ELECTROMAGNETIC TRANSITION STRENGTHS STUDIED WITH DOPPLER-SHIFT TECHNIQUES ACROSS THE CONTOURS OF THE VALLEY OF STABILITY By Charles Robert Loelius A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Physics – Doctor of Philosophy 2017 ABSTRACT ELECTROMAGNETIC TRANSITION STRENGTHS STUDIED WITH DOPPLER-SHIFT TECHNIQUES ACROSS THE CONTOURS OF THE VALLEY OF STABILITY By Charles Robert Loelius The electromagnetic transition strengths between bound states in nuclei provide insight into nuclear structure. On one hand, from a single particle perspective the electromagnetic excitation and de-excitation of nuclei quantify the overlaps of nuclear wavefunctions, probing the internal configuration. On the other hand, in a collective model, the shape and dynamics of the nucleus are reflected in the electromagnetic transition strengths. For example, electric quadrupole transitions are sensitive to the deformation of a nucleus and distinguish between various pictures of collectivity, such as rotors and vibrators. In this work, electromagnetic transition strengths are studied through lifetime and Coulomb-excitation measurements. Nuclei across the contours of the valley of stability are studied to investigate features of nuclear structure and how they change near and far from stability. The first experiment discussed in this work investigates the effect of the N = Z = 28 shell closure on collectivity in 58 Ni. 58 Ni has 28 protons and 30 neutrons, and therefore is not expected to exhibit enhanced collectivity compared to its neutron rich neighbors. However, a previous mea+ + surement of the lifetime of the 4+ 1 state indicates an enhanced B(E2; 41 → 21 ) transition strength, suggesting unexpectedly large collectivity. The present work revisits the lifetime of the 4+ 1 state with a more sensitive technique, namely a Recoil Distance Method measurement at the National Superconducting Cyclotron Laboratory. The GRETINA detector array was employed with the + + S800 Spectrograph to measure the 4+ 1 state lifetime. The model independent B(E2; 41 → 21 ) from the present work supports an unenhanced transition strength, as expected near the shell closure. The second experiment discussed in this work is a study of the electromagnetic transition states in 27 Ne. 27 Ne is a loosely-bound neutron rich nucleus although it does not exhibit a halo structure in its ground state. This work investigates the excited states of 27 Ne that lie closer to the particle threshold for features associated with a halo structure. The lifetimes of the 1/2+ and 3/2− states in 27 Ne and the branching ratio of the 1/2+ state decaying into the 3/2− excited state and 3/2+ gs ground state are measured. These values are used to determine the B(E1; 3/2− → 3/2+ gs ) and B(E1; 1/2+ → 3/2− ) values. It was found that the B(E1) connecting the 1/2+ state to the 3/2− state is at least 50 times larger than that between the 3/2− and 3/2+ gs states, indicating an extended radial component in the 1/2+ state wavefunction. Lifetime measurements of excited states in 28 Ne are also presented. A new setup for performing Coulomb-excitation measurements based on heavy ion inelas+ tic scattering with two targets is presented and employed to measure the B(E2; 3/2+ gs → 1/2 ) strength in 27 Ne. The method is demonstrated through a reference measurement of the B(E2; 0+ gs → 30 Mg. Combined with the lifetime measurement, the B(M1; 1/2+ → 3/2+ ) transition 2+ gs 1 ) of strength is extracted. The measured value is unhindered, indicating that the 1/2+ excited state of 27 Ne is not dominated by an s-wave component. The electromagnetic transition strengths in 27 Ne therefore indicate that the 1/2+ excited state may exhibit behaviors characteristic of deformed halos. In summary, this study of electromagnetic strengths demonstrates the features of nuclei in the valley of stability, and explores new aspects of nuclear structure that arise toward the edges of stability. Concurrently, Doppler-shift methods are shown to be powerful tools for investigating the structure of nuclei across the nuclear landscape. ACKNOWLEDGMENTS This work, which I hope to be a small contribution to the ever growing library of human knowledge, is the product of numerous people’s visions, support, and effort. I am especially grateful to my advisor, Dr. Hironori Iwasaki, who has guided me throughout the work enclosed here from experiment to analysis to written document. I have also been fortunate to have the constant support of my guidance committee, Dr. Remco Zegers, Dr. Morten Hjorth-Jensen, Dr. Lisa Lapidus, and Dr. Norman Birge. Their contributions to this work have been invaluable. This work would have been impossible without the tremendous effort from all the members of the National Superconducting Cyclotron Laboratory who kept the status bar green during the running of the experiments, and provided advice and help throughout. I must also thank my professors at Rutgers University who brought me into this field. In particular, I must thank Dr. Yitzhak Sharon and Dr. Larry Zamick, who introduced me to electromagnetic transition strengths and hooked me in. I am also thankful to my undergraduate thesis advisor Dr. Ronald Gllman, Dr. Noemie Benczer Koller, and Dr. Gulhan Gurdal for their support in my undergraduate research. I would be remiss in not also thanking my undergraduate philosophy advisors Dr. Larry Temkin and the late Dr. Derek Parfit who helped me tremendously in clarifying both my future and my thoughts. I am deeply indebted to my friends and fellow students at Michigan State University. This work is especially enriched by my fellow graduate students at the National Superconducting Cyclotron Laboratory, with whom I have often had numerous productive and entertaining conversations. In particular, I must thank my fellow lifetime group members for their advice and support, especially Dr. Chris Morse and Dr. Kenneth Whitmore. However, all my friends whether discussing ROOT or adventuring in Haumark have helped to make this work possible and meaningful. I am also especially thankful for my many friends with whom I have worked with the Graduate Employees Union, including those I roped into organizing and revitalizing our union and building solidarity across campus. Likewise, I continue to respect my friends from the Council of Graduate iv Students who have sought to make our university a more welcome and inclusive place. I hope to have helped you make Michigan State University at least a little bit more of a truly supportive community for all students, faculty and workers. I cannot express my gratitude to the many amazing people with whom I have had a chance to improve our community. Everything in this work is due to the constant encouragement and support of my family. I am unendingly thankful for my parents, who have helped all along the way in my each and every endeavor. I am deeply grateful to Alexandra Genia for her support throughout this process and her constant faith in me. Finally, I am always thankful for my sisters, who are the most incredible people I know, and who always have my back. Finally, I would like to thank my 7th grade English teacher Ms. Drukten, who introduced me to science-fiction books and got me into this whole mess to begin with. v TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi CHAPTER 1 NUCLEAR STRUCTURE . . 1.1 Atomic Nuclei . . . . . . . . . . . . 1.2 Shell Model . . . . . . . . . . . . . 1.3 Nuclei Far From Stability . . . . . . 1.3.1 Evolution of Shell Structure 1.3.2 Halo Nuclei . . . . . . . . . 1.4 Electromagnetic Transition Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 . 5 . 8 . 9 . 12 . 15 METHODS FOR MEASURING ELECTROMAGNETIC TRANSITION STRENGTHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lifetime Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Doppler-shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Recoil Distance Method . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.1 Feeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.2 Modifications to the Method . . . . . . . . . . . . . . . . . . 2.1.3 Doppler-shift Attenuation Method . . . . . . . . . . . . . . . . . . . . Coulomb-excitation Measurements . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Safe Coulomb-excitation . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Intermediate Energy Coulomb Excitation . . . . . . . . . . . . . . . . 2.2.2.1 Impact Parameter . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.2 Two Target Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 22 23 25 28 29 30 32 34 35 36 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 39 40 42 43 44 45 46 47 48 48 49 50 50 52 52 CHAPTER 2 2.1 2.2 CHAPTER 3 BEAM PRODUCTION AND MEASUREMENT DEVICES 3.1 National Superconducting Cyclotron Laboratory . . . . . . . . . . 3.1.1 Ion Source . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Coupled Cyclotrons . . . . . . . . . . . . . . . . . . . . . 3.1.3 Production and Separation of Fragments . . . . . . . . . . 3.1.3.1 A1900 Fragment Separator . . . . . . . . . . . 3.2 S800 Spectrograph . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Magnets and Optics . . . . . . . . . . . . . . . . . . . . . 3.2.2 Timing Scintillators . . . . . . . . . . . . . . . . . . . . . 3.2.3 Cathode-Readout Drift Chambers . . . . . . . . . . . . . 3.2.4 Ion Chamber . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 IsoTagger . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6.1 Timing Scintillators . . . . . . . . . . . . . . . 3.2.6.2 Ion Chamber . . . . . . . . . . . . . . . . . . . 3.2.6.3 CRDC . . . . . . . . . . . . . . . . . . . . . . vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 55 55 56 58 58 58 59 61 62 64 67 68 69 72 72 74 76 76 77 77 77 78 CHAPTER 4 INVESTIGATION I: 58 NI 4+ 1 LIFETIME MEASUREMENT 4.1 Shell Closure at N=Z=28 . . . . . . . . . . . . . . . . . . . . . . . 4.2 58 Ni 4+ 1 Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Recoil Distance Experiment on 58 Ni . . . . . . . . . . . . . . . . . 4.3.1 Particle Identification . . . . . . . . . . . . . . . . . . . . . 4.4 58 Ni γ-ray Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Laboratory Frame Background . . . . . . . . . . . . . . . . 4.5 58 Ni Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Interpretation of 58 Ni results . . . . . . . . . . . . . . . . . . . . . 4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 81 84 85 88 89 91 94 99 102 INVESTIGATION II: ELECTROMAGNETIC TRANSITION STRENGTHS IN 27 NE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Structure of 27 Ne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Setup I: Branching Ratio and Lifetime Measurement . . . . . . . . . 5.2.1 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Laboratory Frame Background . . . . . . . . . . . . . . . . . . . . . . . . 28 Ne Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Target Only Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 25-mm Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 104 108 110 110 111 112 112 3.4 GRETINA . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Gamma-ray Interactions With Matter . . . . 3.3.1.1 Photoabsorption . . . . . . . . . 3.3.1.2 Compton Scattering . . . . . . . 3.3.1.3 Pair Production . . . . . . . . . . 3.3.2 γ-ray Detection . . . . . . . . . . . . . . . 3.3.2.1 Semi-conductor detectors . . . . 3.3.2.2 High Purity Germanium Detectors 3.3.3 GRETINA Design . . . . . . . . . . . . . 3.3.4 Signal Decomposition . . . . . . . . . . . 3.3.5 Addback . . . . . . . . . . . . . . . . . . . 3.3.6 Doppler Reconstruction . . . . . . . . . . . 3.3.7 Timing Between S800 and GRETINA . . . 3.3.8 Calibration . . . . . . . . . . . . . . . . . TRIPLEX . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Bearing Structure and Motors . . . . . . . 3.4.2 Foil mounting structures . . . . . . . . . . 3.4.3 Support Structure . . . . . . . . . . . . . . 3.4.4 Beam Pipe . . . . . . . . . . . . . . . . . . 3.4.5 Distance Measurements . . . . . . . . . . . 3.4.5.1 Linear encoder . . . . . . . . . . 3.4.5.2 TESA micrometer . . . . . . . . 3.4.5.3 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 5 5.1 5.2 5.3 vii 5.3.3 Lifetime Measurement . . . . . . . 27 Ne Measurement . . . . . . . . . . . . . 5.4.1 Target Only Measurement . . . . . 5.4.2 25-mm Separation . . . . . . . . . 5.4.3 Lifetime Measurement . . . . . . . 5.5 Modified Coulomb-excitation measurement 5.5.1 Coupled Channel Equations . . . . 5.6 Experimental Setup II: Coulomb Excitation 5.6.1 GRETINA Efficiency . . . . . . . . 5.6.2 Number of Projectile Nuclei . . . . 5.6.3 Extraction of γ-ray Counts . . . . . 5.7 30 Mg Measurement . . . . . . . . . . . . . 5.8 27 Ne Coulomb Excitation Measurement . . 5.9 Discussion . . . . . . . . . . . . . . . . . . 5.10 Conclusion . . . . . . . . . . . . . . . . . 5.4 CHAPTER 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 120 120 122 123 127 128 129 130 132 133 133 137 139 144 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 viii LIST OF TABLES Table 3.1 The efficiencies in data and simulation for a 152 Eu source are shown along with error and the ratio of the efficiency in data to the efficiency in simulation. Error is from uncertainties in the source activity and statistical error in the simulation and data. The efficiencies in data are generally 25% − 30% smaller than in simulation. This adjustment is taken into account in the cross section data. 72 Table 4.1 Observed γ-ray counts from neutron induced reactions on 27 Al for 1-mm and 10-mm separations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Table 4.2 The values for bare E2 matrix elements A p and An are presented. The values were calculated using the GXPF1A and KB3G Hamiltonians. Values are + + + shown for the 2+ 1 → 01 and 41 → 21 transitions in the Ni isotopes. In addition, B(E2) values are calculated using two sets of effective charges. Calculations with the standard effective charges e p = 1.5, en = 0.5 are denoted with the superscript 1. Calculations with modified effective charges e p = 1.12, en = 0.67 are denoted with the superscript 2. Table and caption are from [91]. . . 101 Table 5.1 The number of counts in the peak for each neutron induced background is provided. For each case this is for the cut with a 7.5-cm spherical addback and with the timing cut applied. For the 0.05-mm separation data, the counts ◦ are provided for interactions at angles below 40 from the degrader. . . . . . . . 111 Table 5.2 The measured lifetimes for transitions among states lying higher than the 2+ 1 state in 28 Ne are presented. Error presented is statistical only. . . . . . . . . . . 116 Table 5.3 Scaled simulated efficiencies for the γ ray of interest for 30 Mg and 27 Ne are presented. Efficiencies for the two targets are different due to the 25 mm separation between the two and the difference in energy from the different β values before and after the projectile travels through the Ta target. Error is from the uncertainties in the efficiency of GRETINA and the scaling factor. . . . 132 Table 5.4 The purity of the beam and total projectile nuclei impinged on the 9 Be and Ta targets are listed for 30 Mg and 27 Ne along with the purity. The error is from statistics and the systematic error in the purity. . . . . . . . . . . . . . . . . . . 132 Table 5.5 + 30 B(E2; 0+ gs → 21 ) values for Mg determined from Coulomb-excitation measurements are presented along with the technique used in the measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 ix Table 5.6 The number of γ rays observed and the cross sections for each state in 30 Mg are presented for reactions on the 9 Be and Ta targets. Statistical and systematic errors are presented for the cross sections, where the systematic error is the quadratic sum of the individual systematic errors discussed in the text. Error for the number of observed γ rays is statistical only. The cross section of the 2+ 1 state corrects for feeding contributions. . . . . . . . . . . . . . . . . . . . . 137 Table 5.7 The γ-ray counts in each peak are presented along with statistical error. The extracted cross sections are presented along with statistical and systematic error, whose sources are discussed in the text. . . . . . . . . . . . . . . . . . . 139 Table 5.8 The electromagnetic transition strengths of 27 Ne measured in this work are presented along with the partial lifetime corresponding to that strength. Error presented for lifetimes and transition strengths is from the quadratic sum of systematic and statistical error. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 x LIST OF FIGURES Figure 1.1 Figure 1.2 Figure 1.3 The Chart of Nuclides is displayed, with proton number Z increasing along the y-axis, and neutron number N increasing along the x-axis. The nuclei are color coded by decay type. Black represents stable nuclei. Pink decays vis β − , blue through either electron capture or β + , yellow through alpha emission, orange through proton emission, purple through neutron emission, and green through fission. The standard magic numbers are plotted for protons and neutrons. Figure is from [8]. . . . . . . . . . . . . . . . . . . . . . . . . . 3 The chart of nuclei is shown for even-even nuclei. The nuclei are color coded by the energy of the first 2+ excited state. Nuclei with energies up to 200 keV are colored in green. Those with energies up to 1 MeV are colored yellow. Those with higher energy are colored from orange to red. The magic numbers are highlighted, and demonstrate bands of large 2+ 1 energy. Figure and data are from [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The energy levels for the harmonic oscillator (left) are shown and compared with those where an l 2 attractive term is added (center) and where that and a spin-orbit term are added (right). The shell gaps in the harmonic oscillator is compared to that of the magic numbers predicted with the inclusion of the l 2 and spin orbit terms. Figure is from [4]. . . . . . . . . . . . . . . . . . . . . . 7 Figure 1.4 Level energies and spin parity in odd-A isotones with N = 17 near the island of inversion are shown. The 7/2− and 3/2− states arising from excitations over the N = 20 gap into the f7/2 and p3/2 orbitals are shown. As the N/A ratio increases, the energy levels of the 7/2− and 3/2− states lower from 3 MeV in 35 Ar to less than 1 MeV in 27 Ne. Figure is from [21]. . . . . . . . . . . 10 Figure 1.5 The root mean square radii of a nucleon in a square well potential are presented. The binding energy normalized by the size of the square well (R0 ) is presented on the x-axis. The root mean square radius for each energy is displayed on the y-axis, scaled by the radius of the square well potential. Radii as a function of binding energy are shown for angular momenta l = 0, 1, and 2. The radii diverge as the binding energy approaches zero for l = 0 and 1 but converge for l = 2. Figure is from [26]. . . . . . . . . . . . . . . . . . . . 14 Figure 1.6 A focused view of the chart of nuclides is presented. Halo nuclei and potential halo nuclei are colored in. The valence nucleons in the orange colored nuclei are protons. The valence nucleons in the green colored nuclei are neutrons. The lighter shades of green are nuclei for which enhanced interaction cross sections have been measured. Darker shades of green are those nuclei for which both enhanced cross sections and narrow momenta distributions have been measured. Figure and discussion are from [26]. . . . . . . . . . . . . 15 xi Figure 1.7 The transverse momenta distributions for (top) 6 He produced from reactions of 8 He+C and (bottom) 9 Li produced from reactions of 11 Li+C are presented. The incoming beam energy of the 8 He and 11 Li was 790 (20) MeV/u. The solid lines are fitted Gaussian distributions, while (bottom) the dotted line is a wide-momentum contribution to the distribution in 11 Li [30]. Figure is from [30]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Figure 2.1 The lifetime regions accessible by direct and indirect measurement techniques are presented. The x-axis shows the lifetime region in lifetimes (bottom) and energy widths (top). Figure is from [41]. . . . . . . . . . . . . . . . . 23 Figure 2.2 The prototypical Recoil Distance Method setup is displayed. An incoming beam produces excited states at a target. It then travels a distance x over the flight time t f to a stopper foil. Decays occurring while the nucleus is between the target and stopper emit γ-rays with a Doppler-shifted energy. Decays occurring after the nucleus is slowed in the stopper emit with a lower energy. In the figure, the nucleus is stopped in the stopper, while in the present work nuclei are only slowed. For this reason, the foil used for stopping is instead called a degrader in the text. Figure is from [44]. . . . . . . . . . . . . . . . . . 25 Figure 2.3 Gamma-ray spectra for 17 C using the Recoil Distance Method with two degrader foils. Three spectra are shown with different distances between the target and first degrader (d1) and first degrader and second degrader (d2). Simulated spectra are compared incorporating an exponential background. The peaks are identified as corresponding to β f ast (f), βreduced (r) and βslow (s). The lifetime of the 1/2+ → 3/2+ state was measured to be 528+21 −14 ps, +3.4 ps. while that corresponding to the 5/2+ → 3/2+ was found to be 21.8−3.3 The different three peak structures demonstrate the ability to measure two different lifetime regions simultaneously. Figure and results are from [49]. . . . 31 Figure 2.4 A prototypical DSAM setup is presented. The incoming particles A react in the target to induce excited states in the recoiling nucleus B. The γ-ray emitted in the excited state decays in the backing or target, with the location contingent on the lifetime of the state and the stopping time of the nucleus in the backing (target). The energy of the emitted γ ray will be shifted depending on the velocity of the nucleus B at the time of decay. The mean Doppler-shifted energy can be used to extract the mean velocity compared to the incoming velocity (F = v/v ¯ 0 ). This is compared to the stopping time extracted from theory to determine the lifetime. Figure is from [41]. . . . . . . 33 Figure 3.1 The layout of the NSCL showing the ion sources, K500 and K1200 cyclotrons, and the A1900 is shown. Figure is from [59]. . . . . . . . . . . . . . . 40 xii Figure 3.2 The layout of the S800 is presented, showing the analysis line and the spectrograph. The figure is from [71]. . . . . . . . . . . . . . . . . . . . . . . . . . 46 Figure 3.3 Uncorrected (top) and corrected (bottom) graphs are shown for dispersive angle in the focal plane compared with the time of flight from the S800 Object to E1 scintillators. The corrected graph shows no dependence of the time of flight on the angle, compared with the uncorrected graph which shows a substantial correlation. Similar corrections are performed for the A1900 XFP scintillator for both the dispersive angle and dispersive displacement. . . . . . . 51 Figure 3.4 The uncalibrated data (left) and known mask positions (right) are shown. The regions of activity in the data correspond to the positions in the mask, and can be compared by the number of holes in each region to uniquely identify the locations. The arbitrary units for the data (left) are distinguished from the mm units for the mask (right). The mask image is from [78]. . . . . . . . . . . 53 Figure 3.5 The maximum response amplitudes for each pad are histogramed for the uncalibrated (top) and gain matched (bottom) CRDCs. The calibration prevents any one pad from having too much influence over the resulting position measurement, and can be seen by the roughly straight line in the amplitudes plotted out by pad number in the bottom figure. This is performed for both CRDCs to ensure proper calculation of both positions and angles. . . . . . . . . 54 Figure 3.6 The regions where each of the photoelectric, Compton scattering and pair production cross sections are dominant are presented as a function of photon energy and the atomic number of the absorbing matter. The lines where cross sections are equal between the neighboring effects are shown to demarcate three regions. Figure is from [80]. . . . . . . . . . . . . . . . . . . . . . . . . . 56 Figure 3.7 (left) The design of a detector in GRETINA is shown, with four crystals aligned into a cluster to increase detection area. (right) The segmentation of each crystal is shown for the six angular and six depth slices leading to 36 total segments. Figure is from [79]. . . . . . . . . . . . . . . . . . . . . . . . . 62 Figure 3.8 The GRETINA setup used in the 27 Ne experiment in March 2016 is shown. Nine GRETINA detectors were used in the experiment, which are arrayed around the beampipe for the TRIPLEX plunger. . . . . . . . . . . . . . . . . . 63 xiii Figure 3.9 The spherical addback method is shown for a case of five interactions across two crystals. Interactions are labeled with energies in arbitrary units. Raw interactions are represented by a flash, while added back interactions are represented by a many-point star. The energies of the raw interactions are denoted with a number in the subscript (E1 , E2 , ...). Energies of the added back interactions are denoted with a subscript letter (Ea , Eb , ...). (Top-left) In the first step of the process, the highest energy interaction is identified. (Top-right) A spherical cut is made, including any interactions within radius R. (Bottomleft) The interactions within the spherical cut are summed to create a single addback interaction. The interactions in the addback are no longer used in the algorithm, and the process repeats again. (Bottom-right) The remaining two hits have only themselves within the radius of the highest energy hit and the addback interaction is the same as the original interaction. Note that in the first frame the E1 interaction is within the radius R of E2 , but E2 is removed from consideration in the third frame. In this work a radius of 7.5 cm is employed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Figure 3.10 A GEANT4 simulation [87] of a single γ-ray transition measured by GRETINA is shown, with a comparison between singles and a spherical addback. The singles replicate the summing up of all energies within a single GRETINA crystal, while the spherical addback is as described in the text with a radius of 7.5 cm. The state energy is assumed to be 1480 keV, and the Compton-edge can be observed at around 1300 keV, with scattered events extending back towards 0 keV. A clear reduction in the Compton-events and an increase in the peak yield is apparent for the spherical addback compared to the singles. . . 66 Figure 3.11 A γ-ray spectrum of 58 Ni is shown for a Recoil Distance Measurement with three foils each separated by 1 mm, as discussed in Chapter 4. Spherical addback radii of 4, 7.5 and 15 cm are compared, and discussed in the text. . . . 67 Figure 3.12 The difference in GRETINA and S800 timing is plotted as a function of the γ-ray energy. A sample timing cut to reduce the impact of neutron-induced backgrounds is shown in red. Neutron induced backgrounds can be seen at the vertical lines at 511 (electron annihilation energy), 590 (76 Ge), 700 (72 Ge), 850 (27 Al),1040 (70 Ge), 1460 (40 K), and 2614 (208 Tl) keV. The band along the cut between −50 and −70 represents γ rays coincident with ions near the target position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Figure 3.13 Spectra for a Eu-152 source placed on a 0.92 mm Ta Degrader 12 cm upstream of the GRETINA center for (top) data and (bottom) GEANT4 simulation are presented. The efficiencies can be compared to each other to determine the scaling need from the simulation. . . . . . . . . . . . . . . . . . 71 xiv Figure 3.14 (top) A photograph of the TRIPLEX is shown compared to (bottom) a schematic of the TRIPLEX. In the schematic the labeling is as such: (A) The support frame, (B) Motor used to position the target foil, (C) the outermost tube, used to adjust the position of the second degrader, (D) The immobile central tube used to mount the degrader, (E) The inner tube, used to move the target, (F) The foil mounting structure for the target attached to the inner tube (E), (G) The foil mounting structure for the first degrader, which is immobile and connected to the central tube (D), (H) The foil mounting structure for the second degrader, attached to the outermost tube (C). Figure and identification are from [48]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Figure 3.15 The stepping action of the N-381 NEXACT piezoelectric motor is shown in four figures. In (1) the first actuator is made to connect with the guiderail. In (2) the horizontal electric fields are applied, causing the first actuator to expand in the direction contrary to the motion of the motor, but the second actuator expands in the direction of the motion. In (3) the first actuator is removed from the guiderail and the second actuator connected. In (4) the horizontal electric field is reversed, causing the materials to expand against the direction of motion, leading to a net motion for the runner. Figure is from [88]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Figure 3.16 Plots showing the (top) induced voltage and the (bottom) reciprocal of the induced voltage between a beryllium and tantalum foil as a function of the separation between them. The distance was measured using the linear encoder of the TRIPLEX, and is based from the point of electrical contact. In the top figure, saturation near 0 µm can be seen, after which the 1/d behavior dominates. In the bottom figure, the points near saturation are removed. A linear fit to the data is also shown. For this case, the contact derived from the linear fit is within 10 µm of the measured electrical contact. . . . . . . . . . . 80 Figure 4.1 2+ 1 energies for Ca and Si isotopic chains near N = 28. The difference in systematics shows the decreasing energy gap for the N=28 magic number for nuclei far from stability. Figure is from [99]. . . . . . . . . . . . . . . . . . . . 82 Figure 4.2 + (top) The observed state energies for the 4+ 1 and 21 levels in the even-even Nickel isotopes between A=56 and A=68 are shown and compared to calculations from the GXPF1A shell model interaction. (bottom). The observed + B(E2; 2+ 1 → 01 ) transition strengths for the even-even Nickel isotopes between A = 56 and A = 68 compared against predictions from the KB3G interaction and the GXPF1A interaction with standard and modified effective charges. These figures are from [91]. . . . . . . . . . . . . . . . . . . . . . . . 83 xv Figure 4.3 Doppler-shift attenuation method spectrum for 58 Ni from [106]. The lifetime is extracted from the relative yield in the Doppler-shifted and un-shifted parts of the peak and compared to the stopping time in the material. Lifetimes were + + + extracted for each of the 2+ 1 , 41 , 23 and 24 excited states. . . . . . . . . . . . 86 Figure 4.4 A timing signal correlation plot used as a particle identification for the incoming beam of interest is presented. The A1900 XFP scintillator timing signal is shown on the y-axis and S800 OBJ scintillator timing signal is shown on the x-axis. The components of the incoming cocktail beam are identified. 74 Kr is highlighted with a cut, while contaminants 73 Br and 72 Se are also labeled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Figure 4.5 A particle identification plot for the reaction residues is presented. The y-axis shows the Ion Chamber ∆E while the x-axis shows the trajectory-corrected timing signal from the S800 OBJ scintillator. The particle identification plot is gated on the 74 Kr incoming beam. The 58 Ni residue is identified with a cut and labeled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Figure 4.6 A partial level scheme of 58 Ni showing states observed in the Recoil Distance Method experiment. The lifetimes in parentheses are from the present mea+ surement for the 4+ 1 → 21 state and for the remainder are from the NNDC evaluation [109]. The thicknesses of lines represent the relative yield from each transition. Figure is from [91]. . . . . . . . . . . . . . . . . . . . . . . . . 91 Figure 4.7 The Doppler-corrected (top) and lab-frame spectra for the 1-mm separation at forward angles are shown. The arrows in the Doppler-corrected spectrum identify the three-peak structure for observed 58 Ni transitions. In both the Doppler-corrected and lab-frame spectra γ-ray peaks from laboratory frame backgrounds are identified and labeled. . . . . . . . . . . . . . . . . . . . . . 92 Figure 4.8 Results for the Recoil Distance Method for 58 Ni with a 1-mm (top) and 10mm (bottom) separation between the target and degraders. The data is compared to a spectrum with best fit parameters for state population and ratio of target to degrader reactions. The simulated spectrum is decomposed into the GEANT4 simulation, the background from neutron-induced reactions on 27 Al, and the exponential background. Figure is from [91]. . . . . . . . . . . . 95 Figure 4.9 The dta spectrum showing the energy distribution for 58 Ni fragments observed in the S800 Spectrograph is shown. The x-axis shows the percentage difference from the Bρ central value, 1.9 T-m, which corresponds to an energy of 40 MeV per nucleon. The asymmetric shape demonstrates the enhanced S800 selection for reactions occurring in the degraders. . . . . . . . . . 96 xvi Figure 4.10 The energy distributions for 58 Ni fragments in coincidence with the fast and slow components of the 4+ 1 transition energies are shown. The overall dsitribution of the slow component is slightly larger in scale due to the larger yield, but there is no clear difference in the shapes of the distribtuions. . . . . . . . . . 97 Figure 4.11 A zoomed in γ-ray spectrum for the 1-mm data set focused on the 4+ 3 state is presented. Simulations using a fixed exponential background are compared to the data for the best fit lifetime of 3 ps and for the upper and lower bounds of 8 ps and 0 ps. The figure is from [91]. . . . . . . . . . . . . . . . . . . . . . 98 + Figure 4.12 The systematics of the B(E2; 4+ 1 → 21 ) values throughout the Nickel isotopic chain. The experimental values are compared with the shell model values from the GXPF1A and KB3G interactions. For the GXPF1A shell model interactions, results for two sets of effective charges are shown. Figure is from [91]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Figure 5.1 The level scheme for 27 Ne is presented. The adopted values for state energies and J π are from [117]. The neutron separation energy is from [27]. γ-ray transitions are shown and are labeled with the transition energy and corresponding multipolarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Figure 5.2 A histogram of γ-ray interactions with GRETINA is presented. The Dopplercorrected γ-ray energy is plotted on the x-axis, while the angle from the target to the interaction position is plotted on the y-axis. The two nearly parallel lines near 1305 and 1240 keV are from reactions populating of 28 Ne at the target and degrader respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 109 Figure 5.3 A timing signal correlation plot for the incoming 29 Na secondary beam is shown. The data is from the target-only measurement. The incoming 29 Na beam and contaminants are labeled. . . . . . . . . . . . . . . . . . . . . . . . 