 highlights the multipletstructure of the peaks at 3868, 3920, 5753, and 6039 keV with binning of1 keV per division.77(a)(b)Figure 5.19: Number of β-particles as a function of cycle time for the β-decay of 131Cd in (a) laser-on mode and (b) laser-blocked mode. The cycleconsists of the tape move and background measurement (1 s), beam on (10 s)and beam off (1 s).780 200 400 600 800 1000 120002004006008001000120014001600Counts per 1 keV*315*355*433451 (130In)e-/e+590774 (130Sn)779 (131Sn)798 (131Sb)*9881221 (130Sn)Lasers on    0 200 400 600 800 1000 1200Energy [keV]020406080100120140160Counts per 1 keV344e-/e+590779 (131Sn)798 (131Sb)1221 (130Sn)Laser blockedFigure 5.20: Comparison of β-gated γ-ray spectra observed for the decay of 131Cd in ad-dback mode with the lasers on [blue] and laser blocked [red]. No normalization is applied.The peaks with a black asterisk correspond to γ-rays published in Ref. [6].790 200 400 600 800 1000 120002004006008001000120014001600Counts per 1 keV331344451 (130In) e-/e+590774 (130Sn)779 (131Sn)798 (131Sb)*9881221 (130Sn)1500 2000 2500 3000 35000200400600800100012001400Counts per 2 keV1655 (131Sn)1905 (130Sn)19232434 (131Sn)3042*35544000 4500 5000 5500 6000Energy [keV]020406080100120Counts per 2 keV*3868*39203990 (131Sn)40864194 4487 (131Sn)*5525*5753*5796*5825*5958*6003 *6039Figure 5.21: β-gated γ-ray spectra observed for the decay of 131Cd in addback mode. Thepeaks with a black asterisk correspond to γ-rays published in Ref. [6].80(a) 3868 keV (b) 3920 keV(c) 5753 keV (d) 6039 keVFigure 5.22: β-gated γ-ray spectra around peaks with multiplet structures in the decay of 131Cd.All displayed in 1 keV per bin.81β-Delayed Neutron Decay to 130InThe βn-decay of 131Cd is investigated by looking at the four most intensetransitions in 130In based on Ref. [45]. Figure 5.23 shows β-gated γ-rayspectra around 451 keV (Iγ,rel=100%), 1669 keV (Iγ,rel=99.8%), 950 keV(Iγ,rel=22.5%), and 1170 keV (Iγ,rel=20.4%). The relative intensities of thetransitions Iγ,rel are quoted in parentheses [1].The βn-feeding from 131Cd (7/2− ground state) to 130In populates higherspin states than the β-decay of 130Cd (0+ ground state) to 130In and, there-fore, the transition intensities are different in the two decays. The only clearpeak is the one seen at 451 keV. While Ref. [6] listed a 451-keV transition,the peak observed in this work is assumed to be in 130In and not in 131In.This question can only be solved by future inclusion of neutron-tagging, e.g.β-n-γ coincidences with the DESCANT4 detector at TRIUMF [75].The partial decay scheme for the β-decay of 130Cd presented in Fig-ure 5.24 shows the three levels which are involved with the four most intensetransitions. The spin and parity of the states which could be populated in130In depend on the spin and parity of the excited states above the neu-tron separation energy populated in 131In and the angular momentum ofthe emitted neutron. The observed 451-keV transition suggests that theβn-decay of 131Cd (7/2−) populates the 2− level with an l = 1 neutron.The population of the 1+ level via l = 2 neutrons is not observed, so we donot see the intense peaks at 1669, 1170 and 950 keV. The non-observanceof the 950 keV transition in our data also excludes that the 950-keV statehas a spin of 1. Finally, the 3+ isomer at 388 keV should be populated byl = 0 neutrons. However, the 4.4(2)-µs transition [45] mostly decays outsideof the β-γ coincidence window of 200 ns.5.2.2 β-Gated γ-γ Coincidence MeasurementsCoincidences are investigated with the β-gated γ-γ coincidence matrix inFigures 5.25 and 5.26, and in the β-gated background-subtracted projectionsin Figure 5.27. These figures highlight the limited coincidence statistics forthis data set.The coincidence between the 988-keV and 3290-keV transitions, whichwas published in Ref. [6], is confirmed here in both Figure 5.26d at 1 keVper bin and in the projection at 2 keV per bin (Figure 5.27). No newcoincidences can be observed with the most intense transition at 988 keV,while five transitions were observed in coincidence at RIKEN and placed4DEuterated SCintillator Array for Neutron Tagging82(a) 451 keV (Iγ,rel=100%) (b) 1669 keV (Iγ,rel=99.8%)(c) 950 keV (Iγ,rel=22.5%) (d) 1170 keV (Iγ,rel=20.4%)Figure 5.23: β-gated γ-ray spectra around possible transitions in 130In from the βn-decay of 131Cd.The absolute intensities of the transitions Iγ,rel quoted in parentheses are reported from Ref. [45]for the β-decay of 130Cd. All displayed in 1 keV per bin. Only the 451-keV transition can beobserved due to the selection rules.8349130 In811- 0(3+) 388(2-) 451(1-, 2-) 9501+ 2120388(7.6%)451(100%)950(22.5%)1170(20.4%)1669(99.8%)Figure 5.24: Partial decay scheme for the β-decay of 130Cd [45](t1/2=126(4) ms [4]). Energies are displayed in keV.in their level scheme: at 2637 (4 counts), 2777 (3 counts), 3178 (4 counts),3290 (8 counts), and 3417 keV (4 counts). Also, Ref. [6] saw 3 counts around2908 keV, but did not list this energy in their list of transitions and did notplace it in their level scheme. This work observes 50% more counts at 3290-keV (14 counts), but does not see other coincidences with the 988 keV.In addition, Ref. [6] saw coincidences between the transitions at 3555 keVand 315 keV, and between the 5525 keV and 433 keV transitions. Fig-ures 5.25b and 5.25c zoom in at these energies on the coincidence matrix.However, the current data set does not see coincidences between the tran-sitions at 3554 keV and 315 keV, or at 5527 keV and 433 keV. The β-gatedbackground-subtracted gate at 315-keV in Figure 5.27 shows 3 counts at843042 keV and 3 counts at 2648 keV, however these counts are consistentwith the statistical fluctuations of the surrounding background. Therefore,the statistics available with this data set cannot confirm the coincidences ob-served in Ref. [6], except for the coincidences between the 988and 3290-keVtransitions.Advantages of the EURICA dataset are their gates on one isotope intheir particle identification plot as well as on the time of implantation in thedetector, which make their spectrum cleaner than ours. Although this dataset has a ∼50% higher statistics than the EURICA data set, our backgroundconditions do not allow a better identification of coincident states. Theinclusion of the new GRIFFIN detector shields fabricated from the high-density scintillator bismuth germanate (BGO) will help for future low-rateexperiments [66].5.2.3 Decay SchemeFigure 5.28 presents a revised level scheme for 131In with 11 transitions and11 excited states. Taprogge et. al [6] claimed that they saw 23 transitionsand 19 levels, 12 of which were based on single transitions placed as feedingdirectly the ground state without further justification. However, withoutcoincidence data or knowledge about their multipolarity, transitions cannotbe placed with confidence and it is difficult to tell whether they are feedingthe 1/2− isomeric state at 365 keV or the 9/2+ ground state.First, the only coincidences observed were between the transitions at988 