Decay Spectroscopy of Neutron-Rich Cadmium Around the N = 82 Shell Closure

Nikita Bernier

B.Sc., Universit´eLaval, 2011 M.Sc., Universit´eLaval, 2013

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

The Faculty of Graduate and Postdoctoral Studies

(Physics)

THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2018 c Nikita Bernier 2018 The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:

Decay Spectroscopy of Neutron-Rich Cadmium Around the N = 82 Shell Closure submitted by Nikita Bernier in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Physics.

Examining Committee:

Dr Reiner Kr¨ucken, Physics Supervisor

Dr Colin Gay, Physics Supervisory Committee Member

Dr Janis McKenna, Physics University Examiner

Dr Chris Orvig, Chemistry University Examiner

Additional Supervisory Committee Members:

Dr Sonia Bacca, Physics Supervisory Committee Member

Dr Robert Kieﬂ, Physics Supervisory Committee Member

ii Abstract

The neutron-rich cadmium isotopes (Z = 49) near the well-known magic numbers at Z = 50 and N = 82 are prime candidates to study the evolving shell structure observed in exotic nuclei. Additionally, nuclei around the doubly-magic 132Sn have been demonstrated to have direct implications for astrophysical models, leading to the r-process abundance peak at A ≈ 130 and the corresponding waiting-point nuclei around N = 82. The β-decay of the N = 82 isotope 130Cd into 130In was investigated in 2002 [1], but the information for states of the lighter indium isotope 128In is still limited. Detailed β-γ-spectroscopy of 128,131,132Cd was accomplished using the GRIFFIN [2] facility at TRIUMF. In 128In, 32 new transitions and 11 new states have been observed in addition to the four previously observed excited states [3]. The 128Cd half-life has also been remeasured via the time distribution 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 EURICA data [5, 6]. These new results are compared with recent shell model and IMSRG [7, 8, 9] calculations, which highlight the necessity to re-investigate even “well-known” decay schemes for missing transitions.

iii Lay Summary

The discovery of radioactivity (1896) and the atomic nucleus (1911) are fairly recent in the history of mankind, but our understanding of the nucleus has advanced rapidly through numerous experiments. The Earth and its inhabitants are composed of various elements, such as gold and uranium, which are not produced in our Solar system but in massive stars and are transferred into the Solar System via the Interstellar Medium. Thus, every atom around us is made of previous stardust. Such radioactive nuclei could not be studied until we produced them with particle accelerators. These new experiments push the limits of our theories on the nuclear structure: how neutrons and protons work together to make up matter. Nuclear astrophysics works on explaining how these elements are created in stars. This work highlights results from experiments at TRIUMF with radioactive cadmium nuclei, which bring important information on the structure of neutron-rich nuclei.

iv Preface

Chapter 5 is based on work conducted at the TRIUMF laboratory under the supervision of Professor Reiner Krücken [TRIUMF/UBC] and Dr Iris Dill- mann [TRIUMF/University of Victoria, Canada]. I was responsible for the analysis of the data sets for the β-decay of 128,131,132Cd collected in August 2015. The data files were sorted using the analysis framework GRSISort [10], a code written in ROOT [11]. The figures of level schemes for this thesis have been created using the SciDraw scientific figure preparation system [12]. Chapter 6 is based on work conducted at the TRIUMF laboratory under the supervision Professor Reiner Krücken and Dr Jason Holt [TRIUMF]. I was responsible for running the NuShellX@MSU [13] code provided by Professor Alex Brown [Michigan State University/National Superconduct- ing Cyclotron Laboratory, USA] for 128,131In. I was also responsible for running the NuShellX@MSU code with the IMSRG interaction provided by Dr Jason Holt for 127,128,129,130,131Sn, 127,128,129,131In, 125,126,127,128,129Cd and 125,127Ag. Finally, I was responsible for running the effective single particle energy (ESPE) code provided by Dr Jason Holt using the calculations previously mentioned. Section 5.1.5 presents the analysis of the half-life of 128Cd, which was independently extracted from the same data set by Ryan Dunlop [University of Guelph, Canada] and published in 2016 [4]. A manuscript describing the current work on the nuclear structure of 128In is in preparation for submission to Physical Review C.

v Table of Contents

Abstract ...... iii

Lay Summary ...... iv

Preface ...... v

Table of Contents ...... vi

List of Tables ...... ix

List of Figures ...... x

List of Symbols and Acronyms ...... xiii

Acknowledgements ...... xv

Dedication ...... xvi

1 Introduction ...... 1

2 Motivation and Theory ...... 3 2.1 Nuclear Structure ...... 3 2.1.1 Non-Interacting Shell Model ...... 4 2.1.2 Interacting Shell Model ...... 8 2.1.3 Recent Developments ...... 9 2.2 Nuclear Astrophysics ...... 9 2.3 Nuclear Decay ...... 12 2.3.1 Decay Law ...... 12 2.3.2 Beta Decay ...... 14 2.3.3 Gamma Decay ...... 19

vi 3 Review of Literature ...... 24 3.1 128Cd ...... 24 3.2 131Cd ...... 28 3.3 132Cd ...... 33

4 Experiment ...... 36 4.1 Beam Production ...... 36 4.2 Detectors ...... 38 4.3 Data Processing ...... 41

5 Data Analysis and Results ...... 47 5.1 128Cd ...... 47 5.1.1 β-Gated γ-Singles Measurements ...... 47 5.1.2 β-Gated γ-γ Coincidence Measurements ...... 52 5.1.3 Decay Scheme ...... 60 5.1.4 Spin Assignments ...... 67 5.1.5 Half-Life ...... 70 5.1.6 248-keV Isomer ...... 72 5.2 131Cd ...... 77 5.2.1 β-Gated γ-Singles Measurements ...... 77 5.2.2 β-Gated γ-γ Coincidence Measurements ...... 82 5.2.3 Decay Scheme ...... 85 5.3 132Cd ...... 94 5.3.1 β-Gated γ-Singles Measurements ...... 94 5.3.2 β-Gated γ-γ Coincidence Measurements ...... 97

6 Shell Model Calculations ...... 101 6.1 128In ...... 102 6.1.1 Level Energies ...... 102 6.1.2 Configurations ...... 104 6.1.3 Effective Single-Particle Energies ...... 110 6.2 131In ...... 112 6.2.1 Level Energies ...... 112 6.2.2 Configurations ...... 112

7 Conclusions and Outlook ...... 114

Bibliography ...... 117

vii Appendices

A Data Calibration and Processing ...... 123

B Data Analysis ...... 126

viii List of Tables

2.1 Selection rules for β-decay angular momentum and parity . . 18 2.2 Selection rules for γ-decay angular momentum and parity . . 21

128 5.1 γ-ray energies in In, their intensities relative to Iγ(247.96) = 100 % and the initial energy levels are compared to previous work [14]...... 66 5.2 Level energies in 128In, their β-feeding intensities per 100 decays and the log(ft) values ...... 68 131 5.3 γ-ray energies in In, their intensities relative to Iγ(988) = 100 %, absolute intensities per 100 decays, and the initial energy levels are compared to previous work Ref. [6]. . . . . 91 5.4 Level energies in 131In, their β-feeding intensities per 100 decays and the log(ft) values ...... 93

6.1 Single-Particle Energies for the jj45pn model space ...... 102 6.2 Comparison of proton-neutron coupling conﬁgurations in 128In 105 6.3 Orbitals occupancy and conﬁguration in 131In ...... 113

ix List of Figures

2.1 Nuclear shell structure with various potentials ...... 6 2.2 Proton (π) and neutron (ν) valence orbitals for 128In (Z = 49, N = 79) and single-particle energies (SPE) [in MeV] . . . . . 7 2.3 Nuclide chart with one potential rapid neutron capture (r-) process path and r-process solar abundances ...... 12 2.4 N = 82 region of the nuclide chart close to Z = 50 ...... 13 2.5 Number of β-decays as a function of time for 128Cd and 128In 15 2.6 β-decay and β-delayed neutron decay processes ...... 17 2.7 Examples of γ-γ angular correlations ...... 22

3.1 Published decay schemes of 128Cd ...... 26 3.2 Evolution of the ground state, ﬁrst 1+ and isomeric state(s) in even-mass 122−130In ...... 27 3.3 Published decay schemes of 131Cd ...... 29 3.4 Evolution of the 1/2–9/2 states in odd-mass 123−131In . . . . 31 3.5 Single-particle orbitals in the 132Sn region [6] ...... 32 3.6 Published decay schemes of 132Cd...... 34 3.7 Tentative levels energies [in keV] for 132In ...... 35

4.1 TRIUMF ISAC experimental hall layout ...... 37 4.2 Concept of the Ion Guide Laser Ion source (IG-LIS) . . . . . 38 4.3 124−130Cd yields at ISAC using the Ion Guide Laser Ion source 39 4.4 GRIFFIN γ-ray spectrometer ...... 40 4.5 SCEPTAR scintillator array and moving-tape collector . . . . 40 4.6 Comparison of spectra observed for a 60Co source with and without crosstalk correction ...... 42 4.7 Comparison of clover addback [blue] and γ-singles [red] spectra observed for a 60Co source ...... 44 4.8 Time diﬀerence between consecutive triggers as a function of crystal number for a 152Eu source ...... 45 4.9 Absolute γ-ray detection eﬃciency for the GRIFFIN spectrometer ...... 46

x 5.1 Difference between time stamps of β-particles and γ-rays . . . 48 5.2 Comparison of β-gated γ-singles [blue] and γ-singles [red] spectra observed for the decay of 128Cd ...... 49 5.3 Comparison of β-gated γ-ray spectra observed for the decay of 128Cd in addback mode with lasers on [blue] and laser blocked [red] ...... 51 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 53 5.5 Comparison of β-gated γ-ray spectra observed for the decay of 128Cd in addback mode as a function of cycle structure . . 54 5.6 Difference between the time stamp of a γ-ray coincident with a β-particle and the time stamp of a second γ-ray as a function of the energy of the second γ-ray ...... 55 5.7 Symmetrized β-gated γ-γ coincidence matrix for 128Cd data . 56 5.8 β-gated background-subtracted γ-gated spectra observed with the GRIFFIN spectrometer for the β-decay of 128Cd: 68 keV and 173 keV ...... 58 5.9 β-gated background-subtracted γ-gated spectra observed with the GRIFFIN spectrometer for the β-decay of 128Cd: 462 keV and 857 keV ...... 59 5.10 β-gated background-subtracted γ-gated spectra observed with the GRIFFIN spectrometer for the β-decay of 128Cd: 408 keV and 336 keV ...... 61 5.11 β-gated background-subtracted γ-gated spectra observed with the GRIFFIN spectrometer for the β-decay of 128Cd: 305 keV and 619 keV ...... 62 5.12 Energy levels [in keV] and γ-ray transitions in 128In following the β-decay of 128Cd ...... 63 5.13 Coefficients and mixing ratios of γ-γ angular correlations . . 71 5.14 Normalized γ-γ angular correlation data and fit for the 857-68 keV cascade ...... 72 5.15 Fitted activity of selected γ-rays in 128In ...... 73 5.16 Fitted activity of the sum of the 857- and 925-keV γ-rays in 128In...... 74 5.17 Effect of changing the fitting region on the extracted 128Cd half-life ...... 74 5.18 Difference between time stamps of β-particles and γ-rays (zoom in) ...... 76 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 78

xi 5.20 Comparison of β-gated γ-ray spectra observed for the decay of 131Cd in addback mode: lasers on [blue] and laser blocked [red] ...... 79 5.21 β-gated γ-ray spectra observed for the decay of 131Cd in addback mode ...... 80 5.22 β-gated γ-ray spectra around peaks with multiplet structures in the decay of 131Cd ...... 81 5.23 β-gated γ-ray spectra around possible transitions in 130In from the βn-decay of 131Cd ...... 83 5.24 Partial decay scheme for the β-decay of 130Cd ...... 84 5.25 Symmetrized β-gated γ-γ coincidence matrix for 131Cd data . 86 5.26 Symmetrized β-gated γ-γ coincidence matrix for the 988-keV transition 131Cd ...... 87 5.27 β-gated background-subtracted γ-gated spectra observed with the GRIFFIN spectrometer for the β-decay of 131Cd . . . . . 89 5.28 Energy levels [in keV] and γ-ray transitions in 131In following the β-decay of 131In ...... 90 5.29 β-gated γ-ray spectrum observed for the decay of 132Cd in addback mode ...... 95 5.30 Comparison of 131Cd and 132Cd data sets around 988 keV . . 96 5.31 Partial decay scheme for the β-decay of 132Sb ...... 97 5.32 Comparison of the activity of selected γ-rays in the 131Cd and 132Cd data sets ...... 98 5.33 β-γ-γ coincidence matrix for A = 132 data ...... 100

6.1 Comparison of excitation energies [in keV] in 128In ...... 103 6.2 Single-particle energies (SPEs) in the jj45pn model space, and eﬀective single-particle energies (ESPEs) [in MeV] for the four 128 even Z, even N neighbouring isotopes of 49 In79 ...... 111 6.3 Comparison of excitation energies [in keV] in 121In ...... 113

xii List of Symbols and Acronyms

A Total nucleon (mass) number α Internal conversion coefficient (ICC) Bn Binding energy of an electron in the n-shell BRγ Branching ratio of a γ-ray transition δ Mixing ratio E Energy γ Detection efficiency of a γ-ray transition H Hamiltonian H0 Non-interacting Hamiltonian Iγ Absolute/Relative intensity of a γ-ray transition j Total angular momentum of a nucleon J Nuclear spin (Total angular momentum of a nucleus) K Kinetic energy l Orbital angular momentum L Total orbital angular momentum λ Decay constant m Mass of a nucleon me Electron mass mu Atomic mass unit M Mass of the Sun µ(r) Mean-field potential n Radial quantum number N Neutron number n(A, Z) Abundance of element A, Z nn Neutron density ν Neutron pˆ Momentum operator π Proton Π Parity

xiii Ψ, ψ Eigenstates Qβ Q-value of the β-decay r Position of a nucleon s Spin of a nucleon S Total spin of a system Sn Neutron separation energy S2n Two-neutron separation energy S2p Two-proton separation energy T Temperature t1/2 Half-life τ Lifetime V (r) Potential energy Vso(r) Spin-orbit potential WRES Residual interaction Z Proton (atomic) number

3N Three-nucleon forces βn β-delayed neutron emission DAQ Data acquisition system EFT Effective field theory ESPE Effective single particle energy (J ) FWHM Full width at half maximum GRIFFIN Gamma-Ray Infrastructure For Fundamental Investigations of Nuclei HPGe High purity germanium IG-LIS Ion Guide Laser Ion source IMSRG In-Medium Similarity Renormalization Group ISAC Isotope Separator and ACcelerator ISOL Isotope separation on-line MIDAS Maximum Integrated Data Acquisition System NN Two-nucleon forces pp chain Proton-proton chain QCD Quantum chromodynamics r-process Rapid neutron capture process s-process Slow neutron capture process SCEPTAR SCintillating Electron-Positron Tagging ARray SPE Single particle energy TBMEs Two-body matrix elements

xiv Acknowledgements

My sincere thanks to

Reiner Kr¨ucken, for his precise explanations and open-mindedness

Iris Dillmann, for her alternate perspective and advice

Peter C. Bender, for his invaluable support and understanding

Shaun Georges, for his dedication and care of the gamma-ray group

Gordon Ball, for his irreplaceable experience and discussions

Jason Holt, for his patient teachings to an experimentalist

Jens Lassen, for his continued support since my ﬁrst day at TRIUMF

Sonia Bacca, for her continued support since my ﬁrst course at UBC

Tammy Zidar, for her life-changing points of view

Aurelia Laxdal, for her timeless love

Evidently, the support of

xv Pour l’exercice d’urgence nucléaire qui a réveillémes parents six heures avant la naissance de leur première fille.

Il n’y a pas eu d’autre exercice depuis.

For the nuclear emergency drill which woke my parents up six hours before the birth of their ﬁrst daughter.

There has not been another drill since.

xvi Chapter 1

Introduction

The discovery of neutrons in nuclei (1932, J. Chadwick [15]), following the discovery of nuclei in atoms (1911, E. Rutherford [16]), prompted the addition of a new fundamental force. The observation of a bound system composed of only neutral and positively charged particles revealed a force stronger than the well-known electromagnetic force, which was named the strong nuclear force. As we know it today, this nuclear force is only a residual force that is felt outside of the nucleons from the interaction of the quarks inside the nucleons, similarly to the van der Waals interaction between neutral molecules. This fundamental strong interaction is governed by quantum chromodynamics (QCD), which provides the basis of modern nuclear force modeling between two and three nucleons that are then used to describe more complex nuclei. To this day, the understanding of the nuclear force from first principles is still limited and nuclear theory struggles to accurately predict properties of heavy nuclei. Nuclear astrophysicists have been theorizing about locations where the heavy elements could be created, with such candidates including core- collapse supernovae, neutron-star mergers and certain burning phases of low-mass (1-3 solar masses, M ) and massive (M > 8 M ) stars. The Sun can fuse protons into heavier atoms up to 16O [17] (1967 Nobel Prize in Physics, H.A. Bethe). Massive stars can ignite further advanced burning phases 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 to 238U(Z = 92, N = 146). Three different processes in different astrophysical scenarios were described in 1957 [18] (1983 Nobel Prize in Physics, S. Chan- drasekhar and W.A. Fowler): the rapid neutron capture (r-) process, the slow neutron capture (s-) process, and several production mechanisms for proton-rich (neutron-deficient) nuclei summarized as p-process. Each one of the first two processes accounts for ∼50% of the nucleosynthesis of elements heavier than 56Fe, while the p-processes contribute ∼1%. Modern astrophysical simulations calculate the probable chains of nuclear reactions and their corresponding isotopic abundances for the different production mechanisms, which are then compared to the observed abundances in the Solar System.

1 The neutron-rich region of the nuclide chart around A = 132 is of spe- cial interest to both nuclear structure and nuclear astrophysics. For nuclear structure studies, the neighbours of the doubly-magic 132Sn (Z = 50, N = 82) are an ideal test ground for nuclear structure theories, such as the famous 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 is connected to the waiting-point nuclei around N = 82, at which the r-process material within an isotopic chain accumulates and transfers the material to the next isotopic chain, and the corresponding abundance peak at A ' 130. Together, the shell structure and half-lives far oﬀ stability provide critical information on the position and the shape of the abundance peaks for the r-process. The decay of the N = 82 isotope 130Cd into 130In was investigated 15 years ago [1], but puzzling questions remained open. The information for the decay of the lighter, less exotic Cadmium isotopes 128,129Cd was 128 also very limited. For the β-decay of Cd (t1/2 = 246.2(21) ms [4]), only seven transitions were published [3] and the last known level in 128In at 1173 keV is still 4146 keV away from the neutron separation energy (Sn) at 5321(155) keV [19]. For the N = 82 isotope 131In, the EURICA collaboration at RIKEN has recently published results for the proton (π) hole states in 131In from the β-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 of 128,131,132Cd using laser ionization and moving tape cycle methods with the GRIFFIN spectrometer at TRIUMF. The relevant nuclear theory and motivation are presented in Chapter 2, while Chapter 3 details the previous published works on the decay of 128,131,132Cd isotopes. The beam production techniques and the detectors of β- and γ-radiation used are described in Chapter 4. Chapter 5 presents the data analysis based on coincidence analysis and angular correlations. These new results are then compared with shell model calculations in Chapter 6.

