β-DECAY TOTAL ABSORPTION SPECTROSCOPY AROUND A = 100-110 RELEVANT TO NUCLEAR STRUCTURE AND THE ASTROPHYSICAL R PROCESS By Alexander Connor Dombos A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Physics — Doctor of Philosophy 2018 ABSTRACT β-DECAY TOTAL ABSORPTION SPECTROSCOPY AROUND A = 100-110 RELEVANT TO NUCLEAR STRUCTURE AND THE ASTROPHYSICAL R PROCESS By Alexander Connor Dombos This dissertation details the initiation of a new experimental program to study β decay that is now in use at the National Superconducting Cyclotron Laboratory and will be an inte- gral part of the science conducted at the Facility for Rare Isotope Beams. This experimental program studies the β-decay properties of nuclides relevant to the astrophysical r process with the total absorption spectroscopy technique. Descriptions of r-process nucleosynthesis, an overview of β decay and γ decay, the experimental setups, and analysis procedures are included in this dissertation. This dissertation contains the commissioning experiments of this experimental program. These commissioning experiments were performed at the Coupled Cyclotron Facility at the National Superconducting Cyclotron Laboratory and combined charged-particle detection using silicon detectors and γ-ray detection using a segmented total absorption spectrometer called the Summing NaI(Tl) (SuN) detector. The commissioning experiment with a thermalized beam examined the β decay of 76Ga. The extracted β-decay half-life agrees with previously published values. However, the ex- tracted β-decay feeding intensity distribution disagrees with the existing decay scheme at the National Nuclear Data Center. The extracted distribution provided experimental data in the A = 76 mass chain. This experimental data can constrain nuclear structure models that calculate nuclear matrix elements for neutrinoless double-β decay. The commissioning experiment with a fast beam studied neutron-rich nuclides in the A = 100-110 mass region. This experiment was the first-ever application of the total ab- sorption spectroscopy technique with a fast beam produced via projectile fragmentation. β-decay half-lives were extracted for 99Y, 101Zr, 102Zr, 102mNb, 103Nb, 104mNb, and 109Tc. Overall, the extracted half-lives agree with previously published values. Additionally, the β-decay feeding intensity distributions and B(GT) distributions were extracted for 101Zr, 102Zr, and 109Tc. The extracted distributions were compared to QRPA calculations, which are commonly used to provide β-decay properties in r-process reaction network calculations. In these comparisons, none of the QRPA calculations were able to reproduce the extracted distributions. The extracted distributions were compared to another set of QRPA calcula- tions in an attempt to learn about the shape of the ground state of the parent nucleus. For 101Zr and 102Zr, calculations assuming a pure shape configuration (oblate or prolate) were not able to reproduce the extracted distributions. These results may indicate that some type of mixture between oblate and prolate is necessary to reproduce the extracted distributions. For 109Tc, a comparison of the extracted distribution with QRPA calculations suggests a dominant oblate configuration. ACKNOWLEDGMENTS Thanks firstly to Artemis, my advisor, for listening to my ideas with excitement and respect, and treating me as a fellow explorer in the field of nuclear astrophysics. Her office door has always been open and she has graciously allowed me to ask questions (too many to count) at any time. Her openness has created an environment in which I have thrived. Thank you, Artemis! I am incredibly grateful to everyone who I interacted with through using the SuN detector. This includes research physicists (Jorge), SuN postdoctoral researchers (Anna, Farheen, Mallory, Stephanie), SuN graduate students (Steve, Debra, Alicia, Caley), other graduate students (Wei Jia, Becky, Adriana), and undergraduate students (Jason, Antonius, Maya). In addition to being a wonderful friend, Steve deserves special recognition for his mentorship during my early years as a graduate student. Thanks also to Professors Sean Liddick, Morten Hjorth-Jensen, Edward Brown, and Norman Birge for their advice, questions, and offering their valuable time to serve on my guidance committee. I would also like to thank the beam physicists, cyclotron operators, design group, electrical engineers, machinists, and others who made possible the experiments comprising this dissertation. The expertise of the β group, especially Sean Liddick and Chris, has been invaluable. I am additionally grateful to Matthew Mumpower for being my national laboratory mentor through the Nuclear Science and Security Consortium. Matthew, thank you for teaching me how to run r-process reaction network calculations at Los Alamos National Laboratory during the summer of 2017 and for taking me on some truly memorable hikes! Thank you to members of my cohort at the NSCL, especially Chris, Eric, and Sam, for working on iv coursework together and helping me gain the necessary programming skills to succeed in this graduate program. I would also like to thank Arthur Cole, who I worked with as an undergraduate for his continued guidance and friendship. I am so thankful for the love and support of my parents, brother, grandparents, aunts and uncles, and neighbors. Finally, I want to thank my wife, Kelsey, for the love and support she has given me throughout this process while also completing her own doctoral degree in Neuroscience. There has been quite a large synaptic cleft between our neurons these past six years, but soon we will be reunited. I can’t wait to begin our life together and use our hippocampi to make new memories! v TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter 1 1.4 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Nuclides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Nucleosynthesis Beyond the Iron Peak . . . . . . . . . . . . . . . . . r process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Astronomical Observations . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Astrophysical Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2.1 Core-collapse Supernovae . . . . . . . . . . . . . . . . . . . 1.4.2.2 Compact-object Mergers . . . . . . . . . . . . . . . . . . . . 1.4.3 Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 β decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . β-decay Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 β-delayed γ-ray Emission . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Internal Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 β-delayed Neutron Emission . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 β-decay Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.5 1.5.6 Half-life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Pandemonium Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Total Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Summing NaI(Tl) (SuN) detector . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Dissertation Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 2 β-decay Studies with Thermalized Beams . . . . . . . . . . . . . 2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Technical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Physics Motivation: Neutrinoless Double-β Decay . . . . . . . . . . . 2.2 Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Half-Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Total Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . 2.4.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 1 3 5 5 12 12 16 16 17 21 26 26 29 30 33 33 39 43 46 48 50 53 54 54 54 57 61 69 69 71 75 78 Chapter 3 3.1.3 3.1.1 3.1.2 β-decay Studies with Fast Beams . . . . . . . . . . . . . . . . . . 3.1 Experimental End Station . . . . . . . . . . . . . . . . . . . . . . . . . . . . Silicon PIN Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . Implantation Station . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.1 DSSD triggering conditions . . . . . . . . . . . . . . . . . . 3.1.2.2 DSSD Calibrations and Thresholds . . . . . . . . . . . . . . Summing NaI(Tl) (SuN) Detector . . . . . . . . . . . . . . . . . . . . 3.1.3.1 Gain Matching and Calibration . . . . . . . . . . . . . . . . 3.1.3.2 Thresholds 79 79 84 85 91 93 93 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.2.1 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.2.1.1 Momentum Correction to the Time of Flight . . . . . . . . . 102 3.2.1.2 Charge States . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.2.2 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.2.2.1 Random Correlations . . . . . . . . . . . . . . . . . . . . . . 113 3.2.3 Total Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . 118 3.2.3.1 GEANT4 Simulation . . . . . . . . . . . . . . . . . . . . . . 119 3.2.3.2 Known Levels . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.2.3.3 Pseudo Levels . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.2.3.4 Contamination . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.2.3.5 Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 39Y60 → 99 . . . . . . . . . . . . . . . . . . . . . . . . 132 40Zr59 40 Zr61 → 101 . . . . . . . . . . . . . . . . . . . . . . 134 40 Zr62 → 102 . . . . . . . . . . . . . . . . . . . . . . 135 41 Nb61 → 102 42 Mo60 . . . . . . . . . . . . . . . . . . . . . 138 41 Nb62 → 103 42 Mo61 . . . . . . . . . . . . . . . . . . . . . . 139 41 Nb63 → 104 42 Mo62 . . . . . . . . . . . . . . . . . . . . . 142 43 Tc66 → 109 . . . . . . . . . . . . . . . . . . . . . . 143 3.3.2 Total Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . 146 40 Zr61 → 101 . . . . . . . . . . . . . . . . . . . . . . 147 40 Zr62 → 102 . . . . . . . . . . . . . . . . . . . . . . 157 43 Tc66 → 109 . . . . . . . . . . . . . . . . . . . . . . 169 3.3.1.1 3.3.1.2 3.3.1.3 3.3.1.4 3.3.1.5 3.3.1.6 3.3.1.7 41 Nb60 41 Nb61 44 Ru65 3.3.2.1 3.3.2.2 3.3.2.3 41 Nb60 41 Nb61 3.3.1 Half-lives 101 102 109 44 Ru65 99 101 102 102m 103 104m 109 Chapter 4 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 179 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 vii LIST OF TABLES Table 1.1: Classifications of β-decay transitions. Adapted from Ref. [1]. . . . . . . . 29 Table 1.2: Classifications of γ-ray transitions. Adapted from Ref. [1]. . . . . . . . . 30 Table 2.1: High voltages and multiplication factors used to gain match each PMT of SuN for NSCL experiment e13502. For a given PMT label, the letter indicates the top or bottom half of SuN (“B” for bottom and “T” for top), the first number indicates the segment of SuN, and the second number indicates the PMT within the segment. For example, the label T23 means PMT 3 of segment 2 of the top half of SuN. . . . . . . . . . . . . . . . . Table 2.2: The β-decay feeding intensity distribution of 76Ga as a function of excita- tion energy in the daughter nucleus 76Ge. Intensity values below 10−4 % are set to 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 3.1: High voltages and multiplication factors used to gain match each PMT of SuN for NSCL experiment e12001. For a given PMT label, the letter indicates the top or bottom half of SuN (“B” for bottom and “T” for top), the first number indicates the segment of SuN, and the second number indicates the PMT within the segment. For example, the label T23 means PMT 3 of segment 2 of the top half of SuN. . . . . . . . . . . . . . . . . 62 74 96 Table 3.2: Standard radioactive sources used to calibrate the segments of SuN. . . . 97 Table 3.3: Energy deposition (MeV) in different detectors for two isotopes with ap- proximately the same mass-to-charge ratio (A/q). . . . . . . . . . . . . . 105 Table 3.4: Standard deviation (σ) of a Gaussian function fit to various experimental γ-ray peaks, along with the corresponding energy resolution. . . . . . . . 120 viii Table 3.5: The values used for different parameters in dicebox when creating pseudo levels above Ecrit (critical energy) for the three daughter nuclides in the present work. The parameters associated with giant resonances that were needed for the γ-ray strength functions were Er (resonance energy), Γ (width), and σ (peak cross section). The parameters for the E1 γ-ray strength function were from the nearest nuclide of the same type (even Z and even N, even Z and odd N, etc.) for which experimental measurements exist in RIPL-3. However, there were no odd Z and odd N measurements near 102 41 Nb61 and therefore the nearest measurement was used regardless of even/odd proton/neutron numbers. The nearest nuclides for 101 41 Nb60, 102 41 Nb61, and 109 50 Sn67, respectively. The final results of this work were not sensitive to small variations in these parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 42 Mo58, and 117 44 Ru65 were 103 45 Rh58, 100 Table 3.6: Half-lives from the present work along with previous measurements. The selection of events in the TAS spectrum (“Level(s)”) and sum-of-segments spectrum (“γ ray(s)”) to extract the half-life are listed for each nuclide. If a reference cited in the ENSDF file could not be obtained, the ENSDF file is cited along with the original reference. A previous measurement that does not contain any uncertainty will have no uncertainty in this table. . . 133 Table 3.7: QRPA calculations that are compared to the experimental results in this dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Table 3.8: The β-decay feeding intensity distribution of 101Zr as a function of excita- tion energy in the daughter nucleus 101Nb. Intensity values below 10−4 % are set to 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Table 3.9: The β-decay feeding intensity distribution of 102Zr as a function of exci- tation energy in the daughter nucleus 102Nb. Intensity values below 10−4 % are set to 0. As explained in Sec. 3.3.2.2, each level was assumed to be built on top of the β-decaying isomeric state. That is, a value of “x” = 93 keV as determined by Ref. [2] should be added to each level. As explained in Sec. 3.3.2.2, the detector response function for the level at 20+x keV was not used in the TAS analysis. . . . . . . . . . . . . . . . . . . . . . . 