and 3290 keV, which place a level at 4643 keV. While the four othercoincidences observed at RIKEN (2637, 2777, 3178, and 3417 keV) were notseen here, their coincidences are assumed to be correct and levels are placedfor the three transitions which this work saw. Therefore, a level is placed at3920 keV with the 2636-keV transition, one at 3989 keV with the 2778-keVtransition, and one at 4768 keV with the 3414-keV transition.When coincidence data is not available, transitions can be placed in thelevel scheme by looking for differences in level energies which could matchthe energies of particular transitions. Knowing that the 1/2− isomer wasrecently measured at 365(2) keV [53] above the 9/2+ ground state, it islikely that the transitions at 3554 and 3920 keV (which are ∼366 keV apart)depopulate the same level with the first feeding the 1/2− and the second,the 9/2+ ground state. The new excited level would then be at 3920 keVand seems to be the only level connecting both the isomer and ground state.Based on observed coincidences at RIKEN, Taprogge et. al places the315-keV transition in a cascade from levels at 3869 to 3555 keV, and then85(a)(b) (c)Figure 5.25: Symmetrized β-gated γ-γ coincidence matrix for 131Cd data:(a) full matrix, (b) zoomed in on 315 keV and the region around 3554 keV,and (c) zoomed in on 433 keV and the region around 5527 keV. All aredisplayed in 1 keV per bin.86(a) 988 and 2636 keV(b) 988 and 2778 keV (c) 988 and 3178 keV(d) 988 and 3290 keV (e) 988 and 3417 keVFigure 5.26: Symmetrized β-gated γ-γ coincidence matrix for the 988-keV transition in 131Cd:988 keV and (a) 2636 keV, (b) 2778 keV, (c) 3178 keV, (d) 3290 keV, and (e) 3417 keV. Alldisplayed in 1 keV per bin.87via a 3555-keV transition to the ground state. Since the 3554-keV transitionwas placed as feeding the 1/2− isomer at 365 keV, the 315-keV transition isplaced as feeding the 3920-keV level from a level at 4235 keV. The energydifference between the 4235-keV level and the 365-keV isomer is ∼3870,which matches the energy of the transition observed at 3868/3870 keV.Finally, if the transitions at 5958, 6003, and 6038 keV were feeding theisomer at 365 keV, their energy would lie 110, 155, and 190 keV above theSn-value at 6213(38) keV. However, it is possible to see transitions slightlyabove the Sn-value, such as the four transitions in130In with Iγ,rel. ∼ 1%which are seen up to 463 keV above Sn=5120(40) keV [45].The energies and intensities of the γ-rays are listed in Table 5.3. Theintensities were fitted in 2 keV/bin, which combined the multiplet structuresof the lines at 3866, 3920, 5753, and 6039-keV (see Figure 5.22). Therefore,their listed intensities might be overestimated. It is worth noting that notonly there are groups of transitions with similar intensities but also of similarenergies. There is a group of seven transitions with energies in the 3290 to4040-keV range and seven transitions in the 5527 to 6038-keV range whichmight have a similar configuration as discussed in Ref. [6].Several transition and level energies differ from the previous works inRef. [54, 55, 56] by up to 2 keV for the 5727/5725 transitions. The lowstatistics can affect the centroid of the fitted peaks and increase its un-certainty. Also, most transitions observed are above the energy range of53–3450 keV which was used for the energy and efficiency calibrations. Theenergy of the levels were rounded to the closest integer, with an uncertaintyof 1 keV, and 2 keV for doublet structures.There are several discrepancies between the relative intensities calculatedin this work and the ones from Ref. [6]. The absolute intensities rely on thenumber of β-particles from the decays of 131Cd and 131In, the βn-decay of131Cd, and on the background fitted in Figure 5.19a. In addition, the num-ber of β-particles is corrected by subtracting the number of β-particles whichare observed during the same time in laser-blocked mode (see Figure 5.19b).Again, the low statistics add to the uncertainty of this value. For the EU-RICA data, the 131Cd decays are identified in-flight event-by-event basedon energy loss, time-of-flight measurement and magnetic rigidity in a par-ticle identification plot. In addition, the time of the implant is recordedand allows an additional coincidence constraint, so that their implant-β-γ-γcoincidence plots are cleaner than our β-γ-γ spectra in Figure 5.27.882000 2500 3000 3500 4000 45002−1−012345Counts per 2 keV*3290   988 keV gate 2000 2500 3000 3500 4000 45003−2−1−0123Counts per 2 keV  315 keV gate2000 2500 3000 3500 4000 4500 5000 5500 6000Energy [keV]3−2−1−01234Counts per 2 keV  433 keV gateFigure 5.27: β-gated background-subtracted γ-gated spectra observed with the GRIFFINspectrometer for the β-decay of 131Cd. The top panel shows coincidences with the 988-keV transition (1353 → 365 keV) in 131In for Eγ= 1800-4500 keV, the middle panel showscoincidences with the 315-keV transition for Eγ= 1800-4500 keV, and the bottom panelshows the coincident γ-rays with the 433-keV transition for Eγ= 1800-6000 keV.8949131 In82(9/2+) 0(1/2-) 365(3/2-) 1353(21/2-) 3764(17/2-) 3782398941324643476839604237Sn 6213(38)131 Cd0+ Qβ- = 12806(103)0.28(3) s0.35(5) s0.32(6) s630(60) ns988.22635.72778.73290.234183554.43919.8315.43867.8Figure 5.28: Energy levels [in keV] and γ-ray transitions in 131In followingthe β-decay of 131Cd. Previously published levels (365, 1353, and 4643 keV)from Ref. [6] are shown in bold black. The colour of the transitions representsthe intensity of the γ-ray relative to the 988-keV transition: Iγ > 10% inred, and Iγ < 10% in blue. The 17/2− and 21/2− states are reported inRef. [52] and are not populated by β-decay in this work.90Table 5.3: γ-ray energies in 131In, their intensities relative to Iγ(988) = 100 %, absolute intensities per100 decays, and the initial energy levels are compared to previous work Ref. [6].this work Ref. [6]Eγ [keV] Iγ(rel.) [%]a Iγ(abs.) [%] Ei [keV] Eγ [keV] Iγ(rel.) [%] Iγ(abs.) [%] Ei [keV]315.4(8) 28(7) 1.9(4) 4237(1) 315.0(2) 25(12) 2.2(10) 3869(1)354.9(7)b 17(7) 1.2(4) - 355.0(2)c 12(4) 1.1(4) -433.3(7)b 28(7) 2.0(4) 5958(2) or 6323(9) 433.2(2) 27(7) 2.4(6) 5958(1)451(2)c,d 11(5) 1.0(5) -988.2(1)* 100(16) 6.9(8) 1353.2(1) 987.9(2)* 100(15) 8.8(17) 1353(8)2004.8(3)b 14(5) 1.0(3) - 2004.7(7)c 18(5) 1.6(5) -2635.7(7) 9(3) 0.6(2) 3989(1) 2636.7(7) 7(3) 0.6(2) 3990(8)2778(2) 3(3) 0.2(2) 4132(1) 2776.8(7) 12(3) 1.0(5) 4130(8)3177.6(13) 15(5) 1.3(4) 4531(8)3290.2(9) 15(5) 1.0(3) 4643(1) 3290.1(7) 26(8) 2.3(7) 4644(8)3418(2) 9(5) 0.6(3) 4772(8) 3417.0(7) 8(4) 0.7(4) 4770(8)3478.9(5)b 19(6) 1.3(4) 3478(1) or 3844(8) 3480.0(7) 33(8) 2.9(7) 3480(1)3554.4(3)* 114(15) 7.9(6) 3920(2) 3555.2(10)* 98(16) 8.6(15) 3555(1)3867.8(4)d,* 149(20) 10.3(8) 4237(2) 3869(3)d,* 115(19) 10.1(16) 3869(3)3919.8(8)d 46(8) 3.2(5) 3920(2) 3920.6(11) 46(12) 4.1(10) 3921(1)4040(1)b 24(6) 1.7(4) 4040(1) or 4405(8) 4038.0(10) 36(15) 3.2(13) 4038(1)5527.2(4)b 63(11) 4.3(6) 5527(1) or 5892(8) 5525.0(11) 29(9) 2.5(8) 5525(1)5752.8(7)b,d 64(13) 4.4(8) 5752(1) or 6118(8) 5754.7(11) 64(18) 5.6(16) 5755(1)5795(2)b 22(6) 1.5(4) 5795(2) or 6160(8) 5796.0(11) 29(9) 2.5(8) 5796(1)5824.1(8)b 43(9) 3.0(5) 5824(1) or 6189(8) 5824.7(10) 87(24) 7.7(21) 5825(1)5958(2)b 6(4) 0.4(2) 5958(2) or 6323(9) 5958.3(12) 25(9) 2.2(7) 5958(1)6003(3)b 20(5) 1.4(3) 6003(3) or 6368(9) 6002.8(11) 33(13) 2.9(12) 6003(1)6038(2)b,d 56(10) 3.9(6) 6038(2) or 6403(8) 6039.2(10)* 63(20) 5.5(18) 6039(1)a For absolute intensity per 100 decays, multiply by 0.041(1).b Not placed in this work’s decay scheme, see text.c Not placed in the decay scheme from Ref. [6].d Doublet structure.