2 Chapter 2

Motivation and Theory

2.1 Nuclear Structure

Several pieces of experimental evidence observed in the early 20th century exhibit a shell structure for neutrons and protons inside nuclei. For example, the two-proton separation energy (S2p) and the two-neutron separation energy (S2n), which are the energies required to remove two protons and neutrons from a nucleus, respectively, show sharp decreases just above spe- ciﬁc number of protons and neutrons. Moreover, the numbers at which the 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 nuclear charge 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 are reminiscent of the structure of electron shells around the atomic nucleus. A nucleus with both its proton and neutron shells exactly full (closed) is called doubly-magic, such as 132Sn (Z = 50, N = 82). Quantum mechanically, the shell structure at the atomic and nuclear scales is described by the eigenstates Ψ which solve the Schr¨odingerequation:

HΨ = EΨ. (2.1) The Hamiltonian H has the form:

A 2 A X pˆi X H = K + V = + V (~ri,k) , (2.2) 2mi i=1 i6=k where K is the kinetic energy, A is the mass number,p ˆ and m are the momentum operator (ˆp = −i~∇) and the mass of a nucleon, respectively, and V (~ri,k) is the nucleon-nucleon interaction as a function of the coordinates of the i-th and k-th particles. The nuclear potential V (~ri,k) describes how protons and neutrons interact if we neglect three-body and higher order forces. This potential is ultimately related to the interaction of the quarks inside the nucleons. The simple existence of a bound system of neutrons

3 and protons reveals an attractive component of the nuclear force stronger than the repulsion of the electromagnetic force, at least at the nuclear scale.

2.1.1 Non-Interacting Shell Model

By adding and subtracting a mean-ﬁeld potential v (~ri) (omitting spin and isospin 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 interaction WRES:

" A A #  A A  X pˆ2 X X X H = i + v (~r ) + V (~r ) − v (~r ) 2m i  i,k i  i=1 i i6=k i (2.3) 0 = H + WRES, where WRES = 0 for the non-interacting shell model (or independent particle model). An infinite square well potential is a reasonable first-order approximation of the nuclear potential, as shown in Figure 2.1. A specific number of protons/neutrons can occupy each level according to the Pauli exclusion principle for fermions before filling up the next level sequentially. Following the electron nomenclature, the orbital angular momentum l of a level defines its 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 of the orbital angular momentum label simply indicates the major shell. The number of nucleons allowed per level, or degeneracy, is 2(2l + 1). The factor of (2l+1) arises from the ml degeneracy and the factor 2 comes from the ms degeneracy [20]. Groups of levels form shells which are separated by large energy gaps for some total numbers of nucleons. These gaps are called shell closures and the infinite square well reproduces only the first three magic numbers observed experimentally. The second potential, which already explains a large number of phenomena in various fields, is the harmonic oscillator. The potential has the form 2 2 V (ri) = mω0ri /2 with solutions En,l = ~ω0(2n + l − 1/2), where ~ is the reduced Planck’s constant, ω0 is the classical angular frequency of the oscillator, and n = 0, 1, 2, 3... is the radial quantum number and l = 0, 1, 2, ..., n−1 is the angular momentum. Here again, only the first three magic numbers are reproduced. Both these approximations have the considerable flaw of requiring an infinite amount of energy to remove a nucleon from the potential. A realistic nuclear potential would include the flat bottom of the interior well, the

4 parabolic raise of the harmonic oscillator, and also a ﬁnite saturation at the nuclear 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:

−V0 V (ri) = , (2.4) 1 + exp [(ri − R)/a] 1/3 where V0 is the well depth, R is the mean radius following R = 1.25A with a = 0.524 fm, the skin thickness of the nucleus. The Woods-Saxon potential in a one-body problem can only be solved numerically, whereas the harmonic oscillator can be solved analytically. Only the first three magic numbers are reproduced by the Woods-Saxon potential. However, the addition of a spin- orbit component causes the levels to be reordered and all magic numbers to be reproduced. The spin-orbit term was introduced by M. Goeppert-Mayer [21] and H. Jensen [22], who shared the 1963 Nobel Prize in Physics. While the atomic spin-orbit interaction arises from the electron’s magnetic moment interacting 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 the nucleon (s = 1/2). The total angular momentum of a level (labelled as a subscript on the right) is given by ~j = ~l + ~s, such that j = l ± 1/2. The degeneracy of each level is (2j + 1) and its parity is Π = (−1)l. This energy splitting doesn’t affect the magic numbers 2, 8 and 20, however it brings the 1f7/2 level low enough to create a shell closure at 28. The 1g is split into 1g11/2 (12 nucleons) and 1g9/2 (10 nucleons), adding 10 nucleons to the previous magic number of 40 to form a new one at 50 nucleons as observed empirically. In the independent particle shell model, only the unpaired nucleons contribute to the ground state properties of the nucleus. For a nucleus with an even number of neutrons N and an even number of protons Z, all nucleons are paired and therefore the ground state (the configuration with the lowest energy) has a spin-parity of 0+. For an even Z-odd N or odd Z-even N nucleus, the properties of the ground state are defined by the total angular momentum 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 neutron determines the possible spin-parity combinations for the ground state. For example, 128In is made of 49 protons (π) and 79 neutrons (ν). As shown in Figure 2.2, 128In is one π-hole and 3ν-holes from the double shell closure at Z = 50 and N = 82. According to the independent particle shell model, the single proton in π2p1/2 would couple to the neutron in ν1g7/2 to

5 184 168 1 j15/2 16 3 d3/2 4 2g 4 s 1j 4s 1/2 2 4s,3d,2g,1i 3d 2 g7/2 8 2g 1 i11/2 12 138 3 d5/2 6 1i 2 g9/2 10 3p 112 112 126 1 i13/2 14 3p,2f,1h 3 p 1i 3p 1/2 2 2f 3 p3/2 4 2f 2 f5/2 6 70 2 f7/2 8 92 92 1 h9/2 10 3s,2d,1f 1h 82 3s 1 h 1h 11/2 12 40 3s 3 s1/2 2 2d 2 d 2d 3/2 4 2 d 6 58 5/2 2p,1f 58 1 g7/2 8 1g 1g 50 2p 20 40 1 g9/2 10 2 p1/2 2 34 1 f 6 2s,1d 2p 5/2 2 p3/2 4 1f 1f 28 20 1 f7/2 8 8 20 20 2s 2s 1 d3/2 4 1d 1d 2 s1/2 2 1p 1 d5/2 6 8 8 8 1p 2 1p 1 p1/2 2 1 p3/2 4 2 2 2 1s 1s 1s 1 s1/2 2 (a) Square well (b) Harmonic oscillator (c) Woods-Saxon (d)WS+spin-orbit

Figure 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].)

6 82 1 g7/2 5.7402 1 h11/2 2.6795 2 d3/2 2.5148 2 d5/2 2.4422 3 s1/2 2.1738 50 50 2 p 1/2 ν 1.1262 2 p3/2 1.1184 1 g9/2 0.1785 1 f5/2 -0.7166 28 π

Figure 2.2: Proton (π) and neutron (ν) valence orbitals for 128In (Z = 49, N = 79) and single-particle energies (SPE) [in MeV] for the jj45pna interaction 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 observed in nuclei close to magic shell closures, contemporary experiments with radioactive nuclei have produced new results which do not directly agree with these simple nuclear theories. Since the eﬀective potential resulting from the interaction between the nucleons (proton-proton, neutron-neutron and proton-neutron) is responsible for the energy of the levels, the number of nucleons of the same type and the number of nucleons of the other type both 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 (neutron rich) especially test our understanding of the nucleon-nucleon forces. This shell evolution is already seen in Figures 2.1 and 2.2, where the orbitals within the shells of interest are displayed in a diﬀerent order. These

7 differences arise from the fact that large asymmetries in Z and N, such as in the region around Z = 50 and N = 82, produce a nuclear potential with levels which have slightly different energies than stable nuclei. Figures 2.2 shows the orbitals in order of their single-particle energies (SPE) as defined in the shell model calculation code NuShellX [13], which will be discussed in Chapter 6.

2.1.2 Interacting Shell Model

The interacting shell model considers the residual interaction (WRES 6= 0). Calculations for the interacting shell model divide the proton and neutron orbitals in three spaces: a non-interacting core, a valence space and an external space. For calculations in the model space between 28 < Z ≤ 50 and 50 < N ≤ 82, the four π- and five ν-orbitals orbitals shown in Figure 2.2 are included in the valence space and contribute to the mean-field potential. While the lower closed shells form an inert 78Ni core, the empty orbitals above 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 a mean field calculation. They can be derived from many-body perturbation theory or by phenomenologically fitting matrix elements to experimental data from nuclei in the region [23]. Including these orbitals as part of the valence space means that the Hamiltonian takes into account the interactions of the nucleons in these orbitals with all other nucleons in the valence space orbitals via the respective two-body matrix elements (TBMEs). The shell model code diagonalizes the Hamiltonian matrix that is set up by the SPEs and TBMEs. When the effective Hamiltonian is applied, the orbital SPEs shift to effective single-particle energies J (ESPE). The ESPE of an occupied orbit is calculated by taking the average of the one-nucleon separation energies weighted by the probability to reach the corresponding A ± 1 eigenstates by adding/removing a nucleon to/from a single-particle state ΨJ :

X + + X − − J = Si Ei + Sk Ek , (2.5) i k where S± are the spectroscopic probabilities of the single-particle state of energy E± for the A±1 neighbouring isotopes [24]. These calculated ESPEs evolve as a function of the spectroscopic factors, which characterize the occupation of the levels.

8 2.1.3 Recent Developments The most recent development for the derivation of realistic interactions is the connection of nuclear forces to the fundamental theory of quantum chromodynamics (QCD) using effective field theories (EFT). The latter preserves all the symmetries of the underlying fundamental strong interaction, but uses effective degrees of freedom such as neutrons, protons and pions. These theories, such as the chiral EFT [25, 26, 27], focus on the characterization of two-nucleon (NN) and three-nucleon (3N) forces with an expansion to several leading orders. Ideally, calculations would take root in physics first principles without approximations. Those forces have been used in ab initio calculations of nuclear properties for light nuclei and near closed shells. Also, they have lead to the development of interactions that can be used in large-scale shell model calculations, such as the ones used in this work. The advanced com- putational techniques which are required to find solutions to these quantum problems represent one of the main limitations of the theory and also progress quickly with modern technologies [28]. Experiments on nuclei at and around shell closures provide empirical parameters which are indispensable to test and benchmark nuclear structure theories. Data on the odd-odd 128In and 132In, and the odd-even 131In can provide information on the proton-hole structure along the Z = 49 isotopic chain. The configurations of key states and the size of shell gaps are examples of fundamental information required to develop and understand two-body matrix elements of the effective interaction.

2.2 Nuclear Astrophysics

The Big Bang nucleosynthesis predominantly produced hydrogen, helium, deuterium, and small amounts of lithium. Beryllium and boron can be produced by galactic cosmic-ray spallation, which is the bombardment of pre- existing matter by high-energy cosmic-ray particles. Two hydrogen fusion cycles in stars can fuse protons into 4He: the proton-proton chain reactions (pp chains), and the carbon-nitrogen-oxygen (CNO) cycle. In the following helium burning phase, two 4He nuclei are fused in a ﬁrst step to unstable 8Be, and then further to 12C due to the high density of 4He and the existence of a resonant state (“Hoyle state”), and then partially by another α-particle capture to 16O. Stars more than eight times the mass of the Sun go through further advanced nuclear burning phases, such as oxygen, carbon, neon and silicon burning, of which the ashes can reach up to 56Fe (Z = 26,

9 N = 30). Finally, heavy elements beyond iron cannot be created via fusion reactions 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. Within an isotopic chain away from a neutron shell closure, the maximum of the abundance distribution occurs at one neutron number, which is considered a waiting-point nucleus. This local maximum is produced by the equilibrium between radiative neutron capture reactions (n,γ), where the nucleus absorbs a 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 isotopic chain, multiple neutron captures and photodisintegrations happen until a new equilibrium distribution is established. Another β-decay to the next isotopic chain then takes place, most likely from the waiting point within that isotopic chain. The main astrophysics parameters determining the reaction equilibrium for each element are the neutron separation energy Sn, the environment temperature T and the density of neutrons nn. Therefore, there can be several waiting-point nuclei within an isotopic chain, for given T and nn, if the Sn-values are similar. The r-process abundances of element A, Z and element A + 1,Z at nuclear statistical equilibrium are determined by the nuclear Saha equation for neutron capture [29]:

2 3/2 n (A + 1,Z) A + 1 2π~ Sn (A + 1,Z) = nn · · · exp − , (2.6) n (A, Z) A kBT · mu kBT where kB is Boltzmann constant, and mu is the atomic mass unit. This production mechanism requires very neutron-rich environments. In explosive scenarios, typical neutron densities are higher than 1020 neutrons/cm3 and temperatures are higher than 1 GK. The second, “classic” waiting-point nuclei are caused by neutron shell closures and is the most relevant for the cadmium isotopes of interest in this work. At neutron shell closures, the energy required to capture an extra neutron above the neutron magic number rises and the neutron capture cross section drops. This leads to an accumulation of r-process material at the neutron shell closures, and provides the required waiting time for the probability of β-decay to increase and to overcome the waiting point. Once the β-decay and another neutron capture happen, the material is still at

10 the neutron shell closure and thus another classic waiting point is reached. The β-decays followed by neutron captures are repeated until the neutron capture cross section rises due to lower Sn-values, and therefore the r-process follows the neutron shell closure for several nuclei until it is able to break out to more neutron-rich nuclei. Figure 2.3 shows a possible path of the r-process identified with the respective waiting-point nuclei (red boxes) across the nuclide chart (with Z on the y-axis and N on the x-axis). Nuclei which were identified experimentally but for which no physical properties like half-lives or masses are known are shown in grey; isotopes for which the half-life was measured are shown in blue. The r-process path follows the waiting-point nuclei, which accumulate r-process material before β-decaying to stability (freeze-out). Since several such nuclei are located at the neutron shell closures at N = 82 and 126, the shell closures directly translate to peaks at masses A ' 130 and 195 in the solar r-abundance curve. The solar r-abundance (Nr) curve is deduced by subtracting the well-known calculated s-process abundances (Ns) from the observed solar abundances (N ): Nr = N − Ns. For example, the N = 82 isotope 130Cd is responsible for the peak maximum at A ' 130 and provides critical information on the second abundance peak of the r-process. Sensitivity studies of the r-process guide nuclear experiments by determining the nuclear properties and the isotopes which have the most impact on the calculations [32, 33]. Experimental information for isotopes between reaction path and stability affect calculations of the element abundances and astrophysical processes. First, the masses and the half-lives define the position and shape of r-process reaction path. Second, the nuclear structure far off stability defines the position of the abundance peaks. So far, no evidence for any deviation from the known shell structure like shell quenching has been observed for N = 50, 82, 126, and the robust location of the peaks of the r-abundance curve also do not hint to any new phenomena. For waiting-point nuclei, the important nuclear-structure properties are the decay half-life, the Qβ-value, the Sn-value, the excitation energies of the key levels and their comparative half-lives (log(ft) values) [1]. Unique level structures, such as long-lived isomeric states, are characteristic to the decay of the isotopes toward stability. Finally, neutrons from β-delayed neutron emission (βn) or (γ,n) have a smoothing effect on the abundance distribution. The region of the nuclide chart around Z = 50 and N = 82 has been investigated extensively in the past decades. However, the detailed explo- 132 ration of the region “south” and “south-east” of Sn (Z < 50, N > 82) requires the new generation of radioactive beam facilities and powerful γ-ray

11 250

100 200 98 96 94 92 90 88 Solar r abundances 86 84 82 80 78 184 180182 76 178 176 74 164 168 172 166 170 174 150 72 162 70 160 68 158 N=184 66 156 154 64 152 150 62 140 144 148 142 146 1 60 138 134136 58 130132 128 10 0 56 54 126 10 124 −1 52 122 120 116118 N=126 10 −2 50 112114 48 110 46 108 10 106 44 104 100102 42 98 96 Identified 40 92 94 38 86 88 90 36 84 82 Known half−life 34 80 78 32 7476 N=82 30 72 r−process waiting point 70 28 6668 64 26 62 60 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 N=50 Figure 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 Decay

2.3.1 Decay Law Nuclear decay is a random process for which the decay probability within a time interval dt is constant. Therefore, the probability of observing n events in a given dt follows a Poisson distribution with a standard deviation √ σ = n. The number of radioactive nuclei dN to decay within a time interval dt is equal to the decay constant λ times the number of nuclei in the sample N 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 of the isotope: the characteristic time for half the nuclei in a sample to decay.

12 Figure 2.4: N = 82 region of the nuclide chart close to Z = 50. (Adapted from ENSDF with updated half-lives from [34])

Alternatively, the lifetime of an excited state τ is deﬁned as 1/λ, i.e. the time 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): −λt − ln (2)·t/t N(t) = N0e = N(0)e 1/2 . (2.8)

When new nuclei N1 are being produced and added to the sample at a rate R1, with the initial condition N1(0) = 0, the size of the sample grows as: dN1 = R1dt − λ1N1dt (2.9a)

R 1 −λ1t N1(t) = 1 − e (2.9b) λ1 −λ1t A1(t) = λ1N1(t) = R1 1 − e . (2.9c)

The activity of the sample as a function of time A1(t) is given by multiplying N1(t) by the decay constant λ. For a production time t t1/2, secular equilibrium (saturation) is reached, the exponential goes to zero and

13 ∼ A1(t) = R1. When the beam of produced radioactive nuclei is turned oﬀ after reaching saturation, the isotope decays according to: R 1 −λ1t N1(t) = e (2.10a) λ1

−λ1t A1(t) = λ1N1(t) = R1(t)e . (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 nuclei N2(t) with decay constant λ2 and activity A2(t) = λ2N2(t) follows, while the beam is on:

R R 1 −λ2t 1 −λ2t −λ1t N2(t) = 1 − e + e − e . (2.11) λ2 λ2 − λ1 While the beam is oﬀ, the daughter isotope decays following:

R 1 −λ1t −λ2t −λ2t N2(t) = e − e + N2(0)e . (2.12) λ2 − λ1 Figure 2.5 illustrates the decay curves for 128Cd and its ﬁrst daughter 128In (Equations (2.9) to (2.12)).

