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Single particle structure of exotic strontium isotopes Cruz, Steffen James 2017

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Single Particle Structure of ExoticStrontium IsotopesUsing Single Neutron Transfer Reactions as a Tool toStudy the Evolution of Nuclear StructurebySteffen James CruzM.Sci., The University of Birmingham, England, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)June 2017c© Steffen James Cruz 2016AbstractThe sudden onset of ground state deformation and the emergence of shape-coexisting states inthe vicinity of N∼60 and Z∼40 has been a subject of substantial interest for many years. It hasbeen shown that the emergence of deformed low-energy configurations can be explained in theshell model by the evolution of single particle structure and the interaction between protons andneutrons in certain valence orbitals. However, the numerous theoretical models that have beendeveloped for this transitional region are limited by the experimental data that is available. Inparticular, a description of the underlying single particle configurations of low energy states isessential for a detailed description of this region.In this work, the single particle structure of states in 95Sr and 96Sr has been investigatedthrough the one-neutron transfer reactions 94,95Sr(d,p) in inverse kinematics at TRIUMF. Ineach of these experiments, a 5.5 MeV/u Sr beam was impinged on a 5.0 mg/cm2 CD2 target, andemitted particles and γ-rays were detected using the SHARC and TIGRESS detector arrays,respectively. Using an angular distribution analysis, firm spin assignments have been madefor the first time of the low-lying 352 keV, 556 keV and 681 keV excited states in 95Sr from94Sr(d,p), and a constraint has been made on the spin of the higher-lying 1666 keV excited statein 95Sr. Similarly, angular distributions have been extracted for 12 states in 96Sr from 95Sr(d,p),and new experimental constraints have been assigned to the spins and parities of 8 states in96Sr. Additionally, two new states in 96Sr have been identified in this work. A measurementof the mixing strength between the 1229 keV and 1465 keV shape-coexisting states in 96Sr wasalso made, which was found to be a2 = 0.48(17).iiLay SummaryNeutron-rich isotopes of strontium exhibit an abrupt change of structure in their lowest energystates at the neutron number N=60. This sudden transition occurs as a result of the deli-cate interplay between several different nuclear structure phenomena. Such dramatic structuralchanges in nuclei are rarely observed and so they present a unique and exciting opportunityto learn about the nuclear force and to refine our theoretical tools. These exotic strontiumisotopes have been a subject of substantial interest for a number of years, however a precisedescription of this region has been restricted by the very limited experimental data available.In this work, a systematic study has been carried out to elucidate the structure of 17 statesin 95Sr (N=57) and 96Sr (N=58) which are very important isotopes. These measurements cannow be used to advance our understanding of this structural transition and also to benchmarkcurrent theoretical models.iiPrefaceThis dissertation is original intellectual work by the author, S. Cruz. At the time of submission,the content of this thesis is unpublished.All of the experimental work presented henceforth was carried out at TRIUMF, Canada’snational laboratory for particle and nuclear physics and accelerator-based science in Vancou-ver, British Columbia. The proposal for this work (Exp. No. S1389) was written by thespokespeople Dr. K. Wimmer and Dr. R. Kru¨cken, with some contributed calculations fromthis author. The experiments were carried out as part of a larger collaboration, with approx-imately 30 members. The primary contributions of the S1389 collaboration members were tooversee the data acquisition by volunteering for beam shifts and to assist with setting up theexperimental apparatus.All of the calibrations, analysis, results and calculations are the work of this author.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Deformation in Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Onset of Deformation at Z ∼ 40, N ∼ 60 . . . . . . . . . . . . . . . . . . . . . . 41.3 Shape Coexistence in 96Sr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Shell Model Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Transfer Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.5.1 DWBA Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.5.2 Inverse Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.6 Overview of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1 Overview of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27ivTable of Contents2.2 RIB Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.1 Beam Delivery at TRIUMF . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Target Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4 Detector Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4.1 TBragg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4.2 SHARC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.3 TIGRESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.4 Trifoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.5 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1 SHARC Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 TIGRESS Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.1 SHARC Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.1.1 α Source Gain-Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.1.2 Full Energy Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.1.3 SHARC Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 TIGRESS Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2.1 152Eu Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.2.2 Add-Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2.3 Absolute Efficiency Calibration . . . . . . . . . . . . . . . . . . . . . . . 744.2.4 Doppler Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.2.5 TIGRESS Energy Resolution . . . . . . . . . . . . . . . . . . . . . . . . . 794.3 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.3.1 Angular Ranges and Excitation Energy Ranges . . . . . . . . . . . . . . 844.4 Method of Extracting Angular Distributions . . . . . . . . . . . . . . . . . . . . 865 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.1 Elastic Scattering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92vTable of Contents5.2 94Sr(d,p) Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.3 95Sr(d,p) Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.1 Discussion of 94Sr(d,p) results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.1.1 Comparison to 94Sr(d,p) Shell Model Calculations . . . . . . . . . . . . . 1276.2 Discussion of 95Sr(d,p) results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.2.1 Comparison to 95Sr(d,p) Shell Model Calculations . . . . . . . . . . . . . 1336.3 Mixing Between the Excited 0+ States in 96Sr . . . . . . . . . . . . . . . . . . . 1397 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146AppendicesA FRESCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151B SHARC Solid Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152C Analysis Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155D Low Energy Background in SHARC . . . . . . . . . . . . . . . . . . . . . . . . . 159E Calculation of Electromagnetic Transition Rates . . . . . . . . . . . . . . . . . 160viList of Tables1.1 Summary of NushellX model spaces and interactions that were used (more detailsin the text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2 Wavefunction composition and orbital occupation numbers for the 95Sr groundstate. Each of the underlying configurations (i)-(v) are coupled to Jpi = 12+. Theoccupation numbers are the weighted sum of the underlying configurations. . . . 172.1 Summary of 94,95Sr beam delivery . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 SHARC detector summary table. The array is arranged into several sections;DBOX+PAD, UBOX and UQQQ. D(U) prefixes in detector names refer to down-stream(upstream) components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1 Summary of NPTool simulations carried out for 95Sr reactions in SHARC. . . . . 473.2 Simulated states used for efficiency calibration. States marked with † were as-signed 5 ps half-lives (more details in the text). . . . . . . . . . . . . . . . . . . . 544.1 Energies and intensities of 239Pu 241Am 244Cm α-source, taken from [Lab]. Strongα-branches are highlighted in bold text. . . . . . . . . . . . . . . . . . . . . . . . 634.2 Energies and intensities of 152Eu γ-ray calibration source [Lab]. Intensities aregiven relative to the 1408 keV γ-ray transition. The γ-rays highlighted in boldwere used for calibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3 Energies and intensities of 60Co γ-ray calibration source [Lab]. . . . . . . . . . . 755.1 Global optical model parameters that were used to fit 94,95Sr proton elastic scat-tering angular distributions; Becchetti and Greenlees (BG), Chapel Hill (CH)and Perey and Perey (PP). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95viiList of Tables5.2 Optical model parameters used to fit 94,95Sr deuteron elastic scattering angu-lar distributions compared to global parameter fits; Lohr and Haeberli (LH),Daehnick (D) and Perey and Perey (PP) . . . . . . . . . . . . . . . . . . . . . . . 985.3 Normalization constants extracted from dσdΩ fits of (p,p) and (d,d) data to DWBAcalculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.4 Table of spectroscopic factors for directly populated 95Sr states. . . . . . . . . . . 1025.5 Table of spectroscopic factors for directly populated 96Sr states. †Spectroscopicfactors and cross sections determined using a relative γ-ray analysis. . . . . . . . 1125.6 Comparison of experimental and simulated counts ratio for 414 keV and 650 keVγ-rays gated on different angular ranges in TIGRESS. The simulated ratio isdiscussed in section 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.1 Comparison of experimental to calculated spectroscopic factors for 94Sr(d,p),relative to the 12+state spectroscopic factor. Values in parentheses are absolutespectroscopic factors. a©, b© and c© denote the different model spaces that werecalculated with the glek interaction, as discussed in section 1.4. . . . . . . . . . . 1286.2 Experimental spectroscopic factors for N=57 nuclei relative to the ground statespectroscopic factor. Values in parentheses are absolute spectroscopic factors. †no 1d52 analysis was carried out for the96Zr(d,p) experiment, and so all ` = 2transfer states were assumed to be 32+. . . . . . . . . . . . . . . . . . . . . . . . . 1326.3 Comparison of experimental to calculated spectroscopic factors for 0+ states in96Sr, relative to the ground state spectroscopic factor. Values in parentheses areabsolute spectroscopic factors. b© and c© denote the different model spaces thatwere calculated with the glek interaction, as discussed in section 1.4. . . . . . . . 1366.4 Occupation numbers for the calculated ground state of 95Sr and the 0+ states in96Sr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.5 Occupation numbers for the calculated ground state of 97Zr and the 0+ states in98Zr from references [SNL+09] and [HEHJ+00]. . . . . . . . . . . . . . . . . . . . 138viiiList of Tables6.6 Branching ratios for states which are known to feed the 1229 keV 96Sr state,expressed relative to the strongest decay branch. Estimated branching ratios tothe 1465 keV state are also given, assuming a mixing strength of a2 = 0.5, for apure M1 and a pure E2 transition (as labelled). . . . . . . . . . . . . . . . . . . . 141B.1 Summary table of SHARC the 204 strips that were excluded from the 2013 94Sranalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154B.2 Summary table of SHARC the 107 strips that were excluded from the 2013 95Sranalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154E.1 Weisskopf estimates for first few electromagnetic rates. Energy is in units of MeV.161E.2 Weisskopf estimates for the decay of the 2084 keV 96Sr state to states withestablished spin and parity, using measured branching ratios from [Lab]. . . . . . 162E.3 Weisskopf estimates for the decay of the 2084 keV 96Sr state to states withestablished spin and parity, using known branching ratios. . . . . . . . . . . . . . 162ixList of Figures1.1 (a) Finite range drop model (FRDM) calculation of the ground state deformationacross the nuclear chart (taken from [MSI+16]). The solid lines correspond tonuclei with magic numbers of protons and neutrons, which are predicted to havespherical shapes. (b) Isotope shift across the Sr isotopic chain indicates thatthere is a sudden transition from spherical to deformed ground states at N = 60(taken from [WZD+99]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 (a) Experimentally measured energy of the first excited (2+1 ) state in even-evennuclei in the vicinity of Z ∼ 40 and N ∼ 60 (data taken from [Lab]). (b) Beyondmean field calculations of potential energy as a function of quadrupole moment(Q20) in Sr, Zr and Mo isotopic chains (taken from [RGSR+10]). . . . . . . . . . 51.3 Federman and Pittel mechanism for the onset of deformation in 96Sr [FP79]. . . 61.4 Proposed lowering of a deformed structure (β > 0) across Sr and Zr isotopes,which results in a ground state shape transition at N = 60. Adapted from[LPK+94]. Energy levels that are associated with the spherical and deformedconfigurations are also drawn, and the data is taken from [Lab]. . . . . . . . . . . 81.5 (a) Low energy states of 96Sr, indicating the strong monopole transition strength(given in units of ρ2(E0)x103) between excited 0+ states (adapted from [HW11]).(b) Two level mixing model for coexisting states, where the unmixed states areassumed to be pure spherical and deformed configurations, respectively. . . . . . 101.6 Shell model description of a nucleus. The residual interaction between valencenucleons (blue wavy line) causes different configurations to mix. . . . . . . . . . . 13xList of Figures1.7 Selected low-lying 95Sr states compared to shell model calculations using the jj45model space and interaction with various proton valence spaces (more details inthe text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8 Selected low-lying 95Sr states compared to shell model calculations using the glekmodel space and interaction with various proton valence spaces (more details inthe text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.9 Schematic diagram of a transfer reaction. . . . . . . . . . . . . . . . . . . . . . . 191.10 Schematic diagram of a nuclear reaction in scattering theory. . . . . . . . . . . . 211.11 (a) Shape of a Woods-Saxon potential. The surface potential is defined mathe-matically as the derivative of the volume potential. (b) Example DWBA angulardistribution calculations for different orbital angular momentum transfer. . . . . 242.1 Schematic diagram of the experiment, showing the detector set-up. . . . . . . . . 282.2 Diagram of TRIUMF-ISAC facility [BHK16]. The 95Sr16+ beam was deliveredto the TIGRESS experimental station in ISAC-II. . . . . . . . . . . . . . . . . . 302.3 Beam identification plot of energy loss versus total energy as measured in theT-Bragg spectrometer. The beam was 98.5(5)% 95Sr. See section 2.4.1 for moredetails. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4 (a) Measured α-particle energy spectrum, with and without the target foil present.(b) Target thickness fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5 Schematic diagram of a Bragg ionization chamber, adapted from [Nob13]. Ionsenter the chamber and lose energy (example ion paths are drawn), creating freeelectrons (also indicated). The electrons are then drifted to an anode using anapplied electric field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.6 (a) Schematic diagram SHARC detector, indicating main sections and targetposition with respect to beam. (b) A photograph of SHARC detector beinginstalled [DFS+11]. The micro-pitch ribbon cables and PCB feedthroughs arealso visible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36xiList of Figures2.7 The target wheel photographed at the end of the experiment. From left to right:5.0 mg/cm2 CD2, 2mm collimator, 0.5 mg/cm2 CD2 primary target. The primarytarget was burned through after approximately 2.5 days due to the high beamrate delivered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.8 (a) CAD cutaway drawing of TIGRESS surrounding the SHARC detector. (b)TIGRESS clover detector with indication of crystal segmentation [SPH+05]. . . . 392.9 Detector arrangement used for this experiment. TIGRESS can be seen in highefficiency mode surrounding the target chamber, which contains SHARC. Themounted pre-amp rack and cabling setup, suspended above the beam line adja-cent to TIGRESS, is also visible in this photograph. . . . . . . . . . . . . . . . . 402.10 (a) Schematic diagram of trifoil degrader-scintillator arrangement. (b) Trifoildetector, photographed during setup. . . . . . . . . . . . . . . . . . . . . . . . . . 412.11 (a) Damage caused to the BC400 foil during the experiment by the high beamintensity made it difficult to use the trifoil in analyzing the data. (b) Count rateof the trifoil throughout the experiment. . . . . . . . . . . . . . . . . . . . . . . 422.12 Diagram of DAQ logic. Detector signals were sent to the TIG-10 front-end (FE)modules before digitization, logical discrimination and event assembly. . . . . . . 433.1 (a) NPTool implementation of SHARC including DBOX, UBOX and UQQQsections. (b) Beam-line view, showing the CD2 target. . . . . . . . . . . . . . . . 463.2 (a) Simulation results for 95Sr(p,p), 95Sr(d,d) and 95Sr(12C,12C) kinematics. (b)Simulated energy spectrum for a small angular range in SHARC (more detailsin the text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 (a) Simulation results for ` = 0 95Sr(d,p) kinematics. (b) Counts versus lab anglefor all data (black) and for |Eexc| > 500 keV (red). The latter data correspondsto incomplete energy measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4 (a) Simulation results for (d,p) analysis. (b) Excitation energy resolution forSHARC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51xiiList of Figures3.5 (a) Measured counts versus center-of-mass for the simulated 0 keV (black), 815keV (red) and 1792 keV (blue) 96Sr states, compared to efficiency-corrected input.(b) Excitation energy resolution of SHARC for 95Sr(d,p) reactions in inversekinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.6 (a) Geant4 model that was used to study decays in TIGRESS. (b) Beam-lineview, showing the CD2 target. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.7 (a) Simulated γ-ray spectrum using the decays listed in table 3.2. The sumspectrum is drawn in black. (b) Absolute efficiency curve produced using thedecays listed in table 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.8 Decay scheme for 0+2,396Sr states, indicating transition energies (given in units ofkeV) and branching ratios (given as a percentage). The 0+3 → 0+2 and 0+3 → 2+1branching ratios were taken from [Jun80]. . . . . . . . . . . . . . . . . . . . . . . 563.9 (a) Comparison of simulated spectra for states fed by isomeric 0+3 (black) todirect decay (grey). (b) Contributions to total photo-peak (black) from eachring of detectors (more details in the text). . . . . . . . . . . . . . . . . . . . . . 583.10 (a) Simulated γ-ray spectrum for the decay of the 96Sr 0+2 state (blue) and0+3 state (red) for S2 = S3 = 106, using only TIGRESS crystals positioned atθ > 135◦. Total spectrum is drawn in black. (b) Simulated results for the ratioof counts in the 650 keV γ-ray peak to the 414 keV peak, plotted as a function ofthe 0+3 → 2+1 branching ratio for all TIGRESS angles (black), θTIG > 120◦ (blue)and θTIG > 135◦ (red). The green band indicates the experimental branchingratio from [Lab] with its uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . 594.1 (a) Calibrated α-source energy vs. channel matrix. (b) Example α-source spec-trum with total fit (red) and extracted calibration peaks (blue). . . . . . . . . . . 644.2 (a) Full energy calibration of DBOX ∆E detectors using 95Sr(p,p) (blue) and95Sr(d,d) (red) data. (b) An example gain-matched charge spectrum showing95Sr(p,p) and 95Sr(d,d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66xiiiList of Figures4.3 (a) Energy calibration of a pad detector using 95Sr(p,p) (blue) and 95Sr(d,d) (red)data. Solid lines and closed symbols indicate a calibration performed using ex-perimental ∆E values, while broken lines and open symbols show the calibrationresult using only calculated ∆E values (see text for more details). (b) CalibratedDBOX showing kinematics curves for 95Sr(p,p) and 95Sr(d,d) compared to theorycurves. Details of the cuts used are given in section 4.3. . . . . . . . . . . . . . . 684.4 (a) Calibration fit for UBOX detector using 95Sr(d,p) data. (b) Calibrated UBOXshowing kinematics curve of 95Sr(d,p) ground state transfer compared to theorycurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.5 (a) Calibration fit for UQQQ detector using 95Sr(d,p) data. (b) CalibratedUQQQ showing kinematics curve of 95Sr(d,p) ground state transfer comparedto theory curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.6 (a) SHARC solid angle coverage in lab frame with all strips included (blue),strips from table B.2 removed (black) and maximum coverage 2pi sin θdθ (red),(b) SHARC geometrical efficiency in lab frame, with colours indicating the sameas before. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.7 (a) SHARC solid angle coverage in centre-of-mass frame with all strips included(blue), strips from table B.2 removed (black) and maximum coverage (red), (b)SHARC geometrical efficiency in centre-of-mass frame, with colours indicatingthe same as before. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.8 (a) Plot showing the calibrated channels of TIGRESS using a 152Eu source. (b)Calibrated sum spectrum of all crystals using a 152Eu source. . . . . . . . . . . . 734.9 (a) Relative 152Eu efficiency curves made with and without add-back. (b) Ratioof efficiency curves, giving the add-back factor. . . . . . . . . . . . . . . . . . . . 754.10 Absolute efficiency curve of TIGRESS (solid red line) with ±1σ uncertaintybands (broken red lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.11 (a) Comparison of raw experimental γ-ray spectrum (red) to Doppler correctedspectrum (black). (b) Doppler correction of fitted 815 keV γ-ray peak centroidas a function of θTIG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.12 Empirically determined photo-peak width as a function of photo-peak energy. . . 79xivList of Figures4.13 (a) Measured ∆E kinematics drawn with kinematic lines corresponding to various95Sr reactions. Black curves show elastic scattering channels (p,p), (d,d) and(12C,12C), while red curves show (d,p) kinematics for 0 (solid), 2, 4 and 6 MeV(dashed) excitation energy. (b) Measured pad energy kinematics. More detailsare given in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.14 Kinematics of various 95Sr reactions. Black curves show elastic scattering chan-nels, (p,p) and (d,d) and red curves show (d,p) kinematics for 0 (solid), 2, 4 and6 MeV (dashed) excitation energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 824.15 (a) Particle identification (PID) plot for all pads in DBOX section. Protons havelower ∆ energy for a given pad energy than deuterons, forming two separate loci.(b) Same as before, but with an effective thickness correction (see text for details). 834.16 (a) Excitation energy versus centre-of-mass angle for (p,p), showing angular andenergy ranges of the PID cuts. The black line indicates 0 keV excitation energy.(b) Same as (a), but for (d,d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.17 Excitation energy versus centre-of-mass angle for (d,p), showing angular andenergy ranges of the PID cuts. The solid red line indicates 0 MeV excitationenergy and the broken red lines indicate 2, 4 and 6 MeV. . . . . . . . . . . . . . 864.18 Example analysis for 1995 keV 96Sr state (top to bottom, left to right). (a) γ-ray singles spectrum gated on the excitation energy range 1500-2500 keV, withestimated peak counts. (b) Coincident γ-rays with the 1180 keV transition in thesame excitation energy range. (c) Excitation energy spectrum coincident with1180 keV γ-ray rays projected over all center-of-mass angles. (d) Excitationenergy versus center-of-mass angle coincident with 1180 keV γ-ray rays. . . . . . 884.19 Experimental angular distribution for 95Sr(d,p) to the 1995 keV 96Sr state. . . . 895.1 Example fit of small center-of-mass angle (p,p) and (d,d) data using (a) expo-nential background model (b) linear background model. . . . . . . . . . . . . . . 94xvList of Figures5.2 Example of large center-of-mass angle background subtraction for 95Sr (a) (d,d)data and (b) (p,p) data. The green region was taken to be the peak region andso the counts were extracted by subtracting the background curve from the totalcounts in that range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.3 Comparison of various optical model potentials to experimental angular distri-butions for (a) 94Sr(p,p) elastic scattering (b) 95Sr(p,p) elastic scattering. . . . . 955.4 Comparison of various optical model potentials to experimental angular distri-butions for (a) 94Sr(d,d) elastic scattering (b) 95Sr(d,d) elastic scattering. . . . . 965.5 (a) Measured angular distribution for 94Sr(d,d) elastic scattering. (b) Measuredangular distribution for 95Sr(p,p) elastic scattering. . . . . . . . . . . . . . . . . . 975.6 Comparison of global and fitted DWBA differential cross sections for (a) 94Sr(d,p)and (b) 95Sr(d,p). The large difference in calculated angular distributions be-tween the optical model parameter sets for a given reaction indicates that thereis a very large uncertainty on the overall cross sections for the reactions, whichmeans that the absolute value of the resulting spectroscopic factors is not well-known. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.7 Level scheme for 95Sr states populated through 94Sr(d,p). Level energies, life-times and spin assignments are taken from [Lab]. . . . . . . . . . . . . . . . . . . 1005.8 (a) γ-ray spectrum for 95Sr. (b) Excitation energy versus γ-ray matrix for 94Sr. . 1015.9 Example 3 peak fits for (a) θCM = 10◦ (b) θCM = 38◦. . . . . . . . . . . . . . . . 1035.10 Angular distribution for 94Sr(d,p) to the 95Sr ground state compared to DWBAcalculations using (a) fitted optical potential (b) unmodified optical potential. . . 1045.11 Angular distribution for 94Sr(d,p) to the 95Sr 352 keV state extracted using a(a) three peak fit, (b) 352 keV γ-ray gate. . . . . . . . . . . . . . . . . . . . . . . 1045.12 Angular distribution for 94Sr(d,p) to 95Sr the 95Sr 681 keV state extracted usinga three peak fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.13 Angular distribution for 94Sr(d,p) to 95Sr 681 keV state extracted using a (a)329 keV γ-ray gate, (b) 681 keV γ-ray gate. . . . . . . . . . . . . . . . . . . . . . 107xviList of Figures5.14 (a) Measured γ-rays coincident with 95Sr excitation energies of 1-2 MeV usingtransfer protons in the DBOX section of SHARC only. The 427 keV γ-ray indi-cates direct population of the 1666 keV state. (b) Level scheme taken from 252CfSF [HRH+04], indicating three possible band structures in 95Sr. . . . . . . . . . . 1085.15 Level scheme for 96Sr states populated through 95Sr(d,p). Level energies, life-times and spin assignments are taken from [Lab]. Dashed lines represent pro-posed new states and transitions. The 235 keV E0 transition was not seen inthis work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.16 γ-ray spectrum for 96Sr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.17 (a) γ − γ matrix for 96Sr. (b) Excitation energy versus γ-ray matrix for 96Sr. . . 1115.18 (a) Angular distribution for 95Sr(d,p) to the 96Sr 0+1 ground state. (b) An ex-ample exponential background fit, indicating the peak region (green) where thecounts above the background were assigned to the 0+1 state. . . . . . . . . . . . . 1135.19 (a) Angular distribution for 95Sr(d,p) to the 96Sr 1229 keV 0+2 state. (b) Exci-tation energy spectrum gated on the UQQQ and UBOX SHARC sections, coin-cident with 414 keV γ-rays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.20 (a) Angular distribution for 95Sr(d,p) to the 96Sr 0+2 1465 keV state. (b)96Srγ-ray spectrum gated on the excitation energy range 900-1900 keV. . . . . . . . 1155.21 (a) Counts in 414 keV (red) and 650 keV (blue) γ-ray peaks as a function ofexcitation energy. (b) Ratio of counts in 650 keV γ-ray peak to 414 keV γ-raypeak as a function of excitation energy (more details in the text). . . . . . . . . . 1175.22 Angular distribution for 95Sr(d,p) to (a) 96Sr 1507 keV state and (b) 96Sr 1793keV state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.23 (a) Angular distribution for 95Sr(d,p) to the 96Sr 1628 keV state. (b) Excitationenergy γ − γ-gated on the 813 keV and 815 keV γ-rays. . . . . . . . . . . . . . . 1195.24 (a) 96Sr excitation energies in coincidence with a 1240 keV γ-ray. (b) γ-rays incoincidence with a 1240 keV γ-ray. . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.25 Angular distribution for 95Sr(d,p) to (a) 96Sr 1995 keV state and (b) 96Sr 2084keV state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122xviiList of Figures5.26 (a) Angular distribution for 95Sr(d,p) to the 96Sr 2217 keV state. (b) 96Sr γ-raysgated on excitation energy range 1600 keV to 2600 keV. . . . . . . . . . . . . . . 1235.27 Angular distribution for 95Sr(d,p) to (a) 96Sr 2217 keV state and (b) 96Sr 2576keV state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.28 (a) Angular distribution for 95Sr(d,p) to 96Sr 3239 keV state. (b) Comparisonof measured coincidence 978 keV spectrum (black) with simulated coincidencespectrum (red) using known decays and intensities. The simulated spectrum in-cludes appropriate efficiency scaling and energy resolution to generate a realisticphoto-peak spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.29 Angular distribution for 95Sr(d,p) to 96Sr 3500 keV state. . . . . . . . . . . . . . 1266.1 Selected low-lying 95Sr states compared to shell model calculations, where thelength of each line represents the spectroscopic factor. The ground state spectro-scopic factors are normalized to one, and all excited state spectroscopic factorsare drawn relative to the ground state. . . . . . . . . . . . . . . . . . . . . . . . . 1306.2 Selected low-lying 96Sr states compared to shell model calculations, where thelength of each line represents the spectroscopic factor. The experimental groundstate spectroscopic factor is normalized to one, and all experimental excited statespectroscopic factors are drawn relative to the ground state. The calculatedspectroscopic factors are drawn to scale (more details in the text). . . . . . . . . 1356.3 Quadrupole deformation as a function of mixing strength for a monopole tran-sition strength ρ2(E0) = 0.185(50). The drawn bands represent 68% confidenceintervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140A.1 Example FRESCO input file for 95Sr(d,p) to the 96Sr ground state. . . . . . . . . 151B.1 Solid angle of each individual pixel in (a) a DSSD detector and (b) a QQQ detector.153D.1 (a) ∆E energy spectrum for upstream detectors. (b) γ-ray spectrum gated onparticles with different threshold ∆E energy in upstream detectors. . . . . . . . . 