* Transitions previously observed in Ref. [54, 55, 56].91The β-branching ratios and log(ft) values are presented in Table 5.4.Since the absolute γ-ray intensities and the decay scheme are different fromthe published scheme from EURICA [6], it follows that the β-branchingratios and log(ft) values differ as well. In order to conservatively accountfor unseen low-intensity transitions and the Pandemonium effect, limits onbranching ratios and log(ft) values are reported for levels up to 6403 keV.For the 1353-keV level, the intensities of the four transitions feeding thelevel were included in the calculation of the β-branching ratio since theyare seen in γ-singles, even if they are not seen in the coincidence data. The3920-keV and 5527(1)/5892(8) levels are the only two other levels for whichtransitions are placed feeding (with the 315-keV and 433-keV transitions,respectively) and depopulating the level. Decays from 7/2− to 3/2− wouldbe second forbidden and so, the state at 1353 keV is not directly fed by β-decay and the log(ft) value is expected to be much higher (∼10). Therefore,the missing intensity comes from other weaker transitions to this level thathave not yet been observed.For this work, the states at 3920 and 4237 keV have the two lowest log(ft)values below 4771 keV, which are consistent with allowed decays from the7/2− ground state of 131Cd to 5/2− or 9/2− states. The 9/2− state wouldhowever be expected at much higher energy, around 6 MeV. Overall, it is notpossible to firmly make out the spins of these two states from this analysis.The β-feeding to the 9/2+ ground state is estimated by subtractingthe sum of the observed β-feeding intensities and the βn-branching ratio(Pn = 3.5(10)% [50]) from 100%. Here, the sum of the absolute intensi-ties observed is 46(2)%. Therefore, we can place a limit of < 53% for theβ-feeding to the ground state through a first forbidden transition (7/2− to9/2+). By including the 20 transitions in their level scheme, Ref. [6] esti-mated ∼ 30%.92Table 5.4: Level energies in 131In, their β-feeding intensities per 100 decays and the log(ft) valuesare compared to previous work Ref. [6].this work Ref. [6]Elevel [keV] Iβ− [%] log(ft)a Jpi Elevel [keV] Iβ− [%] log(ft) Jpi0 <53 >4.9 (9/2)+ 0 ∼30 ∼5.6 (9/2)+365(8)b - - (1/2)− 365(8)b - - (1/2)−1353.2(1) <3.4 >6.32 (3/2)− 1353(8) 2.9(20) 6.4(3) (3/2)−3478(1) or 3844(8) <1.7 >6.4 3480(1) 2.9(7) 6.0(1)3555(1) 6.4(18) 5.6(1)3869(3)c 12.3(19) 5.3(1)3920(2) <9.4 >5.4 3921(1) 4.1(10) 5.8(1)3989(1) <0.8 >6.45 3990(8) 0.6(2) 6.6(2)4040(1) or 4405(8) <2.1 >6.03 4038(1) 3.2(13) 5.8(2)4132(1) <0.4 >6.5 4130(8) 1.0(5) 6.3(2)4237(1) <12.5 >5.214531(8) 1.3(4) 6.1(2)4643(1) <1.4 >6.09 4644(8) 2.3(7) 5.8(2)4771(1) <0.9 >6.2 4770(8) 0.7(4) 6.3(3)5527(1) or 5892(8) <3.1 >5.47 5525(1) <1.3 >5.95752(1) or 6118(8) <5.2 >5.21 5755(1) 5.6(16) 5.2(1)5795(2) or 6160(8) <1.9 >5.6 5796(1) 2.5(8) 5.5(2)5824(1) or 6189(8) <3.5 >5.37 5825(1) 7.7(21) 5.0(1)5958(2) or 6323(9) <0.7 >6.06 5958(1) 4.6(9) 5.2(1)6003(3) or 6368(9) <1.7 >5.63 6003(1) 2.9(12) 5.4(2)6038(2) or 6403(8) <4.5 >5.19 6039(1) 5.5(18) 5.1(2)a Calculated with the Logft web application [71] using the parent ground state energy, the parent half-life (98(2) ms[34]), the Q-value (12806(103) keV [19]), the daughter level energy and the transition intensity.b Trap measurement of the 1/2− isomer in Ref. [53].c Two close-lying states.935.3 132Cd5.3.1 β-Gated γ-Singles Measurements132Cd data was collected for 18 hours at an intensity of 0.15 pps based onthe yield of the 989-keV peak. A β-gated spectrum of this data can be seenin Figure 5.29.The same timing settings were used as for the 131Cd data set, that is aβ-γ coincidence window of 200 ns and a cycle structure consisting of the tapemove and background measurement (1 s), beam on (10 s) and beam off (1 s).The most dominant peak in Figure 5.29 is at 511 keV, from electron/positronpair production, followed by peaks at 973.9(1), 696.8(1) and 989.6(2) keV.These three peaks are known to be the three most intense transitions inthe β-decay of the 2.79-minute state 132Sb, with respective intensities of99(5)%, 86(5)% and 14.9(15)% [76]. Other strong transitions observed inthe data set are at the five most intense transitions in the β-decay of 132Sn(t1/2=39.7 s): 340.53(5) keV (Iγ,abs=49%), 85.58(8) keV (Iγ,abs=48.2(10)%),899.04(5) keV (Iγ,abs=44.8(25)%), 246.87(5) keV (Iγ,abs=42.3(20)%), and992.66(8) keV (Iγ,abs=36.9(20)%).132Sb and 132In are isobars but closer tostability than 132Cd (see Figure 2.4).From the 2014 RIKEN publication [5], a high branching ratio of βn-decay of 132Cd to 131Cd was expected (Pn-value close to 100%), meaningthat the spectrum should include common transitions with the β-decay of131Cd, which was discussed in the previous section. The comparison ofthe spectra from the 131Cd and 132Cd data sets is presented in Figure 5.30around 988 keV, which is the highest intensity line observed in the β-decayof 131Cd. The top panel shows γ-singles without the β-gate, highlighting theagreement of the energy calibration in both data sets and the amplitude ofthe background suppression enabled by the β-gating with SCEPTAR. Thebottom panel shows that while the 131Cd data (in red) shows a peak at988 keV, the peak at 989 keV in 132Cd (in blue) shows a more complicatedstructure. The peak at 989 keV has contributions from the 989 keV transi-tion in 132Te and from the 993-keV transition in 132Sb. Both the 988 and989-keV peaks have a FWHM of 2.5 keV.It is important to note that 132Sb has two β-decaying states [77]: a 4+ground state (t1/2=2.79(7) min [76]) and a 8− isomer (t1/2=4.10(5) min[76]). Ref. [76] reports that the 989-keV line is only seen in the 2.79-minstate decay with an intensity of 15% (see Figure 5.31). However, Ref. [77]reports that the intensity of the 989-keV line drops to 9% when the two132Sb decaying states are present. The 4.10-min state has three lines which940 200 400 600 800 1000Energy [keV]0100200300400500600700Counts per 1 keVe-/e+697 (132Te)974 (132Te)989 (132Te)341 (132Sn)86 (132Sn)899 (132Sn)247 (132Sn)Figure 5.29: β-gated γ-ray spectrum observed for the decay of 132Cd in addback mode withthe lasers on.are not present in the 2.79-min state decay: 150 keV (Iγ,abs=70%), 496 keV(Iγ,abs=13%) and 1041 keV (Iγ,abs=18%). These lines are not observed inthis data set, suggesting that the 8− isomer was not produced.The characteristic activity curves for the individual γ-rays