163 ix Table 3.10: The β-decay feeding intensity distribution of 102Zr as a function of exci- tation energy in the daughter nucleus 102Nb. Intensity values below 10−4 % are set to 0. As explained in Sec. 3.3.2.2, each level was assumed to be built on top of the β-decaying isomeric state. That is, a value of “x” = 93 keV as determined by Ref. [2] should be added to each level. As explained in Sec. 3.3.2.2, the detector response function for the level at 20+x keV was not used in the TAS analysis. As explained in Sec. 3.3.2.2, the values reported in this table are from the fit in which the ground-state-to-ground- state transition was held fixed between 60 and 61%. . . . . . . . . . . . . 165 Table 3.11: The β-decay feeding intensity distribution of 109Tc as a function of excita- tion energy in the daughter nucleus 109Ru. Intensity values below 10−4 % are set to 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 x LIST OF FIGURES Figure 1.1: Chart of the nuclides. Each shaded cell is an individual nuclide, defined by a unique combination of atomic number (Z) and neutron number (N). The stable nuclides are black, the unstable nuclides that are experimentally known to exist are dark gray, and the unstable nuclides that are predicted to exist according to the FRDM (2012) [3] mass model are light gray. An unstable nuclide is predicted to exist if both the one-proton separation energy (the energy required to remove a proton from the nucleus) and one-neutron separation energy (the energy required to remove a neutron from the nucleus) are greater than zero. In other words, an unstable nuclide is predicted to exist if the spontaneous emission of nucleons is energetically forbidden. The black, dotted lines indicate magic numbers . . . . . . . . . . . . . . . . that correspond to closed shells of nucleons. Figure 1.2: Abundances of nuclides in the solar system and the processes responsible for their production. The top panel shows the decomposition of the abun- dances by both parity of mass number and production mechanism. In the top panel, various nuclides are labeled. The bottom panel shows the decomposition of the abundances beyond the iron peak into the individ- ual abundance patterns of the p process, s process, and r process. In the bottom panel, different features arising from nuclear structure are labeled in the abundance patterns of the s process and r process. Both panels use the cosmochemical scale, which normalizes silicon to 106 atoms. The abundance data is from Ref. [4] (solar system), Ref. [5] (p process), Ref. [6] (s process), and Ref. [7] (r process). . . . . . . . . . . . . . . . . . . Figure 1.3: Schematic illustration of the operation of the p process (green arrows), s process (cyan arrows), and r process (orange arrows) for a region of the chart of the nuclides. The r process shows the operation during the neutron flux (orange, solid arrows) and decay back to the valley of stability after the neutron flux (orange, dashed arrows). The neutron closed shell at the magic number N = 50 is indicated with the red, shaded region. A cell that is labeled and outlined in black indicates a stable nuclide. A cell that is not labeled and outlined in gray indicates an unstable nuclide. This illustration ignores the possibility of “branching points” in the s process. 2 6 8 xi Figure 1.4: Elemental abundances in ten metal-poor halo stars (specifically, r-I and r-II stars). Markers of the same type and color correspond to the same r-I or r-II star. The solid blue lines are the r-process solar system abundance pattern. The only difference between the blue lines is a scaling factor. The scaling factor is obtained by normalizing the europium abundance in the r-process solar system abundance pattern to that in the r-I or r-II star. Europium is chosen due to being an r-process element. Figure adapted from Ref. [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.5: Snapshot of a simulation of a NS-NS merger. The color indicates the magnitude of the magnetic field (the lighter the color, the larger the mag- nitude of the magnetic field). The two neutron stars are in the center, surrounded by dynamical ejecta in the tidal tails. Figure adapted from Refs. [9, 10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.6: Abundance pattern and path of the r process at different times for a NS-NS merger. Each quadrant corresponds to a different time step in a reaction network calculation. The reaction network is called PRISM [11, 12]. Each quadrant contains a top panel and bottom panel. The top panels show the absolute abundance pattern of the r process for the solar system and from the reaction network calculation. The abundances are expressed in the cosmochemical scale, which normalizes silicon to 106 atoms. The bottom panels show the chart of the nuclides, with stable nuclides in black, unstable nuclides that are experimentally known to exist in dark gray, and unstable nuclides that are predicted to exist according to the FRDM (2012) [3] mass model in light gray. The neutron magic numbers (N = 2, 8, 20, 28, 50, 82, 126) are indicated with a black, dashed line. The relative abundances of nuclides produced from the PRISM calculation are shown with shaded cells. Each quadrant has a label for time in units of seconds (t), temperature in units of 109 K (T9), and density in units of g/cm3 (ρ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.7: Abundance weighted timescales for important nuclear processes during . . . . . . . . . . . . . . . . . . . . . the r process for a NS-NS merger. Figure 1.8: Total internal conversion coefficients for ruthenium (Z = 44) for a range of transition energies and multipolarities. The total internal conversion coefficients were obtained with the BrIcc program [13, 14] provided by the National Nuclear Data Center. The inset shows a zoomed-in view of the . . . . . . . . . . . . . . . . . . . . . . . . low transition energy region. 15 18 24 25 32 xii Figure 1.9: A simplified decay scheme for β− decay, with different transitions, Fermi functions, and electron kinetic energy distributions. See main text for details. All functions and distributions are normalized to unity. In the right panels, the red, cyan, and blue lines (both dotted and solid) are on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . top of each other. Figure 1.10: A simplified decay scheme for β+ decay, with different transitions, Fermi functions, and positron kinetic energy distributions. See main text for details. All functions and distributions are normalized to unity. In the right panels, the cyan and blue lines (both dotted and solid) are on top of each other. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 1.11: Definition of the Fermi integral for a single, representative β-decay transi- tion in terms of electron kinetic energy (top panel), electron total energy (middle panel), and electron momentum (bottom panel). The Fermi inte- gral is the area under the corrected phase space distribution. Unlike Fig. 1.9, the functions and distributions are not normalized to unity. . . . . Figure 1.12: Representative comparison of β-decay feeding intensity, Fermi integral, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and B(GT). Figure 1.13: A complex and fragmented β-decay scheme with many β-decay transitions . . . . . . . . . . and γ-ray transitions. Figure adapted from Ref. [15]. Figure 1.14: Example spectra obtained with a 60Co source at the center of a segmented total absorption spectrometer. The spectra are the TAS spectrum (top panel), sum-of-segments spectrum (middle panel), and multiplicity spec- trum (bottom panel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 38 41 44 46 49 Figure 1.15: The Summing NaI(Tl) (SuN) detector. . . . . . . . . . . . . . . . . . . 51 Figure 2.1: Isobaric chain for A = 76, which contains 76Ga. Data for the mass excess [16]. The nuclide 76Ge can undergo two-neutrino double-β is from Ref. (2νββ) decay, and is a candidate for neutrinoless double-β (0νββ) decay. The nuclide 76Ga undergoes β decay to 76Ge. . . . . . . . . . . . . . . . Figure 2.2: The linear gas cell in the N4 vault used for NSCL experiment e13502. The secondary beam from the A1900 fragment separator enters from the right side of the picture. Ions are thermalized though collisions with helium gas . . . . . . . . . . . . and then extracted on the left side of the picture. Figure 2.3: Experimental end station attached to the D Line for NSCL experiment . e13502. The thermalized beam enters from the left side of the picture. 55 58 59 xiii Figure 2.4: The silicon surface barrier detector installed inside the bore hole of SuN . . . . . . . . . . . . . . . . . . . . . . . . for NSCL experiment e13502. 60 Figure 2.5: Calibrations for each segment of SuN for NSCL experiment e13502. For a given segment label, the letter indicates the top or bottom half of SuN (“B” for bottom and “T” for top), and the number indicates the segment number. For example, the label T2 means segment 2 of the top half of SuN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 2.6: The TAS spectrum of SuN for the β decay of 76Ga in singles (green, dashed line), normalized room background (blue, dotted line), and with a coincidence requirement with the silicon surface barrier detector (black, solid line). The energies of prominent sum peaks are labeled in the coin- cidence spectrum. There is also a label for the Q value for the β decay of 76Ga at 6916.2(2.0) keV [17]. . . . . . . . . . . . . . . . . . . . . . . . . Figure 2.7: Visualization of the GEANT4 simulation for the SuN detector (left panel) and the silicon surface barrier detector attached to the target holder at the center of SuN (middle and right panels). The aluminum frame and aluminum foil are shown attached to the front of the silicon surface barrier detector. The sensitive volume of the silicon surface barrier detector (the silicon wafer) is not visible. The green lines are γ-ray tracks and the red . . . . . . . . . . . . . . . . . . . . . . lines are β-decay electron tracks. Figure 2.8: Example spectra from simulations showing the energy deposition in the detectors only from β-decay electrons. No γ rays were emitted in the simulation that created these spectra. The top panel shows the input electron kinetic energy distribution, the middle panel shows the energy deposited in the silicon surface barrier detector, and the bottom panel shows the total energy deposited in SuN. These spectra were obtained from a GEANT4 simulation of 2000000 events of electrons from a (Z, A) = (31, 76) nuclide. The maximum electron kinetic energy was 4000 keV. Electrons were isotropically emitted from the center of the aluminum foil (see Fig. 2.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 65 66 67 Figure 2.9: Comparison of experimental (black, solid line) and simulated (red, dashed- dotted line) spectra using the existing decay scheme [18] to illustrate the discrepancy with the measurements reported in the present work. The spectra are (a) the TAS spectrum, (b) sum-of-segments spectrum, and (c) multiplicity spectrum. All three spectra were created with an energy threshold of 80 keV applied to each SuN segment. There is a label for Q value in the TAS spectrum for the β decay of 76Ga at 6916.2(2.0) keV [17]. 70 xiv Figure 2.10: The experimental decay curve for the β decay of 76Ga (black, solid line) and the exponential fit from 0 to 300 s (red, dashed line). The extracted half-life is 30.6(3) s. The inset shows the history of measurements of the half-life of 76Ga. The measurement from 1961 is from Ref. [19], from 1971 is from Ref. [20], from 1974 is from Ref. [21], from 1985 is from Ref. [22], from 2016 is from the current work [23]. . . . . . . . . . . . . . . . . . . Figure 2.11: Comparison of experimental (black, solid line) and simulated (red, dashed- dotted line) spectra after fitting all three spectra simultaneously with the decay scheme modifications for (a) the TAS spectrum, (b) sum-of- segments spectrum, and (c) multiplicity spectrum. This is an example of one of the different fitting conditions with a specific energy calibration and binning. All three spectra were created with an energy threshold of 80 keV applied to each SuN segment. There is a label for Q value in the TAS spectrum for the β decay of 76Ga at 6916.2(2.0) keV [17]. . . . . . . Figure 2.12: Cumulative β-decay feeding intensity of 76Ga as a function of excitation energy in the daughter nucleus 76Ge for the present work (blue, solid line, with uncertainty in light-blue shading) and calculations with dif- ferent Hamiltonians and different assumptions of the spin and parity of the ground state of 76Ga. Panel (a) contains calculations using the jj44b Hamiltonian and panel (b) contains calculations using the JUN45 Hamil- tonian. The half-life from the present work and theoretical calculations are in parentheses. For the present work, the blue, solid line is the cu- mulative average intensity, and the lower/upper bound of the light-blue uncertainty band is the cumulative minimum/maximum intensity. See text for an explanation of why the 2− calculation and the present work are identical at relatively low excitation energy. . . . . . . . . . . . . . . Figure 2.13: Same as Fig. 2.12, but for cumulative B(GT). The inset shows a zoomed- in view of the low excitation energy region. . . . . . . . . . . . . . . . . . Figure 3.1: Schematic layout of the Coupled Cyclotron Facility at the NSCL. Shown are the K500 cyclotron, K1200 cyclotron, A1900 fragment separator, and the experimental end station in the S2 vault. More details can be seen in Fig. A.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.2: Experimental end station in the S2 vault for NSCL experiment e12001. The secondary beam from the A1900 fragment separator enters from the . . . . . . left side of the picture. More details can be seen in Fig. A.2. Figure 3.3: Overview of the electronics setup for NSCL experiment e12001. All de- tectors are shown along with trigger conditions. More details can be seen in Fig. A.