159xviiiAcknowledgementsFirst of all, thank you to Dr. Reiner Kru¨cken for granting me this opportunity to carry outcutting-edge research at a world-class institution such as TRIUMF, and for your guidance andsupport throughout these years.I would also like offer my most sincere gratitude to Dr. Peter Bender and Dr. Kathrin Wimmerfor their tireless discussions and endless encouragement, particularly when I needed it most.Without you both I would probably still be calibrating SHARC (which will be discussed insection 4).I would also like to extend my thanks to the S1389 collaboration members for their partic-ipation in the experiments, and especially to Dr. Greg Hackman for his contributions to thisproject.To my supportive family and friends, thank you for your love and patience, and for alwaysreminding me that home is just a phone call away.Finally, I would like to thank my new friends in Canada. You have made these years un-forgettable.xixDedicationTo Carole,whose footsteps will always be beside my own.You continue to be my inspiration in life.xxChapter 1IntroductionIt was noted by Meyer and Jensen in 1949 [May49][HJS49] that experimentally observed sys-tematic trends in binding energies and excited states energies across the nuclear landscape couldbe well-described by the existence of shell structure in nuclei. It was known at the time that theatomic shell model successfully predicts many of the chemical properties of atoms. The atomicshell model is based on the assumption that electrons are confined to well-defined orbitals, andthat the motion of the electrons in these orbital is essentially independent. Similarly, in thenuclear shell model it is assumed that the protons and neutrons move in independent single par-ticle orbitals, within a mean field that is generated by the interaction with all the other nucleons.In the nuclear shell model, low-energy states are described by a residual interaction whichacts between nucleons in the mean field [Hey90]. One aspect of the residual interaction is thestrong pairing interaction which acts between like-nucleons within the same orbital. The pair-ing interaction allows like-nucleon pairs to gain additional binding energy by anti-aligning theirintrinsic spins to S = 0. The importance of the pairing interaction is evidenced by the observa-tion that all even-N, even-Z nuclei have total angular momentum (J) and parity (pi) of Jpi = 0+in their ground states. In addition to coupling nucleon pairs to J = 0, the pairing interactionis responsible for the scattering of nucleon pairs across valence orbitals, which in turn tends tocause partial occupancy of orbitals above the Fermi energy [Cas00]. The occupancy of thesehigher orbitals in turn depends on their energy relative to the Fermi energy.One of the major predictions of the shell model is the existence of magic numbers [Kra88]. Inso-called magic nuclei, there are substantial energy gaps between the filled orbitals and higherunfilled orbitals. Magic nuclei are especially bound because these large energy gaps stronglysuppress the scattering of valence nucleons into the higher orbitals [Hey90], which is similar to11.1. Deformation in Nucleithe role of filled electron shells in noble gases.1.1 Deformation in NucleiAn atomic nucleus can deform its shape in order to minimize its energy. This is observed acrossthe nuclear landscape, both in ground states and excited states. Figure 1.1a shows the degree ofpredicted ground state quadrupole deformation across the nuclear chart. The dark blue regionsof figure 1.1a indicate nuclei which are expected to have approximately spherical ground states,and these regions correspond to nuclei with magic numbers of protons and neutrons.Another striking feature of figure 1.1a is that most nuclei are expected to have non-sphericalshapes in their lowest energy states, which can be seen by the rather abrupt colour change infigure 1.1a as we depart from nuclei with magic numbers of protons and neutrons. It seemsthat even a small number of valence protons and neutrons outside of a closed core can drivethe whole nucleus into a deformed shape. Clearly, the underlying shell structure of nuclei playsan important role in the propensity for nuclei to deform.(a) (b)Figure 1.1: (a) Finite range drop model (FRDM) calculation of the ground state deformationacross the nuclear chart (taken from [MSI+16]). The solid lines correspond to nuclei with magicnumbers of protons and neutrons, which are predicted to have spherical shapes. (b) Isotopeshift across the Sr isotopic chain indicates that there is a sudden transition from spherical todeformed ground states at N = 60 (taken from [WZD+99]).21.1. Deformation in NucleiThere are two primary components of nuclear shell structure which must be understood in orderto describe deformation. Firstly, the microscopic mechanism which drives nuclei to deformationis attributed to the long-range attractive proton-neutron (p − n) residual interaction [Cas00].The strongest component of the long-range p − n residual interaction is the quadrupole term,which is why quadrupole deformation is particularly common in nuclei [Cas00]. This residualp − n interaction allows the nucleus to gain additional binding energy by arranging protonsand neutrons in certain ways across the valence orbitals, which in turn causes a departure fromsphericity. Secondly, the size of the energy gaps between single particle orbitals above the Fermienergy plays a central role. If the energy spacing is small, the valence nucleons can scatter intovalence orbitals which are above the Fermi energy and drive the nucleus into a low-energy de-formed configuration. On the other hand, if the energy spacing is large, the valence nucleons areunable to scatter into higher orbitals and this favours spherical shapes. The size of these energygaps is in turn dependent on the number of valence nucleons, due to the monopole componentof the residual interaction (see section 1.4 for further details). In summary, the competitionbetween deformed and spherical shapes in nuclei is governed by the evolution of single particlestructure and the interplay between the short-range pairing and the long-range p−n interactions.Figure 1.1b shows experimental measurements of the isotope shift across the strontium (Sr)isotopic chain. The isotope shift, ∆〈r2〉, is a measure of the change in the nuclear radius be-tween adjacent isotopes, and can be used to study to the evolution of the ground state shapeacross an isotopic chain. It can be seen in figure 1.1b that the radius of the ground state evolvesgradually as a function of increasing neutron number, and reaches its minimum at N = 50.N = 50 is a magic number, and so this minimum can be explained in terms of a large energy gapwhich suppresses the scattering of valence nucleons above the Fermi surface and gives rise to aspherical ground state. The isotope shift continues to rise steadily as the number of neutronsincreases beyond N = 50 due to increased occupation of higher-lying orbitals, up until N = 60,at which point a sudden increase in the ground state radius is observed. This indicates thata shape transition has taken place, and that the lowest energy configuration in Sr becomes astrongly deformed structure.31.2. Onset of Deformation at Z ∼ 40, N ∼ 60In some regions of the nuclear chart, there are spherical and deformed shapes which offeralmost degenerate energy minimizations. This is remarkable, considering that the underly-ing single particle configurations are significantly different for spherical and deformed nuclei.In these cases, the shapes are said to be coexisting and in some cases can mix very strongly[WZD+99][HW11]. In strongly mixed shape-coexisting states, the nucleus is therefore in aquantum superposition of two different many-body wavefunctions. Shape coexistence appearsto be a unique feature of finite many-body quantum systems, and is intimately related to thephysics of phase transitions [Cas00].1.2 Onset of Deformation at Z ∼ 40, N ∼ 60As was shown in figure 1.1b, it has been observed that Sr undergoes a ground state shapetransition at N = 60. This shape transition is in fact observed in several nuclei in the A ∼ 100region, at Z ∼ 40 and N ∼ 60 [EMS88]. Another experimental indicator of deformation in nu-clei can be found in the excited state spectrum. Deformed nuclei often have a very low energyfirst excited state compared to spherical nuclei, which is caused by the additional collectivedegrees of freedom. Figure 1.2a shows the energy of the first excited state, E(2+1 ), for severaleven-even nuclei in the vicinity of Z ∼ 40 and N ∼ 60. It can be seen that there is a suddendrop in the E(2+1 ) energy in Sr (Z = 38), Zr (Z = 40) and to a lesser degree, Mo (Z = 42)at N = 60. It can also be seen in figure 1.2a that the transition to deformation is much moregradual in the Ru (Z = 44) isotopic chain, which indicates that the proton degrees of freedomare important in this transitional region.Given that the ground states of the N = 60 isotones 98Sr and 100Zr have strongly deformedshapes, it is clear that an inversion between sphericity and deformation in the low-lying statesis taking place in this mass region. Beyond mean field calculations have been carried out[RGSR+10], which predict the potential energy as a function of deformation. Figure 1.2bshows the theoretical potential energy curves for several isotopes of Sr, Zr and Mo. It can beseen that close to stability (N = 52 isotopes), the potential energy curves have a single mini-mum which corresponds to a spherical shape. As neutrons are added to these isotopes, multiple41.2. Onset of Deformation at Z ∼ 40, N ∼ 60Number of Neutrons52 54 56 58 60 62 64 66)(MeV)1+E(200.20.40.60.811.21.41.61.8RuMoZrSr(a) (b)Figure 1.2: (a) Experimentally measured energy of the first excited (2+1 ) state in even-evennuclei in the vicinity of Z ∼ 40 and N ∼ 60 (data taken from [Lab]). (b) Beyond mean fieldcalculations of potential energy as a function of quadrupole moment (Q20) in Sr, Zr and Moisotopic chains (taken from [RGSR+10]).minima begin to form in the potential energy curves up until N = 60, where the ground state(global minimum) becomes a deformed nuclear shape. In addition to this, it can be seen thatthe multiple minima in figure 1.2b are almost degenerate in energy and have different shapes,which is to say that they are predicted to be shape-coexisting.It was demonstrated by Federman and Pittel [FP79] that the sudden onset of deformationin this region can be described in the shell model by the residual interaction between protonand neutron spin-orbit partner orbitals. Their proposed mechanism is illustrated in figure 1.3.As the number of neutrons increases beyond N = 50, the ν1d52 orbital is gradually filled (andis approximately full at N = 56) and the residual interaction between the valence neutronsgradually lowers the energy of the higher-lying ν2s12 , ν1d32 and ν0g72 orbitals. As the numberof neutrons approaches N = 60, there is increased occupancy of the neutron ν0g 72 orbital in thelow-lying states due to pair scattering across the Fermi surface. The residual p− n interactionstrongly affects orbitals with a large spatial overlap, and so the nucleus is able to lower its en-ergy by re-arranging the valence protons so that they occupy the pi0g 92 orbital in tandem with51.2. Onset of Deformation at Z ∼ 40, N ∼ 60Z=28pi0f 52pi1p32pi1p12pi0g 92N=50ν1d52ν2s12ν1d32ν0g 72Figure 1.3: Federman and Pittel mechanism for the onset of deformation in 96Sr [FP79].the neutrons in the ν0g 72 orbital. This particular configuration of valence protons and neutronsgenerates deformation, and so a competition between low energy spherical and deformed shapesarises.It was also shown by Federman and Pittel that there is quenching of the proton 1p32 and 1p12orbitals in the Sr and Zr isotopes [FPE84], which is to say they their spin-orbit splitting is re-duced. This is evidenced by the constant E(2+1 ) values for Sr isotopes for N < 60 in figure 1.2a,which is in contrast to the E(2+1 ) values of the Zr isotones. At N = 56, the E(2+1 ) values for Zrisotopes rises abruptly, which indicates that there are fully occupied orbitals with substantialenergy gaps to unfilled orbitals above the Fermi energy. In other words, the increase in theE(2+1 ) value at N = 56 is evidence for a significant shell closure as the ν1d52 orbital is filled.Contrastingly, the constant E(2+1 ) values in Sr indicate that a low energy Jpi = 2+ configura-tion persists, which can be explained by a low energy [pi1p32 ]2J=2[pi1p12 ]2J=0 configuration. Thisquenching of the pi1p orbitals results in a very similar energy gap between the Fermi energyand the pi0g 92 orbital in Sr compared to Zr, and so it also serves to explain why both Sr and Zrundergo such similar shape transitions at N = 60 [Lab].By generalizing the mechanism of Federman and Pittel to other nuclei in this mass region, itis also possible to discuss why the sudden transition to deformation is only seen in Sr and Zr.The energy gap between the valence pi1p12 and pi0g92 orbitals effectively regulates the onsetof deformation for Z ≤ 40 nuclei. For nuclei with Z > 40 such as Mo, there is already sub-stantial occupation of the pi0g 92 orbital and so the ground state shape transition is more gradual.61.2. Onset of Deformation at Z ∼ 40, N ∼ 60The sudden onset of ground state deformation in Sr and Zr at N = 60 is the result of a graduallowering of the deformed configuration across several isotopes, which ultimately becomes thelowest energy state. In 96Sr and 98Zr, evidence for a deformed low-lying excited 0+ state hasbeen found and this is expected to have a similar structure as the deformed ground states inthe N = 60 isotones [WZD+99][HW11][LPK+94]. The proposed evolution of the spherical anddeformed structures is shown in figure 1.4, which compares their relative energy at N = 58and N = 60. For an axially symmetric quadrupole-deformed nucleus, the shape of the surface,R(θ, φ), is described by [Kra88]R(θ, φ) = Rav[1 + βY20(θ, φ)](1.1)where Rav is the average nuclear radius, Y20(θ, φ) is the spherical harmonic for an orbital withorbital angular momentum ` = 2 and z-projection of ` = 0. The quadrupole deformationparameter β, which is used throughout this work is defined asβ =43√pi5∆RRav(1.2)where ∆R is the difference in radius between the semimajor and seminor axes of the ellipse.Nuclei with a deformation parameter of β > 0 (β < 0) have a shape which is extended (con-tracted) along the z axis compared to the average radius and are described as prolate (oblate)spheroids. While the deformation of the ground state of the N = 60 isotones in figure 1.4 areknown to be β ∼ 0.4 [EMS88], less experimental evidence is available regarding the nature ofthe deformed states in the N < 60 isotones. In particular, the precise degree of deformationof the low-lying deformed 0+ state in 96Sr has not been measured. The deformation valueof β ∼ 0.3 that is given in figure 1.4 is therefore assumed, based on level energies (which isfurther discussed in section 1.3). The proposed evolution of spherical and deformed structuresshown in figure 1.4 is supported by a recent experiment at ISOLDE by Cle´ment et al., whichdemonstrated that the spherical structure in 98Sr is indeed similar to the ground state sphericalstructure of 96Sr [CZP+16a][CZP+16b].71.3. Shape Coexistence in 96SrN = 58β ∼ 00+Zr2+0+Sr0+2+β ∼ 0.3Zr0+2+4+Sr0+2+4+N = 60β ∼ 0Zr0+2+Sr2+β ∼ 0.4Zr0+2+4+Sr0+2+4+Esph.-Edef.[MeV]0123Figure 1.4: Proposed lowering of a deformed structure (β > 0) across Sr and Zr isotopes, whichresults in a ground state shape transition at N = 60. Adapted from [LPK+94]. Energy levelsthat are associated with the spherical and deformed configurations are also drawn, and the datais taken from [Lab].Shape coexistence in low-lying states has been identified in the neutron-rich N = 58 and N = 60nuclei 96,98Sr and 98,100Zr, and this region remains a subject of substantial interest, both exper-imentally and theoretically [MXY+12][RGSR+10][SNL+09][HEHJ+00]. Unfortunately, little iscurrently known about the low energy states in this region. In 95Sr, only the ground state spinand parity is firmly established. Similarly, many of the low-lying 96Sr states do not have firmspin and parity assignments. It is therefore the intent of this work to firmly establish the spinsand parities and to measure the single particle structure of the low-lying states, so that theonset of deformation and shape coexistence in this region can be better understood.1.3 Shape Coexistence in 96SrAn experimental signature of shape-coexisting states that mix is an enhanced monopole tran-sition strength, which is measured using conversion electron spectroscopy [WZD+99]. Themonopole transition strength ρ2(E0) between mixed states isρ2(E0) =(3Z4pi)2a2(1− a2)[∆(β2)]2 (1.3)81.3. Shape Coexistence in 96Srwhere a is the mixing amplitude and ∆(β)2 is the difference in deformations of the two config-urations. In the case of mixing between a strongly deformed state and a nearly spherical state,∆(β)2 ' β2. A large ρ2(E0) value is therefore a valuable indicator of shape-coexisting statesas it depends on both the mixing strength and the difference in deformation.Several nuclei in the vicinity of N ∼ 60 and Z ∼ 40 have been found to have very large monopoletransitions strengths with ρ2(E0) ∼ 0.08−0.12, compared to typical values of ρ2(E0) ≤ 0.05 inthis mass region [WZD+99]. One of the largest monopole transition strengths that have beenmeasured in nuclei is between two low energy excited 0+ states in 96Sr, which has a ρ2(E0) valueof 0.185(50) [Jun80]. Figure 1.5a shows the low energy states in 96Sr, with the large ρ2(E0)value between the 1465 keV state and the 1229 keV state drawn as a thick black arrow. Thesequence of levels built on top of the 1465 keV state in figure 1.5a shows a proposed rotationalband, which is a spectroscopic signature of deformed nuclear structures [Cas00]. Rotationalbands are sequences of levels which are interpreted to be the energy spectrum associated withrotating a deformed structure, with E ∝ J(J + 1). The proposed rotational band shown in fig-ure 1.5a would indicate a strongly deformed structure with β ∼ 0.3, and this value was used infigure 1.4. Large quadrupole (E2) transition rates are also commonly observed between stateswithin strongly deformed rotational bands. It should be noted, however, that a common struc-ture underlying these states has not been experimentally verified, and the rotational band builton the 1465 keV state was proposed based on the spins, parities and energy spacing betweenthe states. One can also argue that the relative branching ratio of decays from higher-lyingstates to the excited 0+ states is an indicator of the mixing strength between them. If theyare populated with roughly equal intensity through the decay of higher-lying states, then thissuggests that they have similar transition matrix elements. At the time of this work, no γ-raytransitions to the 1465 keV 96Sr state from higher states have been observed, which indicatesthat feeding from higher states is weak compared to the 1229 keV state. This is discussed morequantitatively in section 6.3.It was also shown in the recent work of Cle´ment et. al. that the 1229 keV 0+2 state appearsto be dominated by a spherical configuration, while an observation of the 1465 keV 0+3 statewas not reported [CZP+16a][CZP+16b]. Taken together, these experimental findings favour a91.3. Shape Coexistence in 96Srsmall mixing strength in equation 1.3.(a)Two Level Mixing ModelBefore Mixing After Mixing∣∣∣0+sph〉∣∣0+def〉∣∣0+2 〉∣∣0+3 〉∆Eu ∆Ep(b)Figure 1.5: (a) Low energy states of 96Sr, indicating the strong monopole transition strength(given in units of ρ2(E0)x103) between excited 0+ states (adapted from [HW11]). (b) Two levelmixing model for coexisting states, where the unmixed states are assumed to be pure sphericaland deformed configurations, respectively.Two Level MixingA straightforward but very useful way to analyze the shape-coexisting 0+ states in 96Sr isthrough a two level mixing model, which is demonstrated in figure 1.5b. In this model, thetwo states initially exist as two distinct configurations with an energy separation of ∆Eu; oneof which is nearly spherical and one of which is strongly deformed. A residual interaction V isthen introduced which acts on the states. E1 VV E2Ψ = EΨ (1.4)101.4. Shell Model CalculationsThe interaction V causes the states to be pushed apart in energy and mixes the spherical anddeformed configurations by an amount a such that∣∣0+2 〉 = a ∣∣∣0+sph〉+√1− a2 ∣∣0+def〉 (1.5)∣∣0+3 〉 = √1− a2 ∣∣∣0+sph〉− a ∣∣0+def〉 (1.6)The energy separation between the two states after mixing, ∆Ep, is given by∆Ep =√(∆Eu)2 + 4V 2 (1.7)From experiment, we know that ∆Ep = 235keV, however the strength of the interaction andthe initial energy separation ∆Eu is not known.By measuring the mixing strength between the excited 0+ states in 96Sr, it will therefore bepossible to determine the β value of the deformed structure for the first time by using equation1.3, and this value can be compared to the ground state deformation of 98Sr. In addition,information about the mixing between the excited 0+ states in 96Sr will also elucidate the levelof degeneracy between the pure spherical and deformed configurations, which will help to refinetheoretical models of this region.1.4 Shell Model CalculationsIn the shell model, the many-body Schro¨dinger equation is solved based on the assumption thatlow energy states in nuclei can be described without explicitly including all of the nucleons inthe nucleus [Cas00]. For this reason, the nucleus is divided into two parts; an inert core anda valence space, as shown in figure 1.6. The core is defined as a set of fully occupied singleparticle orbitals, from which nucleons cannot be excited. Whenever possible, this is usuallytaken to be a full major oscillator shell. All nucleons outside of the core are then treated asvalence nucleons which can couple to form different configurations and occupy different orbitals.The valence space (or model space) is usually limited to several valence orbitals outside of the111.4. Shell Model Calculationsclosed core. The shell model Hamiltonian isHˆ =∑iinˆi +∑i,j,k,lVij,klaˆ†i aˆ†j aˆlaˆk (1.8)where nˆi, aˆ†i and aˆi are the number operator, creation operator and annihilation operator forvalence orbital i. The one-body term of the Hamiltonian describes the binding of each valencenucleon within a mean field potential. The solutions of the one-body part of the Hamiltoniangive rise to the independent particle model, where the eigenvalues i are are simply the boundstates within the potential. The i are normally referred to as the single particle energies(SPEs) of the Hamiltonian. The SPEs are adjusted phenomenologically so that the low energyspectra of nuclei in the vicinity of magic numbers (such as N = 50, see figure 1.3), where theindependent model is an adequate description of low energy states, are well reproduced.The two-body term in the Hamiltonian describes the nucleon-nucleon residual interaction withinthe valence space, and the Vij,kl are called the two-body matrix elements (TBMEs). The TBMEsmake it possible for nucleons in the valence orbits i, j, k and l to interact with each other. Thisprovides additional correlation energy and leads to the mixing of different configurations, whichcauses a departure from the simplistic independent particle model. The TBMEs contribute tothe diagonal terms of the Hamiltonian matrix and also the off-diagonal terms. The diagonalterms contribute to the one-body SPEs, which gives rise to effective single particle energies(ESPEs).The ESPEs describe the evolution of the underlying shell structure within the valence spaceas more nucleons are added and interact with each other. The ESPEs determine the energygaps between the valence orbitals which in turn changes the energy of different single particleconfigurations, and can lead to a competition between different structures across an isotopicchain. The off-diagonal TBMEs act in a way that is closely resembles the two level mixingmodel that was presented in section 1.3, although they couple many nucleon pairs across mul-tiple different orbitals. The TBMEs can be derived using first principles models such as ChiralEffective Field Theory (χEFT) for light nuclei and nuclei closed to magic numbers, but aregenerally determined phenomenologically using nucleon-nucleon scattering data.121.4. Shell Model CalculationsInert CoreValence OrbitalsExcluded OrbitalsFigure 1.6: Shell model description of a nucleus. The residual interaction between valencenucleons (blue wavy line) causes different configurations to mix.The Hamiltonian matrix therefore contains all allowed configurations of the valence nucleonsacross the valence orbitals, and must be solved using efficient matrix diagonalization algorithms.The matrix quickly becomes very large as the valence space increases, which is why a limitednumber of valence nucleons and orbitals are included.The shell model code NushellX [BR14] was used to calculate the wavefunctions and energylevels of for 94,95,96Sr. Two model spaces and interactions are available for the Z∼40, N∼60region; jj45 [EHJH+93] and glek [MWG+90]. The jj45 interaction was developed for studies ofheavier Sn isotopes, and it was found that the SPEs required substantial modification beforethe calculated levels of Z ∼ 40, N = 51 nuclei were in reasonable agreement with the availableexperimental data. N = 51 nuclei were used to benchmark the interaction because these nucleicontain a single neutron outside of a closed shell. The low-energy states of N = 51 nuclei tendto be dominated by single neutron excitations, and so these states provide a reasonable measureof the valence orbital spacing. The glek interaction was developed for Y and Zr studies, andso only a small adjustment to the SPEs was necessary to give reasonable agreement with Srexperimental data.The proton degrees of freedom are expected to play an important role in describing the low-lying states of neutron-rich Sr isotopes. As was discussed in section 1.2, the proposed quenchingof the 1p orbitals indicates that proton excitations into the 1p12 orbital are required to describethe low-lying states. The same is true for the 0g 92 orbital, which was also found to be important131.4. Shell Model Calculationsfor describing the structure of neutron-rich Zr isotopes. In this work, systematic shell modelcalculations were carried out using a series of model spaces for both of the available interactionsso that the effect of adding proton degrees of freedom could be examined. It was not possibleto carry out full model space calculations as the computational requirements exceeded availableresources. The full model spaces associated with the jj45 and glek interactions, along with thetruncated model spaces which were used in this work are summarized in table 1.1. It should alsobe noted that the shell model calculations assume a spherically symmetric nuclear potential,and so are only appropriate for states which are nearly spherical.Interaction Full Model Space This Work : a© This Work : b© This Work : c©jj45 [pi] 0f 521p0g92 [1p32 ]4 up to [1p32 ]2[1p12 ]2 up to [1p32 ]0[1p12 ]2[0g 92 ]2jj45 [ν] 1d2s0g 720h112 1d2s0g72 same as a© same as a©glek [pi] 0f1p0g 92 [1p32 ]4 up to [1p32 ]2[1p12 ]2 up to [1p32 ]0[1p12 ]2[0g 92 ]2glek [ν] 1d2s0g 1d2s0g 72 same as a© same as a©Table 1.1: Summary of NushellX model spaces and interactions that were used (more detailsin the text).As can be seen in table 1.1, the jj45 and glek interactions were developed for larger modelspaces than were used in this work. In particular, the jj45 and glek interactions include theproton 0f orbitals in the model space. The proton 0f orbitals were required to be fully occupiedin this work. It is expected that proton excitations from the 0f orbitals will not contributesignificantly to the underlying configurations of the low energy states as they are significantlybelow the unoccupied valence orbitals. It should be noted that all orbitals which are included inthe full model space and have nonzero occupations will contribute to the ESPEs of the orbitals,including those which are required to be fully occupied and inert.A series of calculations were carried out for the low-lying states of 94,95,96Sr, using three differentmodel spaces which are denoted as a©, b© and c© in table 1.1. Model space a© required thatthe four valence protons outside of the full 0f orbitals were inert and only occupied the 1p32orbital, as denoted by [1p32 ]4. Model space b© included an extended proton valence space, where141.4. Shell Model Calculationsthe protons could scatter in an unconstrained way from the 1p32 into the 1p12 orbital, allowingup to 2p − 2h proton configurations as denoted by [1p32 ]2[1p12 ]2. Model space c© included afurther increased proton valence space, allowing all four 1p32 protons to scatter into the 1p12and 0g 92 orbitals. A maximum of two protons were allowed to occupy the 0g92 orbital so thatthe calculation size remained manageable. This enabled up to 4p− 4h proton configurations asdenoted by [1p32 ]0[1p12 ]2[0g 92 ]2.Similarly, the neutron valence spaces were truncated to make the calculations tractable. Theneutron 0h112 orbital was excluded from the jj45 model space as it has a large SPE compared tothe other valence neutron orbitals, and also has negative parity. In this work, we are only inter-ested in the low lying positive parity states which are populated through one neutron transfer.The 0h112 would only contribute to the structure of these states through configurations such as[0h112 ]2, which would be a very high energy configuration. For these reasons, the 0h112 orbitalis not expected to play a significant role in the low-lying states. Large scale shell model cal-culations that were carried out for the Zr isotopes by Sieja et al. [SNL+09] predict very lowoccupancy of the 0h112 orbital across all of the calculated states, which further indicates thatthis state can be neglected. All of the calculations that were carried out in this work using theglek interaction were required to have a fully occupied neutron 0g 92 orbital. This is reasonableas there is a large energy difference between the 0g 92 and the 1d52 orbitals due to the Z = 50shell closure, and so neutron excitations across the Z = 50 shell gap would be energeticallyexpensive. In addition to a large energy gap, the 1d52 orbital is expected to be almost fullyoccupied for nuclei with N > 56, which would inhibit the scattering of neutrons across theZ = 50 shell gap.Figures 1.7 and 1.8 compare the shell model calculations that were carried out for 95Sr usingmodel spaces a©, a© and a© to experimental data.It can be seen in figure 1.7 that the jj45 interaction incorrectly predicts a 72+ 95Sr ground statewhen any additional proton degrees of freedom are included. Given that the SPEs were opti-mized for N = 51 nuclei, such a discrepancy suggests that the TBMEs of the jj45 interactionmay not be well suited to describing Sr isotopes. On the other hand, the calculated levels in151.4. Shell Model Calculationsa© b©12+32+52+c©72+32+Experiment12+352(32+)556(72+)681(32+, 52+)Figure 1.7: Selected low-lying 95Sr states compared to shell model calculations using the jj45model space and interaction with various proton valence spaces (more details in the text).figure 1.8 show that the 12+ground state of 95Sr is correctly predicted for each of the protonvalence spaces. Including the proton 1p12 orbital in the model space led to an improvementin the calculated energy levels, with particularly good agreement between the excited 32+and72+state energies and the experimental levels. The addition of the 0g 92 orbital in the protonvalence space caused further lowering of the 72+state energy. This indicates that there is astrongly attractive residual interaction between these orbitals. Calculations in model space c©were computationally intensive and very time consuming, and optimizing the SPEs generallyrequires many iterations. For this reason, the SPE of the proton 0g 92 orbital was not modifiedfrom the original glek interaction. As a result, the over-binding effect that can be seen in figure1.8 for the 72+state in model space c© may also be due to the SPE for the 0g 92 orbital.Given that the TBMEs of the glek interaction were developed in this same mass region, they arebetter suited to studying Sr than the jj45 interaction. For this reason, only the glek interactionis compared to experimental data for the remainder of this thesis.A consequence of the residual interaction Vij,kl is that the calculated shell model wavefunctionof a given state consists of many different configurations with the same spin and parity. Ingeneral, these terms do not contribute equally and so in most cases the wavefunction is madeup predominantly from one or a few important configurations. The wavefunction of the 95Srground state, as calculated using the glek interaction in model space a©, is shown in table 1.2.161.4. Shell Model Calculationsa© b© c©32+52+72+32+Experiment12+352(32+)556(72+)681(32+, 52+)Figure 1.8: Selected low-lying 95Sr states compared to shell model calculations using the glekmodel space and interaction with various proton valence spaces (more details in the text).Configuration ν1d52 ν2s12 ν1d32 ν0g72 a2 [%](i) 6 1 0 0 82.63(ii) 4 1 2 0 6.29(iii) 5 1 1 0 5.13(iv) 4 2 1 0 1.50(v) 4 1 0 2 1.14Occupation 5.70 1.01 0.25 0.04 96.69Table 1.2: Wavefunction composition and orbital occupation numbers for the 95Sr ground state.Each of the underlying configurations (i)-(v) are coupled to Jpi = 12+. The occupation numbersare the weighted sum of the underlying configurations.For orbitals which are not fully occupied, the nucleons may also couple to J 6= 0. An example ofthis would be the coupling of a J = 2 neutron pair in the ν1d52 orbital to a J = 2 neutron pairin the ν1d32 orbital to form a configuration with total J = 0, which is possible in configuration(ii). A similar configuration also exists for the ν1d52 and ν0g72 