2.3.2 Beta Decay Beta (β-) decay is a type of nuclear decay involving three particles. Decay processes are energetically possible for positive Q-values, which are defined as the difference between the atomic masses of the parent and the daughter atoms. Neutron-deficient nuclei β+-decay by changing a proton p into a neutron + n and emitting a positron e and an electron neutrino νe :

+ p → n + e + νe, (2.13a)

A A ∗ + Z XN →Z−1 XN+1 + e + νe, (2.13b) + from which the Qβ -value follows:

+ A A ∗ 2 Qβ = m Z XN − m Z−1XN+1 − 2me c , (2.13c)

A A ∗ where m Z XN and m Z−1XN+1 are the neutral atomic masses of parent and daughter nuclei, respectively, me is the mass of the positron and c is the

14 128 Figure 2.5: Number of β-decays as a function of time for Cd (t1/2,parent = 128 0.246 s) and In (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 from t = 0 to 10 s. speed of light. The last equation highlights that β+ decay is only possible 2 if the atomic mass energy diﬀerence is greater than 2mec = 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 atom is left in an excited state (X0) 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)

A − A 0 Z XN + e →Z−1 XN+1 + νe, (2.14b) and A A 0 2 Q = m Z XN − m Z−1XN+1 c − Bn, (2.14c) where Bn is the binding energy of the captured n-shell (n = K, L, ...) electron.

15 Neutron-rich nuclei β−-decay by changing a neutron into a proton and emitting an electron and an electron antineutrino:

− n → p + e +ν ¯e, (2.15a)

A A ∗ − Z XN →Z+1 XN−1 + e +ν ¯e, (2.15b) and − A A ∗ 2 Qβ = m Z XN − m Z+1XN−1 c . (2.15c)

Energy Release in β-decay

The 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 and the daughter atoms. In Figure 2.6, the Qβ-value represents the difference between the ground state energies of the parent and the daughter nuclei. When the electron is released Ee < Qβ, the β-decay typically also populates excited states in the daughter nucleus. These excited states, which are marked by ∗ in Equations (2.13) and (2.15), then emit the extra energy as gamma-radiation in order to reach the ground state of the daughter nucleus or a β-decaying isomeric state. This process is called gamma (γ-) decay and is discussed in the next section. If the Qβ-value is larger than the Sn-value of the daughter, the β-decay can populate states above the Sn-value which will decay either by emission of a neutron or γ-decay. Therefore, the final nucleus from a β-delayed neutron emission (βn) has Z + 1, N − 2 and A − 1. The probability of a βn-decay depends on the difference between the Qβ-value of the parent and the Sn of the daughter.

Selection rules The orbital angular momentum l of the emitted electron defines the type of β− decay transition, or the degree of forbiddenness, which can either be superallowed or allowed (l = 0), first forbidden (l = 1), second forbidden (l = 2), etc. The parity of the transition follows (−1)l. Forbidden transitions are suppressed in decay rate compared to allowed transitions [36]. The change in angular momentum between the parent nucleus (Ji) and the 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 =

16 Figure 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 transitions must carry one unit of angular momentum. It follows that transitions for which ∆J = L + 1 are only possible via Gamow-Teller transitions and are called “unique”. In addition, Fermi decays populate isobaric analog states, which means that a neutron from a neutron orbital decays to a proton in a proton orbital with the same l and J. Since most very exotic nuclei do not have neutron and proton valence orbitals in the same shells, Fermi decays are not possible and only Gamow-Teller decays are expected. For example, considering the 0+ ground state of the even-even 128Cd parent nucleus, it follows from the selection rules that its β-decay only feeds low-spin states in the daughter nucleus 128In. Isobaric analog states do not exist in this very exotic isotope (48 protons and 80 neutrons) and only Gamow-Teller decays are expected. Therefore, allowed decays populate 1+ states, and ﬁrst forbidden decays populate 0−, 1− or 2− states.

17 Table 2.1: Selection rules for β-decay angular momentum and parity. Note: Parentheses indicate transitions which are not possible if either Ji or Jf is 0.

Fermi Gamow-Teller Transition Type l ∆Π ∆J ∆J log(ft) Superallowed 0 No 0 – ∼3.5 Allowed 0 No 0 (0),1 ∼4.0–7.5 First Forbidden 1 Yes (0),1 0,1,2 ∼6.0–9.0 Second Forbidden 2 No (1),2 2,3 ∼10–13 Third Forbidden 3 Yes (2),3 3,4 ∼14–20 Fourth Forbidden 4 No (3),4 4,5 ∼23

Comparative Half-Lives The 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 and ψf and the transition operators τ, στ can be extracted from measurements of the Qβ-value and the half-life t1/2 of the parent nucleus, and the intensity of 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 transitions; gV is the vector coupling constant and gA is the axial-vector coupling constant.

3 7 f(Qβ − Ef ,Z)t1/2 2 ln 2π ~ 1 ft = = 5 4 2 2 (2.16a) Iβ(Ef ) mec gV B(F) + gAB(GT) where ∗ 2 B(F) = |hψf |τ|ψii| , (2.16b)

∗ 2 B(GT) = |hψf |στ|ψii| . (2.16c)

The log(ft) values allow an identiﬁcation of possible types of transition and a comparison of β-decay probabilities in diﬀerent nuclei. Typical ranges of log(ft) values for known decays are listed in Table 2.1.

18 2.3.3 Gamma Decay Similar 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 . Successive γ-ray emissions happen as a cascade until the ground state or a β-decaying isomeric state is reached. The γ-rays carry crucial information on the change in 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 exist with half-lives longer than 10−8 s. Isomeric transitions can have a big impact on nuclear astrophysics simulations as they halt the r-process ﬂow back to stability because of their much longer half-lives. In addition, the suppressed transition probability of isomeric states provides important information on the nuclear conﬁguration of these states.

Electromagnetic Radiation From classical electrodynamics, the energy emitted per unit time by a dipole radiation ﬁeld expends to higher multipole orders with:

2L+1 2(L + 1)c ω 2 P (σL) = 2 [m(σL)] , (2.17) 0L [(2L + 1)!!] c where σ = E or M for electric or magnetic radiation, the multipole order is deﬁned as 2L (L = 1 for dipole, L = 2 for quadrupole, etc.), ω is the angular frequency, and m(σL) is the amplitude of the time-varying electric or magnetic multipole moment. In quantum mechanics, the probability of photon emission per unit time is obtained by integrating the average radiated power over the volume of the nucleus:

2 2L+1 2 8π(L + 1) e E 3 2L λ(EL) ' 2 cR (2.18a) L [(2L + 1)!!] 4π0~c ~c L + 3 for electric transitions, and

2 2 2 8π(L + 1) 1 ~ e λ(ML) ' 2 µp − L [(2L + 1)!!] L + 1 mpc 4π0~c (2.18b) E 2L+1 3 2 × cR2L−2 ~c L + 2

19 for magnetic transitions, where E is the transition energy, R is the nuclear radius, and µp and mp are the proton magnetic dipole moment and mass, 2 respectively. The (e /4π0~c) dimensionless factor is the ﬁne structure constant ('1/137). The Weisskopf estimates of the transition probability for the lower mul- 1/3 2 tipoles are obtained by replacing R = R0A and [µp − 1/(L + 1)] = 10:

λ(E1) ' 1.0 × 1014A2/3E3 λ(E2) ' 7.3 × 107A4/3E5 (2.19a) λ(E3) ' 34A2E7 λ(E4) ' 1.1 × 10−5A8/3E9

λ(M1) ' 5.6 × 1013E3 λ(M2) ' 3.5 × 107A2/3E5 (2.19b) λ(M3) ' 16A4/3E7 λ(M4) ' 4.5 × 10−6A2E9, where λ is in s−1 and E is in MeV [20]. It is obvious from the Weisskopf estimates that the lower multipolarities are 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 as isomeric transitions with multipolarities M2 and E3. Also, in medium and heavy nuclei, electric radiation is more likely than magnetic radiation for a given multipole order. The unobserved transitions contribute to the Pandemonium effect which affects the uncertainty of the measured transitions intensities [37, 38]. There- fore, experimental knowledge of higher energy excited states, which de-excite to lower states with both low and high multipole orders, can reduce the impact of the Pandemonium effect.

Selection rules

Transitions between excited states Ji and Jf are characterized by the change in total angular momentum |Ji−Jf | 6 L 6 Ji+Jf , where L gives the order of the multipole. Electric and magnetic multipoles of same order have opposite parity, such that Π(EL) = (−1)L and Π(ML) = (−1)(L+1). These selection rules are summarized in Table 2.2. Since a photon carries a minimum of

20 L = 1 and J~i = L~ +J~f , it is important to note that a 0 → 0 transition cannot decay 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 internal conversion electrons.

L 0 1 2 3 4 5

ΠiΠf = −1 E1 E1 M2 E3 M4 E5 ΠiΠf = +1 M1 M1 E2 M3 E4 M5

Angular correlations A state of given angular momentum l is also characterized by its magnetic substate m, which runs from -l to l. For a cascade of two γ-ray transitions detected in coincidence by two diﬀerent detectors, an uneven population of m-substates is observed for the distribution of one γ-ray with respect to the direction of the other γ-ray [20]. The probability of observing one γ-ray as a function of the angle to the direction of the other γ-ray, is given by:

W (θ) = A0 + a2P2(cos θ) + a4P4(cos θ) (2.20a) = A0 [1 + a22P2(cos θ) + a44P4(cos θ)] , where 1 P (cos θ) = 3 cos2 θ − 1 , (2.20b) 2 2 1 P (cos θ) = 35 cos4 θ − 30 cos2 θ + 3 (2.20c) 4 8 are the Legendre polynomials [39, 40]. The a22 and a44 coeﬃcients are characteristic to the transition characters and their mixing ratio δ: hψ |E(L + 1)|ψ i δ = f i , (2.21) hψf |M|ψii which compares the relative matrix elements for electric and magnetic transitions. The electric transition is in the numerator since the mixing of higher order electric transitions with lower order magnetic transitions is possible. This mixing is possible since the transition rates of higher order electric transitions and lower order magnetic transitions can be comparable (from

21 Figure 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 transitions with lower order electric transitions is not common, but still possible. Information on the spin and parities of different levels can be extracted by obtaining experimental coefficients and comparing them to theoretical a22 and a44 coefficients for given γ-ray cascades, given that enough statistics are available. For example, Figure 2.7 highlights a variety of γ-γ angular correlations curves [41]. A 4 → 2 → 0 cascade with coefficients a22 = 0.1020 and a44 = 0.0091 is shown. The angular distributions are also shown for a 1 → 2 → 0 cascade with a22 = 0.2829 and a44 = −0.1524 (mixing ratio δ = −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 Conversion It is possible for the excited nucleus to interact electromagnetically with an atomic electron, which is then emitted. This internal conversion pro-

22 cess competes with the γ-decay and the total decay probability of a level λt is then the sum of the γ-emission probability λγ and the internal conversion probability λe. Therefore, a level decays more rapidly than it would from γ-decay only. Internal conversion includes E0 transitions (0 → 0) (see Table 2.2). The internal conversion coeﬃcient (ICC) α is deﬁned as λe/λγ and is estimated by:

3 2 4 2 L+5/2 Z L e 2mec α(EL) ' 3 (2.22a) n L + 1 4π0~c E

3 2 4 2 L+3/2 Z e 2mec α(ML) ' 3 . (2.22b) n 4π0~c E where n is the principal quantum number of the atomic shell and me are the electron mass. Firstly, the Z3 dependence of α means that the internal conversion process is more important for heavy nuclei. Secondly, α decreases rapidly for higher 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 multipole order. The ICCs can be calculated very precisely with the BrIcc code [42].

23 Chapter 3

Review of Literature

Nuclear structure studies on the neighbours of the doubly-magic 132Sn (Z = 50, N = 82) contribute to our knowledge of the shell evolution and of the extreme limits of the nuclear force, as well as the application of the information to calculations of the astrophysical r-process, which is thought to 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 off stability experimentally accessible today, is still not well known. The doubly magic character of 132Sn (Z = 50, N = 82) was established through (d,p) neutron transfer reaction with a radioactive ion beam of 132Sn in Ref. [43]. The spectroscopic factors calculated were consistent with 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. Very recently, the results of a Coulomb-excitation experiment of 132Sn at CERN- ISOLDE were considered to be the first direct verification of the sphericity and double magicity of 132Sn [44]. Their new experimental values for the reduced 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 waiting point 130Cd was published in 2003 from ISOLDE data [1]. It claimed that only mass models including the phenomenon of N = 82 shell quenching 132 (where the shell gap is reduced) below Sn agreed with the high Qβ-value of 8.34 MeV. In 2016, Ref. [45] used high statistics γ-γ coincidences from the EURICA array at RIKEN to revise the decay scheme of 130In. These new results were found to be in good agreement with shell model calculations and do not show shell quenching.

3.1 128Cd

128 The half-life of the ground state of Cd was ﬁrst measured in 1986 (t1/2 = 340(30) ms [46]). The EURICA collaboration recently measured a shorter half-life of 245(5) ms [34]. A consistent result (t1/2 = 246.2(21) ms) was

24 extracted from the data set presented in this work and published in 2016 [4]. 128 The Qβ-value of Cd is reported to be 6.904(153) MeV [19], while the 128 Sn-value in In 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 lightest isotope studied in this work, 128Cd, is very limited. The relevant publications are based on an experiment at the OSIRIS ISOL-facility in Sweden. As shown in Figure 3.1, seven transitions between five states in 128In were published [3]. The large majority of the β-decay of the 0+ ground state of 128Cd feeds the 1+ excited state in 128In at 1173 keV. It is worth noting that this 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) transitions were deduced by conversion electron measurements. It is important to note that the decay scheme presented comes from a work which was not published and therefore only few details are known about the experiment and the analysis. The decay scheme reported in 1988 was re-evaluated in 2015 by ENSDF (Evaluated Nuclear Structure Data File) evaluators [14] and the tentative spin and parity J Π assignment of the 316 keV level was revised from a 2− to a 1− [14]. The neutron-rich even In isotopes (odd N, odd Z) have common properties: a low-lying 1+ excited state and isomeric state(s). The evolution of the important states across the neutron-rich even-mass In isotopes from A = 122 to 130 are shown in Figure 3.2. The energy of the first excited 1+ state is important for r-process simulations because it is the main β-decay branch −1 −1 (GT feeding) [1]. Its predominant configuration is noted as πg9/2 ⊗ νg7/2, indicating the coupling of a proton hole in the πg9/2 orbital with a neutron hole in the νg7/2 orbital. In 2004, the half-life of the 248-keV transition was measured to be 23(2) µs in Ref. [47]. This 1− isomeric state was reported to result pre- −1 −1 dominantly from the coupling πg9/2 ⊗νh11/2. The main configuration of the −1 −1 + ground state is πg9/2 ⊗ νd3/2, 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 not observed in this experiment since it is not populated by the β-decay of the parent 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 assignments of the excited states at 316 keV and 711 keV.

26 (8-) 2900 1+ 2120

1+ 1173

(1+) 688

(8-) 262 (5+) 359 (1)+ 242 (1-) 248 (8-) (10-) 5+ 40 50 (8-) 90 50 1+ 0 (1)+ 0 3(+) 0 (3)+ 0 1(-) 0 122 124 126 128 130 49 In73 49 In75 49 In77 49 In79 49 In81

Figure 3.2: Evolution of the ground state, ﬁrst 1+ and isomeric state(s) in even-mass 122−130In. Level energies [in keV] shown with left and right wings are not to scale. [Data taken from ENSDF]

27 3.2 131Cd

131 The large difference between the Qβ-value of Cd (12806(103) keV [19]) 131 and the Sn of In (6213(38) keV [19]) leaves a large window of 6593 keV for the βn-decay to 130In. However, the half-life and neutron-branching ratio measured at ISOLDE showed surprising results [50]. While Quasi-Particle Random Phase Approximation (QRPA) calculations deduced a half-life of 286 ms and a neutron-branching ratio (Pn) of 39.8% [51], experimental values found 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, which represents 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 energy of 3782 keV and a tentative spin assignment of 17/2+ was identified in a GSI experiment 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 a trap measurement published in Ref. [53]. The β-decays of 131−132Cd have received a lot of attention in the last two decades. The first spectroscopic information was reported in 2000 in a PhD thesis with transitions at 995, 1587, 1737, 1910, 360, and 3870 keV [54]. Information on γ-transitions from an experiment at CERN-ISOLDE is reported in [55] and in a PhD thesis of the same group [56]. However, both references 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 three of the transitions in a single-particle picture: the 988-keV γ-ray corresponds to the transition between the πp3/2 and the πp1/2, the 2428-keV between the π5/2 and the πp1/2, and tentatively the 844-keV transition between the πf5/2 and the πp3/2. The EURICA collaboration at RIKEN has recently published results for the π-hole states (with respect to the 132Sn core) in 131In from the β-decay of 131Cd and the βn-decay of 132Cd, which confirmed the main transition at 988 keV [5]. This transition had also been seen in the ISOLDE data [55], but was not mentioned in the following PhD thesis [56]. A more detailed analysis of the EURICA data identified more than 20 transitions [6]. A tentative level scheme for each data set is shown in Figure 3.3. While the transition of 988 keV was observed again between the πp3/2 and πp1/2 orbitals, the − −1 −1 −1 (5/2) levels were not assigned. While the πg9/2, πp1/2 and πp3/2 states

28 (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.