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 73 76 76 81 82 83 xv Figure 3.4: The two silicon PIN detectors used for NSCL experiment e12001. These were installed in the cross flange (labeled in the picture) with a specific rotation angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.5: The implantation station used for NSCL experiment e12001. Shown are the double-sided silicon-strip detector (DSSD) and the silicon surface bar- rier detector (veto). More details can be seen in Fig. A.5. . . . . . . . . Figure 3.6: Diagram of the circuit board that was an intermediate stage between the DSSD and the dual-gain preamplifiers. More details can be seen in Fig. A.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.7: Traces recorded for strips of the low-gain stage of the DSSD. Left panel: A clipped signal for strip number 4 on the front side. Right panel: A non-clipped signal for strip number 7 on the front side. One clock tick . . . . . . . . . . . . . . . . . . . . . . . . . . . equals 10 nanoseconds. Figure 3.8: DSSD strip and logic signals used to create the front-back coincidence external trigger. The annotations describe either signals of the same color or a timing parameter. See main text for details. . . . . . . . . . . . . . Figure 3.9: The SuN detector during NSCL experiment e12001. Also shown are cables for the implantation station inside of SuN, and the circuit board and dual- gain preamplifiers for the DSSD. More details can be seen in Fig. A.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.10: Calibrations for each segment of SuN for NSCL experiment e12001. For a given segment label, the letter indicates the top or bottom half of SuN (“B” for bottom and “T” for top), and the number indicates the segment number. For example, the label T2 means segment 2 of the top half of SuN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.11: Residual plots for the SuN segment calibrations for NSCL experiment e12001. For a given segment label, the letter indicates the top or bottom half of SuN (“B” for bottom and “T” for top), and the number indicates the segment number. For example, the label T2 means segment 2 of the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . top half of SuN. 86 88 90 91 94 95 98 99 Figure 3.12: Particle identification spectrum for ions implanted in the DSSD for ex- periment e12001. The energy loss is from PIN1 and the time of flight is between PIN1 and I2N. This spectrum has uncorrected time of flight on the x-axis. Therefore only individual elements can be identified, not isotopes of a given element. . . . . . . . . . . . . . . . . . . . . . . . . . 102 xvi Figure 3.13: Top panel: Position at the I2 scintillator vs. time of flight between PIN1 and I2N for Zr isotopes implanted in the DSSD. Each band is a separate isotope of Zr. This spectrum has uncorrected time of flight on the x-axis. Bottom panel: Same as the top panel except the x-axis is corrected time of flight. In this panel, an isotope has the same time of flight regardless of position at the I2 scintillator. The horizontal gap in intensity around the value of 45 that is present in both panels is due to the I2 scintillator being damaged during NSCL experiment e12001. . . . . . . . . . . . . . 103 Figure 3.14: Particle identification spectrum for ions implanted in the DSSD during experiment e12001. The energy loss is from PIN1 and the time of flight is between PIN1 and I2N. This spectrum has corrected time of flight on the x-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Figure 3.15: Example of an isomeric transition used for particle identification. Hydrogen- 40Zr39+ was a charge-state contaminant of the fully stripped ion of like 99 interest 102 40 Zr40+. The nuclide 99Zr has an excited state at 252 keV with a lifetime of 293 ns [24]. The excited state deexcites by emitting two γ rays with energies of 130 keV and 122 keV. The individual γ rays and summed energy were detected in the TAS spectrum (left panel) and sum- of-segments spectrum (right panel) for ions implanted into the DSSD using the particle identification gate for 102Zr. The other counts in the spec- tra are from room background and Bremsstrahlung radiation, which is emitted as the ions slow down and stop in the DSSD. . . . . . . . . . . 105 Figure 3.16: Top panel: Spatial distribution of implantation events in the DSSD. Bot- tom panel: Average time in seconds between consecutive implantations for each pixel of the DSSD. The average time for central pixels is approx- imately 12 seconds. The average time gradually increases when moving from central pixels to peripheral pixels. . . . . . . . . . . . . . . . . . . 109 Figure 3.17: Top panel: Spatial distribution of decay events in the DSSD. Bottom panel: The average number of decay events observed in the DSSD within a one second time interval. For central pixels, there is on average approx- imately 0.25 observed decays per second. The average number gradually decreases when moving from central pixels to peripheral pixels. . . . . . 110 Figure 3.18: Correlation logic used in NSCL experiment e12001. The end result is a correlation event that has the particle identification information associ- ated with the implantation event and the β-delayed radiation information associated with the decay event. A “SuN only” event mostly refers to room background radiation, but may also be from the γ decay of an im- plantation in an isomeric state that γ decays outside of the coincidence time window of the implantation event. . . . . . . . . . . . . . . . . . . 112 xvii Figure 3.19: Spatial distribution of correlation events in the DSSD with a correlation time window of one second. Because a single pixel correlation field was used, the spatial distribution of correlated implantations and correlated decays is the same. This means there is only one spatial distribution of correlation events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Figure 3.20: Sequence of events resulting in a random correlation. A subset of pixels in the DSSD is shown for different events at times t1, t2, and t3 (with t1 < t2 < t3). At time t1, an ion is implanted (Implant A) in a pixel. At time t2, another ion is implanted (Implant B) in the same pixel as Implant A. At time t3, Implant A undergoes β decay. The decay event is localized to the same pixel as Implant A and Implant B. If decay events are only correlated to the most recent implantation event in the correlation field, then the decay is incorrectly correlated to Implant B. This results in a random correlation in which an incorrect, shorter decay time is assigned to Implant B. Any β-delayed radiation from the decay is also incorrectly assigned to Implant B. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Figure 3.21: Sequence of events resulting in a random correlation. A subset of pixels in the DSSD is shown for different events at times t1, t2, and t3 (with t1 < t2 < t3). At time t1, an ion is implanted (Implant A) in a pixel. At time t2, another ion is implanted (Implant B) in a pixel that is different from Implant A. At time t3, Implant A undergoes β decay. The maximum energy deposition of the decay event occurs in a pixel that is different from Implant A but the same as Implant B. If decay events are only correlated to the most recent implantation event in the correlation field, then the decay is incorrectly correlated to Implant B. This results in a random correlation in which an incorrect, shorter decay time is assigned to Implant B. Any β-delayed radiation from the decay is also incorrectly assigned to Implant B. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Figure 3.22: Visualization of the geant4 simulation for the SuN detector (left panel) and the implantation station at the center of SuN (middle and right pan- els). The green lines are γ-ray tracks and the red lines are β-decay electron tracks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Figure 3.23: Standard deviation in the detected energy of SuN’s NaI(Tl) crystals as a function of γ-ray energy. The points correspond to experimental data and the curve is the best fit function that was implemented in the geant4 simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 xviii Figure 3.24: Comparison of experimental and simulated sum-of-segments spectra for 60Co. Normalized room background has been subtracted from the ex- perimental spectrum. The experimental spectrum is shown with a black, solid line. The simulation using the resolution function from Ref. [25] is shown with a red, dashed line. This simulation does not include a gradual threshold for the segments of SuN. The simulation using the resolution function displayed in Fig. 3.23 is shown with a blue, dotted line. This simulation does include a gradual threshold for the segments of SuN. The effect of the gradual threshold can be seen at low energies (less than 100 keV). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Figure 3.25: Analysis pipeline used to extract the β-decay feeding intensity distribution and Gamow-Teller transition strength distribution in NSCL experiment e12001. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Figure 3.26: Decay curve for 99Y. The selection of events in this decay curve is de- scribed in Table 3.6 and Sec. 3.3.1.1. . . . . . . . . . . . . . . . . . . . . 134 Figure 3.27: β-delayed γ-ray spectra for decay events correlated to 101Zr implantations with a correlation time window of one second. The backward-time corre- lations (random background) have been subtracted from the forward-time correlations. The top panel, labeled (a), shows the TAS spectrum, and the red, cross hatches indicate the selection of events used to examine the individual γ rays in the sum-of-segments spectrum in the bottom panel. The bottom panel, labeled (b), shows the sum-of-segments spectrum only for certain events in the TAS spectrum as indicated in the top panel. In- dividual γ rays identified within the energy resolution of SuN are labeled. In (b), the red, cross hatches indicate the selection of events used to create the decay curve for 101Zr, which is shown in Fig. 3.28. . . . . . . . . . . 136 Figure 3.28: Decay curve for 101Zr. The selection of events in this decay curve is described in Table 3.6 and shown in Fig. 3.27. . . . . . . . . . . . . . . . 137 Figure 3.29: Decay curve for 102Zr. The selection of events in this decay curve is described in Table 3.6 and Sec. 3.3.1.3. . . . . . . . . . . . . . . . . . . . 138 Figure 3.30: Decay curve for 102mNb. The selection of events in this decay curve is described in Table 3.6 and Sec. 3.3.1.4. . . . . . . . . . . . . . . . . . . . 140 Figure 3.31: Decay curve for 103Nb. The selection of events in this decay curve is described in Table 3.6 and Sec. 3.3.1.5. . . . . . . . . . . . . . . . . . . . 141 Figure 3.32: Decay curve for 104mNb. The selection of events in this decay curve is described in Table 3.6 and Sec. 3.3.1.6. . . . . . . . . . . . . . . . . . . . 144 xix Figure 3.33: Decay curve for 109Tc. The selection of events in this decay curve is described in Table 3.6 and Sec. 3.3.1.7. . . . . . . . . . . . . . . . . . . . 145 Figure 3.34: Comparison of experimental (black, solid line) and reconstructed (blue, solid line) spectra from the β decay of 101Zr for the TAS spectrum (top panel), sum-of-segments spectrum (middle panel), and multiplicity spec- trum (bottom panel). The experimental spectra were obtained by corre- lating decay events to 101Zr implantations with a correlation time window of one second. Contamination from random correlations and the decay of the daughter has been subtracted from the experimental spectra. There is a label for the ground-state-to-ground-state Q value in the TAS spectrum for the β decay of 101Zr at 5726 keV [16]. . . . . . . . . . . . . . . . . . 151 Figure 3.35: Left panel: Reduced χ2 global as a function of ground-state-to-ground-state transition probability for the β decay of 101Zr. The inset shows an en- larged view of the minimum. Right panel: Initial number of nuclei as a function of ground-state-to-ground-state transition probability for the β decay of 101Zr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Figure 3.36: Comparison of experimental and theoretical cumulative β-decay feeding intensity distributions and cumulative B(GT) distributions for the β decay of 101Zr. The upper panels show cumulative β-decay feeding intensity. The lower panels show cumulative B(GT). The left panels contain QRPA 1 calculations. In the left panels, the present work (black, solid line, with uncertainty in orange shading) is compared to QRPA 1 calculations assuming the shape of the ground state of the parent is oblate (red, dashed line) and prolate (blue, dotted line). The right panels contain QRPA calculations commonly used in r-process reaction network calculations. In the right panels, the present work (black, solid line, with uncertainty in orange shading) is compared to QRPA 2 (cyan, dash-dotted line) and QRPA 3 (green, dotted line). The lower right panel does not contain a QRPA 2 calculation. Some panels may contain an arrow indicating the ground-state-to-ground-state Q value for the β decay of 101Zr at 5726 keV [16]. The experimental and theoretical half-lives T1/2 are provided in parentheses. If the quadrupole deformation parameter β2 was provided with a theoretical calculation, that is provided in the parentheses. . . . 158 xx Figure 3.37: Comparison of experimental (black, solid line) and reconstructed (blue, solid line) spectra from the β decay of 102Zr for the TAS spectrum (top panel), sum-of-segments spectrum (middle panel), and multiplicity spec- trum (bottom panel). The experimental spectra were obtained by corre- lating decay events to 102Zr implantations with a correlation time window of one second. Contamination from random correlations and the decay of the daughter has been subtracted from the experimental spectra. There is a label for the ground-state-to-ground-state Q value in the TAS spectrum for the β decay of 102Zr at 4717 keV [16]. . . . . . . . . . . . . . . . . . 