110 Figure 5.4 The particle identification for reaction products at the S800 is shown. The incoming 29 Na beam and the 28,27 Ne isotopes generated by the reaction are labeled. Data shown is for the target-only measurement and is gated on the incoming 29 Na beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Figure 5.5 The level scheme for 28 Ne is shown for known level states and transitions. The 1127 keV transition is a decay between unknown states represented by the dashed lines. The J π assignments and values for state and transition energies are from [121]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 xvii Figure 5.6 The Doppler-corrected γ-ray spectrum for 28 Ne with only a 1-mm Be target is presented. The observed transitions are labeled, including the 1127 keV transition decaying from an unknown state. The best fit simulation for the relative population of states is presented in comparison, along with an exponential background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Figure 5.7 The spectrum for the 25-mm separation between target and degrader gated on 28 Ne particles is presented. States of interest are labeled and transition energies identified. Two peak structures are apparent for all transitions, and a best fit simulation presented. The best fit simulation is for a ratio of 2.1 reactions on the target for each reaction on the degrader for all states. . . . . . 115 Figure 5.8 The γ-ray spectrum for 28 Ne is presented for GRETINA angles below 40 for a 50 µm separation between the target and degrader. The observed transitions are labeled along with the neutron-induced background. A best fit simulation using the best fit lifetimes is overlayed, along with an exponential background. The lifetimes used for the 1304 keV state was 6.25 ps. The lifetimes for higher-lying states are the best fit values in Table 5.2 as discussed in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Figure 5.9 28 Ne ◦ spectra for the 0.05 mm separation between target and degrader are shown zoomed in to focus on the (Top) 1706 keV and (Bottom) 894 keV and 1127 keV transitions. The data are compared to best fit simulations for each state, which for the 894 keV transition is 0 ps, for the 1127 keV transition is 1.5 ps, and for the 1711 keV transition is 2.5 ps. . . . . . . . . . . . . . . . . . 117 Figure 5.10 The energy distribution for 28 Ne reconstructed for after the degrader is presented. The x-axis shows the energy in terms of percentage above or below the Bρ central value, which in this case was 3.02 T-m, which corresponded to an energy of 54.5 MeV per nucleon. The 28 Ne makes up the shoulder on the high energy side of the spectrum, while the shoulder on the low energy edge is a secondary beam contaminant in the particle identification. Therefore, only the lower energy tail of the 28 Ne distribution is accepted by the S800, and so uncertainty in the lifetime due to uncertainty in the distribution must be constrained. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Figure 5.11 The Doppler-corrected spectrum for 27 Ne with only a 1 mm thick Be target is shown. The transitions observed are labeled. The data is compared with a best fit simulation to reproduce the branching ratio of the 885 keV state. . . . . 121 Figure 5.12 The γ-ray spectrum for 27 Ne with a 25 mm separation between the target and degrader is presented. Spherical addback with a 7.5-cm radius was employed. Only events in which a single added-back event was measured are included. The observed transitions are identified and labeled. . . . . . . . . . . 122 xviii Figure 5.13 γ-ray spectra for 27 Ne with a 25-mm separation are presented focused on the 765 keV state (top) and 885 keV state (bottom). Spherical addback with a 7.5-cm radius was employed. Only events with a single added-back event measured in coincidence with the ion are included. Best fit spectra for each of the target-to-degrader excitation ratios are shown along with the fixed exponential background. The ratio for the 765 keV state is 3.9 reactions in the target per reaction in the degrader. The ratio for the 885 keV state is 5.2 reactions in the target per reaction in the degrader. . . . . . . . . . . . . . . . . 124 Figure 5.14 The 27 Ne γ-ray spectrum for a 0.05-mm separation between the target and ◦ degrader is presented for angles below 40 . Spherical addback with a 7.5 cm radius was employed. Only events with a single added-back event measured in coincidence with the ion are included. Peaks from transitions and neutroninduced backgrounds are identified. . . . . . . . . . . . . . . . . . . . . . . . 125 Figure 5.15 The γ-ray spectra for 27 Ne with a 0.05-mm separation focused on 765 keV state (top) and 885 keV state (bottom) are shown. The fixed exponential background is presented in both figures, and in (top) the small contribution from neutron-induced background is presented. In both figures a χ 2 plot for the fits is presented, where the number of degrees of freedom is 15 for the 765 keV and 19 for the 885 keV. (top) The best fit simulated spectra with a 765 keV state lifetime of 3.1 ps is compared to simulated spectra with lifetimes 0 ps and 6 ps. (bottom) Simulated spectra with the upper bound 885 keV state lifetimes of 2 ps, is compared to simulated spectra with lifetimes of 0 ps and 6 ps.126 Figure 5.16 The particle identification of the secondary cocktail beam in the S800 focal plane is presented. The ion chamber energy loss is plotted against the time of flight from the S800 Object scintillator to the S800 E1 scintillator. The 30 Mg and 27 Ne components are labeled. . . . . . . . . . . . . . . . . . . . . . . . . 131 Figure 5.17 The Coulomb-exctiation spectrum for 30 Mg using a 1.0 mm 9 Be and 0.92 mm Ta target separated by 25 mm is presented. The observed transitions are labeled. The higher (lower) peak components are associated with reactions on the 9 Be (Ta) target. A best fit spectrum is presented, with the ratio of reactions on the 9 Be target to reactions on the Ta target found to be 0.65. The dashed curve is an exponential background included in the fit. . . . . . . . . . 135 Figure 5.18 The partial level scheme of 30 Mg showing states observed in the Coulomb excitation data is presented. The J π assignment and state and transition energies are from [124]. The dashed lines represent transitions not observed in this study because of overlap with Compton events. . . . . . . . . . . . . . . . 136 xix Figure 5.19 The 27 Ne Coulomb-excitation spectrum with a 9 Be target and Ta target separated by 25 mm is presented. The 885 keV and 765 keV transitions are identified. A best fit simulation with a ratio of 0.4 reactions in the 9 Be target per reaction in the Ta target is shown with an exponential background. . . . . . 138 Figure 5.20 Level schemes for the bound states of 27 Ne, 11,12 Be, and 17,19 C are presented. Each level scheme is aligned to the neutron separation energy of the nucleus, represented by the dashed line. The neutron separation energy in keV for the ground state of each nucleus is stated above the dashed line. The J π for each level is labeled, and the energies of excited states in keV are shown. The multipolarity of the transitions are also denoted. The energy of the 0+ state in 12 Be is not to scale and is marked with a dotted line. Beneath the level schemes, a table of the downward electromagnetic transition strengths for the transitions shown in each nucleus are presented in Weisskopf units. The data is from the present work and [27][31][38][49][111][113][125]. . 141 xx CHAPTER 1 NUCLEAR STRUCTURE 1.1 Atomic Nuclei Nearly all matter in everyday life is made up of atoms. As discovered by Rutherford, the atom consists of a positively charged nucleus that is surrounded by one or more electrons [1]. While the electrons are thought to be point particles with no internal structure, atomic nuclei are known to consist of positively charged protons and electrically neutral neutrons. This work seeks to add to our knowledge of the structure of atomic nuclei near and far from stability through the study of the electromagnetic properties of nuclei. Therefore, it is necessary to outline current knowledge of nuclei, including observed trends in stable and unstable nuclei. Near stability, a number of models based on an independent-particle or collective picture of nuclei have been employed to understand nuclear structure. However, these models do not necessarily reproduce features of nuclei far from stability. In this section systematics of atomic nuclei and ways to interpret them will be discussed. The protons and neutrons, or nucleons, interact not only through the electromagnetic force, but through the strong and weak nuclear forces. While the weak force is involved in nuclear decays, the strong nuclear force is involved in binding nuclei. In particular, the nuclear strong force is a short range interaction (with a range of 1-2 fm) that is generally attractive, with a repulsive contribution at very short distances [2][3]. The strong nuclear force in combination with the Pauli-exclusion principle leads to a rich structure in atomic nuclei [4]. Protons and neutrons are spin 1/2 fermions, and thus obey the Pauli exclusion principle. The Pauli exclusion principle is a fundamental restriction on particles with non-integer spin, which requires that the total wavefunction be antisymmetrized. This can be stated as the requirement that [5]: ψ(x1 , ..., xi , x j , , ..., xN ) = −ψ(x1 , ..., x j , xi , ..., xN ) 1 (1.1) where the xi ’s represent the position and quantum numbers of the i-th fermion. One key result of this is that two fermions cannot occupy the same state with the same quantum numbers. If one neglects the Coulomb term, neutrons and protons can be thought of as two states of a nucleon particle, differing in a new quantum number called isospin. The isospin of the nucleon is τ = 1/2, with the proton having τz = −1/2 and the neutron having τz = +1/2 [6][7]. Figure 1.1 shows the Chart of the Nuclides. The nuclei are listed with proton number on the vertical axis, and neutron number on the horizontal axis. Those nuclei with the same number of protons are known as isotopes, and those with the same number of neutrons are isotones. In Figure 1.1, the color represents the decay mode. Unstable nuclei can decay through β decay, emitting an electron or positron, or by capturing an electron. β decay proceeds through the weak interaction and changes a neutron to a proton or vice-versa. Nuclei can also decay via the emission of one or more protons, neutrons, or bound α particles consisting of two neutrons and two protons. It is critical to note, however, that there is a line of stable nuclei known as the valley of stability. These nuclei do not decay and thus make up the nuclei in the majority of ordinary atoms. Stable nuclei have been well described by both single-particle and collective models. However, it can be seen that in fact most nuclei are unstable. One early view of atomic nuclei useful for qualitatively understanding their properties is the liquid drop model. In this model, nuclei are considered to consist of an incompressible fluid in a spherical droplet, because the short range repulsive interaction of the strong force and the Pauli-exclusion principle effectively prevent the compression of the nucleus. This means that, in reasonable agreement with experiment, the radius of the nucleus can be determined as r0 A1/3 , where r0 is generally 1.2 fm and A is the total number of nucleons [4]. From the liquid drop model, one can derive the Bethe-Weizsäcker semi empirical mass formula [9][10]: M(A, Z) = (A − Z)mn + Zm p − a1 A + a2 A2/3 + a3 Z2 A1/3 + a4 (A − 2Z)2 1 ± a5 A A3/4 (1.2) In this equation mn is the neutron mass, m p is the proton mass, and the quantities ai are various constants chosen to fit the data. Each of the terms with coefficients ai are used to reproduce the 2 Figure 1.1 The Chart of Nuclides is displayed, with proton number Z increasing along the y-axis, and neutron number N increasing along the x-axis. The nuclei are color coded by decay type. Black represents stable nuclei. Pink decays vis β − , blue through either electron capture or β + , yellow through alpha emission, orange through proton emission, purple through neutron emission, and green through fission. The standard magic numbers are plotted for protons and neutrons. Figure is from [8]. binding energy of the nuclei, without which the mass would simply be the sum of the proton and neutron masses. As such, each of the terms can be related to the nuclear Hamiltonian. The −a1 A term simply relates to the attractive feature of the strong force, while the +a2 A2/3 2 accounts for the missing binding energy at the surface of the nucleus. The +a3 Z1/3 term represents A the repulsive Coulomb force acting on the protons. While these terms are clear from the liquid drop model alone, the remaining two terms are needed to match the experimental data and must be added. The +a4 (A−2Z)2 A term accounts for the fact that stable nuclei tend to have similar 1 term is somewhat more numbers of protons and neutrons, especially at low masses. The ±a5 3/4 A 3 complicated, and represents a pairing force. This force is understood to lower the energy of states where nucleons pair off into total angular momentum and parity J π = 0+ states [4]. a5 is taken to be positive for odd-odd nuclei, which are consequently less stable, and negative for even-even nuclei which are consequently more stable [9]. While the liquid drop model is a simple model of nuclear interactions, it can be of use for a conceptual understanding of stability in nuclei. Each atomic nucleus has a ground state and may have one or more bound excited states. Each nuclear state can be described by its total angular momentum J, its parity, its energy, and its width. The total angular momentum J is the sum of the orbital angular momentum l and spin s of each of the nucleons. In even-even nuclei, the ground state nearly always has total angular momentum and parity of J π = 0+ . This can be understood as arising from the pairing contribution to the Hamiltonian, that lowers the energy of two identical particles coupled together to form a J π = 0+ state. The first excited state of even-even nucleus is normally a 2+ state [4]. Among the most important features observed in nuclear trends are the behavior of nuclei with certain magic numbers of protons or neutrons [11][12][13]. Figure 1.2 shows the energy of the first excited 2+ state for even-even nuclei across the chart of nuclides. One feature that is highlighted on the chart is the large 2+ 1 state energy for most isotopes with proton or neutron number 2, 8, 20, 28, 50, 82, and 126. The features of these magic numbers show up in the systematics of other quantities as well, including natural nuclear abundances, the large number of stable isotopes within an isotopic (isotonic) chain, and electromagnetic transition strengths [4][11]. The quadrupole electromagnetic transition strengths are known to be correlated with excitation energies, and are found to be reduced at magic nuclei [11][14]. The magic numbers were independently identified as arising from closed shells by analogy to atomic structure by Mayer and Jensen [12][13]. This identification has led to the nuclear shell model, which treats nuclei as free particles in a central potential, leading to distinct energy levels and explaining the shell structure. 4 Figure 1.2 The chart of nuclei is shown for even-even nuclei. The nuclei are color coded by the energy of the first 2+ excited state. Nuclei with energies up to 200 keV are colored in green. Those with energies up to 1 MeV are colored yellow. Those with higher energy are colored from orange to red. The magic numbers are highlighted, and demonstrate bands of large 2+ 1 energy. Figure and data are from [8]. 1.2 Shell Model The nuclear shell model arises from an independent-particle model in which the individual nucleons freely move through a central potential V (r) [4]. That this model is appropriate for nuclei is not immediately apparent. Unlike the Coulomb potential in atoms, the nuclear binding potential is not generated from a central particle. The nuclear binding potential is instead generated by the individual nucleon-nucleon interactions. At the same time, the nucleons are packed to saturation, so that it may seem that they should be incapable of traveling freely in the volume of the nucleus. However, the Pauli exclusion principle means that in order for a given nucleon to be interacted with, it 5 must be able to somehow reach the next highest energy unoccupied state. For most nucleons in a nucleus the energy gap is large compared to the scattering interactions, so that the nucleons cannot be excited by collisions. Therefore, the effects of the dense packing do not result in an impediment to the free movement assumption. At the same time, the dense packing means that a given nucleon will experience only the average interaction of all of the nearby nucleons. This interaction can therefore be modeled as a central potential acting on each nucleon independently [4]. The central potential V (r) used in the independent-particle model can be chosen from a number of options, including the Woods-Saxon and harmonic oscillator potentials. Figure 1.3 illustrates how the harmonic oscillator potential can be used to understand the magic numbers. The harmonic oscillator potential has exact solutions that result in degenerate energies for states with the same principle quantum number. These result in shell closures at 2, 8, 20, 40, 70 and 112. However, only the first three shell closures agree well with the observed nuclear magic numbers. It was the insight of Mayer and Jensen that it was possible to identify the magic numbers with energy gaps in the harmonic oscillator if the degeneracy of the orbital angular momentum and total angular momentum states was broken [12][13]. To accomplish this, an attractive l 2 term can be added to the Hamiltonian. This term lifts the degeneracy of the orbital angular momentum states. This term also makes the central potential more constant across all angular momenta by counteracting the increased harmonic oscillator potential experienced by states with higher l [4]. This is consistent with a constant nuclear density. The addition of a spin-orbit coupling term to the Hamiltonian of the form Vs.o. = −VlsV (r)(l · s) lifts the degeneracy of the total angular momenta states, resulting in two total j states for each nl state [4]. If the magnitude of the spin-orbit term is comparable to the harmonic oscillator strength, the magic numbers can be reproduced. The independent-particle model already accounts for a number of features that can be observed in nuclei. For example, the large 2+ 1 energy at the magic numbers can be understood as the increased energy needed to excite nucleons across the shell gap. The spin and parities of nuclei with one nucleon above or below a shell closure can be described as deriving solely from the single particle level structure. For example, 40 Ca has a magic number of neutrons and protons filling 6 Figure 1.3 The energy levels for the harmonic oscillator (left) are shown and compared with those where an l 2 attractive term is added (center) and where that and a spin-orbit term are added (right). The shell gaps in the harmonic oscillator is compared to that of the magic numbers predicted with the inclusion of the l 2 and spin orbit terms. Figure is from [4]. 7 through the d3/2 orbital. The next highest level is the f7/2 , and it can be seen that the ground state spin and parity of 41 Ca is 7/2− , consistent with the expected total angular momentum and the negative parity arising from the l = 3 orbital angular momentum [15]. It is worth remembering that because protons and neutrons are distinguishable, they occupy separate energy levels. This also means that the energy levels for protons will be increased compared to neutrons to account for the Coulomb contribution. At the same time, the single particle model must account for the impact of nucleon-nucleon interactions, which of course do contribute to nuclear structure and are especially important away from the shell closures. Therefore, in the shell model, a non-diagonal Hamiltonian is constructed of the form H = ∑ (Tk +Uk ) + ∑ Vkl k (1.3) k = l + 1/2. The tensor force between two nucleons with total angular momenta j and j is based on the relative orientations of the orbital and spin angular momenta. The tensor contribution to the potential is attractive between nucleons with oppositely coupled angular momenta, i.e. j< and j> . On the other hand, the tensor force is repulsive between nucleons with the same configuration j< and j< or j> and j> [23]. Therefore, changes in the proton configuration in nuclei can affect neutron shell closures through the tensor interaction, and vice-versa. In particular, if the single particle state for the valence protons is j< , this will serve to lower the neutron single particle states with j> and raise those with j< . This can manifest in sudden changes where the valence protons switch configuration [23]. Of particular interest in this work are the shell closures at N, Z = 28. The magic number at 28 is the first arising from the spin-orbit interaction [12][13]. The proton shell gap at Z = 28 varies between N = 28 and N = 40. However, in this case the shell gap increases with increasing neutron number, ranging between 5 MeV to 6 MeV. This can be understood as arising from an attractive interaction between the protons filling the f7/2 orbital and the neutrons in the f5/2 orbital [21]. On the other hand, along the N = 28 isotones the repulsive interaction between the proton in the d3/2 orbital and the neutron in the f7/2 orbital modifies the gap at N = 28. Therefore, the N = 28 shell closure, which is robust for 48 Ca, vanishes in 36 S and 34 Si [21]. In addition to providing insight into the nature of the nuclear force, the evolution of shell properties away from stability can give rise to other phenomena. For example, the disappearance of the N = 8 shell gap was evidenced by the 1/2+ ground state of 11 Be and was confirmed in observations of 12 Be [24]. The ground state occupation of the 2s1/2 orbital in 11 Be with l = 0 means that the valence neutron is not affected by a centrifugal barrier. This in combination with the low neutron separation energy causes the valence neutron to have a large radial extent, called a halo [25]. 11 1.3.2 Halo Nuclei Halo nuclei are a feature of nuclei far from stability, where there is a long tail in the radial density distribution of the nucleus [26]. The extended nuclear distribution is ascribed to one or more valence nucleons with a large radial extent decoupled from the remaining core of the nucleus [26]. The halo formation can then be understood from the perspective of a valence nucleon in a potential well generated by the core. If we assume that the valence nucleon is an s-wave neutron and approximate the potential as a finite square well, it is easy to conceptualize the halo formation. The wavefunction for the neutron outside of the square well potential can be described as [26]: ψ(r) = 2π k e−kr r ekR 1 + kR (1.5) where R is the potential width and k is related to the neutron separation energy (Sn ) as [26]: √ 2µSn k= h¯ (1.6) where µ is the reduced mass of the system. This means that the nuclear density, which is just |ψ|2 is [26]: ρ(r) ∝ e−2kr r2 (1.7) The factor k then determines the overall slope of the decay in the distribution function. In stable nuclei, Sn is on the order of 6-8 MeV, and the k factor is relatively constant [26]. However, far from stability the Sn can be reduced by orders of magnitude. For example, the neutron rich nucleus 27 Ne has a neutron separation energy of approximately 1.4 MeV, while the two neutron separation energy in 11 Li is only 360 keV [26][27]. Therefore in these nuclei far from stability, the factor k can be reduced rapidly as a function of excess neutrons. This results in the distribution function having a larger radial extent [26]. Halo nuclei formation is also dependent on the angular momentum of the valence nucleon, as shown in Figure 1.5. This is because higher angular momentum states have a centrifugal barrier in their Hamiltonians, which serves to confine the wavefunction [28]. As is demonstrated in Figure 12 1.5 as the binding energy approaches 0, the nuclear density s-wave and p-wave states with l = 0 and 1 respectively diverge towards large radii. States with higher angular momentum on the other hand converge to a finite radius. As such, in addition to a low binding energy, halo nuclei require l = 0 or 1 to form [26]. A survey of observed and candidate ground state halo nuclei is presented in Figure 1.6. Consistent with the discussion above, these nuclei can be found far from stability, near the proton and neutron driplines, where the separation energies are reduced [26]. Experiments on these nuclei have determined certain characteristic behaviors of halo structure. The first halo nucleus discovered was 11 Li, which was identified as having a large total interaction cross section for reactions on carbon at 790 MeV/u compared to nearby nuclei [29]. The large interaction cross section can be related to the radial extent of the nuclear matter distribution and thus serves as an indicator of halo properties. Another technique first employed on 11 Li is to measure the momentum distribution of fragments from reactions where the halo neutrons are removed. Figure 1.7 shows the transverse momenta distributions of 6 He produced via reactions of 8 He on carbon and 9 Li produced from reactions of 11 Li on carbon [30]. The Heisenberg uncertainty relationship between position and momentum is: ∆x∆p ≥ h¯ 2 (1.8) The increased radial extension of the halo nuclei increases the uncertainty in position, resulting in a reduction of the uncertainty of the momentum distribution. This is observed in Figure 1.7, where the momentum distribution of 9 Li fragments produced from 11 Li has a component with a σ of 23 ± 5 MeV/c, as opposed to that of the 6 He, which is 71 ± 15 MeV/c [30]. Thus, the reduced width in momentum distributions also serves as a measurement of the radial extension in the nuclear distribution for halo nuclei [26]. In addition to the above, electromagnetic transition strengths have been used to study halo nuclei. Large dipole electromagnetic transition strengths have been measured in halo nuclei through the use of Coulomb-excitation and Coulomb dissociation measurements [31][32]. This is because 13 Figure 1.5 The root mean square radii of a nucleon in a square well potential are presented. The binding energy normalized by the size of the square well (R0 ) is presented on the x-axis. The root mean square radius for each energy is displayed on the y-axis, scaled by the radius of the square well potential. Radii as a function of binding energy are shown for angular momenta l = 0, 1, and 2. The radii diverge as the binding energy approaches zero for l = 0 and 1 but converge for l = 2. Figure is from [26]. 14 Figure 1.6 A focused view of the chart of nuclides is presented. Halo nuclei and potential halo nuclei are colored in. The valence nucleons in the orange colored nuclei are protons. The valence nucleons in the green colored nuclei are neutrons. The lighter shades of green are nuclei for which enhanced interaction cross sections have been measured. Darker shades of green are those nuclei for which both enhanced cross sections and narrow momenta distributions have been measured. Figure and discussion are from [26]. the electromagnetic transition strengths of interactions with nuclei is in part determined by their spatial extent. 1.4 Electromagnetic Transition Strengths Electromagnetic transition strengths are a useful tool for understanding nuclear structure. The electromagnetic interaction is particularly involved in γ-ray decays of nuclear excited states and in scattering with Coulomb fields. The interaction of the electromagnetic field with a nucleon can be expressed in terms of electric and magnetic components decomposed by angular momenta. Therefore, the total electromagnetic interaction with nuclei can be expressed as [5]: Oˆ = ∑ ˆ ˆ O(Eλ )µ + O(Mλ )µ λ ,µ 15 (1.9) Figure 1.7 The transverse momenta distributions for (top) 6 He produced from reactions of 8 He+C and (bottom) 9 Li produced from reactions of 11 Li+C are presented. The incoming beam energy of the 8 He and 11 Li was 790 (20) MeV/u. The solid lines are fitted Gaussian distributions, while (bottom) the dotted line is a wide-momentum contribution to the distribution in 11 Li [30]. Figure is from [30]. 16 The electric transition operator for a nucleon is given simply as [5]: ˆ O(Eλ )µ = rλ Yµλ (rˆi )ei (1.10) In this equation Yµλ is the spherical harmonic for angular momentum λ and projection µ. The ei term is the effective charge of the nucleon under consideration. While for a bare proton, ei would just be the proton charge e and for a bare neutron ei would be 0, effective charges that vary from the bare charges are used in shell-model calculations with a non-interacting core in order to account for effects from center of mass corrections and core polarization induced by wavefunctions outside the model space [5]. Other no-core shell models exist that use the bare charges directly for certain mass regions [33][34]. The magnetic transition operator is given by the sum of the spin coupling and orbital angular momentum coupling terms as [5]: ˆ O(Mλ )µ = 2gli li + si gsi ∇ rλ Yµλ (rˆi ) µN λ +1 (1.11) In this equation, µN = 2me¯hp c is the nuclear magneton. The gli and gsi are the orbital and spin gfactors for the i-th nucleon. These too may vary in shell-model calculations from the free nucleon g-factors for protons (glp = 1, gsp = 5.586) and neutrons (gln = 0, gsn = −3.826) to account for wavefunctions outside the model space [5]. The total electromagnetic interaction strength is just the sum of these matrix elements over all of the nucleons for each electromagnetic transition type π = E, M and each angular momentum λ . The transition rate between two states with initial and final angular momenta Ji , J f and transition ˆ energy Eγ connected by an electromagnetic transition element O(πλ ) is [5] : Ti, f ,π,λ = 8π (λ + 1) Eγ2λ +1 λ [(2λ + 1)!!]2 h¯ 2λ +2 c2λ +1 2 ˆ J f ||O(πλ )||Ji 2Ji + 1 (1.12) The term on the right hand side of the equation is defined as the reduced transition strength for the interaction as: B(πλ ) = 2 ˆ J f ||O(πλ )||Ji 2Ji + 1 17 (1.13) The reduced matrix element is independent of the angular momentum projection of the states, with the factor of 2Ji + 1 serving to average over those states. This means that the B(πλ ; Ji → J f ) must 2J +1 be multiplied by a factor of 2J i +1 to find the B(πλ ; J f → Ji ) [5]. f Electromagnetic transition strengths are often compared with the strength expected if the transition was solely due to a single independent nucleon changing state within the nucleus [35]. These are called Weisskopf units (W.u.) and are defined as [5]: BW (Eλ ) = BW (Mλ ) = 1 4π 10 π 2 2λ 2 2λ 3 1.2A1/3 e ❢♠ (3 + λ ) (1.14) 2 3 (1.2A1/3 )2λ −2 µN2 ❢♠2λ −2 3+λ (1.15) The electromagnetic transitions between nuclear states are determined by parity and angular momentum selection rules. The operator Eλ is proportional to rλ Yλ µ and the Mλ operator is proportional to ∇(rλ Yλ µ ). The parity of the spherical harmonics are determined by the angular component as (−1)λ . The parity of the radial term is positive. In addition, the operator ∇ has a parity of −1. Therefore, the total parity of any Eλ operator is (−1)λ and for any Mλ operator is (−1)λ −1 [7]. Parity conservation therefore requires that electric transitions between states with the same parity must occur via Eλ transitions with even λ , and transitions between states with opposite parity must proceed via Eλ transitions with odd λ . Parity conservation requires that magnetic transitions between states with the same parity must proceed via Mλ transitions with odd λ , while transitions between states with opposite parities must proceed via Mλ transitions with even λ [7]. The angular momentum selection rules further determine the electromagnetic transitions between nuclear states. The angular momentum conservation requires that λ + J f = Ji (1.16) In quantum mechanics, this conservation results in the triangle rule, namely that [36]: J f − Ji ≤ λ 18 (1.17) This rule means that only transitions with angular momentum λ greater than the magnitude of the difference between the angular momentum of the initial and final state can occur. In addition, the fact that the photon is a massless spin-1 particle means that it must have a spin projection of ±1 [7]. Angular momentum conservation then means that γ-ray transitions between states cannot occur via the E0 or M0 operators. In addition, the πλ transition strength decreases rapidly with increasing λ . Therefore the overall transition strength tends to be dominated by the lowest λ transition strength. One exception to this is the competition between E2 and M1 strengths in transitions between states with the same parity and where J f − Ji ≤ 1 [7]. The electromagnetic transition strengths between states can be useful for understanding nuclear properties. The wavefunction of the nucleus is directly involved in the determination of the transition strength, meaning that changes in structure can manifest in changes to these quantities. + For example, the B(E2; 2+ 1 → 0gs ) values are of interest in even-even nuclei, since in almost all + + cases the ground state is 0+ and the first excited state is 2+ 1 . The B(E2; 0gs → 21 ) value can be related to the deformation parameter β of the nucleus by the equation [37]: |β | = 4π 3ZeR20 + B(E2; 0+ gs → 21 ) (1.18) While the electromagnetic transition strengths are the more fundamental quantity, they can also be understood as a function of the geometry of the nucleus. In a simple liquid drop picture, the deformation parameter β represents the quadrupole deformation of the droplet compared to a spherical shape. Mathematically, this is represented where the quadrupole moment is Q0 as [4]: 3 Q0 = √ ZR20 β (1 + 0.16β ) 5π (1.19) Far from stability, electromagnetic transitions strengths are also useful for examining novel features in nuclei. B(E1) transition strengths are related to the expectation value of the nuclear radius, and therefore are enhanced in halo nuclei [26]. For example halo nucleus 11 Be has the largest B(E1) transition strength observed between bound states in nuclei, with a value of 0.36(3) + W.u. [38]. On the other hand, nuclei at shell closures are spherical and the B(E2; 2+ 1 → 01 ) values 19 + are small [21]. Therefore, the experimental B(E2; 2+ 1 → 0gs ) values are indicators of changes in magic numbers. For example, the relatively large B(E2) in 44 S with 28 neutrons is an indicator that the N = 28 shell closure disappears for neutron rich nuclei [21][39]. The transition strength in nuclei can be measured through the probability of exciting a state through the electromagnetic force or the probability of an excited state decaying via an electromagnetic transition. The transition rate for a particular mode Tπ,λ ,i, f is the inverse of the partial lifetime for decays f → i. In cases where the dominant transition is the Tπ,λ ,i, f , then the mean lifetime of the excited state is simply τ = T1 [5]. The transition strength can also be determined from the cross section of electromagnetic interactions with the nucleus. Methods of measuring these quantities are discussed in the following chapter. 