29 were observed in both experiments, the location of the 5/2−π-hole state is still uncertain. The evolution of the 1/2–9/2 states across the neutron-rich odd-mass In isotopes is shown in Figure 3.4. 131 The ground state of Cd with configuration ν1f7/2 [6] is in very good 130 approximation the ground state of Cd coupled to a 1f7/2 neutron across the N = 82 shell gap. A single-particle decay should convert the 1f7/2 neutron into a proton, which can only occupy a single-particle state that is not occupied in the ground state of 131Cd. For protons, the ground state of 131Cd is equal to the ground state configuration of 130Cd. Figure 3.5 summarizes the possible allowed (GT) and first forbidden (ff) decays as discussed in Ref. [6]. Firstly, the allowed Gamow-Teller decay (L = 0, ∆I = 1, ∆Π = 0) of a 0g7/2 neutron into the 0g9/2 orbital is expected to dominate the decay of 131Cd. This decay, which has been observed in the decay of many other −1 −1 cadmium and indium isotopes, populates states of the π9/2 ⊗ ν(1f7/20g7/2) multiplet and should be high in energy. The transition of a 1f7/2 neutron into a 0f5/2 proton is not expected to be directly populated since the two orbitals are separated by two harmonic oscillator shells [6]. Secondly, several first forbidden decays (L = 1, ∆I = 0, 1, 2, ∆Π = 1) are possible. A 1f7/2 neutron can decay to a 0g9/2 proton, which populates the + 131 9/2 ground state of In. Also, a 1f7/2 neutron can decay to a 0g7/2 proton across the Z = 50 shell. This decay results in a 7/2+ cross-shell excited state at very high energy due to the cross-shell excitation (5-6 MeV). A ν0h11/2 → π0g9/2 transition populates a multiplet of states that results from the coupling of an unpaired 0g9/2 proton with a 1f7/2 neutron and a −1 −1 0h11/2 neutron hole, i.e. π0g9/2 ⊗ν(1f7/20h11/2). According to Ref. [6], these states 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 of −1 −1 states π(1p, 0f5/2) ⊗ ν(1f7/21d3/2) from coupling an unpaired 1f7/2 neutron, a 1d3/2 neutron hole and a 1p1/2,3/2 or 0f5/2 proton hole. However, these decays are possible only to the fractions of the wave function which have −2 −2 −2 130 the appropriate 1p1/2, 1p3/2 or 0f5/2 holes in the Cd ground state. These states are also expected at ∼4 MeV. However, in comparison to the allowed transition, these first forbidden transitions would be even further suppressed and one would expect only larger log(ft) values on the order of ∼6 and higher.

30 (5/2-)(2730)

(5/2-)(2134)

(9/2+) 1686 (7/2+) 1566 (7/2+) 1578 (9/2+) 1586 (9/2+) 1564 (5/2-) 1588 (9/2+) 1512 (5/2-)(1525) (5/2+) 1423 (3/2-) 1353 (5/2-) 1220 (5/2+) 1202 (5/2-) 1137 (5/2+) 1099 (5/2+) 1052 (3/2-)(983) (3/2-) 932 (3/2-) 797 (3/2-) 698

(1/2-) 459 (1/2-) 1/2(-) 408 (1/2-) (1/2)- 327 360 365

(9/2)+ 0 9/2+ 0 (9/2+) 0 (9/2+) 0 (9/2+) 0 123 125 127 129 131 49 In74 49 In76 49 In78 49 In80 49 In82

Figure 3.4: Evolution of the 1/2–9/2 states in odd-mass 123−131In. Level energies [in keV] shown with left and right wings are not to scale [Data taken from ENSDF and Ref. [55]].

31 Figure 3.5: Single-particle orbitals in the 132Sn region [6]. The ground state configuration of 131Cd is shown with open (proton holes) and filled (neutron particle) dots. The allowed (GT) and first forbidden (ff) single-particle decays are indicated by solid and dashed arrows, respectively.

32 3.3 132Cd

132 The large difference between the Qβ-value of Cd (12 150(510) keV [19]) 132 and the Sn of In (2450(60) keV [19]) predicts an important β-delayed neutron (βn) branch. A neutron-branching ratio of Pn = 60(15)% and a 132 half-life for the ground state of Cd of t1/2 = 97(10) ms were reported from a previous ISOLDE experiment [50]. The half-life was remeasured at RIKEN to t1/2 = 82(4) ms [34]. A QRPA calculation published in [50] and shown in Figure 3.6a indicates a 1+ state at ∼1200 keV which would be fed by an allowed Gamow-Teller decay. It is very likely that the transitions from the 1+ state to the tentatively 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+ state is a β-decaying isomer. These two cases could be distinguished and indi- rectly measured. If there is a β-decaying isomeric state in 132In, transitions from excited states in 132Sn would be seen. Otherwise, the appearance of the 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 about any excited states in 132In. The non-observation of any transition in 132In could hint to a Pn-value closer to 100% than the previously measured 60% 132 [50]. The Pn-value of Cd has been remeasured in 2017 with the BRIKEN setup at RIKEN, and a preliminary value of Pn ∼ 88(5)% was presented recently by A. Estrade at Nuclear Structure 2018 [57], which would confirm the EURICA value. The question remaining is where the remaining ∼ 12% of feeding go. The P2n channel can be excluded experimentally from the BRIKEN data (P2n ∼ 0%, Qβ2n=3480(200) keV). Different insight into missing π-hole states in 131In is gained from the 132Cd βn-decay compared to the 131Cd β-decay. 132Cd is an even-even nucleus with a 0+ ground-state and so the βn-decay populates only low-spin states 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, it populates only high-spin states by its β-decay. Additionally, the probability of β-decay to a specific state depends on the energy transition to the fifth power (∝ E5). β-delayed neutron emission tends to populate states at low excitation 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 and

33 (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 energies in 132In and therefore the six γ-rays are believed to form a cascade of M1 − −1 −1 − transitions between the 1 multiplet (πg9/2 ⊗νf7/2) and the 7 ground state (see Figure 3.7).

34 Figure 3.7: Tentative levels energies [in keV] for 132In [Figure from XUNDL].

35 Chapter 4

Experiment

In August 2015, the ﬁve neutron-rich isotopes of Cadmium were delivered to the GRIFFIN spectrometer in the low-energy experimental area of the ISAC-I facility at TRIUMF over seven days of beam time (13 shifts). This chapter covers the production of the beam, the detector hardware as well as the data processing.

4.1 Beam Production

A 480-MeV 9.8-µA proton beam from the TRIUMF main cyclotron was directed to the East production target in the ISAC1 facility [59]. The ISAC experimental hall layout is shown in Figure 4.1. An Isotope Separation On-Line (ISOL) production target of uranium carbide (UCx) was used to produce the neutron-rich cadmium beams studied in this work via proton spallation 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 at temperatures around 1500◦C releases strong surface-ionized isobaric contaminants of the same mass such as cesium, barium and indium, along with the cadmium atoms of interest. First, an electrostatic potential barrier re- pels these contaminants, allowing only neutral atoms to enter the cold region of the resonance ionization laser ion source (RILIS), which is protected by a heat shield. Second, a three-step laser excitation, which depends on the proton number Z, enables to selectively ionize the cadmium atoms [61]. The excitation scheme for cadmium (Z = 48) consists three successive transitions at wavelengths 228.9 nm, 466.3 nm and 1064 nm, respectively. Finally, a linear radio frequency quadrupole (RFQ) guides the ions toward the high voltage exit electrode. Next, the isotope of interest was isolated with the ISAC high resolution mass separator based on the A/q ratio, where A is the mass number and q is the charge of the ions, with a resolving power in the order of 1/2000.

1Isotope Separator and ACcelerator

36 Figure 4.1: TRIUMF ISAC experimental hall layout [59]. The 8π γ-ray spectrometer was replaced by the GRIFFIN array, which was commissioned in 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) enabled to control the length of time for which the beam was received. These steps allow the suppression of surface-ionized indium and cesium to the extent that the spectroscopy of laser-ionized cadmium becomes possible, 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 at ISAC’s yield station after the high-resolution mass-separator isolated the isotope of interest. When the repeller is oﬀ, the yield of laser-ionized cadmium (blue) is drowned in background (pink). When the repeller is on, the cadmium yield is comparable to the remaining contamination (red), making the study of these cadmium beams possible. Because of the low intensity of the 131−132Cd beams, their yields were measured for the ﬁrst time directly

37 Figure 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 eﬃciency than the ISAC yield station.

4.2 Detectors

The delivered beam was implanted on an in-vacuum moving tape collector (MTC) system at the center of both the GRIFFIN spectrometer and the SCEPTAR3 plastic scintillator array [64], shown in Figures 4.4 and 4.5. The aluminum coated Mylar tape removes long-lived activity from the centre of the array to behind a lead shielding wall, reducing the background radiation from such long-lived nuclei. The moving tape is operated in cycles which typically consists of the tape move, a background measurement, a collection time (beam on) and a decay time (beam oﬀ). The transition between the beam on and oﬀ parts of the cycle are controlled by the kicker. The cyclic motion of the tape station enables crucial analysis techniques which will be discussed in Chapter 5, such as the measurement of half-lives and discrimination of background γ-rays. SCEPTAR is an array of 20 scintillating paddles enabling β-particle tagging. While SCEPTAR was positioned inside the vacuum chamber, a low-Z Delrin plastic sphere (10 or 20 mm thick) was installed on the outside of the chamber in order to absorb the high energy electrons and protect the high purity germanium (HPGe) crystals. Details on the geometry of SCEPTAR are given in Ref. [66].

2Gamma-Ray Infrastructure For Fundamental Investigations of Nuclei 3SCintillating Electron-Positron Tagging ARray

38 Figure 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 detectors, for a total of 64 coaxial semiconductor crystals, dedicated to decay spectroscopy of the low-energy radioactive ion beams at TRIUMF. Details on the geometry of the GRIFFIN array are given in Ref. [66]. γ-rays interact with matter through three mechanisms: photoelectric absorption, Compton scattering and pair production. With photoelectric absorption, an atom completely absorbs an incoming γ-ray and emits an electron with energy Ee = hf − Bn, where f is the frequency of the photon and Bn is the binding energy of the electron. The high energy electron then interacts with the charge carriers in the germanium crystal. For an n-type semiconductor, electrons are the majority carriers and holes are the minority carriers. The electron/holes pairs migrate in opposite directions with the high bias voltage applied between the p- and n-contacts, where the total charge deposited is collected. The photoelectric process is the dominant mode of interaction for γ-rays of relatively low energy [67]. With Compton scattering, an incoming γ-ray transfers part of its energy to an electron in the

39 Figure 4.4: The west hemisphere (8 of 16 High Purity Germanium (HPGe) clover detectors) of the GRIFFIN γ-ray spectrometer is shown, with the SCEPTAR 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 array and moving-tape collector inside the vacuum chamber at the centre of the GRIFFIN array. [65]

40 detector 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 a positron, both with masses of 0.511 MeV/c2. The positron then annihilates with 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 a neighbouring detector and form a peak at 511 keV. The GRIFFIN signals are processed by four digitizer modules with 16 channels (GRIF-16). A fifth GRIF-16 collects the SCEPTAR signals. Col- lector modules (GRIF-C Slave) concentrate the 16 outputs of a GRIF-16 into a single output, which is linked to the GRIF-C Master collector. The master clock of the GRIF-C Master collector provides the reference clock to 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 the preset filter and sends the accepted signals to the MIDAS [68] frontend to be written to disk. The current data were collected in “singles” mode, which mean that every γ-ray and every β-particle detected was written. The fully digital data acquisition (DAQ) system is described in [69]. The MIDAS data files were sorted using an in-house analysis framework named GRSISort [10], based on the code ROOT [11] from CERN. GRSISort first sorts the raw data into fragment trees. Then, analysis trees are written by ordering the hits by time stamps and grouping them in events defined by a coarse coincidence time window typically of 2 µs. The code also includes interactive analysis tools which enable rapid visualization and analysis of the data. From this point, the user writes scripts to display the full data set or parts of it by applying one or several constraints as required.

4.3 Data Processing

Crosstalk Correction An important effect of clover-type HPGe detectors is the crosstalk between the four crystals. A γ-ray detected in a crystal can induce signals in the neighbouring crystals and hence decreases the resolution of the detectors. This energy-dependant effect is corrected by applying a clover-by-clover correction 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 energy resolution is recovered.

41 Clover Addback 105

Gamma-ray singles

104

103 Counts per 1 keV

102

1100 1150 1200 1250 1300 1350 1400 Energy [keV] (a) Without crosstalk correction

Clover Addback 105

Gamma-ray singles

104

103 Counts per 1 keV

102

1100 1150 1200 1250 1300 1350 1400 Energy [keV] (b) With crosstalk correction

Figure 4.6: Comparison of clover addback [blue] and γ-singles [red] spectra observed for a 60Co source (a) without and (b) with crosstalk correction.

42 Addback Method Compton scattering of γ-rays between the four crystals or even between neighbouring detectors is detected as background. Any γ-ray entering a crystal can deposit a fraction of its energy before scattering in one or more other crystals. Therefore partial energies are collected and build a continuous background underneath the full-energy photopeaks. The full energy of the scattered γ-ray can be recovered by adding back the partial energies which were detected in neighbouring crystals. Therefore, the addback method enables us to increase the detection efficiency [66]. Obviously, partial energies for γ-rays which were scattered to the outside of the array cannot be recovered. This undesirable effect can be reduced with Compton suppression shields, which were installed on GRIFFIN in 2018, after the neutron-rich Cd data sets were collected. The addback mode is defined with both a time condition and a space condition. The γ-rays added-back have to be detected within 300 ns of each other, and within the same clover detector. Figure 4.7 (which shows the full range of the data in Figure 4.6) presents the effect of addback: the Compton background is reduced and the signal-to-noise ratio is improved.

Summing Correction The summing of γ-rays is another effect seen in arrays with large angular coverage. Since the isotopes are implanted and stopped in the tape, their decays happen isotropically. The isotropic angular distribution is a result of the random orientation of nuclear spins and thus a lack of quantization axis. Therefore, there is a non-zero probability that two γ-rays are emitted in the same direction and detected in the same crystal. γ-rays from different nuclei on the tape can also deposit their energies in a single crystal, which will only detect the total energy deposited. However, the probability of two γ-rays being emitted at 0◦ is the same as being emitted at 180◦. The summing effect can be corrected by measuring the total number of events in the γ-gated coincidence spectrum between two opposite detectors separated by 180◦.

Energy Resolution and Detection Eﬃciency The average energy resolution at full width at half maximum (FWHM) at 122.0 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 a

43 105 Clover Addback

Gamma-ray singles

104

103 Counts per 1 keV

102

0 200 400 600 800 1000 1200 1400 Energy [keV]

Figure 4.7: Comparison of clover addback [blue] and γ-singles [red] spectra observed for a 60Co source. germanium detector, which has a band gap of ∼0.7 eV, is given by the statistical deviation of the number of electrons emitted divided by the number of electrons. For a 1332-keV γ-ray, the number of electrons emitted Ne = 1332 keV/(0.7√ eV/electron) = 1902857 electrons and the theoretical best resolution is Ne/Ne ∗ 1332 = 0.97 keV. The absolute γ-ray detection eﬃciency γ is deﬁned as:

Number of γ-rays detected Nγ,detected γ = = , (4.1) Number of γ-rays emitted by source Iγ · A · t where Iγ is the absolute decay intensity of the γ-ray, A is the activity of the calibration 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, which is the minimum amount of time for which two γ-rays can be distinguished as separate signals [67]. The detection efficiency γ as a function of energy E is determined by fitting the efficiency of all the individual γ-rays γ to:

2 2 p0+p1 log(E)+p2 log (E)+p3/E γ(E) = 10 . (4.2)

44 Figure 4.8: Time diﬀerence between consecutive triggers as a function of crystal number for a 152Eu source. The dead time, i.e. the minimum time between triggers, is found to be 7.5 µs per trigger.

The dead time per trigger is determined by looking at the time difference between consecutive triggers as shown in Figure 4.8. The minimum time between triggers is found to be 7.5 µs per trigger per crystal. The average dead time 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 time difference less than 7.5 µs are called pile-up events and are rejected in the analysis. Newer versions of the DAQ are able to distinguish these pile-up events in to two or more separate events [69]. The absolute γ-ray detection efficiency curves shown in Figure 4.9 for the energy range of 53–3450 keV were determined with calibration sources of 133Ba, 152Eu and 56Co and include both the addback and summing corrections. For the 131,132Cd data sets, the 20 mm Delrin plastic sphere was used with 62 operational crystals. Step-by-step methods for energy and efficiency calibrations 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 sphere

Figure 4.9: Absolute γ-ray detection eﬃciency for the GRIFFIN spectrometer in addback mode with summing correction: (a) with SCEPTAR and 10 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.

46 Chapter 5

Data Analysis and Results

5.1 128Cd

5.1.1 β-Gated γ-Singles Measurements Approximately seven hours of 128Cd data were collected with a beam intensity of ∼1000 pps. With these statistics, a detailed coincidence analysis of weaker lines became possible. Also enabled is the determination of spin and parities with γ-γ angular correlation measurements, where only a few thousand β-γ-γ coincidences are required. The data set was collected in “singles” mode, which means each γ-ray and each β-particle detected was written to disk. Therefore, looking at γ-singles data includes room background radiation in addition to the γ- rays and X-rays emitted by the isotopes of the beam composition. The SCEPTAR scintillator array is the first tool used to isolate the γ-rays of interest, by only looking at the ones which were detected in GRIFFIN within a set coincidence time window relative to the detection of a β-particle in SCEPTAR. Figure 5.1 shows the energy of γ-ray plotted as a function of the difference 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 seen within a 200 ns wide time window. This window is called the prompt coincidence window and points to a time correlation between the β-particle and the γ-ray, meaning they are statistically more likely to come from the same decay event. At lower energies, a positive tail, which is a symptom of the slower charge collection in the germanium crystals for smaller amounts of electrons, can be noticed. γ-rays which are seen consistently with larger time differences, appearing as horizontal lines on the matrix, are most likely uncorrelated, e.g. coming from different isotopes, and are called time-random coincidences. Therefore the gate (or cut) selects the correlated β-γ events and rejects a majority of uncorrelated events. Step-by-step methods for time and energy gates, along further data analysis techniques, can be found in Appendix B.

47 Figure 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 is overlayed on the matrix.

The effect of this β-tagging is shown in Figure 5.2, which shows γ-singles spectra with and without β-tagging. The β-tagged data is defined by the coincidence 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 advantage is the significant reduction of the background at lower energy. Environ- mental background γ-rays include decay products from the natural decay of 235,238U and 232Th (U/Th-series), 40K from concrete, and a continuous background from cosmic-rays mesons [67]; all of which can be detected by GRIFFIN while collecting the data of interest. However, most of the background β-particles will not be detected by SCEPTAR, which is shielded from the experimental hall by the GRIFFIN clovers. Therefore a coincidence spectrum suppresses the room background while also increasing the photopeak-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 emission and their γ-ray(s). The 248-keV isomer seen in Ref. [3] with a half-life of 23 µs is confirmed here. In Figure 5.2, this transition disappears in the

48 ×103 1600 Beta-gated singles 247 1400 * Gamma-ray singles 1200

1000 68 * 857 800 *

600 Counts per 1 keV 400 462 * e-/e+

200 925 1172 * * 0 0 200 400 600 800 1000 1200 Energy [keV]

Figure 5.2: Comparison of β-gated γ-singles [blue] and γ-singles [red] spectra observed for the decay of 128Cd. No normalization is applied. The peaks with 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-γ coincidence window of 200 ns.