161 Figure 3.38: Left panel: Reduced χ2 global as a function of ground-state-to-ground-state transition probability for the β decay of 102Zr. The inset shows an en- larged view of the minimum. Right panel: Initial number of nuclei as a function of ground-state-to-ground-state transition probability for the β decay of 102Zr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Figure 3.39: Comparison of experimental and theoretical cumulative β-decay feeding intensity distributions and cumulative B(GT) distributions for the β decay of 102Zr. The upper panels show cumulative β-decay feeding intensity. The lower panels show cumulative B(GT). The left panels contain QRPA 1 calculations. In the left panels, the present work (black, solid line, with uncertainty in orange shading) is compared to QRPA 1 calculations assuming the shape of the ground state of the parent is oblate (red, dashed line) and prolate (blue, dotted line). The right panels contain QRPA calculations commonly used in r-process reaction network calculations. In the right panels, the present work (black, solid line, with uncertainty in orange shading) is compared to QRPA 2 (cyan, dash-dotted line) and QRPA 3 (green, dotted line). The lower right panel does not contain a QRPA 2 calculation. Some panels may contain an arrow indicating the ground-state-to-ground-state Q value for the β decay of 102Zr at 4717 keV [16]. The experimental and theoretical half-lives T1/2 are provided in parentheses. If the quadrupole deformation parameter β2 was provided with a theoretical calculation, that is provided in the parentheses. . . . 168 Figure 3.40: Comparison of experimental (black, solid line) and reconstructed (blue, solid line) spectra from the β decay of 109Tc for the TAS spectrum (top panel), sum-of-segments spectrum (middle panel), and multiplicity spec- trum (bottom panel). The experimental spectra were obtained by correlat- ing decay events to 109Tc implantations with a correlation time window of one second. Contamination from random correlations has been subtracted from the experimental spectra. There is a label for the ground-state-to- ground-state Q value in the TAS spectrum for the β decay of 109Tc at 6456 keV [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 xxi Figure 3.41: Left panel: Reduced χ2 global as a function of ground-state-to-ground-state transition probability for the β decay of 109Tc. The inset shows an en- larged view of the minimum. Right panel: Initial number of nuclei as a function of ground-state-to-ground-state transition probability for the β decay of 109Tc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Figure 3.42: Comparison of experimental and theoretical cumulative β-decay feeding intensity distributions and cumulative B(GT) distributions for the β decay of 109Tc. The upper panels show cumulative β-decay feeding intensity. The lower panels show cumulative B(GT). The left panels contain QRPA 1 calculations. In the left panels, the present work (black, solid line, with uncertainty in orange shading) is compared to QRPA 1 calculations assuming the shape of the ground state of the parent is oblate (red, dashed line) and prolate (blue, dotted line). The right panels contain QRPA calculations commonly used in r-process reaction network calculations. In the right panels, the present work (black, solid line, with uncertainty in orange shading) is compared to QRPA 2 (cyan, dash-dotted line) and QRPA 3 (green, dotted line and purple, dashed line). The lower right panel does not contain a QRPA 2 calculation. Some panels may contain an arrow indicating the ground-state-to-ground-state Q value for the β decay of 109Tc at 6456 keV [16]. All panels contain an arrow indicating the one-neutron separation energy Sn of the daughter 109Ru at 5148 keV [16]. The experimental and theoretical half-lives T1/2 are provided in parentheses. If the quadrupole deformation parameter β2 was provided with a theoretical calculation, that is provided in the parentheses. . . . 178 Figure A.1: Schematic layout of the Coupled Cyclotron Facility at the NSCL. Shown are the K500 cyclotron, K1200 cyclotron, A1900 fragment separator, and the experimental end station in the S2 vault. . . . . . . . . . . . . . . . 186 Figure A.2: Experimental end station in the S2 vault for NSCL experiment e12001. The secondary beam from the A1900 fragment separator enters from the left side of the picture. . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Figure A.3: Overview of the electronics setup for NSCL experiment e12001. All detec- tors are shown along with trigger conditions. This figure may be viewed together with Fig. A.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Figure A.4: Overview of the NIM crates and other equipment for NSCL experiment e12001. This figure may be viewed together with Fig. A.3. . . . . . . . 189 Figure A.5: The implantation station used for NSCL experiment e12001. Shown are the double-sided silicon-strip detector (DSSD) and the silicon surface bar- rier detector (veto). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 xxii Figure A.6: Diagram of the circuit board that was an intermediate stage between the DSSD and the dual-gain preamplifiers. . . . . . . . . . . . . . . . . . . 191 Figure A.7: The SuN detector during NSCL experiment e12001. Also shown are cables for the implantation station inside of SuN, and the circuit board and dual- gain preamplifiers for the DSSD. . . . . . . . . . . . . . . . . . . . . . . 192 Figure A.8: The chamber in the A1900 fragment separator that contains the Image 2 scintillator. Shown are the Image 2 scintillator, achromatic wedges, and the slits that control the momentum acceptance. . . . . . . . . . . . . . 193 xxiii Chapter 1 Introduction 1.1 Nuclides The atomic nucleus is composed of nucleons (protons and neutrons). A nuclide is defined by a unique combination of proton number (or atomic number, Z) and neutron number (N). The notation for a nuclide is A Z XN where A is the mass number (number of nucleons), Z is the atomic number, X is the chemical symbol for the element (defined by the atomic number), and N is the neutron number. Isotopes are nuclides that contain the same atomic number but different neutron numbers, isotones are nuclides that contain the same neutron number but different atomic numbers, and isobars are nuclides that contain the same mass number. Figure 1.1 shows the chart of the nuclides, which is a common graphic representation to display unstable and stable nuclides. Out of the more than 7000 nuclides that are predicted to exist, only a little more than 3000 have actually been observed. Of those nuclides that have been observed, less than 300 are stable. The rest are unstable and will decay by various processes until reaching a stable nuclide. The stable nuclides form the valley of stability. In atoms, completely filled shells of electrons correspond to enhanced stability. Com- pletely filled electronic shells correspond to the noble gases, where the first ionization energy (the energy required to remove an electron from the atom) is a local maximum. Immedi- ately following (meaning, increasing Z by 1) a noble-gas element, there is a relatively large 1 Figure 1.1: Chart of the nuclides. Each shaded cell is an individual nuclide, defined by a unique combination of atomic number (Z) and neutron number (N). The stable nuclides are black, the unstable nuclides that are experimentally known to exist are dark gray, and the unstable nuclides that are predicted to exist according to the FRDM (2012) [3] mass model are light gray. An unstable nuclide is predicted to exist if both the one-proton separation energy (the energy required to remove a proton from the nucleus) and one-neutron separation energy (the energy required to remove a neutron from the nucleus) are greater than zero. In other words, an unstable nuclide is predicted to exist if the spontaneous emission of nucleons is energetically forbidden. The black, dotted lines indicate magic numbers that correspond to closed shells of nucleons. 2 050100150200Neutron Number, N020406080100120140Atomic Number, ZN = 2N = 8N = 20N = 28N = 50N = 82N = 126Z = 2Z = 8Z = 20Z = 28Z = 50Z = 82Stable NuclideUnstable Nuclide (Observed)Unstable Nuclide (Predicted) decrease in the first ionization energy. A similar phenomenon is also observed for nucleons. Nucleons fill single-particle states, and groups of single-particle states with similar energies form shells. Separate shells exist for protons and neutrons. Completely filled shells corre- spond to enhanced stability. Immediately following a completely filled shell (a closed shell), there is a relatively large decrease in the amount of energy required to remove a nucleon from the nucleus. For example, at a neutron closed shell, the one-neutron separation energy (the energy required to remove a neutron from the nucleus) is a local maximum. Immediately following the neutron closed shell, there is a relatively large decrease in the one-neutron sep- aration energy. Similarly, the neutron capture cross section (the probability for the nucleus and a neutron outside of the nucleus to merge together) at a neutron closed shell is relatively small. The nuclear shell model [26, 27, 28, 29, 30, 31] describes the configuration of single- particle states for nucleons and successfully predicts where closed shells occur for protons and neutrons. Closed shells for protons and neutrons occur at “magic numbers” and are labeled in the chart of the nuclides in Fig. 1.1. The magic numbers for protons and neutrons are 2, 8, 20, 28, 50, and 82. There is an additional magic number for neutrons at N = 126. These magic numbers can change and evolve far from stability. 1.2 Abundances One goal within the field of nuclear astrophysics is to explain the origin of the nuclides in the solar system. A necessary first step in achieving that goal is making a detailed inventory of the nuclides that exist in the solar system. One example of an inventory that scientists use is an abundance distribution (or abundance pattern). The solar system abundance pattern 3 displays the amount of each element or isobar in the solar system. The amounts (or abundances) of the elements in the solar system are typically obtained from two independent and complementary sources [32]. One source for elemental abun- dances is absorption spectroscopy of the Sun’s photosphere. In this case, the presence of an absorption line in the absorption spectrum indicates the presence of an element (different elements have different absorption lines), and the magnitude of the absorption line leads to inference of the abundance (for example, a relatively large magnitude indicates a relatively large abundance). Because the Sun accounts for almost all of the mass in the solar system, the abundance pattern of the Sun is considered to be representative for the entire solar system. In addition, the current abundances from absorption spectroscopy are believed to reflect the abundances at the formation of the solar system [32]. Another source for elemen- tal abundances is mass spectrometry of meteorites called CI chondrites (the “C” stands for “carbonaceous” and “I” indicates the geological type locality). Out of the different types of meteorites, CI chondrites have been modified the least by chemical and physical processes since the formation of the solar system [32]. Only five CI chondrites have been identified [4]. The elemental abundances obtained from CI chondrites are generally more accurate than those obtained from absorption spectroscopy [32, 4]. Nevertheless, the abundances obtained from both sources are generally in good agreement. For example, the abundances of 56 elements can be obtained from both sources. Out of the common 56 elemental abundances, the relative abundances of 41 elements from both sources agree within 15% [4]. For each element, a recommended abundance is chosen from one of the two sources, an average value of both sources, or a theoretical value [4]. Once the elemental abundances have been obtained, isotopic abundances of the solar system are obtained using isotopic compo- sitions as found on Earth (see Sec. 2.5 of Ref. [4]). For example, Ref. [33] contains these 4 isotopic compositions. Finally, one way of expressing abundances is with the cosmochemical scale, which normalizes silicon to 106 atoms (see, for example, Fig. 1.2). 1.3 Nucleosynthesis Nucleosynthesis refers to all the different processes that produce nuclides. Many nucle- osynthesis processes were first outlined in 1957 by Burbidge, Burbidge, Fowler, and Hoyle (referred to as the B2FH paper) [34]. While the nucleosynthesis processes described in the B2FH paper have been revised since 1957, this seminal work provides a foundation for the current theory of nucleosynthesis. The top panel of Fig. 1.2 shows the solar system abundance pattern. All features in the abundance pattern can be explained with different nucleosynthesis processes. Big Bang nucleosynthesis produced mostly hydrogen and helium, and small amounts of 2H, 3He, and 7Li. Nuclear fusion inside of stars is mostly responsible for the production of nuclides with 12 ≤ A ≤ 56. Material undergoing nuclear statistical equilibrium and then cooling (as happens in type Ia supernovae and core-collapse supernovae) is mostly responsible for the production of nuclides within the iron peak (50 (cid:46) A (cid:46) 62). The nucleosynthesis processes that produce heavier nuclides will be discussed in the following section. 1.3.1 Nucleosynthesis Beyond the Iron Peak Beyond the iron peak, the Coulomb barrier is insurmountably large and fusion is endother- mic. Nucleosynthesis beyond the iron peak instead proceeds with a γ-induced process and two neutron-induced processes. These three processes do not have a Coulomb barrier because γ rays and neutrons are electrically neutral. Historically, these three processes have received 5 Figure 1.2: Abundances of nuclides in the solar system and the processes responsible for their production. The top panel shows the decomposition of the abundances by both parity of mass number and production mechanism. In the top panel, various nuclides are labeled. The bottom panel shows the decomposition of the abundances beyond the iron peak into the individual abundance patterns of the p process, s process, and r process. In the bottom panel, different features arising from nuclear structure are labeled in the abundance patterns of the s process and r process. Both panels use the cosmochemical scale, which normalizes silicon to 106 atoms. The abundance data is from Ref. [4] (solar system), Ref. [5] (p process), Ref. [6] (s process), and Ref. [7] (r process). 