20 CHAPTER 2 METHODS FOR MEASURING ELECTROMAGNETIC TRANSITION STRENGTHS The electromagnetic transition strengths between bound states of nuclei can be determined through a number of experimental techniques. The measurement of the lifetimes of excited states can be used to determine the dominant transition strengths allowed by selection rules. In general, for a given state primarily decaying to one lower level, the electromagnetic transition strength would be dominated by the lowest angular momentum transition allowed by ∆L and parity considerations. If states decay to different levels or through multiple electromagnetic moments with similar strengths, the partial lifetimes of the states must be extracted. The partial lifetime between decays to different states can be found from the measured state lifetime and the branching ratio of decays to each of the lower states. Other methods to determine the relative contributions of different B(πλ ) strengths include measurements of γ-ray distributions [40]. Another set of methods used to extract B(πλ ) values are those that relate inelastic cross sections to electromagnetic transition strengths. In particular, Coulomb-excitation is the inelastic scattering of a nucleus in the electric field generated by another nucleus. The cross section for a nucleus to undergo Coulomb-excitation is determined in part by the electric transition strengths (B(Eλ )) between its ground and excited states. In general, measurements of the cross section of the nuclei can be compared to coupled channel calculations from reactions theory to extract the B(Eλ ) of interest. A number of complications present themselves in the measurement of Coulomb-excitation cross sections in rare-isotopes, because the beam energies available at rare-isotope facilities tend to be above the Coulomb-barrier. Therefore, there may be a nuclear contribution to the cross section that must be accounted for. These considerations have been addressed by intermediate energy Coulomb-excitation techniques that account for potential nuclear scattering. In this section, techniques used to measure lifetimes and Coulomb-excitation cross sections are described. Additional techniques not used in this work are also briefly discussed. These techniques not currently used are of interest for understanding choices made in the current experiments and 21 for future measurements of excited states 27 Ne. 2.1 Lifetime Measurements The excited states of atomic nuclei can decay to a lower energy state through various channels, including electromagnetic transitions. It is worth noting that by the Heisenberg uncertainty principle, the intrinsic energy width (Γ) of the state can be related to the lifetime of the state by [41]: τΓ = h¯ (2.1) The overall energy width, and hence lifetime, may be made up in part by different partial widths corresponding to different transition strengths. In this case Γ = ∑ Γi , where each Γi is related to the transition strength of one decay channel. It is also worth noting that the lifetime τ is the mean lifetime of the state, i.e. the amount of time that it takes for the total number of excited states to be reduced by a factor of e. This should not be confused with the half-life, which is the amount of time needed to reduce the total number of excited states by a factor of 2. ˆ the decay For a given transition between initial and final states Ii and I f , and for an operator O, probability is proportional to the overlap of the initial and final states as mediated by the operator. The transition strength amplitude is inversely related to the lifetime of the state, and thus can be related to the overlap mediated by the operator [41]. That is: ˆ i Γi ∝ ψ f |O|ψ 2 (2.2) In the cases of interest in this work, these decays occur via γ-ray transitions mediated by the electromagnetic force. Therefore the lifetimes can be related to the reduced transition strengths B(πλ ). The lifetimes of excited states can be measured through a number of methods suited to different lifetime regions. Figure 2.1 shows the regions accessible to each technique [41]. In this work, lifetimes between 0 and 20 ps are probed. In this region the most appropriate techniques are the Recoil Distance Method (RDM) and the Doppler-shift Attenuation method (DSAM). Both 22 Figure 2.1 The lifetime regions accessible by direct and indirect measurement techniques are presented. The x-axis shows the lifetime region in lifetimes (bottom) and energy widths (top). Figure is from [41]. methods rely on the use of the Doppler-effect to determine lifetimes, which will be discussed briefly. In this work, the Recoil Distance Method was exclusively used to measure lifetimes, and will be the focus of the section. However, the Doppler-shift attenuation method may present future opportunities with rare isotope beams at low energies. 2.1.1 Doppler-shifting The Doppler-effect is fundamental to the Recoil Distance and Doppler-shift Attenuation methods. The classical Doppler-effect is a simple phenomenon whereby the observed frequency of a wave is increased (decreased) for a source moving towards (away from) an observer. As photons propagate as waves, with energy E = hν, the observed energy of a photon is affected by the Doppler-effect. For in-beam nuclear physics experiments, the observed γ-ray energies are Doppler-shifted. This is 23 because the energy of the photon emitted from de-excitation of a nuclear excited state will have the energy of the difference between the levels in the projectile frame of the nucleus. But as in these experimental techniques the nuclei are moving relative to the detectors, the observed energy will be modified by the Doppler-effect. However, as the beams used in this work are at velocities near 30% − 50% of the speed of light, the relativistic Doppler-effect must be used. This is given as: EObs = Ecm γ (1 − β cos θlab ) (2.3) where EObs is the measured energy in the γ-ray detectors, Ecm is the transition energy of the nucleus in the projectile frame, β is the velocity of the nucleus in the lab frame relative to the speed of light, γ is the Lorentz factor 1 1−β 2 , and θlab is the angle between the vector from the source to the position the γ ray is observed at and the velocity of the ion. The relativistic Dopplereffect is a simple consequence of the Lorentz transformation applied to the energy-momentum vector of the photon [42][43]. There are two key features of relativistic Doppler-shifting worth noting for lifetime measurement techniques. The first is that the Doppler-shifting is largest for small θlab , where the cos θlab attains a maximum. The second is that for forward angles the larger the β is, the higher the observed energy is. These two facts determine much of what is required for successful Recoil Distance and Doppler-shift Attenuation Method experiments. One final thing worth noting is the relativistic effect on γ-ray distributions. If the nucleus decays with an isotropic distribution of γ rays in the projectile frame, in the lab frame an observer will see an excess of γ rays along the direction of the motion of the nucleus, and a corresponding decrease in the direction opposite the motion. This can be understood as a Lorentz-focusing of γ rays emitted from nuclei moving at relativistic speeds. Therefore, the geometric efficiency increases for detectors located downstream from the nucleus, and decreases for upstream detectors. In the present experiments, the detectors are placed downstream at forward angles to benefit from the increased efficiency. 24 Figure 2.2 The prototypical Recoil Distance Method setup is displayed. An incoming beam produces excited states at a target. It then travels a distance x over the flight time t f to a stopper foil. Decays occurring while the nucleus is between the target and stopper emit γ-rays with a Doppler-shifted energy. Decays occurring after the nucleus is slowed in the stopper emit with a lower energy. In the figure, the nucleus is stopped in the stopper, while in the present work nuclei are only slowed. For this reason, the foil used for stopping is instead called a degrader in the text. Figure is from [44]. 2.1.2 Recoil Distance Method The basic Recoil Distance Method setup is shown in Figure 2.2. The Recoil Distance Method was first described in [45] and more details including application to rare isotope beams can be found in [44]. The Recoil Distance Method is performed by producing a beam of excited nuclear states at a production target and sending that beam to a degrader that slows down the nuclei. The position between the production target and degrader is measured, along with the velocity, so that the total 25 travel time for nuclei from the target to the degrader is known. The nucleus will have its highest velocity β f ast when traveling between the target and degrader. After passing through the degrader, it will be slowed by interaction with the material and therefore will have a lower velocity βslow . In the original technique, a thick metal stopper was used so that the βslow was 0 [45]. However, in the present work with fast-moving rare-isotope beams, particle identification in a spectrograph is vital. A degrader foil is thus used which only slows but does not stop the beam. The γ-ray energy observed for decays taking place between the target and degrader will be E f ast = Ecm γ f ast 1 − β f ast cos θ f ast (2.4) The energy for γ rays emitted after the degrader will be Eslow = Ecm γslow (1 − βslow cos θslow ) (2.5) It is worth noting that the θ f ast and θslow will be different because of the different locations of emission. ◦ If one assumes an observer at 0 , the Eslow and E f ast will form two distinct peaks. In actual experiments, detectors are set at forward angles, so that the difference in cos θ is minimized. In fast beam experiments, to identify the two peaks, the measured energy is Doppler-reconstructed to account for the in-flight emission. However, to preserve the separation between the two peaks, the β and θlab used in reconstruction are chosen to correspond to only one emission location, generally those corresponding to just after the degrader. In an ideal scenario, the lifetime of the excited state can be found from the ratio of counts in each peak. This is because the total emissions from the state of interest (Iγ ) over a given time range ti to t f can be modeled as [44]: Iγ = tf N0 t /τ (1 − e−t/τ ) = N0 e−ti /τ − e f ti τ (2.6) This equation can be applied to determine the total counts in each of the two peaks. The overall time of decays in the fast peak can be taken as t f = v x starting from ti =0 at the production target. f ast 26 Here, t f is the flight time between the target and degrader, x is the distance between the target and degrader, and v f ast is the velocity of the ion between the target and degrader. The overall time for observation of decays in the slow peak can, in an ideal case, be taken as starting at t f . The total observation time must be finite for the present work, as the traveling nucleus eventually passes out of range of the γ-ray detector. However, if the lifetime of the state is short enough compared to the total travel time, the time of observation of decays after the degrader can be approximated as infinite. This is similar to the original method in which the beam is stopped after the target, allowing observations to take place over an extended period [45]. In this case the total counts observed in the fast peak (I f ast ) and slow peak (Islow ) will be tf I f ast = N0 1 − e− τ Islow = N0 tf − e τ (2.7) (2.8) Then it can be seen that the ratio of counts in the slow peak to overall counts is related to the lifetime by [44]: tf x Islow − R= = e τ = e− vτ I f ast + Islow (2.9) Since the velocity (v) and the distance (x) between the target and degrader are known, the τ value can be extracted. In general it is required to take measurements with various distances between the target and degrader to account for feeding contributions, finite thickness effects, and decays occurring in the foils. The resulting curve is then fit to extract the lifetime τ. In the present work, due to experimental considerations, only two measurements are taken. A short distance measurement is taken to constrain the lifetime, while a long distance measurement is taken to constrain the direct population of states from reactions in the degraders. The lifetimes accessible by this method are therefore related to the velocity of the ions and the distances between target and degrader, quantities which can be measured. As such, over the range of accessible distances and energies, lifetimes on the order of 1 ps to 10 ns are measurable [41]. While the basic principles of the recoil distances are straightforward to understand, there are a number of complications that must be accounted for. Primary among these considerations is 27 feeding, discussed below. Additionally, there are experimental limitations for the efficiency of detectors, the velocity distribution of the beam, finite size and thickness of the target and degrader, which must be accounted for. Additional considerations from γ-ray distributions, relativistic effects on γ-ray distributions, and more are discussed in [44]. In general these additional complications are accounted for by comparison of the measured spectral shapes with simulations incorporating the experimental setup [44][46]. 2.1.2.1 Feeding The decay of higher-lying states can populate the state of interest. This adds an additional component to the analysis, known as feeding. For example, consider a three state system with a ground state, a first excited state with lifetime 1 ps, and a second excited state with lifetime 10 ps, where each state decays into the decay below it. If in the experiment the second excited state is populated along-side the first, it will decay into the first excited state later than the initial population. This nucleus will then decay from the first excited state to the ground state in a time around 1 ps. However, as the first excited state was not populated until on average 10 ps after the initial reaction, the apparent lifetime of the first state will be lengthened to appear closer to 10 ps. This can be modeled as a series of differential equations for each excited state populated in the reaction. The differential equations can be written as [44]: N d ni (t) = −λi ni (t) + ∑ λk nk (t)bki dt k=i+1 (2.10) In this equation ni (t) is the number of nuclei in the excited state i at time t. The decay constants are denoted by λi , where λi = τ1 . The sum on the right hand of the equation runs over all higher-lying i states through the last state considered in the analysis, denoted N. The bki term represents the branching ratio from state k to state i, and is zero for any decays where the k-th state has energy below the i-th state. This differential equation can be solved for the ratio of emitted γ rays before the degrader to 28 the total number of decays as [44]: Ri (t) = Pi e−tλi + N Mki ∑ k=i+1 λi −tλ e k − e−tλi λk (2.11) where Ri is the ratio of of decays before the degrader to the total, λi is the decay constant for the ith state, Pi is the direct feeding intensity of the level i, and where [44]: N k−1 λi λm Mk i − 1 = bki Pk − bki ∑ Mmk + ∑ Mkm bmi λk λk m=i+1 m=k+ (2.12) This equation demonstrates the potential importance of feeding since each feeding state contributes to all of the states below it. Generally speaking, the observed feeding states along with branching ratios can be incorporated into the analysis, but require additional fitting for each τi and branching ratio bik for each state i for those values not previously measured. 2.1.2.2 Modifications to the Method In the present work, the Recoil Distance Method was used to determine lifetimes. In the case of 58 Ni, a modified Recoil Distance Method using two degraders was performed. Modifications to the Recoil Distance Method are discussed below, to explain the interest in using two degraders in an experiment. The standard Recoil Distance Method has been extended to include more than one degrader (stopper). In this case, three regions of velocity exist, denoted as β f ast , βreduced and βslow . This setup enables the use of the Differential Recoil-Distance Method (DRDM), first proposed in [47]. In DRDM, the separation of the three peaks makes it possible to measure both the function R(t f ) for a given time of flight t f , but also the derivative of the function with respect to the time of flight. This allows for a direct measurement of the lifetime as [47]: τi = − Ri j − bi j ∑ Rhi (t) / dRi j (t)/dt (2.13) h where Ri j is the total number of transitions from state i to state j which occur after time t. In addition, the DRDM provides a way to decouple the lifetime of the state of interest from short lived feeding times, as most feeding decays will occur in the β f ast region before the first 29 degrader. Then the ratio of decays in the regions after the first and after the second degrader will have a peak ratio determined primarily by the lifetime of the state of interest [47]. In addition to the Differential Recoil-Distance Method, the use of a target and two degraders makes it possible to use the Recoil Distance Method to simultaneously measure lifetimes in different regions of sensitivity [48]. An example spectrum using this technique is shown in Figure 2.3. If the first degrader is positioned a short distance from the target, and the second degrader a much larger distance, the ratio of peaks between the fast and reduced components can be used to determine a shorter lifetime. On the other hand the ratio of counts in the reduced and slow components can be used to determine the lifetime of a longer lived state. In the present work, the two degraders were kept at the same distance, which was sensitive to the lifetime of the excited state of interest. 2.1.3 Doppler-shift Attenuation Method The Recoil Distance Method does not provide sensitivity to lifetimes below 1 ps [41]. For experiments with fast rare isotopes beams, this is because the minimum thickness for a degrader foil is around 100 µm to provide peak separation between β f ast and βslow . In addition, thick target sizes are needed due to the small production cross sections of rare nuclei. However, the ion transit time over 100 µm is 1 ps, limiting the sensitivity. At the same time, the ion stopping time in matter for standard experiments tends to be on the order of 1 ps, which hinders the Recoil Distance Method. Therefore, in order to measure lifetimes in this region additional techniques must be applied. Among these, the Doppler-shift Attenuation Method is of particular interest for the work presented here. The Doppler-shift Attenuation Method has been used previously to study 58 Ni, and may provide opportunities in the future for studying 27 Ne. As this technique is not used in the current work, however, it will only be briefly discussed. The Doppler-shift Attenuation Method (DSAM) provides information on lifetimes on the order of 10 fs to 1 ps [41]. Figure 2.4 shows a typical DSAM setup. In this method, a target is made, often through implantation, with a backing material behind it. A beam of nuclei is impinged on the target, where it undergoes a reaction into an excited state of the nucleus of interest. The nucleus 30 Figure 2.3 Gamma-ray spectra for 17 C using the Recoil Distance Method with two degrader foils. Three spectra are shown with different distances between the target and first degrader (d1) and first degrader and second degrader (d2). Simulated spectra are compared incorporating an exponential background. The peaks are identified as corresponding to β f ast (f), βreduced (r) and βslow (s). The lifetime of the 1/2+ → 3/2+ state was measured to be 528+21 −14 ps, while that +3.4 + + corresponding to the 5/2 → 3/2 was found to be 21.8−3.3 ps. The different three peak structures demonstrate the ability to measure two different lifetime regions simultaneously. Figure and results are from [49]. 31 then travels through the remainder of the target and the backing material, losing energy along the way, and eventually stopping. The excited state of the nucleus may decay while it is stopping or after it has stopped. If it decays before it has fully stopped, the γ-ray energy will be shifted by the Doppler-effect, but if it has stopped the energy will correspond to the transition energy. The velocity of the nucleus after the mean lifetime (v) ¯ can be determined from the Dopplershift observed in the average energy of the observed γ ray (E¯s ). DSAM measurements in general are performed using beams with non-relativistic velocities, and so the energy can be related by the equation [41]: E¯s = E0 (1 − vc¯ cos θγ ) (2.14) As such, the velocity can be determined and compared to the incoming velocity of the nucleus v0 . The Doppler shift attenuation factor is then defined as F = vv¯ . The lifetime of the nucleus can 0 then be extracted by comparison to a theoretical distribution of F(τ) as a function of the lifetime of the nucleus. The distribution F(τ) is a function of the stopping time of the recoiling nucleus in the stopping material (and the target). The stopping times of ions in matter in DSAM experiments tends to be on the order of 10−13 s, and therefore provides sensitivity to lifetimes in that range [41]. The stopping functions and corresponding Doppler-shifted spectra can be simulated using MonteCarlo methods to determine the lifetimes [50]. More information on DSAM and other lifetime measurement techniques can be found in [41]. 2.2 Coulomb-excitation Measurements The reduced transition strengths B(πλ ) determined from lifetimes are related to the overlap of the excited state wavefunction (ψ1 ) with the wavefunction of the state reached by the decay (ψ0 ) as ˆ 0 . Due to the Hermitian nature of observable operators, mediated by the relevant operator, ψ1 |O|ψ ˆ 1 , representing now the overlap for excitations from ψ0 to ψ1 . this must be equal to ψ0 |O|ψ Therefore the probability of excitation of the ground state of a nucleus into an excited state by the electromagnetic field can be related to the reduced matrix elements B(πλ ). The primary means of 32 Figure 2.4 A prototypical DSAM setup is presented. The incoming particles A react in the target to induce excited states in the recoiling nucleus B. The γ-ray emitted in the excited state decays in the backing or target, with the location contingent on the lifetime of the state and the stopping time of the nucleus in the backing (target). The energy of the emitted γ ray will be shifted depending on the velocity of the nucleus B at the time of decay. The mean Doppler-shifted energy can be used to extract the mean velocity compared to the incoming velocity (F = v/v ¯ 0 ). This is compared to the stopping time extracted from theory to determine the lifetime. Figure is from [41]. 33 determining this interaction directly is through the inelastic scattering of nuclei in the electric field generated by other nuclei, known as Coulomb-excitation [41][51]. 2.2.1 Safe Coulomb-excitation The use of Coulomb-excitation to measure electromagnetic transition strengths was first performed using beam energies below the Coulomb-barrier. The Coulomb barrier is the energy needed for projectile nucleus to overcome the electric repulsion of the target nucleus and allow for direct nuclear-nuclear interactions. This energy is classically [51]: Z Z e2 EB = 1 2 R (2.15) where Z1 , Z2 are the proton number of the projectile and the target nucleus respectively, e is the electron charge, and R is the effective radius of the interaction. R in general can be approximated 1/3 as R = 1.2(A1 1/3 + A2 ) fm, which is just the sum of the radii of the two nuclei. In standard kinematics, Coulomb-excitation is generally performed by using the projectile to excite the target nucleus, which is the formalism adopted here. However, in the present work experiments are performed in inverse kinematics, where the projectile is excited in the electromagnetic field of the target. At safe energies, the only interaction affecting the target nuclei is the electromagnetic field generated by the projectile. In general, the electric component of the field is dominant, as the magnetic field generated by the moving charged particle is reduced in strength from the electric component by a factor of β 2 [51]. Thus the B(Eλ ) transition elements can be extracted from the cross section of the interaction. Coulomb-excitation is in particular frequently used to measure + B(E2) values, due to the prominent transition from 0+ gs → 21 in even-even isotopes. In the non-relativistic approximation the cross section can be calculated as [51]: σEλ = Z1 e 2 h¯ vi Z1 Z2 e2 m0 vi v f −2λ +2 B(Eλ ) fEλ (ηi , ξ ) (2.16) where vi is the velocity of the beam, v f is the outgoing velocity of the projectile, m0 is the reduced mass of the projectile and target ,and fEλ (ηi , ξ ) is a function of the strength of the interaction and 34 change in velocity over the collision calculated in [51]. For a B(E2) this can be simplified to [41]: σE2 (E) = CE2 (E − ∆E )B(E2) fE2 (ηi , ξ ) (2.17) where CE2 is a function of the charge and mass of the target, E is the energy of the beam and ∆E is excitation energy of the level multiplied by (1 + A1 /A2 ). As such, a measurement of the cross section of Coulomb-excitation gives direct access to the transition strength B(Eλ ). Cross sections can be measured by γ-ray detection, where the overall count of measured γ rays corresponding to the energy of interest, scaled by the efficiency of the detector can be compared to the total number of projectiles and the number of nuclei in the target interacted with. In an experiment where the γ rays are measured in coincidence with ions of interest, this can be written as: σexp = Ncoin Ninc Ntarget ε(γ)ε(p) (2.18) where Ncoin is the number of coincident measurements between the projectile and γ ray, Ninc is the number of projectiles, Ntarget is the number of target atoms per unit area, and ε(γ) is the efficiency of the γ-ray detectors, and ε(p) is the efficiency of the ion detectors. 2.2.2 Intermediate Energy Coulomb Excitation In safe Coulomb-excitation, there is no contribution from the nuclear-nuclear interaction which is hindered by the Coulomb-barrier. However, beam energies at rare-isotope facilities are generally above the Coulomb barrier, including those used in this work. To study such nuclei, intermediate energy Coulomb-excitation techniques have been developed. Intermediate energy Coulombexcitation is relevant for nuclei with the relativistic energies of secondary beams produced at rare isotope facilities, meaning that there is no loss of yield from degrading beam energies to produce beams below the Coulomb barrier. The intermediate energy Coulomb excitation technique furthermore uses thick targets that offset the low intensity of rare isotope beams. Finally, the re- 35 action products can be identified in downstream detectors to tag coincident γ-ray signals, which is important for secondary beams with limited purity. The first of two methods discussed here involves selection of the impact parameter to exclude nuclear contributions. The other involves determining the nuclear contribution to the reaction through measurement on a low Z target and then using this to constrain the nuclear contribution from a high-Z target in order to extract Coulomb-excitation cross sections. 2.2.2.1 Impact Parameter In intermediate Coulomb energy measurements, the projectile nucleus is excited in the electromagnetic field of the target [52]. A selection for a minimum impact parameter (bmin ) can be made by selecting events below a maximum scattering angle in the center of mass frame as [52]: a cm /2) bmin = 0 cot (θmax γ (2.19) Z pro Ztar e2 a0 = m0 v2 (2.20) where This relationship allows for the selection of interactions between the projectile and target corresponding to an impact parameter larger than the nuclear interaction radius. This is done by finding the scattering angle θlab between the photon and the beam particle, which is related to the center of mass angle as [52]: tan θlab = where γ is the relativistic factor 1 1−β 2 sin θ cm γ (cos θ cm + β cm /β pro ) (2.21) , β cm is the center of mass velocity, and β pro is the pro- jectile velocity in the laboratory frame. The restricted cross section for only impact parameters greater than the selected bmin can be related to the B(πλ ) by [52][53]: σπλ ≈ Z pro e2 h¯ c 2 π −2 e2 b2λ min 36 B(πλ ) ×C(λ ) (2.22) where C(λ ) is 1/(λ − 1) for λ ≤ 2 and is 2 ln(ba /bmin ) for λ = 1. ba is defined as γ h¯ cβ /Eγ . As such, the electromagnetic transition strength can be directly calculated from the cross section [52]. 2.2.2.2 Two Target Measurement While the selection of impact parameters is well established for measuring electromagnetic transition strengths, it cannot be used in all cases. For example, yield or experimental constraints can prevent the use of the angular selection, which is the case in the present work where the maximum scattering angle is too small to constrain bmin . In this case, it is possible to constrain or determine the nuclear contribution through the use of two targets, one with a large atomic number Z and one with a small Z. The low-Z material, for example carbon, can be used to determine the nuclear contribution. This is because the relative impact of the Coulomb interaction should be minimal and the cross section should be dominated by the nuclear contribution. The nuclear contribution measured from the low-Z target can provide a good estimate of the nuclear contribution for reactions in the high-Z target through reaction theory. The high-Z target, for example lead, should on the other hand have a substantial Coulomb contribution, and the cross section should be dominated by the Coulomb component. For example, this technique has been performed in [54][55]. In general, the relative contribution of nuclear and Coulomb interactions to the cross section is constrained by using a coupled-channel equation program [56][57]. These calculations are performed to determine the cross section of the Coulomb-excitation into one or more states of interest, and include couplings and excitations between the states of interest. Additionally, the interactions into channels not explicitly included in the calculations are approximated using an optical potential derived from measurements of other nuclei [58]. In these coupled-channel calculations, the structure of the nucleus of interest can be treated in a collective mode or by setting matrix elements directly. These allow for the calculation of the nuclear and Coulomb contributions independently. The calculations can then be performed to vary the interaction strength of the nuclear component to match the observed cross section on the light Z target. With this transition strength fixed, 37 the calculations can then be performed varying the Coulomb contribution to the transition strength, the B(Eλ ), to reproduce the cross section of the high Z material. This allows for the extraction of the B(Eλ ) transition strength. It is worth noting that this method, unlike lifetime measurements requires a theoretical treatment of the scattering process. The extracted B(Eλ ) will depend on the choice of optical potential and the method of determining the relative nuclear contribution to the interaction. The uncertainty in theory can be found by comparing against various optical potentials, and trying various assumptions for the relationship between the Coulomb and nuclear interaction strengths [55]. More on these calculations can be found in [56][57]. 38 CHAPTER 3 BEAM PRODUCTION AND MEASUREMENT DEVICES In this work, electromagnetic transition strengths were determined by lifetime and intermediate energy Coulomb-excitation measurements. The Recoil Distance Method and Coulomb-excitation experiments employed in this work require the use of specialized equipment. In particular, these experiments involve the production of fast rare-isotope beams, the measurement of γ-ray energies and yields, the precise placement of target and degrader foils in the beamline, and the identification of the reaction products of interest. These experiments were performed at the National Superconducting Cyclotron Laboratory, which hosts equipment that make these measurements possible. In this chapter, the devices used at the National Superconducting Cyclotron Laboratory will be discussed. The rare isotope beams were provided by the Coupled Cyclotron facility and the A1900 fragment separator, which are discussed first. The reaction products were identified using the S800 Spectrograph, so that γ rays emitted from those products could be identified in coincidence. Gamma rays were measured using the Gamma Ray Energy Tracking In-flight Nuclear Array (GRETINA) high purity germanium detector array. Finally, the target and degrader foils were mounted on the TRIPLEX plunger, which was used to mount and precisely separate the foils. 3.1 National Superconducting Cyclotron Laboratory The experiments discussed in this text were performed at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University (MSU). Figure 3.1 from [59] is provided to show the outline of the layout at the NSCL for accelerating a primary beam from an original ion source and producing a secondary rare isotope beam. The NSCL is a NSF sponsored user-facility that provides rare-isotope beams at energies around 60 − 120 MeV/u. The NSCL has recently installed a reacceleration facility that produces rare isotope beams at energies of 3 − 5 MeV/u [60]. The 39 Figure 3.1 The layout of the NSCL showing the ion sources, K500 and K1200 cyclotrons, and the A1900 is shown. Figure is from [59]. facility also hosts experimental devices used in measurements presented here. This section will describe the process by which rare isotope beams are produced at the NSCL [61]. The primary beam is produced from a stable or very long lived isotope. The stable isotope is ionized and sent to the coupled cyclotrons. There, it is accelerated to approximately 50% of the speed of light. The primary beam is then sent to a production target, where the beam is fragmented into various nuclei. The secondary rare isotope beam is produced by filtering the reaction products in the A1900 fragment separator [59]. This results in a rare isotope beam with energies corresponding to approximately 40% of the speed of light to be delivered to the experimental vault. The beam production process is discussed in greater detail in the sections below. 