β-gated spectrum since most of the decays only happen outside of the β-γ coincidence window of 200 ns. Hence, few γ-rays were observed to be in coincidence with the 248-keV isomeric transition. Finally, this comparison also displays the eﬃciency of the SCEPTAR array. By taking the ratio of the area of a photopeak in β-gated γ-ray singles (Nβγ) and in γ-ray singles (Nγ):

Nβγ β · γ · Iγ = = β, (5.1) Nγ γ · Iγ 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 further

49 discrimination of isobaric background [55]. When the lasers are on, the spectrum is composed of γ-rays from surface- and laser-ionized species (cesium, 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 comparing subsequent spectra taken with lasers on and with lasers blocked. The comparison of the γ-spectra for 128Cd with the lasers on and blocked can be seen in Figure 5.3. Approximately 20% of the nine hours of 128Cd beam time was used to collect background data while one of the three lasers was blocked. 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 asterisk are newly observed lines. In both the lasers-on and laser-blocked data sets, there is a transition of 831 keV which is fed by the 8− isomer (0.72(10) s) in 128In only and not the 3+ ground state of 128In. However, when scaled to the 1460 keV room background line (from 40K) in γ-ray singles, the 831 keV transition is 5.5 times 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 state of 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 highly unlikely. Since the isomer cannot be populated by the β-decay of 128Cd, the 128In isomer has to be part of the beam composition. The first excitation step of cadmium with the 228.9 nm laser has enough energy to excite an indium electron into a Rydberg state, from which any of the two other lasers can non-resonantly ionize the indium. Some neutral indium can pass through the repeller electrode or, depending on the operation mode, into the ionization volume inside the ion guide, where it will then be ionized and extracted. Based on the intensity of the 831-keV gamma-ray, 45(2) pps for the 8− isomer were delivered with the lasers on, while only 8(1) pps were delivered with the first laser blocked. In the case of a heavily contaminated spectrum, i.e. with beam contaminants or decay chains of short-lived daughters, peaks of interest can also be identified by looking at counts as a function of the time within the tape cycle. 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 drops rapidly has the used tape is moved behind the shielding bricks. With the laser blocked (Figure 5.4b), the lower activity highlights the noise caused by the moving tape collector system. Then, the background is measured for 1.5 s before the beam is implanted on the tape (10 s). During the subsequent

50 ×103 500 Lasers on 68 857 *

400 *

300

200 462 Counts per 1 keV * 1172 100 925 1089 + (128Sn) 173 221 315 619 * * e-/e+ * * 408 * * * * 0 0 200 400 600 800 1000 1200

900 Laser blocked 800 e-/e+ 700 600 743 (128Te) 753 (128Te) 500 313 (128Te) 400 75 (128Sb) 300 121 (128Sn) Counts per 1 keV 168 442 (128Xe) 482 (128Sb) 200 589 788 (128Te) 808

100 1168 (128Sn) 0 0 200 400 600 800 1000 1200 Energy [keV]

Figure 5.3: Comparison of β-gated γ-ray spectra observed for the decay of 128Cd in addback mode with lasers on [blue] and laser blocked [red]. The laser-on spectrum was collected in cycle mode for ∼6.5 hours and the laser-blocked spectrum, for ∼2.5 hours. The peaks with a black asterisk correspond to γ-rays published in Ref. [3], whereas those with an orange asterisk are newly observed lines.

51 time window (2 s), data is collected with the beam off. One can see that due to the difference in half-lives, the first seconds of the beam-on window are dominated by the Cd activity (t1/2= 246.2(21) ms [4]), whereas the In activity (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 counted once. The plotted curves show the number of decays for the cadmium isotopes (red line), the first daughter indium (yellow line), the indium present in the 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 similar to how the laser on/blocked comparison highlighted the cadmium lines, the spectra for the beam-on (Cd + In) and beam-off (In) time windows also point to peaks of interest likely to be related to the Cd decay, as shown in Figure 5.5. The discrepancy between the fit and the data for the decay part of the cycle comes from a small fraction (∼5%) of the cycles which do not include a decay part. The transitions intensities were calculated from the beam-on part of the cycle only, which is not affected by the missing decay parts.

5.1.2 β-Gated γ-γ Coincidence Measurements Similar to the β-γ coincidence gate in Figure 5.1, a β-γ-γ coincidence gate can be defined by looking only at β-particles which were detected in coincidence with at least two γ-rays, as shown in Figure 5.6. This further cuts an order of magnitude of statistics and enables us to extract information on cascades 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 the energy of the first γ-ray on the x-axis and the energy of the second γ-ray on the y-axis. To ensure the symmetry of the matrix, the energy of the second 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, including horizontal, vertical and diagonal lines. Continuous horizontal and vertical straight lines consist of γ-rays coincident with every γ-ray energy and suggest random coincidences, as seen before in Figure 5.1. Diagonal lines show incomplete charge deposition when γ-rays scatter between crystals. Some of the scattered γ-rays were already corrected with the addback mode, however, some scatter events cannot be recovered and appear on the coincidence matrix 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 cycle consists of the tape move, background measurement (1.5 s), beam on (10 s) and beam oﬀ (2 s).

53 ×103 450 Beam on 68 857

400 * * 350 300 250 200 462

Counts per 1 keV 150 * 1172 100 925 1089 173 221 315 619 * * * * 408 * e-/e+

50 * * * 0 0 200 400 600 800 1000 1200

35000 Beam off 30000 68 857

25000 * * 20000

15000 1168 (128Sn) Counts per 1 keV 121 (128Sn) 462

10000 1089 + (128Sn) 257 (128Sn) e-/e+ 935 (128Sn) * * 925 538 (128Sn) 5000 831 (128Sn) *

0 0 200 400 600 800 1000 1200 Energy [keV]

Figure 5.5: Comparison of β-gated γ-ray spectra observed for the decay of 128Cd in addback mode as a function of cycle structure. The beam-on spectrum was collected for 10 s per cycle 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 asterisk are identiﬁed as newly observed lines.

54 Figure 5.6: Diﬀerence between the time stamp of a γ-ray coincident with a β-particle and the time stamp of a second γ-ray (γ1 − γ2) as a function of the energy of the second γ-ray. The 2-dimensional time cut is overlayed on the 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 is the second most intense transition in 128In, shows two spots: at 173 keV and 221 keV. In Figure 5.7c, there is a spot at the intersection of 316 and 857 keV, with the latter being the third most intense transition in 128In. The increased number of counts suggest a higher probability of observing γ-rays of these two energies at the same time when compared to γ-rays of neighbouring 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 the projection, which should include the coincidence spots, and two background gates on either sides of the photopeak in the projection spectrum. The background and scatter included in the second and third cuts can then be subtracted from the first cut of interest. However, since the diagonal scatter lines are seen at different energies in the projection, they create dips in the

55 (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 keV and (c) 857 keV and the region around 280 keV. All are displayed in 1 keV per bin.

56 background-subtracted projection. Firstly, the top and middle panels in Figure 5.8 show the gate on the third most intense transition in 128In: 68-keV between the 316 and 248-keV levels. The high intensity of the transition causes a lot of uncorrelated coincidences from different nuclei decays on the tape. Also, the projection includes a high level of low energy background, which includes X-rays and several Compton scatter lines from all transitions. The background subtraction is sensitive to these different sources of background, which can be over-subtracted (dips) or under-subtracted (peak). Therefore, it is useful to gate on higher energy transitions which show cleaner coincidences to extract convincing coincidence. The 2782, 3707 and 4432-keV transitions are placed directly 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 transition was placed on top of the 68-keV transition and results in a new level at 489 keV, which feeds the level at 316 keV and is directly fed by the 221-keV transition, which is fed by the top 462-keV transition. The 1097-peak is seen in this gate only, which results in a level at 1585 keV and directly feeds the 173-keV transition and 489-keV level. The subtraction of scatter lines creates 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 middle panels of Figure 5.9. The peak at 711 keV corresponds to the transitions from the level at 711 keV to the ground state. By comparing with the 857-keV gate (bottom panel), transitions can be placed either feeding the top 462-keV transition (1173 → 711 keV) or the bottom 462-keV transition (711 → 248 keV). The 1552, 2387 and 3313-keV transitions, which are not seen in the 857-keV gate, are placed directly feeding the 711-keV level. The 1089, 1924 and 2486-keV transitions, which are seen in both the 462 and 857-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-keV transition, which is not seen in coincidence with the 857-keV, places a level at 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 and middle panels) and on the 336-keV transition (bottom panel). The 408- keV transition is seen in coincidence with peaks at 211, 336, 516, 1608 and 2441 keV, and are placed directly feeding the 408-keV transition. The total energy of the cascade formed by the 408 and 516-keV transitions is ∼924keV, which corresponds to the energy difference between the 248 and 1173-keV

57 2500 68 keV gate (low) 221

2000 173 857 * * * 1500 75 (Xray) 462 * 1000 Scatter Counts per 1 keV 500 Scatter 1089 * 00 200 400 600 800 1000 1200

50 68 keV gate (high) 40 2782 Scatter

30 * 3707 *

20 Scatter Scatter

10 4432 Scatter *

Counts per 1 keV 0 −10 2600 2800 3000 3200 3400 3600 3800 4000 4200 4400

2000 173 keV gate 68 221 462 * 1500 * *

1000 Scatter Scatter Scatter

500 1097 Counts per 1 keV

0 *

0 200 400 600 800 1000 Energy [keV]

Figure 5.8: β-gated background-subtracted γ-gated spectra observed with the GRIFFIN spectrometer for the β-decay of 128Cd. The top and middle panels show the coincidences with the 68-keV transition (316 → 248 keV) in 128In, and the bottom panel shows the coincident γ-rays with the 173-keV transition (489 → 316 keV). The black asterisks correspond to transitions published in Ref. [3], whereas the orange asterisks mark newly observed lines.

58 * 315 68 *

2500 462 keV gate (low)

2000 173 221 462 * * *

1500 68 *

1000 711 Scatter *

Counts per 1 keV 500 1089 * 0 0 200 400 600 800 1000 1200

60 462 keV gate (high) 50 2387 40 1552 *

30 * 3313 * 1642

20 1797 1924 2486 Scatter 10 * *

Counts per 1 keV 0 −10 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400

140 857 keV gate 120

100 1089 * 80 1924

60 * 2486 40 *

Counts per 1 keV 20 0 − 20 1000 1200 1400 1600 1800 2000 2200 2400 2600 Energy [keV]

Figure 5.9: β-gated background-subtracted γ-gated spectra observed with the GRIFFIN spectrometer for the β-decay of 128Cd. The top and middle panels show the coincidences with the 462-keV doublet (1173 → 711 keV, and 711 → 248 keV) in 128In, and the bottom panel shows the coincident γ-rays with the 857-keV transition (1173 → 316 keV). The black asterisks correspond to transitions published in Ref. [3], whereas the orange asterisks mark newly observed lines.

59 levels. Therefore, the 408-keV transition is placed on top of the 248-kev level, 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 the 336-keV transition places a level at 991 keV. Also, this places the 1608-keV transition depopulating the level previously placed at 2263 keV, and the 2441-keV transition depopulating the level previously placed at 3097 keV. Since the 1395-keV transition is also seen in coincidence with the 211-keV transition (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-keV gate (992 → 656 keV) sees a transition at 1270 keV, which again supports the position of the 2263-keV level. Fourthly, we look at gates on the 305-keV and 619-keV transitions (see Figure 5.11). The total energy of the cascade formed by the 305, 211 and 408-keV transitions is ∼924keV, which again corresponds to the energy difference between the 248 and 1173-keV levels. The low-energy range of the 305-keV gate (top panel) highlights the diﬀerence in intensity between the 211-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 transitions, and the 305 and the 1924-keV transitions, which were both already placed as feeding the 1173-keV level. Finally, the 619-keV gate (bottom panel) shows coincidences with the 305-keV transition. The sum of their energies 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 transitions. The 619-keV transition is also fed by transitions at 1395 keV (from the 2263-keV level) and 2230 keV (from the 3097-keV level).

5.1.3 Decay Scheme The 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 shows 32 new transitions and 11 new levels in 128In. The highest level was found at 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 transition given in Table 5.1 were measured by diving the efficiency-corrected area of a peak in β-gated γ-singles by the area of the 248-keV peak:

Areaγ/γ Iγ,rel. = . (5.2) Area248/248

60 800 408 keV gate (low) 516 336

600 * * Scatter 400 211 1608 Scatter 200 1395 * * * Counts per 1 keV 0

0 200 400 600 800 1000 1200 1400 1600

408 keV gate (high) 40

30 2871 Scatter * Scatter 2441

20 3187 4092 3376 Scatter *

10 *

0 Counts per 1 keV

−10

2400 2600 2800 3000 3200 3400 3600 3800 4000 4200

500 336 keV gate 400 408 *

300 Scatter Scatter 200 Scatter Scatter 1270 Scatter

100 Scatter * 0 Counts per 1 keV −100 −200 0 200 400 600 800 1000 1200 Energy [keV]

Figure 5.10: β-gated background-subtracted γ-gated spectra observed with the GRIFFIN spectrometer for the β-decay of 128Cd. The top and middle panels show the coincidences with the 408-keV transition (656 → 248 keV) in 128In, and the bottom panel shows the coincident γ-rays with the 336-keV transition (992 → 656 keV). The black asterisks correspond to transitions published in Ref. [3], whereas the orange asterisks mark newly observed lines.

61 800 305 keV gate (low)

600 Scatter 619 400 * Scatter 200 211 *

Counts per 1 keV 0

−200 200 300 400 500 600 700

305 keV gate (high) 50

40 1089 *

30 Scatter Scatter 1924 20 * 10 0 Counts per 1 keV −10 −20 1000 1200 1400 1600 1800 2000 2200 2400 2600

800 619 keV gate

700 305

600 * Scatter 500 400 300 1395 * 200 Scatter Counts per 1 keV Scatter 2230 100 * 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 Energy [keV]

Figure 5.11: β-gated background-subtracted γ-gated spectra observed with the GRIFFIN spectrometer for the β-decay of 128Cd. The top and middle panels show the coincidences with the 305-keV transition (1173 → 868 keV) in 128In, and the bottom panel shows the coincident γ-rays with the 619-keV transition (868 → 248 keV). The black asterisks correspond to transitions published in Ref. [3], whereas the orange asterisks mark newly observed lines.

62 0+ Qβ-= 6904(153) 128 Cd

Sn= 5321(155)

(1+) 4432.8 4091.8 4748.5

(1+) 4023.3 3708.3 3312.7 4023.8 (1+) 3659.0 2486.6

(1+) 3526.4 2871.2 3527.5

(1+) 3097.4 2781.4 2441.7 2386.5 2230.1 1923.37 3097.2

(1+) 2015.5 1607.5 1552.5 1395.5 1270.8 1090.2 2263.2

(1+) 1097.7 1587.2 1+ 1172.9

(0-,1-,2-) 1173.0 924.99 856.92 516.5 461.8 305.6 992.4

(0-,1-,2-) 743.8 336.6 867.8 (1-,2-) 619.50 211.8 710.9 710.5 462.52 221.97

(0-,1-,2-) 408.31 656.3

(0-,1-,2-) 173.36 489.5

(1-,2-) 316.14 68.16 316.1 247.96 (1-) 248.0 23(2)μs (3+) 0 810(30) ms 128 49 In79

Figure 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 transitions represents 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 published in Ref. [3].

63 It 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 the 248-keV isomeric transition is suppressed in β-gated γ-singles, its intensity relative to the strong 462-keV peak was fitted in γ-singles and then scaled with the intensity of the 462-keV peak in β-gated γ-singles. The uncertainties on the fitted areas and centroids are determined by GRSISort, while the uncertainties on the detection efficiency is obtained from the fit shown in Figure 4.9a. The uncertainties are propagated through the calculations by addition in quadrature. Also, the energy of the levels are determined by taking the average of the transitions and cascades depopulating the given levels. The standard deviation of the cascades energy give the uncertainties on the level energies. The areas for 17 of the 39 transitions were calculated from γ-ray branching ratios because the statistics in β-gated γ-singles were too low to be seen or a close background peak prevented a clean fit. These ratios represent the fraction of decays which happen through a particular γ-transition when a particular level is fed and they are obtained in β-tagged background subtracted γ-gated spectra. The γ-ray branching ratio BR for a transition γb depopulating a level is found by gating on a strong transition γa feeding the same level (i.e. gating from above):

I N /γ BR = γb(γa) = γb(γa) b , (5.3) γb k k P P Iγi(γa) Nγi(γa)/γi i=1 i=1 where Nγb(γa) is the area of the γb peak in the γa gate, and γb and γi are the detection eﬃciency at the energies of γb and γi, respectively. When no strong transition is available to gate on from above, the intensity of a transition γa feeding a level is calculated by gating on a transition γb depopulating the same level (i.e. gating from below):

Nγa(γb) Iγa = . (5.4) γa · γb · BRγb The intensity of the bottom 462-keV transition (710 → 248) was calculated from its branching ratio in the gate on the 1552-keV transition feeding the 710-keV level. The intensity of the top 462-keV transition (1173 → 710) was calculated by subtracting the intensity of the bottom 462-keV transition from the total intensity of the 462-keV doublet in β-gated γ-singles. The absolute intensity of a transition represents the number of times it occurs per 100 decays and is calculated by dividing the intensity of the

64 transition by the number of 128Cd decays, which is calculated by fitting the number of β-particles from 128Cd decays in Figure 5.4a. In addition, the number of β-particles is corrected by subtracting the number of β-particles which 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-keV transitions have to be corrected for internal conversion. The multipolarities and conversion coefficients were used as reported in Ref. [3] (see Table 5.1). The absolute correction factor in the footnote a was determined by dividing the conversion-corrected absolute intensities by the relative intensities. The uncertainty on the correction factor was propagated from the calculations of 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 from Ref. [3]. There are no significant discrepancies for the level energies, and the transitions energies and their intensities, except for the 462-keV doublet. Fogelberg et al. measured both transitions at 462.7(3) keV, while the bottom transition was measured at 462.52(1) keV in the 1552-keV gate and the doublet 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 branching ratio to the 710-keV level by evaluators [14]. However, looking back at Figure 3.1a, intensities of 8.7% are given for both 462-keV transitions by Ref. [3], which is very close to our new value of 8(3)%. The β-decay branching ratios, which represent the fraction of the parent decays 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 absolute γ-ray intensities which feed and depopulate a given level. The uncertainties are propagated through the calculations by addition in quadrature. In addition, since the β-decay of the 0+ ground state to the 3+ ground state of 128In would be a second forbidden, direct β-feeding to the 3+ ground state is assumed to be zero. The sum of the absolute γ-ray intensities of the seven transitions decaying to the ground state is 86(3)%, which gives an estimate of ∼14% for all unobserved ground state transitions. The sum of the β-decay branching ratios to all states observed is 86(12)%. Also given are the log(ft) values, which represent an important piece of information for assigning spins and parities to the levels. They are calculated using the parent ground state energy, the parent half-life, the Qbeta-value, the daughter level energy, the transition intensity and the transition uniqueness. In order to conservatively account for unseen low-intensity transitions and the Pandemonium effect, limits on branching ratios and log(ft) values are reported for levels below 1173 keV. Decays to 0− or 1− states below 1173 keV

65 128 Table 5.1: γ-ray energies in In, their intensities relative to Iγ(247.96) = 100 % and the initial energy levels are compared to previous work [14].

this work Ref. [14]

a b c Ei(level) [keV] Eγ [keV] Iγ [%] Ei(level) [keV] Eγ [keV] Iγ [%] Mult. α 248.0(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) 1.55 316.14(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) 710.37 462.52(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) 1172.88 462.20(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). 66 c Uncertainties are assumed as 10% by evaluators [14]. d Divided on the basis of Iβ(to 710 level) by evaluators [14]. are first forbidden, while decays to 2− states are first forbidden unique. Unique transitions have larger log(ft) values, which are consistent with the lower limits listed in Table 5.2. Table 5.2 also compares the current results to those from Ref. [3]. The previous discrepancy on the intensities of the 462-keV doublet propagates to the β-decay branching ratios for the 710- and 1173-keV levels. Elekes and Timar 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 the 1+ assignment for the level. For the 710-keV level, our values increase the limit to log(ft) > 6.4 from the published log(ft) > 5.8. The fact of only comparing limits, along with the significant uncertainties on the several new low-intensity transitions feeding the level, make it more difficult to come to clear conclusions.