orbitals. The number of unpairednucleons is called the seniority (ν). These configurations will usually play a small role in theoverall wavefunctions due to the energy expense of breaking multiple nucleon pairs.For model spaces b© and c©, there are many more components in the wavefunctions, such as thecoupling between proton and neutron excitations. This makes it less straightforward to rep-resent the underlying configurations of the many-body nuclear wavefunction. In model spaceswhich include proton and neutron degrees of freedom, it is common to discuss the total orbitaloccupation numbers instead. The occupation numbers are the expectation value for number ofnucleons in each of the valence orbitals, and can have a value of up to 2j + 1 for fully occupied171.5. Transfer Reactionsorbitals. Table 1.2 shows that in model space a©, the 12+ 95Sr ground state wavefunction ismostly made up of configuration (i), in which the neutron 1d52 orbital is full and there is anunpaired neutron in the 2s12 orbital. This is also reflected in the occupation numbers presentedat the bottom of table 1.2, which predict that the 1d52 orbital is effectively fully occupied andthat there is a single neutron in the 2s12 orbital.The shell model wavefunctions can also be used to calculate the overlap between differentstates. In this way, the predicted cross section for 94,95Sr(d,p) can be compared to the ex-perimental data. A quantity of particular importance for this work are spectroscopic factors,which are described at the end of the next section. In section 6, spectroscopic factors that werecalculated using the shell model are compared to the experimental results.1.5 Transfer ReactionsTransfer reactions are a well established tool for studying the single particle structure of nu-clei [Fes92]. In these reactions, a small number of valence nucleons are transferred from onenucleus to another and populate unfilled valence orbitals. By measuring the outgoing flux ofparticles from these reactions, detailed information of the populated orbitals can be obtained.This makes them a very useful tool for measuring the single particle configurations of nuclearstates.Transfer reactions are direct reactions, which means that they are peripheral collisions betweennuclei that occur very quickly and are well-described as a single-step process. A single-stepprocess has no intermediate state, and so the final state is related to the initial state in astraightforward way, which will be discussed later in this section. Consequently, other thanthe exchange of a small number of valence nucleons at the surface there is minimal rearrange-ment of the constituent nucleons within each of the nuclei. In contrast to this, nuclei can alsoundergo compound reactions where many nucleons participate in the process, forming a long-lived, highly excited state. In compound reactions, information of the initial state is lost andso they cannot be used to study single particle structure. Compound reactions tend to havelarger cross sections than transfer reactions and can take place under the same experimental181.5. Transfer ReactionsEntrance Channel, αAa = b+ xbxExit Channel, βB = A+ xAxbFigure 1.9: Schematic diagram of a transfer reaction.conditions, which is often substantial source of background.Transfer reactions are denoted bya+A→ b+B (1.9)where B = A + x is the target nucleus plus the transferred nucleons, a = b + x is the beamnucleus plus the transferred nucleons and x represents the transferred nucleons. It is also com-mon to write equation 1.9 as A(a, b)B. A and b are referred to as the core nuclei as they remainunchanged throughout the reaction, while B and a are referred to as composite nuclei as theycontain the inert core nuclei plus the exchanged nucleons. The left side of equation 1.9 is calledthe entrance channel and is labelled by α, while the right side of equation 1.9 is called the exitchannel and is labelled by β.Figure 1.9 gives a schematic representation of a transfer reaction. As can be seen, the nu-cleons (x) are transferred from the initial composite system a and bind to the target corenucleus A to form a final composite system B. Both the initial and final states of this reactioninvolve two bodies, which means that the kinematics of the final state can be fully describedby measuring either b or B.The most commonly used transfer reactions involve the addition or subtraction of a single191.5. Transfer Reactionsnucleon from a nucleus of interest, which is called one-nucleon transfer. These reactions areespecially useful as they avoid the complications that arise from the exchange of many-nucleonconfigurations. The cross sections for one-nucleon transfer reactions are also considerably largerthan for many-nucleon transfer, which is an important consideration when studying exotic nu-clei.In this work, a series of Sr(d,p) reactions were carried out. In a (d,p) reaction, the neutronis stripped from the deuterium and populates an unfilled valence orbital in the target nucleus.The remaining proton is then measured, and from this measurement the details of the trans-ferred neutron can be inferred. Deuterium is an ideal tool for one-nucleon transfer reactions asit is a well understood nucleus, it is loosely bound and is also readily available to laboratories.1.5.1 DWBA TheoryNuclear reactions are described mathematically using scattering theory. In this framework, thereaction is modelled as the scattering of a travelling wave upon an interaction potential. Thewave subsequently scatters off the potential, and this induces a transition from the entrancechannel α to the exit channel β. The scattered wave then propagates radially away from thetarget and interferes with the unreacted component of the incident wave, producing an interfer-ence pattern. The resulting interference pattern describes the expected intensity distributionof particles as a function of scattering angle in the exit channel, which is proportional to theangular distribution (or differential cross section) of the reaction. This process is illustrated infigure 1.10.When the beam and target nuclei collide, there are numerous different reactions that can occur.The most important, and dominant, of these is elastic Coulomb scattering. The long-rangedCoulomb interaction distorts the waves far away from the interaction region. These asymptoticsolutions are called Coulomb waves and are denoted by χ. As the cross sections for transferreactions are smaller than elastic scattering, the transfer process can be approximated as aperturbation on the elastic scattering channel. This approximation is the basis of the DistortedWave Born Approximation (DWBA) [Sat83].201.5. Transfer ReactionsBeam Scattering off targetInterferencePatternFigure 1.10: Schematic diagram of a nuclear reaction in scattering theory.The elastic scattering channel is modelled mathematically through the use of a complex poten-tial, where the real component of the potential describes elastic scattering and the imaginarycomponent of the potential describes inelastic processes which remove flux from the reaction.This is called the optical model (or optical potential) [Hod71], as it is analogous to the scatteringof light through a semi-opaque medium. The general form of the optical potential isU(r) = Uc(r,Rc)− V0f(r,R0, A0)− iWD ddrf(r,RD, AD)− VSO ddrf(r,RD, AD)~` · ~S (1.10)Where Uc is the long-ranged (repulsive) Coulomb potential and f describes the shape of theshort-ranged interaction potential. The optical potential normally includes derivative terms toaccount for interactions which take place at the surface of the nucleus, such as the spin orbitinteraction between the incident and target nuclei. The short-ranged potential f is commonlydescribed using a spherically symmetric Woods-Saxon shape [Sat83], which has the formf(r,R,A) =11 + exp(r−RA) (1.11)where A = Z+N is the mass number of the nucleus. This potential has the approximate shapeof the nucleus. Figure 1.11a shows the shape of the Woods-Saxon potentials that are used inthe optical model. The numerous V (W ), R and A terms in equation 1.10 are known as theoptical model parameters, which are determined phenomenologically. There are readily avail-211.5. Transfer Reactionsable global parameters such as those of Perey and Perey [PP76] which come from global fits tomany data sets, however it is standard practice to use the global parameters as a starting pointand optimize the potential further using experimental elastic scattering data. In this work, thelatter experimental optimization procedure was used, which is discussed further in section 5.1.In scattering theory, the transition matrix (or T -matrix) describes the amplitude of a re-action. The cross-section for the reaction is therefore proportional to the square modulus ofthe T -matrix element for α→ β. A full derivation of the T -matrix can be found in reference[Sat83], and so will not be given in this thesis. The T -matrix element for α → β in DWBAtheory isT DWBAβα =〈χ(−)β Φβ∣∣∣V ∣∣∣Φαχ(+)α 〉 (1.12)where χ(±) are the incoming and outgoing asymptotic (Coulomb) distorted waves for r → ∞and Φα,β are intrinsic wavefunctions which describe the bound state wavefunction of the neutronin the entrance channel α and the exit channel β, respectively.The V term in equation 1.12 describes the total interaction potential and can be expressed interms of the entrance channel αV = VνA + VbA − Uα (1.13)or equivalently, in terms of the exit channel βV = Vνb + VbA − Uβ (1.14)where VνA and Vνb are called the binding potentials for the composite systems B and a, re-spectively. The binding potentials are required to calculate the bound state wavefunction ofthe transferred neutron in the entrance and exit channels. The Uα,β correspond to the opticalpotentials that were defined in equation 1.10, which describe elastic scattering of the nuclei inthe α and β channels, respectively. VbA is called the core-core interaction potential, and repre-sents the interaction between the cores b and A during the reaction. VbA is usually describedusing an optical potential.221.5. Transfer ReactionsIn summary, there are several inputs that are required to calculate the cross section for areaction. It is necessary to define optical potentials for the entrance and exit channels, as theseproduce the distorted waves in equation 1.12. Ideally, these potentials are optimized by fittingto experimental elastic scattering data. Given that the reaction cross section is dominated byelastic scattering, a good description of the optical potentials is essential. As was previouslymentioned, the core-core potential is also well-described by the optical model. For (d,p) reac-tions, the core-core potential is taken to be approximately the same as the the optical potentialfor the outgoing (exit) channel. This approximation is especially good for (d,p) reactions in-volving heavy nuclei, since the difference between the (N,Z) core and the (N +1, Z) compositeis very small.It is also necessary to describe the overlap wavefunction between the core and composite nucleusin both the entrance and exit channel. It is standard practice to obtain the overlap wavefunc-tions by solving the radial Schro¨dinger equation within a Woods-Saxon potential and matchingthe bound state wavefunction to the asymptotic distorted wave. The binding Woods-Saxonpotential is usually assigned standard R and A values which approximate the size and shape ofthe nucleus, while the potential depth is adjusted manually to reproduce the experimental sep-aration energy of the bound neutron [TN09]. With these physical inputs defined, the reactioncalculations can be carried out using standard codes.In this work, cross sections for 94,95Sr(p,p), 94,95Sr(d,d) and 94,95Sr(d,p) were calculated us-ing the DWBA code FRESCO [Tho88]. An example FRESCO input file is shown in appendixA.The final state overlap wavefunctions were assumed to be pure single particle states, wherethe transferred neutron populates a single valence orbital. For example, the 12+ground statewavefunction of 95Sr is assumed to be made up of the Jpi = 0+ ground state wavefunction of94Sr coupled to a neutron in the 2s12 (` = 0) orbital, which is written as∣∣∣95Sr; 12 +〉 = ∣∣94Sr; 0+g.s.〉⊗ ∣∣ν2s12〉 (1.15)231.5. Transfer ReactionsThis particular description of the reaction denotes an ` = 0 transfer, and leads to a distinctiveshape in the differential cross section (or equivalently angular distribution) curve as a function ofthe center-of-mass scattering angle. The DWBA angular distributions for 94Sr(d,p) for different` transfers can be seen in figure 1.11b. Other descriptions, such as ` = 4 (which denotes thetransfer of the neutron into the 0g 72 orbital), predict very different angular distributions andtotal reaction cross sections. It is these unique characteristics of the DWBA curves which allowthe angular momentum transfer, `, of the experimental states to be determined. The shape ofthe angular distribution and total cross section for a reaction is also dependent on the Q-valueof the reaction, and so different excited states would have different calculated curves. Moredetails on how the DWBA calculations were carried out can be found at the end of section 4.4. R [fm]0 2 4 6 8 10 V [MeV]50−40−30−20−10−0Woods-Saxon PotentialsVolumeSurface(a)]° [CMθ 0 20 40 60 80 100 120 140 160 180 [mb/sr]Ωdσd 110Sr(d,p)94Angular Distributions for 21L=0, 2s23L=2, 1d27L=4, 0g(b)Figure 1.11: (a) Shape of a Woods-Saxon potential. The surface potential is defined mathe-matically as the derivative of the volume potential. (b) Example DWBA angular distributioncalculations for different orbital angular momentum transfer.As was discussed in section 1.4, the wavefunctions of the 94,95,96Sr states are expected to consistof many different configurations. For this reason, the experimental cross sections will be lessthan those predicted by the pure single particle configurations that are assumed in the DWBAcalculations. By comparing experimental cross sections to DWBA calculations it is thereforepossible to obtain information about the purity of the single particle configuration in the real241.5. Transfer Reactionswavefunction. In mathematical terms[δσδΩ(θ)]exp= SF[δσδΩ(θ)]DWBA(1.16)where SF is the experimental spectroscopic factor, which is defined as the overlap betweenthe real wavefunction with the pure single particle wavefunction that is calculated [Sat83]. Inother words, the spectroscopic factor gives the extent to which a populated state is describedby the ground state of the core nucleus plus a neutron in a single particle orbital, as shown inequation 1.15. Clearly, states that are not populated, or that are populated very weakly, in a(d,p) reaction do not have a strong component of these single particle configurations in theirwavefunctions.An important consideration for this work is the population strength of the excited 0+2,3 states in96Sr through 95Sr(d,p). Both 0+ states contain a mixture of spherical and deformed configura-tions, as is described by equations 1.5 and 1.6. Here we assume that the deformed configuration0+def will not be directly populated through95Sr(d,p) as the reaction cannot be described as asingle step process. The relative population strength of the 0+2,3 states in96Sr through 95Sr(d,p)will therefore measure the relative amounts of spherical configurations 0+sph in each of the states.This can be formulated asSF(0+3 )SF(0+2 )=1− a2a2(1.17)1.5.2 Inverse KinematicsFor many years, (d,p) reactions have been successfully carried out using deuterium beams tostudy the single particle structure of stable and long-lived nuclei. In more recent times, thenuclei to be studied are short-lived and so cannot be fabricated into target materials. It is there-fore necessary to carry out (d,p) reactions in inverse kinematics [Jon13], where the deuteriumis the target nucleus and the exotic nucleus under study is produced as a radioactive beam.Inverse kinematics experiments are more challenging than normal kinematics experiments fora number of reasons that will briefly be discussed.251.6. Overview of ThesisInverse kinematics (d,p) experiments have much larger center-of-mass motion than normalkinematics experiments, which leads to less favourable angular coverages for detection of theejected protons [Cat14]. Although there have been many significant advances in the productionof radioactive ion beams, the beam intensities which are currently achievable are many ordersof magnitude less than those which are possible for stable beams (such as deuterium). Conse-quently, much thicker targets must be used to produce comparable experimental yields. Thispresents a challenge, as the heavy ion beam loses a large amount of energy as it passes throughthe thick target and this leads to a substantial degradation of the energy resolution. Whentransfer reactions are carried out in inverse kinematics it is important to be able to separatethe different populated states, and so γ-ray detectors are commonly used to identify the states.1.6 Overview of ThesisIn summary, the aim of this experimental work is to measure the spins and parities, and tostudy the single particle structure of low-lying states in 95,96Sr through 94,95Sr(d,p). It is also aprimary goal of this work to measure the relative population strengths of the excited 0+ statesin 96Sr through 95Sr(d,p) so that the mixing strength of the underlying configurations can bedetermined.In the next chapter of this thesis, the 94,95Sr(d,p) experiments are described in detail. Importantresults from simulations of the experiments are given in chapter 3. The analysis proceduresthat were carried out in this work are explained in chapter 4. The experimental results arepresented in chapter 5, and these are compared to shell model calculations in chapter 6. Asummary and overview is given in section 7.26Chapter 2Experiment2.1 Overview of ExperimentsStates in 95,96Sr were investigated through the reactions 94,95Sr(d,p)95,96Sr in inverse kinematicsusing a 5.5 MeV/u beam impinged on a deuterated polyethylene (CD2) target. A combinationof position-sensitive particle detection (SHARC) and γ-ray detection (TIGRESS) systems madeit possible to carry out proton angular distribution and cross section measurements and also touse γ-particle coincidences to extract information about the excited 95,96Sr states which werepopulated. Figure 2.1 shows a schematic diagram of the experiment.Both experiments were carried out at TRIUMF, Canada’s national laboratory for particle andnuclear physics research. The 94Sr(d,p) and 95Sr(d,p) experiments were carried out in June2013 and June 2014, respectively, and were the first high mass (A>30) experiments with a re-accelerated secondary beam to be performed at TRIUMF. Consequently, this campaign marksan important step forward for TRIUMF and serves as a demonstration of the enhanced ca-pabilities of the laboratory to study heavier exotic nuclei through nuclear reactions involvingaccelerated exotic beams.In this chapter, the general discussion of beam delivery and experimental setup will focusprimarily on the 95Sr(d,p) experiment, as both studies were carried out under very similar con-ditions. Some details of 94Sr(d,p) experiment will also be presented throughout the subsequentchapters, so that significant differences and important experimental details can be highlighted.272.2. RIB ProductionFigure 2.1: Schematic diagram of the experiment, showing the detector set-up.2.2 RIB ProductionThere are two main techniques that are used to produce radioactive ion beams (RIBs); in-flightfragmentation and isotope separation online (ISOL). In-flight fragmentation facilities such asNSCL, GSI and RIKEN are driven by heavy ion beams which are impinged upon thin targetsat high energies (> 50 MeV/u). The beam loses very little energy in the target but can undergofragmentation as a result of its interaction with the target nuclei, resulting in a fast cocktailbeam which requires mass separation online. This makes it possible to study nuclei with veryshort half-lives but production cross sections tend to be low and beam energy distributions canbe broad. Contrastingly, the ISOL technique which is used at TRIUMF and ISOLDE makesuse of a very thick target which is bombarded with a light primary beam such as protons orhelium nuclei. The primary beam loses a large amount of its energy within the target, whichmaximizes the production yield but introduces an inherent challenge to promptly and efficientlyextract the produced isotopes. The production target is operated at typical temperatures of282.2. RIB Productionseveral thousand kelvins so that the produced radioisotopes diffuse to the surface to decreasethe extraction time. It remains a limitation of the ISOL method that nuclides with very shorthalf-lives (.10 ms) cannot be presently studied using this approach. For nuclear reactionstudies it is also necessary to re-accelerate the RIB produced using the ISOL method. Whilere-acceleration of the RIB introduces additional inefficiencies which lower the secondary beamyield, it is possible to produce precise final beam energies using this technique.2.2.1 Beam Delivery at TRIUMFThe TRIUMF 500 MeV proton cyclotron was used to produce a high intensity beam of protonswith a beam current of up to 10 µA [DKM14]. The beam was then sent to the TRIUMFIsotope Separator and Acceleration (ISAC) facility where it impinged on a thick Uranium Car-bide (UCx) target. Proton-induced uranium fission and spallation within the target produceda yield consisting of a wide range of nuclei. The isotopes were extracted from the UCx targetand numerous stripping and filtering techniques were utilized in order to optimize the purityand rate of the 95Sr beam.Surface ions were ionized into a singly charged (1+) state using the TRIUMF Resonant Ion-ization Laser Ion Source (TRILIS) [DKM14]. TRILIS uses multiple high-intensity laser beamsto ionize atoms using multistep resonant photon absorption, which makes possible efficient andelement selective ionization. TRILIS was used to enhance the extraction rate of 95Sr comparedto contaminants which are also produced within the production target. The cocktail beam wasthen sent through the ISAC mass separator, which has a resolution of ∆AA ∼ 10−4 [DKM14],to produce a beam with only A = 95 isotopes. The beam was then transported to the ChargeState Booster (CSB) where the isotopes were charge-bred by an Electron Cyclotron Resonance(ECR) plasma source to a 16+ charge state. This was necessary so that the beam could nextbe sent to the Radio-Frequency Quadrupole (RFQ), which accepts a maximum mass-to-chargeratio (A/q) of 30 [DKM14]. Inside the RFQ, time-dependent electric fields were tuned to ac-celerate the specific A/q of 95Sr ions. Contaminant isotopes in the beam were mismatchedwith the acceleration phase of the RFQ and so did not undergo any acceleration. Following theRFQ, these contaminants were deflected out of the beam using the bending dipole magnets inthe accelerator chain.292.2. RIB ProductionThe highly charged 95Sr beam was then transported to the ISAC-II facility where its kineticenergy was increased to 524 MeV (5.515 MeV/u) using the superconducting linear accelerator[DKM14] before finally being delivered to the TIGRESS experimental station in the ISAC-IIexperimental hall. An overview of the ISAC-I and ISAC-II facilities is shown in figure 2.2.Figure 2.2: Diagram of TRIUMF-ISAC facility [BHK16]. The 95Sr16+ beam was delivered tothe TIGRESS experimental station in ISAC-II.The beam composition at the experimental station was monitored using the TRIUMF Bragg(TBragg) detector [Nob13] and was found to be composed of ∼98% 95Sr, with some small con-tamination due to 95Rb and 95Mo. The TBragg detector is described in section 2.4.1. The mass95 beam composition as measured by the TBragg detector is shown in figure 2.3.The beam intensity was monitored periodically throughout the experiment using the ISAC-IFC4 Faraday cup, which was positioned upstream of the TIGRESS experimental station. Thetotal running time of the 95Sr experiment was approximately 2.5 days, with an average 95Srbeam intensity of approximately 106 s−1. This sustained high beam rate caused a hole to be302.3. Target CompositionFigure 2.3: Beam identification plot of energy loss versus total energy as measured in theT-Bragg spectrometer. The beam was 98.5(5)% 95Sr. See section 2.4.1 for more details.burned through the CD2 target, at which point the experiment was ended. The burned-outtarget can be seen in figure 2.7. In both experiments, there was substantial fluctuation of thebeam intensity. In the 2013 94Sr experiment, there was also some fluctuation in the beamcomposition throughout the beam time, although the main constituent of the beam remained94Sr. The Table 2.1 gives a summary of the beam delivery for the 94,95Sr experiments.Beam Average Rate [s−1] Total Duration [days] Purity [%]94Sr ∼3x104 3 ∼8095Sr ∼1x106 2.5 ∼98Table 2.1: Summary of 94,95Sr beam delivery2.3 Target CompositionThe deuterated polyethylene (CD2) target was manufactured with a specified nominal thick-ness of 5.0µm. This thickness was chosen as a satisfactory trade-off between total reaction crosssection and energy broadening due to energy loss effects, both of which increase with the targetthickness. The presence of elastic proton scattering in the data indicated that the target was312.3. Target Compositionnot fully deuterated, and so the ratio of hydrogen to deuterium in the target was determinedusing the (d,d) and (p,p) elastic scattering data. It was found that the target deuteration inthe 2014 95Sr experiment was 92(1)%, which is discussed further in section 5.1.It was possible to determine the thickness of the target, ∆x, in a consistent way using theknown deuteration factor by measuring the energy loss of α-particles using a simple experi-mental setup and comparing this to stopping power calculations that were carried out usingTRIM [ZZB10] in order to find a corresponding thickness. A triple-α ( 239Pu 241Am 244Cm)calibration source was positioned behind the target in a zero-degree configuration. The energiesand intensities of the α-particles emitted from the source are summarized in table 4.1. Energy [keV]4000 4500 5000 5500 60000100200300400500No TargetWith Target Energy Spectrumα Energy [keV]α Source 5100 5200 5300 5400 5500 5600 5700 5800 Energy Shift [keV]α Measured 375380385390395400405410415  / ndf 2χ  1.811 / 2Prob   0.4044Thickness  0.02067±  4.39  Energy After Targetα(a) Energy [keV]4000 4500 5000 5500 60000100200300400500No TargetWith Target Energy Spectrumα Energy [keV]α Source 5100 5200 5300 5400 5500 5600 5700 5800 Energy Shift [keV]α Measured 375380385390395400405410415  / ndf 2χ  1.811 / 2Prob   0.4044Thickness  0. 2067±  4.39  Energy After Targetα(b)Figure 2.4: (a) Measured α-particle energy spectrum, with and without the target foil present.(b) Target thickness fit.Figure 2.4a shows measured α-particle energies, both with and without the target foil. Figure2.4b shows the fit of the energy loss measurements to TRIM data, including the measured ratioof 2H to 1H. The nominal thickness was used as the free parameter to be minimized in the fit.An additional estimated uncertainty of 10% was also added to account for theoretical uncer-322.4. Detector Systemstainty of the ion stopping power. Best agreement between the measurements and theory wasachieved for ∆x = 4.4(4)µm. In the 94Sr experiment, the target thickness was not measuredafter the experiment, and so was taken to be 5.0µm.2.4 Detector SystemsThe various detectors used for the experiments covered by this thesis are described in thefollowing sections together with their basic operational principles.2.4.1 TBraggThe energy loss of charged particles in matter is well described by the Bethe-Bloch formula,details of which can be found in [KAW77]. The features of the Bethe-Bloch formula which arerelevant for this work are summarized in equation 2.1.dEdx∝ −mz2E(2.1)The energy loss of an ion in a given medium as a function of distance (or stopping power)depends on its kinetic energy E, mass m, and atomic number z.The TBragg detector [Nob13] is a gas-filled ionization chamber used for identifying heavy ionswhich are present within the radioactive cocktail beam. As the beam ions enter the chamberthey lose energy in the gas and produce many free electrons, which are then collected at ananode at the end of the chamber under the action of an applied longitudinal electric field. Thegas pressure controls the penetration depth of the beam into the chamber, and must be chosenso that the beam is completely stopped within the chamber so a full energy measurement canbe made. The strength of the electric field determines how quickly and efficiently the electronscan be collected. A pulse shape analysis of the electrical signal is used to extract informationabout the incident ion: the slope of the signal is related to the stopping power of the ion, dEdx ,and the total signal amplitude is proportional to the total ion energy, E.332.4. Detector SystemsAnodeFigure 2.5: Schematic diagram of a Bragg ionization chamber, adapted from [Nob13]. Ionsenter the chamber and lose energy (example ion paths are drawn), creating free electrons (alsoindicated). The electrons are then drifted to an anode using an applied electric field.2.4.2 SHARCThe Silicon Highly-segmented Array for Reactions and Coulex (SHARC) is a multi-purposecompact array of double sided silicon detectors (DSSDs) which is designed to have almost 4piacceptance in its full instrumentation [DFS+11]. It is optimized for angular distribution mea-surements due to its excellent angular resolution. Figure 2.6a shows a schematic diagram ofSHARC as it was used in this experiment.The array was designed and built by the University of York and the University of Surrey,and the silicon detectors are all manufactured by Micron. Each detector unit of the array ismounted on a separate PBC backing which is then attached to an aluminium frame. The mod-ular design of SHARC makes it possible to configure different detector arrangements so thatspecific experimental needs can be satisfied.There are several distinct detector types within the array; DSSD ∆E detectors, quadrant ∆Edetectors and pad (E) detectors. The arrangement of these detectors within the array is shownschematically in figure 2.6a.• The BB11 DSSDs (Double-Sided Strip Detectors) are rectangular detectors with dimen-sions of 72mm x 48mm. The front face is divided into 24 strips which run parallel tothe longer dimension with 3mm pitch and the back face is divided into 48 strips which342.4. Detector Systemsrun parallel to the shorter dimension with 1mm pitch. The DSSDs are configured so thatthe side with highest segmentation runs perpendicular to the beam axis, giving the bestpossible resolution in θ. The largest θ range subtended by any strip was for BS0, with∆θBS0 ∼ 1.3 ◦. Four DSSDs are then combined into φ-like rings to form open-endedboxes, also known as barrel detectors. The compact arrangement of the DSSDs withinbarrel detectors is such that they are mounted approximately 42mm from the beam axis.The target is placed between two barrel detectors, which have a small separation at 90◦.In this work, the upstream and downstream DSSD barrel sections are referred to as UBOXand DBOX respectively.• The QQQ2 quadrant detectors (or QQQ for short) are also double sided. The front face isdivided into 16 annular strips which span from 9-41 mm and the back face is divided into24 annular strips which span a φ angular range of 81.6◦. The largest θ range subtendedby any strip was for FS15, where ∆θFS15 ∼ 1.7 ◦. The quadrant detectors are mountedin groups of four in SHARC to produce CD detectors which cover almost all φ anglesand act as end-caps on both the upstream and downstream sections of SHARC. Thenaming conventions for the upstream and downstream end-caps are UQQQ and DQQQrespectively.