3.1.1 Ion Source The production of the primary beam at the NSCL begins with the ionization of atoms, which is accomplished using one of the available ion sources, the Superconducting Source for Ions (SuSI) [62] and ARTEMIS [63]. SuSI and ARTEMIS are electron cyclotron resonance ion sources. These produce a charged plasma that can be accelerated in the cyclotron. The ion source used in this work was SuSI, which uses superconducting magnets to produce the magnetic fields needed for electron 40 cyclotron resonance [62]. In an electron cyclotron resonance (ECR) ion source, the method of ionization is to remove electrons one at a time from an atom until the desired charge state is reached through collisions with free electrons [64]. This is accomplished by creating a gas of the isotope to be ionized, generally at or below 10−6 torr [64] to avoid ion-atom collisions which serve to lower the overall ionization of the plasma. A magnetic field is generated across the gas, which serves both to confine the plasma and to enable the ECR heating. As the ions and electrons in the plasma are charged particles, they will be confined by the magnetic field as the Lorentz force, FL = qv × B, will cause them to rotate in circles at the cyclotron frequency ωc = qB m. However, to trap the ion in both the transverse and parallel direction, a magnetic mirror configuration is used [65]. The magnetic field in a magnetic mirror varies in magnitude along the length of the trap, attaining a maximum at the edges and a minimum in the center. However, particles with directions of velocity below a critical angle from the magnetic field will not be trapped. The overall confinement time is finite due to ion-ion collisions which can change the velocity of the ion [65]. An electric field is then used to guide the ions that become unconfined to their destination, in this case the K500 cyclotron [64]. One additional consideration that motivates the use of ECR ion sources is that higher energy free electrons, with energy on the order of 1 keV, have larger cross sections for removal from an atom. At the same time, higher energy ions decrease overall beam quality by decreasing the confinement time and also can increase the energy spread of the resulting beam [64]. It is therefore necessary to add energy selectively to the system, preferentially energizing the electrons. This is possible by applying an RF field at the cyclotron frequency of the electrons in the plasma. The cyclotron frequency of the electrons is much larger than that of the ions due to the small mass of the electron (0.5 MeV) compared to ions (> 1000 MeV). Hence, the energy is imparted selectively to the electrons. Therefore the RF field preferentially increases the temperature of the electrons in the plasma without exciting the ions [64]. This increases the free electron-ion collisions and thus the rate of ionization. 41 A final detail of the ion production at the NSCL worth noting is that the ECR method can be applied directly to a gas. However, for a metallic source, a plasma must be produced by another means [64]. At the SuSI source this is accomplished using an external oven, which heats the metal until it produces a gaseous form which can then be used in the ion source directly. The isotopes used in the experiments discussed here were gaseous 78 Kr and metallic 48 Ca. For these experiments, the ionization at SuSI was not complete, for example the ionization of the 48 Ca was only to 8+ . Instead, the ions were fully stripped at the coupled cyclotrons. 3.1.2 Coupled Cyclotrons The K500 and K1200 coupled cyclotrons are used to accelerate the primary beam to energies corresponding to nearly 50% of the speed of light. As cyclotron accelerators, these operate on the principles first developed by Lawrence in [66]. The cyclotron accelerator has an advantage that particles can be accelerated in a compact area, with the voltage applied repeatedly to a single ion, thus allowing for a large energy transfer without the need for prohibitively high voltages [67]. A cyclotron consists of two fundamental devices, a magnetic field over the entirety of the device, and electrodes called dees which induce an rf electric field that can be used to accelerate the particles [67]. The magnetic field is applied perpendicular to the electric fields, so that the ions will experience a Lorentz force. As in the ion source, the particles will rotate in a circular orbit with radius ρ and frequency qB m . The radius of the circle can be derived for non-relativistic particles as : F = qv × B = mv mv2 →ρ = ρ qB (3.1) where m is the mass of the ion, q is the charge of the ion, B is the applied magnetic field, v is the velocity, and F is the force. The ions are accelerated by the electric field between the dees. The rf frequency for the applied electric field is set to the cyclotron frequency of the ion in the constant magnetic field. Therefore, as the ion travels between the dees, it will experience an electric field that is always constant relative 42 to its direction of motion. This means that once the ion is traveling in the correct direction, it will continuously experience an increase of energy, and thus velocity. The dependence of the radius on the velocity relates the maximum energy of the beam to the radius of the cyclotron. As the ion is accelerated, the radius of the ion’s path through the magnetic field will increase. At the maximum energy it reaches the outermost part of the cyclotron and is then guided into a beamline. As such, the energy of the beam is effectively determined by the ion’s charge and mass, the magnetic field strength and radius of the cyclotron. The overall increase in energy from acceleration of the cyclotron can be calculated as [67]: ∆E = q2 B2 R2 mc2 (3.2) The coupled cyclotron facility at the NSCL enables the production of higher energy and intensity beams than the use of the K1200 Cyclotron alone. The ion source can be used to select more abundant charge states and send those to the K500. The K500 Cyclotron is used to accelerate the partially stripped ions to velocities near 10% of the speed of light in approximately 230 turns. After reaching their final velocity in the K500, the ions are guided to the K1200 Cyclotron and injected through a stripper to fully strip the ions [61]. The fully stripped ions are then accelerated in the K1200 cyclotron to energies corresponding to velocities of 50% of the speed of light and sent to production target and A1900 fragment separator, where the radioactive secondary beam is produced [61]. It is worth noting that the beams produced by the Coupled Cyclotrons are necessarily not continuous, but are discrete in timing. This is due to the rf electric pulses needed to accelerate the ions, which creates discrete windows of acceleration and hence transfer of ions from the cyclotrons to the beamline. 3.1.3 Production and Separation of Fragments The secondary beam of radioactive ions is produced by impinging the primary beam on a target to induce fragmentation reactions. In a fragmentation reaction at the NSCL, the fast beam interacts 43 with the target nucleus, causing the beam to split into multiple nuclei, including free nucleons. The fragments from this reaction continue at a significant portion of the energy of the incoming beam, allowing them to be used as a secondary beam of interest [61][68][69]. The target generally used at the NSCL is beryllium, which is chosen due to its high number density, which makes fragmentation reactions more likely. The fragmentation reaction produces many of the possible nuclear products that can result with fewer neutrons or protons than the incoming particle [68]. This allows for the production of radioactive isotopes, as nuclei closer to the driplines can be produced in reactions that remove more protons than neutrons or vice-versa. This however leads to the outgoing product of the fragmentation containing a combination of numerous isotopes, only a very few of which may be of interest at any time. The A1900 fragment separator is therefore employed, which allows for the selection of the reaction products of interest. 3.1.3.1 A1900 Fragment Separator The A1900 fragment separator consists of 28 superconducting iron-dominated magnets [59]. Twentyfour of these are focusing quadrupole magnets. The remaining four are dipole magnets [59]. The magnetic field in each magnet can be selected by changing the applied currents. In addition an aluminum wedge is placed between the first and second pairs of dipole magnets [59]. The first set of dipole and quadrupole magnets before the wedge are used to separate out the fragments by their momentum to charge ratio. This is because the bending radius is determined by ρ = mv qB . Since only particles with a certain radius will make it through the dipoles and subsequent slits, particles of constant momentum to charge ratio can be selected by taking a particular magnetic field and applying it. The rigidity of the beam, or Bρ, is given by mv q . Since the velocity tends to be similar for each reaction product from fragmentation, the rigidity is determined by the ratio of mass (A) to charge (Z) of the fragments. The wedge is used to slow the particles down differentially by a factor proportional to the square of the atomic number (Z 2 ) of the ion. This second level of selection means that ions with 44 the same selected incoming Bρ but with different atomic numbers will have different outgoing momenta. Therefore, the rigidity of the outgoing ions will be differentiated according to their atomic numbers. The second set of quadrupole and dipole magnets is then used to select a new rigidity determined by the energy loss in the fragments of interest. This allows for the purification of the rare isotope beams used in experiments [70]. After the production of the secondary beam, the ions are sent to a scintillator called the extended focal plane scintillator (xfp). The scintillator is used for timing measurements of the secondary beam at the A1900 focal plane. These timing measurements are combined with other timing measurements for the beam to determine a time of flight. The time of flight can be used to identify the incoming ions. The selected ions are then sent to the rest of the facility through dedicated beamlines. In the following experiments, the secondary beam was sent to the S3 vault, where the S800 Spectrograph was used in conjunction with the GRETINA array and the TRIPLEX. 3.2 S800 Spectrograph The S800 Spectrograph is a piece of equipment at the NSCL that enables the identification of reaction products. Figure 3.2 from [71] shows the layout of the S800 Spectrograph in the S3 vault. As a spectrograph, the S800 provides position and momentum information for particles at the target position reconstructed from the focal plane particle information using inverse transfer maps [71]. The S800 Spectrograph can also be set up in conjunction with other detectors, making correlated data sets. In the experiments that are discussed here, the secondary beams are impinged on another target, causing them to undergo various reactions to produce the isotope of interest. This has the secondary effect of creating other isotopes, which may contaminate the results of the experiment. In addition, the beam profile information can be used to improve Doppler-correction. Therefore the S800 was employed to identify particles, reconstruct their trajectory at the position where the reaction of the secondary beam on target occurs, and tag coincident γ-ray measurements. 45 Figure 3.2 The layout of the S800 is presented, showing the analysis line and the spectrograph. The figure is from [71]. In this section, the features of the S800 Spectrograph are described along with calibrations common to the experiments discussed in this work. 3.2.1 Magnets and Optics The S800 is made up of a series of superconducting magnets in the analysis line and in the spectrograph itself. The magnets are used to guide the beam through the S800. In the spectrograph, as in the A1900, a magnetic field is used to separate out ions by their magnetic rigidity. This creates a position dependence on momentum at the S800 focal plane. Furthermore, since the time of flight through the S800 is therefore determined by the cyclotron frequency ω = qB m , the time of flight can be used to separate particles out by their charge to mass ratio. The S800 is thus equipped with a number of detectors which allow for the extraction of time-of-flight, position, and other quantities. The S800 can be operated in two modes: the focus mode, and the dispersion matching mode. In the focus mode, the S800 is achromatic in the analysis line. The magnets do not correct for intrinsic momentum distribution at the focal plane. This decreases the overall resolution of the S800, but provides for a momentum acceptance of d p/p = 5%. This can be compared to the 46 dispersion matching mode, in which the S800 is achromatic in the focal plane. This reduces the impact of the initial momentum distribution of the incoming beams. However, this also leads to a more limited momentum acceptance of d p/p = 1%. In this work, the focus mode is used because the yield is more important than the energy resolution [71]. In addition, although most information from the S800 comes from detectors installed in the spectrograph, which is downstream from the target location, it is possible to reconstruct position and energy information at the target position. The fringe effects of the magnets are modeled using a function B(z) = 1 1+Σai zi fit to the magnetic fields. This information is then used in the code COSY Infinity [72], which calculates a transfer map for particles through the S800. The transfer map is a 4 × 4 matrix that relates the quantities at the focal plane (x f , a f , y f , b f ) to the quantities at the target position (dt , at , yt , bt ). Here x is the displacement of the ion in the dispersive plane, y is the displacement of the ion in the non-dispersive plane, a is the trajectory angle in the dispersive plane, b is the trajectory angle in the non-dispersive plane, and d is the energy of the ion. In all cases the f subscript denotes the quantity at the focal plane, and the t subscript denotes the quantity at the target position [71][72]. These quantities are also sometimes denoted as afp, xfp, ata, xta, etc. to represent quantities at the focal plane ( f p) and target (ta) positions. 3.2.2 Timing Scintillators The S800 Spectrograph incorporates plastic scintillators to provide time of flight measurements through the S800. When ions interact with the scintillator, they excite the molecules of an organic scintillator material dissolved into the plastic. The excited molecules de-excite by emission of florescent light. Photo-multiplier tubes are positioned on the scintillators [73]. The photo-multiplier tubes convert the fluorescent photons into electrons, and then amplify the current. The measured pulse time can then be taken as the timing of the interaction. As plastic scintillators have a short decay constant, generally on the order of 1 ns or less, they are suited for measuring time of flight measurements [74]. There are two locations where plastic scintillators are used in the S800, at the object position 47 in the analysis line (OBJ) and at the end of the spectrograph (E1) [71]. At the E1 position, a 1 cm thick plastic scintillator is placed. The time of flight between the OBJ and E1 scintillators can thus be calculated from the difference in the timing between the two scintillators. In addition, the E1 scintillator is used as the S800 trigger, from which a coincidence window is available to correlate events between secondary devices and the S800. 3.2.3 Cathode-Readout Drift Chambers The S800 has two Cathode-Readout Drift Chambers (CRDC) within the spectrograph portion, CRDC 1 at the focal plane and CRDC 2 located 1 m downstream, which are used to provide position and trajectory information for the particles. The CRDCs are detectors filled with gas, a mixture containing 80% of CF4 and 20% of C4 H1 O [73]. They have an active depth of 1.5 cm and an active area of 30 cm in the non-dispersive plane by 59 cm in the dispersive plane. When an ion traverses the CRDC, it ionizes the gas. The resulting free electrons are collected on an anode wire, which is located between cathode pads behind and in front. There are 224 pads, each of which is 2.54 mm wide. The induced charge on the cathodes is read out and fit to a Gaussian to provide position information in the dispersive direction [73]. The position information in the non-dispersive direction is determined from the drift time relative to the E1 scintillator timing. It takes longer to get to the wire from farther away, with drift times varying between 0 to 20 µs [75]. CRDC 1 is used for determining the x f and y f positions directly, due to its location, while CRDC 2 is used to determine the trajectory by comparing the x and y positions downstream 1 meter to the x and y positions at the S800 focal plane. 3.2.4 Ion Chamber The S800 Spectrograph uses an Ionization Chamber to identify nuclei by their atomic number (Z), as the energy loss in the chamber is proportional to the square of the atomic number of the nucleus. The ionization chamber is located downstream of the CRDCs and upstream of the scintillators. 48 The ionization chamber is a Frish grid design and consists of sixteen parallel-plate ion chambers, which are filled with P10 gas of 90% argon and 10% methane [73][75]. The ions passing through the chamber will cause electrons to be ejected from the gas, ionizing it. As described by the Bethe formula, the total energy deposited is proportional to the square of the atomic number as [74]: − dE 4πe4 Z 2 NB(v) = dx m0 v2 (3.3) In this equation, Z is the atomic number of the ion, e is the electron charge, m0 is the electron mass, v is the velocity of the ion, N is the number density of the absorbing material, and B is a function of the velocity of the ion and the ionization properties of the absorbing material [74]. The electrons and ions are collected at anode-cathode pairs in the 16 chambers. As the amount of free-electrons and ions excited is proportional to the energy deposited by the incoming nuclei, this can be expressed as an energy loss of ions in the detector, and thus provide separation based on their atomic number. 3.2.5 IsoTagger The S800 Spectrograph has been equipped with the IsoTagger configuration to identify long-lived isomeric states in nuclei [76]. The IsoTagger device consists of a stopper at the end of the S800 and detectors used to identify β particles and γ rays. The stopper is a 6.35 mm thick Aluminum plate [76]. The detectors are 32 closely packed CsI(Na) crystals coupled to photomultipliers. The CsI detectors can be used independently as a hodoscope to provide charge state identification [77]. The travel time for ions from the target position to the IsoTagger is 150 ns. Therefore, the IsoTagger can be used to identify states with lifetimes comparable to or longer than 150 ns. The IsoTagger is used with two different triggers. The first trigger is a coincidence trigger with a 20 µs window to record γ rays measured in the CsI coincident with ions measured at the E1 scintillator. The second trigger condition is for the individual CsI crystals, which are not tagged, but are instead timestamped to allow for offline correlation analysis [76]. The latter trigger is necessary to observe isomers with long lifetimes. 49 3.2.6 Calibrations Each of the detectors used in the S800 Spectrograph must be calibrated for use in the experiments. The calibrations from the experiment e14045 investigating 27 Ne are demonstrated below to describe the calibration procedure used in both experiments. 3.2.6.1 Timing Scintillators The time of flight between the scintillators at the object position of the S800 and the E1 scintillator at the end of the spectrograph are used for particle identification. However, the fragments identified in the S800 have a momentum distribution that means they do not all take the same path in traveling through the S800. If the fragment has a large trajectory angle, the flight time will be larger than for an ion with the same magnitude of momentum traveling a straight line. Therefore a lower (higher) magnitude of momentum will increase (decrease) the time of flight. This means that the x position and angle of trajectory a, which are reflective of the momentum distribution, are correlated with the time of flight. In order to identify the nuclei unambiguously, the position and trajectory dependence of the time of flight must be factored out. Therefore, corrected times of flight are calculated using a linear function to relate the displacement and trajectory of the ions in the dispersive direction in the focal plane (xfp and afp respectively) to the various timing scintillators. These corrections are of the form: Tcorr = Tuncorr +C1 × ❛❢♣ +C2 × ①❢♣ (3.4) The coefficients Ci are chosen to make each corrected time-of-flight independent of the xfp and afp. These corrections are done for the time of flight from the E1 scintillator to the A1900 XFP scintillator and to the S800 OBJ scintillator. An example of the calibration for the time-of-flights is presented below in Figure 3.3. 50 Figure 3.3 Uncorrected (top) and corrected (bottom) graphs are shown for dispersive angle in the focal plane compared with the time of flight from the S800 Object to E1 scintillators. The corrected graph shows no dependence of the time of flight on the angle, compared with the uncorrected graph which shows a substantial correlation. Similar corrections are performed for the A1900 XFP scintillator for both the dispersive angle and dispersive displacement. 51 3.2.6.2 Ion Chamber The ion chamber, consisting of sixteen individual parallel plate chambers, needs to be gain matched to ensure that no one of the sixteen sub-components is over-represented in determining the energy loss in the chamber, which could lead to less accurate particle identification. To address this, the 16 ion chamber pads are calibrated to provide equal contributions to the energy measurement based on a sample from the incoming secondary beam. The charge deposited and collected in the ion chamber is position dependent, as the number of electrons and ions collected depends in part on the momentum of the incoming ion, which is correlated with the dispersive position. The ion chamber output must therefore be corrected for the position of the ions in the chamber to ensure sensitivity to the atomic number. These corrections are incorporated using a simple exponential to model the relationship between the energy deposited to the energy detected as Ecorr = Euncorr × e−C1 (x−x0 ) (3.5) The parameters C1 and x0 can then be fit to a correlation graph of energy detected vs position for a particular isotope to improve the particle identification. 3.2.6.3 CRDC The CRDCs are used to measure the position and trajectory of fragments in the S800. However, the raw CRDC signals are sensitive to changes in the temperature and pressure of the gas, which must be corrected for. The CRDCs are calibrated for position and trajectory by using two metal masks, each of which is placed in front of one of the CRDCs during a portion of the experiment. A comparison of the calibration data to the expected mask positions is shown in Figure 3.4. The masks have holes with known positions in them, which allow the beam through to interact with the CRDCs. These interactions can then be calibrated against the known positions of the holes in the masks. This calibration allows for a reconstruction of positions and trajectories in the dispersive and non-dispersive 52 Figure 3.4 The uncalibrated data (left) and known mask positions (right) are shown. The regions of activity in the data correspond to the positions in the mask, and can be compared by the number of holes in each region to uniquely identify the locations. The arbitrary units for the data (left) are distinguished from the mm units for the mask (right). The mask image is from [78]. planes. This allows for a reconstruction of those positions at the target and a correlation between the energy of ions at the target and their position in the dispersive plane. Because of changes to the pressure and temperature of the gas among other reasons, the positions in the CRDC may drift over the course of the measurement. This is accounted for on a run by run basis, where it is assumed that the beam spot is stable over the course of the experiment. The values for each of the calculated positions and angles at the target position are then centered at zero by adjusting the parameters relating the CRDC readouts to the positions. In addition, as in the ion chamber, the 224 pads need to be gain matched to ensure that each pad is equally weighted. An example of the gain matching for the CRDC pads is shown in Figure 3.5 for the first CRDC at the focal plane. The uncalibrated and calibrated spectra are compared, and the clear improvement in consistency after gain matching is apparent particularly at the highest numbered pads. 53 Figure 3.5 The maximum response amplitudes for each pad are histogramed for the uncalibrated (top) and gain matched (bottom) CRDCs. The calibration prevents any one pad from having too much influence over the resulting position measurement, and can be seen by the roughly straight line in the amplitudes plotted out by pad number in the bottom figure. This is performed for both CRDCs to ensure proper calculation of both positions and angles. 54 3.3 GRETINA The precise measurement of γ-ray energies and interaction points is vital to the Recoil Distance Method and the Coulomb excitation experiments described in this work. These precise measurements are enabled by the Gamma Ray Energy Tracking In-Flight Nuclear Array (GRETINA). GRETINA is a high purity germanium detector array designed for in-beam measurements to provide high efficiency and excellent energy and position resolution [79]. In this section, the basics of γ-ray interactions with matter and the principles of high purity germanium detectors (HPGe) will be discussed. The GRETINA array will then be described in greater detail. 3.3.1 Gamma-ray Interactions With Matter To comprehend the γ-ray spectra from Germanium detectors, it is important to understand the three primary types of interactions between γ rays and matter. Namely, these are photoabsorption, Compton scattering, and pair production. The energy regions where each mechanism has the largest cross section is shown in Figure 3.6. Each of these γ-ray interactions result in the production of high energy-electrons. It is also worth noting that Rayleigh scattering is a form of elastic scattering between photons and atoms. However, Rayleigh scattering is unlikely at energies near 1 MeV and does not impart a significant amount of energy into the material. Therefore, Rayleigh scattering is not generally important for γ-ray detection [74]. 3.3.1.1 Photoabsorption Photoabsorption occurs when a γ-ray is completely absorbed by an atom. In this case, the energy of the γ ray is converted into the excitation of a bound atomic electron into the continuum. As such, the total kinetic energy of the electron released will be Ee = Eγ − W where W is the work function of the electron needed to excite it into the continuum. Because of energy and momentum conservation, photoabsorption can only take place with bound electrons [74]. 55 Figure 3.6 The regions where each of the photoelectric, Compton scattering and pair production cross sections are dominant are presented as a function of photon energy and the atomic number of the absorbing matter. The lines where cross sections are equal between the neighboring effects are shown to demarcate three regions. Figure is from [80]. Photoabsorption tends to have the largest cross section for the lowest energy γ rays, but its cross section is still significant for even higher energy γ rays [74]. Because photoabsorption converts all of the energy of a photon into measurable electron kinetic energy, it leads to a clear peak at the energy of the transition, representing the most important process in γ-ray spectroscopy. This is in contrast to the more complicated Compton-scattering and pair production, which may result in the emission of a lower-energy γ ray to undergo another reaction. 3.3.1.2 Compton Scattering Compton scattering is an interaction of a photon with a free electron, where momentum and energy is transferred to the electron. While the electrons in germanium atoms are bound, the bind- 56 ing energy is small compared to the energy of the γ ray and the interaction is well described by Compton-scattering [74]. As this is effectively a form of inelastic scattering, it becomes significant at energies comparable that of the electron mass, at 511 keV. The reaction produces two outgoing particles, an energetic electron and a deflected γ-ray with reduced energy. The electron will be detected in the germanium detector. Meanwhile, the scattered γ ray is free to continue travelling through the detector, either interacting again with the material or escaping the detector volume. If the photon does escape, it leaves only a partial energy deposit in the detector [74]. The energy of the scattered γ ray is determined by the scattering angle as [74]: Eγ = Eγ 1+ Eγ (1 − cos θ ) me c2 (3.6) The remaining energy is imparted to the electron and measured in the detector. To the lowest order the scattering cross section of photons on electrons can be described by the Klein-Nishina formula [74]: 2 1 dσ 2 = Zr0 dθ 1 + α(1 − cos θ ) 1 + cos2 θ 2 1+ α 2 (1 − cos θ )2 1 + cos2 θ (1 + α(1 − cos θ )) (3.7) 1 ). where α is the fine structure constant (≈ 137 The Compton-scattered photons, when measured, cause a step-like discontinuity in the spectrum, called a Compton-edge. The Compton-edge is located at the maximum energy imparted into the detector at θ = π. Therefore, the Compton edge will occur at [74]: EC.E. = Eγ 1+ 2Eγ me c2 (3.8) One advantage of the GRETINA system to be discussed in the later section on addback is that the position resolution afforded allows for the reconstruction of multiple interactions to sum up multiple Compton scatterings and recover the energy of the emitted γ ray. 57 3.3.1.3 Pair Production In quantum electrodynamics, photons can be considered as eigenstates of an uncoupled photon field, while electrons and positrons are eigenstates of an uncoupled electron field. However, in reality these fields are coupled, and the eigenstate of an electromagnetic propagation is not a pure photon, but rather a superposition of photons and a virtual electron-positron pair, among other higher order Feynman-diagrams [81]. In free-space, it is impossible for the photon to become an electron-positron pair because there is no way to conserve both momentum and energy in the conversion. However, in the presence of a charged particle, generally atoms, photons can conserve energy and momentum by imparting a fraction of the energy into the atom [74]. Therefore, it is possible for photons with more than twice the energy of an electron mass to undergo pair-production in the detector. In this case, the positron will almost certainly annihilate on an electron and thus re-emit two γ rays around 511 keV, which will either escape or undergo Compton-scattering or photo-absorption. The electron will interact with the material and thus be measured directly. As with Compton-scattering, the multiple interactions of the photon can be summed together using GRETINA’s position resolution. 3.3.2 γ-ray Detection γ rays, being high energy photons, can be measured through interactions with matter. There are a number of detector types used to measure γ rays, including scintillating materials such as sodium iodide (NaI) detectors and semiconductor detectors. High purity Germanium detectors, including those used in GRETINA, are semi-conductor detectors, which are discussed briefly here. 3.3.2.1 Semi-conductor detectors In solid state physics, electrons can be considered to form a band structure in many materials, as opposed to the individual molecular wavefunctions throughout the material. In a simple picture ignoring higher energy bands and deeply bound states, electrons can occupy one of two bands, a 58 valence band and a conduction band. The valence band consists of electrons at the outer shells in the molecules, and are bound to particular locations in the material [74]. The conduction band consists of electrons that are capable of traveling throughout the material [74]. There may be an energy gap between the valence and conduction bands, called the band gap [74]. Insulators have large band gaps, above 5 eV, while conductors effectively have no band gap, with the valence band serving as the conduction band. Semiconducting materials have a small but finite band gap, less than 5 eV. Semiconductors are useful as they can be doped with other materials or have electric fields applied to change the band gap and induce different electrical properties [74]. Semiconductors may be doped with electron donor or accepting materials to create an imbalance in the electron to hole ratio. Semiconductor materials with more of an electron donor impurity are designated as n-type, while those with more of an electron accepting impurity are p-type. Where a p-type and n-type semiconductor are joined, a junction is formed where electrons and holes recombine and there is a region with no excess electrons or holes. An electric field manifests across this zone, which is called the depletion region [74]. γ-rays are detected in semi-conductors through the measurement of free electrons and holes produced in the depletion region by the interaction. A reverse bias voltage is applied to the detector to make an extended depletion region in the semiconductor [74]. As discussed below, the primary interactions between γ rays and matter involve the production of high energy electrons. These high energy electrons, when interacting at the depletion region, excite electrons into the conduction band as they stop in the material. The electron-hole pairs then drift apart under the influence of the electric field, and the total voltage induced by the electrons (holes) is measured. This total voltage is directly related to the number of electron-hole pairs excited, and hence the total energy deposited by the electron. 