provide ad-ditional information on the isotope they were emitted from. In Figure 5.32,the activity of selected γ-rays are plotted as a function of time within a cycle.In the A = 131 data, the line at 988 and 433 keV follow the same behaviour:a fast increase when the beam is turned on. This points to a short half-life,consistent with the 131Cd decay (t1/2=98(2) ms [34]). In the A = 132 data,the lines at 974 and 697 keV follow a different behaviour: a slow increasewhen the beam is turned on, suggesting a longer half-life, consistent with132Sb (t1/2=2.79 min). However, the 989-keV line (Figure 5.32e) might showa faster increase than the 974- and 697-keV transitions, but there is no ev-95950 960 970 980 990 1000 1010 1020 10301520253035310×Counts per 1 keV964 (228Ac)968 (228Ac)10001014Gamma-ray singles950 960 970 980 990 1000 1010 1020 1030Energy [keV]50100150200250300350Counts per 1 keV 974 (132Te)988 (131In)989 (132Te)Beta-gated singlesFigure 5.30: Comparison of 131Cd [red] and 132Cd [blue] data sets around988 keV in γ-singles [top] and β-gated γ-singles [bottom panel].9652132 Te800+ 02+ 9744+ 16716+ 1774(2+, 3, 4+) 24882764974(99%)697(86%)103(13.9%)816(10.9%)989(14.9%)Figure 5.31: Partial decay scheme for the β-decay of 132Sb (t1/2=2.79 min)[76, 77]. Energies are displayed in keV.idence of a decrease when the beam is turned off. This suggests that the989-keV peak from 132Sb might be mixed with a shorter-lived line.5.3.2 β-Gated γ-γ Coincidence MeasurementsCoincidence data can provide extra information on which isotopes contributeto the peak at 989 keV. Figure 5.33 shows four regions of the coincidencematrix around the 974 and 989 keV lines, displayed on the x-axis. Thereis a faint increase in coincidence between the 974- and 989-keV transitions,which can be seen in the symmetrized matrix at (974,989) and (989,974)(Figure 5.33a). Figures 5.33b and 5.33d show coincidences between 974 and816 keV, and 974 and 696 keV. There are no coincidence counts observed97(a) 988 keV in 131Cd (b) 974 keV in 132Cd(c) 433 keV in 131Cd (d) 697 keV in 132Cd(e) 989 keV in 132CdFigure 5.32: Comparison of the activity of selected γ-rays in the 131Cd and 132Cddata sets.98between 989 keV and 816 keV or 697 keV.These coincidence relationships agree with the decay scheme of 132Sb(2.79 min) in literature (see Figure 5.31). While our data sees coincidencesbetween the ground state transition at 974 keV and 697, 816 and 989-keV,the 989-keV transition should also be seen as coincident with the 697-keV.However, it would be reasonable to miss the coincidence between the 989-and 103-keV transitions considering the low statistics of the data set andhigh background at low energies (see Figure 5.33c).In conclusion, there is no clear evidence of 132Cd in the data, howeverthere are open questions remaining about the 989-keV peak. Mainly, thatit is only seen in coincidence with only two of three gamma-rays in 132Sb.Because we see lines from the decays of 132Sb and 132Sn, the beam seemsto have been highly contaminated with several isobars. While it cannot beruled out that there is a marginal amount of 132Cd with a smaller intensitythan we see the longer-lived A = 132 isobars, it cannot be confirmed thatthe transition at 988 keV from the βn-decay of 132Cd was observed.99(a) 974 and 988 keV (b) 974 and 816 keV(c) 974 and 103 keV (d) 974 and 696 keVFigure 5.33: Symmetrized β-γ-γ coincidence matrix for A = 132 data: (a) 974 and 988 keV, (b)974 and 816 keV, (c) 974 and 103 keV, and (d) 974 and 696 keV. All displayed in 1 keV per bin.100Chapter 6Shell Model CalculationsNuShellX is a set of computer codes that can calculate energies, eigenvectorsand spectrosopic overlaps for low-lying states in shell model Hamiltonianmatrix calculations [13]. NuShellX@MSU is another set of codes used togenerate input for NuShellX using data files for model spaces (.sp inputfiles) and Hamiltonians (.int input files). The latter codes also convert theNuShellX output into figures and tables for energy levels, γ-decay and β-decay.The NuShellX code uses a proton-neutron basis with Hamiltonians ofthe form:H = Hνν +Hpipi +Hpiν , (6.1)where Hνν is the neutron-neutron interaction, Hpipi is the proton-protoninteraction, and Hpiν is the proton-neutron interaction.For the neutron-rich In isotopes, the calculations allow the nucleons tointeract in the model space composed of four pi- and five ν-orbitals above a78Ni core, as shown in Figure 2.2: pi(1f5/2, 2p3/2, 2p1/2, 1g9/2) and ν(1g7/2,2d5/2, 2d3/2, 3s1/2, 1h11/2). This model space is called jj45pn (jj-coupling,4pi-5ν orbitals, proton-neutron coupling).The first of two interaction files which are considered in this work is thejj45pna interaction included in the NuShellX@MSU package. The singleparticle energies (SPE) of this interaction, which are displayed in MeV onthe right in Figure 2.2, were adjusted to experimental data available in the132Sn region [78, 79, 80].A recent ab initio framework for the nuclear shell model uses the In-Medium Similarity Renormalization Group (IMSRG) [7, 8, 9]. While theIMSRG uses the same model space as jj45pna, the valence shell modelHamiltonians are constructed from first principles which result in a differentinteraction and SPEs.Since physics can be shifted between SPEs and two-body matrix elementsin phenomenological fits, and SPEs themselves are not observable, we wouldnot necessarily expect the ESPEs from the jj45pna interaction to be the sameas those from the IMSRG. More meaningful would be the spacings between101Table 6.1: Single-Particle Energies for the jj45pn model space for the NuShellX (jj45pna) andIn-Medium Similarity Renormalization Group (IMSRG) interactions.1pif5/2 1pig9/2 2pip3/2 2pip1/2 1νg7/2 2νd5/2 2νd3/2 3νs1/2 1νh11/2jj45pna -0.71660 1.11840 1.12620 0.17850 5.74020 2.44220 2.51480 2.17380 2.67950IMSRG -20.9571 -18.1975 -17.6453 -12.6203 -3.14284 -4.1184 -3.23028 -3.77358 2.6652levels, and Table 6.1 shows that there is general agreement between the twointeractions.This chapter compares the experimental results for 128In with calculatedexcitation energies, level occupancy and effective single particle energies(ESPE) from the jj45pna and IMSRG interactions. Finally, the experimen-tal results for 131In are compared with calculated excitation energies andlevel occupancy from both interactions.6.1 128In6.1.1 Level EnergiesThe excited states in 128In found experimentally are compared to levelscalculated (.lpt output file) in Figure 6.1. Since 128In is an odd Z, oddN nucleus with one pi-hole and three ν-holes in the jj45pn model space,more excited states are expected when compared to even-odd, odd-even oreven-even nuclei. The unpaired nucleons can be in different single-particlestates and for each of these different proton and neutron configurations, amultiplet of states results from the spin coupling of unpaired protons andneutrons.First, one can see that the 3+ ground state is reproduced in the jj45pnacalculation. In the IMSRG calculation, the first 3+ state, which should bethe ground state, is seen