6 050100150200250Mass Number, A1061041021001021041061081010Abundance (Si = 106 atoms)11H21H32He42He63Li94Be105B115B126C168O4020Ca5626Fe"iron peak"23290Th23592U23892Usolar system (even A)solar system (odd A)Big Bangcosmic ray spallationstellar fusionnuclear statistical equilibrium- and n-induced processes80100120140160180200Mass Number, A106105104103102101100101102Abundance (Si = 106 atoms)A88A138A208A80A130A195rare-earth peaksolar system p processsolar system s processsolar system r process (residuals) the most attention because they are most likely the dominating nucleosynthesis processes beyond the iron peak. However, there may be additional processes. The γ-induced process is the p process [5], and the two neutron-induced processes are the slow neutron-capture process (s process) [35, 36] and rapid neutron-capture process (r process) [7]. The solar system abundance patterns of these three processes are shown in the bottom panel of Fig. 1.2. A schematic illustration of the operation of these three processes in the chart of the nuclides is shown in Fig. 1.3. Regarding possible additional processes, some authors have proposed, for example, the intermediate neutron-capture process (i process) [37]. The p process is responsible for producing the 35 neutron-deficient stable nuclides that cannot be produced by the s process or r process. During the p process, photodisintegration reactions occur on existing seed nuclei. The photodisintegration reactions include (γ, p), (γ, n), and (γ, α) reactions. Because these γ-induced reactions are integral to the p process, some authors instead use the term γ process instead of p process. The seed nuclei are produced in the s process and/or r process. As shown in Fig. 1.3, two of the 35 nuclides that can only be produced in the p process are 78Kr and 84Sr. In this case, 78Kr is produced by a series of (γ, n) reactions on the seed nucleus 80Kr. Similarly, 84Sr is produced by a series of (γ, n) reactions on the seed nucleus 86Sr. The schematic illustration in Fig. 1.3 neglects the possibility of 78Kr and 84Sr being produced in a more complex series of photodisintegration reactions and β+ decay. As shown in the bottom panel of Fig. 1.2, the contribution from the p process to the solar system abundance pattern is much less than the contribution from the s process or r process. However, the p process is an active area of research because uncertainties exist in the astrophysical site as well as the nuclear physics input. Concerning the nuclear physics, uncertainties exist in the cross sections (the probability for a reaction to occur) for the 7 Figure 1.3: Schematic illustration of the operation of the p process (green arrows), s process (cyan arrows), and r process (orange arrows) for a region of the chart of the nuclides. The r process shows the operation during the neutron flux (orange, solid arrows) and decay back to the valley of stability after the neutron flux (orange, dashed arrows). The neutron closed shell at the magic number N = 50 is indicated with the red, shaded region. A cell that is labeled and outlined in black indicates a stable nuclide. A cell that is not labeled and outlined in gray indicates an unstable nuclide. This illustration ignores the possibility of “branching points” in the s process. 8 424446485052Neutron Number, N28303234363840Atomic Number, Z74Ge76Ge75As76Se77Se78Se80Se82Se79Br81Br78Kr80Kr82Kr83Kr84Kr86Kr85Rb87Rb84Sr86Sr87Sr88Sr89Y90Zr91Zr92Zr photodisintegration reactions. Researchers are currently trying to reduce these uncertainties by measuring cross sections for relevant (p, γ) and (α, γ) capture reactions [38, 39, 40, 41, 42, 43, 44]. From the capture-reaction measurements, the cross sections for the inverse reactions (the photodisintegration reactions) are obtained using the reciprocity theorem (also known as detailed balance). The bottom panel of Fig. 1.2 shows that the s process and r process contribute approx- imately equally to the solar system abundance pattern. Because the p-process contribution to the solar system abundance pattern is relatively small, the s process and r process each contribute approximately half to the total abundance of stable nuclides beyond the iron peak. However, these two neutron-induced processes occur in different astrophysical environments and on different timescales. The s process occurs in asymptotic giant branch (AGB) stars. This type of star is formed in a late phase of stellar evolution for low-mass stars. During this phase, neutrons are primarily produced from two sources. One source is the 13C(α, n)16O reaction, which results in a neutron density between 106 and 108 neutrons/cm3 [36]. The other source is the 22Ne(α, n)25Mg reaction, which results in a neutron density up to 1010 neutrons/cm3 [36]. The s process occurs on the order of thousands of years and is divided into a “weak” component (producing nuclides with A (cid:46) 90), a “main” component (producing nuclides with 90 (cid:46) A (cid:46) 205), and a “strong” component (producing nuclides with A (cid:38) 205). These three components differ in the average number of neutrons captured per seed nucleus. In the s process, the timescale for neutron capture τn is generally much longer than the timescale for β− decay τβ. That is, τn (cid:29) τβ. produced from neutron capture will undergo β− decay before capturing another neutron. In In other words, an unstable nuclide this sense, neutron capture is “slow” compared to β− decay, hence the name slow neutron- 9 capture process. A representative path for the s process is shown in Fig. 1.3. Some nuclides, such as 82Kr, can only be produced in the s process. These nuclides are shielded from the r process by stable nuclides. For 82Kr, the stable nuclide 82Se acts as a shield from the r process (see Fig. 1.3). The path of the s process proceeds close to the valley of stability, and is never more than one unit away from stability. As mentioned in Sec. 1.1, magic numbers of nucleons correspond to enhanced stability. At the magic number N = 50 in Fig. 1.3, the neutron capture cross section is relatively small. This means the probability of capturing a neutron is relatively small at N = 50. As a consequence, the abundance accumulates at magic numbers such as N = 50, resulting in peaks in the solar system abundance pattern. The neutron magic numbers at N = 50, 82, and 126 produce local maxima in the solar system abundance pattern that are attributed to the s process at A ≈ 88, ≈ 138, and ≈ 208, respectively (Fig. 1.2). Unlike the s process, in the r process the timescale for neutron capture τn is much shorter than the timescale for β− decay τβ. That is, τn (cid:28) τβ. In other words, an unstable nuclide may capture many neutrons before undergoing β− decay. In this sense, neutron capture is “rapid” compared to β− decay, hence the name rapid neutron-capture process. Rapid neutron capture requires a relatively large neutron density, with typical values ranging from 1024 to 1028 neutrons/cm3. A representative path for the r process is shown in Fig. 1.3. The path of the r process proceeds far from the valley of stability and involves many neutron- rich nuclides. Creating heavy, unstable nuclides with neutron capture during the r process occurs on the order of seconds. When there are no more neutrons to be captured, the unstable nuclides produced during the r process will undergo β− decay back to the valley of stability. An example of a nuclide that can only be produced in the r process is 76Ge (see Fig. 1.3). 10 As happens in the s process, the matter accumulates at neutron magic numbers during the r process. However, the reason for the accumulation in the two processes is different. In the environment in which the r process takes place, there are high-energy γ rays that can cause photodisintegration reactions. When a nuclide at N = 50 captures a neutron, a photodisintegration reaction has a very large probability to occur, bringing the nuclide back to the neutron magic number. This is because the photodisintegration cross section immediately after a neutron magic number is relatively large. As a result, the nuclides at neutron magic numbers act as “waiting points,” in that the r process must wait for β− decay in order to continue onto the next isotopic chain. At a given neutron magic number, the r process encounters a larger range of atomic numbers at smaller mass numbers compared to the s process (for example, see Fig. 1.3). This results in local maxima in the solar system abundance pattern that are broader and at smaller mass numbers compared to the s process. The neutron magic numbers at N = 50, 82, and 126 produce local maxima in the solar system abundance pattern that are attributed to the r process at A ≈ 80, ≈ 130, and ≈ 195, respectively (Fig. 1.2). Another structure that appears in the solar system abundance pattern that is attributed to the r process is the “rare-earth peak” at A ≈ 160 [45]. The formation of this structure is sensitive to physics at the late stages of the r process when nuclides undergo β− decay back to the valley of stability. Because the s process occurs close to the valley of stability, the relevant neutron capture cross sections can be experimentally measured. The nuclear physics for the s process is therefore generally well understood and theoretical models can successfully reproduce the s-process solar system abundance pattern. Given this success, the r-process contribution to the solar system abundance pattern is obtained by subtracting the s-process contribution from the solar system abundance pattern [46]. Due to this subtraction, the r-process solar 11 system abundance pattern is actually a “residual” abundance pattern. 1.4 r process Of all the nucleosynthesis processes described in Sec. 1.3, this dissertation focuses on the r process. The r process is one of the nucleosynthesis processes described in the B2FH paper [34]. In the six decades that have passed since this pioneering work was published, theoretical models are unable to reproduce the r-process solar system abundance pattern (bottom panel of Fig. 1.2). This inability is due to uncertainties in the astrophysical environment and the nuclear physics properties of nuclides relevant to the r process. 1.4.1 Astronomical Observations Astronomical observations of old stars in certain parts of the Milky Way Galaxy provide important information about the r process. One way to classify stars is by their composition. Specifically, stars may be classified by their metal content (or “metallicity”). In this case, a metal is any element heavier than hydrogen and helium. The first stars formed approximately 100 million years after the Big Bang [47]. These stars formed out of the hydrogen and helium from the Big Bang, were massive, quickly underwent stellar evolution, and exploded as supernovae [48]. The lifetime for these hypothetical “Population III” stars was only a few million years. When these stars exploded as supernovae, they enriched the interstellar medium with metals. Forming out of the enriched interstellar medium were “Population II” stars. Compared to Population III stars, Population II stars are less massive and have longer lifetimes (greater than 10 billion years) [48]. Due to their long lifetimes, some of these stars can still be observed this current day. Finally, there are “Population I” stars, 12 such as the Sun, which formed out of the interstellar medium further enriched by multiple nucleosynthesis events from Population II stars. Population I stars therefore have a higher metal content than Population II stars. In general, metallicity correlates with the age of a star. The older the star, the lower the metallicity for that star. The younger the star, the higher the metallicity for that star. Furthermore, stars are found in different parts of the Milky Way Galaxy, which is organized into a flat disk, a spherical bulge at the center, and a surrounding spherical halo. The halo of the Milky Way Galaxy contains Population II stars. These stars are called metal-poor halo stars [6, 48, 49, 47, 50]. They are metal-poor because they contain less than 1% of the Sun’s iron abundance. A small subset of these metal-poor halo stars show an enrichment or enhancement in the abundances of neutron-capture elements (Z > 30) relative to non-neutron-capture elements (Z < 30). For example, approximately 3-5% show a strong enhancement of r-process elements (known as r-II stars) and approximately 14% show a mild enhancement of r-process elements (known as r-I stars) [51]. Figure 1.4 shows the elemental abundances for ten metal-poor halo stars (specifically, r-I and r-II stars) compared to the r-process solar system abundance pattern. For 56 ≤ Z < 83, the relative abundance patterns for these r-I and r-II stars agree with each other and contain the same relative abundances as the r-process solar system abundance pattern. That is, the r-process solar system abundance pattern can be scaled to match the abundance patterns for the r-I and r-II stars. More examples of this phenomenon for r-I stars may be found in Ref. [52]. This similarity is remarkable given that these stars have different formation histories. Recall that the Sun is relatively young and formed out of the interstellar medium that was enriched by many nucleosynthesis events. Meanwhile, the r-I and r-II stars are relatively old, scattered throughout the halo, and formed out of the interstellar medium that was enriched by only 13 one or two nucleosynthesis events. The similarity in the abundance patterns suggests that the r process is a “universal” process and produces a “universal” abundance pattern (for 56 ≤ Z < 83). In other words, regardless of when and where the r process occurs, the r process operates in a consistent manner and always produces the same elements in the same relative amounts (for 56 ≤ Z < 83). While the relative elemental abundances for r-I and r-II stars agree with each other and the r-process solar system abundance pattern for 56 ≤ Z < 83, there is more scatter for the lighter neutron-capture elements (Z < 49). Possible explanations for this scatter include observational uncertainties, multiple sites or components for the r process [53, 54], or additional nucleosynthesis processes [55, 56]. Regarding the possibility of multiple sites or components for the r process, a “weak r process” would produce nuclides with A (cid:46) 130 (corresponding to the lighter neutron-capture elements) and a “main r process” would produce nuclides with A (cid:38) 130 (corresponding to the heavier neutron-capture elements) [6]. This dissertation focuses on the mass region relevant to the weak r process. The r-I and r-II stars provide other important information about the r process. One detail is that because r-I and r-II stars are relatively old, the r process was happening early in the history of the universe in order to enrich the interstellar medium out of which those stars formed. None of the enrichment in r-I and r-II stars could have been from the s process because not enough time had passed to reach the AGB phase necessary for s process nucleosynthesis. Another detail is that because only approximately 3-5% of metal-poor halo stars are r-II stars, the r process is a relatively rare process. 