3.3.2.2 High Purity Germanium Detectors High purity Germanium (HPGe) detectors are well established devices for detecting γ-ray energies, and are especially useful for their resolution of approximately a few keV, which allows for the 59 separation of peaks close in energy. This is necessary for the experimental methods used here to measure lifetimes and Coulomb-excitation, as the differences in the Doppler-shifted energies used is on the order of 50 keV for a 1-MeV γ ray. HPGe detectors work by taking advantage of the semiconducting properties of germanium. Germanium requires 2.9 eV on average to excite an electron-hole pair, meaning that interactions of γ-rays with energies on the order of 1 MeV will produce approximately 350,000 electronhole pairs [74]. This is higher than the band gap of 0.7 eV due to momentum conservation and other modes of energy deposition that do not excite electron-hole pairs [82]. The statistical error √ from n is then approximately 0.15% [74]. Silicon has a similar energy to excite an electron-hole pair, 3.7 eV. However silicon has a smaller atomic number and cannot be made larger than a few mm in thickness due in part to the impractically large voltage that would need to be applied [83]. Therefore, γ rays cannot easily be completely absorbed in silicon detectors. By contrast, the low impurity in the germanium detector allows for the creation of depletion zones of thickness on the order of centimeters. The thickness of the depletion region is [74]: d= 2εV 1/2 eN (3.9) where ε is the dialectic constant of the Ge, V is the reverse bias voltage, and N is the concentration of impurities throughout the material. Hence, reducing N makes it possible to get a larger depletion region, useful for measuring γ-rays which at energies around 1 MeV have attenuation lengths on the order of 1 cm through germanium [84]. While high purity germanium is generally extremely pure, with approximately 1010 atoms/cm3 of impurities in the material, the impurities still induce a slight bias in the material as with other semi-conductors [74]. These slight biases are called π-type for a slight advantage for electron absorbers and ν-type for a slight advantage to electron donors. As with other semi-conductor detectors, a reverse bias must then be applied across the germanium to enable its use as a detector. This is accomplished in germanium by implanting high concentrations of absorber or donor ions into the surface of the detector. These result in regions with large numbers of free electrons or 60 holes, and are called p+ or n+ contacts [74]. The need to apply a reverse bias to the detector leads to two natural configurations, a parallel plate-like planar setup, and a cylindrical coaxial setup. As crystal growth restricts the overall size of planar detectors, GRETINA consists of a coaxial design which is discussed here. In a coaxial setup, the germanium crystal is grown and shaped to be a cylinder in shape, and the center of the crystal is removed to create an inner cylinder. For an n-type detector, which GRETINA consists of, generally the outer portion of the cylinder is made to be a p+ contact, and the inner core is made a n+ contact. A positive voltage is applied to the n+ contact and the outer contact is grounded, so that the detector is reverse biased to make a large depletion region [74]. One consequence of the small band gap in high purity germanium is that HPGe detectors must be kept cold to prevent thermal noise from swamping the detector. This is accomplished for GRETINA through the use of a dewar which stores liquid nitrogen attached to each of the detectors. These dewars are connected to the liquid nitrogen lines at the NSCL so that the detectors can be kept cold. 3.3.3 GRETINA Design GRETINA is an array of individual coaxial HPGe detectors arranged to cover a large solid angle. Each detector consists of four HPGe crystals, each of which has a central contact [79]. The outer contact of the crystals is segmented into six angular strips and six depth strips for a total of 36 segments, as shown in Figure 3.7. Each crystal is approximately 90 mm in length and 80 mm in diameter, while each segment is on the order of 10 mm thick [79]. Figure 3.8 shows the GRETINA setup used in the 27 Ne experiment during March 2016. GRETINA was originally designed to have seven detectors but is now planned to have twelve total detectors. At the time of the measurement of 58 Ni during the first GRETINA campaign at the NSCL, seven of the detectors had been fabricated and were employed. During the second campaign in 2016 when 27 Ne was studied, two additional detectors had been made and a total of nine detectors were 61 Figure 3.7 (left) The design of a detector in GRETINA is shown, with four crystals aligned into a cluster to increase detection area. (right) The segmentation of each crystal is shown for the six angular and six depth slices leading to 36 total segments. Figure is from [79]. employed. In the long run, GRETINA is planned to be converted into a 4π array, Gamma Ray Energy Tracking Array (GRETA), which will consist of 30 total detectors [85]. The segmented crystals used in GRETINA allow for a position resolution of 2 mm for the interactions. The segmentation provides position information up to the segment size on its own, but this is on the order of 10 mm or larger for each segment. The additional sub-segment resolution is enabled by signal decomposition. 3.3.4 Signal Decomposition The use of signal decomposition to achieve sub-segment resolution is a defining feature of the GRETINA array. When a γ ray interacts with a GRETINA crystal, it will induce a voltage pulse not only in the segment the interaction occurs in, but also in the neighboring segments through the 62 Figure 3.8 The GRETINA setup used in the 27 Ne experiment in March 2016 is shown. Nine GRETINA detectors were used in the experiment, which are arrayed around the beampipe for the TRIPLEX plunger. image charge of the electric field. The overall timing of the pulse, and consequently its shape, is dominated by the drift time of electrons and holes in the detectors, which in turn is sensitive to the position of the interaction. Therefore the pulse shape and amplitude in each of the segments can be used to determine the position to a sub-segment level [79]. An additional complication occurs because in most cases, except photoabsorption of the emitted γ ray, multiple interactions will occur within the timing window of GRETINA. Therefore, the total pulse signals will consist of the sum of pulses induced by each interaction. In order to decompose this superposition into individual interactions, a basis composed of simulated signals from point charge interactions on a non-uniform grid with average spacing of 1 mm has been developed. 63 GRETINA has an auxiliary computer system that processes each GRETINA signal, and performs a χ 2 best fit analysis on each of the signals to the simulated basis. This decomposes the signal into multiple interactions with a position resolution better than 2 mm [79]. 3.3.5 Addback The signal decomposition of GRETINA enables the use of addback to improve the measurements of γ rays. Addback is a means of recovering the energies of γ rays affected by Compton-scattering or pair-production. The energies of γ rays interacting in neighboring segments can be summed up, so that the total energy deposited in the region is reconstructed. To use the method, it is necessary to determine what type of region the summing should be performed over. A number of methods are possible, for example the region can be the four crystals of a detector, every GRETINA detector, or a region based on the expected kinematics of the photons, as is done in tracking algorithms [86]. However, in the results shown here, the addback is performed using a spherical addback technique, which is also shown in Figure 3.9. Spherical addback is a recursive method, which works first by looking at all of the hits in GRETINA. The highest energy interaction is chosen as the first interaction point, which may have photo-absorbed, scattered, or been part of a pair-production. This highest energy interaction is also used to determine the angle of emission of the γ ray. Then, all points within a sphere of a chosen radius, which is a parameter of the method, are added together. The radius is chosen based on a comparison of yields and signal-to-noise ratio of the γ ray of interest. The interactions with that sphere are then considered as one event, and removed from the list of events. Then, recursively, the remaining hits are searched for the highest energy and a sphere is drawn around it, and so on until all of the points have been added together. In a simple picture with only one γ ray involved, the effect of addback is to increase the yield in the peak, and to reduce the lower energy yields in the Compton-edge. A simulated example using the spherical addback is shown in Figure 3.10 and compared to the singles without addback. However, in the data there are a number of compounding effects that complicate the spectrum 64 E1=2 E1=2 E2=1 E3=3 E2=1 E3=3 E4=1 E4=1 R E5=1 E5=1 Eb=2 E1=2 Ea=5 Ea=5 R R E5=1 Ec=1 Figure 3.9 The spherical addback method is shown for a case of five interactions across two crystals. Interactions are labeled with energies in arbitrary units. Raw interactions are represented by a flash, while added back interactions are represented by a many-point star. The energies of the raw interactions are denoted with a number in the subscript (E1 , E2 , ...). Energies of the added back interactions are denoted with a subscript letter (Ea , Eb , ...). (Top-left) In the first step of the process, the highest energy interaction is identified. (Top-right) A spherical cut is made, including any interactions within radius R. (Bottom-left) The interactions within the spherical cut are summed to create a single addback interaction. The interactions in the addback are no longer used in the algorithm, and the process repeats again. (Bottom-right) The remaining two hits have only themselves within the radius of the highest energy hit and the addback interaction is the same as the original interaction. Note that in the first frame the E1 interaction is within the radius R of E2 , but E2 is removed from consideration in the third frame. In this work a radius of 7.5 cm is employed. 65 Figure 3.10 A GEANT4 simulation [87] of a single γ-ray transition measured by GRETINA is shown, with a comparison between singles and a spherical addback. The singles replicate the summing up of all energies within a single GRETINA crystal, while the spherical addback is as described in the text with a radius of 7.5 cm. The state energy is assumed to be 1480 keV, and the Compton-edge can be observed at around 1300 keV, with scattered events extending back towards 0 keV. A clear reduction in the Compton-events and an increase in the peak yield is apparent for the spherical addback compared to the singles. compared to the simple picture in the simulation. This is partly because of possible accidental coincidences with background, which can lead to the unwanted summing of background events with the γ rays of interest and a loss of peak yield. In addition, γ rays of higher energy than the peaks of interest may Compton-scatter into the region. The γ ray of interest is denoted γ0 and the higher energy γ ray is denoted γH . When the addback is applied, the Compton events from γH will be summed up into the peak. This consequently reduces the background in the γ0 peak arising from the Compton scattering of γH . This may decrease the overall yield in the γ0 peak, but increases the signal to background ratio. In Figure 3.11, three different addback comparisons are shown, a spherical addback with radii of 4 cm, 7.5 cm, and 15 cm is applied to a 58 Ni spectrum. It is worth observing that at energies below 1-MeV, there is a significant amount of background from the Compton-scattering of high energy γ-rays. Thus, larger radii lower the total counts in the energy region. So, for the 7.5 cm 66 Figure 3.11 A γ-ray spectrum of 58 Ni is shown for a Recoil Distance Measurement with three foils each separated by 1 mm, as discussed in Chapter 4. Spherical addback radii of 4, 7.5 and 15 cm are compared, and discussed in the text. case, the peak at around 650 keV has roughly the same yield as for the 4 cm case, but with less background. However, once the radii become too large, accidental coincidences can be summed in with the actual photopeaks, pair production, and Compton-scattered events. This leads to a reduction of the peaks, as can be seen in the 15 cm radius addback, where the background is lowered but so is the yield for the peaks between 500 and 1200 keV. In the experiments discussed here, the spherical addback radius determined is 7.5 cm. This is roughly twice the attenuation length of a 1 MeV γ ray in Germanium [84]. 3.3.6 Doppler Reconstruction All of the measurements in this work rely on the Doppler-effect to succeed, making the ability to reconstruct the projectile-frame energies of γ rays important. To reconstruct the γ-ray energy, the lab-frame energy must be corrected for the relativistic Doppler-effect as: ECor = EObs γ (1 − βCor cos θlab ) 67 (3.10) In this equation, βCor is the ion velocity used for the correction, γ is the corresponding Lorentz factor for βCor and θlab is the laboratory frame angle between the γ-ray emission and the beam axis. To reconstruct the energy would require the velocity, ion position and trajectory, and γ-ray interaction position, which are not known exactly. However, through the coupled use of GRETINA and the S800, it is possible to extract Doppler-correction factors. As discussed in Section 3.2.1, the S800 inverse map can be used to determine the non-dispersive displacement, the dispersive and non-dispersive angles of trajectory, and the energy of the ion at an upstream position. The inverse map is generally determined for the downstream degrader face. The S800 rigidity is used to determine a central β value for the ions of interest. The difference from the central value is incorporated by using the ion energy information from the S800 to determine the βCor . The interaction position information from GRETINA is used to calculate the angle relative to the position of the ion at the degrader face, which is taken as θlab . 3.3.7 Timing Between S800 and GRETINA One complication when using fragmentation reactions is the creation of laboratory frame γ rays from neutron reactions on the beampipe and the GRETINA detectors themselves. These γ rays, when a Doppler-reconstruction is applied, can overlap with peaks of interest and complicate the resulting spectrum. However, as a number of these neutron induced γ rays come from states with lifetimes on the order of nanoseconds or longer, a timing cut between GRETINA and the S800 can be used to filter out the unwanted background. Figure 3.12 shows a plot of γ-ray energy plotted against the difference in timing between GRETINA and the S800 E1 scintillator, with a sample timing cut drawn in red on top. The key features of this figure are that for energies above 500 keV most counts are located between channels −50 and −65, which corresponds to the time that the ion is at the target position. Above and below this band, at energies around 511 keV, 600 keV, 700 keV and 1460 keV, vertical lines can be observed in the plot. These are the γ-ray energies from the products of certain neutron reactions, the electron annihilation energy (511 keV), and natural background 40 K (1460 keV). The vertical 68 Figure 3.12 The difference in GRETINA and S800 timing is plotted as a function of the γ-ray energy. A sample timing cut to reduce the impact of neutron-induced backgrounds is shown in red. Neutron induced backgrounds can be seen at the vertical lines at 511 (electron annihilation energy), 590 (76 Ge), 700 (72 Ge), 850 (27 Al),1040 (70 Ge), 1460 (40 K), and 2614 (208 Tl) keV. The band along the cut between −50 and −70 represents γ rays coincident with ions near the target position. lines are caused by the independence of these γ rays from the beam timing signal. A cut on the timing between the S800 and GRETINA can then selectively remove these backgrounds. 3.3.8 Calibration The efficiency of GRETINA is incorporated into simulations used for both lifetime measurements and Coulomb-excitation measurements. In the lifetime measurements only the relative efficiency of higher and lower energy γ rays is critical. The energy difference between different Dopplershifted peaks corresponding to the same transition in a lifetime measurement is on the order of 100 keV so that the differences in efficiency are well reproduced in the simulation. On the other hand, for Coulomb-excitation, the absolute efficiency of GRETINA is vital. The geometric efficiency of GRETINA is determined from simulations that incorporate the detector geometry as well as attenuation through the detectors, foils and other components. In the simulation, an isotropic distribution of γ rays is assumed in the projectile frame, and then the boosted frame angular dis- 69 tribution is calculated. As the projectiles are at energies between a third and half of the speed of light, the relativistic shift in geometry changes the expected efficiency by forward-focusing the γ rays. However, the intrinsic efficiency of the GRETINA detectors and the attenuation of the γ rays through the target and degrader foils must be accounted for. This is done by using γ ray sources that are placed in various configurations with the plunger device. For example, a 152 Eu source was used in the March 2016 experiment in two settings, and was used with only the target foil mounted, and with both the target and degrader foil mounted and separated by 25 mm. In the latter case, measurements were made with the source on the target and with the source on the degrader to account for different geometric efficiencies and attenuation lengths. The γ ray sources have known intensities and a known branching ratio for the emission of each γ ray with different energy per 152 Eu decay. Therefore, the efficiency could be calculated and compared to the simulated efficiency. In this case, the efficiency in the simulation was found to be 25% larger than in the data. This discrepancy may be due to the upstream position of the target which can lead to attenuation through the TRIPLEX and material surrounding the detectors not included in the simulations. In addition details in the structure of the GRETINA detectors may have an impact on efficiency not taken into account. A comparison of the data and simulation for the 152 Eu source on the degrader face for selected γ-ray peaks is shown in Table 3.1. The total emissions expected are determined in the data from the activity of the source, the probability distribution for γ-ray emissions, and the length of time of the run. The total emissions for the simulation are determined from the number of simulated events and the branching ratio of the simulated 152 Eu. In the data the efficiency must take into account the deadtime of the system, which was approximately 72%. 70 Figure 3.13 Spectra for a Eu-152 source placed on a 0.92 mm Ta Degrader 12 cm upstream of the GRETINA center for (top) data and (bottom) GEANT4 simulation are presented. The efficiencies can be compared to each other to determine the scaling need from the simulation. 71 γ-ray Energy (keV) Efficiency in Data (%) Efficiency in Simulation (%) Scaling Factor 1086 5.6± 0.2 7.8 ± 0.1 0.71±0.04 1112 4.8 ± 0.2 6.6± 0.1 0.72 ±0.04 1408 4.4 ± 0.2 5.9 ± 0.1 0.75 ±0.04 Table 3.1 The efficiencies in data and simulation for a 152 Eu source are shown along with error and the ratio of the efficiency in data to the efficiency in simulation. Error is from uncertainties in the source activity and statistical error in the simulation and data. The efficiencies in data are generally 25% − 30% smaller than in simulation. This adjustment is taken into account in the cross section data. 3.4 TRIPLEX The Recoil Distance Method and Coulomb-excitation techniques require the precise placement and control of target foils in the beamline. This precise movement is afforded by the Triple PLunger for Exotic Beams (TRIPLEX) [48]. The TRIPLEX is a plunger device that can hold up to three foils, two of which can be moved independently with respect to a middle foil placed at a fixed position. The TRIPLEX layout is shown in Figure 3.14. The TRIPLEX consists of a bearing structure that enables the separation of the foils, a foil mounting structure that enables the mounting and aligning of foils, and a support apparatus to place the TRIPLEX into a beam-line and connect its electronics. The TRIPLEX makes use of two motors and measurement devices to ensure precise and accurate placements of foils. 3.4.1 Bearing Structure and Motors The TRIPLEX is able to precisely position the three foils relative to one another through its bearing structure. The main structure consists of three concentric stainless steel tubes. The middle tube is connected to the support structure and is immobile. The outer and inner tubes are separated from the middle tube and held by four sliding bearings. The outer and inner tubes are each connected to the spindle of a piezoelectric motor (one on each side of the plunger). The connection is made using an elastic wire to connect the motors to rings clamped to the movable tubes to avoid any radial forces on the tubes, which ensures a smooth parallel motion of the foils [48]. 72 Figure 3.14 (top) A photograph of the TRIPLEX is shown compared to (bottom) a schematic of the TRIPLEX. In the schematic the labeling is as such: (A) The support frame, (B) Motor used to position the target foil, (C) the outermost tube, used to adjust the position of the second degrader, (D) The immobile central tube used to mount the degrader, (E) The inner tube, used to move the target, (F) The foil mounting structure for the target attached to the inner tube (E), (G) The foil mounting structure for the first degrader, which is immobile and connected to the central tube (D), (H) The foil mounting structure for the second degrader, attached to the outermost tube (C). Figure and identification are from [48]. 73 The TRIPLEX is driven by two N-381 NEXACT linear actuators from Physik Instrumente, which is a piezoelectric motor. Piezoelectric motors work by application of electric fields to piezoelectric materials that expand in the direction of an applied electric field as illustrated in Figure 3.15. The piezoelectric material is set up as a number of actuator pairs attached to a guided runner. Each actuator consists of two blocks, a vertically aligned material attached to the moving rail and a horizontally aligned material that can be attached to a guiderail if a vertical electric field is applied to the vertical group. The electric field for the horizontal components in each pair are oppositely oriented, so that the applied electric field will cause one to expand to the left and the other to the right. The actuators can be made to step across the runner by applying a field to cause the first actuator to connect with the guide rail. Then the horizontal components are activated, causing an extension in the horizontal direction opposite that of the direction to be traveled. Then the actuators are switched vertically, so the second actuator is now connected to the guide rail, and the horizontal field is inverted, causing a net shift into the direction of motion. The target and second degrader tubes can be moved over a range of approximately 25 mm. The target can be moved upstream 25 mm from the fixed first degrader position, while the second degrader can be moved up to 25 mm downstream. 3.4.2 Foil mounting structures The target, first degrader, and second degrader foils are mounted onto aluminum frames attached to the three tubes by brass rings. The foils to be used with the TRIPLEX must be glued to specialized mounting cones, which can then be attached to the aluminum frames by three screws. These cones are insulated from the TRIPLEX and one another so that the capacitance between the two foils can be measured. The target cone and degrader cone face each other directly, while the second degrader cone is designed to fit within the first degrader, which requires that the second degrader foil be a circular shape [48]. The target and first degrader foils are generally square foils with a size dimension of 50 × 50 mm2 . The second degrader foil has a radius of around 47.5 mm. To ensure that the separation is consistent across the foils, it is important to make sure that 74 Figure 3.15 The stepping action of the N-381 NEXACT piezoelectric motor is shown in four figures. In (1) the first actuator is made to connect with the guiderail. In (2) the horizontal electric fields are applied, causing the first actuator to expand in the direction contrary to the motion of the motor, but the second actuator expands in the direction of the motion. In (3) the first actuator is removed from the guiderail and the second actuator connected. In (4) the horizontal electric field is reversed, causing the materials to expand against the direction of motion, leading to a net motion for the runner. Figure is from [88]. the foils are parallel to one another. This is accomplished by the use of springs attached to the mounting screws for the target and second degrader. The screws can be adjusted on each of the three mounting points for the cone. First the target and degrader are mounted and brought near contact. The capacitance is measured between the two foils, and the screws in the target cone are adjusted until the maximum capacitance is reached. When this is accomplished, the foils are parallel to each other. When three foils are used, the alignment must be done in three steps. First the target is aligned with the first degrader. Then the first degrader is unmounted, and the second degrader is mounted. The second degrader and target are brought near contact. The screws in the 75 second degrader are then adjusted until the second degrader is aligned with the target. Then the second degrader and target are separated and the first degrader is remounted. Then the alignment of the target with the first degrader is checked by measuring the capacitance. If the measured capacitance is consistent with the previous alignment of the target and first degrade, the three foils are considered aligned [48]. 3.4.3 Support Structure The unmoving parts of the TRIPLEX are attached to a support ring and an outer support frame. The outer frame connects to six screws that are used to align the TRIPLEX within the beam pipe. The motors used to drive the inner and outer tubes are directly connected to the support ring, as are two TESA inductive-working micrometers used to measure the distances driven by the motors [89]. Furthermore, wires are connected from the upstream side of the support frame which can be attached to the mounting screws for the target, first degrader and second degrader. The wire from the first degrader can be connected to a pulse generator, while the wires from the target or second degrader can be connected to either an oscilloscope or a multi-channel analyzer to determine the induced voltage. This allows for a measurement of the capacitance of the foils. 3.4.4 Beam Pipe The TRIPLEX requires the use of a specialized beam pipe to align the TRIPLEX in the beamline and to allow for manipulation and readout of the electronic components. The TRIPLEX beam pipe is designed for use with HPGe detector arrays such as GRETINA, and is made from aluminum in order to minimally attenuate γ rays. The downstream section of the beam pipe has a flange designed to be attached to a downstream spectrometer, in these experiments the S800. The beampipe is also long enough to allow for the plunger to be placed upstream from the center of the array, as many measurements made in these experiments benefit from a larger detector efficiency for events emitted at small angles, which can be accomplished by moving the target upstream of the center 76 of the array. In these experiments, the target was placed approximately 13 cm upstream of the GRETINA center. The upstream portion of the beam pipe has a diameter of six inches to accommodate the cabling of the TRIPLEX electronics. A number of electric feedthroughs are installed in the beampipe that can be connected to the TRIPLEX device, to allow for readout and control of the motors, readout of the TESA micrometers, connection of the pulse generator, and readout of the induced pulses. The upstream portion of the beam pipe also includes six alignment screws which can be adjusted to align the TRIPLEX with the beam axis. 3.4.5 Distance Measurements The precise placement and adjustment of the three foils in the beamline is central to the methods used in this work. The measurement of the separation between the foils is afforded by three different tools that can be cross-checked for consistency. 3.4.5.1 Linear encoder The first measurement available is directly from the driving motors. Each actuator is equipped with a linear encoder which provides information on the relative difference between the starting position of the motor and the final position of the motor after being driven. In principle, the linear encoder provides sub-µm resolution, but cross-checks are needed. 3.4.5.2 TESA micrometer The second measurement is made via the an inductive-working micrometer, GTL21 from the TESA group [48]. The micrometer can measure compression over the range of ± 2 mm, and can be used for sub-µm distance measurements across that region. The TESA micrometer can be moved along the support frame to change the sensitive region. The TESA precision is considered to be better 77 than that of the linear encoder, but it also provides only a relative distance measurement. The TESA measurements can be compared to the linear encoder measurements to ensure consistency. 3.4.5.3 Capacitance While the linear encoder and TESA micrometer provide precise measurements of the relative distances between foils, they do not on their own provide a measurement of the absolute distance between the foils. To obtain the absolute distance, the relative distance tools are calibrated using a capacitance measurement of the foils. As mentioned above, each foil can be connected to either a pulse generator or an oscilloscope/multi-channel analyzer after processing through a spectroscopic amplifier. In fact, in these experiments both the multi-channel analyzer and the oscilloscope were used to check consistency. Therefore, an exponential voltage from a pulse generator can be applied to one foil and the induced voltage read out. The two foils can be treated as a parallel plate capacitor, for small distances between the foils. Therefore the capacitance can be modeled as: ε A C pp = 0 d (3.11) where C pp is the capacitance of two conductive parallel plates, ε0 is the permittivity of free space, A is the area of the foil and d is the distance between the two foils. The pulse generator in connection with the two foils and the readout can be modeled as an RC circuit between the internal resistance of the system and the capacitance of the two foils. For an RC circuit with an AC voltage with frequency ω, the voltage drop across the capacitor can be modeled as [90]: VC 1 = V0 1 + iωRC (3.12) where VC is the voltage drop across the capacitor, V0 is the voltage of the pulse generator, C is the capacitance of the two foils, and R is the internal resistance of the circuit. Therefore, the voltage 78 on the second foil relative to the ground is simply VC2 = V0 −VC which leads to: VC2 V0 = ωRC 1 + ω 2 R2C2 ∝ 1 d 2 +1 (3.13) where d = 1/ωRC is a dimensionless term proportional to the separation of the foils. While the applied voltage in this case is not AC, the exponential voltage can be Fourier decomposed into a sum of such terms so that the discussion is still generally valid. For distances d << 1, the capacitance reaches a saturation value and the induced voltage VC2 = V0 . At distances d >> 1 the equation is no longer valid, as the parallel plate model does not hold. In this case, the induced voltage becomes near zero and fluctuations cannot be distinguished from other electronic noise. In the region between these two extremes, where d is on the order of 10-100 µm in actual cases, the capacitance is inversely proportional to the distance between the foils. The absolute distance between the two foils is calibrated by applying a voltage to one foil and measuring the induced voltage on the second foil over a range of distances. For example, for a target-degrader combination, the target is driven towards the degrader until the induced voltage saturates. This position is taken as the electrical contact position. From this position, the target is separated from the degrader in steps. At each position, the induced voltage is measured. The distance from the contact position is measured using the linear encoder and the TESA micrometer as well. A calibration curve is then created by plotting the inverse of the voltage as a function of the distance from the electrical contact position measured by the TESA (linear encoder). An example of a voltage plot and a calibration curve employed in the 27 Ne experiment is shown in Figure 3.16. A linear fit is performed, and the x intercept, where the voltage would diverge, is determined. This would be expected contact position for the linear fit, and can be compared to the electrical contact position. The electrical contact position was found in the 27 Ne experiment to agree within 30 µm of the actual contact position. 79 Figure 3.16 Plots showing the (top) induced voltage and the (bottom) reciprocal of the induced voltage between a beryllium and tantalum foil as a function of the separation between them. The distance was measured using the linear encoder of the TRIPLEX, and is based from the point of electrical contact. In the top figure, saturation near 0 µm can be seen, after which the 1/d behavior dominates. In the bottom figure, the points near saturation are removed. A linear fit to the data is also shown. For this case, the contact derived from the linear fit is within 10 µm of the measured electrical contact. 80 CHAPTER 4 INVESTIGATION I: 58 NI 4+ 1 LIFETIME MEASUREMENT The first experiment discussed as part of this work is a measurement of the lifetime of the first 4+ state of 58 Ni [91]. This measurement at stability probes collectivity in the vicinity of the shell + gap at N = Z = 28. A previous measurement of the B(E2; 4+ 1 → 21 ) has indicated enhanced collectivity in 58 Ni relative to expectations near a shell closure. A new measurement is described and the results are discussed. The present measurement is shown to resolve the discrepancy. 4.1 Shell Closure at N=Z=28 The Nickel isotopes are of significance for nuclear structure as Nickel has a magic number of protons with Z = 28 and spans over three doubly magic nuclei. These doubly magic nuclei include the most neutron deficient doubly magic nucleus 48 Ni [92], the N=Z nucleus 56 Ni [93], and the neutron rich 78 Ni [94][95]. Furthermore, 68 Ni is of interest as a semi-magic nucleus at the subshell closure at N = 40, a magic number for the harmonic oscillator [96]. This wide range of shell closures covered by the isotopic chain has led to much interest in the Nickel isotopes as benchmarks for shell-model calculations [91][97][98]. From a shell-model perspective, the closed-shell occurring at 28 nucleons is significant as the first closure arising from spin-orbit effects instead of from harmonic-oscillator levels. The f7/2 orbit is lowered by the spin-orbit coupling, separating it from the remainder of the p f -shell. This results in an energy gap that manifests as a magic number. The evidence of the (sub)shell closures for neutrons at N = 28 and 40 comes in part from the systematics of isotopic chains. For example, for the Nickel isotopic chain, the energies of the first 2+ and 4+ states attain a maximum at 56 Ni, where N = Z = 28, indicating that the energy of the gap between the f7/2 and the remainder of the p f shell is significant. The energy of the 2+ 1 and + + 68 4+ 1 states in Ni also significantly increase. Similarly, the B(E2; 21 → 01 ) attains a minimum at 56,68 Ni, indicating a substantial decrease in collectivity arising from the same large energy gap. 81 Figure 4.1 2+ 1 energies for Ca and Si isotopic chains near N = 28. The difference in systematics shows the decreasing energy gap for the N=28 magic number for nuclei far from stability. Figure is from [99]. The N = 28 shell closure disappears in neutron-rich nuclei [99]. The reduction of the N = 28 shell gap in neutron rich nuclei is also indicated in the systematics, as shown in Figure 4.1. The 42 40 energy of the 2+ 1 state in Si, with the neutron number N = 28 is lower than in nearby Si with N = 26. This is contrary to the expectation of a maximum 2+ 1 energy at a shell closure, and is unlike the case of 48 Ca, where the N = 28 shell closure is still significant. Even in nuclei near stability the N = 28 shell closure is known to be modest. For example, shellmodel calculations of 56,68 Ni lead to ground state wavefunctions with substantial contributions from excitations across the N = 28, 40 and Z = 28 energy gaps [96][100][101]. Furthermore, the + B(E2; 2+ 1 → 01 ) in the Nickel isotopic chain can not be reproduced using shell-model calculations without including proton and neutron excitations from the f7/2 orbital into the remaining p f shell. A comparison across the Nickel isotopes from A=56 to A=68 of the state energy levels and 82 Energy (keV) Adopted 4+ Energy Adopted 2+ Energy GXPF1A 4+ Energy GXPF1A 2+ Energy 5000 4000 3000 2000 1000 0 56 58 60 62 64 66 68 70 B (E 2;2+1 → 0+1) e 2fm4 A for Ni Isotopes Adopted Values GXPF1A en=0.5, ep=1.5 GXPF1A en=0.67, ep=1.12 KB3G en=0.5, ep=1.5 250 200 150 100 50 0 56 58 60 62 64 66 68 70 A for Ni Isotopes + Figure 4.2 (top) The observed state energies for the 4+ 1 and 21 levels in the even-even Nickel isotopes between A=56 and A=68 are shown and compared to calculations from the GXPF1A + shell model interaction. (bottom). The observed B(E2; 2+ 1 → 01 ) transition strengths for the even-even Nickel isotopes between A = 56 and A = 68 compared against predictions from the KB3G interaction and the GXPF1A interaction with standard and modified effective charges. These figures are from [91]. 