CALCULATION OF BETA-DECAY RATES IN HEAVY DEFORMED NUCLEI AND IMPLICATIONS FOR THE ASTROPHYSICAL r PROCESS Thomas R. Shafer A dissertation submitted to the faculty at the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics and Astronomy. Chapel Hill 2016 Approved by: Jonathan Engel Charles Evans Carla Fröhlich Christian Iliadis John Wilkerson c 2016 Thomas R. Shafer ALL RIGHTS RESERVED ii ABSTRACT Thomas R. Shafer: Calculation of Beta-decay Rates in Heavy Deformed Nuclei and Implications for the Astrophysical r Process (Under the direction of Jonathan Engel) The rapid neutron-capture process (r process) is responsible for synthesizing approximately half of the heavy elements in the solar system, but after decades of work the astrophysical site where it occurs is unknown. Because a very large number of heavy, neutron-rich nuclei are populated during the r process, theoretical efforts to locate the r-process site require a tremendous amount of nuclear physics input. However, most nuclei populated during the r process are very neutron-rich and unstable, and many r-process nuclei cannot be studied experimentally. As a result, the basic properties of a large number of heavy, deformed, short-lived nuclei must be calculated with reliable nuclear models. β-decay half-lives are among the most important properties that affect r-process abundances. In this work we present a new, efficient method for calculating β-decay half-lives of heavy, deformed nuclei: the proton-neutron finite amplitude method (pnFAM). We apply the pnFAM to calculate half-lives of nuclei that demonstrate the strongest effects on r-process abundances in the A'80 and A'160 mass regions, including even-even, odd-A, and odd-odd nuclei. We find that our calculations result in half-lives very similar to those reported in previous studies, with correspondingly small effects on r-process abundances. iii For Martha, Mom, & Dad. iv ACKNOWLEDGEMENTS None of the work presented here would have been possible without the continuous guidance and assistance of my advisor, Jon Engel. I am tremendously grateful for the opportunity to study nuclear physics, but even more, I am grateful for his direction, insight, patience, and encouragement. I am also indebted to the postdoctoral researchers with whom I have worked during my time at UNC, particularly Mika Mustonen and Matt Braby. I believe it was Mika who first suggested attempting to build what became the pnFAM, and I greatly enjoyed working closely with him these last few years to develop this method. Many thanks are also due to my collaborators on this project, Carla Fröhlich, Gail McLaughlin, Matt Mumpower, and Rebecca Surman, who have provided astrophysical insight and r-process calculations for this work. I also gratefully acknowledge funding from the U.S. Department of Energy Topical Collaboration for Neutrinos and Nucleosynthesis in Hot and Dense Matter. I especially thank the members of my committee: Charles Evans, Carla Fröhlich, Christian Iliadis, and John Wilkerson. I greatly appreciate the time and effort you have dedicated to this project. I thank Nicolas Schunck, P. G. Reinhard, Markus Kortelainen, and Dong-Liang Fang for helpful notes, conversations, and computational programs during the course of this project. Also, my gratitude to Pedro Sarriguren for allowing me to reproduce his figures in Chapter 8 for a comparison between calculations. Thanks to Derek Vermeulen and Laurie McNeil for supervising my experimental research project. I am tremendously grateful for the support and encouragement of my family and friends, including my parents, Rick and Liz; my brother and sister-in-law, Daniel and Brenda; the Godwins; my friends in Wilmington and from The Summit Church in the Raleigh-Durham area; and my friends from Archer Lodge Middle School. Life is so much better because of these people and many others. Finally, I cannot thank my wife, Martha, enough for her ceaseless encouragement and support during this season. Only she knows how vital her wisdom and encouragement have been over these several years. I love you, Martha, and I am excited for what comes next (Romans 8:28–32)! v TABLE OF CONTENTS LIST OF TABLES . ix LIST OF FIGURES . xi LIST OF ABBREVIATIONS AND SYMBOLS . xiv 1 INTRODUCTION AND MOTIVATION . 1 1.1 The origin of the heavy elements . 1 1.2 The r process . 2 1.2.1 Abundance features . 4 1.2.2 The search for the r-process site . 4 1.3 The importance of β decay in r-process calculations . 6 1.3.1 Outline of this work . 7 2 β DECAY IN NUCLEI . 8 2.1 Allowed and forbidden β decay................................... 