at 290 keV. Since the gap observed between thepositive and negative parity spectra changes with increasing the number ofthree-nucleon (3N) matrix elements, this means that the calculations havenot yet converged. However, since the spacings between levels of similarparity are largely converged, energy difference between same parity statescan be compared. Calculations in a bigger space for the three-nucleon (3N)forces are required to converge the gap, however such calculations are beyondthe available computing resources at this time. The truncation of the chiralEFT expansion and of the many-body operators in the IMSRG formalism102Experimental 49128 In79(3+) 0(1-) 248(8-) 262(1-,2-) 316(0-,1-,2-) 489(0-,1-,2-) 656(1-,2-) 711(0-,1-,2-) 868(0-,1-,2-) 9921+ 1173(1+) 1587(1+) 2263(1+) 3097(1+) 3528(1+) 3659(1+) 4024(1+) 4749jj45pna3+ 01- 3352- 4381- 6031+ 6592- 6760- 8072- 9638- 10250- 11021+ 22421+ 23711+ 25971+ 27651+ 28171+ 30981+ 3284IMSRG (-)1- 02- 1128- 4532- 7940- 8490- 10131- 12441- 1442IMSRG (+)3+ 2901+ 20461+ 24541+ 28731+ 35701+ 38921+ 40871+ 44121+ 4463Figure 6.1: Comparison of excitation energies [in keV] in 128In between thiswork [left], NuShellX (jj45pna) [center], and In-Medium Similarity Renor-malization Group (IMSRG) [right]. Positive parity states are shown in redand negative parity states in blue.are the main sources of uncertainty in these calculations, and efforts areongoing to quantify their impact [81].For the jj45pna interaction, the first 1+ state, at 659 keV, is 514 keVlower than the experimental one, at 1173 keV. On the IMSRG side, the first1+ state is 1756 keV above the first 3+ state, which is 583 keV higher thanthe experimental level. Therefore, both interactions are shifted by ∼500 keV103relative to the experimental level, with a discrepancy of 1097 keV betweenthem.The experiment saw seven states with tentative assignments 1+ abovethe highly populated 1173 leV level and so Figure 6.1 compares the secondto eighth 1+ calculated states. The main feature is the density at which thestates are seen: they all lie between 2242 and 3284 keV with the jj45pnainteraction, and spread out between 2164 and 4173 keV with IMSRG. Thestructure of the IMSRG level scheme is closer to the experimental scheme.For example, the first and second 1+ states are seen 408 keV apart, whilethe experiment saw a difference of 414 keV.A few spin assignments are possible for the seven negative parity statesobserved experimentally. On the calculations side, only the spins of the firsttwo negative parity states agree, whereas the five next states have differentspins. Both interactions see two 0− states; however jj45pna calculates one1− and two 2− states, and IMSRG, one 2− state and two 1− states.Finally, there is the 8− β-decaying isomer at 262(13) keV [49], whichwas not populated in this decay spectroscopy experiment. It is calculatedat 1025 keV with the jj45pna interaction and at 453 keV with the IMSRGinteraction. However, since the gap between both parity spectra is notconverged, the energy of the 8− state should be at least 290 keV higher,which would bring it to over 743 keV. Even with this gap, the IMSRG valueis closer to the experimental value than the jj45pna.6.1.2 ConfigurationsInformation on the configurations of the excited states is found in the decom-position of the wave functions of the states (.ls output files). In Table 6.2,the excited states are decomposed by listing the total spin of the protons Jpi,the total spin of the neutrons Jν , and the fraction of the particular couplingbetween Jpi and Jν for this state. Therefore, the wave functions follow theequation:|Ψ(JΠn )〉 =∑ici|ψ(Jpi⊗Jν)〉, (6.2)where c2i are the coefficients listed in Table 6.2 for both interactions.For the 3+1 ground state, both calculations agree that the main config-uration comes from the coupling of Jpi = 9/2 and Jν = 3/2 with 48.66%(jj45pna) and 66.39% (IMSRG). This is consistent with the pig−19/2 ⊗ νd−13/2configuration published in Ref. [47].Both calculations also agree on the main configuration of the 1+1 state,which comes from the coupling of Jpi = 9/2 and Jν = 7/2 with 80.33%104(jj45pna) and 69.60% (IMSRG). This is consistent with the pig−19/2 ⊗ νg−17/2configuration published in Ref. [1]. One can see that a transition from 1+1to the 3+ ground state represents a neutron moving from 1g7/2 to 2d3/2.Two configurations have high representations for the 1+2 state: (Jpi =1/2)⊗(Jν = 3/2) and (Jpi = 9/2)⊗(Jν = 7/2). For the 1+3 state, the mainconfiguration is again pig−19/2⊗νg−17/2, like the 1+1 state. The next five 1+1 statesshow mixed configurations, for which the highest contribution to the wavefunction arises from different couplings with the Hamiltonians from jj45pnaand IMSRG.A similar conclusion is reached for the 8−1 state, where jj45pna calculatesa representation of 52.58% for (Jpi = 9/2)⊗(Jν = 7/2) and IMSRG, 78.22%for (Jpi = 9/2)⊗(Jν = 11/2).Both calculations describe the main configuration of the 2−1 state as(Jpi = 9/2)⊗(Jν = 11/2), and the main configuration of the 2−2 state as(Jpi = 1/2)⊗(Jν = 3/2). For the third 2− state, which is not displayed forIMSRG in Figure 6.1, configurations are mixed and no configuration has afraction higher than 29.33%.The main configuration of the first 1− state is described by the coupling(Jpi = 9/2)⊗(Jν = 11/2) by both calculations, which then disagree on thenext two 1− states. The isomeric transition between the first 1− state andthe 3+ ground state represents the transition of a neutron from 1h11/2 to2d3/2.Finally, the coupling configurations for the two first 0− states are largelymixed and the highest representation of each does not agree between thetwo calculations.Table 6.2: Comparison of proton-neutron coupling configurations in 128Inbetween the NuShellX (jj45pna) and In-Medium Similarity RenormalizationGroup (IMSRG) interactions. The subscript in JΠn indicates the major shellof the spin-parity combination. The main configuration is shown in bold.JΠn Jpi Jν jj45pna [%] IMSRG [%]3+1 1/2 7/2 7.67 3.663/2 3/2 4.64 1.123/2 5/2 1.663/2 7/2 2.335/2 11/2 1.35Continued on next page105JΠn Jpi Jν jj45pna [%] IMSRG [%]9/2 3/2 48.66 66.399/2 5/2 19.51 16.499/2 7/2 8.48 6.439/2 9/2 2.41 1.829/2 11/2 1.031+1 1/2 3/2 3.09 11.663/2 1/2 1.853/2 3/2 1.74 1.293/2 5/2 1.695/2 5/2 1.125/2 7/2 1.84 1.939/2 7/2 80.33 69.609/2 9/2 8.63 9.349/2 11/2 1.02 2.771+2 1/2 3/2 28.30 41.643/2 3/2 2.57 2.133/2 5/2 7.61 2.515/2 5/2 4.01 1.55/2 7/2 9.76 3.479/2 7/2 22.09 30.239/2 9/2 12.00 5.719/2 11/2 13.19 1.061+3 1/2 3/2 7.46 3.853/2 3/2 3.523/2 5/2 3.295/2 5/2 1.805/2 7/2 6.72 2.359/2 7/2 68.87 84.659/2 9/2 3.46 4.919/2 11/2 4.21 3.101+4 1/2 3/2 8.96 8.113/2 3/2 3.745/2 3/2 1.165/2 5/2 1.12 2.545/2 7/2 2.72 1.509/2 7/2 52.62 13.93Continued on next page106JΠn Jpi Jν jj45pna [%] IMSRG [%]9/2 9/2 12.18 5.429/2 11/2 16.92 66.541+5 1/2 1/2 1.231/2 3/2 2.503/2 3/2 3.963/2 5/2 1.06 8.135/2 3/2 2.115/2 5/2 1.05 1.129/2 7/2 8.83 35.129/2 9/2 5.87 25.039/2 11/2 78.18 23.801+6 1/2 3/2 8.86 1.773/2 3/2 3.50 2.153/2 5/2 1.245/2 7/2 22.76 