83 + B(E2; 2+ 1 → 01 ) transition strengths from experimental values is presented in Figure 4.2 [102][103]. + 56 The systematics of the energies of the 2+ 1 and 41 states indicate an increase at Ni. This is con- sistent with the expectation if the N = 28 shell closure is present. Furthermore, at a shell closure it is expected that the nucleus should be spherical and the B(E2) transition strengths should be + reduced. It can be seen in the systematics that the B(E2; 2+ 1 → 01 ) values attain a maximum at + midshell between 56 Ni and 68 Ni. The B(E2; 2+ 1 → 0gs ) values smoothly decrease towards minima in the values is observed at 56 Ni and 68 Ni. The data are also compared to shell-model calculations in the full p f -shell model space using the KB3G and GXPF1A interactions [102][103]. The KB3G interaction is a modification of the earlier KB3 interaction intended to reproduce the appropriate energy gap for the N = Z = 28 magic number [103]. The GXPF1A shell model interaction is a modified version of the GXPF1 interaction, which was fit to experimental values of the binding and state energies in 87 nuclei in the p f shell [104]. In GXPF1A five two-body matrix elements have been adjusted to match energies in neutron-rich Ca, Ti, and Fe [102]. As expected, the shell-model calculations also predict the in+ creased state energies at 56 Ni. The B(E2; 2+ 1 → 0gs ) predictions anticipate a maximum at midshell which gradually decreases towards the closed (sub)shells at 56,68 Ni. 4.2 58 Ni 4+ 1 Lifetime As 58 Ni has just two neutrons more than the doubly magic nucleus 56 Ni, it is expected to be sensitive to core excitations across the proton and neutron f7/2 shells. As a stable nucleus, the + energy levels of 58 Ni have been well established, and the B(E2; 2+ 1 → 01 ) has been studied through Coulomb-excitation and lifetime measurements [105][106]. Indeed, shell-model calculations can + only reproduce the values fo the B(E2; 2+ 1 → 01 ) if excitations above the N = Z = 28 shell closure are permitted [105]. + 58 The B(E2; 4+ 1 → 21 ) has also been measured once before in Ni. A Doppler-shift attenuation method measurement resulted in a lifetime of 5.4 ± 0.6 ps [106]. The DSAM spectrum from 84 Kenn’s measurement of 58 Ni is shown in Figure 4.3 [106]. This corresponds to a B(E2; 4+ 1 → +16 2 4 + + 2+ 1 ) value of 148−15 e ❢♠ . This value is significantly larger than the B(E2; 41 → 21 ) value for nearby 60 Ni which is 77(34) e2 fm4 . This is somewhat surprising, as it might be anticipated + + + that the B(E2; 4+ 1 → 21 ) values should follow the behavior of the B(E2; 21 → 01 ) values and attain a maximum at midshell with a smooth trend towards minima at the shell closures at 56,68 Ni. + 58 The apparently enhanced B(E2; 4+ 1 → 21 ) value in Ni may be a result of the scarcity of data for lifetime measurements of 4+ 1 states in the Nickel isotopes, or may be caused by additional contributions from higher-lying sdg orbitals or excitations over the N = Z = 20 shell closure. The only previous measurement was performed using DSAM, which is primarily sensitive to lifetimes of 1 ps or less as shown in Figure 2.1 [41]. Assuming that the sensitivity of the Recoil Distance and Doppler Shift Attenuation methods overlap near 5 ps, the upper bound for the + 2 4 B(E2; 4+ 1 → 21 ) from a lifetime measured with DSAM would then be near 150 e fm . However, + the systematics of the B(E2; 4+ 1 → 21 ) would be restored with a lifetime longer than approximately 10 ps. Furthermore, the data cannot be reproduced by the previously mentioned shell-model calculations which predict a lifetime between 10 and 20 ps and a consequent reduction in collectivity consistent with the shell gap [91][102][103]. Therefore, in order to confirm or refute this potential + + enhancement in B(E2; 4+ 1 → 21 ), a new measurement of the lifetime of the 41 excited state in 58 Ni was performed to distinguish between the literature value of 5.4 ps and the 10-20 ps value expected for an unenhanced B(E2). 4.3 Recoil Distance Experiment on 58 Ni 58 The lifetime of the 4+ 1 excited state of Ni was measured using the Recoil Distance Method. The Recoil Distance Method can be set up to be sensitive to lifetimes on the order of 10 ps. Therefore, it is able to distinguish between the previous experimental value of 5.4 ps and the 10-20 ps consistent with an unenhanced B(E2). The experiment e11013 was performed at the National Superconducting Cyclotron Laboratory on campus at Michigan State University and employed the 85 Figure 4.3 Doppler-shift attenuation method spectrum for 58 Ni from [106]. The lifetime is extracted from the relative yield in the Doppler-shifted and un-shifted parts of the peak and + compared to the stopping time in the material. Lifetimes were extracted for each of the 2+ 1 , 41 , + 2+ 3 and 24 excited states. 86 S800 Spectrograph in coincidence with the GRETINA detector array [61][71][79]. The experiment was run with the same setup as in [107]. The coupled-cyclotron facility at the NSCL was used to produce a primary beam of 78 Kr at 150 MeV per nucleon. This beam was impinged on a 9 Be target to produce a secondary cocktail beam, which included 74 Kr. The cocktail beam was sent to the A1900 fragment separator to produce a secondary beam of 74 Kr at 93 MeV per nucleon at a purity of 40% with a total intensity of approximately 105 particles per second [107]. The excited states of interest in 58 Ni were produced from multi-nucleon removal reactions on the target, and recoil particles were identified and tagged using the S800 [71]. The TRIPLEX Plunger device was employed to place the target and degrader foils into the beam-line and set the distances between them precisely [48]. The TRIPLEX was used with a three foil configuration. The target was a 750-µm 9 Be foil. The first degrader was a 125-µm Ta foil. The second degrader was a 90-µm Ta foil. This results in a three peak structure in the γ-ray spectrum where the relative peak yield is sensitive to the lifetime. To maximize the sensitivity of the measurement to lifetimes near 10 ps, a 1-mm distance between the target and the first degrader and a 1-mm distance between the distance between the first degrader and second degrader were chosen. This is because the flight time over 1 mm of the 74 Kr beam at 30% of the speed of light is approximately 10 ps. However, as the excited states of 58 Ni could be produced by fragmentation of the secondary beam on the degraders, it is necessary to take measurements with the foils separated at a larger distance to constrain these reactions. This was done by setting the separation between the target and the first degrader to 10 mm, while maintaining the distance between the second degrader and first degrader at 1 mm. This means that any excited states that are produced on the target will decay back to the ground state before the beam is slowed in the degraders. Therefore the ratio of the counts in each of the three Doppler-shifted peaks provides a measurement of the relative amount of excited states produced by reactions in the target compared to those produced by reactions in the degraders. 87 XFP Time-of-Flight (Arb. Unit) 8000 −950 7000 −1000 6000 74 Kr −1050 5000 4000 −1100 73 Br −1150 3000 72 Se 2000 −1200 1000 −1700 −1650 −1600 −1550 −1500 0 −1450 OBJ Time-of-Flight (Arb. Unit) Figure 4.4 A timing signal correlation plot used as a particle identification for the incoming beam of interest is presented. The A1900 XFP scintillator timing signal is shown on the y-axis and S800 OBJ scintillator timing signal is shown on the x-axis. The components of the incoming cocktail beam are identified. 74 Kr is highlighted with a cut, while contaminants 73 Br and 72 Se are also labeled. The γ rays emitted from the de-excitation of the 58 Ni nuclei were measured using the GRETINA array [79]. The GRETINA array was used with seven detector modules. In order to maximize the efficiency of GRETINA to γ-rays emitted at a small angle from the beam axis at the first degrader, the TRIPLEX target was positioned 13 cm upstream of the pivot point. As a result, four of the ◦ ◦ detector modules were placed at laboratory angles between 20 and 50 , which were used for the ◦ ◦ lifetime measurement. The remaining three detectors were arranged at between 70 and 90 , and were used to identify higher-lying states and coincidence relations. 4.3.1 Particle Identification The incoming beam of 74 Kr was identified using a correlation plot between the timing signals from the XFP and OBJ scintillators. The resulting particle identification (PID) plot is shown in Figure 4.4. The particle identification of the reaction residues was done using the energy deposited in the 88 74 Kr Beam Particle Identification Ion Chamber Energy Loss (arb. unit) 3000 74 2500 Kr 102 2000 10 58 Ni 1500 1000 1 500 −180 −160 −140 −120 −100 −80 −60 −40 Time of Flight (arb. unit) −20 0 20 40 Figure 4.5 A particle identification plot for the reaction residues is presented. The y-axis shows the Ion Chamber ∆E while the x-axis shows the trajectory-corrected timing signal from the S800 OBJ scintillator. The particle identification plot is gated on the 74 Kr incoming beam. The 58 Ni residue is identified with a cut and labeled. S800 ion chamber and the corrected time-of-flight for the OBJ scintillator. The resulting particle identification plot for this experiment is shown in Figure 4.5. The PID shown in Figure 4.5 contains many nuclei in the upper p f shell. The 74 Kr secondary beam can be easily identified by the large yield near the top of the graph. Each nearly horizontal line below that identifies a different isotopic chain with constant Z. At the far right of the graph a vertical straight line identifies the N = Z line of constant A/Z = 2. From this information, the other nuclei, including the labeled 58 Ni nucleus can be identified. 4.4 58 Ni γ-ray Spectra 58 In order to constrain the lifetime of the 4+ 1 state of Ni, the observed transitions and background in the γ-ray spectra tagged for 58 Ni need to be understood. In order to identify these states and backgrounds it is necessary to incorporate Doppler-correction to account for the Doppler-shifted energies from in-beam gamma-ray emission. 89 In this experiment, the Doppler-correction was optimized for decays that occur after the first degrader and before the second degrader. As the target foil in the TRIPLEX was located upstream of the GRETINA detector, the Doppler-correction angles were determined by assuming decays occurred at the downstream face of the first degrader. The β for Doppler-correction was set for the average velocity of 58 Ni particles between the two degrader foils. The β value was calculated based on the measured Bρ value from the S800 rigidity and accounting for energy loss in the second degrader. The program LISE++ was used to calculate the expected energy loss through the foil to calculate the appropriate β , which was found to be 0.324 [108]. The Doppler-reconstruction will thus result in three peaks in the spectrum when a cut is applied ◦ for γ-ray emission angles below 40 . The highest energy peak is from decays between the target and first degrader and called the fast peak. The peak centered on the actual transition energy (shown in Figure 4.6) is from decays between the first and second degrader and called the reduced peak. Finally, the lowest energy peak is for decays occurring after the second degrader and is called the slow peak. The Doppler-corrected and uncorrected γ-ray spectra for 58 Ni are presented in Figure 4.7 for the 1-mm setting. Forward-angle data was selected by introducing a cut on GRETINA angles ◦ less than 40 . In addition, a spherical addback was employed with a radius of 7.5 cm to increase the signal-to-background ratio. The 7.5-cm radius was chosen to balance the gain from summing Compton-scattered photons with the losses from accidental coincidence, which were not insignificant due to the large background from the multi-nucleon removal reaction employed in this study. In order to reduce background, only γ rays detected within a restricted timing window are included. + + + In the Doppler-corrected spectrum, the 4+ 1 → 21 transition at 1005 keV and the 21 → 01 transition at 1454 keV manifest a clear three peak structure, as expected. Hints of a three peak+ structure for the 4+ 3 → 41 transition at 1161 keV are also apparent. A partial level scheme for 58 Ni presented in Figure 4.6 shows the transitions observed in the data. A small excess of yield at + 58 870 keV is consistent with the fast component of the 7+ 2 → 62 transition in Ni. A γ − γ analysis was performed to investigate possible coincidences between the 870-keV γ rays and other known 90 4+ 3 1161 3620 keV (0.159 ps) 1005 2459 keV (16 ps) 4+ 1 2+ 1 1454 keV (0.94 ps) 1454 0+ 1 0 keV Figure 4.6 A partial level scheme of 58 Ni showing states observed in the Recoil Distance Method + experiment. The lifetimes in parentheses are from the present measurement for the 4+ 1 → 21 state and for the remainder are from the NNDC evaluation [109]. The thicknesses of lines represent the relative yield from each transition. Figure is from [91]. 58 Ni transitions. However, the 870 keV γ ray was found not to be in coincidence with the other observed transitions and thus could not be identified as part of the decay scheme. 4.4.1 Laboratory Frame Background In this work, a number of clear discrete peaks are identified which originate from the laboratory frame background γ rays, as shown in Figure 4.7. The laboratory frame background was primarily generated by neutron-induced reactions on Aluminum and Germanium in the beam pipe and detectors. The manifestation of a clear peak structure in the Doppler-corrected spectra is a consequence of the cut on the GRETINA angles. The laboratory frame γ-rays are identified through comparison to well known background for 91 Counts/4 keV 1000 (a) 4+1 → 2+1 800 4+3 → 4+1 58 Ni Projectile Frame 2+1 → 0+1 600 Slow 400 Fast Reduced 200 0 400 600 800 1000 1200 1400 1600 1800 Counts/4 keV γ -ray Energy (keV) 1200 58 (b) Ni Lab Frame e +e 76 Ge(n,n’ ) 1000 74 Ge(n27,Al(n, n’)γ ) 800 72 Ge(n,n’) Al reactions 27 600 70 Ge(n,n’) 400 200 0 400 600 800 1000 1200 1400 1600 1800 γ -ray Energy (keV) Figure 4.7 The Doppler-corrected (top) and lab-frame spectra for the 1-mm separation at forward angles are shown. The arrows in the Doppler-corrected spectrum identify the three-peak structure for observed 58 Ni transitions. In both the Doppler-corrected and lab-frame spectra γ-ray peaks from laboratory frame backgrounds are identified and labeled. 92 Target-Degrader Separation 1-mm 10-mm Counts in 1369 keV Peak 355 248 Counts in 1808 keV Peak 513 319 Table 4.1 Observed γ-ray counts from neutron induced reactions on 27 Al for 1-mm and 10-mm separations. in-beam experiments using Germanium detectors [74]. The identified reactions are labeled in the uncorrected γ-ray spectrum in Figure 4.7 and include e+ e− interactions at 511 keV and inelastic neutron scattering on Germanium isotopes in GRETINA. The most important reactions for this experiment are the neutron induced reactions on 27 Al in the beam-pipe and GRETINA frame and detectors. In particular the 27 Al(n, α) reaction results in a γ ray at 1369 keV and the 27 Al(n, dγ) results in a γ ray at 1808 keV. When Doppler-corrected in this experimental setup, these energies overlap with those of the three-peak structures coming from decays of the excited states of 58 Ni. In particular the 1369 keV transition results in a peak near 1070 keV that overlaps with the fast + part of the 4+ 1 → 21 peak, while the 1808 background results in a peak around 1400 keV, near the + slow component of the 2+ 1 → 01 transition. The laboratory frame background γ rays that when Doppler-corrected overlap with the transition energies must be accounted for. Counts in the background peaks were estimated by taking an integral of the background energy region and subtracting off a constant background. A summary of the results is shown in Table 4.1. A GEANT4 simulation package was employed to generate simulated spectra incorporating the beam properties, GRETINA geometry and efficiencies, and the lifetimes of excited states [46]. The simulations were then fit to the data to determine the lifetime. To account for the laboratory frame background, the 1369 keV and 1808 keV signals were simulated for the entire array. The resulting γ-ray spectrum was then scaled to reproduce the appropriate number of counts in each peak. This histogram was then added to simulations for the 10-mm and 1-mm spectra when fitting to data in order to account for the background. 93 4.5 58 Ni Data Analysis The GEANT4 based simulation package incorporates the S800, TRIPLEX and GRETINA to make simulated γ-ray spectra [46][87]. These simulated spectra were fit to data using a χ 2 minimization, where the amplitude of the simulated spectrum, an exponential background, and the lifetime of the 4+ 1 state are variable parameters. In addition, other parameters had to be constrained to extract a lifetime. These parameters were the contribution from reactions in the degraders and effects from feeding of higher-lying states. The GEANT4 simulation incorporates reaction kinematics, velocity changes in the foils, and γ-ray interactions with matter, including attenuation and detection in GRETINA. Best fit spectra incorporating an exponential background, and contributions from neutron-induced background are compared to the data for the 1-mm and 10-mm data sets in Figure 4.8. The 10-mm data set was used to constrain both the relative population of states in 58 Ni and the ratio of population in the target to the degraders. The best fit proportion of populations was + + found to be 25 ± 5% for the 2+ 1 , 50 ± 5% for the 41 state and 25 ± 5% for the 43 state. The ratio of populations in the target to the degraders was determined to be 13+4 −2 . The population ratio between the first and second degraders was estimated as being the ratio of their thicknesses, 1.389. The errors determined here are of statistical origin from a χ 2 minimization. It is worth noting that the apparent ratio between the population in the target and the degraders is smaller than the best fit of 13+4 −2 . This is because the S800 rigidity is optimized for N = Z nuclei, which means that only the lower momentum tail of the 58 Ni distribution can be accepted. This can be understood by examination of the spectrum in Figure 4.9, which shows the spread of energy at the exit of the second degrader (dta). It is apparent from the spectrum that the lower energy tail of the distribution is being selected for by the S800. As the 58 Ni fragments are made from a 74 Kr beam, and as energy loss in materials is proportional to the square of the atomic number, it follows that there will be more energy loss in the degrader for the 74 Kr beam than for a 58 Ni particle. Therefore, the low energy side of the distribution is expected to contain more reactions where the 94 Counts/4 keV 600 500 58 (a) Ni 1mm Separation 4+1→2+1 400 Data 16 ps Fit 5 ps Fit Background Simulation 27 4+3→4+1 Al n-induced reactions + 21→0+1 300 200 100 0 1000 1100 1200 1300 1400 1500 1600 Counts/4 keV γ -Ray Energy (keV) 400 350 300 250 200 150 100 50 0 58 (b) Ni 10mm Separation 4+1 → 2+1 Data Fit Background Simulation 27 Al n-induced reactions 2+1 → 0+1 4+3 → 4+1 1000 1100 1200 1300 1400 1500 1600 γ -Ray Energy (keV) Figure 4.8 Results for the Recoil Distance Method for 58 Ni with a 1-mm (top) and 10-mm (bottom) separation between the target and degraders. The data is compared to a spectrum with best fit parameters for state population and ratio of target to degrader reactions. The simulated spectrum is decomposed into the GEANT4 simulation, the background from neutron-induced reactions on 27 Al, and the exponential background. Figure is from [91]. 95 Figure 4.9 The dta spectrum showing the energy distribution for 58 Ni fragments observed in the S800 Spectrograph is shown. The x-axis shows the percentage difference from the Bρ central value, 1.9 T-m, which corresponds to an energy of 40 MeV per nucleon. The asymmetric shape demonstrates the enhanced S800 selection for reactions occurring in the degraders. 58 Ni was produced in the degraders. There may be a difference in the energy distributions for ions emitting γ rays in each of the three peaks due to the location of the reaction. A comparison of the dta spectra gated on the fast and slow peaks of the 4+ 1 transition is shown in Figure 4.10. It can be seen that aside from a difference in scale, the distributions are similar and do not indicate a strong correlation to the γ-ray peak. The corresponding distribution gated on the reduced peak was found to also show minimal signs of biasing. The γ-ray spectrum for the 1-mm separation is presented in Figure 4.8 (a) and compared to the best fit simulation. The simulation is fit over a range from 900 to 1700 keV, which includes three observed transitions. The lifetime of the 2+ 1 state has been measured previously as 0.94 (3) ps, which is too short to be observed with the setup [106][109]. Due to the large feeding contribution + + + of the 4+ 1 state to the 21 , the apparent lifetime of the 21 state is lengthened by the 41 lifetime. 96 Figure 4.10 The energy distributions for 58 Ni fragments in coincidence with the fast and slow components of the 4+ 1 transition energies are shown. The overall dsitribution of the slow component is slightly larger in scale due to the larger yield, but there is no clear difference in the shapes of the distribtuions. + Therefore, the 2+ 1 spectral shape is also sensitive to the lifetime of the 41 lifetime. The three peak + + structure for both the 4+ 1 and 21 states are thus incorporated to measure the lifetime of the 41 state. Feeding effects from higher-lying states increase the apparent lifetime of the 4+ 1 state. This is accounted for by assuming that all higher-lying feeding states decay through the observed 4+ 3 state focused on in Figure 4.11. Indeed, the three peak structure in the 4+ 3 state is consistent with a lifetime in the region of 0 to 10 ps, longer than the previously measured lifetime of 0.16+0.11 −0.07 ps [109]. This indicates that indeed there are higher-lying states with finite lifetimes populating the 4+ 3 state. In order to constrain this effective lifetime, the three-peak structure at 1161 keV was analyzed separately between 1080 and 1300 keV. The amplitude of the simulation was taken as a free parameter, while the exponential background was fixed to that of the full 900 to 1700 keV region. The results of a χ 2 minimization for these spectra led to a best fit lifetime of 3+5 −3 ps. This 97 58 Counts/4 keV 4+3 → 4+1 Ni 1mm Separation Data 3 ps 300 0 ps 8 ps 200 1100 1150 1200 1250 γ -ray Energy (keV) 1300 Figure 4.11 A zoomed in γ-ray spectrum for the 1-mm data set focused on the 4+ 3 state is presented. Simulations using a fixed exponential background are compared to the data for the best fit lifetime of 3 ps and for the upper and lower bounds of 8 ps and 0 ps. The figure is from [91]. comparison is shown in Figure 4.11, where the fits for lifetimes of 0 ps, 3 ps, and 8 ps are shown and compared to the 1-mm data. Linear and quadratic backgrounds were also fit to the full 900 to 1700 regions and fixed to account for systematic error from the choice of background shape. The systematic error was found to be marginal. The best fit lifetime of the 4+ 1 state was found to be 16 ± 1 ps, where the error is statistical. The systematic error in the 4+ 1 lifetime comes from ambiguities in the ratio of reactions in the target to the degrader, and to the feeding contributions from higher-lying states. By analyzing the + effects on the 4+ 1 lifetime from varying the reaction ratio and the lifetime of the feeding 43 state, the systematic errors were quantified. The uncertainty from the reaction ratio was found to be ±1 ps. The uncertainty due to feeding was constrained with the uncertainty in the apparent lifetime +1 of the 4+ 3 state. This was found to be −2.5 ps. The systematic error from other sources, including uncertainties in the direct population of states was negligible. The statistical error was ±1 ps, and by adding the systematic and statistical uncertainties in quadrature, the overall lifetime of the 4+ 1 98 B (E 2;4+1 → 2+1) e 2fm4 Current Data Kenn et al. Adopted Values GXPF1A en=0.5, ep=1.5 GXPF1A en=0.67, ep=1.12 KB3G en=0.5, ep=1.5 350 300 250 200 150 100 50 0 56 58 60 62 64 66 68 70 A for Ni Isotopes + Figure 4.12 The systematics of the B(E2; 4+ 1 → 21 ) values throughout the Nickel isotopic chain. The experimental values are compared with the shell model values from the GXPF1A and KB3G interactions. For the GXPF1A shell model interactions, results for two sets of effective charges are shown. Figure is from [91]. + + +11 2 4 state was found to be 16+2 −3 ps. The resulting B(E2; 41 → 21 ) is then 50−6 e fm . 4.6 Interpretation of 58 Ni results + The experimental values of the B(E2; 4+ 1 → 21 ) across the Nickel isotopic chain are presented in Figure 4.12, including both the present measurement of 58 Ni and the previous measurement by Kenn et al. [106]. The experimental values for the other isotopes are from the NNDC eval2 4 uation [109]. The present result of 50+11 −6 e fm can be seen to restore the expected unenhanced + 58 60 B(E2; 4+ 1 → 21 ) in Ni relative to Ni, consistent with the shell closure at N = 28. Indeed, the current result restores the smooth parabolic trend between the shell closures at 56,68 Ni, consistent + with the systematics of the B(E2; 2+ 1 → 01 ) values presented in Figure 4.2. To provide a comparison on a quantitative basis, results from the GXPF1A and KB3G shell 99 model interactions are also presented. The calculations were performed in the full p f shell model space, and a portion of the calculations were performed using the code MSHELL64 [102][103][110]. Two sets of proton and neutron effective charges are used, namely the standard values of e p =1.5, en = 0.5 and e p = 1.12, en = 0.67 which were fit to the mirror 27/2− to 23/2− transitions in 51 Fe and 51 Mn in [105]. The B(E2) values for the shell model comparisons were determined by calculating the bare E2 matrix elements for protons and neutrons, A p and An , values. These values are presented in + + + Table 4.2. The bare E2 matrix elements, the B(E2; 2+ 1 → 01 ) and the B(E2; 41 → 21 ) were calculated with the GXPF1A and KB3G interactions with standard and modified effective charges. The calculations were performed in the p f model space and radial integrals were evaluated using harmonic oscillator wave functions, with the harmonic oscillator energy quantum h¯ ω = 45A−1/3 − 25A−2/3 . The B(E2) is obtained from the bare E2 matrix elements through the relation: B(E2; Ji → J f ) = (A p e p + An en )2 /(2Ji + 1) (4.1) + +11 As can be seen in Figure 4.12, the B(E2; 4+ 1 → 21 ) value of 50−6 ps from the Recoil Distance Method discussed here agrees more closely with the shell-model calculations than with the DSAM results from [106]. It can also be seen that in the systematics, the result here restores the single enhancement of collectivity at 62 Ni, which is also reproduced by the shell-model calculations. This strong peak can be understood to arise from the increased occupation of the neutron f5/2 orbital relative to 58 Ni, as the neutron f5/2 contribution to the E2 matrix element adds coherently with those from the neutron p3/2 and p1/2 orbitals [91]. 100 GXPF1A Nucleus 56 Ni 58 Ni 60 Ni 62 Ni 64 Ni Ji → J f + 2+ 1 → 01 + 2+ 1 → 01 + + 21 → 01 + 2+ 1 → 01 + 2+ 1 → 01 56 Ni + 4+ 1 → 21 + + 41 → 21 + 4+ 1 → 21 + + 41 → 21 + 4+ 1 → 21 58 Ni 60 Ni 62 Ni 64 Ni KB3G A p [efm2 ] An [efm2 ] B(E2)1 [e2 fm4 ] B(E2)2 [e2 fm4 ] Ap An 14.19 14.19 161 129 13.33 13.33 10.11 17.97 117 109 6.13 17.49 12.88 22.17 185 171 7.45 22.72 15.00 23.38 234 211 7.71 22.58 10.50 21.08 138 134 5.16 20.29 18.58 9.88 15.40 22.54 7.44 18.58 19.01 24.74 33.38 9.87 153 66 140 283 29 123 63 127 252 25 11.59 11.59 6.11 17.81 5.63 14.13 7.86 19.98 1.28 6.04 B(E2)1 [e2 fm4 ] B(E2)2 [e2 fm4 ] 142 114 64 69 102 111 104 113 64 75 60 36 27 53 2.7 48 39 28 55 3.3 Table 4.2 The values for bare E2 matrix elements A p and An are presented. The values were calculated using the GXPF1A and + + + KB3G Hamiltonians. Values are shown for the 2+ 1 → 01 and 41 → 21 transitions in the Ni isotopes. In addition, B(E2) values are calculated using two sets of effective charges. Calculations with the standard effective charges e p = 1.5, en = 0.5 are denoted with the superscript 1. Calculations with modified effective charges e p = 1.12, en = 0.67 are denoted with the superscript 2. Table and caption are from [91]. 101 4.7 Conclusions 58 The present Recoil Distance Measurement of the 4+ 1 excited state in Ni is a model-independent measurement of the lifetime sensitive to the 10 ps region, in contrast to the previous DSAM measurement sensitive to lifetimes on the order of 1 ps or less. The resulting lifetime of 16+2 −3 ps + +11 2 4 results in a B(E2; 4+ 1 → 21 ) value of 50−6 e fm . This value restores the smooth parabolic curve + of B(E2; 4+ 1 → 21 ) values for Nickel between the N = 28 shell closure and N = 40 subshell clo- sure. This indicates that there is no enhancement of collectivity in 58 Ni. Shell model calculations with the KB3G and GXPF1A interactions are consistent with this picture. The predicted values of 36e2 fm4 and 66e2 fm4 calculated respectively with the KB3G and GXPF1A interactions are consistent with the unenhanced collectivity indicated by the present 2 4 40 measurement of 50+11 −6 e fm . Therefore, contributions from Ca core breaking or higher-lying sdg orbitals may not be needed to interpret the structure of 58 Ni within the Nickel isotopic chain. 102 CHAPTER 5 INVESTIGATION II: ELECTROMAGNETIC TRANSITION STRENGTHS IN 27 NE Electromagnetic transition strengths are useful in studying features of nuclei far from stability. In particular, nuclear states with low orbital angular momentum close to the particle-decay threshold can exhibit extended radial wavefunctions, called halos [26]. The electric dipole operator (E1) is proportional to the operator rY1 (ˆr), and consequently is sensitive to the radial extension of a nucleus. As such, large E1 transition strengths are a known feature of halo nuclei [32]. It is also the case that the E2 transition strength with the operator r2Y2 is anticipated to be sensitive to extended nuclear radii, but this has not yet been observed. The deformation of nuclei away from spherical shapes is also accessible through electromagnetic transition strengths. The electric quadrupole (E2) operator r2Y2 (ˆr) is sensitive to the quadrupole deformation or collectivity of nuclei. Furthermore, the magnetic dipole operator (M1) is sensitive to the nuclear spin and thus affected by the configuration of the nuclear wavefunction. It has recently been demonstrated that the M1 transition strength can be used to identify the dominance of the s-wave component in a valence nucleon, which is the basis of halo formation [111]. For a nucleus with a single particle in a pure s-wave state, the M1 transition cannot occur because there is no spin-flip partner available. Therefore, a hindered M1 transition is expected, as demonstrated in 19 C [111]. Deformed nuclei must have contributions from angular momentum states above l = 0, breaking the spherical symmetry in its intrinsic wavefunction. Therefore, the M1 strength can also be used to probe deformation in nuclei. The interplay of loosely bound effects and deformation is of particular interest in nuclei far from stability. 27 Ne has a low one-neutron separation energy and is one neutron above 26 Ne with a large deformation [27][112]. Therefore, 27 Ne is a candidate for probing a possible interplay between deformation and loosely bound effects. Furthermore, E1, M1, and E2 transitions among low lying states of 27 Ne are all accessible via experiment. In this work, the transition strengths between bound states of 27 Ne have been measured. The E1 103 and M1 transition strengths were determined through a lifetime and branching ratio measurement of excited states in 27 Ne. The E2 transition strength was measured by Coulomb-excitation. In this chapter, the physics motivation for studying 27 Ne is expanded upon, and the experimental setup and results are discussed. In addition to the study of 27 Ne, a lifetime measurement of nearby 28 Ne was performed. Furthermore, a Coulomb-excitation measurement on previously studied 30 Mg was undertaken as a reference to validate the present setup. The measurements were all performed in March 2016 during the second GRETINA campaign at the National Superconducting Cyclotron Laboratory. 5.1 27 Ne Nuclear Structure of 27 Ne with 10 protons and 17 neutrons has five more neutrons than the most neutron-rich stable isotope 22 Ne. 27 Ne has a one-neutron separation energy of 1430(110) keV, and therefore serves as a typical example of a loosely bound neutron rich nucleus [27][113]. Furthermore, as an even-odd nucleus, 27 Ne is useful for investigating neutron single-particle states at the border between the sd and p f shells. 