14 Figure 1.4: Elemental abundances in ten metal-poor halo stars (specifically, r-I and r-II stars). Markers of the same type and color correspond to the same r-I or r-II star. The solid blue lines are the r-process solar system abundance pattern. The only difference between the blue lines is a scaling factor. The scaling factor is obtained by normalizing the europium abundance in the r-process solar system abundance pattern to that in the r-I or r-II star. Europium is chosen due to being an r-process element. Figure adapted from Ref. [8]. 15 1.4.2 Astrophysical Sites The r process requires a neutron-rich environment. Of the many sites that have been pro- posed for the site of the r process, two sites have received the most attention. These two sites are core-collapse supernovae and the compact-object mergers [57]. 1.4.2.1 Core-collapse Supernovae A core-collapse supernova occurs when the iron core of a massive star undergoes gravitational collapse. The remnant from this event is a proto-neutron star, which cools by releasing a large amount of energy in the form of neutrinos. The neutrinos deposit energy on the surface of the proto-neutron star, which drives material off the surface of the proto-neutron star. This outflow of neutrinos and material from the surface of the proto-neutron star is called the neutrino-driven wind [58, 59]. The material driven from the surface is initially in the form of free nucleons. As the material expands and cools, some nucleons combine into α particles, which in turn combine to form seed nuclei for eventual neutron capture reactions. As the material further expands and cools, the seed nuclei rapidly capture neutrons from the large abundance of free neutrons, forming r-process nuclides. The neutrino-driven wind from a core-collapse supernova initially was favored as the site of the r process. One reason for being favored concerns the observations of metal-poor halo stars described in Sec. 1.4.1. The massive Population III stars quickly underwent stellar evo- lution, and some would have become core-collapse supernovae. If the r process occurred in the neutrino-driven winds from these core-collapse supernovae, then the interstellar medium would have quickly been enriched with r-process elements. This would explain the abun- dances that are observed in r-I and r-II stars. However, recent simulations of neutrino-driven winds have had difficulty in producing the most neutron-rich nuclides associated with the r 16 process [60]. These simulations have shown that the conditions do not appear to be suffi- ciently neutron-rich to produce nuclides associated with the third peak at A ≈ 195 (see Fig. 1.2). Instead, these simulations only produce nuclides with A (cid:46) 130 (associated with the lighter neutron-capture elements). Therefore, the neutrino-driven winds from core-collapse supernovae could be a site for the weak r process mentioned in Sec. 1.4.1. 1.4.2.2 Compact-object Mergers A compact object may be either a neutron star or a black hole. While the merging of a neutron star (NS) and black hole could be a potential site of the r process [61], this section will only consider the merging of two neutron stars (hereafter referred to as a NS-NS merger) [62]. Figure 1.5 shows a NS-NS merger. As the two neutron stars approach each other, they become deformed from gravity and neutron-rich material is ejected from the merging system. Gravitational waves are also generated. The ejected material is called dynamical ejecta [63]. One origin of dynamical ejecta is the tidal tails, as shown in Fig. 1.5. As the name implies, the tidal tails form as a result of tidal forces. Another origin of dynamical ejecta is the contact interface of the two neutron stars. In any case, the dynamical ejecta is flung out into space, and, similar to the case described for core-collapse supernovae in Sec. 1.4.2.1, the ejecta expands and cools, and seed nuclei are formed which can then rapidly capture neutrons [64]. The dynamical ejecta is very neutron-rich. Therefore, simulations of the r process in NS-NS mergers can easily produce nuclides associated with the third peak at A ≈ 195 (see Fig. 1.2). In addition, simulations show even heavier nuclides are produced. These heavier nuclides are susceptible to fission (spontaneous fission, neutron-induced fission, and β-delayed fission). When these nuclides fission, the resulting fragments can themselves capture neutrons 17 Figure 1.5: Snapshot of a simulation of a NS-NS merger. The color indicates the magnitude of the magnetic field (the lighter the color, the larger the magnitude of the magnetic field). The two neutron stars are in the center, surrounded by dynamical ejecta in the tidal tails. Figure adapted from Refs. [9, 10]. 18 and eventually become susceptible to fission. This repeating cycle of fission and neutron capture is called “fission recycling” or “fission cycling.” Simulations show that if fission cycling occurs, the r-process abundance pattern is “robust.” That is, the abundance pattern is insensitive to variations in properties of the merging system [65]. Initially, NS-NS mergers were not favored as the dominant site of the r process [66]. This was due to the presumably long time (approximately 100-1000 million years [66, 67]) to merge (or coalesce). Within this amount of time, the interstellar medium would not have been enriched with r-process elements to explain the abundances that are observed in r-I and r-II stars. However, recent studies have shown that even with long times to merge, NS-NS mergers can explain these observed abundances [68, 69]. Furthermore, a recent discovery made while studying ultra-faint dwarf galaxies provides evidence [70], and a recent discovery of a kilonova confirms [71] that NS-NS mergers are a site of the r process. These two recent discoveries are discussed below. The first recent discovery concerns ancient and ultra-faint dwarf galaxies, which orbit the Milky Way Galaxy. Reticulum II [70] is an ancient and ultra-faint dwarf galaxy, and the tenth such galaxy for which elemental abundances were obtained. While all the stars in the other nine galaxies do not show any enrichment in neutron-capture elements, seven of the nine stars in Reticulum II are r-II stars. These seven stars have abundances that follow the universal r-process abundance pattern [70]. These observations suggest that a rare and prolific r-process event was responsible for the enrichment observed in Reticulum II. The event was rare in that only one of ten of these galaxies have any form of enrichment. The event was prolific in that a large yield of r-process elements was produced in order for seven of the nine stars to be r-II stars. The rate and yield for this r-process event are incompatible with core-collapse supernovae as the site of the r process, but compatible with a NS-NS 19 merger [70]. Compared to NS-NS mergers, core-collapse supernovae are more frequent and produce smaller amounts of r-process elements. Therefore, if core-collapse supernovae were the dominant site of the r process, one would expect the amount of r-process enrichment to be the same in all ten ultra-faint dwarf galaxies. The second recent discovery concerns the observation of a kilonova. The r process pro- duces many unstable neutron-rich nuclides far from the valley of stability, which will even- tually undergo β− decay back to the valley of stability. As the nuclides undergo β− decay, energy will be released in different forms of radiation, such as electrons and γ rays. This radiation will deposit energy in the surrounding material, which will affect the black-body radiation from the surrounding material. The black-body radiation that is powered by the radioactive decay of nuclides created in the r process is called a kilonova [72]. A kilonova is therefore the electromagnetic counterpart to the emission of gravitational waves from the merging of two neutron stars. Recently, a kilonova was observed by various observatories [71] in coincidence with the detection of gravitational waves from the merging of two neutron stars [73]. The gravitational waves were detected by the LIGO-Virgo detector network. This event confirmed that NS-NS mergers are a site of the r process. One observable from a kilonova is the light curve. The light curve displays the luminosity of the kilonova as a function of time. Any observed light curve will be affected by the possible presence of lanthanides (57 ≤ Z ≤ 71) and actinides (89 ≤ Z ≤ 103) [74]. This is because lanthanides and actinides have a complex atomic structure, which creates a large number of absorption lines. The large number of absorption lines increases the opacity of the material, which increases the time for photons to diffuse out of the material and be observed. If the material contains lanthanides and actinides, the observed light curve will be relatively dim and peak at red wavelengths on the timescale of weeks. If the material does not contain 20 lanthanides and actinides, the observed light curve will be relatively bright and peak at blue wavelengths on the timescale of days. In the recent kilonova observation, both components were observed in the light curve [75]. The blue component was associated with dynamical ejecta from the collision interface, which is hot and less neutron-rich. The red component was associated with dynamical ejecta from the tidal tails, which is cold and very neutron- rich. The red component was also associated with a wind from an accretion disk that formed around the remnant from the merger. 1.4.3 Nuclear Physics Understanding whether NS-NS mergers are the only site, the dominant site, or one of multiple sites of the r process will require reducing the uncertainty in the nuclear physics. In this case, nuclear physics refers to all the nuclear physics properties of nuclides that participate in the r process. The nuclear physics properties include masses, fission properties, neutron capture cross sections, and β-decay properties. One way of assessing the uncertainty in the nuclear physics is with a reaction network calculation. Figure 1.6 shows the result of a reaction network calculation for the r process. The reac- tion network is called PRISM (Portable Routines for Integrated nucleoSynthesis Modeling) [11, 12]. The input to a reaction network is the astrophysical environment and the nuclear physics. The astrophysical environment determines important quantities such as the tem- perature and density as a function of time, which in turn affect the rate at which different nuclear processes occur during the r process. The astrophysical environment used in the reaction network calculation that produced Fig. 1.6 is a NS-NS merger. The nuclear physics also determines the rate at which different nuclear processes occur during the r process. At each time step in the reaction network calculation, the relative abundances of all the nuclides 21 are calculated, from which a relative abundance pattern can be created. The relative abun- dance pattern is then normalized and compared to the r-process solar system abundance pattern. If the astrophysical environment and nuclear physics are correct, the abundance pattern from the reaction network calculation should agree with the r-process solar system abundance pattern. Unlike the illustration in Fig. 1.3, Fig. 1.6 shows the full path of the r process. Each quadrant in Fig. 1.6 corresponds to a different time step during the reaction network calcu- lation. Each quadrant contains a top panel and bottom panel. The top panel compares the reaction network calculation abundance pattern with the r-process solar system abundance pattern. The bottom panel shows how the r process proceeds in the chart of the nuclides by showing the relative abundance of each nuclide. Each time step in Fig. 1.6 emphasizes the main stages of the r process that were described Sec. 1.3.1. In the upper left quadrant of Fig. 1.6, seed nuclei are exposed to an extreme flux of neutrons. In the upper right quadrant of Fig. 1.6, nuclei rapidly capture many neutrons before undergoing β− decay, and therefore move far from the valley of stability. This quadrant shows the accumulation of abundance at the neutron magic numbers, as discussed in Sec. 1.3.1. Because the astrophysical en- vironment used in this reaction network calculation is very neutron-rich, the path of the r process reaches the neutron drip line. The lower left quadrant of Fig. 1.6 shows that once the neutron flux ends, all the nuclei produced during the r process will undergo β− decay back to the valley of stability. Note that the neutron flux is exhausted in less than one second (the time is labeled in each quadrant), as discussed in Sec. 