2.329/2 7/2 50.81 25.019/2 9/2 11.47 46.219/2 11/2 20.751+7 1/2 3/2 9.77 6.063/2 3/2 6.19 2.583/2 5/2 1.06 2.225/2 3/2 1.475/2 5/2 1.265/2 7/2 4.209/2 7/2 36.52 4.749/2 9/2 27.27 41.439/2 11/2 12.47 40.641+8 5/2 5/2 2.495/2 7/2 1.559/2 7/2 13.20 49.369/2 9/2 48.15 35.889/2 11/2 31.85 12.058−1 1/2 17/2 1.399/2 7/2 52.58 3.239/2 9/2 9.029/2 11/2 25.78 78.22Continued on next page107JΠn Jpi Jν jj45pna [%] IMSRG [%]9/2 13/2 1.689/2 15/2 5.93 10.369/2 19/2 1.11 1.712−1 1/2 3/2 7.27 3.331/2 5/2 4.08 3.143/2 3/2 3.19 1.173/2 5/2 6.24 1.903/2 7/2 1.445/2 5/2 2.045/2 7/2 1.489/2 7/2 2.91 2.749/2 9/2 19.14 30.859/2 11/2 47.81 53.359/2 13/2 2.732−2 1/2 3/2 45.30 71.931/2 5/2 3.19 3.863/2 3/2 5.76 4.613/2 5/2 2.21 1.253/2 7/2 9.08 2.905/2 3/2 2.165/2 5/2 1.195/2 7/2 1.529/2 7/2 1.239/2 11/2 5.64 1.649/2 13/2 20.01 10.162−3 1/2 5/2 15.96 7.833/2 1/2 2.65 1.543/2 3/2 8.50 2.523/2 5/2 2.77 1.113/2 7/2 2.315/2 9/2 2.429/2 5/2 1.92 6.219/2 7/2 26.41 24.999/2 9/2 8.23 23.079/2 11/2 19.63 29.339/2 13/2 6.83Continued on next page108JΠn Jpi Jν jj45pna [%] IMSRG [%]1−1 1/2 1/2 3.29 3.671/2 3/2 3.86 4.833/2 3/2 2.053/2 5/2 7.62 3.165/2 5/2 1.655/2 7/2 1.35 1.269/2 7/2 1.89 2.449/2 9/2 11.22 26.359/2 11/2 66.23 55.621−2 1/2 1/2 14.191/2 3/2 26.02 4.883/2 1/2 6.333/2 3/2 9.14 2.193/2 5/2 2.10 1.395/2 3/2 1.605/2 7/2 3.519/2 7/2 33.17 3.509/2 9/2 5.56 49.619/2 11/2 11.28 21.871−3 1/2 1/2 7.25 56.081/2 3/2 9.08 3.913/2 1/2 1.593/2 3/2 25.61 6.953/2 5/2 13.32 2.425/2 3/2 2.93 1.025/2 5/2 1.675/2 7/2 1.229/2 7/2 1.36 1.819/2 9/2 22.01 7.449/2 11/2 14.59 17.710−1 1/2 1/2 31.03 25.763/2 3/2 48.14 9.635/2 5/2 5.83 1.129/2 9/2 14.99 63.490−2 1/2 1/2 37.36 48.723/2 3/2 25.03 16.50Continued on next page109JΠn Jpi Jν jj45pna [%] IMSRG [%]5/2 5/2 5.55 2.159/2 9/2 32.06 32.636.1.3 Effective Single-Particle EnergiesThe diagonal matrix elements of the Hamiltonian govern the evolution ofSPEs throughout the region, and resulting ESPEs are shown in Figure 6.2for the four even Z, even N isotopes closest to 128In (Z = 49, N = 79):126Cd (Z = 48, N = 78), 128Cd (Z = 48, N = 80), 128Sn (Z = 50, N = 78)and 130Sn (Z = 50, N = 80).ESPEs are calculated for even-even isotopes by multiplying the separa-tion energies and the spectroscopic strength of the Z, N ± 1 and Z ± 1, Nneighbouring nuclei [24]. Since the spectroscopic strength is often spreadamong high-lying states, a large number of states must be calculated. Inorder to calculate the ESPE for the ν-orbitals in 126Cd, 150 states of eachspin-parity were calculated and averaged for the two even Z, odd N ± 1neighbours: 125Cd and 127Cd. Similarly, calculations were performed on127Cd and 129Cd to extract the ν-orbitals ESPEs in 128Cd; 127,129Sn for the128Sn ESPEs, and 129,131Sn for the 130Sn ESPEs. The order of the five ν-orbitals calculated for the four even-even isotopes agree and show a largergap between the 2νd5/2 and 3νs1/2 orbitals.The ESPEs for the pi-orbitals in 126Cd were calculated and averaged from150 states of each spin-parity in the two odd Z±1, even N neighbours: 125Agand 127In. Similarly, calculations were run on 127Ag and 129In to extract thepi-orbitals ESPEs in 128Cd. For 128,130Sn, the Z+1 = 51 neighbours 129,131Sbare outside of the jj45pn model space and therefore could not be calculatedwith this Hamiltonian.By comparing the ESPEs to the SPEs (left), one can see that the in-teraction lowered the 1νg7/2 down by ∼6 MeV, to the bottom of the shell.Also, the 1pig7/2 and 1νh11/2 orbitals were lowered by ∼2 MeV and ∼9 MeV,respectively. Finally, the gap between the 2pip3/2 and 2pip1/2 orbitals is ∼5times larger, at almost ∼3 MeV. The order of the four pi-orbitals calculatedfor the cadmium and tin isotopes agree and show a larger gap between the2νd5/2 and 3νs1/2 orbitals.110IMSRG SPE2πf5/2 -20.8982πp3/2 -18.1082πp1/2 -17.5761πg9/2 -12.5571 νg7/2 -3.2072 νd5/2 -4.2603 νs1/2 -4.0302 νd3/2 -3.4191 νh11/2 2.52448126 Cd782πf5/2 -20.4852πp3/2 -18.5452πp1/2 -15.7711πg9/2 -14.5271 νg7/2 -9.2622 νd5/2 -9.0993 νs1/2 -7.0702 νd3/2 -6.7471 νh11/2 -6.24148128 Cd802πf5/2 -22.1072πp3/2 -19.3232πp1/2 -16.3521πg9/2 -15.1621 νg7/2 -9.0712 νd5/2 -8.7843 νs1/2 -6.7502 νd3/2 -6.4161 νh11/2 -5.83750128 Sn781 νg7/2 -9.9702 νd5/2 -9.3543 νs1/2 -6.8762 νd3/2 -6.5511 νh11/2 -5.29750130 Sn801 νg7/2 -9.9792 νd5/2 -9.2183 νs1/2 -6.8312 νd3/2 -6.6221 νh11/2 -5.752Figure 6.2: Single-particle energies (SPEs) in the jj45pn model space, and effective single-particleenergies (ESPEs) [in MeV] for the four even Z, even N neighbouring isotopes of 12849 In79 fromIn-Medium Similarity Renormalization Group (IMSRG). Proton orbitals are shown in red andneutron orbitals in blue.1116.2 131In6.2.1 Level EnergiesThe excited states in 131In found experimentally are compared to calculatedlevels in Figure 6.3. 131In is an odd Z, even N (magic N = 82) nucleus witha single pi-hole in the jj45pn model space. Since NuShellX does not allow thenucleon to excite across the shell gap to higher orbitals, only single-particleexcited states can be calculated.The energy of the first three levels calculated with the jj45pna interac-tion are in good agreement with the experiment. The 3/2− state is just141 keV above the experimental value of 1353 keV. The 988-keV transitionof a proton between the 2p3/2 and the 2p1/2 was calculated to be 1131 keV.This is to be expected as the SPEs were adjusted to the 132Sn region and131In is just one pi-hole away from the double shell closure and doubly magic132Sn.For IMSRG, the gap between the positive and negative parities is notconverged. While there were enough states to compare levels of same parityin 128In, there is only one positive parity state in 131In and we cannot sayhow large the gap is with the 1/2+ state.6.2.2 ConfigurationsInformation on the configurations and the occupation numbers of the ex-cited states in 131In is found in Table 6.3. The occupation numbers (.occoutput file) are the weighted sum of the contributions of the different Jpi⊗Jνcouplings to a state wave function.The four levels calculated in 131In are known very well to be single-particle states. This is confirmed with the occupation numbers, which allshow a single configuration (100%). The spin of the state is determined bythe orbital which is occupied by the unpaired proton or unpaired proton-hole. The five neutron orbitals in the shell are fully occupied and are notcontributing to the calculated excited states. Hence, the higher-spin 21/2−and 17/2− states observed in experiments [6] arise from more complex ex-citation modes, such as 1pi-2ν and 2pi-2ν mixing across the shell gap, whichare not included in the current model space. Finally, the only 5/2− statecalculated here is the pif−15/2 state, of which the observation has been moti-vating several recent experiments [55, 6] and our analysis did not find anyconclusive evidence for it.112Experimental 49131 In82(9/2+) 0(1/2-) 365(3/2-) 1353(21/2-) 3764(17/2-) 3782399041304643477039214237jj45pna9/2+ 01/2- 3633/2- 14945/2- 2994IMSRG (-)1/2- 93/2- 24315/2- 5451IMSRG (+)9/2+ 0Figure 6.3: Comparison of excitation