• The MSX-35 pad detectors are thick silicon detectors which are not segmented. Theyare mounted immediately behind the wafer DSSDs and so the position of the measuredparticles can be taken from the DSSDs. The pad detectors are designed to enable fullparticle energy measurements at forward angles where the kinetic energy is largest. Theuse of ∆E-E detector arrangements also allows for particle identification (PID).The specifications of each SHARC detector are summarized in table 2.2 and the naming con-ventions that will be used in this thesis are also introduced.This configuration was chosen to optimize sensitivity to small center-of-mass angles for (d,p)and maximize overall geometrical coverage, while minimizing the risk of damage to the de-tectors. The downstream QQQ detectors were not instrumented as this was at risk of hardradiation damage due to high energy carbon nuclei elastically scattering from the deuterated352.4. Detector Systems(a) (b)Figure 2.6: (a) Schematic diagram SHARC detector, indicating main sections and target po-sition with respect to beam. (b) A photograph of SHARC detector being installed [DFS+11].The micro-pitch ribbon cables and PCB feedthroughs are also visible.polyethylene target at small laboratory angles. A thick (∼ 1 mm) aluminium foil was placed infront of the downstream CD section (DQQQ) to protect the detectors. It was later found duringthe analysis that a large amount of beam-like nuclei were implanted into the aluminium foil as aresult of elastic scattering off carbon in the target, which led to a large amount of β-decay dataalso being taken during the experiment. This is discussed further in appendix D. There wasapproximately 90% φ coverage over the following theta ranges; 35◦-80◦ in the downstream box(DBOX), 95◦-142◦ in the upstream box (UBOX) and 148◦-172◦ in the upstream CD (UQQQ)section.The SHARC detector signals were read out with micro-pitch ribbon cables which are con-nected to PCB feedthroughs. These can be seen in figure 2.6b, which is a photograph takenduring installation into the target chamber. Each channel was connected to fixed-gain pre-amplifier boards. The gain was set to limit the maximum energy to be 25 MeV. This is wellabove the kinematic energy of the transfer protons, although carbon elastic scattering wouldcause saturation.A custom pre-amplifier rack was built for SHARC in this experiment which was hung fromthe overhead TIGRESS cable track. This can be seen in figure 2.9. It was built to allow the362.4. Detector SystemsName Type Dimension [mm] Segmentation Thickness [µm] Deadlayer [µm]DBOX 5 BB11 72 x 48 24(J) x 48(O) 141 0.1DBOX 6 BB11 72 x 48 24(J) x 48(O) 142 0.1DBOX 7 BB11 72 x 48 24(J) x 48(O) 133 0.1DBOX 8 BB11 72 x 48 24(J) x 48(O) 143 0.1PAD 5 MSX-35 72 x 48 1(J) x 1(O) 1534 1.0PAD 6 MSX-35 72 x 48 1(J) x 1(O) 1535 1.0PAD 7 MSX-35 72 x 48 1(J) x 1(O) 1535 1.0PAD 8 MSX-35 72 x 48 1(J) x 1(O) 1539 1.0UBOX 9 BB11 72 x 48 24(J) x 48(O) 999 0.1UBOX 10 BB11 72 x 48 24(J) x 48(O) 1001 0.1UBOX 11 BB11 72 x 48 24(J) x 48(O) 1001 0.1UBOX 12 BB11 72 x 48 24(J) x 48(O) 1002 0.1UQQQ 13 QQQ2 9-41 x 81.6◦ 16(J) x 24(O) 390 0.7UQQQ 14 QQQ2 9-41 x 81.6◦ 16(J) x 24(O) 390 0.7UQQQ 15 QQQ2 9-41 x 81.6◦ 16(J) x 24(O) 383 0.7UQQQ 16 QQQ2 9-41 x 81.6◦ 16(J) x 24(O) 385 0.7Table 2.2: SHARC detector summary table. The array is arranged into several sec-tions; DBOX+PAD, UBOX and UQQQ. D(U) prefixes in detector names refer to down-stream(upstream) components.pre-amplifiers be positioned as close as possible to SHARC, and to minimize the cable lengths.This is beneficial as longer cables have higher capacitance than short cables.Several targets and a 2mm collimator were mounted on a target wheel to facilitate onlinebeam alignment and target changing. Figure 2.7 shows the target holder in SHARC at the endof the 2014 95Sr experiment. Heavy damage is clearly visible on the 0.5 mg/cm2 CD2 primarytarget.2.4.3 TIGRESSThe TRIUMF-ISAC Gamma-Ray Escape Suppressed Spectrometer (TIGRESS) is an array ofHyper-Pure Germanium (HPGe) clover detectors [SPH+05]. When it is fully implementedthere are 16 clover detectors in the array. However, when it used in conjunction with SHARCa maximum of 12 detectors can be used. This is due to space being taken up by the SHARCpre-amplifiers. TIGRESS is purpose-built for use in reaction studies where photons are emitted372.4. Detector Systems7/1/2014 target.jpeg (734×979)https://elog.triumf.ca/Tigress/TIGRESS/140624_141429/target.jpeg 1/1Figure 2.7: The target wheel photographed at the end of the experiment. From left to right:5.0 mg/cm2 CD2, 2mm collimator, 0.5 mg/cm2 CD2 primary target. The primary target wasburned through after approximately 2.5 days due to the high beam rate delivered.from moving nuclei and so an important design feature of the array is good angular resolution.Each TIGRESS clover detector is made up of four closed-ended n-type coaxial HPGe crystals.The individual crystals contain an electrical core contact and an eight-fold electrical segmenta-tion; four quadrants and a lateral divide. This produces an overall 32-fold segmentation withineach clover. This provides very good sensitivity to the emission angle of the γ-ray, which meansthat precise Doppler reconstruction is possible. The segmented design also makes it possibleto improve the quality of the data taken in TIGRESS by using addback algorithms, as will bedescribed in section 4.2.2. Figure 2.8b depicts the segmentation within a single clover detector.The clover detectors are arranged into constant θ rings surrounding the target chamber, withtheir centres at 45◦, 90◦ and 135◦. There is close to full φ coverage within each ring. As waspreviously mentioned, TIGRESS was restricted to 12 of the 16 HPGe detectors, and so the 45◦ring (with respect to the beam dump) was not implemented in this experiment. Each cloveris surrounded by a set of four Bismuth Germanate (BGO) Compton suppressor shields, whichcan be used as a veto for γ-rays Compton scattering in the detector and escaping from the Gecrystal [SPH+05]. Figure 2.9 shows a photograph of the TIGRESS array as it was used in this382.4. Detector Systems(a) (b)Figure 2.8: (a) CAD cutaway drawing of TIGRESS surrounding the SHARC detector. (b)TIGRESS clover detector with indication of crystal segmentation [SPH+05].work. The 90◦ ring of clovers can be seen alongside one of the four retracted BGO shields whichaccompanies each detector. The SHARC pre-amp rack is also visible, as is the beam line andtarget chamber.There are two operational modes for the array; optimized peak-to-total and high-efficiency. Inoptimized peak-to-total mode, the clovers are pulled back and the BGO shields are broughtforward so that they are flush with the front face of the clover detectors at a distance of 14cmfrom the reaction target. This increases the peak-to-total efficiency of the array as partial en-ergy measurements are detected by the shields and can then be removed from the data.In high-efficiency mode, all of the clover detectors are brought forward so that their front facesform a continuous detector surface (a rhombicuboctohedron) surrounding the target chamberat a distance of 11cm from the reaction target. The Compton suppressor shields must be with-drawn in high-efficiency mode, and so suppression cannot be fully implemented. The advantageof this operational mode is that it maximizes the geometrical efficiency of the array and there-fore maximizes the total statistics. The 22.5◦ tapering at the front of each clover detector allowsthe front segments of adjacent clovers to pack together tightly. The clover design makes it pos-sible to add together the energies of Compton scattered γ-rays between neighbouring crystals,392.4. Detector Systems7/1/2014 IMG_20140622_113226.jpg (2448×3264)https://elog.triumf.ca/Tigress/TIGRESS/140627_123158/IMG_20140622_113226.jpg 1/1Figure 2.9: Detector arrangement used for this experiment. TIGRESS can be seen in highefficiency mode surrounding the target chamber, which contains SHARC. The mounted pre-amp rack and cabling setup, suspended above the beam line adjacent to TIGRESS, is alsovisible in this photograph.which increases the photo-peak efficiency of the array. This will be described in more detail insection 4.2.2. Figure 2.8a shows a cut-away CAD drawing of TIGRESS set-up in high efficiencymode. The shields, which would be pulled back in this configuration, are not included in thediagram.Each TIGRESS clover contains an in-built pre-amplifier and high voltage supply for everysegment, and also one for the core contact. These are all connected to a shared cryostat whichis maintained at 77K using a liquid nitrogen LN2 reservoir.2.4.4 TrifoilThe trifoil scintillator is an auxiliary detector system which was operates as a zero-degree scin-tillator [W+12]. It is comprised of a 10 µm BC400 foil held within a plexiglass frame andconnected to a set of three photomultipliers. The very fast scintillation light signals from the402.4. Detector SystemsBC400 foil allows for very high counting rates, enabling this detector to be used as a RIB beamcounter. Figure 2.10a shows a schematic diagram of the trifoil as it was used in this experimentand figure 2.10b is a photograph of the trifoil detector before the experiment. This detectorwas set-up downstream of the target chamber, as can be seen in figure 2.1.(a) (b)Figure 2.10: (a) Schematic diagram of trifoil degrader-scintillator arrangement. (b) Trifoildetector, photographed during setup.The mass-95 radioactive beam can undergo fusion within the target and then quickly evaporateseveral light particles into SHARC. These particles can be indistinguishable from the transferprotons of interest and so this can make the analysis very difficult. It is possible to suppressthese fusion reactions using the trifoil by mounting an aluminium degrader foil in front of thetrifoil which stops the heavy fusion recoils but not the light 95Sr and 96Sr nuclei. The thicknessof aluminium degrader foil was selected to be 40 µm, and this was mounted on a rotating flangeso that the effective thickness could be adjusted to be up to 56.6 µm by adjusting the angle ofthe foil up to 45◦.The trifoil was installed approximately 30% into the beam time during both experiments. Dur-ing the 95Sr experiment, the BC400 foil sustained heavy radiation damage due to the intensebeam, gradually lowering the light collection efficiency around the beam spot. This made itdifficult to determine the efficiency of the trifoil, which is essential for a quantitative analysis.412.4. Detector SystemsFor this reason the trifoil was used as a qualitative tool, primarily to check if γ-ray peaks werecoincident with beam-like nuclei or fusion events. Figure 2.11a shows the heavily damagedtrifoil caused by the 2014 RIB.7/1/2014trifoil_24-06-14.jpg (1536×2048)https://elog.triumf.ca/Tigress/TIGRESS/140624_134521/trifoil_24-06-14.jpg1/1(a) (b)Figure 2.11: (a) Damage caused to the BC400 foil during the experiment by the high beamintensity made it difficult to use the trifoil in analyzing the data. (b) Count rate of the trifoilthroughout the experiment.The trifoil operates using a logical discriminator which requires two of the three photomultipliersto trigger in order for the signal to be read out. This was implemented using a NIM coincidencemodule, which had a 6.5 ns resolving time. The fast NIM output from the coincidence unit wassent directly to the TIG-10 input in the data acquisition system.2.4.5 Data Acquisition SystemThe pre-amplified signals from SHARC, TIGRESS and the trifoil were processed in the dataacquisition system (DAQ) before the data could be written to disk. TIGRESS and SHARCuse a novel custom DAQ that was built in-house at TRIUMF to serve the specific requirementsof the highly-segmented arrays. The advantage to using such a system is that every outputsignal from the detector only needs a single channel on a single module to create an energy,time and trigger logic. By comparison, older systems, where traditional NIM electronics are422.4. Detector Systemsused, require a number of modules to perform the same task. The custom digital DAQ allowsfor the thousands of TIGRESS and SHARC channels to be practically readout. Nevertheless,the simpler wiring diagram comes at the cost of the complexity of the firmware running onthe individual digitizers and associated software on the DAQ computers needed to write thedata to disk. Firmware problems during the setup of the 2014 95Sr experiment created issueswith the SHARC triggering time signals, which prevented the use of SHARC timing gates inthe analysis. Timing gates are a very powerful tool for discriminating between correlated andrandom events, and are especially useful for determining whether charged particles and γ-raysare emitted simultaneously from a source. It was possible, however, to use the RF phase to dis-tinguish between subsequent beam bunches which was the basis of the DAQ triggering system.Figure 2.12 shows a diagram of the DAQ system logic. The front-end (FE) of the TIGRESSand SHARC DAQ architecture was made up of TIG-10 (10-channel) and TIG-64 (64-channel)front end modules, respectively. The TIG-10 modules were used to digitize a sample of eachFigure 2.12: Diagram of DAQ logic. Detector signals were sent to the TIG-10 front-end (FE)modules before digitization, logical discrimination and event assembly.pulse shape using a 10 ns timing window. The time stamp assigned to the measured pulsesin the TIG-10 modules was determined using a constant-fraction discriminator (CFD) circuit.Each TIG-10 output data stream was then sent to a slave port in the TIG-C collector module,432.4. Detector Systemswhich builds sub-events from the input streams. As was briefly mentioned earlier, it was notpossible to determine the time stamp of the SHARC signals due to firmware issues. Instead, theSHARC time stamps were bootstrapped to the RF timing signals. In the collector module, atriggering condition was used which required that all SHARC events had a simultaneous DSSDfront-back hit. Upon passing this condition, the good SHARC hit was read out and used as amaster trigger for the DAQ. All TIGRESS and trifoil signals that were measured within 1 µsof the silicon trigger were then taken to be associated with the same event, and so were alsowritten to disk.44Chapter 3SimulationsSimulations are an essential part of many physics experiments, especially in nuclear and particlephysics. They provide a valuable tool for understanding experimental data, devising new exper-iments and also designing detectors. For data analysis, simulations allow analysis techniques tobe developed and tested using user-specified input data. Geant4 is a powerful simulation soft-ware package developed by CERN and others [Ago03] which allows users to build sophisticatedmodels of detector systems and carry out monte-carlo simulations of experiments. Geant4 useshigh-quality descriptions of physics processes such as the interactions of particles with matterto generate realistic data.At the time of this work, a Geant4 implementation of the integrated SHARC and TIGRESS de-tector system was not available and so simulations including both particles and γ-ray were notpossible. Instead, separate simulations were carried out for the detection of charged particlesand γ-rays.3.1 SHARC SimulationsExperiments using heavy ion beams in inverse kinematics are limited to rather poor energy res-olution in the measured ejectile particles. The use of a thick target contributes to substantialenergy broadening due to considerable beam energy loss within the target. A limiting factorin the analysis of this experiment was therefore the energy resolution of measured particles inSHARC, which was studied using NPTool.NPTool, developed by Adrian Matta [Mat], is a Geant4-based framework for nuclear physicswhich contains a detailed implementation of SHARC. This can be seen in figures 3.1a and 3.1b.In NPTool, an experiment is modelled through two stages, simulation and analysis.453.1. SHARC SimulationsFirstly, a Geant4 simulation is carried out which models the various processes that take placein a reaction: Firstly, the beam loses energy in the target before a reaction takes place at somerandomly selected point. An input angular distribution is used to define the center-of-massangle of emission for the reaction products. The angular distribution is multiplied by sin θCMand then normalized to produce a probability density function (PDF). The reaction productsare boosted into the laboratory frame using Lorentz vectors and propagated until there is noremaining energy or the particles have reached the edge of the simulated volume. The lattersituation arises if particles pass through the detector but are not stopped entirely, or if theparticle misses the detector.(a) (b)Figure 3.1: (a) NPTool implementation of SHARC including DBOX, UBOX and UQQQ sec-tions. (b) Beam-line view, showing the CD2 target.The second stage of the modelling process is an analysis of the calculated detector hits so thatthe reaction can be studied in the same way as an experiment. In reality, the reactions takeplace approximately equally at all positions in the target (within the beam profile) althoughthis position information cannot be determined experimentally. Instead, each measured particlein SHARC must be assumed to have been created in a reaction at the center of the target. Thesimulated data was also analyzed in this way and so all energy losses of the particle were added463.1. SHARC Simulationsback to reconstruct the total energy of the particle, assuming it propagated from the center ofthe target along the beam axis.A series of reactions were simulated using a 4.5µm CD2 target and a 5.515 MeV/u95Sr beam,which are summarized in table 3.1. A Gaussian function with σ ∼0.5 mm was used to modelthe beam profile, which is a reasonable description of the beam profile delivered in this exper-iment. Simulations of the equivalent reactions involving a 94Sr were also carried out and hadvery similar results, however for the sake of brevity only the 95Sr results are presented.Several reactions were studied using NPTool. Elastic scattering was simulated assuming pureRutherford scattering angular distributions so that the energy losses and energy resolutioncould be investigated. Each of the elastic scattering reactions were simulated using 106 events.Realistic 95Sr(d,p) simulations were also carried out for a series of different states in 96Sr, wherethe excitation energy Eexc of these states is given in table 3.1. FRESCO [Tho88] was used togenerate the DWBA angular distributions and the number of simulated reactions was chosento reflect the relative total cross section for each reaction, assuming a spectroscopic factor ofunity.Reaction Eexc [keV] Angular Distribution Number of Reactions95Sr(p,p) 0 Pure Rutherford 1 x 10695Sr(d,d) 0 Pure Rutherford 1 x 10695Sr(12C,12C) 0 Pure Rutherford 1 x 10695Sr(d,p) 0 Flat 1 x 1060 `=0 DWBA 3.6 x 105815 `=2 DWBA 1 x 10695Sr(d,p) 1229 `=0 DWBA 4.9 x 1051465 `=0 DWBA 5.1 x 1051792 `=4 DWBA 3.4 x 105Table 3.1: Summary of NPTool simulations carried out for 95Sr reactions in SHARC.Elastic Scattering SimulationsEach of the elastic scattering simulations listed in table 3.1 was carried out and analyzed sepa-rately which ensured that every particle and reaction was identified correctly. This is an ideal473.1. SHARC Simulationscase. Under real experiment conditions, different reaction products can be measured at thesame position with the same energy energy which can lead to ambiguities in particle identifi-cation.Figure 3.2a shows the reconstructed kinematic energy of the elastically scattered particles.The increasing number of counts as θ approaches 90◦ is characteristic of Rutherford scatteringin inverse kinematics. It can be seen that the elastically scattered carbon has a much greaterspread of energies than the protons and deuterons. This is because the energy lost by a carbonion is much more than for a proton or deuteron, and so the approximation that every reactionoccurred in the center of the target induces a much bigger error. The lower statistics of carbondata is also a result of the large stopping power. For angles close to θLAB=90◦, the differentialcross section is maximum but the amount of target material that the ion travels through is alsolarger and so most of the carbon is stopped within the target.(a) (b)Figure 3.2: (a) Simulation results for 95Sr(p,p), 95Sr(d,d) and 95Sr(12C,12C) kinematics. (b)Simulated energy spectrum for a small angular range in SHARC (more details in the text).Figure 3.2b shows the raw energy spectrum gated on a small angular range. The elasticallyscattered protons and deuterons form well-defined peaks while the carbon data forms a verybroad distribution which is very difficult to fit. For this reason, the carbon data will not be483.1. SHARC Simulationsanalyzed in this thesis. It is also clear that the elastically scattered protons and deuterons havesimilar energies and so their peaks increasingly overlap as θ approaches 90◦. In this angularregion their low kinetic energy means that they do not punch through the ∆E detectors and sothey cannot easily be separated. It is necessary to fit them together in this region so that theangular distribution can be extracted. Moreover, inelastic scattering of deuterons will not beresolvable from elastically scattered protons in this region.Transfer Reaction SimulationsThe expected energy resolution and angular coverage for 95Sr(d,p) was investigated by simu-lating an ` = 0 ground state transfer reaction. Figure 3.3a shows the reconstructed kinematicenergy of the protons. The gap in the data at 80◦ < θLAB < 100◦ in figure 3.3a is due tothe spacing between the DBOX and UBOX sections in SHARC, which can be seen in figure3.1a. At small laboratory angles, the protons punch through the pad detector and so do notdeposit all of their energy, which can be seen through the departure of the data from the ex-pected kinematic line at θLAB <60◦. This reduces the effective angular coverage of SHARC,since a full energy measurement is required for a correct description of the reaction. Figure3.3b shows the number of measured protons as a function of lab angle which have a completeenergy measurement compared to those with incomplete energy measurements. The protonswith incomplete energy measurement were selected using a graphical cut in figure 3.3a.Figure 3.4a shows the reconstructed excitation energy of the simulated reactions as a functionof center-of-mass angle. The gap in the data at 45◦ < θCM < 65◦ in figure 3.4a is again dueto the spacing between the DBOX and UBOX sections in SHARC. The intensity pattern iscaused by the probability density function combined with the geometrical detector efficiency(including the previously mentioned pad punch-through). It can be seen that the distributionof excitation energies is centered close to the appropriate energy (0 keV) with a small negativeoffset, but has a significant width. This width is different for each section of SHARC, andincreases with θCM .Figure 3.4b shows a projection of the data for the individual sections of SHARC. The pro-493.1. SHARC Simulations(a) (b)Figure 3.3: (a) Simulation results for ` = 0 95Sr(d,p) kinematics. (b) Counts versus lab anglefor all data (black) and for |Eexc| > 500 keV (red). The latter data corresponds to incompleteenergy measurement.jections were fitted using a Gaussian function so that the width of the distribution could bedescribed quantitatively. The UQQQ section has the best energy resolution but the lowest totalcounts, even though the input angular distribution file has maximum cross section at θCM=0◦.This is because angles close to 0◦ and 180◦ have a smaller weight in the probability densityfunction due to the sin θCM factor, as mentioned earlier in this section. The UBOX section hasa slightly larger distribution width and contains most of the data, while the DBOX section hasmuch worse energy resolution and a centroid offset. The combined energy resolution of eachof the SHARC sections was found σ ∼140 keV for an ` = 0 angular distribution, however thisvalue depends on the amount of data in each of the detector sections. If the input angulardistribution instead has a maximum cross section at a large angle (or equivalently, describes alarger angular momentum transfer) there will be more weighting towards larger center-of-massangles, and so the total resolution would reflect this.The simulations were used to demonstrate that the angular momentum transfer and total crosssection (or equivalently the spectroscopic factor) could be extracted from the SHARC measure-ments. The measured counts were plotted against center-of-mass angle for three states; 0 keV503.1. SHARC Simulations(a) (b)Figure 3.4: (a) Simulation results for (d,p) analysis. (b) Excitation energy resolution forSHARC.(` = 0), 815 keV (` = 2), 1792 keV (` = 4). An excitation energy window of 1000 keV was usedto extract the counts from each state, which account for the energy resolution. The measuredcounts at each angular position is the scaled probability density function (PDF) multiplied bythe efficiency of SHARC at that angle. The scaled PDF is the PDF multiplied by the totalnumber of simulated reactions. The same SHARC efficiency curve was used in the simulationsand the experimental analysis, which is described in section 4.1.3.Figure 3.5a shows the angular distribution analysis, where the data points represent the mea-sured counts and the error bands represent the expected distributions. Overall, the results werein good agreement with the expected measurements. Small deviations between the expectedand measured counts were primarily due to a slight discrepancy in the description of the geom-etry and position of SHARC in the NPTool model versus the real detector. Protons punchingthrough the DBOX pad detectors can be seen as the sudden drop in measured counts at largecenter-of-mass angles. Figure 3.5a represents a situation where the measured states can beidentified and analyzed separately. This is possible to do using p−γ coincidence measurementsas long as the states can be identified using unique γ-ray transitions. It is also useful to considersituations where the states cannot be selected using γ-rays, such as when the statistics are verylow or when states that are close in energy decay via the same γ-ray. The latter is true in the513.2. TIGRESS Simulationscase of the excited 96Sr 0+2,3 states, as both states are coincident with a 414 keV γ-ray.(a) (b)Figure 3.5: (a) Measured counts versus center-of-mass for the simulated 0 keV (black), 815 keV(red) and 1792 keV (blue) 96Sr states, compared to efficiency-corrected input. (b) Excitationenergy resolution of SHARC for 95Sr(d,p) reactions in inverse kinematics.Figure 3.5b shows the projected excitation energy spectrum for the 0+1,2,3 states96Sr. It wasfound that the 1229 keV state and 1465 keV state could be fitted together in a reliable wayonly by constraining the width of the Gaussian functions to be σ ∼135 keV. The ground stateis also featured in figure 3.5b to show that the choice of σ was appropriate to describe theindividual peaks. Figure 3.5b shows that analyzing states which are less than approximately250 keV apart in excitation energy (that cannot be separated using γ-rays) will be challenging,even with a large amount of statistics. This is because of the limiting resolution of SHARC inthis experiment. A γ-ray analysis of the 650 keV 0+3 → 2+1 transition would therefore be veryuseful in identifying the 0+3 state.3.2 TIGRESS SimulationsSimulations of γ-ray decays within TIGRESS were also carried out. In this way, the depen-dency of the Doppler correction and photo-peak resolution on the life-time of a given statecould be studied. The Doppler reconstruction in both the simulation and experiment employed523.2. TIGRESS Simulationsthe approximation that the nucleus decayed at the reaction point, which is discussed further insection 4.2.4. For isomers or states with long lifetimes, this approximation becomes invalid asthe fast moving recoil nucleus may decay far from this point. The isomeric 0+3 state in96Sr hasa half-life of 6.7(10) ns [Lab], and given the kinematics of the reaction it would travel approx-imately 20 cm in that time. Clearly, the Doppler reconstruction for this state would performpoorly. The 0+2 state in96Sr has a half-life of 115 ps, which would mostly decay within 1 cm.In addition, most of the isomer decays would happen outside of the TIGRESS array, causingthe geometrical efficiency to be substantially smaller. By using a simulation it was possible toinvestigate these effects so that the sensitivity of TIGRESS to the γ-ray decay of the 1465 keV0+3 state in96Sr could be better understood.(a) (b)Figure 3.6: (a) Geant4 model that was used to study decays in TIGRESS. (b) Beam-line view,showing the CD2 target.A fully implemented model of TIGRESS was not available at the time of this work. Instead,simulations of γ-ray detection were carried out using a model of GRIFFIN, which is geomet-rically equivalent but does not have segmentation within the clover crystals [SG14]. For thisreason, crystal positions were used in Doppler reconstruction of the simulated decays. This led533.2. TIGRESS SimulationsState Energy [keV] γ-ray Energy [keV] t 12[ps] Photo-peak Counts Efficiency [%]815 815 4.8 1.253 x 105 12.531229 414 5† 1.917 x 105 19.171465 650 5† 1.448 x 105 14.481506 692 6.2 1.372 x 105 13.721995 1180 5† 1.019 x 105 10.19Table 3.2: Simulated states used for efficiency calibration. States marked with † were assigned5 ps half-lives (more details in the text).to a slight broadening of the simulated peaks compared to fully segmented crystals.Five low energy 96Sr state decays were simulated so that the photo-peak efficiency could bedetermined for the simulated array. Table 3.2 lists the γ-ray transitions that were used forthe photo-peak efficiency calibration. 106 events were simulated for each decay. Each statewas simulated using a short half-life so that all of the decays occurred within the target. Thestates marked with a † symbol were assigned artificially short half-lives so that the Dopplerreconstruction would perform optimally and the detector acceptance was not reduced due todecays outside the array. In this way, the Doppler reconstruction and efficiency of the array forlong-lived states could be benchmarked against the ideal case.The γ-ray spectra for these states are shown in figure 3.7a. The absolute photo-peak efficiencyof the array at each of the γ-ray energies are shown in the table, and the corresponding efficiencycurve is shown figure 3.7b.Simulating Long-Lived StatesThere are two decay scenarios that must be considered in order to measure the mixing strengthbetween the excited 0+ states in 96Sr. Firstly, the 1229 keV 0+2 state in96Sr can be populateddirectly (with strength S2) through95Sr(d,p), and then decay via the 815 keV 2+1 state to the96Sr ground state. Alternatively, the 1465 keV 0+3 state in96Sr can populated directly (withstrength S3) through95Sr(d,p), which would then go on to feed the 1229 keV state. Eventhough both scenarios have similar decay signatures, the latter case involves the decay of anisomeric state and so it would be measured with a different efficiency and photo-peak shape.543.2. TIGRESS SimulationsSimulated Efficiency Spectrum(a) (b)Figure 3.7: (a) Simulated γ-ray spectrum using the decays listed in table 3.2. The sum spectrumis drawn in black. (b) Absolute efficiency curve produced using the decays listed in table 3.2.The 235 keV 0+3 → 0+2 electromagnetic transition is of pure monopole character, and so cannottake place through the emission of a γ-ray due to angular momentum selection rules (which arediscussed in appendix E). The two decay scenarios can be seen in figure 3.8.The expected number of 0+3 → 2+1 650 keV γ-rays observed in TIGRESS, Nγ3 , can be writ-ten asNγ3 = ε3εgB3S3 (3.1)where S3 is the total direct population strength of the 1465 keV96Sr state, ε3 is the efficiencyof TIGRESS for a decay at the reaction point which emits a 650 keV γ-ray, B3 is the branchingratio of the 0+3 → 2+1 transition and εg is the lowered acceptance of TIGRESS for an isomerwhich has a half-life of 6.7 ns and is travelling at v = 0.1c. εg can be determined by simulatingan isomer decay and a prompt decay using a realistic model of the detector system and takingthe ratio of the values. εg is independent of energy, and simply reflects the decreased geometricalefficiency of the array as the fast moving recoil decays along the z axis.Similarly, the total number of 0+2 → 2+1 414 keV γ-rays observed in TIGRESS, Nγ2 , can be553.2. TIGRESS Simulations0 keV0+18152+11229 (115 ps)0+21465 (6.7 ns)0+3815100%414100%65063(16)%23533(8)%S3S2Figure 3.8: Decay scheme for 0+2,396Sr states, indicating transition energies (given in units ofkeV) and branching ratios (given as a percentage). The 0+3 → 0+2 and 0+3 → 2+1 branchingratios were taken from [Jun80].written asNγ2 = ε3(S2 + εgB2S3) (3.2)where S2 is the total direct population strength of the 1229 keV96Sr state and B2 = 1 − B3.The additional term in equation 3.2 explicitly includes feeding from the 0+3 state. Combiningequations 3.2 and 3.1 provides an expression which relates the population strengths of the 0+2,3states to the observed counts, Nγ2,3.S3S2=Rεg[B3(1 +R)−R] (3.3)where R = ε2Nγ3ε3Nγ2.The 0+2 state decay was simulated using its known half-life of 115 ps [Lab]. It was foundthat this lifetime caused a small shift in the photo-peak energy of approximately -1 keV due to563.2. TIGRESS Simulationsthe Doppler reconstruction error. There was also a decrease in photo-peak counts by less than1% compared to a decay that occurred within the target. The 0+2 state decays to the groundstate through the 2+1 state, and so this increased lifetime also affected the 2+1 decay peak in thesame way. These increased lifetime effects were considered to be negligibly small, and so theexperimental γ-ray data associated with the direct population of the 0+2 state did not need tobe analyzed differently to other short-live states.The decay of the 0+2,3 and 2+1 states were also simulated using a life-time of 6.7 ns. This repre-sents the feeding of states from the long-lived 0+3 state. In actuality, there would be an increasedlifetime for each of the lower states, but their lifetimes were very small compared to the 0+3and so they were considered negligible. 