1.3.1. During this stage, energy is released during β− decay in different forms of radiation. This radiation powers the kilonova described in Sec. 1.4.2.2. Finally, the lower right quadrant of Fig. 1.6 shows that the nuclei will continue to undergo β− decay until reaching a stable or relatively long- 22 lived nuclide. Overall, as time progresses, the temperature and density (which are labeled in each quadrant) decrease because the material is expanding and cooling as described in Sec. 1.4.2.2. The last time step in the reaction network calculation is the lower right quadrant of Fig. 1.6. In the last time step, there are discrepancies between the abundance pattern from the reaction network calculation and the r-process solar system abundance pattern. A significant factor in the discrepancies is the uncertainty in the nuclear physics properties of nuclides that participate in the r process. Different nuclear processes that can occur during the r process are shown in Fig. 1.7 in terms of the abundance weighted timescale [12]. The abundance weighted timescale is defined as where τ is the timescale, λ is the rate, Y is the abundance, i runs over all nuclides, and j is a specific reaction or decay channel. Inspecting the abundance weighted timescale as a function of time reveals which reaction and decay channels are most important at any given time during the r process. The smaller the abundance weighted timescale, the more important that particular process is at that specific time during the r process. For example, at the beginning of the r process, neutron-induced reactions are the dominating processes and therefore have the lowest abundance weighted timescales. Meanwhile, at the end of the r process, β− decay is the dominant process and therefore has the lowest abundance weighted timescale. Figures 1.6 and 1.7 illustrate the complexity of modeling the r process and trying to reproduce the abundance pattern observed in the solar system. Figure 1.6 shows the r 23 τj = (cid:80)i (cid:80)i  , Yi Yiλij (1.1) Figure 1.6: Abundance pattern and path of the r process at different times for a NS-NS merger. Each quadrant corresponds to a different time step in a reaction network calculation. The reaction network is called PRISM [11, 12]. Each quadrant contains a top panel and bottom panel. The top panels show the absolute abundance pattern of the r process for the solar system and from the reaction network calculation. The abundances are expressed in the cosmochemical scale, which normalizes silicon to 106 atoms. The bottom panels show the chart of the nuclides, with stable nuclides in black, unstable nuclides that are experimentally known to exist in dark gray, and unstable nuclides that are predicted to exist according to the FRDM (2012) [3] mass model in light gray. The neutron magic numbers (N = 2, 8, 20, 28, 50, 82, 126) are indicated with a black, dashed line. The relative abundances of nuclides produced from the PRISM calculation are shown with shaded cells. Each quadrant has a label for time in units of seconds (t), temperature in units of 109 K (T9), and density in units of g/cm3 (ρ). 24 120130140150160170180190200Mass Number, A103102101100101Abundance (Si = 106 atoms)PRISMsolar system r process (residuals)050100150200250Neutron Number, N050100150Atomic Number, Zt = 4.24e-03 s, T9 = 2.00e+00, = 8.86e+07 g/cm31010109108107106105104103Abundance, Y(Z,N)120130140150160170180190200Mass Number, A103102101100101Abundance (Si = 106 atoms)PRISMsolar system r process (residuals)050100150200250Neutron Number, N050100150Atomic Number, Zt = 2.60e-01 s, T9 = 2.97e-02, = 4.89e+02 g/cm31010109108107106105104103Abundance, Y(Z,N)120130140150160170180190200Mass Number, A103102101100101Abundance (Si = 106 atoms)PRISMsolar system r process (residuals)050100150200250Neutron Number, N050100150Atomic Number, Zt = 9.54e-01 s, T9 = 8.12e-03, = 9.96e+00 g/cm31010109108107106105104103Abundance, Y(Z,N)120130140150160170180190200Mass Number, A103102101100101Abundance (Si = 106 atoms)PRISMsolar system r process (residuals)050100150200250Neutron Number, N050100150Atomic Number, Zt = 1.00e+08 s, T9 = 1.28e-03, = 3.95e-02 g/cm31010109108107106105104103Abundance, Y(Z,N) Figure 1.7: Abundance weighted timescales for important nuclear processes during the r process for a NS-NS merger. 25 102101100101102103104Time (s)108106104102100102104106, Abundance Weighted Timescale (s)neutron capture (n,)photodisintegration (,n) decay decay(n,2n)neutron-induced fission-delayed fissionspontaneous fission process involves thousands of nuclides, many of which are far from the valley of stability and have not been experimentally studied. Figure 1.7 shows the reaction and decay channels that must be known for each nuclide. Experimentalists will never be able to measure all the relevant nuclear physics properties of all nuclides that participate in the r process. This situation creates a reliance on theoretical models to accurately calculate the nuclear physics properties where experimental data are nonexistent and experiments are currently unfeasible. Of all the important nuclear physics properties that play an important role in the r process, this dissertation focuses on experimental measurements related to β− decay. These experimental measurements will constrain theoretical models and provide more confidence in their extrapolation far from the valley of stability. 1.5 β decay 1.5.1 β-decay Classification β decay is governed by the weak interaction and refers to three processes: electron capture : A Z XN + e− → A Z XN → A Z XN → A Z−1YN+1 + νe Z−1YN+1 + e+ + νe Z+1YN−1 + e− + ¯νe β+ decay : A β− decay : A (1.2) (1.3) (1.4) where A Z XN is the notation for a nuclide described in Sec. 1.1, e− is an electron, e+ is a positron, νe is an electron neutrino, and ¯νe is an electron antineutrino. In β decay, a neutron is converted to a proton, or vice versa. All three processes conserve the number of nucleons (A remains constant), so β decay connects isobars. Conventionally, the original nuclide 26 (cid:16)A Z XN(cid:17) is called the parent or mother, and the final nuclide (cid:16) A Z−1YN+1 or called the daughter. A Z+1YN−1(cid:17) is For a given nuclide, there are multiple levels with different amounts of energy (one ground state and multiple excited states). Therefore, β decay can connect different levels in the parent with different levels in the daughter. These different connections are called β-decay transitions. For a given β-decay transition, energy is released and shared between the final products (the daughter and leptons) as kinetic energy. The amount of energy released is electron capture : Q + Ex,p − Ex,d =(cid:2)m(cid:0)A β+ decay : Q + Ex,p − Ex,d =(cid:2)m(cid:0)A decay : Q + Ex,p − Ex,d =(cid:2)m(cid:0)A Z XN(cid:1) − m(cid:0) A Z XN(cid:1) − m(cid:0) A Z XN(cid:1) − m(cid:0) A Z−1YN +1(cid:1)(cid:3) c2 + Ex,p − Ex,d Z−1YN +1(cid:1) − 2me(cid:3) c2 + Ex,p − Ex,d Z+1YN−1(cid:1)(cid:3) c2 + Ex,p − Ex,d − β (1.5) (1.6) (1.7) where Q is the ground-state-to-ground-state Q value, Ex,p is the energy of the level in the parent, Ex,d is the energy of the level that is populated in the daughter, m(cid:16)A Z XN(cid:17) is the atomic mass of a nuclide, me is the mass of the electron, and c is the speed of light. A β-decay transition can only occur if Q + Ex,p − Ex,d is positive. Electron capture occurs for neutron-deficient nuclei. In electron capture, a proton cap- tures an electron from an atomic orbital. The proton is converted into a neutron and a monoenergetic neutrino is emitted. The electron that is captured is usually from one of the inner electronic shells. The vacancy from the captured electron is filled by an electron from an outer shell, resulting in the emission of X rays or Auger electrons. β+ decay also occurs for neutron-deficient nuclei. In β+ decay, a proton is converted into a neutron, and a positron and neutrino are emitted. The three-body final state (daughter, positron, neutrino) of β+ decay leads to a continuous distribution of positron (and neutrino) 27 kinetic energy that extends from zero up to Q + Ex,p − Ex,d. The kinetic energy of the recoiling daughter nucleus is negligible. Note that Eq. 1.6 implies that β+ decay is only possible if(cid:104)m(cid:16)A Z XN(cid:17) − m(cid:16) A Z−1YN +1(cid:17)(cid:105) c2 > 2mec2. β− decay occurs for neutron-rich nuclei and is therefore relevant to the r process and this dissertation. In β− decay, a neutron is converted into a proton, and an electron and antineutrino are emitted. The three-body final state (daughter, electron, antineutrino) of β− decay leads to a continuous distribution of electron (and antineutrino) kinetic energy that extends from zero up to Q + Ex,p − Ex,d. The kinetic energy of the recoiling daughter nucleus is negligible. This dissertation involves the study of only neutron-rich nuclei, and therefore any future reference to β decay will mean β− decay. The total angular momentum must be conserved in a β-decay transition (cid:126)Jp = (cid:126)Jd + (cid:126)Lβ + (cid:126)Sβ (1.8) where (cid:126)Jp is the total angular momentum of the parent, (cid:126)Jd is the total angular momentum of the daughter, (cid:126)Lβ is the orbital angular momentum of the leptons, and (cid:126)Sβ is the intrinsic spin angular momentum of the leptons. The electron and antineutrino both have an intrinsic spin angular momentum of 1/2. For a given β-decay transition, this leads to two possible values for Sβ. If the intrinsic spin angular momenta of the electron and antineutrino are antiparallel, then those spins couple to create Sβ = 0. If this occurs, the transition is a Fermi transition. If the intrinsic spin angular momenta of the electron and antineutrino are parallel, then those spins couple to create Sβ = 1. If this occurs, the transition is a Gamow-Teller transition. A β-decay transition is classified by the amount of orbital angular momentum carried off by the leptons. If Lβ = 0, the transition is classified as an allowed 28 Table 1.1: Classifications of β-decay transitions. Adapted from Ref. [1]. Superallowed Allowed Transition Type Lβ ∆π 0 No 0 No 1 Yes 2 No 3 Yes First forbidden Second forbidden Third forbidden ∆J 0 0, 1 0, 1, 2 1, 2, 3 2, 3, 4 transition. These transitions are most probable. Other transitions exist such as forbidden transitions. The word “forbidden” is a misnomer because these transitions do occur, but with a smaller probability compared to allowed transitions. The degree of being forbidden increases as Lβ increases. Whether or not there is a change in parity between the initial level in the parent and the final level in the daughter is determined with ∆π = (−1)Lβ . The selection rules for β-decay transitions are summarized in Table 1.1. 1.5.2 β-delayed γ-ray Emission A β-decay transition may populate an excited state in the daughter nucleus. Emission of a γ ray may occur when the excited state decays to a lower-energy state. This is an example of β-delayed γ-ray emission. γ-ray transitions connect an initial state to a final state in the same nucleus (in this case, the daughter). There may be multiple γ-ray transitions as the excited state decays to the ground state. With each γ-ray transition, the γ ray carries off an integer unit of angular momentum λ that can range from |(Ji − Jf )| ≤ λ ≤ |(Ji + Jf )| (1.9) where Ji is the total angular momentum of the initial state, Jf is the total angular momentum 29 Table 1.2: Classifications of γ-ray transitions. Adapted from Ref. [1]. Name Electric dipole Magnetic dipole Electric quadrupole Magnetic quadrupole Electric octupole Magnetic octupole Electric hexadecapole Magnetic hexadecapole Radiation Type λ ∆π 1 Yes 1 No 2 No 2 Yes 3 Yes 3 No 4 No 4 Yes E1 M1 E2 M2 E3 M3 E4 M4 of the final state, and λ is called the multipolarity. For a given γ-ray transition there can be a range of values for the multipolarity, but typically the lowest multipolarity is most probable. For a given multipolarity, whether or not there is a change in parity between the initial and final states determines the type (electric or magnetic) of the transition. The selection rules for γ-ray transitions are summarized in Table 1.2. 1.5.3 Internal Conversion Internal conversion is a process that may also occur when an excited state decays to a lower- energy state in the same nucleus (in this case the daughter). During this process, the excited nucleus interacts electromagnetically with an electron from an atomic orbital, which causes the electron to be emitted. The vacancy from the emitted electron is filled by an electron from an outer shell, resulting in the emission of X rays or Auger electrons. The energy of the emitted electron is EIC = (Ei − Ef ) − EBE (1.10) where EIC is the energy of the emitted electron (the internal conversion electron), Ei is the energy of the initial state, Ef is the energy of the final state, and EBE is the binding energy of the internal conversion electron. Unlike β-decay electrons, which have a continuous 30 distribution of kinetic energies, conversion electrons are monoenergetic. Internal conversion is characterized by an internal conversion coefficient α = TIC Tγ (1.11) where α is the internal conversion coefficient, TIC is the emission rate of the conversion electron, and Tγ is the emission rate of the γ ray. The internal conversion electron may be ejected from different atomic shells (for example, the K, L, or M shells), and the total internal conversion coefficient is defined as αtotal = αK + αL + αM + ... (1.12) Internal conversion coefficients may be calculated using the BrIcc program [13, 14] provided by the National Nuclear Data Center. Approximate values for the internal conversion coef- ficients may be calculated with α(Eλ) = Z3 n3 (cid:18) λ λ + 1(cid:19)(cid:18) e2 4π0c(cid:19)4(cid:18)2mec2 E (cid:19)λ+5/2 α(M λ) = Z3 n3 (cid:18) e2 4π0c(cid:19)4(cid:18)2mec2 E (cid:19)λ+3/2 (1.13) (1.14) where Z is the atomic number of the nucleus, n is the principal quantum number of the electron that is ejected, λ is the multipolarity of the transition, e is the charge of the electron, 0 is the permittivity of free space,  is the reduced Planck constant, c is the speed of light, me is the mass of the electron, and E is the transition energy [1]. Equation 1.13 is used when 31 Figure 1.8: Total internal conversion coefficients for ruthenium (Z = 44) for a range of tran- sition energies and multipolarities. The total internal conversion coefficients were obtained with the BrIcc program [13, 14] provided by the National Nuclear Data Center. The inset shows a zoomed-in view of the low transition energy region. the transition type is electric, while Eq. 1.14 is used when the transition type is magnetic. Inspecting Eqs. 1.13 and 1.14 shows that internal conversion will be significant for heavier (large Z) nuclei, lower-energy transitions, and higher-multipolarity transitions. The latter two features can be seen in Fig. 1.8, which shows the total internal conversion coefficients for ruthenium (Z = 44) for a range of transition energies and multipolarities. For a given type and multipolarity, as the transition energy decreases, the total internal conversion coefficient increases. For a given type and transition energy, as the multipolarity increases, the total internal conversion coefficient increases. 