9 2.2 The total β-decay half-life . 11 3 NUCLEAR SHAPE DEFORMATION . 13 3.1 Deformation and angular momentum symmetry breaking . 13 3.1.1 Angular momentum projection and the collective rotor model . 14 3.2 Matrix elements and transition strengths in the collective rotor model . 15 3.3 β-decay half-lives of axially-deformed nuclei . 17 4 DENSITY FUNCTIONAL THEORY AND THE LINEAR RESPONSE OF THE NUCLEUS . 18 4.1 The Skyrme energy-density functional . 20 4.2 Time-dependent DFT . 21 4.3 Linear response theory . 22 5 THE PROTON-NEUTRON FINITE AMPLITUDE METHOD . 24 vi 5.1 Mean field theory with pairing correlations: the HFB approximation . 25 5.1.1 HFB calculations with both protons and neutrons . 27 5.1.2 The ground state solver HFBTHO . 28 5.2 Overview of the finite amplitude method . 29 5.2.1 Differences between the charge-changing and like-particle FAM . 31 5.3 Solution of the pnFAM equations in the quasiparticle basis . 32 5.3.1 Strength functions, cross-terms, and β-decay half-lives . 33 6 β DECAY OF ODD NUCLEI WITH THE PNFAM . 36 6.1 Ground states of odd nuclei and the equal filling approximation . 37 6.2 Excited states of odd nuclei . 39 6.2.1 The pnFAM equations for odd nuclei . 40 6.3 Interpretation of our results for odd nuclei . 42 7 β-DECAY HALF-LIVES OF RARE-EARTH ISOTOPES . 44 7.1 The rare-earth r-process nuclei . 44 7.2 Overview of the calculation . 45 7.3 Adjustments to and evaluation of ground state properties . 49 7.3.1 Adjustment of the pairing functional . 50 7.3.2 Calculated ground state properties . 52 7.3.3 Treatment of odd nuclei . 55 7.4 β-decay properties . 55 7.4.1 The time-odd functional . 56 7.4.2 The isoscalar proton-neutron pairing . 57 7.5 Evaluation of fits, results, and conclusions . 60 7.6 Consequences for rare-earth r-process nucleosynthesis . 66 8 β-DECAY HALF-LIVES OF ISOTOPES WITH A ' 80 ........................ 69 8.1 Ground state properties . 70 8.1.1 Adjusting the pairing strengths . 71 vii 8.1.2 Shape deformations and ground state pairing properties . 72 8.2 Excited states . 73 8.2.1 Fitting the proton-neutron isoscalar pairing . 74 8.3 Fit evaluation, results, and conclusions . 76 8.3.1 Half-lives of A ' 80 nuclei . 76 8.3.2 Consequences for the weak r process . 80 9 CONCLUSIONS AND IMPLICATIONS FOR FUTURE r-PROCESS AND β-DECAY STUDIES 83 9.1 New calculations of β-decay half-lives of deformed nuclei . 83 9.2 Consequences for the r process . 84 9.3 Future directions . 85 A DETAILED EXPRESSIONS FOR ALLOWED AND FIRST-FORBIDDEN β-DECAY RATES . 87 A.1 Overview of the β-decay rate . 87 A.2 Allowed decay . 88 A.3 First-forbidden decay . 88 B NUCLEAR SHAPE DEFORMATION . 92 B.1 Nuclear matrix elements in the laboratory rest frame . 92 B.2 Reduced matrix elements, transition strengths, and cross terms . 94 C ADDITIONAL DETAILS INVOLVING THE DERIVATION OF THE PNFAM . 97 C.1 Derivation of the standard pnFAM equations . 97 C.1.1 Determination of the perturbed mean fields δH(ω) . 100 C.2 Modifications of the pnFAM equations for odd nuclei . 101 C.2.1 The perturbed mean fields δH(ω) in the pnFAM for odd nuclei . 103 REFERENCES . 104 viii LIST OF TABLES 7.1 The seventy rare-earth r-process nuclei for which we compute β-decay half-lives, including forty-five even-even nuclei and an additional twenty-five proton-odd nuclei. 45 7.2 Even-even nuclei used to fit the pairing strengths Vp and Vn. The odd-even mass indicators ∆˜ (3) are described in the text. 51 7.3 Values of the pairing strength and density dependence used in our rare-earth calculations. The value of α for SV-min, as well as all three pairing parameters for unedf1-hfb, were not changed. 51 7.4 Non-zero time-odd coupling constants used in our calculations. All couplings are in units of 5 s 3 0 0 MeV fm except C10, which has units MeV fm . The functional SkO -Nd differs from SkO s only in its value of the coupling C10................................. 58 7.5 Isotopes used to fit the proton-neutron isoscalar pairing and, later, to evaluate our complete fitting procedure. The experimental half-lives were taken from Ref. [165]. The labels a–e in the “Excluded?” column note which isotopes were excluded from our pairing fits for the functionals (a) SkO0, (b) SkO0-Nd, (c) SV-min, (d) SLy5, and (e) unedf1-hfb. Isotopes were excluded when our calculated half-lives were shorter than experimental values and the proton-neutron isoscalar pairing was disabled. 59 7.6 Fitted proton-neutron isoscalar pairing strengths describing rare-earth nuclei. We have included the dimensionless parameter gpp = V0/V1 = 2V0/(Vp + Vn), which measures the strength of the T = 0 pairing relative to its T = 1 counterpart.