106 decays were simulated for the 0+2,3 and 2+1 states(which emit 414 keV, 650 keV and 815 keV γ-ray rays respectively). The simulated spectrumfor a 650 keV → 414 keV → 815 keV γ-ray cascade with 6.7 ns half-life is shown in figure3.9a. The simulated spectrum for a 414 keV → 815 keV γ-ray cascade with 115 ps half-life isalso shown in 3.9a for comparison. It can be seen that the total number of measured γ-rays inTIGRESS drops as a result of the long lived isomer state. The total counts in the 414 keV, 650keV and 815 keV γ-ray peaks were found to decrease by approximately 50% compared to table3.2. This suggests that the total geometrical coverage for these long-lived states dropped byapproximately 50%. The shape of the Doppler corrected γ-ray peak also becomes very skewedand is shifted to lower energy as a result of the poor Doppler correction.The clover detectors in TIGRESS are arranged into rings of constant θ, called coronas. Fur-thermore, the crystals within each clover allow each corona ring to be divided into two sub-ringsof constant θ, giving a total of four unique θ rings of TIGRESS crystals. The four rings arecentered at approximately θTIG = 80◦, 100◦, 124◦ and 144◦. The performance of the Dopplercorrection was found to be very dependent on the position of the corona ring. This is shownin figure 3.9b, where the black line is the total peak and the coloured lined are the Dopplerreconstructed peaks from individual θTIG rings. The θTIG ∼80◦ ring, drawn in red, shows thatthe Doppler correction performed poorly at this position. This is because the error on theangle of γ-ray emission is very large, and as a result the estimated Doppler shift was not wellknown. In addition, γ-rays that were emitted outside the array could also be detected through573.2. TIGRESS Simulations(a) (b)Figure 3.9: (a) Comparison of simulated spectra for states fed by isomeric 0+3 (black) to directdecay (grey). (b) Contributions to total photo-peak (black) from each ring of detectors (moredetails in the text).the side of the crystal as this provides a large additional detector surface area. The θTIG ∼100◦ring, drawn in blue, shows a slight improvement in resolution. This is mostly because γ-rayscould not enter the side of the crystals since this was obstructed by the θTIG ∼80◦ ring. TheθTIG ∼124◦ ring, drawn in green, shows further improvement in resolution and less asymme-try in the peak. The θTIG ∼144◦ ring, drawn in pink, shows that the Doppler reconstructionworked reasonably well for large TIGRESS angles. This is because the error in the angle ofγ-ray emission is smallest in this region.A matter of practical concern for the measurement of the 650 keV γ-rays is the position ofthe 815 keV γ-ray Compton edge, which is drawn in grey in 3.9b. Since the 2+1 state is a strongcollecting state for the excited 96Sr nucleus, it is to be expected that the 815 keV peak will havemore statistics than the 650 keV peak from the 0+3 state. It would therefore be very difficultto extract the counts in the 650 keV peak for TIGRESS detectors close to 90◦ as this wouldoverlap with the Compton edge of the 815 keV peak. The simulation shows that the 650 keVγ-ray from the 0+3 state would be most separated from the 815 keV γ-ray Compton edge atθTIG ∼144◦.583.2. TIGRESS SimulationsFigure 3.10a shows the simulated γ-ray spectrum for the θTIG ∼144◦ ring, using equal di-rect population strengths S2 = S3 = 106. Feeding effects are also included. A 62.5% branchingratio and a 37.5% branching ratio were used for the 0+3 → 2+1 and 0+3 → 0+2 transitions respec-tively, as given in figure 3.8. The blue spectrum in figure 3.10a corresponds to the decay ofthe 0+2 state, and so both an 414 keV and an 815 keV γ-ray are emitted. The red spectrum infigure 3.10a correspond to the decay of the 0+3 state, which emits a 650 keV γ-ray in additionto the 0+2 decay transitions. The black spectrum is the total, and so is what we would observein the experiment. [keV]γ E0 100 200 300 400 500 600 700 800 900 1000 Counts / keV10210310 States+-Rays From Excited 0γSr 96(a) (b)Figure 3.10: (a) Simulated γ-ray spectrum for the decay of the 96Sr 0+2 state (blue) and 0+3 state(red) for S2 = S3 = 106, using only TIGRESS crystals positioned at θ > 135◦. Total spectrumis drawn in black. (b) Simulated results for the ratio of counts in the 650 keV γ-ray peak to the414 keV peak, plotted as a function of the 0+3 → 2+1 branching ratio for all TIGRESS angles(black), θTIG > 120◦ (blue) and θTIG > 135◦ (red). The green band indicates the experimentalbranching ratio from [Lab] with its uncertainty.The ratio of counts in the 650 keV γ-ray peak compared to the 414 keV γ-ray peak is plottedas a function of the 0+3 → 2+1 branching ratio in figure 3.10b. The simulation results presentedin figure 3.10b include an efficiency correction to account for the lower efficiency of TIGRESSat 650 keV compared to 414 keV. The black data points in figure 3.10b represent the efficiency-corrected counts ratio for all of TIGRESS, and the blue and red data points represent the593.2. TIGRESS Simulationssame ratio using TIGRESS detectors positioned at θTIG ≥ 124◦ and θTIG ≥ 144◦, respectively.Given the measured branching ratio of 62.5±15.6% from [Jun80], the expected ratio of countsis 0.20(6) for θTIG ≥ 124◦ and 0.19(6) for θTIG ≥ 144◦. This means that an experimentallyobserved ratio of counts in the 650 keV γ-ray peak compared to the 414 keV γ-ray peak of0.19-0.20, after efficiency correction, would be consistent with equal direct population of theexcited 0+2,3 states.60Chapter 4Analysis MethodsIn this chapter, the analysis procedure will be outlined. Both data sets were analyzed withinthe same framework, and so the 2014 95Sr(d,p) analysis will be the primary focus. Importantdifferences between this and the 94Sr(d,p) analysis will be highlighted.The software package GRSISort [BBD+], developed by Peter Bender and others, was usedas the basis for this analysis. Details of the specific programs that were developed for this workcan be found in appendix C.4.1 SHARC CalibrationsBefore each of the experiments, calibration data was taken using a triple-α source.It was found that during the 2014 95Sr(d,p) experiment, the pre-amplifier gain values changedabruptly in the time between α-source calibrations and beam delivery. After this gain shifteach of the detector channels within the SHARC units (DBOX, UBOX and UQQQ) were stillwell gain-matched but no longer had the correct energy scaling. The underlying cause wasdetermined to be a floating common ground for the pre-amplifier rack. Because of this issue,the full energy calibration was carried out using the 95Sr elastic scattering data. The (d,d) and(p,p) elastic channel kinematic lines were used to fully calibrate all DBOX detectors, includ-ing the pads. These kinematic lines were also used to determine the nominal target positionwithin the array, which was fixed to a rotating mount. At angles greater than 90◦ there wasno available elastic scattering data, so the ground state of 95Sr(d,p)96Sr was used for calibration.For the 2013 94Sr experiment, the α-source data was sufficient to fully calibrate the ∆E detec-tors in each section of SHARC. The pad detectors were calibrated using in-beam data, in the614.1. SHARC Calibrationssame way as was done for the 2014 pad calibrations.Each step in the energy calibrations of SHARC is described in the following sections, whichfocuses on the 2014 data set.4.1.1 α Source Gain-MatchingEach SHARC detector (with the exception of the pads) was gain matched using a double-sided239Pu 241Am 244Cm α-source placed in the target position. The energies and intensities ofα-particles from the α-source are shown in table 4.1. The strongest α-decay branch for eachisotope was used (which are highlighted in bold) and a 3-point calibration was carried out oneach front strip in SHARC.The observed α-particle energy (Emeas) in SHARC is the source energy (E0) minus energydeposited in insensitive regions (δEi) such as the target and detector deadlayers. A generaldescription of this is given in equation 4.1, although this simplifies considerably for a calibrationsource.Emeas(θ, φ) = E0(θ, φ)−i∑δEi(θ, φ) (4.1)Where the angle-dependence of the energy loss terms δEi is a result of the changing effectivethickness of the detectors and their deadlayers with angle. For a BOX detector the effectivethickness d(θ, φ) isd(θ, φ) =d0sin θ cosφ′(4.2)where d0 is the perpendicular thickness (the values quotes in table 2.2) and φ′ is limited to therange −pi4 < φ′ < pi4 . Note that any detector can be rotated to express φ in this range withoutloss of generality.For a QQQ detector this is simplified tod(θ, φ) =d0cos θ(4.3)as there is no φ dependence for a cylindrically-symmetric geometry.624.1. SHARC CalibrationsIsotope α-particle energy [keV] Intensity [%]5105 11.5239Pu 5143 15.15155 73.45388 1.4241Am 5442 12.85486 85.25763 23.3244Cm 5805 76.7Table 4.1: Energies and intensities of 239Pu 241Am 244Cm α-source, taken from [Lab]. Strongα-branches are highlighted in bold text.In the case of the α-calibration source, E0 is independent of angle and the only unseen en-ergy loss is due to the deadlayer of the ∆E detector. The energy lost due to SHARC deadlayerswas determined using table 2.2 and TRIM stopping power tables [ZZB10] to predict the energylost in the nominal deadlayer thicknesses. For α-particles with approximately 5 MeV of energypropagating through silicon, δE ∼150 keV per 0.1µm. As the energy variation within a frontstrip due to the effective deadlayer thickness was very small, the calibration was done on a frontstrip basis instead of for each individual pixel. An effective deadlayer was calculated for everyfront strip, reflecting the deadlayer thickness in the centre of each strip.The calibrated channels can be seen in figure 4.1a. The energy axis represents the measuredenergy, including deadlayer energy loss. The discontinuity of energies at approximately channel310 is due to a different nominal deadlayer in the UQQQ compared to the BOX detectors.Some missing channels can also be seen, as can some bad channels. These were removed in thefull analysis, as will be discussed later in this chapter. Each charge spectrum was fitted withtwo gaussian curves for each isotope listed in table 4.1. Figure 4.1b shows a typical spectrumfit, indicating the extracted peaks for the strong α branches.It was found that the calibration coefficients from the α-source did not reproduce expectedkinematic energies in the 95Sr beam data. The pre-amplifier settings had shifted after the α-source was taken out. The reason for this was determined to be a floating common groundshared by the pre-amplifiers.634.1. SHARC Calibrations(a) (b)Figure 4.1: (a) Calibrated α-source energy vs. channel matrix. (b) Example α-source spectrumwith total fit (red) and extracted calibration peaks (blue).It was possible, however, to use the α-coefficients to gain match all of the front strips withinthe DBOX, UBOX and UQQQ detectors. The channels within each unit retained a good rel-ative calibration, which was investigated by plotting the energy spectrum of a single channeland comparing this to the sum energy spectrum of each channel within that unit. The energyresolution was the same in the case of many overlaid spectra, which indicated a good relativecalibration. It was therefore necessary to carry out further energy calibrations on each of theSHARC units.4.1.2 Full Energy CalibrationsDBOX CalibrationsAfter the α-source calibration coefficients were applied, the downstream box (DBOX) unit wasanalyzed as a single channel. It was then necessary to fully calibrate the DBOX ∆E detec-tors using 95Sr(p,p) and 95Sr(d,d) elastic channel kinematics data. 95Sr(12C,12C) data was alsotaken, but this was not analyzed because the energy resolution was very poor due to significantenergy losses. The 95Sr(12C,12C) data will be discussed further in section 4.3.Calibrations using reaction data are more challenging than calibrations using mono-energeticsources as there is a strong angular dependence of the ion energy with θ and φ and worse energyresolution due to the inverse kinematics of the reaction and the target thickness. The segmen-644.1. SHARC Calibrationstation and compact geometry of SHARC made it possible to produce calibration spectra usingvery small θ and φ opening angles so that the kinematic energy broadening could be mitigated.In order to calibrate the DBOX ∆E detectors it was necessary to have a full energy mea-surement of the elastically scattered particles. It was observed that the protons and deuteronspunched through the DBOX ∆E detectors at θ ∼62◦ and θ ∼67◦ respectively, and so data be-yond these cut-offs was not used. Additionally, the calibrations were carried out using only thecentral front strips 11 and 12. This was done in order to restrict the opening angle of the cali-bration region which improved the resolution of the spectra. A gain-matched charge spectrumwas produced for each back strip up to the point where the particles began punching throughthe ∆E detectors. The approximate angular range covered by each back strip was θ ∼1.3◦. Thespectra were fitted using two Landau functions convoluted with a Gaussian kernel. The Landaufunctions describe the characteristic distribution of ion energies due to energy loss effects andthe convoluted Gaussian simulates the detector response. An example spectrum fit is shownin figure 4.2b. The low energy background was caused by β-decay and also evaporated lightparticles from fusion events. This background was fitted using a linear function, which was agood approximation over this energy range. For the smallest back strips, the kinematic lines ofthe (p,p) and (d,d) and (12C,12C) reactions were closest together and so the peaks overlappedsignificantly. The carbon elastic scattering data was fitted as a background in these regions.The two peaks correspond to protons and deuterons, with deuterons having a broader energydistribution and higher peak energy. The centroids from the fitted proton and deuteron Landaudistributions were extracted. The energy of the elastic peaks ∆E was calculated using equation4.4∆E(θ, φ) = E0(θ)− δEtarget(θ)− δE∆(θ, φ) (4.4)where the kinematic energy is E0(θ) and the energy lost in the target and deadlayer of the ∆Edetector are δEtarget(θ, φ) and δE∆(θ, φ) respectively. The energy losses were calculated usingTRIM stopping power tables.The position of the reaction point within SHARC was also examined by comparing the numberof counts in each of the DBOX detectors, and it was found to be well-centered within the array.654.1. SHARC Calibrations(a) (b)Figure 4.2: (a) Full energy calibration of DBOX ∆E detectors using 95Sr(p,p) (blue) and95Sr(d,d) (red) data. (b) An example gain-matched charge spectrum showing 95Sr(p,p) and95Sr(d,d).Pad CalibrationsThe DBOX pad detectors were also calibrated using 95Sr(p,p) and 95Sr(d,d) data. This wasdone in a very similar way to the DSSD ∆E detectors.As the charged particles must pass through the target, the ∆E detector (including deadlayer)and pad detector deadlayer before they are measured in the pad detector, the explicit calibrationenergy is given as equation 4.5Epad(θ, φ) = E0(θ)− δEtarget(θ)− δE∆(θ, φ)−∆E(θ, φ)− δEpad(θ, φ) (4.5)where the kinematic energy is E0(θ), the energy lost in the target and deadlayer of the ∆Edetector are δEtarget(θ, φ) and δE∆(θ, φ) respectively, ∆E(θ, φ) is the energy deposited in the∆E detector and δEpad(θ, φ) is the energy lost in the pad detector deadlayer. The energy losseswere calculated using TRIM stopping power tables.The calibration was carried out using two separate approaches. In one approach, ∆E(θ, φ)was calculated using the tabulated detector thicknesses in table 2.2. In the other approach, the664.1. SHARC Calibrationscalibrated ∆E detectors were used to provide an experimentally measured value. Aside fromuncertainties due to TRIM, the two methods were expected to produce very similar results.A discrepancy, especially one which varied across the four pad detectors would indicate thatthe detector thicknesses or deadlayers were not consistent with the quoted values in table 2.2.Figure 4.3a shows the calibration result for PAD 7 using both approaches. The fully calculatedapproach, drawn using broken lines and open symbols, shows a slight discontinuity between the(p,p) data and (d,d) data. This signifies that the exact thicknesses of the DSSDs are not wellknown. Instead, the experimental value of the ∆E was used in the calibration, which is drawnusing solid lines and solid symbols. The quality of the pad calibration was very good using thisapproach.The installation of the trifoil scintillator resulted in a sudden unexpected shift of pre-amplifiersettings for pad 5, and so a piecewise calibration was also required so that this data was notlost. This shift is believed to be caused by the same floating ground issue as was noted betweenthe α-source calibration and 95Sr beam time. The piecewise calibration was carried out by fol-lowing the aforementioned calibration procedure twice; once using only data before the suddenshift and then a second time using only data after the shift.The measured energy Emeas(θ, φ) can be used to determine the kinematic energy E0(θ) byadding each δE(θ, φ) energy loss back in the appropriate way. Figure 4.3b shows the fullyreconstructed kinematic energy E0(θ) of the SHARC DBOX after all energy calibrations.UBOX and UQQQ CalibrationsAt backwards lab angles (θ > 90◦) it was not possible to calibrate SHARC using 95Sr(p,p) and95Sr(d,d) data. Instead, the clearly visible ground state kinematic line of 95Sr(d,p) was used.With the known target position offset of (0,0,0) mm, it was possible to produce a kinematicline using the α-calibrated charge. A series of α-calibrated charge spectra were produced byprojecting θ-slices. The well-defined ground state peak was fitted using a Gaussian function.Equation 4.4 was used to calculate the measured energy of the transfer protons.674.1. SHARC Calibrations(a) (b)Figure 4.3: (a) Energy calibration of a pad detector using 95Sr(p,p) (blue) and 95Sr(d,d) (red)data. Solid lines and closed symbols indicate a calibration performed using experimental ∆Evalues, while broken lines and open symbols show the calibration result using only calculated∆E values (see text for more details). (b) Calibrated DBOX showing kinematics curves for95Sr(p,p) and 95Sr(d,d) compared to theory curves. Details of the cuts used are given in section4.3.The calibration fit for the UBOX detectors is shown in figure 4.4a and the calibrated data isshown compared to the ground state kinematic line of 95Sr(d,p) in figure 4.4b.(a) (b)Figure 4.4: (a) Calibration fit for UBOX detector using 95Sr(d,p) data. (b) Calibrated UBOXshowing kinematics curve of 95Sr(d,p) ground state transfer compared to theory curve.684.1. SHARC CalibrationsThe calibration fit for the UQQ detectors is shown in figure 4.5a and the calibrated data isshown compared to the ground state kinematic line of 95Sr(d,p) in figure 4.5b.(a) (b)Figure 4.5: (a) Calibration fit for UQQQ detector using 95Sr(d,p) data. (b) Calibrated UQQQshowing kinematics curve of 95Sr(d,p) ground state transfer compared to theory curve.4.1.3 SHARC EfficiencyThe geometrical efficiency of SHARC was determined using a Monte-Carlo simulation. Alarge number of detector hits were simulated using a uniform distribution across every pixel inSHARC, and the solid angle of the pixels were used as weight factors. The randomized positionsand pixel solid angles were combined into a weighted histogram. The weighted histogram wasthen divided by the number of particles simulated within each pixel in order to produce theefficiency curve, which is shown in figure 4.6a. Details on how the solid angle of each pixel wascalculated is given in appendix B.The geometrical coverage and efficiency of SHARC are shown in figures 4.6a and 4.6b re-spectively. The total geometrical efficiency of SHARC is calculated to be approximately 90%within the DBOX, UBOX and UQQQ sections. There is a slightly larger geometrical efficiencyin the DBOX section because the DSSDs must be mounted closer to the beam-line so that thepad detector can also be instrumented.694.1. SHARC Calibrations107 of the 736 instrumented pre-amplifier channels (14.5%) were excluded from the 2014 95Sranalysis, either because they were absent or because they were performing inconsistently withother channels. These strips are listed in table B.2. The geometrical efficiency of SHARC withthese strips removed is also shown in figure 4.6a. Similarly, 204 channels (27.7%) were excludedfrom the 2013 94Sr analysis, and these strips are listed in table B.1.(a) (b)Figure 4.6: (a) SHARC solid angle coverage in lab frame with all strips included (blue), stripsfrom table B.2 removed (black) and maximum coverage 2pi sin θdθ (red), (b) SHARC geometricalefficiency in lab frame, with colours indicating the same as before.For a reaction such as 95Sr(d,p) the intensity distribution becomes more complicated. Even ifthe protons are emitted isotropically in the center-of-mass (CM) frame there is an additionalkinematic boost which skews the intensity distribution in the lab frame, as was seen in theNPTool simulation results in chapter 3.1. There is also a non-linear conversion between CMframe angles and lab frame angles which also creates a non-uniform intensity distribution. TheCM frame geometrical coverage was calculated by converting the lab frame geometrical cov-erage simulation pixel by pixel. The resulting geometrical coverage and efficiency curves for95Sr(d,p) are shown in figures 4.7a and 4.7b respectively.704.2. TIGRESS Calibrations(a) (b)Figure 4.7: (a) SHARC solid angle coverage in centre-of-mass frame with all strips included(blue), strips from table B.2 removed (black) and maximum coverage (red), (b) SHARC geo-metrical efficiency in centre-of-mass frame, with colours indicating the same as before.4.2 TIGRESS Calibrations4.2.1 152Eu CalibrationA 152Eu γ-ray source was used to calibrate energies and efficiencies of the TIGRESS array. Theenergies and intensities of the strongest transitions are given in table 4.2, and the peaks usedin this calibration are highlighted in boldface.A charge spectrum was produced for each clover crystal in TIGRESS and the γ-ray peaks werefitted. The centroid of each peak was extracted and a linear calibration fit was carried out.Figure 4.8a shows the calibrated energy for each crystal in TIGRESS and figure 4.8b shows thesum spectrum of all the crystals in TIGRESS. The continuous horizontal lines in the energymatrix indicate that each channel was well calibrated. This was further tested by comparingthe resolution of the sum spectrum to that of an individual crystal. It was found that the sumspectrum peaks had the same width as those in a single spectrum, further confirming the goodcalibration quality.714.2. TIGRESS Calibrationsγ-ray energy [keV] Relative Intensity [%]121.8 141 (4)244.7 36.6 (11)344.3 127.2 (13)367.8 4.19 (4)411.1 10.71 (11)444.0 15.00 (15)488.7 1.984 (23)586.3 2.24 (5)678.6 2.296 (28)688.7 4.12 (4)778.9 62.6 (6)867.4 20.54 (21)964.0 70.4 (7)1005.1 3.57 (7)1085.8 48.7 (5)1089.7 8.26 (9)1112.1 65.0 (7)1212.9 6.67 (7)1299.1 7.76 (8)1408.0 100.0 (1)Table 4.2: Energies and intensities of 152Eu γ-ray calibration source [Lab]. Intensities aregiven relative to the 1408 keV γ-ray transition. The γ-rays highlighted in bold were used forcalibrations.The 152Eu data was also used to determine the relative photo-peak efficiency of TIGRESS.The sum spectrum from figure 4.8b was used to produce a relative efficiency curve by dividingthe counts within each fitted peak by the respective intensity listed in table 4.2. The efficiencydata was fitted using a standard formula [Rad] which is given in equation 4.6.log10(ε(Eγ)) = A+B log10(Eγ) + C log10(Eγ)2 +DE2(4.6)where the free parameters A, B, C and D are fitted to the data. The fit is shown as the redcurve in figure 4.9a. The slope of the curve contains an important physics result: the decreasingphoto-peak efficiency with energy is due to the decreasing cross section of γ-rays depositing allof their energy in a single interaction (photo-electric effect).724.2. TIGRESS Calibrations(a) (b)Figure 4.8: (a) Plot showing the calibrated channels of TIGRESS using a 152Eu source. (b)Calibrated sum spectrum of all crystals using a 152Eu source.4.2.2 Add-BackA γ-ray will often scatter multiple times before being completely absorbed. This is measuredas a series of simultaneous energy signals distributed across different detector channels. Theinteractions are usually localized within a single crystal or clover detector, however adjacentclovers can also share full energy measurements. The photo-peak efficiency of TIGRESS cantherefore be improved by adding together the energies of these separate signals to reconstruct asingle event which contains the total energy. This technique is called add-back. The expectednumber of interactions increases with γ-ray energy, and so the improvement offered by add-backalso depends on the total γ-ray energy. Low energy photo-peaks at ∼ 500 keV or less will belargely unaffected as the dominant interaction process in this energy regime is the photo-electriceffect, which results in a single interaction. At higher energies the efficiency improvement avail-able through the use of add-back depends on the specific details of the algorithm used, and isultimately limited by the segmentation and the geometry of the detector system.The add-back algorithm that was used in this experiment used the following conditions todetermine whether to add together different hits: In cases where multiple segments detected asignal within a single clover crystal, the total energy of the hit was taken from the core contact.734.2. TIGRESS CalibrationsThe segment hits were then ordered from largest to smallest energy deposited and this wasassumed to be the order of interactions effectively describing the γ-ray track. In this approach,the angle of incidence of the γ-ray was determined using the position of the first (largest energy)hit and the final position was taken from the last (lowest energy) hit. A single add-back hitwas thus created with the total core energy and an initial and final position.If two or more crystals fired, the hits were ordered from largest to smallest core energy and thefirst hit in this sorted list was taken as the first add-back hit. The final interaction point in thefirst add-back hit was compared to the first interaction point of the other hits. If the interactionpoints were both in front or back segments, their separation was required to be less than 54mm. If the interaction points were in some front to back segment combination, their separationwas required to be less than 105 mm. This effectively forbids a γ-ray to travel across an activevolume of germanium without scattering. It was also required that the absolute time differencebetween the first add-back hit and any other hit was less than 200 ns. Upon satisfying all ofthese conditions, the two hits were combined into a single add-back hit. The two core contactenergies were summed and the timestamp of the hit was set to be that of the highest energyhit. The list of sorted segment hits were concatenated to give a new final γ-ray position. Hitsthat did not match the add-back criteria were left as individual hits.The add-back algorithm was benchmarked by producing a calibrated 152Eu sum spectrumand extracting a relative efficiency curve, as described in the previous section. The ratio ofefficiencies between the add-back curve and the no-add-back curve was used to determine theimprovement in photo-peak efficiency (also known as the add-back factor). It was found thatthe photo-peak efficiency of TIGRESS could be increased by over 40% by adding together hitswith the following conditions;The corresponding add-back curve is compared to the no-add-back efficiency curve in figure4.9a. The ratio of these curves is shown in figure 4.9b.4.2.3 Absolute Efficiency CalibrationA low activity 60Co source was placed in the TIGRESS array after the experiment for absoluteefficiency calibrations. 60Co β− decays into 60Ni with a half-life of 5.2 years and produces a very744.2. TIGRESS Calibrations(a) (b)Figure 4.9: (a) Relative 152Eu efficiency curves made with and without add-back. (b) Ratio ofefficiency curves, giving the add-back factor.γ-ray energy [keV] Intensity [%] N(γi) N(γi|γj)346.93 (7) 0.0076(5)826.28 (7) 0.0076(8)1173.237 (4) 99.9736 (7) 4.592(7) × 105 2.31(2) × 1041332.501 (5) 99.9856 (4) 4.267(7) × 105 2.32(2) × 1042158.77 (9) 0.0011 (2)Table 4.3: Energies and intensities of 60Co γ-ray calibration source [Lab].simple γ-ray spectrum with two coincident γ-rays of similar energy. These coincident γ-rayscan be used to extract the absolute efficiency of TIGRESS. Details of the 60Co decay schemeare given in figure 4.3.γ-ray coincidences are an effective way of determining the absolute efficiency. This is becausethey are independent of detector dead time and source activity.Consider a situation where two γ-rays, γ1 and γ2, with energies E1 and E2 are emitted in quicksuccession from a source. One can extract the absolute efficiency at energy E2 by comparingthe number of γ-rays in the coincidence spectrum photo-peak to the number of γ-rays in the754.2. TIGRESS Calibrationssingles spectrum photo-peak, as is described by equation 4.7.ε(E2) =N(γ2|γ1)N(γ1)×BR (4.7)Where N(γ2|γ1) is the total number of γ-rays at energy E2 given that a γ-ray was measuredwith energy E1, N(γ1) is the number of γ-rays with with energy E1 in the singles spectrumand BR is the branching ratio (which is effectively 100% for 60Co).When a γ-ray is measured in TIGRESS, the associated detector becomes briefly inactive whilethe signal is read out. This is called dead time. γ-rays in a cascade are generally emitted withinpicoseconds (discounting isomeric state decays) and so one of the TIGRESS detectors wouldbe inactive as the second γ-ray is detected. This reduces the effective number of detectors inthe array by one, as the two γ-rays must interact with different detectors or no coincidenceevent will be measured. The absolute efficiency εabs of TIGRESS is therefore the apparent effi-ciency (that which is observed) multiplied by a factor which accounts for the reduced numberof detectors.εabs = εappηη − 1 (4.8)Where η is the total number of detectors (12) and εapp is the efficiency that was determinedusing equation 4.7. The absolute efficiency that was extracted using the 60Co data was used toscale the 152Eu relative efficiency curve, producing an absolute efficiency curve. This absoluteefficiency curve is shown in figure 4.10.4.2.4 Doppler Correctionγ-rays emitted from the 96Sr recoil nuclei will be Doppler shifted due to the motion of theirsource. This effect depends on the angle of emission of the γ-ray with respect to the recoil, andalso the angle of the recoil with respect to TIGRESS. The measured energy E′ is related to theenergy of emission in the 96Sr rest frame, E, by equation 4.9.E = E′γ(1− β cos θ) (4.9)764.2. TIGRESS Calibrations [keV]γ E0 200 400 600 800 1000 1200 1400 Absolute Addback Efficiency [%]8101214161820TIGRESS Absolute Efficiency Curve for 2014 Data SetFigure 4.10: Absolute efficiency curve of TIGRESS (solid red line) with ±1σ uncertainty bands(broken red lines).Where β = vc is the speed of recoil nucleus as a fraction of the speed of light and γ =1√1−β2 ,both of which are known from the reaction kinematics. θ is defined as the angle between themotion of the recoil nucleus and the position of measurement in TIGRESS.In this experiment, the maximum angle of the 96Sr recoil nucleus with respect to the beamaxis was θmax ∼ 1◦. This means that, to a good approximation, the recoil nucleus can bedescribed as travelling parallel to the beam (along the z axis) after the reaction. In this case,θ is simply the angle of the detector, θTIG. This angle of incidence was determined using thesegment position of the largest γ-ray interaction in TIGRESS. It should be noted that the angleof emission of the 96Sr recoil nucleus affects the kinematic energy and the β value.Using an approximate value of ∼ 0.1c for the recoil velocity, the distance travelled per pi-cosecond is 30 µm. Since excited nuclear states have typical lifetimes in excess of a picosecondit is reasonable to assume that the 96Sr recoil nuclei would travel through the remaining targetmaterial before subsequently emitting a γ-ray. The speed of the recoil nucleus as it leaves thetarget is affected by the position in the target where the reaction happens. This is because thebeam loses energy in the target before the reaction and the recoil loses energy in the targetafter the reaction. The range of possible recoil speeds due to different reaction positions was774.2. TIGRESS Calibrationscalculated to be less than 0.5% and so this effect was considered negligible. All reactions wereassumed to take place in the center of the target.In this work, the Doppler correction was carried out on an event-by-event basis. The measuredproton ejectile angle and energy were used to reconstruct the momentum vector of the heavyrecoil. The recoil energy lost in the target was calculated, giving a final β value which was thenused to Doppler correct the γ-ray energy.Figure 4.11a shows the effect of the Doppler correction on the data. Without a Doppler correc-tion the separate TIGRESS detector positions measure different energies from the 96Sr γ-raydecays. The result is a smeared-out spectrum without clearly defined peaks. The broad rawenergy peaks which surround the corrected 815 keV peak are from the ring of clover detectorscentered at θ = 90◦. The separate feature at roughly 90% of the peak energy is due to theθ = 135◦ clover ring. Each detector in TIGRESS has an angle dependent energy correctionapplied which makes all of the separate spectra consistent. The separate peaks from each de-tector can be seen to merge into a single peak under the action of the Doppler correction.