5.1.4 Spin Assignments The tentative spin assignments of levels presented in Table 5.2 are based on γ-ray spectroscopy and rely on several pieces of experimental information, including log(ft) values and angular correlations, together with the selection rules of β-decay and γ-decay. First, considering the 0+ ground state of the even-even parent nucleus 128Cd, its β-decay only feeds low-spin states in the daughter. Isobaric analog states do not exist below the neutron separation energy in this very exotic isotope (48 protons and 84 neutrons) and only Gamow-Teller decays are expected. Therefore, log(ft) values point to allowed decays, which would correspond to a spin of 1+, or at least first forbidden decays, which would correspond 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 forbidden unique and the log(ft) values are larger. According to shell model calculations, which will be discussed further in Chapter 6, only one positive-parity excited state is expected at low energies and this 1+ state would be fed by an allowed β-decay, which agrees with the previous assignment of 1+ for the 1173-keV state. Table 5.2 shows a log(ft) value of 4.06(6) for this state. This value is smaller than what would be expected for any forbidden transition and thus we conclude it is an allowed decay. Higher log(ft) values could also be allowed decays but since they are in the same range than first forbidden transitions, they cannot be distinguished. For the states above the strongly populated 1+ state at 1173 keV, first

67 Table 5.2: Level energies in 128In, their β-feeding intensities per 100 decays and the log(ft) values are compared to previous work Ref. [14], which include the multipolarity and conversion coeﬀcient α for the 68 and 248-keV transitions.

this work Ref. [14]

a π π Elevel [keV] Iβ− [%] log(ft) J Elevel [keV] Iβ− [%] log(ft) J 248.0(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 the transition 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, only allowed transitions leading to a spin-parity of 1+ are considered. Assignments of J = 0 were ruled out for the levels at 248, 316 and 710 keV based on γ-γ angular correlation analysis, which is described in the next section. In addition, a tentative multipolarity of (M2,E3) was obtained from conversion electron measurements for the 248-keV transition in Ref. [3] (see Table 5.1), which suggests ∆J = 2 and a change of parity from the 3+ ground state. This constraint rules out J π = 2− and leaves a tentative assignment of J π = 1− for the 248-keV level.

68 857-68 keV Cascade The cascade of the 857-keV (1173 → 316) and 68-keV (316 → 248) transitions has the potential of restricting the spin of the intermediate state at 316 keV. The spin assignments of the 1173-kev level (1+) and of the 248-keV level (1−) are established, which leaves only the middle state at 316 keV with two 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 predominantly M1 (as reported in Ref. [3]). Our log(ft) values agree with both a 1 → 2 → 1 cascade, as originally suggested in Ref. [3], and a 1 → 1 → 1 cascade, as revised in Ref. [14]. We are trying to answer this question with γ-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 independent mixing ratio. These multiple degrees of freedom increase the complexity of this analysis. All combinations of the two mixing ratios are summarized in Figure 5.13. By looking at a fixed mixing ratio, one can see the range of possible values for the a22 and a44 coefficients and hopefully find a range of values for which they do not overlap for different spins. Here, the obvious difference is that a44 = 0 for a 1 → 1 → 1 cascade with any combination of mixing 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-overlap for 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 shown in Figure 5.14. Details on the normalization of the γ-γ angular correlation data can be found in Appendix B. Fitting to Equation (2.20) extracts experimental coefficients of a44 = −0.012 ± 0.007, which is consistent with zero within two standard deviations σ, and a22 = 0.168 ± 0.005, which is far from the regions of non-overlap. While this cascade of the most intense transitions provided sufficient statistics (∼14000 counts) to perform an angular correlation analysis of the levels involved, the characters and coefficients of the possible spins for the 316-keV level are very similar and cannot be distinguished within uncertainty. Therefore γ-γ angular correlations are not well suited in this case to discriminate between 1 → 1 → 1 and 1 → 2 → 1 cascades. It is important to note that the theoretical values shown in Figure 5.13 do not take into account some experimental systematic factors, such as the opening angle of the detectors. In measuring angular correlations, the finite size of the detectors attenuates the observed correlations with coefficients Q22 and Q44. The a22 and a44 coefficients become Q22a22 and Q44a44. These

69 attenuation coeﬃcients have been determined experimentally for GRIFFIN by measuring the angular correlations for well known cascades [73, 74]. The tentative spin assignments of 1− or 2− still hold for the 316-keV level and 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 0 can only have an m-substate equal to zero, which translates to an isotropic γ-γ distribution in space. Therefore, an anisotropic angular distribution between 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-Life The half-life of the ground state of 128Cd is measured by fitting the time distribution of the β-gated γ-ray transitions in the daughter 128In during the beam-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 exponential decay (see Equation (2.10)) and a constant background. In Figure 5.16, the summed activity of the 857- and 925-keV γ-rays is fitted to:

− ln 2·t/t − ln 2·t/t Atotal(t) = Ae 1/2,parent + Be 1/2,daughter + C (5.5) A previous analysis of the 128Cd half-life from this data set was carried independently and published in 2016 [4]. The current analysis finds a half-life of 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 higher uncertainties. The different parameters of the fit equation were studied with the summed activity of the most intense γ-rays (see Figure 5.16). Assuming that 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 the daughter 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 per bin 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 time bins included in the fitting range. For the fit of the sum of the 857- and 925-keV transitions, the range of bins does not have an obvious effect on

70 (a) a22 for a 1 → 1 → 1 cascade (b) a44 for a 1 → 1 → 1 cascade

Figure 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 the x-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].

71 Figure 5.14: Normalized γ-γ angular correlation data and fit for the 857-68 keV 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 to 11.5 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 the statistical uncertainty on the final value: 245.4 (23)(stat.) (0.7)(syst.) ms = 245.4(23) ms.

5.1.6 248-keV Isomer In order to identify long-lived states and isomers, one needs to look again at the time diﬀerence between β-particles and γ-rays as previously seen in Figures 5.1 and zoomed-in in Figure 5.18. The 248 keV isomer was observed in 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 line around t = 0 which extends to longer times, with orders of magnitude more statistics on the positive side. This happens when the β-particle is detected before the γ-ray, i.e. the γ-ray time stamp is higher than the β-particle time stamp. Transitions which are correlated in time are seen as spots centered around t = 0, whereas γ-rays which are seen consistently with larger time

72 (a) 68 keV (b) 857 keV

Figure 5.15: Fitted activity of selected γ-rays in 128In collected during the beam-oﬀ part of the tape cycle. The plotted curves show the activity of the parent nuclei 128Cd (red line), a constant background (green line) and the sum of the two curves (light blue line). The ﬁtting parameters and 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 rate

Figure 5.16: Fitted activity of the sum of the 857- and 925-keV γ-rays in 128In collected during the beam-off part of the tape cycle. The fitting function (light blue line) considers the activities of parent nuclei (red line), the indium present in the beam (with t1/2 = 0.810(30) s [34]) (violet line), and a constant background (green line). The fit parameters and reduced χ2 are shown in the respective insets.

(a) Number of the ﬁrst bin included (b) Number of the last bin included

Figure 5.17: Eﬀect of changing the ﬁtting region on the extracted 128Cd half-life.

74 differences 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 extend on 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-decay of the 8− isomer in 128In (0.72(10) s). As discussed in Section 5.1, this 8− isomer cannot be populated by the β-decay of 128Cd, but it can be non-resonantly ionized by the first of the three lasers. None of these nine lines are seen when the first laser is blocked, which is consistent with the laser excitation of the isomer. In addition, the fact that only the 248-keV transition is more intense during the beam-on part of the cycle relative to the beam-off part suggests that the eight other lines are most likely not populated by the β-decay of 128Cd.

75 Figure 5.18: Diﬀerence between time stamps of β-particles and γ-rays (γ − β) as a function of the energy of the γ-ray (zoom in). Long-lived transitions are seen at 121, 207, 247, 321, 426, 626, 1053, 1061 and 1279 keV.

76 5.2 131Cd

5.2.1 β-Gated γ-Singles Measurements Data from the decay of 131Cd were collected for 32 hours in August 2015 with a beam intensity of 0.6-0.8 pps. For the 131Cd data set, the same β-γ coincidence window of 200 ns has been used. The cycle structure shown in Figure 5.19 consists of the tape move and background measurement (1 s), beam on (10 s), and beam off (1 s). The noise caused by the moving tape collector system during the tape move is clearly visible. The peak before the beam is turned on is noise from the tape move. The plotted curves show the number of decays for 131 131 the Cd (t1/2 = 98(2) ms [34]) isotopes (red line), the first daughter In 131 (t1/2 = 261(3) ms [34]) (yellow line), the βn-decaying Cd isotopes into 130 In (Pn = 3.5(10)% [50]) (violet line), the constant background (green line), and the sum of the four curves (light blue line). The comparison of β-gated γ-ray spectra observed with lasers on and laser blocked, which enables the identification of transitions in 131In, is shown in Figure 5.20. Several published transitions from ISOLDE [55] and RIKEN [6] were confirmed, including the transition of 988 keV. 21 of 23 lines seen 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-keV lines observed at ISOLDE. It is suspected that the 844 keV line was from 131 the decay of the Te 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, and 4194 keV. The 331-keV line could possibly be the one in 131Sn, while the 344 and 1923-keV lines could be from 131I. The 590-keV line was observed in the 128Cd data set with the laser blocked as well, but it remained unidentified. It might originate from a long-lived beam contaminant which was still present in the vacuum chamber. Finally, the 3042 and 4086-keV lines are seen in the 131Cd laser-blocked spectra. Figure 5.22 highlights the multiplet structure of the peaks at 3868, 3920, 5753, and 6039 keV with binning of 1 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 cycle consists of the tape move and background measurement (1 s), beam on (10 s) and beam oﬀ (1 s).

78 1600 Lasers on

1400 e-/e+ 1200

1000

800 315 355 590 * 600 * Counts per 1 keV 433 451 (130In) 988

400 * * 1221 (130Sn) 798 (131Sb) 774 (130Sn) 779 (131Sn) 200

0 0 200 400 600 800 1000 1200

160 Laser blocked 140 e-/e+ 120

100

80 344

60 590 Counts per 1 keV 798 (131Sb) 40 779 (131Sn) 1221 (130Sn) 20

0 0 200 400 600 800 1000 1200 Energy [keV]

Figure 5.20: Comparison of β-gated γ-ray spectra observed for the decay of 131Cd in addback 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].

79 1600 1400 e-/e+ 1200 1000 800 590

600 331 344 988 451 (130In) Counts per 1 keV 400 1221 (130Sn) 798 (131Sb) * 774 (130Sn) 779 (131Sn) 200

00 200 400 600 800 1000 1200

1400 1200 1000 800 600 2434 (131Sn) 400 Counts per 2 keV 3554 1905 (130Sn) 1923 200 1655 (131Sn) * 3042

0 1500 2000 2500 3000 3500 * 120

100 3868 5825 * 6039

80 3920 5525 * 5753 *

60 * 4487 (131Sn) 5796 3990 (131Sn) 6003 4086 4194 * 5958 *

40 * * Counts per 2 keV 20

0 4000 4500 5000 5500 6000 Energy [keV]

Figure 5.21: β-gated γ-ray spectra observed for the decay of 131Cd in addback mode. The peaks with a black asterisk correspond to γ-rays published in Ref. [6].

80 (a) 3868 keV (b) 3920 keV

Figure 5.22: β-gated γ-ray spectra around peaks with multiplet structures in the decay of 131Cd. All displayed in 1 keV per bin.

81 β-Delayed Neutron Decay to 130In The βn-decay of 131Cd is investigated by looking at the four most intense transitions in 130In based on Ref. [45]. Figure 5.23 shows β-gated γ-ray spectra 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 the transitions Iγ,rel are quoted in parentheses [1]. The βn-feeding from 131Cd (7/2− ground state) to 130In populates higher spin states than the β-decay of 130Cd (0+ ground state) to 130In and, therefore, the transition intensities are diﬀerent in the two decays. The only clear peak 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 intense transitions. The spin and parity of the states which could be populated in 130In depend on the spin and parity of the excited states above the neutron separation energy populated in 131In and the angular momentum of the emitted neutron. The observed 451-keV transition suggests that the βn-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 do not see the intense peaks at 1669, 1170 and 950 keV. The non-observance of the 950 keV transition in our data also excludes that the 950-keV state has a spin of 1. Finally, the 3+ isomer at 388 keV should be populated by l = 0 neutrons. However, the 4.4(2)-µs transition [45] mostly decays outside of the β-γ coincidence window of 200 ns.

5.2.2 β-Gated γ-γ Coincidence Measurements Coincidences are investigated with the β-gated γ-γ coincidence matrix in Figures 5.25 and 5.26, and in the β-gated background-subtracted projections in Figure 5.27. These figures highlight the limited coincidence statistics for this data set. The coincidence between the 988-keV and 3290-keV transitions, which was published in Ref. [6], is confirmed here in both Figure 5.26d at 1 keV per bin and in the projection at 2 keV per bin (Figure 5.27). No new coincidences can be observed with the most intense transition at 988 keV, while five transitions were observed in coincidence at RIKEN and placed

4DEuterated SCintillator Array for Neutron Tagging

82 (a) 451 keV (Iγ,rel=100%) (b) 1669 keV (Iγ,rel=99.8%)

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 be observed due to the selection rules.

83 + 1 1669 ( 99.8 %) 1170 ( 20.4 %) 2120

(1-,2-) 950 ( 22.5 %) 950

(2-) 451 451 ( 100 %)

(3+) 388 ( 7.6 %) 388

1- 0 130 49 In81

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 around 2908 keV, but did not list this energy in their list of transitions and did not place 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 keV and 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 transitions at 3554 keV and 315 keV, or at 5527 keV and 433 keV. The β-gated background-subtracted gate at 315-keV in Figure 5.27 shows 3 counts at

84 3042 keV and 3 counts at 2648 keV, however these counts are consistent with the statistical fluctuations of the surrounding background. Therefore, the statistics available with this data set cannot confirm the coincidences observed in Ref. [6], except for the coincidences between the 988and 3290-keV transitions. Advantages of the EURICA dataset are their gates on one isotope in their particle identification plot as well as on the time of implantation in the detector, which make their spectrum cleaner than ours. Although this data set has a ∼50% higher statistics than the EURICA data set, our background conditions do not allow a better identification of coincident states. The inclusion of the new GRIFFIN detector shields fabricated from the high- density scintillator bismuth germanate (BGO) will help for future low-rate experiments [66].

5.2.3 Decay Scheme Figure 5.28 presents a revised level scheme for 131In with 11 transitions and 11 excited states. Taprogge et. al [6] claimed that they saw 23 transitions and 19 levels, 12 of which were based on single transitions placed as feeding directly the ground state without further justification. However, without coincidence data or knowledge about their multipolarity, transitions cannot be placed with confidence and it is difficult to tell whether they are feeding the 1/2− isomeric state at 365 keV or the 9/2+ ground state. First, the only coincidences observed were between the transitions at 988 and 3290 keV, which place a level at 4643 keV. While the four other coincidences observed at RIKEN (2637, 2777, 3178, and 3417 keV) were not seen here, their coincidences are assumed to be correct and levels are placed for the three transitions which this work saw. Therefore, a level is placed at 3920 keV with the 2636-keV transition, one at 3989 keV with the 2778-keV transition, and one at 4768 keV with the 3414-keV transition. When coincidence data is not available, transitions can be placed in the level scheme by looking for differences in level energies which could match the energies of particular transitions. Knowing that the 1/2− isomer was recently measured at 365(2) keV [53] above the 9/2+ ground state, it is likely 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 keV and seems to be the only level connecting both the isomer and ground state. Based on observed coincidences at RIKEN, Taprogge et. al places the 315-keV transition in a cascade from levels at 3869 to 3555 keV, and then

85 (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 are displayed 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 keV

Figure 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. All displayed in 1 keV per bin.

87 via a 3555-keV transition to the ground state. Since the 3554-keV transition was placed as feeding the 1/2− isomer at 365 keV, the 315-keV transition is placed as feeding the 3920-keV level from a level at 4235 keV. The energy difference 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 the isomer at 365 keV, their energy would lie 110, 155, and 190 keV above the Sn-value at 6213(38) keV. However, it is possible to see transitions slightly 130 above the Sn-value, such as the four transitions in In 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. The intensities were fitted in 2 keV/bin, which combined the multiplet structures of 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 not only there are groups of transitions with similar intensities but also of similar energies. There is a group of seven transitions with energies in the 3290 to 4040-keV range and seven transitions in the 5527 to 6038-keV range which might have a similar configuration as discussed in Ref. [6]. Several transition and level energies differ from the previous works in Ref. [54, 55, 56] by up to 2 keV for the 5727/5725 transitions. The low statistics can affect the centroid of the fitted peaks and increase its uncertainty. Also, most transitions observed are above the energy range of 53–3450 keV which was used for the energy and efficiency calibrations. The energy of the levels were rounded to the closest integer, with an uncertainty of 1 keV, and 2 keV for doublet structures. There are several discrepancies between the relative intensities calculated in this work and the ones from Ref. [6]. The absolute intensities rely on the number of β-particles from the decays of 131Cd and 131In, the βn-decay of 131Cd, and on the background fitted in Figure 5.19a. In addition, the number of β-particles is corrected by subtracting the number of β-particles which are 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 based on energy loss, time-of-flight measurement and magnetic rigidity in a particle identification plot. In addition, the time of the implant is recorded and allows an additional coincidence constraint, so that their implant-β-γ-γ coincidence plots are cleaner than our β-γ-γ spectra in Figure 5.27.