32 0100020003000400050006000Transition Energy (keV)1031011011031051071091011Total Internal Conversion Coefficient, totalRu (Z = 44)E1M1E2M2E3M3E4M4E5M50501001502001051021011041071010 1.5.4 β-delayed Neutron Emission β-delayed neutron emission occurs when a neutron-rich parent nucleus undergoes β decay and populates a neutron-unbound excited state in the daughter nucleus. This is energetically possible whenever the ground-state-to-ground-state Q value for the β decay is greater than the one-neutron separation energy in the daughter. The daughter may then emit a neutron and populate a level in the one-neutron daughter A−1 Z YN−1. This process is written as A Z XN → A−1 Z YN−1 + n (1.15) The two-body final state leads to a monoenergetic neutron, but note that different levels populated in the one-neutron daughter will lead to different monoenergetic neutrons. Any excited level populated in the one-neutron daughter will decay by emitting radiation. 1.5.5 β-decay Scheme Important information regarding the β decay of a nuclide is collected in a “decay scheme.” A decay scheme displays information about the β-decaying state in the parent and the states populated in the daughter from β decay. An example of a decay scheme is shown in the left panels of Fig. 1.9. This decay scheme will illustrate some of the topics that have already been mentioned regarding β decay. This decay scheme shows the β decay of a parent with Z protons and A nucleons to a daughter with Z+1 protons and A nucleons. In this example, only the ground state of the parent undergoes β decay (and not an excited state, such as a β-decaying isomeric state). The ground state of the parent has an energy (Ex), spin (J), and parity (π). The subscript “p” in (Ex, J π)p refers to the parent. For the ground state, Ex is zero. The ground-state-to-ground-state Q value is shown with the red arrows. The 33 left panels differ in that each one shows a different β-decay transition from the parent to the daughter. These different transitions are labeled with the blue arrows. The different states in the daughter have subscripts i, j, k, l, and m. Each β-decay transition has a β-decay feeding intensity Iβ. This is the probability of populating the state in the daughter during the β decay. As a probability distribution, the β-decay feeding intensity distribution is normalized to unity (1.0 or 100%). The bottom left panel in Fig. 1.9 is an example of a ground-state-to-ground-state tran- sition. All other panels on the left side show a β-decay transition that populates an excited state in the daughter. Any excited state will deexcite by emitting γ rays or internal con- version electrons. Decay schemes also show information about the deexcitation of excited states (such as branching ratios for γ-ray transitions between states), but that information is not shown in Fig. 1.9. The energy released in a given β-decay transition that is shared between the final products (daughter, electron, and antineutrino) as kinetic energy is shown with the green arrows. The green arrows represent the quantity Q − Ex where Q is the ground-state-to-ground-state Q value (red arrows) and Ex is the energy of the populated state in the daughter. In other words, the green arrows represent the quantity given by Eq. 1.7 (where, as mentioned earlier, the initial energy of the parent is assumed to be zero). As already mentioned regarding Eq. 1.7, for a given β-decay transition the kinetic energy shared between the final products is Q − Ex =(cid:104)m(cid:16)A Z XN(cid:17) − m(cid:16) A Z+1YN−1(cid:17)(cid:105) c2 = Ke + K¯νe + Kd ≈ Ke + K¯νe (1.16) (1.17) where Q is the ground-state-to-ground-state Q value (red arrows), m is the atomic mass, c 34 is the speed of light, Ke is the kinetic energy of the electron, K¯νe is the kinetic energy of the antineutrino, and Kd is the kinetic energy of the daughter nucleus. Equation 1.16 ignores the mass of the antineutrino and the difference in electron binding energies of the parent and daughter, while Eq. 1.17 ignores the kinetic energy of the daughter nucleus Kd (the daughter nucleus is massive compared to the electron and antineutrino and therefore has a negligible kinetic energy). The quantity Q − Ex (green arrows) is the maximum kinetic energy available to the electron or antineutrino. As already mentioned, the three-body final state of β decay leads to a continuous dis- tribution of electron (and antineutrino) kinetic energies. The continuous distribution of electron kinetic energies is obtained from Fermi’s theory of β decay (for example, see Sec. 8.3 of Ref. [1]). For a given β-decay transition on the left panels of Fig. 1.9, the right panels show the continuous distribution of electron kinetic energies. The distribution of electron kinetic energies is given by dN dKe = C(cid:16)K2 e + 2Kemec2(cid:17) 1 2 ((Q − Ex) − Ke)2(cid:16)Ke + mec2(cid:17) , (1.18) where dN dKe is the number of electrons per kinetic energy, and C is a constant. The distribution of electron kinetic energies is shown with the black solid lines in the right panels of Fig. 1.9. As already mentioned, the distribution is continuous and extends from 0 to Q − Ex (green arrows). These distributions are commonly called phase space distributions. Equation 1.18 neglects many effects in β decay, one being the Coulomb attraction of the positively charged daughter nucleus and negatively charged electron. This effect is accounted for with a Fermi function. Different expressions exist in the literature for the Fermi function (for example, Refs. [76, 77]) with all of them yielding approximately the same result (red, cyan, and blue 35 dotted lines in the right panels of Fig. 1.9). Multiplying the Fermi function and the phase space distribution yields the “corrected” electron kinetic energy distribution (red, cyan, and blue solid lines in the right panels of Fig. 1.9). In terms of physics, the Fermi function distorts the phase space distribution by shifting the electron kinetic energy distribution to smaller values. Each distribution has an average electron kinetic energy (cid:104)E(cid:105) (orange arrow in the right panels of Fig. 1.9). As an aside, these averages and the antineutrino kinetic energy spectra (not shown in the right panels of Fig. 1.9) are needed to calculate the decay heat [78, 79, 80, 81, 82, 83] and total antineutrino energy spectrum [84, 85, 86, 80, 81, 82, 83] from nuclear reactors. As already mentioned, β− decay is the decay mode that is relevant to this dissertation. However, for completeness, a decay scheme for β+ decay is shown in the left panels of Fig. 1.10. For a given β-decay transition in the left panels of Fig. 1.10, the right panels show the continuous distribution of positron kinetic energies. The quantities labeled in Fig. 1.10 have already been defined in the explanation of Fig. 1.9. The energy released in a given β-decay transition that is shared between the final products (daughter, positron, and neutrino) as kinetic energy is Q−Ex−2mec2. The quantity Q−Ex−2mec2 is the maximum kinetic energy available to the positron or neutrino. The average of each distribution in the right panels refers to the average positron kinetic energy. In β+ decay, the Fermi function takes into account the Coulomb repulsion of the positively charged daughter nucleus and positively charged positron. In terms of physics, the Fermi function distorts the phase space distribution by shifting the positron kinetic energy distribution to larger values. 36 Figure 1.9: A simplified decay scheme for β− decay, with different transitions, Fermi func- tions, and electron kinetic energy distributions. See main text for details. All functions and distributions are normalized to unity. In the right panels, the red, cyan, and blue lines (both dotted and solid) are on top of each other. 37 (Z, A)(Z+1, A)(Ex, J)p(Ex, J)i(Ex, J)j(Ex, J)k(Ex, J)l(Ex, J)mQQ-ExmIm0.00.10.20.30.4Electron Kinetic Energy (units of mec2)012345dN/dEkineticQ-ExmEPhase Space dN/dEFermi Function AFermi Function BFermi Function CPhase Space dN/dE with Fermi Function APhase Space dN/dE with Fermi Function BPhase Space dN/dE with Fermi Function C(Z, A)(Z+1, A)(Ex, J)p(Ex, J)i(Ex, J)j(Ex, J)k(Ex, J)l(Ex, J)mQQ-ExlIl0.00.51.01.52.02.5Electron Kinetic Energy (units of mec2)0.00.10.20.30.40.50.6dN/dEkineticQ-ExlEPhase Space dN/dEFermi Function AFermi Function BFermi Function CPhase Space dN/dE with Fermi Function APhase Space dN/dE with Fermi Function BPhase Space dN/dE with Fermi Function C(Z, A)(Z+1, A)(Ex, J)p(Ex, J)i(Ex, J)j(Ex, J)k(Ex, J)l(Ex, J)mQQ-ExkIk01234567Electron Kinetic Energy (units of mec2)0.000.050.100.150.200.25dN/dEkineticQ-ExkEPhase Space dN/dEFermi Function AFermi Function BFermi Function CPhase Space dN/dE with Fermi Function APhase Space dN/dE with Fermi Function BPhase Space dN/dE with Fermi Function C(Z, A)(Z+1, A)(Ex, J)p(Ex, J)i(Ex, J)j(Ex, J)k(Ex, J)l(Ex, J)mQQ-ExjIj0246810Electron Kinetic Energy (units of mec2)0.000.020.040.060.080.100.120.140.16dN/dEkineticQ-ExjEPhase Space dN/dEFermi Function AFermi Function BFermi Function CPhase Space dN/dE with Fermi Function APhase Space dN/dE with Fermi Function BPhase Space dN/dE with Fermi Function C(Z, A)(Z+1, A)(Ex, J)p(Ex, J)i(Ex, J)j(Ex, J)k(Ex, J)l(Ex, J)mQQ-ExiIi02468101214Electron Kinetic Energy (units of mec2)0.000.020.040.060.080.100.12dN/dEkineticQ-ExiEPhase Space dN/dEFermi Function AFermi Function BFermi Function CPhase Space dN/dE with Fermi Function APhase Space dN/dE with Fermi Function BPhase Space dN/dE with Fermi Function C Figure 1.10: A simplified decay scheme for β+ decay, with different transitions, Fermi func- tions, and positron kinetic energy distributions. See main text for details. All functions and distributions are normalized to unity. In the right panels, the cyan and blue lines (both dotted and solid) are on top of each other. 38 (Z, A)(Z-1, A)(Ex, J)p(Ex, J)i(Ex, J)j(Ex, J)k(Ex, J)l(Ex, J)mQQ-ExmIm0.0000.0250.0500.0750.1000.1250.1500.1750.200Positron Kinetic Energy (units of mec2)0246810dN/dEkineticQ-Exm-2mec2EPhase Space dN/dEFermi Function BFermi Function CPhase Space dN/dE with Fermi Function BPhase Space dN/dE with Fermi Function C(Z, A)(Z-1, A)(Ex, J)p(Ex, J)i(Ex, J)j(Ex, J)k(Ex, J)l(Ex, J)mQQ-ExlIl0.00.51.01.52.02.53.0Positron Kinetic Energy (units of mec2)0.00.10.20.30.40.50.6dN/dEkineticQ-Exl-2mec2EPhase Space dN/dEFermi Function BFermi Function CPhase Space dN/dE with Fermi Function BPhase Space dN/dE with Fermi Function C(Z, A)(Z-1, A)(Ex, J)p(Ex, J)i(Ex, J)j(Ex, J)k(Ex, J)l(Ex, J)mQQ-ExkIk012345Positron Kinetic Energy (units of mec2)0.000.050.100.150.200.250.300.35dN/dEkineticQ-Exk-2mec2EPhase Space dN/dEFermi Function BFermi Function CPhase Space dN/dE with Fermi Function BPhase Space dN/dE with Fermi Function C(Z, A)(Z-1, A)(Ex, J)p(Ex, J)i(Ex, J)j(Ex, J)k(Ex, J)l(Ex, J)mQQ-ExjIj02468Positron Kinetic Energy (units of mec2)0.0000.0250.0500.0750.1000.1250.1500.1750.200dN/dEkineticQ-Exj-2mec2EPhase Space dN/dEFermi Function BFermi Function CPhase Space dN/dE with Fermi Function BPhase Space dN/dE with Fermi Function C(Z, A)(Z-1, A)(Ex, J)p(Ex, J)i(Ex, J)j(Ex, J)k(Ex, J)l(Ex, J)mQQ-ExiIi024681012Positron Kinetic Energy (units of mec2)0.000.020.040.060.080.100.120.140.16dN/dEkineticQ-Exi-2mec2EPhase Space dN/dEFermi Function BFermi Function CPhase Space dN/dE with Fermi Function BPhase Space dN/dE with Fermi Function C 1.5.6 Half-life A fundamental property of a β-decaying nuclide is the β-decay half-life. This is the time required for half of the nuclei in a sample to undergo β decay. The number of β-decaying nuclei at any time is given by − N (t) = N0e ln(2) T1/2 t = N0e−λt (1.19) where N (t) is the number of β-decaying nuclei at time t, N0 is the number of nuclei at time t = 0, T1/2 is the β-decay half-life, and λ is the decay constant (λ = ln(2)/T1/2). If an unstable parent nucleus decays to an unstable daughter nucleus, the number of daughter nuclei at any time is given by N2(t) = N2,t=0e−λ2t + λ1 λ2 − λ1 N1,t=0(cid:16)e−λ1t − e−λ2t(cid:17) (1.20) where N2(t) is the number of daughter nuclei at time t, N1,t=0 is the number of parent nuclei at time t = 0, N2,t=0 is the number of daughter nuclei at time t = 0, λ1 is the decay constant of the parent, and λ2 is the decay constant of the daughter. The β-decay half-life is an important nuclear physics property that is needed for r process reaction network calculations (Sec. 1.4.3). In the r process, half-lives determine the timescale for producing the heaviest nuclides. This will in turn affect the abundance pattern from the reaction network calculation. As mentioned in Sec. 1.4.3, experimentalists will never be able to measure the β-decay half-lives of all nuclides that participate in the r process. This situation creates a reliance on theoretical models to accurately calculate the β-decay half-lives where experimental data 39 are nonexistent and experiments are currently unfeasible. Theoretical models calculate the β-decay half-life according to the following equation [87, 88] 1 T1/2 = (gA/gV )2 K (cid:88)0