(a) (b)Figure 4.11: (a) Comparison of raw experimental γ-ray spectrum (red) to Doppler correctedspectrum (black). (b) Doppler correction of fitted 815 keV γ-ray peak centroid as a function ofθTIG.784.2. TIGRESS CalibrationsThe Doppler correction can be examined in more detail by plotting the γ-ray energies versusTIGRESS angle. This was done by fitting a specific photo-peak in each angular slice and plot-ting the fitted peak centroid as a function of cos θTIG. The quality of the Doppler correctionused in this analysis is demonstrated in figure 4.11b, which shows that the fitted centroid ofthe prominent 815 keV 96Sr γ-ray is flat versus angle to within 0.5%. The approximate widthof the 815 keV peak in each of the underlying angular slices was σ ∼ 6 keV, and so the totalincrease in resolution due to the small Doppler correction error was on the order of 1σ acrossthe entire θTIG range. This was found to give satisfactory overall γ-ray energy resolution.4.2.5 TIGRESS Energy ResolutionThe intrinsic energy resolution of the TIGRESS HPGe detectors is very good, with typicalFWHM ∼ 3 keV. In this experiment, the width of the γ-ray photo-peaks in TIGRESS waslimited by the Doppler correction.The energy resolution curve of TIGRESS was determined by examining a series of prominentpeaks in the γ-ray singles spectrum across a large energy range. A Gaussian function with alinear background fit was found to be adequate for estimating the peak width, σ. The peakwidth was then plotted as a function of photo-peak energy. Figure 4.12 shows the energyresolution of TIGRESS.Figure 4.12: Empirically determined photo-peak width as a function of photo-peak energy.794.3. Particle IdentificationThe resolution curve is a useful tool for differentiating between real γ-ray peaks and falsepeaks. Examples of false peaks are γ-ray decays from reactions such as fusion evaporation, de-cays from nuclei at rest (such as e+e− annihilation γ-rays) and also statistical fluctuations. Theeffect of the Doppler correction on the different peaks versus θTIG is also useful for identifyingfalse peaks.It should also be noted that this curve assumes that each γ-ray was emitted by a state that isnot isomeric. States with long half-lives will have larger photo-peak widths.4.3 Particle IdentificationFigures 4.13a and 4.13b show the measured ∆E and pad energy of particles in SHARC versusθLAB respectively. There are different reactions that can be seen, such as95Sr elastic scatteringand 95Sr(d,p). It can also be seen that the different reaction channels cannot be separatedwithout the simultaneous measurement of both the ∆E and pad energy, as multiple reactionsoverlap in figures 4.13a and 4.13b. Several kinematic lines are drawn in figure 4.13a as a visualaid. The data is systematically lower in energy than these lines because of energy losses in thetarget and dead-layers. Punch-through can also be seen in the (p,p) and (d,d) data, beyondwhich point the ∆E energy decreases in figure 4.13a and the pad energy increases in figure4.13b.Figure 4.13a also shows that there is considerable low energy background which is mostly fromβ-decay of the radioactive beam constituents. This is discussed in more detail in appendix D.Fusion evaporation products also contribute to the background. The 95Sr(12C,12C) data canbe seen to have much worse energy resolution than the (p,p) and (d,d) data, which was pre-viously discussed in section 3.1. The very high energy of the scattered carbon also caused thepre-amplifiers to saturate which is shown as the intense line at ∆E∼18000 keV with constantenergy. The carbon data was not analyzed for these reasons.Figure 4.14 shows the kinetic energy of various particles measured in SHARC as a functionof the lab angle θLAB. This plot was made by reconstructing the full kinematic energy of the804.3. Particle Identification(a) (b)Figure 4.13: (a) Measured ∆E kinematics drawn with kinematic lines corresponding to various95Sr reactions. Black curves show elastic scattering channels (p,p), (d,d) and (12C,12C), whilered curves show (d,p) kinematics for 0 (solid), 2, 4 and 6 MeV (dashed) excitation energy. (b)Measured pad energy kinematics. More details are given in the text.particles by adding the measured energy to the unmeasured energy lost, as is described in equa-tion 4.1. This procedure is the same as the calibration procedure. In order to appropriatelyreconstruct the full kinematic energy of each measured particle, the associated nuclear reactionmust be identified.For backward lab angles (θLAB > 90◦), all data was taken to be (d,p) data. Some fusionevaporation products would also be measured in this angular range, although these particlesform a smooth continuum instead of sharp individual peaks and can therefore be treated as abackground. The only cut that was used at backward angles was a low energy threshold ∆E >1 MeV.At forward lab angles (θLAB < 90◦) it can be seen that there is elastic scattering data in ad-dition to the (d,p) data, which necessitates some form of particle identification. Moreover, the(d,d) data has the same total energy as the (d,p) data as indicated by the overlapping curves infigure 4.14. One important difference between these reaction channels is that the (d,p) protons,owing to their large energy, punch through the ∆E detector and deposit energy in the pad814.3. Particle IdentificationFigure 4.14: Kinematics of various 95Sr reactions. Black curves show elastic scattering channels,(p,p) and (d,d) and red curves show (d,p) kinematics for 0 (solid), 2, 4 and 6 MeV (dashed)excitation energy.detector across the entire DBOX section. Contrastingly, for angles closest to θLAB = 90◦ thekinematic energy of the elastically scattered particles is much lower and so they are completelystopped within the ∆E detectors. In this region, a kinematic (∆E versus θLAB) cut was usedto select only the elastically scattered protons and deuterons. Figure 4.14 shows that the onlyDBOX data which was used with total energy less than ∼5 MeV is from (p,p) and (d,d) events,and the sharp cut-off around these data indicate the cut shapes. The carbon data and largelow energy background was removed through the use of these cuts.Beyond this point, they punch through the ∆E detectors and deposit energy in the pad (E)detectors. The transition from ∆E-only data to ∆E plus E data can be identified through thesudden change in width of the (p,p) and (d,d) kinematic line in figure 4.14.From this point it was possible to identify the protons and deuterons using the ∆E-E de-tector arrangement and so particle identification (PID) plots were made for this purpose.The geometry of the DBOX detectors causes uneven energy loss across the ∆E detector due824.3. Particle Identificationto angular dependence of the effective thickness, which is formulated in equation 4.2. This istrue for both mono-energetic sources and reaction products. The measured ∆E, divided bythe effective thickness angular terms removed the angular spread of energies across the DBOXdetectors. This allowed for a much cleaner PID plot that could then be used to distinguishbetween the different particles. Figure 4.15a shows the PID plot produced using only the ∆Eand pad energies, while figure 4.15b shows the improvement caused by the effective thicknesscorrection.   Particle Identification - Uncorrected(a)Particle Identification - Corrected(b)Figure 4.15: (a) Particle identification (PID) plot for all pads in DBOX section. Protons havelower ∆ energy for a given pad energy than deuterons, forming two separate loci. (b) Same asbefore, but with an effective thickness correction (see text for details).Without the effective thickness correction, the protons from (p,p) and (d,p) lie on separate PIDcurves due to their distinct kinematics. Figure 4.15a shows that this effect is visible in thePID spectrum. The large distribution of proton energies caused proton identification curve tooverlap with the deuteron identification curve, which makes particle identification ambiguousin some cases. The PID spectrum shown in figure 4.15b was used to make graphical cuts whichunambiguously identified the particles as protons or deuterons.834.3. Particle Identification4.3.1 Angular Ranges and Excitation Energy RangesIn addition to the geometrical efficiency of SHARC that was described in section 4.1.3, the par-ticle cuts that were used introduced further limitations to the effective detector coverage. Theangular and excitation energy sensitivity for each reaction was shaped by the energy thresholdsof SHARC, the kinematic cuts and also the PID cuts that were used. These cuts were optimizedto maximize the sensitivity of SHARC to (d,p) data, while preserving important informationabout the elastic scattering data.Once the measured particles were identified, the excitation energy of the states which werepopulated was reconstructed. This was done by converting the measured lab frame kineticenergy of these particles into the centre-of-mass frame. In this frame, the kinetic energy of thefinal state particles is directly related to the reaction Q value, or equivalently, the excited statewhich was populated in the reaction.Figures 4.16a and 4.16b show reconstructed excitation energy versus centre-of-mass angle ofthe identified (p,p) and (d,d) elastic scattering data respectively.Care was taken to include as much (p,p) and (d,d) data close to the target (small centre-of-massangles) as possible. In the centre-of-mass frame, small angles are dominated by pure Rutherfordscattering. This region is important for cross section studies as it is less sensitive to the opticalmodel used in the DWBA fit (described in section 1.5), which helps to reduce uncertainty inthe absolute normalization. Ensuring a large excitation energy range for the (p,p) and (d,d)inelastic scattering data was not considered to be important for this analysis. Only the groundstate (elastic) angular distribution for this data was required for normalization and to optimizethe DWBA optical model. Note that the discontinuity in counts versus centre-of-mass anglewhich is present in each of the two figures corresponds to particles punching through the ∆Edetector and depositing energy in the pad detectors. The pad detector energy thresholds resultin a region where protons have punched through the ∆E but no pad signal is measured, whichcan be seen as a sharp, short rise in excitation energy followed by a decrease in counts.Figure 4.17 shows reconstructed excitation energy versus centre-of-mass angle of the identified844.3. Particle Identification(a) (b)Figure 4.16: (a) Excitation energy versus centre-of-mass angle for (p,p), showing angular andenergy ranges of the PID cuts. The black line indicates 0 keV excitation energy. (b) Same as(a), but for (d,d).(d,p) protons.For the (d,p) data, an effort was made to include all available excitation energy and angleswherever possible. At large backward lab angles (θLAB > 140◦) the kinematic energy of (d,p)protons was close to the detector threshold and so reactions which populated highly excitedstates could not be studied in this angular range. At small forward angles (θLAB < 60◦) theprotons punched through the pad detectors (as was seen in the NPTool simulation results dis-cussed in section 3.1) and so this data was also irretrievable. This introduced an upper angularlimit for studying reactions which populated low-lying 96Sr states. The conversion betweenlaboratory frame and canter-of-mass frame angles is dependent on excitation energy, which canalso be seen in figure 4.17.Protons associated with highly excited 96Sr states (above ∼8 MeV) could not be distinguishedfrom (p,p) protons as can be seen in figure 4.14. It was also not possible to cleanly distinguishbetween these reactions using PID cuts. This was not an issue because the upper excitationenergy limit was beyond the one neutron separation energy (Sn =5.879 MeV), and states abovethis energy decay by neutron emission and only partially via γ-ray emission. The excited 96Srstates in this region are currently under analysis by the Lawrence Livermore group as part of854.4. Method of Extracting Angular DistributionsFigure 4.17: Excitation energy versus centre-of-mass angle for (d,p), showing angular and energyranges of the PID cuts. The solid red line indicates 0 MeV excitation energy and the brokenred lines indicate 2, 4 and 6 MeV.a study of (n,γ) surrogate reactions.4.4 Method of Extracting Angular DistributionsIn this experiment a number of angular distribution measurements were made.The large quantity of low energy β decay background that was measured in the UBOX andUQQQ sections of SHARC during both experiments restricted the angular distribution analysisfor laboratory angles greater than 90◦. In the 94Sr(d,p) experiment, 95Sr states above 1 MeVcould not be analyzed at center-of-mass angles less than approximately 60◦. In the 95Sr(d,p)experiment, 96Sr states above 2.7 MeV could not be analyzed at center-of-mass angles less thanapproximately 15◦. The larger excitation energy range that could be analyzed in the 95Sr(d,p)experiment is primarily a result of the larger Q-value for the reaction.Each angular distribution was produced using the same procedure, including the elastic scatter-ing data. An excitation energy versus center-of-mass angle matrix was made and then divided864.4. Method of Extracting Angular Distributionsinto angular bins of equal size, producing an excitation energy spectrum for each angular bin.The counts associated with the state under analysis were extracted from each excitation energyspectrum. A number of different methods were used to extract the counts from the excitationenergy spectra. The chosen of method extraction depended on the character of the backgroundand whether the spectra contained unresolved peaks. Figures 4.16a and 4.16b show the excita-tion energy versus center-of-mass angle matrices for the 95Sr elastic scattering data and figure4.17 shows the excitation energy versus center-of-mass angle matrix for measured protons from95Sr(d,p).For excited states, detected γ-rays in TIGRESS were used in conjunction with the particleenergies and positions measured in SHARC. The populated excited states were first identifiedusing a matrix of excitation energy versus γ-ray energy. The matrix was projected to producea γ-ray spectrum gated on a narrow excitation energy window. The size of excitation energywindow was chosen to be 800 keV as this was large enough to collect all particles associatedwith a given state in the UQQQ and UBOX sections of SHARC. The window did not generallycollect all contributions from the DBOX section due to the slight energy offset in this region,however most of the measured γ-rays were in coincidence with protons measured in the UBOXsection and so this was not a problem for simply identifying states. The energy window wasshifted and each γ-ray peak that was observed was compared to known transitions so that alevel scheme could be made. A γ-gated γ-ray spectrum for protons within the excitation energywindow was also produced to help elucidate feeding.The angular distributions for excited states were extracted using γ-gated excitation energyversus center-of-mass angle matrices. Three matrices were made for each γ-gate. A matrixwhich contained the particles in coincidence with the γ-ray peak was made. Two backgroundmatrices were also made which sampled the γ-ray background immediately above and belowthe peak. The background matrices were added together and weighted to account for any dif-ferences in statistics due to window size. The background matrices were both subtracted fromthe peak matrix, which effectively removed the background under the γ-ray peak. This γ-gatedexcitation energy versus center-of-mass angle matrix was then analyzed in the same way as the874.4. Method of Extracting Angular Distributionsparticle-only matrices; by producing a set of excitation energy spectra corresponding to equal-sized angular bins. The counts were taken from the γ-gated excitation spectra by integratingthe total counts under the peak. Figure 4.18 gives an example of how the 1995 keV 96Sr statewas analyzed.Figure 4.18: Example analysis for 1995 keV 96Sr state (top to bottom, left to right). (a)γ-ray singles spectrum gated on the excitation energy range 1500-2500 keV, with estimatedpeak counts. (b) Coincident γ-rays with the 1180 keV transition in the same excitation energyrange. (c) Excitation energy spectrum coincident with 1180 keV γ-ray rays projected over allcenter-of-mass angles. (d) Excitation energy versus center-of-mass angle coincident with 1180keV γ-ray rays.Each angular distribution was obtained using equation 4.10dσdΩ(θ) = N N(θ)εp∆θ sin θ1Bεγ(4.10)where N is the normalization constant, εp is the particle detection and identification efficiency,N(θ) is the total number of counts measured at angle θ and ∆θ is the size of the angular bins.884.4. Method of Extracting Angular DistributionsThe sin θ factor arises from the definition of the solid angle element in equation B.1 in appendixB. For γ-ray gated angular distributions, the absolute efficiency of TIGRESS and the branchingratio of the gated transition were also corrected for, and are given by the additional factors εγand B respectively.The angular bin size was chosen to be as small as possible given the available statistics ofthe specific reaction. Small angular bins of ∆θ = 1◦ were used for the elastic scattering as therewas a lot of statistics and so that the oscillatory features of the differential cross section couldbe clearly seen in the data. Most of the (d,p) states were analyzed using ∆θ = 5◦ angular binsas this made it possible to extract several points from within the UQQQ and UBOX sectionsof SHARC. This made it possible to compare the shape of the angular distribution to DWBAcalculations more precisely. Smaller angular bin sizes ∆θ = 3◦ or 4◦ were used for states thatwere strongly populated in the 95Sr(d,p) experiment. Figure 4.19 shows the angular distribu-tion that was extracted for 95Sr(d,p) to the 1995 keV 96Sr state.]° [Cmθ 0 20 40 60 80 100 [mb/sr]Ωdσd 1−101=1180 keVγ=1995 keV excSr, E96Sr(d,p)95Figure 4.19: Experimental angular distribution for 95Sr(d,p) to the 1995 keV 96Sr state.A number DWBA calculations were carried out using the coupled-channels reaction codeFRESCO [Tho88] and these were compared to the extracted angular distributions. The elasticscattering data in each data set was used to optimize the optical model potentials and deter-894.4. Method of Extracting Angular Distributionsmine the normalization constant N . This is discussed in section 5.1.For the (d,p) reactions, a single-step DWBA calculation was carried out for each neutronconfiguration, using the excitation energy of the state to determine the Q-value and neutronseparation energy.This single step description of 94Sr(d,p) allows 95Sr to be populated adding a single neutroninto one of four different single particle orbitals. The one neutron configurations correspondingto adding a neutron to these orbitals are; [3s12 ] through ` = 0, [2d32 ] and [2d52 ] through ` = 2and [1g 72 ] through ` = 4. These configurations correspond to final state Jpi values of 12+, 32+,52+and 72+respectively. The 1h112 orbital was omitted from the DWBA calculations as transferreaction cross sections strongly favour low orbital angular momentum transfer. Because of this,the population of high spin states would not be significant in this data.The theoretical treatment of 95Sr(d,p) as a single step process is slightly more complicated thanin the 94Sr(d,p) process, owing to the unpaired valence neutron in 95Sr. This valence neutroncan couple to the transferred neutron to give two possible spins for each orbital populated inthe reaction. If we assume a spherical seniority 1 description of the 95Sr 12+ground state, theunpaired neutron is in the 3s12 orbital as this is the only possible way to create the required spinand parity. As was described for 94Sr(d,p), the added neutron populates one of four differentsingle particle orbitals. The two neutron configurations corresponding to adding a neutron tothese orbitals are; [3s12 ]2 through ` = 0, [3s122d32 ] and [3s122d52 ] through ` = 2 and [3s121g72 ]through ` = 4. Each of these configurations except for [3s12 ]2 can result in two distinct spinscorresponding to the two neutrons aligning or anti-aligning their intrinsic spins. Adding thetransferred neutron to the 2d32 orbital would populate 1+ and 2+ states while adding the trans-ferred neutron to the 2d52 orbital would populate 2+ and 3+ states. Similarly, 3+ and 4+ statescan be populated by adding a neutron into the 1g 72 and coupling it to the95Sr ground state.In the case of an [3s12 ]2 configuration, only 0+ states can be populated.Each of the calculated angular distributions were fitted to the experimental data using χ2minimization, and these results were used to judge the configuration which is most consistentwith the data. It must be noted, however, that the DWBA model is a simple description of904.4. Method of Extracting Angular Distributionsthe reaction and while the shape of the DWBA curve should approximate the experimental an-gular distribution, the quantitative agreement is unlikely to be perfect. Instead, the χ2 valuesobtained from fitting the data to the different calculated configurations were used to judge themost likely ` transfer and to determine the spectroscopic factor of the populated states. TheDWBA calculation which best fit the data is drawn as a solid line in the angular distributionplots, although it can be seen that there were frequently multiple configurations which pro-duced almost identical χ2 values. The MINUIT [Jam] minimizer was used to perform singleparameter χ2 minimization fits of the spectroscopic amplitudes (SA) to the data, which aredefined as the square root of the spectroscopic factors. The uncertainty of the spectroscopicamplitude was determined by using χ2 + 1 confidence intervals. The minimized values weretherefore squared to produce the extracted spectroscopic factors that are presented in table5.4. The uncertainties of the spectroscopic factors were determined by combining the relativeuncertainties of all absolute scale factors that were applied to the experimental data with therelative uncertainty of the spectroscopic factor, as is shown in equation 4.11. The errors ofthese absolute scaling factors were added in quadrature.δSFSF =√(δNN)2+ 4(δSASA)2+(δεγεγ)2+(δBB)2(4.11)where N is the normalization constant, SA is the spectroscopic amplitude (note the additionalfactor of four), εγ is the absolute efficiency of TIGRESS at the γ-ray gated energy and B is thebranching ratio for the corresponding γ-ray transition. For states that were analyzed withoutγ-ray gating, the last two terms in equation 4.11 did not apply.91Chapter 5Results5.1 Elastic Scattering DataThe normalization constant N in equation 4.10 was determined by fitting the elastic scatteringdata to DWBA calculations. This data was also used to optimize the optical potentials thatwere introduced in section 1.5.1 for the (d,p) calculations.Equation 5.1 relates the measured angular distribution of elastically scattered particles to theDWBA calculation. [δσδΩ(θ)]DWBA=1Lnt∆x[δN(θ)δΩ(θ)]exp(5.1)where the constants L, nt and ∆x are the integrated beam current, the number density oftarget nuclei and the target thickness respectively. The normalization constant N is defined asN = 1Lnt∆x (5.2)and is determined by scaling the experimental angular distribution to the DWBA calculation.When two electrically charged point-like objects undergo elastic scattering, the resulting an-gular distribution is described by the Rutherford scattering formula which varies as sin−4 θ2 .When one or both of the scattered objects has an extended size, a diffraction pattern is pro-duced. This is true even if the distance of closest approach is larger than the combined radiiof the objects. If viewed in normal kinematics, the incident kinetic energy of the elasticallyscattered protons would be ∼5.4 MeV in both experiments. At this energy, the distance ofclosest approach between the beam and target nuclei is greater than their combined radii andso the angular distribution would be dominated by pure Coulomb scattering. The kinetic en-925.1. Elastic Scattering Dataergy of the elastically scattered deuterons was ∼10.8 MeV in both experiments, when viewedin normal kinematics. In this case, the distance of closest approach between the beam andtarget nuclei is less than their combined radii and so the nucleus size plays a significant rolein the scattering process. Other nuclear reactions can also occur when the nuclei are at closerange, and this causes a decrease in cross section in the elastic channel. The elastic scatteringof two nuclei can be described as the scattering of distorted waves off a potential which has areal and an imaginary component. This is called the optical model. It is standard practice tomodify the parameters of the optical potential so that the calculated angular distribution is ingood agreement with data. In this way the optimized potential provides a good description ofthe physics underlying the scattering process, including inelastic effects that remove flux. Thedeuteron elastic scattering data was used to optimize the optical model (OM) parameters usedin the calculated (d,p) reactions.At small center-of-mass angles, the proton and deuteron kinematic lines were close in energyand so the counts were extracted by fitting both peaks together using Gaussian functions plusa background fit. Fits in this angular range were carried out using a linear background modeland an exponential background model, as are shown in figures 5.1a and 5.1b respectively. Thecounts and uncertainties were taken to be the average and difference of these models respec-tively. At larger center-of-mass angles the particles could be identified using the pad detectorand so the counts were extracted by subtracting a fitted background from the total counts. Thebackground was estimated by excluding the peak region and fitting the remaining data using apolynomial model. An example of this is is shown for (d,d) data and (p,p) data in figures 5.2aand 5.2b respectively, where the deuteron data used a linear model and the proton data used aquadratic model.Figures 5.3a and 5.3b show the experimental proton elastic scattering angular distributionscompared to various optical models, which are described in table 5.1. The global optical modelparameters V0 and WD given in table 5.1 have a weak dependence on the neutron numberN and the mass number A of the strontium nucleus. The difference in these values for the94Sr(p,p) and 95Sr(p,p) was very small, and so the same potential was used in each case. All of935.1. Elastic Scattering Data(a) (b)Figure 5.1: Example fit of small center-of-mass angle (p,p) and (d,d) data using (a) exponentialbackground model (b) linear background model.(a) (b)Figure 5.2: Example of large center-of-mass angle background subtraction for 95Sr (a) (d,d)data and (b) (p,p) data. The green region was taken to be the peak region and so the countswere extracted by subtracting the background curve from the total counts in that range.945.1. Elastic Scattering Datathe optical models predict almost pure Rutherford scattering, and are in generally reasonableagreement with the data. Given that the data does not offer sensitivity to discriminate betweenthe different parameter sets, the Perey and Perey [PP76] optical model potential was chosen todescribe this reaction channel for the (d,p) calculations.Parameter BG [BG69] CH [V+91] PP [PP76]Rc 1.25 1.266 1.25V0 60.41 57.314 58.725R0 1.17 1.201 1.25A0 0.75 0.69 0.65W0 0.00 0.558 0.00WD 12.856 10.292 13.5RD 1.32 1.238 1.25AD 0.65 0.69 0.47Vso 6.2 5.9 7.5Rso 1.01 1.077 1.25Aso 0.75 0.63 0.47Table 5.1: Global optical model parameters that were used to fit 94,95Sr proton elastic scatteringangular distributions; Becchetti and Greenlees (BG), Chapel Hill (CH) and Perey and Perey(PP).]° [CMθ 0 20 40 60 80 100 120 140 160 180 [Ratio to Rutherford]Ωdσd 0.50.60.70.80.911.11.21.31.41.5Sr(p,p) @ 5.341 MeV/u94Experimental DataPerey and PereyChapel HillExperimental DataPerey and PereyChapel HillBecchetti Greenlees(a)]° [CMθ 0 20 40 60 80 100 120 140 160 180 [Ratio to Rutherford]Ωdσd 0.50.60.70.80.911.11.21.31.41.5Sr(p,p) @ 5.378 MeV/u95Experimental DataPerey and PereyChapel HillBecchetti Greenlees(b)Figure 5.3: Comparison of various optical model potentials to experimental angular distribu-tions for (a) 94Sr(p,p) elastic scattering (b) 95Sr(p,p) elastic scattering.Figures 5.4a and 5.4b show the experimental deuteron elastic scattering angular distributions955.1. Elastic Scattering Datacompared to various optical models, which are described in table 5.2. As was discussed earlier,the very small difference in V0 and WD between94Sr(d,d) and 95Sr(d,d) due to the differentneutron number and mass number was considered negligible. All of the optical models predictsimilar angular distributions, however none of them reproduced the data to a high quality.The calculated curves predict a cross section maximum at approximately 95◦, however bothdata sets showed that this maximum is below 90◦. For the 95Sr(d,d) data, the gradient in theangular range 53-58◦ was also not well reproduced. The small angle 94Sr(d,d) data was alsounderestimated by the curves. Several optical model parameters were adjusted to improve thefit to the 94Sr(d,d) and 95Sr(d,d) data.]° [CMθ 0 20 40 60 80 100 120 140 160 180 [Ratio to Rutherford]Ωdσd 0.20.40.60.81Sr(d,d) @ 5.341 MeV/u94Experimental dataLohr HaeberliDaehnickPerey and Perey(a)]° [CMθ 0 20 40 60 80 100 120 140 160 180 [Ratio to Rutherford]Ωdσd 00.20.40.60.81Sr(d,d) @ 5.378 MeV/u95Experimental DataLohr HaeberliDaehnickPerey and Perey(b)Figure 5.4: Comparison of various optical model potentials to experimental angular distribu-tions for (a) 94Sr(d,d) elastic scattering (b) 95Sr(d,d) elastic scattering.The Lohr and Haeberli [PP76] global optical model potential was used as a starting point forthe modified optical model as it was fitted exclusively to data in the energy range 8-13 MeVwhich is best suited to this work. The inconsistency between calculated curves and the dataat large angles signifies that the radius of the potential was too large, and so in both data setsthe parameters R0 and RD were adjusted.In the case of the 94Sr(d,d) data, decreasing the radius of the optical potential was the onlymodification necessary in order to get excellent agreement. The set of optimized radii was965.1. Elastic Scattering Dataobtained by performing a two parameter minimization fit using R0 and RD as free parameters.SFRESCO [Tho88], which uses the MINUIT [Jam] minimization package, was used to carryout a χ2 minimization fit. SFRESCO is part of the FRESCO software package that is designedto fit the DWBA calculation to experimental data. This is achieved by adjusting the fit pa-rameters and updating the DWBA calculation until χ2 minimization is achieved between thecalculated curve and experimental data.In order to achieve good agreement with the 95Sr(d,d) data, the potential depth parametersV0 and WD were adjusted in addition to the potential radius parameters R0 and RD. Theoptimized R0 and RD values from the94Sr(d,d) data were used as a starting point for the fit,however it was found that the four parameter minimization was not reliable using SFRESCO,even with reasonable parameter starting points and constrained upper and lower bounds. In-stead, each parameter was varied in turn by hand until satisfactory agreement between theoryand experiment was obtained. Using this approach, the hand-fitted optical model required onlya small adjustment from the Lohr and Haeberli parameter set, in a way that was also consistentwith the 94Sr(d,d) optical model fit. Figures 5.5a and 5.5b show the resulting fit of the modifiedoptical model potentials to both data sets.]