energies [in keV] in 131In between thiswork [left], NuShellX (jj45pna) [center], and In-Medium Similarity Renor-malization Group (IMSRG) [right]. Positive parity states are shown in redand negative parity states in blue.Table 6.3: Orbitals occupancy and configuration in 131In with the NuShellX (jj45pna) and In-Medium Similarity Renormalization Group (IMSRG) interactions.JΠ 1pif5/2 2pip3/2 2pip1/2 1pig9/2 1νg7/2 2νd5/2 2νd3/2 3νs1/2 1νh11/2 Main Conf. [%]9/2+ 6.00 4.00 2.00 9.0 8.0 6.0 4.00 2.00 12.00 pi1g−19/2 (100)1/2− 6.00 4.00 1.00 10.0 8.0 6.0 4.00 2.00 12.00 pi2p−11/2 (100)3/2− 6.00 3.00 2.00 10.0 8.0 6.0 4.00 2.00 12.00 pi2p−13/2 (100)5/2− 5.00 4.00 2.00 10.0 8.0 6.0 4.00 2.00 12.00 pi1f−15/2 (100)113Chapter 7Conclusions and OutlookDetailed data sets for the β-decay of 128−131Cd were successfully obtainedwith the GRIFFIN γ-ray spectrometer at TRIUMF. This experimental cam-paign with exotic neutron-rich radioactive beams would not have been pos-sible without the discriminating power of IG-LIS and the high detectionefficiency of the GRIFFIN array. This careful analysis of the neutron-richcadmium isotopes is of great current interest with respect to advancing ourunderstanding of nuclear forces and shell evolution in a region that is essen-tial for the understanding of the astrophysical r-process and which only re-cently became accessible for more extensive studies. The new data providesimportant inputs for theoretical calculations and models aiming to repro-duce the element abundances of the Solar system and the nuclear structureof exotic isotopes around the N = 82 shell closure.The decay spectroscopy of 128Cd was successfully performed and hasrevealed 32 new transitions and 11 new levels, highlighting the high sensi-tivity of the GRIFFIN spectrometer. Eight allowed Gamow-Teller β-decayswere observed to tentative 1+ states. First forbidden Gamow-Teller decayswere observed to feed seven states. The tentative spin assignments for threeof these seven states were restricted using γ-γ angular correlation analysis,which ruled out the spin 0−. Finally, this work confirmed the half-life of245.4(30) ms for 128Cd which was extracted from this data set and previ-ously published in 2016 (t1/2 = 246.2(21) ms) [4].The extended decay scheme was compared to theoretical calculations us-ing NuShellX (jj45pna) and IMSRG. Both interactions calculated the energyof the first 1+ excited state shifted by ∼500 keV relative to the experimen-tal level at 1173 keV, with a discrepancy of 1097 keV between them. Inaddition, the IMSRG calculates the 3+ ground state at 290 keV. This gapcould be addressed by calculating the three-nucleon (3N) forces in a biggermodel space, which requires computing resources which are not available atthis time. Information on the configurations of the states in 128In is alsoextracted. The 3+ ground state shows a pig−19/2 ⊗ νd−13/2 configuration, whichis consistent with Ref. [47]. The configuration of the first 1+ excited state ispig−19/2⊗νg−17/2, as published in Ref. [1]. The 1− isomer at 248 keV is described114by the coupling pig−19/2 ⊗ νh−111/2 by both calculations. The 248 keV isomerictransition between the first 1− state and the 3+ ground state represents thetransition of a neutron from 1h11/2 to 2d3/2. ESPEs were also calculatedfor the four even N , even Z neighbouring isotopes. A manuscript describ-ing the current work on the nuclear structure of 128In is in preparation forsubmission to Physical Review C.The detailed γ-spectroscopy of the 131Cd enabled the confirmation of21 transitions and the revision of the decay scheme previously published inRef. [6], which placed 12 high-energy levels directly feeding the ground state.However, there was no clear information indicating if these 12 ground statetransitions were feeding the 1/2− isomeric or 9/2+ ground states. This workplaced only 9 transitions and 8 excited states in the decay scheme becauseof the low coincidence statistics. The new excited level at 3920 keV seemsto be the only level connecting both the isomeric and ground states, via a3554-keV transition and a 3920-keV transition. Thirty more hours of 131Cddata were obtained in 2016 with a beam intensity of ∼0.7 pps, which willimprove the statistical uncertainty on the values obtained in this work.Calculations for 131In with the jj45pna interaction are in good agreementwith the experiment since the SPEs were adjusted to the 132Sn region: the3/2− state was just 141 keV above the experimental value of 1353 keV,and the 988-keV transition was calculated to be 1131 keV. For IMSRG, thegap between the positive and negative parities was again not converged andmore computing power will be required. The four levels calculated in 131Inare known very well to be single-particle states, which were confirmed withthe occupation numbers (100%) of the 9/2+ ground state and of the 1/2−,3/2− and 5/2− states. Both these calculations work in a model space whichdo not allow the nucleon to excite across the shell gap to higher orbitalsand therefore, only single-particle excited states can be calculated. Hence,calculations for the higher-spin 21/2− and 17/2− states observed [6] and forthe configurations of the observed mixed states will require a larger modelspace.The low-statistics data set collected during the low-rate A = 132 beamtime was inconclusive with regards to the β-decay and βn-decay of 132Cd.Open questions remained about the important 988-keV transition seen inthe β-decay 131Cd, which was observed very close to the 989-keV transitionin the beam contaminant 132Sb. For neutron-rich nuclei with large neutron-branching ratios like 132Cd, a γ-ray spectrometer alone is not sufficient andneutron-tagging with neutron detectors, such as DESCANT at TRIUMF,will become more important. Many open questions remain for the level115structure of 132In which hopefully can be tackled in the future with higheryields and cleaner beams, for example with the use of photofission targetsfrom the new Advanced Rare Isotope Laboratory (ARIEL) at TRIUMF.These new studies combined highlights unanswered questions, whichshould point the community in the direction of what to probe next to furtherits understanding of nuclear structure and astrophysics theories. The sumof all studies addressing these questions allow to verify assumptions and in-duce changes in the main input for the calculations. This new experimentalknowledge highlights the need for larger-scale shell model calculations andreliable models of exotic nuclear structure, which will be enabled as new cal-culation methods and technologies are developed. Further comparisons withshell model calculations may lead to a deeper or different understanding ofthe structure across the neutron-rich indium isotopic chain.116Bibliography[1] I. Dillmann et al. N = 82 Shell Quenching of the Classical r-Process“Waiting-Point” Nucleus 130Cd. Phys. Rev. Lett., 91:162503, 2003.[2] C.E. Svensson and A.B. Garnsworthy. The GRIFFIN spectrometer.Hyperfine Interact., 225:127, 2016.