27 Ne is also near the boundary of the island of inversion centered at 31 Mg where the shell closure at N=20 nearly disappears [17]. This causes the ground state configurations of nuclei in the region to occupy the p3/2 and f7/2 orbitals. This manifests as a change in the spin and parity of the ground states for odd-A nuclei, which are sensitive to single-particle structure. A related tendency for the N = 17 isotones is depicted in Figure 1.4. Additionally, the presence of intruder states with negative parity is an indicator of the narrowing shell gap at N=20. Previous experiments on 27 Ne have uncovered many properties of its nuclear structure, including the identification of the energies, spins, and parity for bound and unbound states, as well as the neutron separation energy (Sn ). The adopted values are presented in the level scheme in Figure 5.1. The 765 keV first excited state was first observed in 27 Ne produced through fragmentation[114]. In that experiment, the 765 keV transition was the only one observed, consistent with there being only 104 1430(110) keV Sn (1/2+ ) − (3/2 ) 885(2) keV 120 (E1) 765(1) keV 885 (M1+E2) 765 (E1) 3/2+ gs 0 keV 27 Ne Figure 5.1 The level scheme for 27 Ne is presented. The adopted values for state energies and J π are from [117]. The neutron separation energy is from [27]. γ-ray transitions are shown and are labeled with the transition energy and corresponding multipolarity. one excited state [114]. Later work using neutron knockout reactions established that an 885 keV state was also present in 27 Ne, indicating the potential existence of bound intruder states [115]. In addition, a 119 keV transition was observed between the 885 keV and 765 keV transition, and the level ordering and branching ratio determined [116]. The 765 keV state and 885 keV state were studied to determine their spin and parity, as the pres- 105 ence of a negative parity first excited state would support the intrusion of single particle structure from the p f -shell, as opposed to the positive parity expected from excitations within the sd-shell. Evidence from a (d, p) transfer reaction on 26 Ne supports assigning the 765 keV state with a negative parity [118]. Shell model predictions indicate that if the 765 keV state is negative parity, the 885 keV state should have a J π of 1/2+ [118]. Angular distributions of protons from another (d, p) reaction provide additional evidence that the J π of the 765 keV state is 3/2− [119]. Additionally, the existence of more than one bound excited state for 27 Ne could not be understood without crossshell effects [118]. These states were compared to Monte Carlo Shell Model calculations which indicate a 3/2+ ground state and 3/2− and 1/2+ excited states [116]. This evidence supports the lowering of the N = 20 shell gap in 27 Ne approaching the island of inversion. In this work, 27 Ne is studied to gain insight into loosely-bound effects that may manifest in the lower lying excited states. The 885 keV state with spin (1/2+ ) is consistent with a large l = 0 contribution. Furthermore the 885 keV state is less than one MeV from the neutron separation energy. This combination of low angular momentum and small binding energy is the prerequisite for halo-formation. 27 Ne’s excited states can be studied to measure the E1, M1 and E2 transition strengths between them. The 885 keV state decays via an E1 transition to the 765 keV state, and in another branch decays via the M1 and E2 transitions to the ground state. Additionally, the 765 keV state decays via an E1 transition to the ground state. The E1 transition directly samples over the radial extent of the wavefunction and hence increases with a larger wavefunction. A large E1 transition strength is a signature of halo nuclei, and therefore can be measured in 27 Ne to test if the 885 keV state is consistent an extended wavefunction. Besides this, other transitions add further interest in the structure of 27 Ne. The M1 transition in 27 Ne can be used to investigate the dominance of the l = 0 contribution in the 1/2+ excited state. If the state is primarily l = 0, a hindered M1 strength is expected. The E2 transition strength in nuclei is related to the deformation of the nucleus and can provide information on systematics in the region. An enhanced E2 strength may also be an indicator of a radially extended wavefunction. 106 In this work, the lifetimes of the 885 keV and 765 keV states of 27 Ne were constrained using the Recoil Distance Method. The branching ratio from the 885 keV state has been measured previously, and was confirmed in this measurement [116]. The B(E1; 3/2− → 3/2+ gs ) is directly determined by the lifetime of the 765 keV state. However, the lifetime of the 885 keV state is determined by the sum of the decay rates associated with the B(E1; 1/2+ → 3/2− ), B(M1; 1/2+ → + + + − 3/2+ gs ) and B(E2; 1/2 → 3/2gs ) transition strengths. The B(E1; 1/2 → 3/2 ) is determined from the lifetime of the state and the branching ratio. The sum of the M1 and E2 transition strengths is determined by the partial lifetime of the 885 state decaying to the ground state. To separate the E2 and M1 components, the B(E2; 3/2+ gs → 1/2+ ) was measured using a Coulomb-excitation technique. Thus the M1 transition strength can be found from the other constrained partial lifetimes as: 1 1 1 1 = + + τ τM1 τE1 τE2 (5.1) where τ is the lifetime of the 885 keV state, τπλ is the partial lifetime from the B(πλ ) transition strength. The τE2 is constrained directly from the Coulomb-excitation measurement. The τE1 is found by taking the lifetime of the 1/2+ state and dividing by the branching ratio for decays to the 3/2− state. The τM1 is then extracted from the state lifetime and the other two partial lifetimes. In addition to 27 Ne, a lifetime measurement of 28 Ne was performed as a reference. The + B(E2; 0+ gs → 21 ) has been measured previously through heavy ion inelastic scattering [55][120], and therefore the present method serves as a confirmation with a model-independent measure+ ment. Similarly, 30 Mg with a previously measured B(E2; 0+ gs → 21 ) was revisited in the Coulomb- excitation measurement to demonstrate the present Coulomb-excitation setup. The branching ratio measurement, lifetime measurements, and Coulomb-excitation measurements are described in the next section. First the lifetime and branching ratio measurement setup will be described. The results for 28 Ne are shown to validate the present measurement, followed by results for 27 Ne. The modified Coulomb-excitation setup is then described. The results for 30 Mg are presented and discussed, and is followed by the Coulomb-excitation results for 27 Ne. A 107 discussion of the implications of the measurements in 27 Ne is presented last. 5.2 Experimental Setup I: Branching Ratio and Lifetime Measurement The branching ratio and lifetime of 27 Ne were measured at the National Superconducting Cyclotron Laboratory. A γ-ray measurement with only a target was used to constrain the branching ratio. The Recoil Distance Method was employed to measure the lifetimes. The experiment was performed using the S800 Spectrograph in coincidence with the GRETINA array. A 48 Ca primary beam was developed at the coupled-cyclotron facility at 140 MeV per nucleon. This beam was impinged on a 9 Be target to make a secondary beam of 29 Na which was separated in the A1900. The 29 Na secondary beam delivered to the S3 vault had an energy of 90.5 MeV per nucleon with a purity of 48%. Reactions on the target were used to produce fragments of interest, including 27 Ne and 28 Ne, which were identified in the S800. The γ rays in this experiment were measured using the GRETINA array [79], using nine de◦ tector modules. In this experiment, four detectors were located at forward angles between 20 and ◦ ◦ ◦ 50 while five were placed at angles between 70 and 90 . The forward detectors were used in the lifetime measurement while all detectors were used in the branching ratio and 25-mm separation measurements. The TRIPLEX Plunger was used to place target and degrader foils into the beamline. The TRIPLEX was placed upstream of GRETINA’s center by 13 cm to maximize the geometric efficiency of γ rays emitted at small angles from the beam axis. In the branching ratio measurement only a target foil was used. In the lifetime measurement both a target and degrader foil were used. The target was a 1.0-mm 9 Be foil and the degrader was a 0.92 mm-Ta foil. For the lifetime measurement, the target and degrader were positioned near contact to measure the lifetime expected to be near 1 ps. This setting resulted in a separation distance of 50 µm. The experiment was run for three days in this setting. The excited states in 27 Ne and 28 Ne can be populated by reactions in the degrader, which must 108 Figure 5.2 A histogram of γ-ray interactions with GRETINA is presented. The Doppler-corrected γ-ray energy is plotted on the x-axis, while the angle from the target to the interaction position is plotted on the y-axis. The two nearly parallel lines near 1305 and 1240 keV are from reactions populating of 28 Ne at the target and degrader respectively. be accounted for in the lifetime measurement. Therefore, a second measurement with a distance between the target and degrader of 25 mm was performed for one day to constrain the population due to reactions in the degrader. The travel time for a nucleus across 25 mm is approximately 250 ps. Since the states of interest have lifetimes expected to be shorter than 10 ps, decays that occur after the degrader must have been populated in the degrader. In addition, the 25-mm separation measurement results in two distinct peaks for decays occurring before and after the degrader at all angles. Therefore, the entirety of GRETINA can be used to make the measurement. A two-dimensional spectrum showing the separation over all angles is shown in Figure 5.2 for the case of 28 Ne. It can be seen that there are two vertical lines corresponding to decays before and after the degrader. The reaction residues were identified with the S800 Spectrograph. The S800 Spectrograph was used in coincidence with GRETINA to tag γ rays. 109 Figure 5.3 A timing signal correlation plot for the incoming 29 Na secondary beam is shown. The data is from the target-only measurement. The incoming 29 Na beam and contaminants are labeled. 5.2.1 Particle Identification The secondary beam Particle Identification plot is shown in 5.3. The incoming beam of 29 Na was identified using a timing correlation plot between the signals of the S800 OBJ scintillator and the A1900 XFP scintillator. The residues of interest were found by gating on the incoming spectrum on the 29 Na and then looking at the energy deposited in the S800 ion chamber against the corrected time-of-flight for the S800 Object scintillator. This is shown in Figure 5.4 for the target only data, where the 29 Na incoming beam and fragmentation products are identified. 5.2.2 Laboratory Frame Background A number of laboratory frame backgrounds can be observed in the present experiment. As the laboratory frame background can overlap with the peaks of interest, the background was measured for each setting and simulated to determine their Doppler-reconstructed energies. These reconstructed energies were then added to the fit in order to account for these effects. A table of the 110 Figure 5.4 The particle identification for reaction products at the S800 is shown. The incoming 29 Na beam and the 28,27 Ne isotopes generated by the reaction are labeled. Data shown is for the target-only measurement and is gated on the incoming 29 Na beam. γ-ray Energy (keV) 511 596 832 842 1014 1039 Source e− , e+ 74 Ge (n, n ) 72 Ge (n, n ) 27 Al (n, n ) 27 Al (n, n ) 70 Ge (n, n ) 27 Ne 190 470 220 180 120 120 25-mm 27 Ne 0.05-mm 260 500 280 230 160 100 28 Ne 410 630 180 260 170 150 25-mm 28 Ne 0.05-mm 240 310 70 140 70 10 Table 5.1 The number of counts in the peak for each neutron induced background is provided. For each case this is for the cut with a 7.5-cm spherical addback and with the timing cut applied. ◦ For the 0.05-mm separation data, the counts are provided for interactions at angles below 40 from the degrader. background peaks measured for each of the settings is provided in Table 5.1. For the target-only data, the background is minimal and not included. 5.3 28 Ne Measurement + 28 The B(E2; 0+ gs → 21 ) of Ne has been measured before through heavy ion inelastic scattering + measurements [55][120]. In the measurement reported in [120] the B(E2; 0+ gs → 21 ) was found 111 + 2 4 to be 269 (136) e2 fm4 . In [55], a more precise value for the B(E2; 0+ gs → 21 ) of 132(23) e fm was determined. The lifetime corresponding to the transition strength is 8.2(14) ps. The present measurement of 28 Ne serves as a model-independent confirmation of the inelastic scattering measurements. This measurement also demonstrates the sensitivity of the Recoil Distance Method in the current configuration to lifetimes on the order of 1 to 10 ps. The level scheme for 28 Ne is shown in Figure 5.5. The known states include the 2+ state at 1304 keV, the 4+ state at 3010 keV, and another 4+ state at 3904 keV. The 3010 keV state decays via a 1706(5) keV γ ray and the 3904 through a 894(4) keV γ ray [121]. An 1127(4) keV transition has also been previously observed, though it is not known what level it decays from. 5.3.1 Target Only Measurement The 28 Ne γ-ray spectrum from reactions on only the 1.0 mm Be target is shown in Figure 5.6. As there is no low energy γ ray of interest, the 7.5 cm spherical addback is used. Each of the 894, 1127, 1304, and 1710 keV energy γ rays are observed and labeled in the figure. The population of states was determined by comparison to the GEANT4 based simulation [87]. The cascade of states was simulated, and a direct population to each state determined by varying the relative populations until a best fit was found. This resulted in a population of 14% for the 1127 keV transition state, 57% for the 1304 keV state, 13% for the 3010 keV state and 16% for the 3904 keV state, where the statistical error is 3%. 5.3.2 25-mm Separation The 28 Ne spectrum for the 25-mm separation between the 1.0 mm 9 Be target and 0.92 mm Ta degrader is shown in Figure 5.7. Two peak structures for the 894 keV, 1127 keV, 1307 keV and 1710 keV can be observed. GEANT4 simulations incorporating the setup were performed over a range of target-to-degrader ratios, and fit to the data using a scaling factor, an exponential background and the neutron-induced background [46]. As no angle cut is applied, the Doppler-shifted 112 (4+ ) 3904(7) keV 894 (4) (4+ ) 3010(6) keV 1706 (5) 1304(3) keV X+1127(4) keV 2+ 1304(3) 1127(4) 0+ gs 0 keV 28 Ne Figure 5.5 The level scheme for 28 Ne is shown for known level states and transitions. The 1127 keV transition is a decay between unknown states represented by the dashed lines. The J π assignments and values for state and transition energies are from [121]. 113 Figure 5.6 The Doppler-corrected γ-ray spectrum for 28 Ne with only a 1-mm Be target is presented. The observed transitions are labeled, including the 1127 keV transition decaying from an unknown state. The best fit simulation for the relative population of states is presented in comparison, along with an exponential background. laboratory frame background is continuous. The best fit simulation for a target-to-degrader ratio of 2.1 is shown, where the same ratio was used for all transitions. The error in the target-to-degrader ratio is ±0.1 which is statistical only. 5.3.3 Lifetime Measurement The γ-ray spectrum for 28 Ne with a 50-µm separation between the target and degrader and for ◦ GRETINA angles below 40 is shown in Figure 5.8. As in the target only data, the 894 keV, 1127 keV, 1304 keV and 1706 keV transitions are visible. However, the measurement of the lifetime of the excited states of 28 Ne in this experiment poses a number of complications that needed to be accounted for, in particular the presence of feeding states and the S800 acceptance. The presence of feeding states above a state of interest decaying into the state of interest can affect the apparent lifetime. Figure 5.9 shows a comparison of the lifetimes for the 894 keV, 1127 114 Figure 5.7 The spectrum for the 25-mm separation between target and degrader gated on 28 Ne particles is presented. States of interest are labeled and transition energies identified. Two peak structures are apparent for all transitions, and a best fit simulation presented. The best fit simulation is for a ratio of 2.1 reactions on the target for each reaction on the degrader for all states. keV, and 1706 keV transitions. The feeding lifetimes are listed in Table 5.2, where the error is from statistics only. As shown in Figure 5.5, the observed 1706 keV transition from the 3010 keV state decays into the 1304 keV state, while the 894 keV transition decays into the 3010 keV state. This means that the lifetimes of these states can change the apparent lifetime of the 1304 keV state through their feeding, and must be taken into account. This was accomplished by incorporating the feeding states into the GEANT4 simulation used to analyze the data, and determining the lifetime for each state according to where it fit in the decay scheme. The simulations are varied over the lifetime of the state and incorporate an exponential background and a scaling parameter as free parameters. The 894 keV transition corresponds to the highest state measured in the cascade and consequently its lifetime was constrained first. Then the lifetime of the 1706 keV transition was measured using the feeding lifetime for the 894 keV transition found previously. As the 1127 keV transition is 115 ◦ Figure 5.8 The γ-ray spectrum for 28 Ne is presented for GRETINA angles below 40 for a 50 µm separation between the target and degrader. The observed transitions are labeled along with the neutron-induced background. A best fit simulation using the best fit lifetimes is overlayed, along with an exponential background. The lifetimes used for the 1304 keV state was 6.25 ps. The lifetimes for higher-lying states are the best fit values in Table 5.2 as discussed in the text. Transition + 4+ 1 → 21 + 4+ 2 → 41 Unknown Energy (keV) Lifetime (ps) Error (ps) 1711 2.5 0.25 894 0 0.5 1127 1.5 0.25 Table 5.2 The measured lifetimes for transitions among states lying higher than the 2+ 1 state in 28 Ne are presented. Error presented is statistical only. independent of the other levels, it was measured independently of the cascade. Finally, the full spectrum was fit with the fixed feeding lifetimes and the lifetime of the 1304 keV state was varied. In order to account for the feeding to the 2+ 1 state, it is assumed that all feeding proceeds via the 4+ 1 and is accounted for by the measured lifetime of 2.5 ± 0.25 ps, where the error is statistical only. It is worth noting that the 2.5 ps lifetime is consistent with the expectation for this 4+ 1 state if + + + + the B(E2) is constant for the 4+ 1 → 21 and 21 → 0gs transitions. The lifetime of the 41 state will 5 1304 then be different from the 2+ ≈ 0.26. As the lifetime extracted from 1 state by a factor of 1706 116 Counts/8 keV 180 160 140 120 100 80 60 40 20 0 28 Ne 0.05 mm Separation 4+1→2+1 1706 keV Data Fit Background 1550 1600 1650 1700 1750 1800 1850 1900 1950 Counts/8 keV γ -ray Energy (keV) 400 350 300 250 200 150 100 50 0 4+2 →4+1 1127 keV 894 keV 800 900 1000 1100 γ -ray Energy (keV) 1200 Figure 5.9 28 Ne spectra for the 0.05 mm separation between target and degrader are shown zoomed in to focus on the (Top) 1706 keV and (Bottom) 894 keV and 1127 keV transitions. The data are compared to best fit simulations for each state, which for the 894 keV transition is 0 ps, for the 1127 keV transition is 1.5 ps, and for the 1711 keV transition is 2.5 ps. 117 the previous inelastic scattering measurement is 8.2 ps, the expected lifetime of the 4+ 1 state would be approximately 2 ps. To determine the lifetime of the 1304 keV state, the cascade was simulated with the lifetimes extracted above. The 1304 keV state lifetime was then varied in the GEANT4 simulations, which were compared incorporating an exponential background, background from neutron induced γ rays, and a scaling factor. The best fit lifetime was found to be 6.25 ± 0.25 ps. Figure 5.8 shows the best fit simulation compared to the data over all of the measured transitions. The systematic error in the 2+ 1 lifetime came from three sources, namely the uncertainty in the target to degrader population ratio, ambiguity in the feeding lifetime, and effects arising from the momentum distribution. The systematic error from the uncertainty in the target-to-degrader population ratio was constrained by comparing simulations with the ratio modified to the upper and lower statistical bounds of 2.2 and 2.0 respectively. The lifetime of the 2+ 1 state was varied as above, while feeding lifetimes were kept constant. This resulted in a lower bound of 6 ps and an upper bound of 7 ps. The systematic error arising from uncertainty in the feeding lifetime of the 4+ 1 state was accounted for by changing the lifetime of the feeding state to the lower and upper limits of 2.25 ps and 2.75 ps in the simulated cascade and again fitting the lifetime of the 1304 keV transition. This error was found to be negligible compared to statistics. The last source of systematic error in this measurement is due to uncertainty in the energy distribution of the 28 Ne reaction products. As the S800 Spectrograph was not optimized to accept 28 Ne, only the lower tail of the energy distribution was actually accepted by the spectrograph. The cutoff on the energy distribution corresponds to a cut on the reaction position in the target, because reactions that occur further in the target will have greater energy loss than those occurring earlier due to the higher Z of the 29 Na secondary beam compared to the 28 Ne reaction product. As the target width of 1.0 mm dominates the region where emissions corresponding to the fast peak in the spectrum occur, this can affect the lifetime. The energy distribution gated on coincidences with GRETINA for 28 Ne is shown in Figure 5.10. The lifetime analysis was performed independently 118 2500 Counts 2000 1500 1000 500 −6 −4 −2 0 2 4 6 Projectile Energy at Degrader Position(%) Figure 5.10 The energy distribution for 28 Ne reconstructed for after the degrader is presented. The x-axis shows the energy in terms of percentage above or below the Bρ central value, which in this case was 3.02 T-m, which corresponded to an energy of 54.5 MeV per nucleon. The 28 Ne makes up the shoulder on the high energy side of the spectrum, while the shoulder on the low energy edge is a secondary beam contaminant in the particle identification. Therefore, only the lower energy tail of the 28 Ne distribution is accepted by the S800, and so uncertainty in the lifetime due to uncertainty in the distribution must be constrained. on the energies above and below the midpoint of the distribution, which was found at 1.0% above the rigidity setting, to evaluate the systemic uncertainty due to the energy distribution. This was done for the 25-mm and 0.05-mm data to determine the target-to-degrader excitation ratios and the lifetimes independently. The ratio for the lower half of the energy distribution was found to be 2.4, while the ratio for the upper half was found to be 0.9. The corresponding lifetime for the upper half of the energy distribution was 6.5 ps, while the lower half of the energy distribution corresponded to 5 ps. +0.75 Therefore, the lifetime of the 2+ 1 state was found to be 6.25 ps ±0.25 ps (stat) −0.25 ps (Ratio) +0.25 −1.25 ps (Energy Distribution), or combining the systematic errors in quadrature, 6.25 ps ±0.25 + +45 2 4 + ps (stat) +0.8 −1.3 ps (systemic). This corresponds to a B(E2; 0gs → 21 ) of 175−20 e fm . This mea- surement is consistent with the Coulomb-excitation measurement of 269 (136) e2 fm4 measured 119 in [120]. The current result is larger than the previously measured value of 132 (23) e2 fm4 from heavy ion inelastic scattering in [55], although the two measurements marginally overlap within error. Indeed, in the previous inelastic scattering experiment, B(E2) values were extracted using ECIS with two optical potentials, determined respectively from 20 Ne+208 Pb and 40 Ar+208 Pb interactions [55]. The 20 Ne+208 Pb optical potential with equal nuclear and Coulomb deformation parameters resulted in a B(E2) of 116 (12) e2 fm4 , while the 40 Ar+208 Pb optical potential with equal nuclear and Coulomb deformation parameters resulted in a B(E2) of 145 (22) e2 fm4 . The present result indicates that the 40 Ar+208 Pb optical potential may more accurately model reactions in 28 Ne. 5.4 27 Ne Measurement The lifetimes of the 1/2+ and 3/2− states in 27 Ne were constrained with the Recoil Distance Method. The target only measurement was used to determine the direct population of states for the measurement. In addition, the target only measurement was used to determine the branching ratio of the 1/2+ state which decays by either a 119 keV or an 885 keV γ ray. The branching ratio allows for the extraction of partial lifetimes and is needed in the Coulomb-excitation data where only the yield of the 885 keV γ-ray is measured. In the Recoil Distance Method measurement, the 25-mm separation was used to constrain the population of 27 Ne due to reactions in the degrader. The near contact 50-µm separation was used to determine the lifetimes of the excited states. 5.4.1 Target Only Measurement The branching ratio for 27 Ne has previously been measured as 19.3% ± 2.5% for the 119 keV transition, and 81.7% ± 2.5% for the 885 keV transition [116]. To confirm this measurement, the γ-ray transitions for the reaction on the target with no degrader were measured. As there is only one velocity region in this measurement, the γ rays are measured using all nine GRETINA detectors. 120 Figure 5.11 The Doppler-corrected spectrum for 27 Ne with only a 1 mm thick Be target is shown. The transitions observed are labeled. The data is compared with a best fit simulation to reproduce the branching ratio of the 885 keV state. As the energy of the 119 keV transition is small and more likely to be dominated by the photopeak, the branching ratio was measured using only the singles data from the GRETINA crystals. The γ-ray spectrum for the target only data for 27 Ne is shown in 5.11. The 119 keV and 885 keV transitions from the 1/2+ state are identified along with the 765 keV transition from the 3/2− state. The branching ratio of the 885 keV state was determined by comparison to a GEANT4 based simulation that incorporates the response function of GRETINA to the Doppler-shifted energies of the emitted γ rays. For simplicity, the γ rays were considered as coming from three states without a cascade and the relative population used to determine the branching ratio. First the 885 keV and 765 keV γ-rays were considered, finding a yield ratio of 52% for the 885 keV transition and 48% for the 765 keV transition. Then this ratio was fixed and the 119 keV transition was considered separately, with the relative yield of the 119 keV state varied. The best fit was found to be for a yield of 17% ± 2% for the 119 keV transition, and 83% ± 2% for the 885 keV transition, where 121 Figure 5.12 The γ-ray spectrum for 27 Ne with a 25 mm separation between the target and degrader is presented. Spherical addback with a 7.5-cm radius was employed. Only events in which a single added-back event was measured are included. The observed transitions are identified and labeled. the error is statistical only. This is consistent with the previous measurement, and is used for the extraction of the partial lifetimes and Coulomb excitation data [116]. 5.4.2 25-mm Separation As in 28 Ne, the population ratio due to reactions in the target compared to reactions in the degrader was constrained. This was done by measuring the γ rays with a 25-mm separation between the target and degrader. The 25-mm spectrum for 27 Ne is presented in Figure 5.12, and the observed transitions identified. In this data the 7.5-cm spherical addback was employed. In addition, only events that had a single addback sphere were included, which reduced the background at the 885 and 765 keV reconstructed energies. The population due to reactions in the degrader was independently measured for the 765 keV and 885 keV states. The spectra focused on the 885 and 765 keV states compared to best fit 122 simulations for the ratio of population due to reactions in the target to reactions in the degrader are presented in Figure 5.13. It was found there is a significantly larger yield in the 25-mm data for the population of the 765 keV state in the degrader compared to the 885 keV. To individually constrain the population of the 765 and 885 keV states in the degrader, GEANT4 simulations incorporating the current setup were generated varying the ratio. An exponential background was found by fitting it over the spectrum from 500 to 2000 keV. This background was then fixed and the simulations were fit in a limited region for each of the 765 and 885 keV states. The best fit value for the target to degrader population ratio for 885 keV state was 5.2 ±0.6. The best fit value for the 765 keV state was 3.9+6.1 −0.9 . The apparent ratio of population in the target to population in the degrader is even larger than the simulated values because the S800 Spectrograph was focused on the fragments produced in the target. 5.4.3 Lifetime Measurement The γ-ray spectrum for the 0.05-mm separation is presented in Figure 5.14. As before, a cut on ◦ the GRETINA angle below 40 is applied. Spherical addback is applied with a 7.5-cm radius. Only events with a single added-back event measured are included to reduce background. The 765 keV and 885 keV levels are identified and labeled, as are the Doppler-shifted laboratory frame backgrounds. In addition neutron induced backgrounds at 1014 and 1040 keV in the Dopplercorrected spectrum overlap with the 765-keV transition energy. However, the overall yield from these background contributions is only 35 counts total, which was found to have a marginal effect on the result as shown in the shaded histogram in Figure 5.14. The 765 keV peak has a significantly larger low-energy tail compared to the 885 keV peak, which indicates that the 3/2− state has a long lifetime compared to the 1/2+ state. The 27 Ne spectrum was compared to GEANT4 simulations independently for each of the 765 and 885 keV transitions. Figure 5.15 shows the data along with the best fit simulations and comparisons to a 6 ps and 0 ps lifetime to demonstrate sensitivity. The best fit target-to-degrader population ratios measured previously for the 765 and 885 keV states were used for the corre- 123 Figure 5.13 γ-ray spectra for 27 Ne with a 25-mm separation are presented focused on the 765 keV state (top) and 885 keV state (bottom). Spherical addback with a 7.5-cm radius was employed. Only events with a single added-back event measured in coincidence with the ion are included. Best fit spectra for each of the target-to-degrader excitation ratios are shown along with the fixed exponential background. The ratio for the 765 keV state is 3.9 reactions in the target per reaction in the degrader. The ratio for the 885 keV state is 5.2 reactions in the target per reaction in the degrader. 124 Figure 5.14 The 27 Ne γ-ray spectrum for a 0.05-mm separation between the target and degrader ◦ is presented for angles below 40 . Spherical addback with a 7.5 cm radius was employed. Only events with a single added-back event measured in coincidence with the ion are included. Peaks from transitions and neutron-induced backgrounds are identified. sponding lifetime measurement. The χ 2 distributions are presented for each in insets at the top left of each spectrum. The lifetime of the 885 keV state was determined to have an upper limit of 2.0 ps from the χ 2 distribution in Figure 5.15, where statistical error is included in the determination of the upper bound. Ambiguity in the lifetime due to the ratio of population from reactions in the degrader was found by simulating the spectra using the upper bound of 5.8 and varying the lifetime. This was found to have a negligible effect on the upper bound of 2.0 ps. The lifetime of the 765 keV state was constrained to be 3.1 ± 0.5 ps, where the error is purely statistical. Systematic error in the measurement came from ambiguity in the ratio of population from reactions on the degrader. The uncertainty was determined by comparison to simulated spectra with an upper bound of 10 and lower bound of 3. The lifetime was varied and the ambiguity was found to be +0.6 −0.8 ps. The lifetime of the 885 keV state which feeds into the 765 keV state was 125 Figure 5.15 The γ-ray spectra for 27 Ne with a 0.05-mm separation focused on 765 keV state (top) and 885 keV state (bottom) are shown. The fixed exponential background is presented in both figures, and in (top) the small contribution from neutron-induced background is presented. In both figures a χ 2 plot for the fits is presented, where the number of degrees of freedom is 15 for the 765 keV and 19 for the 885 keV. (top) The best fit simulated spectra with a 765 keV state lifetime of 3.1 ps is compared to simulated spectra with lifetimes 0 ps and 6 ps. (bottom) Simulated spectra with the upper bound 885 keV state lifetimes of 2 ps, is compared to simulated spectra with lifetimes of 0 ps and 6 ps. 126 assumed to have a 0 ps lifetime in this analysis. The error due to this is considered to be marginal because of the small branching ratio for decays into the 765 keV state, the upper bound lifetime of 2 ps, and the selection of only events with a single coincident added-back γ-ray interaction. +0.6 ps, where the error is the quadratic sum of the The lifetime of the 765 keV state is then 3.1−0.9 statistical and systematic errors. 5.5 Modified Coulomb-excitation measurement E2 transition strengths can be determined from Coulomb-excitation, but at energies above the Coulomb barrier the inelastic scattering can have substantial contributions from nuclear interactions. As such, it is necessary to constrain the impact of nuclear contributions to determine the E2 strength. One way in which to do this is to measure inelastic scattering on two targets, one with a low Z, and one with a high Z. The low Z target will have a minimal Coulomb-field and therefore will be sensitive to direct nuclear interactions, while the high Z material will be sensitive to Coulomb-excitation. The downside of measuring the inelastic scattering on both high and low Z materials is that it would require a longer running time, or a reduction in statistics for the two separate materials, which is especially difficult for rare-isotope experiments. However, the method performed in this experiment allows for the simultaneous measurement of reactions on two targets. In this modified Coulomb-excitation measurement, two foils are mounted in the beamline, separated by a distance determined by the geometry of the detectors. For the present experiment, the distance chosen was 25 mm. Furthermore, the two targets are placed upstream of the γ-ray detectors, in this case by 13 cm. The first target (T1 ) is chosen to be the low-Z target, so as to minimize the stopping power. The second target (T2 ) is chosen to be the high-Z material, to maximize stopping power. The γ-ray spectra measured in this setup have two distinct peaks for each transition, as in the 25-mm separation measurement of 28 Ne as presented in Figure 5.2. The lower energy peak is due to decays that occur after the degrader, while the higher energy peak is due to decays that occur between the target and degrader. 127 In this experiment, the states of interest are expected to have lifetimes below 10 ps, and the distance corresponds to a flight time of approximately 250 ps. This means that effectively all states that are excited at T1 decay before T2 and the two resulting peaks correspond exactly to reactions in T1 and T2 . As such the relative contribution of Coulomb and nuclear excitations can be found from the peak yields. The integrated cross section of each reaction can therefore be found as: σR = Nγ N N 1 1 = Coinc = Coinc NP × σT εCoinc NP × σT εCoinc NB × PB × σT (5.2) where σR is the cross section of the reaction integrated over all angles, Nγ is the number of γ rays emitted, σT is the areal number density of the target, NP is the number of incoming projectiles of interest, NCoinc is the number of counts in the γ-ray peak measured in coincidence with the ion, εCoinc is the efficiency for measuring a coincidence. This is the combined geometrical and intrinsic efficiency of the γ-ray detector along with the deadtime of the combined spectrographdetector system. NB is the number of particles in the beam, PB is the purity of the beam for the projectile of interest. Therefore, to determine the cross section and extract the B(E2), it is necessary to determine the total incoming counts from the beam, the purity of the beam, the number of detected gammas in each peak, the areal density of the target, and the efficiency of the detectors. 