88 5 988 keV gate 4 * 3290 3 2 1 0 Counts per 2 keV −1 −2 2000 2500 3000 3500 4000 4500

3 315 keV gate 2

−1

Counts per 2 keV −2

−3 2000 2500 3000 3500 4000 4500

4 433 keV gate 3 2 1 0 −1 Counts per 2 keV −2 −3 2000 2500 3000 3500 4000 4500 5000 5500 6000 Energy [keV]

Figure 5.27: β-gated background-subtracted γ-gated spectra observed with the GRIFFIN spectrometer for the β-decay of 131Cd. The top panel shows coincidences with the 988- 131 keV transition (1353 → 365 keV) in In for Eγ= 1800-4500 keV, the middle panel shows coincidences with the 315-keV transition for Eγ= 1800-4500 keV, and the bottom panel shows the coincident γ-rays with the 433-keV transition for Eγ= 1800-6000 keV.

89 0+ Qβ-= 12806(103) 131 Cd

Sn 6213(38)

4768 3418

3290.2 4643

3867.8 315.4 4237 4132 2778.7 2635.7 3919.8 3554.4 3960 3989 630(60) ns (17/2-) 3782 0.32(6)s (21/2-) 3764

(3/2-) 988.2 1353

(1/2-) 365 0.35(5)s

(9/2+) 0 0.28(3)s 131 49 In82

Figure 5.28: Energy levels [in keV] and γ-ray transitions in 131In following the 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 represents the intensity of the γ-ray relative to the 988-keV transition: Iγ > 10% in − − red, and Iγ < 10% in blue. The 17/2 and 21/2 states are reported in Ref. [52] and are not populated by β-decay in this work.

90 131 Table 5.3: γ-ray energies in In, their intensities relative to Iγ(988) = 100 %, absolute intensities per 100 decays, and the initial energy levels are compared to previous work Ref. [6].

this work Ref. [6]

a Eγ [keV] Iγ(rel.) [%] 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)

91 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]. The 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 from the published scheme from EURICA [6], it follows that the β-branching ratios and log(ft) values differ as well. In order to conservatively account for unseen low-intensity transitions and the Pandemonium effect, limits on branching 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 the level were included in the calculation of the β-branching ratio since they are seen in γ-singles, even if they are not seen in the coincidence data. The 3920-keV and 5527(1)/5892(8) levels are the only two other levels for which transitions are placed feeding (with the 315-keV and 433-keV transitions, respectively) and depopulating the level. Decays from 7/2− to 3/2− would be 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 that have 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 the 7/2− ground state of 131Cd to 5/2− or 9/2− states. The 9/2− state would however be expected at much higher energy, around 6 MeV. Overall, it is not possible 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 subtracting the 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 intensities observed is 46(2)%. Therefore, we can place a limit of < 53% for the β-feeding to the ground state through a first forbidden transition (7/2− to 9/2+). By including the 20 transitions in their level scheme, Ref. [6] estimated ∼ 30%.

92 Table 5.4: Level energies in 131In, their β-feeding intensities per 100 decays and the log(ft) values are compared to previous work Ref. [6].

this work Ref. [6]

a π π Elevel [keV] Iβ− [%] log(ft) J Elevel [keV] Iβ− [%] log(ft) J 0 <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 >5.21 4531(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 >5.9 5752(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.

93 5.3 132Cd

5.3.1 β-Gated γ-Singles Measurements 132Cd data was collected for 18 hours at an intensity of 0.15 pps based on the yield of the 989-keV peak. A β-gated spectrum of this data can be seen in Figure 5.29. The same timing settings were used as for the 131Cd data set, that is a β-γ coincidence window of 200 ns and a cycle structure consisting of the tape move 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/positron pair 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 in the β-decay of the 2.79-minute state 132Sb, with respective intensities of 99(5)%, 86(5)% and 14.9(15)% [76]. Other strong transitions observed in the 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)%), and 132 132 992.66(8) keV (Iγ,abs=36.9(20)%). Sb and In are isobars but closer to stability than 132Cd (see Figure 2.4). From the 2014 RIKEN publication [5], a high branching ratio of βn- 132 131 decay of Cd to Cd was expected (Pn-value close to 100%), meaning that the spectrum should include common transitions with the β-decay of 131Cd, which was discussed in the previous section. The comparison of the spectra from the 131Cd and 132Cd data sets is presented in Figure 5.30 around 988 keV, which is the highest intensity line observed in the β-decay of 131Cd. The top panel shows γ-singles without the β-gate, highlighting the agreement of the energy calibration in both data sets and the amplitude of the background suppression enabled by the β-gating with SCEPTAR. The bottom panel shows that while the 131Cd data (in red) shows a peak at 988 keV, the peak at 989 keV in 132Cd (in blue) shows a more complicated structure. The peak at 989 keV has contributions from the 989 keV transition in 132Te and from the 993-keV transition in 132Sb. Both the 988 and 989-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-min state 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 two 132Sb decaying states are present. The 4.10-min state has three lines which

94 700

600 e-/e+ 86 (132Sn)

500

400 247 (132Sn) 341 (132Sn) 300 Counts per 1 keV 697 (132Te) 974 (132Te) 200 899 (132Sn)

100 989 (132Te)

0 0 200 400 600 800 1000 Energy [keV]

Figure 5.29: β-gated γ-ray spectrum observed for the decay of 132Cd in addback mode with the 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 in this data set, suggesting that the 8− isomer was not produced. The characteristic activity curves for the individual γ-rays provide additional 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, 131 consistent with the Cd 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 increase when the beam is turned on, suggesting a longer half-life, consistent with 132 Sb (t1/2=2.79 min). However, the 989-keV line (Figure 5.32e) might show a faster increase than the 974- and 697-keV transitions, but there is no ev-

95 ×103 Gamma-ray singles 35

30 964 (228Ac) 968 (228Ac)

25 1000 Counts per 1 keV

20 1014

15 950 960 970 980 990 1000 1010 1020 1030

Beta-gated singles 350

300 974 (132Te) 250 988 (131In) 989 (132Te) 200

Counts per 1 keV 150

100

50 950 960 970 980 990 1000 1010 1020 1030 Energy [keV]

Figure 5.30: Comparison of 131Cd [red] and 132Cd [blue] data sets around 988 keV in γ-singles [top] and β-gated γ-singles [bottom panel].

96 989 ( 14.9 %) 2764

(2+, 3, 4+) 816 ( 10.9 %) 2488

6+ 1774 103 ( 13.9 %) + 4 697 ( 86 %) 1671

+ 2 974 ( 99 %) 974

0+ 0 132 52 Te80

132 Figure 5.31: Partial decay scheme for the β-decay of Sb (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 the 989-keV peak from 132Sb might be mixed with a shorter-lived line.

5.3.2 β-Gated γ-γ Coincidence Measurements Coincidence data can provide extra information on which isotopes contribute to the peak at 989 keV. Figure 5.33 shows four regions of the coincidence matrix around the 974 and 989 keV lines, displayed on the x-axis. There is 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 and 816 keV, and 974 and 696 keV. There are no coincidence counts observed

97 (a) 988 keV in 131Cd (b) 974 keV in 132Cd

(e) 989 keV in 132Cd

Figure 5.32: Comparison of the activity of selected γ-rays in the 131Cd and 132Cd data sets.

98 between 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 coincidences between 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 and high background at low energies (see Figure 5.33c). In conclusion, there is no clear evidence of 132Cd in the data, however there are open questions remaining about the 989-keV peak. Mainly, that it 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 seems to have been highly contaminated with several isobars. While it cannot be ruled out that there is a marginal amount of 132Cd with a smaller intensity than we see the longer-lived A = 132 isobars, it cannot be conﬁrmed that the transition at 988 keV from the βn-decay of 132Cd was observed.

99 (a) 974 and 988 keV (b) 974 and 816 keV

Figure 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.

100 Chapter 6

Shell Model Calculations

NuShellX is a set of computer codes that can calculate energies, eigenvectors and spectrosopic overlaps for low-lying states in shell model Hamiltonian matrix calculations [13]. NuShellX@MSU is another set of codes used to generate input for NuShellX using data files for model spaces (.sp input files) and Hamiltonians (.int input files). The latter codes also convert the NuShellX output into figures and tables for energy levels, γ-decay and β- decay. The NuShellX code uses a proton-neutron basis with Hamiltonians of the form: H = Hνν + Hππ + Hπν, (6.1) where Hνν is the neutron-neutron interaction, Hππ is the proton-proton interaction, and Hπν is the proton-neutron interaction. For the neutron-rich In isotopes, the calculations allow the nucleons to interact in the model space composed of four π- and five ν-orbitals above a 78 Ni core, as shown in Figure 2.2: π(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, 4π-5ν orbitals, proton-neutron coupling). The first of two interaction files which are considered in this work is the jj45pna interaction included in the NuShellX@MSU package. The single particle energies (SPE) of this interaction, which are displayed in MeV on the right in Figure 2.2, were adjusted to experimental data available in the 132Sn 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 the IMSRG uses the same model space as jj45pna, the valence shell model Hamiltonians are constructed from first principles which result in a different interaction and SPEs. Since physics can be shifted between SPEs and two-body matrix elements in phenomenological fits, and SPEs themselves are not observable, we would not necessarily expect the ESPEs from the jj45pna interaction to be the same as those from the IMSRG. More meaningful would be the spacings between

101 Table 6.1: Single-Particle Energies for the jj45pn model space for the NuShellX (jj45pna) and In-Medium Similarity Renormalization Group (IMSRG) interactions.

1πf5/2 1πg9/2 2πp3/2 2πp1/2 1νg7/2 2νd5/2 2νd3/2 3νs1/2 1νh11/2 jj45pna -0.71660 1.11840 1.12620 0.17850 5.74020 2.44220 2.51480 2.17380 2.67950 IMSRG -20.9571 -18.1975 -17.6453 -12.6203 -3.14284 -4.1184 -3.23028 -3.77358 2.6652

levels, and Table 6.1 shows that there is general agreement between the two interactions. This chapter compares the experimental results for 128In with calculated excitation energies, level occupancy and eﬀective single particle energies (ESPE) from the jj45pna and IMSRG interactions. Finally, the experimental results for 131In are compared with calculated excitation energies and level occupancy from both interactions.

6.1 128In

6.1.1 Level Energies The excited states in 128In found experimentally are compared to levels calculated (.lpt output file) in Figure 6.1. Since 128In is an odd Z, odd N nucleus with one π-hole and three ν-holes in the jj45pn model space, more excited states are expected when compared to even-odd, odd-even or even-even nuclei. The unpaired nucleons can be in different single-particle states and for each of these different proton and neutron configurations, a multiplet of states results from the spin coupling of unpaired protons and neutrons. First, one can see that the 3+ ground state is reproduced in the jj45pna calculation. In the IMSRG calculation, the first 3+ state, which should be the ground state, is seen at 290 keV. Since the gap observed between the positive and negative parity spectra changes with increasing the number of three-nucleon (3N) matrix elements, this means that the calculations have not yet converged. However, since the spacings between levels of similar parity are largely converged, energy difference between same parity states can be compared. Calculations in a bigger space for the three-nucleon (3N) forces are required to converge the gap, however such calculations are beyond the available computing resources at this time. The truncation of the chiral EFT expansion and of the many-body operators in the IMSRG formalism

102 (1+) 4749 1+ 4463 1+ 4412

1+ 4087 (1+) 4024 1+ 3892 (1+) 3659 + (1+) 3528 1 3570

1+ 3284 (1+) 3097 1+ 3098 + 1 2817 1+ 2873 1+ 2765 1+ 2597 1+ 2454 1+ 2371 (1+) 2263 1+ 2242 1+ 2046

(1+) 1587 - 0 1102 1- 1442 - 1+ 1173 8 1025 - - 1 1244 (0-,1-,2-) 992 2 963 - (0-,1-,2-) 868 0 807 0- 1013 - (1-,2-) 711 2 676 0- 849 + (0-,1-,2-) 656 1 659 2- 794 - (0-,1-,2-) 489 1 603 (1-,2-) 316 2- 438 8- 453 (8-) 262 1- 335 3+ 290 (1-) 248 2- 112 (3+) 0 3+ 0 1- 0 128 Experimental 49 In79 jj45pna IMSRG(-) IMSRG (+)

Figure 6.1: Comparison of excitation energies [in keV] in 128In between this work [left], NuShellX (jj45pna) [center], and In-Medium Similarity Renor- malization Group (IMSRG) [right]. Positive parity states are shown in red and negative parity states in blue. are the main sources of uncertainty in these calculations, and efforts are ongoing to quantify their impact [81]. For the jj45pna interaction, the first 1+ state, at 659 keV, is 514 keV lower than the experimental one, at 1173 keV. On the IMSRG side, the first 1+ state is 1756 keV above the first 3+ state, which is 583 keV higher than the experimental level. Therefore, both interactions are shifted by ∼500 keV

103 relative to the experimental level, with a discrepancy of 1097 keV between them. The experiment saw seven states with tentative assignments 1+ above the highly populated 1173 leV level and so Figure 6.1 compares the second to eighth 1+ calculated states. The main feature is the density at which the states are seen: they all lie between 2242 and 3284 keV with the jj45pna interaction, and spread out between 2164 and 4173 keV with IMSRG. The structure of the IMSRG level scheme is closer to the experimental scheme. For example, the first and second 1+ states are seen 408 keV apart, while the experiment saw a difference of 414 keV. A few spin assignments are possible for the seven negative parity states observed experimentally. On the calculations side, only the spins of the first two negative parity states agree, whereas the five next states have different spins. Both interactions see two 0− states; however jj45pna calculates one 1− 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], which was not populated in this decay spectroscopy experiment. It is calculated at 1025 keV with the jj45pna interaction and at 453 keV with the IMSRG interaction. However, since the gap between both parity spectra is not converged, 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 value is closer to the experimental value than the jj45pna.

6.1.2 Configurations Information 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 Jπ, the total spin of the neutrons Jν, and the fraction of the particular coupling between Jπ and Jν for this state. Therefore, the wave functions follow the equation: X |Ψ Π i = c |ψ i, (6.2) (Jn ) i (Jπ⊗Jν ) i 2 where ci are the coefficients listed in Table 6.2 for both interactions. + For the 31 ground state, both calculations agree that the main configuration comes from the coupling of Jπ = 9/2 and Jν = 3/2 with 48.66% −1 −1 (jj45pna) and 66.39% (IMSRG). This is consistent with the πg9/2 ⊗ νd3/2 configuration published in Ref. [47]. + Both calculations also agree on the main configuration of the 11 state, which comes from the coupling of Jπ = 9/2 and Jν = 7/2 with 80.33%

104 −1 −1 (jj45pna) and 69.60% (IMSRG). This is consistent with the πg9/2 ⊗ νg7/2 + configuration published in Ref. [1]. One can see that a transition from 11 + to the 3 ground state represents a neutron moving from 1g7/2 to 2d3/2. + Two configurations have high representations for the 12 state: (Jπ = + 1/2)⊗(Jν = 3/2) and (Jπ = 9/2)⊗(Jν = 7/2). For the 13 state, the main −1 −1 + + configuration is again πg9/2⊗νg7/2, like the 11 state. The next five 11 states show mixed configurations, for which the highest contribution to the wave function arises from different couplings with the Hamiltonians from jj45pna and IMSRG. − A similar conclusion is reached for the 81 state, where jj45pna calculates a representation of 52.58% for (Jπ = 9/2)⊗(Jν = 7/2) and IMSRG, 78.22% for (Jπ = 9/2)⊗(Jν = 11/2). − Both calculations describe the main configuration of the 21 state as − (Jπ = 9/2)⊗(Jν = 11/2), and the main configuration of the 22 state as − (Jπ = 1/2)⊗(Jν = 3/2). For the third 2 state, which is not displayed for IMSRG in Figure 6.1, configurations are mixed and no configuration has a fraction higher than 29.33%. The main configuration of the first 1− state is described by the coupling (Jπ = 9/2)⊗(Jν = 11/2) by both calculations, which then disagree on the next two 1− states. The isomeric transition between the first 1− state and + the 3 ground state represents the transition of a neutron from 1h11/2 to 2d3/2. Finally, the coupling configurations for the two first 0− states are largely mixed and the highest representation of each does not agree between the two calculations.

Table 6.2: Comparison of proton-neutron coupling conﬁgurations in 128In between the NuShellX (jj45pna) and In-Medium Similarity Renormalization Π Group (IMSRG) interactions. The subscript in Jn indicates the major shell of the spin-parity combination. The main conﬁguration is shown in bold.