[3] B. Fogelberg. Systematic Trends in the Level Structure of Neutron RichNuclei. Proceedings of the International Conference Nuclear Data forScience and Technology, Mito, Japan, 93:837, 1988.[4] R. Dunlop et al. Half-lives of neutron-rich 128−130Cd. Phys. Rev. C,93:062801(R), 2016.[5] J. 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Holt. private communication, 30 August 2018.122Appendix AData Calibration andProcessingγ-ray Energy Calibration Identify the probable good runs, the calibration runs and the garbageruns Sort the fragment and analysis trees with the online calibration file For each run:– Build a matrix of charge/rough energy vs. channel number– Check the counting rates of each crystal– Locate the ancillary detector channels (i.e. SCEPTAR) For each crystal and each source run (e.g. 152Eu, 133Ba and 56Co):– Get the number of counts for well-known transitions in γ-raysingles– Repeat for different sources– Plot the centroid channel of the peaks vs. their energy and fit:Energy = gain · channel number + offset (A.1)– Compile the gain and offset for each crystal in a new .cal file– Sort the source analysis trees using the new calibration file andcheck for non-linear effects For each crystal and each data run:– Sort the data analysis trees using the new .cal file and check forgain drifts between different runs– Identify and fit background lines and previously published tran-sitions123– Plot the centroid channel of the peaks vs. their energy and fit toEquation (A.1)– Compile the gain and offset for each crystal in a second calibrationfileAbsolute γ-ray Detection Efficiency Calibration For each source run (e.g. 152Eu, 133Ba and 56Co):– Get the number of counts for well-known transitions in γ-raysingles– Get the number of counts for well-known transitions in γ-ray sin-gles for crystals separated by 180◦ to correct for summing effects– Get the run time t and the source activity at the time of the run– Get the dead time d per hit by plotting the time difference be-tween consecutive hits– Get the average number of hits per crystal N by plotting theγ-ray singles per crystal– Get the absolute γ-ray detection efficiency : =Number of γ-rays detectedNumber of γ-rays emitted by source=Nγ,detectedIγ ·A · (t− d ·N)(A.2) Fit to the germanium detector efficiency [67]:(E) = 10p0+p1 log(E)+p2 log2(E)+p3/E2 (A.3) Repeat in addback mode Repeat for each array configuration (i.e. Delrin spheres)SCEPTAR Settings Check the β-particle energy thresholds and offset Check the β-particle detection efficiency as a function of γ-ray energy– Get the number of counts for strong transitions in γ-ray singles124– Get the number of counts for strong transitions in β-gated γ-raysingles– Divide the number of counts in β-gated γ-ray singles by the num-ber of counts in γ-ray singlesTime coincidence gates Check the event-building window: fixed or moving window, ∼ 2 µs Build a matrix of β-γ time difference vs. γ-ray energy– Draw the prompt time coincidence 2-D (banana) gate Build a matrix of γ-γ time difference vs. γ-ray energy– Draw the prompt time coincidence 2-D (banana) gate Sort the analysis trees to analysis histograms and matricesConstruction of Addback Events [66] Crosstalk correction– For 60Co events with multiplicity 2 within one clover, build amatrix of γ1-energy vs. γ2-energy– Extract the offset with a linear fit for each pair of crystals withinthe clover– Build a crosstalk correction matrix for the clover– Repeat for each clover and for each array configuration Check the prompt β-γ and γ-γ time coincidence gates in addback mode125Appendix BData AnalysisIdentification of new transitions Compare the γ-ray singles spectra with the lasers on and the laserblocked Compare the γ-ray singles spectra for the beam-on and beam-off partsof the cycle– Build a matrix of the γ-ray energy vs. time stamp– Check and filter out bad cycles– Build a matrix of γ-ray energy vs. cycle time by taking themodulus of the cycle length– Build a matrix of γ-ray energy vs. cycle time for the beam-onand beam-off parts of the cycleCoincidence analysis Build and inspect a matrix of prompt γ1-energy vs. γ2-energy Gate on γ1 [1] and on γ1-background [2] Get a γ1-background subtracted energy spectrum: [3] = [1] - [2] Repeat for different γ1-energy Identify coincidences relationships Build a logical decay schemeIntensities and Branching Ratios• Get the transition relative intensities– Get the area of the transition in β-gated γ-ray singles– Divide by the γ-ray detection efficiency126– Divide by the intensity of the most intense γ-ray transition– Propagate the uncertainties in quadrature• Get the transition absolute intensities– Get the number of β-particles γ-ray singles– Divide by the γ-ray detection efficiency– Divide by the intensity of the most intense γ-ray transition– Get the absolute correction factor by dividing the absolute inten-sities by the relative intensities– Propagate the uncertainties in quadrature– Estimate the unobserved ground state transitions by adding ab-solute intensities for all ground state transitions• Get the β-decay branching ratios– Calculate the difference between the sum of the absolute γ-rayintensities which feed and depopulate each level– Estimate the ground state branching ratio by adding the branch-ing ratios for all states– Propagate the uncertainties in quadrature– Calculate the log(ft) value of each state with the Logft web ap-plication [71]Angular correlations [74] Determine the possible spin assignments from the selection rules andthe log(ft) values Make the experimental angular correlations plots to determine thepreliminary the a22/a44 coefficients– Build a matrix of prompt γ1-gated γ-energy vs. angle [w]– Build a matrix of event-mixed γ1-gated γ-energy vs. angle [y]– Gate on γ2 in [w] and in [y]– Divide the gate in [w] by the gate in [y]: [W (θ)] = [gatew]/[gatey]– Fit the Legendre polynomials and extract the correlation coeffi-cients:W (θ) = A0 [1 + a22P2(cos θ) + a44P4(cos θ)] (B.1)127 Compare the theoretical a22/a44 ellipses to determine if the possiblespin assignments overlap at the preliminary a22/a44 values Run GEANT4 simulations for given cascades/mixing ratios to accountfor solid angle corrections Compare data to simulated template, perform χ2 analysis to get mix-ing ratios and spins Repeat for other cascades with over ∼10000 countsHalf-life measurement Project the matrix of γ-ray energy vs. cycle time for the beam-off partof the cycle Gate on γ1 [1] and on γ1-background [2] Get a background-subtracted γ-ray energy vs. cycle time spectrum:[3] = [1] - [2] Fit the activity of the γ-ray:Atotal(t) = Ae− ln 2·t/t1/2,parent +Be− ln 2·t/t1/2,daughter + C (B.2) Check for in-beam contaminants and systematic errors (i.e. binningand chop plot analysis)Isomer hunting Build a matrix of γ-energy vs. β-γ time difference Find long-lived lines Gate on γ1 [1] and on γ1-background [2] Get a background-subtracted γ-ray energy vs. β-γ time differencespectrum: [3] = [1] - [2] Fit the activity of the γ-ray to Equation (B.2)128

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