° [CMθ 0 20 40 60 80 100 120 140 160 180 [Ratio to Rutherford]Ωdσd 00.20.40.60.81Experimental dataFitted DWBASr(d,d) @ 5.341 MeV/u94(a)]° [CMθ 0 20 40 60 80 100 120 140 160 180 [Ratio to Rutherford]Ωdσd 00.20.40.60.81Experimental dataFitted DWBASr(d,d) @ 5.378 MeV/u95(b)Figure 5.5: (a) Measured angular distribution for 94Sr(d,d) elastic scattering. (b) Measuredangular distribution for 95Sr(p,p) elastic scattering.975.1. Elastic Scattering DataParameter 94Sr 95Sr LH [PP76] D [DCV80] PP [PP76]Rc 1.3 1.3 1.3 1.3 1.3V0 109.45 130.0 109.45 93.032 95.28R0 0.903 0.97 1.05 1.17 1.15A0 0.86 0.86 0.86 0.727 0.81WD 10.417 12.0 10.417 12.336 16.99RD 1.11 1.25 1.43 1.3 25 1.34AD 0.771 0.771 0.771 0.849 0.68Vso 7.0 7.0 7.0 7.0 0Rso 0.75 0.75 0.75 1.07 0Aso 0.5 0.5 0.5 0.66 0Table 5.2: Optical model parameters used to fit 94,95Sr deuteron elastic scattering angulardistributions compared to global parameter fits; Lohr and Haeberli (LH), Daehnick (D) andPerey and Perey (PP)It was found that the optimized optical potentials led to a significant change in calculated dif-ferential cross section compared to the global parameters. This can be seen in figures 5.6a and5.6b as the large difference between the global optical model calculations and the fitted opticalmodel calculations. The resulting (d,p) spectroscopic factors were therefore strongly dependenton the choice of optical model parameters. For this reason the spectroscopic factors presentedin the following sections can be compared within a given model (relative) but the absolute valueis not well known. It is important to note that the modified optical potentials did not introduceambiguity in the (d,p) calculations between different orbital angular momentum transfers. Theposition of the differential cross section maxima and minima for different ` transfers were allshifted in unison, so that it remained possible to distinguish between them. It was importantto ensure that this was true, otherwise the spins of the populated states could not be assignedusing these DWBA calculations.The normalization constants for 94,95Sr(p,p) and 94,95Sr(d,d) are given in table 5.3. The nor-malization constants for (p,p) and (d,d), Np,d, were combined using equation 5.2 to estimatethe ratio of protons to deuterons in the target, as is shown in equation 5.3.NdNp =npnd(5.3)985.1. Elastic Scattering Data]° [CMθ 0 20 40 60 80 100 120 140 160 180 [mb/sr]Ωdσd  110Sr(d,p) DWBA Calculations Between Global and Fitted OM94Comparison of L=0, Global OML=0, Fitted OML=2, Global OML=2, Fitted OML=4, Global OML=4, Fitted OM(a)]° [CMθ 0 20 40 60 80 100 120 140 160 180 [mb/sr]Ωdσd  110Sr(d,p) DWBA Calculations Between Global and Fitted OM94Comparison of L=0, Global OML=0, Fitted OML=2, Global OML=2, Fitted OML=4, Global OML=4, Fitted OM(b)Figure 5.6: Comparison of global and fitted DWBA differential cross sections for (a) 94Sr(d,p)and (b) 95Sr(d,p). The large difference in calculated angular distributions between the opticalmodel parameter sets for a given reaction indicates that there is a very large uncertainty onthe overall cross sections for the reactions, which means that the absolute value of the resultingspectroscopic factors is not well-known.This number of proton and deuterons in the target is expressed in table 5.3 as the targetcomposition fraction.Reaction Normalization [mb] Target Composition [%]94Sr(p,p) 4.5(1) x 10−1 3.8(1)94Sr(d,d) 1.79(2) x 10−2 96(2)95Sr(p,p) 5.87(6) x 10−3 8.2(1)95Sr(d,d) 5.25(4) x 10−4 92(1)Table 5.3: Normalization constants extracted from dσdΩ fits of (p,p) and (d,d) data to DWBAcalculations.995.2. 94Sr(d,p) Results0 keV, t 12= 23.90(14) s12+352(32+)681(32+, 52+)1238(92+)1666(112+)556, t 12= 21.9(5) ns(72+)4271109683887125329681204352Figure 5.7: Level scheme for 95Sr states populated through 94Sr(d,p). Level energies, lifetimesand spin assignments are taken from [Lab].5.2 94Sr(d,p) ResultsFigure 5.7 shows the level scheme that was built from this experiment. Spins and paritiesthat are given in figure 5.7 are from previous work. Those given in parentheses are tentativeassignments. The γ-rays that were measured in this experiment are consistent with availableliterature for 95Sr [Lab]. Figure 5.8a shows the measured 95Sr γ-rays coincident with 94Sr(d,p)protons. The labelled peaks indicate known transitions in 95Sr. Figure 5.8b shows the γ-rayenergy plotted against 95Sr excitation energy. It can be seen through the constant strength ofthe 329 keV, 352 keV and 681 keV γ-ray energies as a function of excitation energy that the352 keV and 681 keV are strong collecting states from higher lying levels. This means thatmany of the excited states cascade through these states. A γ − γ coincidence was not possiblefor any of the states in this experiment due to low statistics.It was found that the 352 keV and 681 keV 95Sr excited states were populated directly with1005.2. 94Sr(d,p) Results(a) [keV]γ E0 200 400 600 800 1000 1200 1400 [keV]exc E1000−01000200030004000500000.511.522.533.54Sr95-Ray Energy γExcitation Energy Versus (b)Figure 5.8: (a) γ-ray spectrum for 95Sr. (b) Excitation energy versus γ-ray matrix for 94Sr.enough strength to extract angular distributions. The 1666 keV 95Sr state was also identifiedthrough the 427 keV γ-ray line, although the limited angular range of SHARC at that exci-tation energy meant an angular distribution measurement was not possible. Significant 95Srground state population was also observed. Results for of these states will be discussed in thesubsequent sections.Angular distributions were simultaneously extracted for the ground state, 352 keV and 681keV states using a constrained three (Gaussian) peak plus exponential background fit. Theseparation between the peaks was fixed to preserve the energy spacing 0-352-681 keV. Addi-tionally, the peaks were required to have the same width and the centroid positions could beadjusted in unison by up to 50 keV to allow the fit to reproduce the data well. The backgroundwas also constrained to be the same shape within each of the sections of SHARC to modelthe observed background distribution. Example fits for two different center-of-mass angles areshown in figures 5.9a and 5.9b. Spectroscopic factors for the 352 keV and 681 keV states werealso extracted using γ-ray gates.The 21.9 ns half-life of the isomeric 556 keV state [Lab] substantially lowered the γ-ray detec-tion efficiency for the 204 keV transition (see figure 5.7), and so it was difficult to estimate thedirect population strength of this states. As can be seen in figure 5.8a, there was a small excess1015.2. 94Sr(d,p) Resultsof counts at 204 keV in the TIGRESS γ-ray spectrum which have been labelled. However, itwas difficult to determine whether this was in fact due to the direct population and subsequentdecay of the 556 keV state or whether it is a statistical fluctuation, or some combination of both.Given the low statistics, it was not possible to rule out the signal as a statistical fluctuationbased on the width of the distribution at 204 keV. It was found that there was good quantita-tive agreement between spectroscopic factors obtained using a particle analysis and particle-γanalysis for the 352 keV and 681 keV states, which suggests that the direct population of the556 keV state was negligible. A strongly deformed configuration of this state has been proposed[HRH+04], which would predict weak single particle character and thus weak population crosssection through (d,p). This interpretation is supported by our measurements. Nevertheless,it was possible to assign a spin of 72+to this state based on results of this experiment, as ispresented in table 5.4. Further discussion of this state can be found later in this chapter.The angular distribution analysis results for 94Sr(d,p) are summarized in table 5.4. The tableshows multiple results for the 681 keV 95Sr state as the angular distribution analysis producedtwo possible spin values for the state. The Jpi for each state is given, and those drawn inbold typeface indicate new spin assignments. The previous limits on the spins and parities foreach state are outlined in the following sections, and new constraints from this work are alsodiscussed. The γ-ray gate that was used, if any, is given in the table. The DWBA spectroscopicfactor that was found to best reproduce the experimental data is also presented. The finalcolumn gives the total cross section for each state, which is the integral of the differential crosssection over all θ values. The total cross sections were also calculated using FRESCO.95Sr State [keV] Jpi ` γ-ray gate [keV] Spectroscopic Factor σtot [mb]0 12+0 fit 0.087(15) 8.0(14)352 32+2 352 0.122(13) 11.5(12)556 72+- - - -681 32+, 52+2 fit, 329, 681 0.072(15), 0.045(9) 7.0(14), 6.6(15)Table 5.4: Table of spectroscopic factors for directly populated 95Sr states.1025.2. 94Sr(d,p) Results(a) (b)Figure 5.9: Example 3 peak fits for (a) θCM = 10◦ (b) θCM = 38◦.95Sr Ground StateThe angular distributions shown in 5.10a and 5.10b were produced by using a constrained three(Gaussian) peak plus exponential background fit, as was previously described. Figures 5.10aand 5.10b compare the data to DWBA calculations produced using the fitted deuteron opticalpotential and an unmodified optical potential, respectively. The agreement with the data issignificantly improved using the modified optical potential. The shape of the ground stateangular distribution in figure 5.10a is in excellent agreement with the ` = 0 DWBA calculation,which is consistent with the known spin of 12+.95Sr 352 keV StateFigure 5.11a was produced using the same three peak fit, as was previously described. A 352keV γ-ray gated angular distribution, corrected for the γ-ray ray efficiency, was also producedand is shown in figure 5.11b. The spectroscopic factors that were extracted using each of thesemethods are 0.119(24) and 0.128(29) respectively, which are consistent within errors. This in-dicates that there was not significant feeding from the higher lying 681 keV 95Sr state. Theweighted average of the two spectroscopic factors is given in table 5.4.1035.2. 94Sr(d,p) Results]° [CMθ 0 20 40 60 80 100 [mb/sr]Ωdσd 1 = 0 keVexcSr E95Sr(d,p): 94Experimental Data/N=1.36 2χ +21 Jf=+21L=0 Jo=/N=3.56 2χ +23 Jf=+23L=2 Jo=/N=3.02 2χ +25 Jf=+25L=2 Jo=/N=8.46 2χ +27 Jf=+27L=4 Jo=(a)]° [CMθ 0 20 40 60 80 100 [mb/sr]Ωdσd 1 = 0 keVexcSr E95Sr(d,p): 94Experimental Data/N=4.62 2χ +21 Jf=+21L=0 Jo=/N=3.27 2χ +23 Jf=+23L=2 Jo=/N=3.43 2χ +25 Jf=+25L=2 Jo=/N=5.62 2χ +27 Jf=+27L=4 Jo=(b)Figure 5.10: Angular distribution for 94Sr(d,p) to the 95Sr ground state compared to DWBAcalculations using (a) fitted optical potential (b) unmodified optical potential.The shape of the extracted angular distributions in figures 5.11a and 5.11b are both in goodagreement with the ` = 2 DWBA calculation, constraining the spin and parity of this stateto be Jpi = 32+or 52+. The transferred angular momentum, together with the established M1character of the 352 keV γ-ray transition to the 95Sr ground state allows a firm spin and parityassignment of 32+for this state.]° [CMθ 0 20 40 60 80 100 [mb/sr]Ωdσd 1Experimental Data/N=10.96 2χ +21 Jf=+21L=0 Jo=/N=3.13 2χ +23 Jf=+23L=2 Jo=/N=3.92 2χ +25 Jf=+25L=2 Jo=/N=14.50 2χ +27 Jf=+27L=4 Jo= = 352 keVexcSr E95Sr(d,p): 94(a)]° [CMθ 0 20 40 60 80 100 [mb/sr]Ωdσd 1 = 352 keVexcSr E95Sr(d,p): 94Experimental Data/N=7.24 2χ +21 Jf=+21L=0 Jo=/N=1.70 2χ +23 Jf=+23L=2 Jo=/N=1.85 2χ +25 Jf=+25L=2 Jo=/N=8.80 2χ +27 Jf=+27L=4 Jo=(b)Figure 5.11: Angular distribution for 94Sr(d,p) to the 95Sr 352 keV state extracted using a (a)three peak fit, (b) 352 keV γ-ray gate.1045.2. 94Sr(d,p) Results95Sr 556 keV StateAlthough direct population of the long-lived 95Sr 556 keV state in this experiment could notbe confirmed due to the low γ-ray detection efficiency for the 204 keV transition, its spin andparity can be constrained by combining the assigned spin and parity of the 352 keV state inthis work with previous measurements.The 204 keV γ-ray transition from the 556 keV to 352 keV state was previously measured tohave E2 character using conversion electron spectroscopy [Lab]. The firmly assigned 32+spinand parity of the 352 keV combined with the multipolarity of the 204 keV γ-ray constrains thespin and parity of the 556 keV 95Sr state to be 52+or 72+. For a spin and parity of 52+, the204 keV γ-ray would contain a mixture of M1 and E2 multipolarities. One can estimate thetransition rate using the so-called Weisskopf approximation, which is discussed in appendix E.A Weisskopf estimate for the M1 transition strength predicts that it would be large enough tohave been observed in other studies. As no M1 component of the transition has been reported,the 556 keV state spin and parity is assigned to be 72+.Moreover, the absence of any observed 556 keV γ-rays in previous studies indicates that thetransition rate for a direct decay to the 95Sr ground state is very low. The 72+assignment ofthe 556 keV state would make the 556 keV γ-ray an M3 transition, which is consistent with itsnon-observation.95Sr 681 keV StateFigure 5.12 was produced using the same three peak fit as for the ground state and 352 keVstate. Spectroscopic factors were also determined for this state using 329 keV and 681 γ-raygates. The γ-ray gated angular distributions are shown in figures 5.13a and5.13b. All threeanalyses give consistent spectroscopic factors and favour ` = 2 transfer, which constrains thespin and parity of this state to be 32+or 52+. The spectroscopic factors for a spin and parity of32+were found to be 0.080(21), 0.074(29) and 0.060(26) using the three peak fit, 329 keV γ-rayand 681 keV γ-ray respectively. Similarly, the spectroscopic factors for a spin and parity of 52+were 0.050(13), 0.046(18) and 0.037(15) using the three methods of extraction. The weightedaverage of these values for both 32+and 52+are given in table 5.4.1055.2. 94Sr(d,p) ResultsInformation about the multipolarities of transitions from the 681 keV 95Sr state to lower-lyingstates can also be used to constrain the spin and parity assignment of this state. A possibleM1 component in the 681 keV γ-ray transition to the 95Sr ground state would rule out a spinand parity of 52+, however the existence of the M1 component has not been confirmed [Lab].Another result from 95Rb β decay was a measurement of the log ft value for the decay of theof 95Rb 52−ground state to this state. A log ft value is essentially a measurement of transitionstrength and can be used to constrain the change in spin and parity between the parent anddaughter nuclear states. The measured log ft value of 6.0 is most consistent with a first forbid-den transition. A first forbidden transition causes a parity change (in this case - to +) and alsoa spin change of ∆I = 0, 1, 2 units of angular momentum, which would require the spin andparity of the 681 keV 95Sr state to be 12+to 72+.All of these measurements taken together constrain the spin and parity to be 32+or 52+.]° [CMθ 20 30 40 50 60 70 80 90 100 [mb/sr]Ωdσd 1Experimental Data/N=2.28 2χ +21 Jf=+21L=0 Jo=/N=0.99 2χ +23 Jf=+23L=2 Jo=/N=0.89 2χ +25 Jf=+25L=2 Jo=/N=5.13 2χ +27 Jf=+27L=4 Jo= = 680 keVexcSr E95Sr(d,p): 94Figure 5.12: Angular distribution for 94Sr(d,p) to 95Sr the 95Sr 681 keV state extracted usinga three peak fit.Higher Lying 95Sr StatesThe γ-ray spectrum shown in figure 5.14a was produced by gating on the excitation energyrange 1-2 MeV, using only the particles measured in the DBOX section of SHARC. The strong427 keV peak indicates that the 1666 keV 95Sr state was populated in this reaction. It was not1065.2. 94Sr(d,p) Results]° [CMθ 20 40 60 80 100 [mb/sr]Ωdσd 1Experimental Data/N=3.31 2χ +21 Jf=+21L=0 Jo=/N=1.98 2χ +23 Jf=+23L=2 Jo=/N=2.03 2χ +25 Jf=+25L=2 Jo=/N=2.45 2χ +27 Jf=+27L=4 Jo= = 680 keVexcSr E95Sr(d,p): 94(a)]° [CMθ 20 40 60 80 100 [mb/sr]Ωdσd 1−101Experimental Data/N=4.71 2χ +21 Jf=+21L=0 Jo=/N=2.51 2χ +23 Jf=+23L=2 Jo=/N=2.65 2χ +25 Jf=+25L=2 Jo=/N=4.29 2χ +27 Jf=+27L=4 Jo= = 680 keVexcSr E95Sr(d,p): 94(b)Figure 5.13: Angular distribution for 94Sr(d,p) to 95Sr 681 keV state extracted using a (a) 329keV γ-ray gate, (b) 681 keV γ-ray gate.possible to determine the extent of direct population to the 1238 keV 95Sr state in this workas the strongest transition from this state produces a 682 keV γ-ray which could not be distin-guished from the 681 keV γ-ray line given the γ-ray spectrum energy resolution and availablestatistics. The observed 329 keV γ-ray in this excitation energy range suggests that there isfeeding of the 681 keV 95Sr state from directly populated states in the excitation energy range1-2 MeV. The intensity of the observed 329 keV γ-ray is not consistent with the suggestedbranching ratios.The 1238 keV 95Sr state was observed in 95Rb β decay and 96Rb β-n decay. The measuredlog ft ≥ 7.6 from the decay of the 95Rb 52−ground state to the 1238 keV state is consistentwith a first forbidden decay which changes the total angular momentum by ∆I = 0, 1, 2 andchanges the parity of the daughter state. This constrains the 1238 keV spin and parity to be12+to 92+. The measured multipolarity of the γ-ray transition from the 1238 keV state to the72+556 keV isomer is M1+E2, which leads to a tentative spin assignment of 92+for this state.The 1666 keV 95Sr state was not reported in either of the decay studies.Both the 1238 keV and 1666 keV 95Sr states were observed in 252Cf spontaneous fission (SF)decay [HRH+04], a reaction which preferentially populates high spin states. The proposed levelscheme from the 252Cf SF work is shown in figure 5.14b. The 1666 keV state was assigned a1075.2. 94Sr(d,p) Resultstentative spin and parity of 112+based on the strong branching ratio to the 1238 keV state,however the substantial population strength of the 1666 keV state in this experiment makes theassignment unlikely. The addition of a single neutron to the 94Sr ground state can directly pop-ulate 95Sr states with spins and parities of 12+, 32+, 52+, 72+and 112−, although the cross sectionfor ` = 5 transfer would be very low, and so it would be less favourable.(a) (b)Figure 5.14: (a) Measured γ-rays coincident with 95Sr excitation energies of 1-2 MeV usingtransfer protons in the DBOX section of SHARC only. The 427 keV γ-ray indicates directpopulation of the 1666 keV state. (b) Level scheme taken from 252Cf SF [HRH+04], indicatingthree possible band structures in 95Sr.1085.3. 95Sr(d,p) Results0 keV, t 12= 1.07(1) s0+1815, t 12= 4.8(28) ps2+11229, t 12= 115(12) ps0+21628(2+3) 17934+1211324812576289932393500(10)0+31465, t 12= 6.7(10) ns2+2 1507, t 12≤ 6.2 ps(1+, 2+) 19952055(4)(1+, 2+) 2084(4)2120(2)22173407581761140213051299208485511809783998131628692150765023541481578011063616881240(4)3500(10)Figure 5.15: Level scheme for 96Sr states populated through 95Sr(d,p). Level energies, lifetimesand spin assignments are taken from [Lab]. Dashed lines represent proposed new states andtransitions. The 235 keV E0 transition was not seen in this work.5.3 95Sr(d,p) ResultsFigure 5.16 shows a spectrum of the measured 96Sr γ-rays coincident with 95Sr(d,p) protons.The labelled peaks indicate known transitions in 96Sr, with the exception of the 538 keV peakwhich is not known, and could not be placed in the level scheme. Figure 5.15 shows the levelscheme that was built from this experiment. Spins and parities that are given in figure 5.15 arefrom previous work. Two new candidate states were observed in this experiment, with energies2055(4) keV and 3500(10) keV. These are discussed later in this chapter.1095.3. 95Sr(d,p) ResultsFigure 5.17b shows the γ-ray energy plotted against 96Sr excitation energy. As was also seenin 95Sr, the 815 keV first excited state in 96Sr is a strong collecting state from higher levels.This can be seen by the very strong γ-ray line coincident with 96Sr states all the way up tothe neutron separation energy at 5.88 MeV. Some direct transitions from excited states to theground state were observed, although most of the states which were analyzed used γ-ray gateson strong transitions to the 2+1 state. It was found that many of the weaker observed γ-raytransitions in figure 5.16 were produced by cascades from higher excited states rather thandirect population. For this reason, it was not possible to extract angular distributions for eachof the populated states by γ-ray gating on the associated transition. As can be seen in figure5.17b, 95Sr(d,p) directly populated 96Sr states up to the neutron separation energy. However, itwas not possible to identify individual states above 3.5 MeV as unique γ-ray transitions couldnot be found. Figure 5.17a shows the γ−γ coincidence matrix, where the strongest coincidenceis for 813 keV and 815 keV γ-rays which are from the 1628 keV 96Sr state.Figure 5.16: γ-ray spectrum for 96Sr.1105.3. 95Sr(d,p) Results(a) [keV]γ E500 1000 1500 2000 2500 3000 3500 4000 [keV]exc E0100020003000400050006000110210Sr96-Ray Energy γExcitation Energy Versus (b)Figure 5.17: (a) γ − γ matrix for 96Sr. (b) Excitation energy versus γ-ray matrix for 96Sr.The unpaired valence neutron in 95Sr can couple to the transferred neutron to produce twodifferent possible spins. For this reason, it is not possible to firmly assign the spin of statesin 96Sr by measuring only the angular momentum (`) transfer. Results from this work arecombined with those from previous measurements wherever possible, to further constrain thespins and parities of the 96Sr states. The DWBA calculations predict very similar shapes forthe different spin couplings (such as ` = 2,j = 1+ and ` = 2,j = 2+). Spin values whichcorrespond to aligned neutron spin vectors were calculated to have higher cross sections thanthose with anti-aligned spins due to the presence of a spin-spin interaction in the calculations,and so significantly different spectroscopic factors were obtained using the two couplings. Allpossible spins and parities that are consistent with the DWBA fits in this work and with previ-ous constraints are given in results table 5.5. For states with more than one possible spin andparity, spectroscopic factors are given for each Jpi so that each possible interpretation could becompared to shell model calculations in chapter 6. For cases when the spin was already firmlyestablished, a single spin coupling for each ` value is compared to the data to contrast theirshapes.Table 5.5 gives the spectroscopic factors and constraints on the spin and parity of states thatwere analyzed. Each individual state is discussed in the subsequent sections.1115.3. 95Sr(d,p) Results96Sr State [keV] Jpi l γ-ray gate [keV] Spectroscopic Factor σtot [mb]0 0+ 0 fit 0.114(16) 1.17(16)815 2+ - 815 0.014(5)† 0.6(2)†1229 0+ 0 414 0.19(2), 0.16(3)† 2.0(2), 1.7(3)†1465 0+ - 650 0.18(7)† 1.9(7)†1507 2+ 2 792 0.025(8) 0.7(2)1628 2+ 2 813 & 815 0.050(18) 1.4(5)1793 4+ 4 978 0.049(11) 0.52(12)1995 1+, 2+ 2 1180 0.178(30), 0.100(17) 3.1(5), 2.9(4)2084 1+, 2+ 2 2084 0.214(47), 0.121(27) 3.7(8), 3.5(8)2120 4+ 4 1305 0.136(29) 1.5(3)2217 2+ 2 1402 0.034(7) 1.0(2)2576 1+, 2+ 2 1761 0.062(12), 0.036(7) 1.0(2), 0.97(19)3239 3+, 4+ 4 978 0.72(18), 0.54(14) 6.0(15), 5.8(15)3500(10) 1+, 2+ 2 3500 0.075(20), 0.042(12) 0.76(20), 0.74(20)Table 5.5: Table of spectroscopic factors for directly populated 96Sr states. †Spectroscopicfactors and cross sections determined using a relative γ-ray analysis.96Sr 0+ StatesFigure 5.18a was produced by projecting ∆θ = 2◦ slices and using an exponential functionto fit the background shape over a large energy range with an exclusion window around thewell-defined ground state peak. The background function was constrained to have the sameslope within each of the SHARC sections. Figure 5.18b shows an example fit for this state. Thecounts were taken to be the excess above fitted background within the excitation energy win-dow, as indicated in green in figure 5.18b. The angular distribution is in very good agreementwith the calculated ` = 0 DWBA curve.Figure 5.19a was produced by gating on the 0+2 → 2+1 414 keV γ-ray. Significant feeding fromthe 2084 keV to the 1229 keV state was observed. This was identified through the stronglycoincident 414 keV and 854 keV γ-rays. The excitation energy spectrum coincident with 4141125.3. 95Sr(d,p) Results]° [CMθ 0 10 20 30 40 50 60 70 80 90 [mb/sr]Ωdσd 1−101 = 0 keVexcSr E96Sr(d,p): 95Experimental Data/N=2.14 2χ + Jf=0+21L=0 Jo=/N=10.81 2χ + Jf=2+23L=2 Jo=/N=10.54 2χ + Jf=3+25L=2 Jo=/N=27.52 2χ + Jf=4+27L=4 Jo=(a) (b)Figure 5.18: (a) Angular distribution for 95Sr(d,p) to the 96Sr 0+1 ground state. (b) An exampleexponential background fit, indicating the peak region (green) where the counts above thebackground were assigned to the 0+1 state.keV γ-rays in the UQQQ and UBOX sections of SHARC (θCM < 50◦) is shown in figure 5.19b,which clearly indicates the direct population of the 1229 keV and 2084 keV states. The DBOXsection of SHARC was excluded in figure 5.19b as the energy resolution of the projected ex-citation energy spectrum is depreciated significantly by adding the DBOX section. The 1229keV state and the 2084 keV state can clearly be distinguished in the projection when only theUQQQ and UBOX sections of SHARC are used.A fixed excitation energy window of 800-1650 keV was used for the angular range correspondingto the UBOX and UQQQ detectors to remove any feeding from the 2084 keV state. Significantdirect population of the 1628 keV 96Sr state was also observed through coincident 815 keV and813 keV γ-rays, which has a 4.5(3)% branching ratio to the 1229 keV state. The excitationenergy window that was used would include approximately 50% of the protons associated withthe direct population of the 1628 keV state, effectively lowering the total efficiency for detectingthe 1628 keV state to 2.3(2)%. The spin and parity of the 1628 keV state is established to be2+ and so the angular distribution of the background arising from feeding from the 1628 keVstate would be predominantly of ` = 2 character. The very good agreement of the 1229 keVexperimental angular distribution with the ` = 2 DWBA calculation confirms that the amountof 1628 keV feeding is negligibly small.1135.3. 95Sr(d,p) Results]° [CMθ 0 20 40 60 80 100 [mb/sr]Ωdσd 1−101 = 1229 keVexcSr E96Sr(d,p): 95Experimental Data/N=4.42 2χ + Jf=0+21L=0 Jo=/N=19.68 2χ + Jf=2+23L=2 Jo=/N=19.41 2χ + Jf=3+25L=2 Jo=/N=45.52 2χ + Jf=4+27L=4 Jo=(a) (b)Figure 5.19: (a) Angular distribution for 95Sr(d,p) to the 96Sr 1229 keV 0+2 state. (b) Excitationenergy spectrum gated on the UQQQ and UBOX SHARC sections, coincident with 414 keVγ-rays.Figure 5.20a was produced by gating on the 0+3 → 2+1 650 keV γ-ray. As was discussed insection 3.2, the long lifetime of this state resulted in a lower γ-ray detection efficiency. It wasnot possible to produce an angular distribution for this state using only TIGRESS detectorspositioned at θTIG > 120◦ because the statistics were too low. The poor Doppler reconstructionof long-lived states also increased the photo-peak width and caused a large amount of skew-ness in the photo-peak of γ-rays emitted from the long-lived isomer, which made it difficult todistinguish between the 0+3 → 2+1 650 keV γ-ray and the Compton edge of the strong 815 keV2+1 → 0+1 γ-ray.The 1465 keV state angular distribution shown in figure 5.20a has a large amount of scatter,particularly for small center-of-mass angles where the particle statistics were lowest in SHARC.The characteristic maximum at θCM ∼ 30◦ is in quite good agreement with an ` = 0 transferreaction, however this could also be caused by unwanted gating on the large 815 keV γ-rayCompton edge which would include protons associated with ` = 0 transfer to the 1229 keVstate. Equal amounts of γ-ray background were subtracted from above and below the 650 keVγ-ray peak so that the contributions form the 815 keV γ-ray Compton edge would be mini-1145.3. 95Sr(d,p) Results]° [CMθ 0 20 40 60 80 100 [mb/sr]Ωdσd 2−101−10Experimental Data/N=3.10 2χ + Jf=0+21L=0 Jo=/N=3.91 2χ + Jf=2+23L=2 Jo=/N=3.87 2χ + Jf=3+25L=2 Jo=/N=4.39 2χ + Jf=4+27L=4 Jo= = 1465 keVexcSr E96Sr(d,p): 95(a) (b)Figure 5.20: (a) Angular distribution for 95Sr(d,p) to the 96Sr 0+2 1465 keV state. (b)96Sr γ-rayspectrum gated on the excitation energy range 900-1900 keV.mized. Moreover, the γ-ray efficiency correction that was applied to the angular distributionin figure 5.20a did not include the lowered efficiency of detecting long lived states as this isnot well known, and so the spectroscopic factor that was obtained from the fit was not mean-ingful. Figure 5.20b shows the γ-ray spectrum gated on excitation energies of 900-1900 keV,using TIGRESS detectors at all angles. The labelled 650 keV peak can be seen in the spectrum.A γ-ray analysis was used to determine the relative population strengths of the two excited0+ states in 96Sr. The 414 keV 0+2 → 2+1 transition and the 650 keV 0+3 → 2+1 transition areboth pure E2, and so the ratio of the number of γ-rays emitted from these states is expectedto form isotropic γ-ray angular distribution.A series of excitation energy-gated γ-ray spectra were made using only the 135◦ TIGRESScorona ring and the UBOX and UQQQ SHARC sections in order to improve the energy res-olution of both TIGRESS and SHARC. A 1 MeV excitation energy window was used so thatboth the 1229 keV and 1465 keV 96Sr states could be fully included within the energy window,given the resolution of SHARC. The center of the energy window was increased from 500 keVto 2300 keV in steps of 100 keV and a γ-ray spectrum was made for each excitation energyrange. Using this approach, it was possible to plot the counts in the 414 keV and 650 keVγ-ray peaks as a function of excitation energy, and compare them. The shape of the resulting1155.3. 95Sr(d,p) ResultsθTIG Rexp Rsim S3S2 SF(0+3 ) [x10−2]>120◦ 0.22(4) 0.20(6) 1.1(4) 18(7)>135◦ 0.27(7) 0.19(6) 1.4(6) 23(11)Table 5.6: Comparison of experimental and simulated counts ratio for 414 keV and 650 keV γ-rays gated on different angular ranges in TIGRESS. The simulated ratio is discussed in section3.2.curve for a given peak is expected to increase with excitation energy, as the overlap betweenthe energy window and the excitation energy peak becomes larger, and then saturate when thepeak is fully inside the window.Figure 5.21a shows the counts in the 414 keV and 650 keV γ-ray peaks as a function of the centerof excitation energy window. The counts in both peaks can be found to increase as a functionof energy and then saturate at around 1300 keV, which corresponds to the energy where all ofthe direct population of the two states is included in the excitation energy window. The feedingof the 1229 keV state from higher lying 96Sr states, as shown in figure 5.19b, caused a furtherincrease in counts in the 414 keV γ-ray peak with energy. Similarly, the constant number ofcounts in the 650 keV γ-ray from the saturation energy up to beyond 2000 keV indicates thatthis state is also fed from higher lying 96Sr states. The ratio of counts in the 650 keV peak tothe 414 keV peak is given as a function of energy in figure 5.21b. The ratio of counts in the 650keV peak to the 414 keV peak was obtained by averaging the ratio at 1300 keV and 1400 keV,which corresponds to the energy at which the excitation energy window is centered between1229 keV and 1465 keV. This value was determined to be 0.22(6). This analysis was repeatedusing TIGRESS detectors positioned at θTIG > 120◦, and the ratio was measured to be 0.18(3).These results, adjusted to account for the different photo-peak efficiency of TIGRESS at 414keV versus 650 keV, are compared to the simulation results from section 3.2 in table 5.6.The experimental results for each TIGRESS angular range indicate that the measured 414 keVand 650 keV γ-ray rays are consistent with equal population of both excited 0+ states.Given these relative population strengths, the feeding of the 1229 keV 0+2 state from the 1465keV 0+3 state in the 1229 keV angular distribution can also be estimated. Using a population1165.3. 