Π Jn Jπ Jν jj45pna [%] IMSRG [%] + 31 1/2 7/2 7.67 3.66 3/2 3/2 4.64 1.12 3/2 5/2 1.66 3/2 7/2 2.33 5/2 11/2 1.35 Continued on next page

105 Π Jn Jπ Jν jj45pna [%] IMSRG [%] 9/2 3/2 48.66 66.39 9/2 5/2 19.51 16.49 9/2 7/2 8.48 6.43 9/2 9/2 2.41 1.82 9/2 11/2 1.03 + 11 1/2 3/2 3.09 11.66 3/2 1/2 1.85 3/2 3/2 1.74 1.29 3/2 5/2 1.69 5/2 5/2 1.12 5/2 7/2 1.84 1.93 9/2 7/2 80.33 69.60 9/2 9/2 8.63 9.34 9/2 11/2 1.02 2.77 + 12 1/2 3/2 28.30 41.64 3/2 3/2 2.57 2.13 3/2 5/2 7.61 2.51 5/2 5/2 4.01 1.5 5/2 7/2 9.76 3.47 9/2 7/2 22.09 30.23 9/2 9/2 12.00 5.71 9/2 11/2 13.19 1.06 + 13 1/2 3/2 7.46 3.85 3/2 3/2 3.52 3/2 5/2 3.29 5/2 5/2 1.80 5/2 7/2 6.72 2.35 9/2 7/2 68.87 84.65 9/2 9/2 3.46 4.91 9/2 11/2 4.21 3.10 + 14 1/2 3/2 8.96 8.11 3/2 3/2 3.74 5/2 3/2 1.16 5/2 5/2 1.12 2.54 5/2 7/2 2.72 1.50 9/2 7/2 52.62 13.93 Continued on next page

106 Π Jn Jπ Jν jj45pna [%] IMSRG [%] 9/2 9/2 12.18 5.42 9/2 11/2 16.92 66.54 + 15 1/2 1/2 1.23 1/2 3/2 2.50 3/2 3/2 3.96 3/2 5/2 1.06 8.13 5/2 3/2 2.11 5/2 5/2 1.05 1.12 9/2 7/2 8.83 35.12 9/2 9/2 5.87 25.03 9/2 11/2 78.18 23.80 + 16 1/2 3/2 8.86 1.77 3/2 3/2 3.50 2.15 3/2 5/2 1.24 5/2 7/2 22.76 2.32 9/2 7/2 50.81 25.01 9/2 9/2 11.47 46.21 9/2 11/2 20.75 + 17 1/2 3/2 9.77 6.06 3/2 3/2 6.19 2.58 3/2 5/2 1.06 2.22 5/2 3/2 1.47 5/2 5/2 1.26 5/2 7/2 4.20 9/2 7/2 36.52 4.74 9/2 9/2 27.27 41.43 9/2 11/2 12.47 40.64 + 18 5/2 5/2 2.49 5/2 7/2 1.55 9/2 7/2 13.20 49.36 9/2 9/2 48.15 35.88 9/2 11/2 31.85 12.05 − 81 1/2 17/2 1.39 9/2 7/2 52.58 3.23 9/2 9/2 9.02 9/2 11/2 25.78 78.22 Continued on next page

107 Π Jn Jπ Jν jj45pna [%] IMSRG [%] 9/2 13/2 1.68 9/2 15/2 5.93 10.36 9/2 19/2 1.11 1.71 − 21 1/2 3/2 7.27 3.33 1/2 5/2 4.08 3.14 3/2 3/2 3.19 1.17 3/2 5/2 6.24 1.90 3/2 7/2 1.44 5/2 5/2 2.04 5/2 7/2 1.48 9/2 7/2 2.91 2.74 9/2 9/2 19.14 30.85 9/2 11/2 47.81 53.35 9/2 13/2 2.73 − 22 1/2 3/2 45.30 71.93 1/2 5/2 3.19 3.86 3/2 3/2 5.76 4.61 3/2 5/2 2.21 1.25 3/2 7/2 9.08 2.90 5/2 3/2 2.16 5/2 5/2 1.19 5/2 7/2 1.52 9/2 7/2 1.23 9/2 11/2 5.64 1.64 9/2 13/2 20.01 10.16 − 23 1/2 5/2 15.96 7.83 3/2 1/2 2.65 1.54 3/2 3/2 8.50 2.52 3/2 5/2 2.77 1.11 3/2 7/2 2.31 5/2 9/2 2.42 9/2 5/2 1.92 6.21 9/2 7/2 26.41 24.99 9/2 9/2 8.23 23.07 9/2 11/2 19.63 29.33 9/2 13/2 6.83 Continued on next page

108 Π Jn Jπ Jν jj45pna [%] IMSRG [%] − 11 1/2 1/2 3.29 3.67 1/2 3/2 3.86 4.83 3/2 3/2 2.05 3/2 5/2 7.62 3.16 5/2 5/2 1.65 5/2 7/2 1.35 1.26 9/2 7/2 1.89 2.44 9/2 9/2 11.22 26.35 9/2 11/2 66.23 55.62 − 12 1/2 1/2 14.19 1/2 3/2 26.02 4.88 3/2 1/2 6.33 3/2 3/2 9.14 2.19 3/2 5/2 2.10 1.39 5/2 3/2 1.60 5/2 7/2 3.51 9/2 7/2 33.17 3.50 9/2 9/2 5.56 49.61 9/2 11/2 11.28 21.87 − 13 1/2 1/2 7.25 56.08 1/2 3/2 9.08 3.91 3/2 1/2 1.59 3/2 3/2 25.61 6.95 3/2 5/2 13.32 2.42 5/2 3/2 2.93 1.02 5/2 5/2 1.67 5/2 7/2 1.22 9/2 7/2 1.36 1.81 9/2 9/2 22.01 7.44 9/2 11/2 14.59 17.71 − 01 1/2 1/2 31.03 25.76 3/2 3/2 48.14 9.63 5/2 5/2 5.83 1.12 9/2 9/2 14.99 63.49 − 02 1/2 1/2 37.36 48.72 3/2 3/2 25.03 16.50 Continued on next page

109 Π Jn Jπ Jν jj45pna [%] IMSRG [%] 5/2 5/2 5.55 2.15 9/2 9/2 32.06 32.63

6.1.3 Eﬀective Single-Particle Energies The diagonal matrix elements of the Hamiltonian govern the evolution of SPEs throughout the region, and resulting ESPEs are shown in Figure 6.2 for 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 separation energies and the spectroscopic strength of the Z, N ± 1 and Z ± 1, N neighbouring nuclei [24]. Since the spectroscopic strength is often spread among high-lying states, a large number of states must be calculated. In order to calculate the ESPE for the ν-orbitals in 126Cd, 150 states of each spin-parity were calculated and averaged for the two even Z, odd N ± 1 neighbours: 125Cd and 127Cd. Similarly, calculations were performed on 127Cd and 129Cd to extract the ν-orbitals ESPEs in 128Cd; 127,129Sn for the 128Sn 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 larger gap between the 2νd5/2 and 3νs1/2 orbitals. The ESPEs for the π-orbitals in 126Cd were calculated and averaged from 150 states of each spin-parity in the two odd Z±1, even N neighbours: 125Ag and 127In. Similarly, calculations were run on 127Ag and 129In to extract the π-orbitals ESPEs in 128Cd. For 128,130Sn, the Z+1 = 51 neighbours 129,131Sb are outside of the jj45pn model space and therefore could not be calculated with this Hamiltonian. By comparing the ESPEs to the SPEs (left), one can see that the interaction lowered the 1νg7/2 down by ∼6 MeV, to the bottom of the shell. Also, the 1πg7/2 and 1νh11/2 orbitals were lowered by ∼2 MeV and ∼9 MeV, respectively. Finally, the gap between the 2πp3/2 and 2πp1/2 orbitals is ∼5 times larger, at almost ∼3 MeV. The order of the four π-orbitals calculated for the cadmium and tin isotopes agree and show a larger gap between the 2νd5/2 and 3νs1/2 orbitals.

110 1νh 11/2 2.524

1νg 7/2 -3.207 2νd 3/2 -3.419 3νs 1/2 -4.030 2νd 5/2 -4.260

1νh 11/2 -5.837 1νh 11/2 -5.297 1νh 11/2 -6.241 1νh 11/2 -5.752 2νd 3/2 -6.416 2νd -6.551 2νd 3/2 -6.747 3/2 2νd 3/2 -6.622 3νs 1/2 -6.750 3νs -6.876 3νs 1/2 -6.831 3νs 1/2 -7.070 1/2

2νd 5/2 -8.784 2νd 5/2 -9.099 1νg 7/2 -9.071 1νg 7/2 -9.262 2νd 5/2 -9.354 2νd 5/2 -9.218 1νg 7/2 -9.970 1νg 7/2 -9.979

1πg 9/2 -12.557

1πg 9/2 -14.527 1πg 9/2 -15.162 2πp 1/2 -15.771 2πp 1/2 -16.352

2πp 1/2 -17.576 2πp 3/2 -18.108 2πp 3/2 -18.545 2πp 3/2 -19.323

2πf 5/2 -20.485 2πf 5/2 -20.898 126 IMSRG SPE 48 Cd78 2πf 5/2 -22.107 128 128 130 48 Cd80 50 Sn78 50 Sn80

Figure 6.2: Single-particle energies (SPEs) in the jj45pn model space, and eﬀective single-particle 128 energies (ESPEs) [in MeV] for the four even Z, even N neighbouring isotopes of 49 In79 from In-Medium Similarity Renormalization Group (IMSRG). Proton orbitals are shown in red and neutron orbitals in blue.

111 6.2 131In

6.2.1 Level Energies The excited states in 131In found experimentally are compared to calculated levels in Figure 6.3. 131In is an odd Z, even N (magic N = 82) nucleus with a single π-hole in the jj45pn model space. Since NuShellX does not allow the nucleon to excite across the shell gap to higher orbitals, only single-particle excited states can be calculated. The energy of the ﬁrst three levels calculated with the jj45pna interaction are in good agreement with the experiment. The 3/2− state is just 141 keV above the experimental value of 1353 keV. The 988-keV transition of 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 and 131In is just one π-hole away from the double shell closure and doubly magic 132Sn. For IMSRG, the gap between the positive and negative parities is not converged. While there were enough states to compare levels of same parity in 128In, there is only one positive parity state in 131In and we cannot say how large the gap is with the 1/2+ state.

6.2.2 Configurations Information on the configurations and the occupation numbers of the excited states in 131In is found in Table 6.3. The occupation numbers (.occ output file) are the weighted sum of the contributions of the different Jπ ⊗Jν couplings 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 all show a single configuration (100%). The spin of the state is determined by the orbital which is occupied by the unpaired proton or unpaired proton- hole. The five neutron orbitals in the shell are fully occupied and are not contributing to the calculated excited states. Hence, the higher-spin 21/2− and 17/2− states observed in experiments [6] arise from more complex excitation modes, such as 1π-2ν and 2π-2ν mixing across the shell gap, which are not included in the current model space. Finally, the only 5/2− state −1 calculated here is the πf5/2 state, of which the observation has been moti- vating several recent experiments [55, 6] and our analysis did not find any conclusive evidence for it.

112 5/2- 5451

4770 4643 4237 4130 3990 3921 (17/2-) 3782 (21/2-) 3764

5/2- 2994

3/2- 2431

3/2- 1494 (3/2-) 1353

(1/2-) 365 1/2- 363

(9/2+) 0 9/2+0 1/2-9 9/2+0 131 Experimental 49 In82 jj45pna IMSRG(-) IMSRG (+)

Figure 6.3: Comparison of excitation energies [in keV] in 131In between this work [left], NuShellX (jj45pna) [center], and In-Medium Similarity Renor- malization Group (IMSRG) [right]. Positive parity states are shown in red and 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 1πf5/2 2πp3/2 2πp1/2 1πg9/2 1νg7/2 2νd5/2 2νd3/2 3νs1/2 1νh11/2 Main Conf. [%] + −1 9/2 6.00 4.00 2.00 9.0 8.0 6.0 4.00 2.00 12.00 π1g9/2 (100) − −1 1/2 6.00 4.00 1.00 10.0 8.0 6.0 4.00 2.00 12.00 π2p1/2 (100) − −1 3/2 6.00 3.00 2.00 10.0 8.0 6.0 4.00 2.00 12.00 π2p3/2 (100) − −1 5/2 5.00 4.00 2.00 10.0 8.0 6.0 4.00 2.00 12.00 π1f5/2 (100)

113 Chapter 7

Conclusions and Outlook

Detailed data sets for the β-decay of 128−131Cd were successfully obtained with the GRIFFIN γ-ray spectrometer at TRIUMF. This experimental cam- paign with exotic neutron-rich radioactive beams would not have been possible without the discriminating power of IG-LIS and the high detection efficiency of the GRIFFIN array. This careful analysis of the neutron-rich cadmium isotopes is of great current interest with respect to advancing our understanding of nuclear forces and shell evolution in a region that is essential for the understanding of the astrophysical r-process and which only recently became accessible for more extensive studies. The new data provides important inputs for theoretical calculations and models aiming to reproduce the element abundances of the Solar system and the nuclear structure of exotic isotopes around the N = 82 shell closure. The decay spectroscopy of 128Cd was successfully performed and has revealed 32 new transitions and 11 new levels, highlighting the high sensitivity of the GRIFFIN spectrometer. Eight allowed Gamow-Teller β-decays were observed to tentative 1+ states. First forbidden Gamow-Teller decays were observed to feed seven states. The tentative spin assignments for three of these seven states were restricted using γ-γ angular correlation analysis, which ruled out the spin 0−. Finally, this work confirmed the half-life of 245.4(30) ms for 128Cd which was extracted from this data set and previously published in 2016 (t1/2 = 246.2(21) ms) [4]. The extended decay scheme was compared to theoretical calculations using NuShellX (jj45pna) and IMSRG. Both interactions calculated the energy of the first 1+ excited state shifted by ∼500 keV relative to the experimental level at 1173 keV, with a discrepancy of 1097 keV between them. In addition, the IMSRG calculates the 3+ ground state at 290 keV. This gap could be addressed by calculating the three-nucleon (3N) forces in a bigger model space, which requires computing resources which are not available at this time. Information on the configurations of the states in 128In is also + −1 −1 extracted. The 3 ground state shows a πg9/2 ⊗ νd3/2 configuration, which is consistent with Ref. [47]. The configuration of the first 1+ excited state is −1 −1 − πg9/2 ⊗νg7/2, as published in Ref. [1]. The 1 isomer at 248 keV is described

114 −1 −1 by the coupling πg9/2 ⊗ νh11/2 by both calculations. The 248 keV isomeric transition between the first 1− state and the 3+ ground state represents the transition of a neutron from 1h11/2 to 2d3/2. ESPEs were also calculated for the four even N, even Z neighbouring isotopes. A manuscript describing the current work on the nuclear structure of 128In is in preparation for submission to Physical Review C. The detailed γ-spectroscopy of the 131Cd enabled the confirmation of 21 transitions and the revision of the decay scheme previously published in Ref. [6], which placed 12 high-energy levels directly feeding the ground state. However, there was no clear information indicating if these 12 ground state transitions were feeding the 1/2− isomeric or 9/2+ ground states. This work placed only 9 transitions and 8 excited states in the decay scheme because of the low coincidence statistics. The new excited level at 3920 keV seems to be the only level connecting both the isomeric and ground states, via a 3554-keV transition and a 3920-keV transition. Thirty more hours of 131Cd data were obtained in 2016 with a beam intensity of ∼0.7 pps, which will improve the statistical uncertainty on the values obtained in this work. Calculations for 131In with the jj45pna interaction are in good agreement with the experiment since the SPEs were adjusted to the 132Sn region: the 3/2− 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, the gap between the positive and negative parities was again not converged and more computing power will be required. The four levels calculated in 131In are known very well to be single-particle states, which were confirmed with the 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 which do not allow the nucleon to excite across the shell gap to higher orbitals and therefore, only single-particle excited states can be calculated. Hence, calculations for the higher-spin 21/2− and 17/2− states observed [6] and for the configurations of the observed mixed states will require a larger model space. The low-statistics data set collected during the low-rate A = 132 beam time was inconclusive with regards to the β-decay and βn-decay of 132Cd. Open questions remained about the important 988-keV transition seen in the β-decay 131Cd, which was observed very close to the 989-keV transition in the beam contaminant 132Sb. For neutron-rich nuclei with large neutron- branching ratios like 132Cd, a γ-ray spectrometer alone is not sufficient and neutron-tagging with neutron detectors, such as DESCANT at TRIUMF, will become more important. Many open questions remain for the level

115 structure of 132In which hopefully can be tackled in the future with higher yields and cleaner beams, for example with the use of photoﬁssion targets from the new Advanced Rare Isotope Laboratory (ARIEL) at TRIUMF. These new studies combined highlights unanswered questions, which should point the community in the direction of what to probe next to further its understanding of nuclear structure and astrophysics theories. The sum of all studies addressing these questions allow to verify assumptions and induce changes in the main input for the calculations. This new experimental knowledge highlights the need for larger-scale shell model calculations and reliable models of exotic nuclear structure, which will be enabled as new calculation methods and technologies are developed. Further comparisons with shell model calculations may lead to a deeper or diﬀerent understanding of the structure across the neutron-rich indium isotopic chain.

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122 Appendix A

Data Calibration and Processing

γ-ray Energy Calibration

Identify the probable good runs, the calibration runs and the garbage runs

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)

152 133 56 For each crystal and each source run (e.g. Eu, Ba and Co): – Get the number of counts for well-known transitions in γ-ray singles – 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 and check for non-linear effects

For each crystal and each data run: – Sort the data analysis trees using the new .cal file and check for gain drifts between different runs – Identify and fit background lines and previously published transitions

123 – Plot the centroid channel of the peaks vs. their energy and fit to Equation (A.1) – Compile the gain and offset for each crystal in a second calibration file

Absolute γ-ray Detection Eﬃciency Calibration

152 133 56 For each source run (e.g. Eu, Ba and Co): – Get the number of counts for well-known transitions in γ-ray singles – Get the number of counts for well-known transitions in γ-ray singles 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 between consecutive hits – Get the average number of hits per crystal N by plotting the γ-ray singles per crystal – Get the absolute γ-ray detection efficiency : Number of γ-rays detected = Number of γ-rays emitted by source (A.2) N = γ,detected Iγ · A · (t − d · N)

Fit to the germanium detector eﬃciency [67]:

2 2 (E) = 10p0+p1 log(E)+p2 log (E)+p3/E (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 singles

124 – Get the number of counts for strong transitions in β-gated γ-ray singles – Divide the number of counts in β-gated γ-ray singles by the number of counts in γ-ray singles

Time 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 matrices

Construction of Addback Events [66]

Crosstalk correction

– For 60Co events with multiplicity 2 within one clover, build a matrix of γ1-energy vs. γ2-energy – Extract the offset with a linear fit for each pair of crystals within the 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 mode

125 Appendix B

Data Analysis

Identiﬁcation of new transitions

Compare the γ-ray singles spectra with the lasers on and the laser blocked

Compare the γ-ray singles spectra for the beam-on and beam-oﬀ parts of 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 the modulus of the cycle length – Build a matrix of γ-ray energy vs. cycle time for the beam-on and beam-oﬀ parts of the cycle

Coincidence 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 scheme

Intensities 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 eﬃciency

126 – 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 intensities by the relative intensities – Propagate the uncertainties in quadrature – Estimate the unobserved ground state transitions by adding absolute intensities for all ground state transitions • Get the β-decay branching ratios – Calculate the diﬀerence between the sum of the absolute γ-ray intensities which feed and depopulate each level – Estimate the ground state branching ratio by adding the branching ratios for all states – Propagate the uncertainties in quadrature – Calculate the log(ft) value of each state with the Logft web application [71]

Angular correlations [74]

Determine the possible spin assignments from the selection rules and the log(ft) values

Make the experimental angular correlations plots to determine the preliminary 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 possible spin assignments overlap at the preliminary a22/a44 values

Run GEANT4 simulations for given cascades/mixing ratios to account for solid angle corrections

2 Compare data to simulated template, perform χ analysis to get mixing ratios and spins

Repeat for other cascades with over ∼10000 counts

Half-life measurement

Project the matrix of γ-ray energy vs. cycle time for the beam-oﬀ part of 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:

− ln 2·t/t − ln 2·t/t Atotal(t) = Ae 1/2,parent + Be 1/2,daughter + C (B.2)

Check for in-beam contaminants and systematic errors (i.e. binning and 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 diﬀerence spectrum: [3] = [1] - [2]

Fit the activity of the γ-ray to Equation (B.2)

128