5.5.1 Coupled Channel Equations The cross section determined from the measurement can be converted into a B(E2) using a CoupledChannel calculation code, in this case ECIS97 [57]. The program Equations Couplées en Itérations Séquentielles (ECIS) takes as input an optical potential which models the nuclear scattering between the target and projectile nuclei. The Coulomb potential is incorporated using the proton numbers of the target and projectile nuclei. In addition, the energy levels and deformation parameter for the target nucleus are input to calculate the cross section of an excitation into a particular state at a given beam energy. The deformation parameter can be separated into a nuclear compo- 128 nent (βNuclear ) and a Coulomb component (βCoulomb ). The optical potential used for both 27 Ne and 30 Mg was determined from reactions of 17 O + 208 Pb at 84 MeV per nucleon [58]. To determine the relative importance of nuclear and Coulomb contributions to the reaction cross sections in this experiment, it was assumed that the entire cross section for the 9 Be target came from the nuclear contribution. Thus the βCoulomb was set to 0 and the βNuclear varied until the calculated cross section matched the experimental values. Then with the βNuclear held at the fixed value, the βCoulomb was varied to determine the B(E2) value, which is related to the deformation parameter by: B(E2 ↑) = 3ZeR2 β 4π Coulomb 2 (5.3) where in this equation, Z is the atomic number of the nucleus, e is the electron charge, and R is the radius of the nucleus, generally calculated as 1.2 × A1/3 fm. Error in the B(E2) comes primarily from uncertainty in the measured cross section, with additional error due to uncertainty in the βNuclear . Ambiguity in the reaction energy due to loss in the foils was found to have a marginal impact on the results. 5.6 Experimental Setup II: Coulomb Excitation The lifetime and branching ratio of the 27 Ne 885 keV state cannot directly provide information on the B(M1; 1/2+ → 3/2+ gs ), as there may be a significant component of the lifetime that is from the E2 transition strength. The second setup is designed to measure the Coulomb-excitation of 27 Ne at energies above the Coulomb barrier to extract the E2 directly and therefore constrain the M1 strength. The modified Coulomb-excitation setup is used for the first time in this work. Therefore, + + 27 + 30 in addition to the B(E2; 3/2+ gs → 1/2 ) of Ne, the previously studied B(E2; 0gs → 21 ) of Mg was also measured to confirm the method’s reliability. ◦ ◦ In this experiment, 9 GRETINA detectors were used, four at angles between 20 and 50 ◦ ◦ and 5 at angles between 70 and 90 . GRETINA was employed in coincidence with the S800, which was used to identify and count the incoming particles. The S800 was operated to record 129 incoming particles downscaled by a factor of 100 in addition to measuring particles in coincidence with GRETINA. The S800 trigger condition was stored in the data, and therefore the incoming beam could be determined free from the bias from the coincidence condition. Raw scalar numbers of counts at each scintillator were also recorded for the experiment using the S800, from which the total number of beam particles could be extracted. The measurement was performed over the course of two days. In the experiment, a secondary cocktail beam including 27 Ne and 30 Mg was produced by impinging a 48 Ca beam accelerated to 140 MeV per nucleon at the coupled cyclotrons on a 9 Be target. The fragments were selected for at the A1900 and the secondary beam was sent to the S3 vault. The 30 Mg secondary beam was at 113 MeV per nucleon, while the 27 Ne secondary beam was at 97 MeV per nucleon. The purity of the beam was measured at the S800 by using the ion-chamber and time signal of the S800 Object scintillator. The trigger condition chosen was for the downscaled singles, with coincidences excluded. The purity of the beam through the S800 was then determined by comparison of the total ions measured in the singles to those identified in the PID as the nucleus of interest. This resulted in a purity of the 27 Ne of 28.3 (6)% and 30 Mg of 34.7 (7)%. The error in the purity was due to statistical uncertainty and contaminants in the beam gates. The particle identification for the secondary beam at the S800 is presented in Figure 5.16. The secondary beam was sent through the TRIPLEX plunger which held two foils, a 1.0 mm 9 Be target and a 0.92 mm Ta target separated by 25 mm. The 9 Be target had an areal density of 21 ❛t♦♠s . The TRIPLEX was 1.23 × 1022 ❛t♦♠s 2 . The Ta target had an areal density of 5.08 × 10 2 cm cm located 13 cm upstream from the GRETINA center to enable the inelastic-scattering measurement. 5.6.1 GRETINA Efficiency Because the γ rays emitted in-beam are shifted to forward angles, the combined geometrical and intrinsic efficiencies of GRETINA were determined by comparison to a GEANT4 simulation. GRETINA was calibrated using an 152 Eu source located on the tantalum target. The efficiency 130 Figure 5.16 The particle identification of the secondary cocktail beam in the S800 focal plane is presented. The ion chamber energy loss is plotted against the time of flight from the S800 Object scintillator to the S800 E1 scintillator. The 30 Mg and 27 Ne components are labeled. of a simulated 152 Eu source in the same configuration was compared to the measured efficiencies. A scaling factor was then introduced and determined to account for the difference between the simulated and experimental values. A scaling factor of 0.75 ± 0.05 was adopted based on the efficiency of the highest energy γ ray, and where the error is from the uncertainty in the relative efficiencies for each γ-ray energy, the extrapolation to non-measured energies, and comparison with previous work. Values for the simulated and measured efficiencies for selected 152 Eu γ-ray energies are shown in Table 3.1. The efficiency depends on both the decay position and the energy of the γ-ray, meaning that there are different efficiencies for each of the measured γ rays and for each of the two peaks. To determine the efficiency of the γ rays emitted in-flight, decays from the states of interest were simulated at the first and second target positions. The efficiency was determined from this simulation by counting the amount of detected γ-rays in the related peak and comparing against the total number emitted. This number was then scaled by 0.75 to account for differences between the simulation and data. The efficiencies used for each γ-ray from interactions occurring at each position are shown in Table 5.3. 131 Nucleus γ-ray Energy (keV) Target Efficiency 30 Mg 9 Be 5.6 (4) % Ta 6.2 (4) % 9 Be 7.0 (4)% Ta 7.1(4)% 27 Ne 1483 885 Table 5.3 Scaled simulated efficiencies for the γ ray of interest for 30 Mg and 27 Ne are presented. Efficiencies for the two targets are different due to the 25 mm separation between the two and the difference in energy from the different β values before and after the projectile travels through the Ta target. Error is from the uncertainties in the efficiency of GRETINA and the scaling factor. Nucleus Purity Total 30 Mg 34.7(6) % 27 Ne 7.6(2) × 107 28.3 (7) % 6.2(1) × 107 Table 5.4 The purity of the beam and total projectile nuclei impinged on the 9 Be and Ta targets are listed for 30 Mg and 27 Ne along with the purity. The error is from statistics and the systematic error in the purity. In addition, the number of detected γ rays is affected by the dead time of the S800 for coincidence measurements. As the system had a measured dead time of 16.7%, the efficiency of γ-ray collection is decreased by a factor of 0.83, which is corrected for. 5.6.2 Number of Projectile Nuclei The number of incoming ions incident on the 9 Be and Ta targets over the experiment was determined from the S800 singles measured at the E1 scintillator. This was determined to be 2.1 × 108 . The importance of transmission losses from the targets through the S800 Spectrograph is minimized because the E1 scintillator is used as the trigger, and thus γ rays corresponding to ions not reaching the E1 scintillator will not be included in the measurement. The number of incoming projectile nuclei is therefore the product of the purity and the incoming beams. The total number of incoming nuclei are listed in Table 5.4. 132 5.6.3 Extraction of γ-ray Counts Ideally, the γ-ray peaks between the reactions on the target and reactions on the following second target would be clear with no overlap. However, due to the 2% momentum spread in the beam, the difference in beam energies for de-excitations at various depths in the targets and uncertainties in the Doppler-correction, the resulting γ-ray spectra consist of two well defined peaks with an overlap at the energies in between. To account for this overlap, the data was compared to GEANT4 simulations incorporating the beam properties, energy loss in the targets, and the detectors. The ratio of excitations on the first target to excitations on the second target was varied, and the best fit for the two peaks found, with an exponential background and a scaling factor as free parameters in the fit. The simulated spectrum was then decomposed into the components from the first and second targets. A small linear background was subtracted from the simulated peaks to account for Compton-scattering from the two peaks. The counts in the peak with the background subtracted were then scaled by the parameter used in the fit to determine the overall number of counts in the peaks from reactions on each of the targets. This was compared to a method where a linear background was estimated and the integral of the peaks was used to determine the counts due to the first and second targets to determine systematic error in the background subtraction. The ratio of reactions on each target in the simulations was also varied around the best fit to determine the impact of ambiguity in the assignment of γ rays with energies between the two peaks. 5.7 30 Mg Measurement 30 The B(E2; 0gs → 2+ 1 ) for Mg has been well studied via Coulomb-excitation, and the present work serves as a benchmark for the modified Coulomb-excitation technique used here [120][122][123]. + The B(E2; 0+ gs → 21 ) values measured in these experiments are presented in Table 5.5, along with the technique used in the measurement. The B(E2) values have been measured using safe and intermediate energy Coulomb-excitation. The intermediate energy Coulomb-excitation was performed in one case using two separate targets to fit the nuclear and Coulomb contributions to 133 Method Intermediate-energy Coulomb excitation with cut on impact parameter Intermediate-energy Coulomb excitation with two targets Safe Coulomb excitation 30 Mg B(E2; 0+ gs 295 (26) [120] 435 (58) [122] 241(31) [123] 2 4 → 2+ 1 ) e fm + 30 Table 5.5 B(E2; 0+ gs → 21 ) values for Mg determined from Coulomb-excitation measurements are presented along with the technique used in the measurement. coupled channel calculations. In the other intermediate energy Coulomb-excitation measurement, a cut on the ion angle was employed to restrict the impact parameters to those larger than the nuclear radius. The safe Coulomb-excitation and intermediate-energy Coulomb-excitation using the cut on im+ 2 4 pact parameters indicate a B(E2; 0+ gs → 21 ) between 200 and 300 e fm . The previous intermediate- energy Coulomb-excitation measurement using two targets resulted in a significantly larger value of 453 (58) e2 fm4 , and therefore the present measurement validates the modified setup used here and also serves to help resolve this discrepancy among existing data. The 30 Mg spectrum for inelastic-scattering on the 9 Be and Ta foils with a 25-mm separation is shown in Figure 5.17 compared to a best fit simulation. Two peaks 1400 and 1485 keV correspond 9 to the de-excitations of the 2+ 1 at the Ta and Be targets respectively. Two smaller peaks at 1890 + 9 and 1980 keV correspond to de-excitations of the 4+ 2 → 21 transition at the Ta and Be targets. A partial level scheme for 30 Mg is presented in Figure 5.18. The presence of the 1978 keV γ-ray transition complicates the results, as it corresponds to a state transition feeding into the 2+ 1. As can be seen in the figure, the 1978 keV transition from the 3461 keV state directly feeds the 2+ 1 state. Furthermore, the other branch from the 3461 keV state populates the 2+ 1 state via a state at a 2468.4 keV, which in turn decays to the 1482 keV state. As any direct excitation of the higherlying state will lead to feeding of the 2+ 1 state, this must be accounted for. This is done by finding the cross section of this excitation, and subtracting it directly from the apparent cross section of the 2+ 1 . Additionally, the adopted value for the branching ratio of the decay via the 1978 keV transition (0.58 ± 0.15) must be accounted for in determining the cross section. The 978 keV transition is potentially present in the spectrum, but buried in the background from Compton-scattering. 134 Figure 5.17 The Coulomb-exctiation spectrum for 30 Mg using a 1.0 mm 9 Be and 0.92 mm Ta target separated by 25 mm is presented. The observed transitions are labeled. The higher (lower) peak components are associated with reactions on the 9 Be (Ta) target. A best fit spectrum is presented, with the ratio of reactions on the 9 Be target to reactions on the Ta target found to be 0.65. The dashed curve is an exponential background included in the fit. The measured counts in each of the peaks, the cross section, and the error in the cross section for each of the transitions at each target are provided in Table 5.6. The counts are scaled by the adjusted efficiency determined from simulations and the deadtime of the S800 to determine the total emissions. The total counts impinging on the target are determined from the scalars and purity, and together the cross section is calculated. The systematic error for each comes from error in determining the γ-ray efficiency with GRETINA using the 152 Eu source (4%), error in scaling the simulated efficiency to the scaled efficiency (5%), error in background estimation (2%), error in the purity of the 30 Mg beam (2%), ambiguity in the assignment of the counts between the two peaks (5%). Additionally, there is an error associated with the branching ratio for the 4+ 2 state (25%). The uncertainty in the cross section of the 2+ 1 state is affected by the uncertainty in the feeding state cross section (5.5%). The systematic errors in Table 5.6 are the quadratic sum of each of the individual systematic errors. The βNuclear for the excitation to the 2+ 1 state was determined 135 (4+ ) 3461.1(4) keV 1978.1 (5) 990.0 (5) (2+ ) 2468.4(5) keV 985.1(4)) 1482.8(3) keV 2+ 1482.1(5) 0+ gs 0 keV 30 Mg Figure 5.18 The partial level scheme of 30 Mg showing states observed in the Coulomb excitation data is presented. The J π assignment and state and transition energies are from [124]. The dashed lines represent transitions not observed in this study because of overlap with Compton events. 136 State 2+ 1 2+ 1 4+ 2 4+ 2 Transition + 2+ 1 → 0gs + 2+ 1 → 0gs + 41 → 2+ 1 + 4+ 1 → 21 Target γ-ray Counts 1260 (35) Ta 2100 (45) 9 Be 85 (9) Ta 140 (12) 9 Be σ (mb) Statistical Error (mb) Systematic Error (mb) 24 1 3 88 2 9 4 0.5 1 16 1.5 4.5 Table 5.6 The number of γ rays observed and the cross sections for each state in 30 Mg are presented for reactions on the 9 Be and Ta targets. Statistical and systematic errors are presented for the cross sections, where the systematic error is the quadratic sum of the individual systematic errors discussed in the text. Error for the number of observed γ rays is statistical only. The cross section of the 2+ 1 state corrects for feeding contributions. by ECIS from the measured cross section of 23 ± 3 mb for the reaction on 9 Be. This resulted in a βNuclear value of 0.45+2 −3 . Using the βNuclear value of 0.45, the ECIS calculations were run to reproduce the excitation cross section on the tantalum, which was 88 ± 9 e2 fm4 . This resulted in a βCoulomb of 0.40 ± 0.03(Uncertainty in Cross Section) ± 0.01 (Uncertainty in βNuclear ). This + corresponds to a B(E2; 0+ gs → 21 ) of 254 ± 39 (Uncertainty in Cross Section) ±13 (Uncertainty in βNuclear ) e2 fm4 . Thus in total the result was found to be 254 ± 41 e2 fm4 . + 2 4 This B(E2; 0+ gs → 21 ) agrees with the safe Coulomb excitation value of 241 (31) e fm [123] and is consistent with the 295(26) e2 fm4 from the Intermediate-energy Coulomb-excitation with a selection on impact parameter [120]. This agreement indicates that the method of measuring two targets simultaneously successfully allows for the measurement of Coulomb excitation at intermediate energies. 5.8 27 Ne Coulomb Excitation Measurement The spectrum for the 27 Ne Coulomb excitation measurement taken with the 1.0 mm 9 Be target and 0.92 mm Ta target separated by 25 mm is presented in Figure 5.19. The 885 keV and 765 keV transitions are identified and labeled along with the best fit simulation using an exponential background and with a population ratio of 0.4 reactions in the 9 Be target per reaction in the Ta target. 137 Figure 5.19 The 27 Ne Coulomb-excitation spectrum with a 9 Be target and Ta target separated by 25 mm is presented. The 885 keV and 765 keV transitions are identified. A best fit simulation with a ratio of 0.4 reactions in the 9 Be target per reaction in the Ta target is shown with an exponential background. The total counts in each peak along with the extracted cross section are presented in Table 5.7. The cross section was extracted from the yield of the 885 keV peak using the simulated GRETINA efficiency of 7%. Additionally, as the 885 keV state has two modes, via the 119 keV transition and the 885 keV transition, the cross section needed to be increased by the inverse of the branching ratio. As measured earlier, this branching ratio is 83(2)%. The statistical error and systematic error are also presented in Table 5.7. Due to the small yield, the statistical error is 8% for the Ta cross section and 13% for the Be cross section. The sources of systematic error include uncertainty in the measured γ-ray efficiency of GRETINA (4%), ambiguity in the scaling of the simulated efficiency (5%), uncertainty in background subtraction (14%), uncertainty in the purity (2%), uncertainty in the assignment of the counts between the two peaks to excitations on the Be or Ta target (6.5% for the Ta, 27% for the Be). In addition, the uncertainty in the branching ratio provides another 2.5% ambiguity in the cross section. The resulting error of 17% for the Ta excitation and 31% for the Be 138 State 1/2+ Target Be Ta Counts 60(8) 155 (12) σ (mb) Statistical Error (mb) Systematic error (mb) 1.7 0.2 0.5 10 0.8 1.7 Table 5.7 The γ-ray counts in each peak are presented along with statistical error. The extracted cross sections are presented along with statistical and systematic error, whose sources are discussed in the text. excitation is determined by adding systematic error in quadrature. The B(E2; 3/2+ → 1/2+ ) for 27 Ne was determined using ECIS97 [57]. The βNuclear was constrained using the cross section for scattering on the 9 Be and found to be 0.12 ± 0.02 where the error is from uncertainty in the cross section. The βCoulomb was calculated using ECIS to reproduce the cross section for the Tantalum by setting the βNuclear to 0.12. This resulted in a βCoulomb of 0.175 ±0.025 (Uncertainty in Cross Section) ±0.005 (Uncertainty in βNuclear ). This corresponds +9 2 4 +11 + to a B(E2; 3/2+ gs → 1/2 ) of 29−8 e fm . This result indicates an E2 partial lifetime of 26−6 ps. As the measured lifetime for the 1/2+ state is less than 2.0 ps, the overall E2 contribution to the lifetime is small as expected, and the lifetime of the 885 keV state provides information on the M1 transition strength. 5.9 Discussion The results of the lifetime, branching ratio, and Coulomb-excitation measurements were used to extract B(E1), B(E2) and B(M1) transition strengths between bound states in 27 Ne. These values for electromagnetic transition strengths and their corresponding partial lifetimes are presented in Table 5.8. The lifetime of the 1/2+ state is determined by the sum of the partial lifetimes E1, M1, and E2 as: 1 1 1 1 = + + τ τE2 τE1 τM1 (5.4) where the τ is the total lifetime of the state, and τX is the partial lifetime for each of the states. The lower limits for the B(E1; 1/2+ → 3/2− ) and B(M1; 1/2+ → 3/2+ gs ) were determined from the upper limit of the partial lifetime deduced from the experiment. The B(E2; 1/2+ → 139 Electromagnetic Transition Strength B(E1; 1/2+ → 3/2− ) B(E2; 1/2+ → 3/2+ gs ) + B(M1; 1/2 → 3/2+ gs ) − + B(E1; 3/2 → 3/2gs ) Partial Lifetime (ps) < 13 26+11 −6 < 2.6 +0.6 3.1−0.9 Value > 0.03 e2 fm2 2 4 58+18 −15 e fm > 0.03µN2 −4 e2 fm2 4.5+2 −1 × 10 Value (W.u.) > 0.05 12(3) > 0.02 −4 8+3 −1.5 × 10 Table 5.8 The electromagnetic transition strengths of 27 Ne measured in this work are presented along with the partial lifetime corresponding to that strength. Error presented for lifetimes and transition strengths is from the quadratic sum of systematic and statistical error. 3/2+ gs ) was determined directly from the Coulomb-excitation data. The lower limit for the B(E1; 1/2+ → 3/2− ) amounts to 0.05 W.u. This value was extracted from the partial lifetime of the 885 keV state decaying to the 765 keV state. The partial lifetime was determined from the lifetime and branching ratio measurement of the 1/2+ state of 27 Ne. The lower limit of the B(M1; 1/2+ → 3/2+ ) is 0.02 W.u. This value was determined from the corresponding partial lifetime. The partial lifetime of the 1/2+ state due to the M1 transition was determined from the state lifetime and the previously calculated partial lifetimes of the E1 and E2 transitions. Figure 5.20 shows the electromagnetic transition strengths measured in 27 Ne and those observed in other comparable nuclei. The similarities and discrepancies provide insight into the structure of 27 Ne. In particular, the 885 keV state with an expected s-wave contribution and neutron separation energy less than 1 MeV may be expected to manifest halo-like properties. The 2 2 measured B(E1; 1/2+ → 3/2+ gs ) value with a lower limit of 0.03 e fm or 0.05 W.u. can be com- pared with the enhanced B(E1) transition strengths observed in 11,12 Be. The present B(E1; 1/2+ → 3/2− ) of 27 Ne with a lower bound of 0.03 e2 fm2 or 0.05 W.u. 11 can be compared with the B(E1; 1/2− → 1/2+ gs ) of Be, with a value of 0.36 (3) W.u. [38][126]. 11 Be has a 1/2+ ground state, a 1/2− excited state at 320 keV and a neutron separation energy of 501.25 (58) keV [38][125][127]. These features can be understood from the narrowing of the N = 8 shell gap in 11 Be, in which the 2s1/2 and 1p1/2 single particle states become nearly degenerate. 11 Be has been well established as a halo nucleus through interaction cross sections and momentum 140 1430 Sn 500 1/2- 885 765 E1 580 320 E1 1/2+ 3/2- 730 3170 1- 1/2+ 2680 1/2+ 3/2+ 218 M1 3/2+ 1/2+ 209 M1 E1 M1+E2 3/2+ 0+ 27 Ne 11 12 Be Be 17 19 C C B(E1) >0.05 0.36 (3) 0.05 (2) X X B(M1) >0.02 X X 1.79 (14) × 10−3 B(E2) 4.3+1.3 −1.1 5.8+0.2 −0.7 × 10−3 X X X X Figure 5.20 Level schemes for the bound states of 27 Ne, 11,12 Be, and 17,19 C are presented. Each level scheme is aligned to the neutron separation energy of the nucleus, represented by the dashed line. The neutron separation energy in keV for the ground state of each nucleus is stated above the dashed line. The J π for each level is labeled, and the energies of excited states in keV are shown. The multipolarity of the transitions are also denoted. The energy of the 0+ state in 12 Be is not to scale and is marked with a dotted line. Beneath the level schemes, a table of the downward electromagnetic transition strengths for the transitions shown in each nucleus are presented in Weisskopf units. The data is from the present work and [27][31][38][49][111][113][125]. 141 11 distribution measurements [128][129][130]. The large B(E1; 1/2− → 1/2+ gs ) strength in Be has been understood as arising from the large radial extent of the halo neutron [38][131]. In a simple picture, the valence neutron of 11 Be occupies the 2s1/2 orbital in the ground state, and occupies the 1p1/2 orbital in its 1/2− excited state. This is similar to the single particle structure in 27 Ne, where the valence neutron occupies the 2p3/2 orbital in the 3/2− excited state, and where a neutron excitation out of the 2s1/2 orbital forms the 1/2+ state. Therefore from a single particle perspective the 1/2+ → 3/2− state in 27 Ne involves transition between s and p 11 orbitals similar to that in the 1/2− → 1/2+ gs transition in Be. Furthermore, the neutron separation energy in 27 Ne is approximately 1.4 MeV from the ground state, but only approximately 500 − 600 keV for the 1/2+ , 3/2− excited states, which is comparable to the separation energy in 11 Be [27]. Indeed, the phenomenon of certain enhanced B(E1) strengths in nuclei has been previously understood to arise from one or more decoupled loosely-bound halo neutrons [131][132][133][134]. In a simple picture, the decoupled neutrons can oscillate against the core, and this can create a large E1 strength among transitions between bound states. In a shell model context, the use of extended single particle wavefunctions can be shown to reproduce an increased E1 transition strength for transitions between bound states [131]. For reference, 12 Be has the second largest E1 strength observed in transitions between bound states in light mass nuclei, with a B(E1; 0gs → 1− ) value of 0.051(13) e2 fm2 or 0.15 (4) W.u. [31]. This B(E1) strength is comparable with the lower limit of 0.05 W.u. for the measured B(E1; 1/2+ → 3/2− ) of 27 Ne. Thus, the enhanced B(E1; 1/2+ → 3/2− ) supports the model in which 27 Ne is made up of a 26 Ne core and a valence neutron with a radially extended wavefunction. This is consistent for a state with a large s-wave contribution and only 500 keV from the neutron separation energy [27]. This is also consistent with the angular momentum components of the 1/2+ state being l = (0, 1) as determined from momentum distributions [116]. + The B(E1; 3/2− → 3/2+ gs ) provides additional evidence for the E1 enhancement of the 1/2 state. The 3/2− state is common to both the B(E1; 1/2+ → 3/2− ) and B(E1; 3/2− → 3/2+ gs ). If the valence neutrons in the ground state do not have an extended wavefunction, the B(E1; 3/2− → 142 3/2+ ) can be taken as a reference strength. Indeed, the B(E1; 3/2− → 3/2+ gs ) with a value of −4 e2 fm4 or 8 ± 3 × 10−4 W.u. is smaller than the B(E1; 1/2+ → 3/2− ) by a factor 4.5+2 −1 × 10 of at least 50. This supports the picture of the 1/2+ state having a unique nature which can be ascribed to a radially extended wavefunction. The M1 and E2 transition strengths can be understood in part through comparison to stable nucleus 33 S which is an isotone of 27 Ne with 6 more protons. With an even number of protons and a single valence neutron, it can be expected that at low energies 33 S still is dominated by the single particle structure of the neutron. The 841 keV state with spin and parity 1/2+ has previously been + + measured to determine the B(M1; 1/2+ → 3/2+ gs ) and B(E2; 1/2 → 3/2gs ) transition strengths of 0.031(4) W.u. and 6(4) W.u. respectively [135][136]. These can be seen to be close to the measured values in 27 Ne from decays of the 885 keV state. The B(M1; 1/2+ → 3/2+ gs ) has a lower limit of 0.02 W.u., consistent with the 0.031(4) W.u. in 33 S. Furthermore, the B(E2; 1/2+ → 3/2+ gs ) of 12(3) W.u.is consistent within error with the 6(4) W.u. in 33 S. The B(E2) strengths in 33 S may indicate moderate deformation, or could be due to band mixing [137]. The B(M1) transition strength in loosely-bound nuclei has been investigated previously in 19 C 19 C was found to be 1.79(14) × 10−3 [111]. In that work, the B(M1; 3/2+ → 1/2+ gs ) of W.u, representing one of the smallest observed in the mass region A < 40 [111]. Furthermore, +0.7 17 −3 W.u. [49]. the B(M1; 1/2+ → 3/2+ gs ) has been measured in halo nucleus C as 5.8−0.2 × 10 The hindered M1 transition strengths can be understood in part due to the lack of spin-flip transitions within the sd-shell. For that reason, if the valence neutron s-wave component were to be entirely dominant in the 885 keV state of 27 Ne, the M1 transition strength would be expected to be hindered. However, the present data do not support this picture of 27 Ne. The interest in the contribution of the s-wave to the 1/2+ excited state of 27 Ne needs to be discussed in terms of an interaction between loosely-bound effects and deformation because neighboring 26,28 Ne are deformed. Indeed, at first glance s-wave dominance would be somewhat at odds with the large deformation in neighboring 26,28 Ne [55][112]. The deformed 26 Ne core would be expected to induce higher angular momentum states in the valence neutron [138]. However, it has 143 been predicted in theory that for loosely-bound states the norm of the s-wave ( s|s ) is expected to diverge as Sn → 0. As such, at extreme binding energies the overall contribution of the s-wave will dominate the structure despite the intrinsic deformation [138]. Since the B(M1) would then be dominated by the overlap of the s-wave component, the transition strength would be hindered. However, the present value with a lower bound of 0.02 W.u. is ten times larger than the hindered transition in 19 C, suggesting that there is an admixture of l > 0 angular momentum states in the 885 keV wavefunction. This indicates that the 885 keV state may have the characteristics of a deformed halo, such as that suggested in 31 Ne [139][140] and 37 Mg [141]. Comparison of the present data to shell-model calculations may provide an additional indicator + + of halo formation. The B(M1; 1/2+ → 3/2+ gs ) and B(E2; 1/2 → 3/2gs ) were calculated using the SDPF-M and USD interactions [142]. These calculations predict a B(M1; 1/2+ → 3/2+ gs ) value of 0.033 W.u. and 0.042 W.u. for the SDPF-M and USD interactions respectively. These are consistent with the lower limit of 0.02 W.u. established by the present measurements. The 2 4 predicted B(E2; 1/2+ → 3/2+ gs ) is between 25 and 35 e fm , only half that of the measured tran- sition strength. This indicates possible enhancement of collectivity in 27 Ne, providing a consistent picture for a deformed halo structure. 5.10 Conclusion In conclusion, electromagnetic transition strengths between bound states of 27 Ne have been measured in this work. The 1/2+ state at 885 keV is expected to decay by a combination of M1 and E2 contributions to the ground state along with E1 contributions to the 765 keV state. The 3/2− state at 765 keV is expected to decay via a E1 transition to the ground state. The B(E1; 1/2+ → 3/2− ) was found to be at least 0.05 W.u., consistent with halo formation. The B(E1; 3/2− → 3/2+ ) value −4 W.u. supports the influence of loosely-bound effects on the 885 keV state as it is of 8+3 −1.5 × 10 50 times weaker than the B(E1; 1/2+ → 3/2− ). The B(E2; 1/2+ → 3/2+ ) has been measured to decompose the M1 strength and indicates the potential interplay of deformation with the loosely- 144 bound effects in the 1/2+ state. The unhindered B(M1; 1/2+ → 3/2+ ) transition strength with a lower limit of 0.02 as well as the favored E2 transition are also consistent with a picture of the 885 keV state as a deformed halo. 145 CHAPTER 6 CONCLUSION This work investigates the richness of nuclear structure at the floor and edge of the valley of stability through measurements of electromagnetic transition strengths. The electromagnetic transition strengths between bound states of nuclei offer a unique window into the dynamics and internal configuration of the nuclear wavefunction. The nuclear structure accessible to these transition strengths can only be probed through measurements of lifetimes and cross sections. The Recoil Distance Method and modified Coulomb-excitation setup developed and employed in this work have been demonstrated as powerful tools that give us insight into nuclear structure. The continued development of new devices such as GRETINA and the TRIPLEX for precise γ-ray and lifetime measurements is opening up new opportunities for the study of novel features in nuclei. 58 Ni demonstrates the utility of electromagThe lifetime measurement of the 4+ 1 state of netic transition strengths in studying nuclei at the floor of the valley of stability. The measured + B(E2; 4+ 1 → 21 ) restores the systematics of the Nickel isotopes to the behavior expected near the shell closure at N = Z = 28. The model-independent lifetime measurement enabled by GRETINA + and the TRIPLEX plunger could distinguish between an enhanced B(E2; 4+ 1 → 21 ) indicated by a previous measurement and the unenhanced value expected from nuclei near a shell closure. The lifetime and Coulomb-excitation measurements of the excited states of 27 Ne have demonstrated the usefulness of the Recoil Distance Method and inelastic scattering techniques. In this work, the major electromagnetic transition strengths in 27 Ne have been constrained, providing direct insight into the nuclear wavefunctions of all the bound states of 27 Ne. The enhanced B(E1; 1/2+ → 3/2− ) strength relative to the B(E1; 3/2− → 3/2+ gs ) indicates a unique structure in the 1/2+ excited state that does not manifest in the 3/2− state. The measured B(M1; 1/2+ → 3/2+ gs ) value is considered to originate from the admixture of non-spherical components to the wavefunction of the valence neutron, indicating that the 1/2+ excited state of 27 Ne is consistent with a deformed halo. This work has demonstrated that rich variety of physics can be extracted 146 from the electromagnetic transition strengths of nuclei far from stability. The present experiments also point to new horizons in studying nuclear structure. On the one hand, the modified Coulomb-excitation setup presented here may be of use in future experiments. On the other hand, the present measurement determined only lower bounds for the B(E1; 1/2+ → 3/2− ) and B(M1; 1/2+ → 3/2+ gs ) values. A precision measurement of the exact lifetime would be of much interest in determining the scope of the loosely-bound effects in 27 Ne. The 27 Ne lifetime in the region of less than two picoseconds would be best measured using the Dopplershift Attenuation Method [41]. However, as 27 Ne is a rare isotope, it is difficult to create an appropriate beam at existing facilities. Future developments at the Facility for Rare Isotope Beams may make it possible to develop intense beams of 26 Ne [143]. The ReA re-accelerator facility could then be used to produce beams with energies suitable for Doppler-shift Attenuation Method measurements, and allow for a precise determination of the lifetime of the 1/2+ state, and hence the electromagnetic transition strengths [60]. 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