95Sr(d,p) Results [keV]exc E600 800 1000 1200 1400 1600 1800 2000 2200 Counts210 and UBOX+UQQQ°>135TIGθCounts Gated on (a) [keV]exc E600 800 1000 1200 1400 1600 1800 2000 2200 Counts0.10.150.20.250.30.350.4 and UBOX+UQQQ°>135TIGθCounts Ratio Gated on (b)Figure 5.21: (a) Counts in 414 keV (red) and 650 keV (blue) γ-ray peaks as a function ofexcitation energy. (b) Ratio of counts in 650 keV γ-ray peak to 414 keV γ-ray peak as afunction of excitation energy (more details in the text).strength ratio of S3S2 = 1.1, approximately 15% of the total number of 414 keV γ-rays in theexcitation energy range that was used for the 1229 keV 96Sr state angular distribution would beexpected to be fed from the 1465 keV state. In other words, the ` = 0 spectroscopic factor thatwas extracted for the 1229 keV 96Sr state must be reduced by 15% in order to remove feedingeffects from the 1465 keV 96Sr state. The reduced spectroscopic factor for the 1229 keV stateis given in table 5.5. The spectroscopic factor that is given for the 1465 keV state in table 5.5is taken from the θTIG >120◦ measurement as this contained the most statistics.96Sr 815 keV StateIt was not possible to produce an angular distribution for this state due to the strong feedingfrom the 1229 keV state and the resolution of SHARC. This feeding was removed using a γ-ray analysis instead. An excitation energy range of 400-1200 keV, gated on the UBOX andUQQQ sections of SHARC, was used so that all contributions from the 815 keV 96Sr statewere included. The counts in the 414 keV γ-ray peak were corrected for the efficiency ofTIGRESS and subtracted from the efficiency-corrected counts in the 815 keV peak, giving theestimated number of particles associated with the direct population of the 815 keV 96Sr state.This value was compared to the number of counts in the same angular range associated with1175.3. 95Sr(d,p) Resultsdirect population of the 1628 keV 2+ state, and the ratio was found to be 0.34(6). The sameanalysis was carried out in the DBOX section of SHARC and the ratio was found to be 0.5(1).The spectroscopic factor for the direct population of the 815 keV 96Sr state in table 5.5 is theweighted average of the counts ratios multiplied by the spectroscopic factor of the 1628 keV96Sr state.96Sr 1507 keV StateFigure 5.22a was produced by gating on the 2+2 → 0+1 1507 keV γ-ray. It was found that thisstate was weakly populated through 95Sr(d,p). The measured angular distribution is consistentwith the established spin of 2+ [JPA+80].]° [CMθ 0 10 20 30 40 50 60 70 80 90 [mb/sr]Ωdσd 1−10 = 1507 keVexcSr E96Sr(d,p): 95Experimental Data/N=1.97 2χ + Jf=0+21L=0 Jo=/N=1.61 2χ + Jf=2+23L=2 Jo=/N=1.38 2χ + Jf=3+25L=2 Jo=/N=3.33 2χ + Jf=4+27L=4 Jo=(a)]° [CMθ 10 20 30 40 50 60 70 80 90 [mb/sr]Ωdσd 1−10Experimental Data/N=4.58 2χ + Jf=0+21L=0 Jo=/N=6.76 2χ + Jf=2+23L=2 Jo=/N=6.32 2χ + Jf=3+25L=2 Jo=/N=1.32 2χ + Jf=4+27L=4 Jo= = 1793 keVexcSr E96Sr(d,p): 95(b)Figure 5.22: Angular distribution for 95Sr(d,p) to (a) 96Sr 1507 keV state and (b) 96Sr 1793keV state.96Sr 1628 keV StateFigure 5.23a was produced by γ − γ-gating on the strongly coincident 813 keV and 815 keVγ-rays. The almost identical γ-ray energies allowed the same γ-gate to be used for both tran-sitions, which effectively doubled the sensitivity of TIGRESS to this state. Figure 5.23b showsthe excitation energy spectrum for all sections of SHARC that was measured in coincident withboth 813 keV and 815 keV γ-rays, clearly indicating that the 1628 keV was populated directlyin 95Sr(d,p). This state is fed by states above ∼2 MeV, however it was possible to remove most1185.3. 95Sr(d,p) Resultsof the feeding data by using a narrow excitation energy window in each of the SHARC sections.It is estimated that less than 5% of the data in the angular distribution is from other states,and so the spectroscopic factor in table 5.5 is an upper bound.The measured angular distribution is in very good agreement with the ` = 2 DWBA cal-culation, which constrains the spin and parity to be 1+, 2+ or 3+. A suggested spin and parityof 2+ was assigned to this state though β decay studies of 96Rb [JPA+80] using γ-γ angularcorrelation between the 813 keV and 815 keV γ-rays, although 1+ could not be firmly ruled outgiven the available statistics. The observed branching ratios of this state to the 0+1,2 states makeit is highly unlikely that this state has spin and parity 3+. If this state was 1+, the decay to the0+1,2 states would be purely M1 transitions. The Weisskopf estimates for the strength of theseM1 transitions indicate that they would be similar in strength to the 813 keV M1+E2 γ-ray,which is not observed as they are both an order of magnitude less intense. These observationsfavour a spin and parity assignment of 2+ for the 1628 keV 96Sr state.]° [CMθ 0 20 40 60 80 100 [mb/sr]Ωdσd 1−101Experimental Data/N=2.44 2χ + Jf=0+21L=0 Jo=/N=1.01 2χ + Jf=2+23L=2 Jo=/N=0.95 2χ + Jf=3+25L=2 Jo=/N=5.42 2χ + Jf=4+27L=4 Jo= = 1628 keVexcSr E96Sr(d,p): 95(a) (b)Figure 5.23: (a) Angular distribution for 95Sr(d,p) to the 96Sr 1628 keV state. (b) Excitationenergy γ − γ-gated on the 813 keV and 815 keV γ-rays.96Sr 1793 keV StateFigure 5.22b was produced by gating on the 4+1 → 2+1 978 keV γ-ray. It was found that thisstate was weakly populated, with most of the strength coming from feeding from higher levels.1195.3. 95Sr(d,p) ResultsThe measured angular distribution is consistent with the established spin of 4+ [JPA+80].96Sr 1995 keV StateFigure 5.25a was produced by gating on the 1180 keV γ-ray. This state was strongly populateddirectly through the transfer reaction, and feeding from higher states was negligible. Thisstate can be clearly seen in figure 5.17b as the strong 1180 keV γ-ray in coincidence with theexcitation energy range 1600-2400 keV.The measured angular distribution shows strong ` = 2 character which constrains the spin andparity to be 1+, 2+ or 3+. A spin and parity of 3+ is a very unlikely assignment based on theobserved decay branches to the 0+1,296Sr states as these would correspond to M3 transitionswhich, when calculated using standard matrix elements, are predicted to be extremely weaktransitions. A suggested spin and parity of 1+ was assigned to this state though β decaystudies of 96Rb [JPA+80] using γ-γ angular correlation between the 1180 keV and 815 keVγ-rays, although 2+ could not be firmly ruled out given the available statistics. It was arguedin reference [JPA+80] that Jpi = 2+ is unlikely since the 1180 keV transition would connect2+ states with a low E2 admixture which has been observed infrequently, and so Jpi = 1+ is afavourable spin assignment.96Sr 2055 keV StateA candidate 1240(3) keV γ-ray was observed in the γ-ray singles spectrum gated on excitationenergies 1600-2600 keV spectrum, which is shown in figure 5.26b. Figures 5.24a and 5.24b showthe excitation energies and γ-ray rays which were measured in coincidence with the 1240(3)keV γ-ray respectively. The 1240(3) keV γ-ray was found to be in strong coincidence with the815 keV γ-ray, and also possibly with an unknown 830 keV γ-ray. Given the known resolutionof TIGRESS at 815 keV, it is more likely that the 830 keV peak is a fluctuation associatedwith the 815 keV state. The width of the peak at 1270 keV is not consistent with the knowndetector resolution at that energy, and so is likely a statistical fluctuation.These observations support a proposed 96Sr state at 2055(3) keV. No angular distributionanalysis was possible for this state due to low statistics.1205.3. 95Sr(d,p) Results(a) (b)Figure 5.24: (a) 96Sr excitation energies in coincidence with a 1240 keV γ-ray. (b) γ-rays incoincidence with a 1240 keV γ-ray.96Sr 2084 keV StateFigure 5.25b was produced by gating on the 2084 keV direct γ-ray decay to the ground state.This state was strongly populated directly through the transfer reaction, and feeding fromhigher states was negligible. The direct ground state decay of the 2084 keV state can be clearlyseen in figure 5.17b as the strong 2084 keV γ-ray in coincidence with the excitation energy range1600-2400 keV. The measured angular distribution shows clear ` = 2 character, and the strongpopulation of this state confirms that it has positive parity. The spin and parity is constrainedto be 1+, 2+ or 3+. Using similar arguments as for the 1995 keV 96Sr state, the strong decaybranch to the 0+1,296Sr states effectively rules out 3+.The log ft value of the β decay of the 96Rb 2(−) ground state to the 2084 keV 96Sr state wasmeasured to be 6.3 [JPA+80]. Given the change in parity and taking the parity of the 96Rbground state to be negative, the decay is most likely first forbidden which gives ∆I = 0, 1, 2.Combining the angular distribution measurements with the log ft value constrains the 2084 keV96Sr state to have spin and parity 1+ or 2+.1215.3. 95Sr(d,p) Results]° [CMθ 0 20 40 60 80 100 [mb/sr]Ωdσd 1−101Experimental Data/N=14.38 2χ + Jf=0+21L=0 Jo=/N=3.66 2χ + Jf=1+23L=2 Jo=/N=4.20 2χ + Jf=2+23L=2 Jo=/N=4.47 2χ + Jf=2+25L=2 Jo=/N=4.78 2χ + Jf=3+25L=2 Jo=/N=18.47 2χ + Jf=3+27L=4 Jo=/N=19.54 2χ + Jf=4+27L=4 Jo= = 1995 keVexcSr E96Sr(d,p): 95(a)]° [CMθ 0 20 40 60 80 100 [mb/sr]Ωdσd 1−101Experimental Data/N=8.03 2χ + Jf=0+21L=0 Jo=/N=2.30 2χ + Jf=1+23L=2 Jo=/N=2.63 2χ + Jf=2+23L=2 Jo=/N=2.78 2χ + Jf=2+25L=2 Jo=/N=2.86 2χ + Jf=3+25L=2 Jo=/N=11.63 2χ + Jf=3+27L=4 Jo=/N=12.28 2χ + Jf=4+27L=4 Jo= = 2084 keVexcSr E96Sr(d,p): 95(b)Figure 5.25: Angular distribution for 95Sr(d,p) to (a) 96Sr 1995 keV state and (b) 96Sr 2084keV state.96Sr 2120 keV StateFigure 5.26a was produced by gating on the ∼1300 keV γ-ray transition to the 815 keV 2+1 state.The 2113 and 2120 keV 96Sr states decay most strongly via a (39%) 1299 keV and a (91%)1305keV γ-ray respectively. This difference in γ-ray energies could not be resolved. The centroid ofthe fitted γ-ray peak at ∼1300 keV was determined to be 1302.7(5) keV. Figure 5.26b showsthe γ-rays in coincidence with the excitation energy range 1600-2600 keV. A γ − γ analysis ofthe different 2113 keV state decay branches was was not possible due to the available statistics.The weak 328 keV (10%) γ-ray associated with the 2120 keV and the weak 321 keV (6%) γ-rayassociated with the 2120 keV state were not observed. The (21%) 485 keV and (35%) 607 keVγ-rays associated with the 2113 keV state were observed, however it was not possible to extractangular distributions using these γ-ray gates based on the statistics. The counts in the 485keV and 607 keV γ-ray peaks were approximately consistent with the tabulated intensities andindicate that the population strength of the two states are 25(20)% 2113 keV and 75(20)% 2120keV.The angular distribution shown in figure 5.26a has strong ` = 4 character, although it couldalso contain a mixture of ` = 2 and ` = 4. Spontaneous fission decay studies of 248Cm haveestablished the spin of the 2120 keV 96Sr state to be J = 4 [WHC+04]. This is in agreement1225.3. 95Sr(d,p) Resultswith the observed angular distribution, which would support a 4+ spin and parity assignmentfor the 2120 keV 96Sr state. The 2113 keV was populated weakly in 96Rb β and β-n decayand has not been assigned a spin or parity, however a 0+ assignment is very unlikely due tothe observed decay to the 1793 4+1 state. The angular distribution for the 2113 keV state istherefore constrained to have ` = 2 or ` = 4 character, or a mixture of both.The angular distribution shown in figure 5.26a is assumed to be mostly in association withthe 2120 keV state. The spectroscopic factor given in table 5.5 assumes 100% 2120 keV and0% 2113 keV, and so it is an upper limit. A nominal uncertainty of 10% was also assigned tothe branching ratio of this state.]° [CMθ 0 20 40 60 80 100 [mb/sr]Ωdσd 1−10Experimental Data/N=6.99 2χ + Jf=0+21L=0 Jo=/N=9.46 2χ + Jf=1+23L=2 Jo=/N=9.29 2χ + Jf=2+23L=2 Jo=/N=9.00 2χ + Jf=2+25L=2 Jo=/N=9.15 2χ + Jf=3+25L=2 Jo=/N=2.22 2χ + Jf=3+27L=4 Jo=/N=2.71 2χ + Jf=4+27L=4 Jo= = 2120 keVexcSr E96Sr(d,p): 95(a) (b)Figure 5.26: (a) Angular distribution for 95Sr(d,p) to the 96Sr 2217 keV state. (b) 96Sr γ-raysgated on excitation energy range 1600 keV to 2600 keV.96Sr 2217 keV StateFigure 5.27a was produced by gating on the 1402 keV γ-ray transition to the 815 keV 2+1 state.A suggested spin of 2 was assigned to this state though β decay studies of 96Rb [JPA+80] usingγ-γ angular correlation between the 1402 keV and 815 keV γ-rays. The state was not seen inother 96Sr measurements. The measured angular distribution in this work shows clear ` = 2character, and so it is expected that this state has Jpi = 2+.1235.3. 95Sr(d,p) Results96Sr 2576 keV StateFigure 5.27b was produced by gating on the 1761 keV γ-ray transition to the 815 keV 2+1 state.This state has been observed only though β decay studies of 96Rb [JPA+80] and no suggestedspin was given. The angular distribution measurement in this work has strong ` = 2 character,constraining the spin and parity to be 1+, 2+ or 3+.]° [CMθ 0 20 40 60 80 100 [mb/sr]Ωdσd 2−101−101 Experimental Data/N=13.31 2χ + Jf=0+21L=0 Jo=/N=8.04 2χ + Jf=1+23L=2 Jo=/N=8.54 2χ + Jf=2+23L=2 Jo=/N=8.68 2χ + Jf=2+25L=2 Jo=/N=8.45 2χ + Jf=3+25L=2 Jo=/N=19.06 2χ + Jf=3+27L=4 Jo=/N=19.85 2χ + Jf=4+27L=4 Jo= = 2217 keVexcSr E96Sr(d,p): 95(a)]° [CMθ 20 40 60 80 100 [mb/sr]Ωdσd 1−10Experimental Data/N=9.17 2χ + Jf=0+21L=0 Jo=/N=2.47 2χ + Jf=1+23L=2 Jo=/N=2.69 2χ + Jf=2+23L=2 Jo=/N=2.69 2χ + Jf=2+25L=2 Jo=/N=2.42 2χ + Jf=3+25L=2 Jo=/N=11.61 2χ + Jf=3+27L=4 Jo=/N=11.82 2χ + Jf=4+27L=4 Jo= = 2576 keVexcSr E96Sr(d,p): 95(b)Figure 5.27: Angular distribution for 95Sr(d,p) to (a) 96Sr 2217 keV state and (b) 96Sr 2576keV state.96Sr 3239 keV StateA very strong 978 keV 4+1 → 2+1 γ-ray peak was observed for excitation energies in the range2800-3600 keV, however it was not possible to confidently identify the state which was populatedin this range. A small number of 688 keV and 1107 keV γ-rays were observed in coincidencewith the 978 keV γ-ray, indicating that the 2481 keV and 2899 keV 96Sr states were both fedfrom this state (or collection of states). A ∼1300 keV γ-ray peak was also seen in the γ-raysingles spectrum, which is consistent with the known feeding of the 2120 keV state from the2481 keV and 2899 keV 96Sr states. Evidence was also found for the feeding of the 2113 keVstate from the 2481 keV and 2899 keV states, based on a weak 321 keV γ-ray in coincidencewith the 978 keV γ-ray.The 3239 keV 96Sr state was seen in 248Cm studies and is currently the only known state which1245.3. 95Sr(d,p) Resultsdecays to both the 2481 keV and 2899 keV states, through 758 keV and 340 keV γ-rays re-spectively. The maximum of the 978 keV γ-gated excitation energy distribution was found tobe at approximately 3200 keV which is consistent with this state. A small number of 340 keVand 758 keV γ-rays were observed in the γ-ray singles spectrum gated on excitation energies of2800-3600 keV, which correspond to the decay of the 3239 keV state to the 2481 keV and 2899keV states respectively. These transitions were not seen in coincidence with the 978 keV γ-ray,as is shown in figure 5.28b. A 538(3) keV γ-ray was also seen in coincidence with the 978 keValthough this could not be placed in the level scheme.The angular distribution for this state (or collection of states) shows ` = 4 character, con-straining the spin to be 3+ or 4+.]° [CMθ 20 40 60 80 100 [mb/sr]Ωdσd 1 Experimental Data/N=9.64 2χ + Jf=0+21L=0 Jo=/N=3.16 2χ + Jf=1+23L=2 Jo=/N=3.92 2χ + Jf=2+23L=2 Jo=/N=3.77 2χ + Jf=2+25L=2 Jo=/N=4.91 2χ + Jf=3+25L=2 Jo=/N=1.76 2χ + Jf=3+27L=4 Jo=/N=2.01 2χ + Jf=4+27L=4 Jo= = 3239 keVexcSr E96Sr(d,p): 95(a) [keV]γ E200 400 600 800 1000 1200 1400 Counts / 4.0 keV20−020406080100120140Sr State96-Ray Gated Spectrum for 3239 keV γ978 keV (b)Figure 5.28: (a) Angular distribution for 95Sr(d,p) to 96Sr 3239 keV state. (b) Comparisonof measured coincidence 978 keV spectrum (black) with simulated coincidence spectrum (red)using known decays and intensities. The simulated spectrum includes appropriate efficiencyscaling and energy resolution to generate a realistic photo-peak spectrum.96Sr 3500 keV StateFigure 5.29 was produced by gating on the direct transition to the ground state of 96Sr. Thistransition can be seen in figure 5.17b. The measured angular distribution is consistent with` = 2 transfer.1255.3. 95Sr(d,p) ResultsThis state is not consistent with any previously measured 96Sr state and so we propose a newlyidentified excited state in 96Sr at 3500(10) keV, where the error indicated the uncertainty on themeasured γ-ray. No other new transitions were observed in this energy range, indicating thatthe branching ratio for the 3500 keV γ-ray is 100%. An uncertainty of 10% was assumed for thebranching ratio from this state to account for the limited sensitivity and statistics available inthis experiment. Due to the observed direct transition to the ground state, it is highly unlikelythat this transition is of higher multipole order than 2. This constrains the spin and parity tobe 1+, 2+.]° [CMθ 20 40 60 80 100 [mb/sr]Ωdσd 1−10Experimental Data/N=5.92 2χ + Jf=0+21L=0 Jo=/N=2.62 2χ + Jf=1+23L=2 Jo=/N=2.78 2χ + Jf=2+23L=2 Jo=/N=2.73 2χ + Jf=2+25L=2 Jo=/N=2.81 2χ + Jf=3+25L=2 Jo=/N=5.12 2χ + Jf=3+27L=4 Jo=/N=5.06 2χ + Jf=4+27L=4 Jo= = 3500 keVexcSr E96Sr(d,p): 95Figure 5.29: Angular distribution for 95Sr(d,p) to 96Sr 3500 keV state.126Chapter 6InterpretationIn this chapter, the experimental results for 94Sr(d,p) and 95Sr(d,p) are discussed. Both datasets are compared to energy levels and spectroscopic factors that were calculated using the shellmodel. The results are also compared to other experiments and calculations, where available.As was discussed in section 5.1, the absolute experimental spectroscopic factors were foundto be strongly dependent on the optical potential that was used in the DWBA calculations.The shell model calculations predict significantly larger spectroscopic factors than were mea-sured in both data sets, which also indicates that the absolute scaling of the experimental valuesis not well known. However, since the ratio of spectroscopic factors within a given data set areindependent of the unknown absolute scaling factor, the relative spectroscopic strengths withineach data set are still meaningful.For this reason, the experimental spectroscopic factors are discussed relative to the ground statespectroscopic factor when they are compared to the shell model calculations.6.1 Discussion of 94Sr(d,p) results6.1.1 Comparison to 94Sr(d,p) Shell Model CalculationsThe shell model code NushellX [BR14] was used to calculate the energy levels and wavefunctionsof 94,95,96Sr using two model spaces and interactions; jj45 [EHJH+93] and glek [MWG+90]. Aswas discussed in section 1.4, the glek interaction and model space was found to be more suitablefor describing the energy levels in Sr isotopes than the jj45 interaction. For this reason, onlythe glek calculation results are compared to the results from this experimental work.Spectroscopic factors for 94Sr(d,p) were calculated in the shell model so that the experimental1276.1. Discussion of 94Sr(d,p) resultsresults could be compared to shell model configurations. This is carried out by calculating thewavefunction overlap between the 94Sr ground state plus a neutron in a single valence orbitalwith numerous states in 95Sr. In this section, the experimental spectroscopic factors are com-pared to shell model results for each of the model spaces that were discussed in section 1.4, alsoallowing for the investigation of the effect of the proton degrees of freedom on the spectroscopicfactors.Table 6.1 compares the 94Sr(d,p) experimental spectroscopic factors to the shell model re-sults.E [keV] Jpi SF [This Work] SF [glek] : a© SF [glek] : b© SF [glek] : c©0 12+1.00 (0.087) 1.00 (0.553) 1.00 (0.453) 1.00 (0.413)352 32+1.40±0.24 (0.122) 1.56 (0.865) 1.69 (0.767) 1.80 (0.744)681 52+0.52±0.14 (0.045) 0.26 (0.146) 0.39 (0.177) 0.49 (0.202)Table 6.1: Comparison of experimental to calculated spectroscopic factors for 94Sr(d,p), relativeto the 12+state spectroscopic factor. Values in parentheses are absolute spectroscopic factors.a©, b© and c© denote the different model spaces that were calculated with the glek interaction,as discussed in section 1.4.The model spaces a©, b© and c© have the same meaning as in section 1.4 and so they cor-respond to a closed proton 1p32 core, an intermediate 1p32 + 1p12 proton valence space and alarger 1p32 +1p12 +0g92 valence space, respectively. Once again, a maximum of two protons werepermitted to occupy the 0g 92 orbital in model space c©. Each of the three models predictedsimilar spectroscopic factors of the excited 95Sr states relative to the 12+ground state spectro-scopic factor, as can be seen in table 6.1. It can be seen in table 6.1 that calculated groundstate spectroscopic factor decreases from 0.553 in model space a© to 0.453 in model space b©to 0.413 in model space c©. This indicates that the added proton degrees of freedom causefurther fragmentation of the wavefunctions which reduces the spectroscopic factors. In a fullmodel space calculation it is expected that the calculated spectroscopic factors would be evensmaller. Once again, while the absolute values of the experimental spectroscopic factors are1286.1. Discussion of 94Sr(d,p) resultsconsiderably lower than the shell model calculations it may be that this is due to uncertaintyin the absolute scaling factor and the truncated space of the calculations.The calculated spectroscopic factor of the first excited 32+ 95Sr state relative to the calcu-lated 12+ground state calculated spectroscopic factor is consistent with the experimental valueof 1.40(24) for all of the model spaces. The larger spectroscopic factor for the 32+excited statecompared to the 12+ground state that can be seen in table 6.1 simply reflects that there aremore available substates within the 1d32 orbital than in the 2s12 orbital that can be populated bythe transferred neutron. In other words, the total spectroscopic strength for an empty valenceorbital is equal to 2j + 1, which is called the transfer reaction sum rule. In reality, the mixedwavefunction of the 94Sr ground state contains nonzero occupations of valence orbitals whichare above the Fermi energy and so the total spectroscopic strength for each of the orbitals is2j + 1 − 〈nˆ〉. Similarly, the mixed wavefunction contains partially unfilled orbitals below theFermi energy such as the 1d52 , and so there is also some spectroscopic strength which is availablefor this orbital.The calculations predict a low-lying 52+state with substantial spectroscopic strength. Thespectroscopic strength of this state relative to the ground state steadily increases as more pro-ton orbitals are included in the model space, which is a result of the wavefunctions furtherfragmenting within the larger model spaces. The calculations predict a spectroscopic factorfor the low-lying 52+state to be 0.39 and 0.49 of the ground state strength in model spaces b©and a©, respectively. These values are in good agreement with the experimental spectroscopicfactor of the 681 keV 95Sr state, which was found to be 0.52(14) of the ground state for ` = 2,j = 52+. The shell model calculations did not predict significant spectroscopic strength forhigher lying 32+states in any of the model spaces. The combined spectroscopic strength ofall of the higher lying 32+states up to 2 MeV was found to be on the order of 20% of theground state spectroscopic factor, which indicates that these states would not be observed inthis experiment. In addition, large scale shell model calculations that have been carried out for97Zr (N=57) also predict a low-lying 52+state and a second 32+state at a much higher energy[HEHJ+00] [SNL+09]. Taken together, it is therefore very likely that the 681 keV 95Sr state,1296.1. Discussion of 94Sr(d,p) results12+Exp.32+52+12+glek32+52+E [MeV]00.250.50.7510.1SF Scale0.52s121d321d52Figure 6.1: Selected low-lying 95Sr states compared to shell model calculations, where thelength of each line represents the spectroscopic factor. The ground state spectroscopic factorsare normalized to one, and all excited state spectroscopic factors are drawn relative to theground state.which has a previous assignment of (32+, 52+) has spin and parity of 52+.The spectroscopic factors and level energies that were calculated using model space b© give thebest overall agreement with 94Sr(d,p) experimental data. Figure 6.1 compares the 94Sr(d,p)experimental spectroscopic factors to the shell model results using model space b©.The substantial spectroscopic strength of a low energy 52+in 95Sr is an interesting experimentalresult. A simplistic, independent particle model description of the ground state of 94Sr wouldpredict that the neutron 1d52 orbital is fully occupied. In this case, a neutron could not beadded to the already full 1d52 orbital and so the spectroscopic factor for a52+state in 95Srwould be zero. The significant spectroscopic strength of the 681 keV 52+state in 95Sr suggeststhat the ground state wavefunction of 94Sr does not consist of a fully occupied 1d52 orbital,and so contains a substantial component of [2s12 ]2, [1d32 ]2 and possibly [0g 72 ]2 configurations.Substantial occupation of the neutron 2s12 orbital in the ground state wavefunction of94Srwould also inhibit the spectroscopic strength of ` = 0 transfer, which would lower the absolutespectroscopic factor for the 12+ 95Sr ground state. This is consistent with the experimentalfindings.1306.1. Discussion of 94Sr(d,p) resultsIn conclusion, it can be seen in table 6.1 that adding proton degrees of freedom in the va-lence space of the calculation leads to more fragmented wavefunctions, which reduces the puresingle particle character and spectroscopic factors of the states. Further fragmentation of theshell model wavefunctions is expected to occur as the valence space (and hence the numberof possible configurations) of both protons and neutrons increases in the calculations. Thedependence of the calculated spectroscopic factors on proton degrees of freedom indicates thatproton excitations cannot be neglected when describing the low energy states in 95Sr. Suchfragmentation of the wavefunctions may also explain the small absolute spectroscopic factorsthat were extracted in this experiment.Large scale shell model calculations which include fully optimized SPEs and more proton andneutron degrees of freedom, such as the full proton 0g 92 orbital, would be an important nextstep towards an improved description of 95Sr, however such calculations are beyond the scopeof this thesis.Comparison to Other ExperimentsWe can also compare the relative 94Sr(d,p) spectroscopic factors from this work to other (d,p)experiments which populated states in N=57 nuclei close to A=100. At this time, the onlyneutron transfer studies that have been performed in this mass region are 96Zr(d,p)97Zr (Z=40)[BF73] and 98Mo(d,p)99Mo (Z=42) [MM69], and so the results of this work are compared tothese experiments in this section.As 95Sr, 97Zr and 99Mo are all N=57 nuclei, their neutron configurations can be comparedby using the spectroscopic factors of the first few states. Differences that arise between thespectroscopic factors of these experiments can be used to discuss the changing effective singleparticle energies and proton degrees of freedom in the low-lying states.In 97Zr (Z=40), the relative energy spacing and branching ratios of the 32+, 72+and (32+,52+)low-lying excited states are very similar to the low-lying states states in 95Sr [Lab]. However,1316.1. Discussion of 94Sr(d,p) resultsthe energy of the first excited (32+) state is at 1103 keV which is quite large for an even-odd nu-cleus. This can be explained by a Z=40 proton sub-shell closure, due to a significant energy gapbetween the full 1p12 proton orbital and the empty 0g92 proton orbital. The residual interactionbetween nucleons provides additional correlation energy by mixing different configurations. Alarge energy gap between the 1p12 and 0g92 orbitals would suppress the scattering of protonsfrom the 1p12 orbital into the 0g92 and correlation energy associated with these mixed configu-rations would not be available. This would therefore increase the energy of the low-lying 97Zrstates compared to those in 95Sr.Contrastingly, in 99Mo (Z=42) there are 18 excited states in 99Mo below 1 MeV [Lab]. Twovalence protons occupy the 0g 92 orbital above the full 1p12 (Z=40) sub shell, which allows strongconfiguration mixing to take place between valence neutrons and protons. The interactions be-tween the valence nucleons provides a large amount of correlation energy and results in a highdensity of low energy states. This is further supported by the very low-lying 112−state in 99Moat 684 keV, which is above 2 MeV in excitation energy in 95Sr and 97Zr. Table 6.2 comparesthe relative spectroscopic factors from this work to results for 96Zr(d,p) and 98Mo(d,p).Jpi SF [this work] SF , 96Zr(d,p) [BF73] SF , 98Mo(d,p) [MM69]12+1.00 (0.087) 1.00 (1.02) 1.00 (0.67)32+1.40±0.24 (0.122) 0.67 (0.68) 0.64 (0.43)52+0.52±0.14 (0.045) † 0.31 (0.21)Table 6.2: Experimental spectroscopic factors for N=57 nuclei relative to the ground statespectroscopic factor. Values in parentheses are absolute spectroscopic factors. † no 1d52 analysiswas carried out for the 96Zr(d,p) experiment, and so all ` = 2 transfer states were assumed tobe 32+.The relative spectroscopic factors of the excited 32+states compared to the ground state in96Zr(d,p) and 98Mo(d,p) are significantly different to 94Sr(d,p). In this work, the spectroscopicfactor of the 352 keV 32+ 95Sr state was found to be 1.40(24) relative to the ground state. Con-trastingly, the spectroscopic strength of the first excited 32+state in 97Zr and 99Mo was foundto be less than that of the ground state. In the 96Zr(d,p) experiment, it was assumed that the1326.2. Discussion of 95Sr(d,p) resultsd52 orbital was fully occupied and so every ` = 2 orbital angular momentum transfer that wasmeasured was assumed to be from population of the d32 orbital.Most of the spectroscopic strength for the valence neutron orbitals was found to reside inthe first state of each spin and parity in 97Zr. This suggests that these low energy states arewell described by pure single particle excitations. In 99Mo, the spectroscopic strength was foundto be more fragmented across several states, again indicating that the configurations were moremixed.The significant discrepancies between the measured spectroscopic factors in 94Sr(d,p), 96Zr(d,p)and 98Mo(d,p) further supports the conclusion that proton degrees of freedom play an importantrole even at low energies.6.2 Discussion of 95Sr(d,p) results6.2.1 Comparison to 95Sr(d,p) Shell Model CalculationsSpectroscopic factor calculations were carried out for 95Sr(d,p) in NushellX using the sameinteraction and valence space as the 94Sr(d,p) calculations. In this section, model spaces b©and c© are compared to the 95Sr(d,p) experimental results.Figure 6.2 compares the experimental spectroscopic factors to the calculation results usingboth model spaces. Only calculated spectroscopic factors greater than 0.03 (approximately 5%of the ground state strength) are shown in figure 6.2. There are many more calculated stateswith smaller (1-5%) spectroscopic factors. The calculations did not predict any significantspectroscopic strength for the 1d52 orbital, which is in contrast to the calculation results for94Sr(d,p). This can be interpreted as a mostly full 1d52 orbital in the ground state of95Sr, ascan be seen in table 6.4. This is because the 12+ground state of 95Sr contains an unpairedneutron in the 2s12 orbital, which prevents neutron pairs from being scattered into the 2s12orbital from the lower lying 1d52 orbital. Instead, they must scatter into the higher-lying 1d32orbital to vacate the 1d52 which is a less energetically favourable configuration.1336.2. Discussion of 95Sr(d,p) resultsThere are substantial differences between the calculated states and the known low-lying 96Srstates and so it is difficult to quantitatively compare the experimental results to the calculations.Additionally, many of the excited 96Sr states do not have firm spin and parity assignments andso choosing which experimental state corresponds to each calculated state can be ambiguous.For this reason, figure 6.2 presents the calculated levels which have substantial spectroscopicfactors alongside the experimental data so that qualitative similarities can be seen. The cal-culations correctly predict that there is substantial spectroscopic strength for the 1d32 and 0g72orbitals at approximately 2 MeV, which is distributed across several states. Both of the modelspredict a strongly populated ` = 2, Jpi = 1+ state at ∼2 MeV which is in agreement with theexperimental result for the (1+, 2+) 1995 keV 96Sr state. There is, however, no prediction forany strongly populated ` = 2 or ` = 4 states above 3 MeV in either calculation which is indisagreement with the experimental results. At higher excitation energies, the truncated modelspaces will have an increasingly important effect on the calculated states. This is becauseconfigurations that are explicitly excluded, such as proton excitations from the 0f 52 orbital[TTO+16] and neutron excitations into the 0h112 orbital, will become increasingly important[RUSU+09]. Furthermore, the limited occupation of two protons in the 0g 92 orbital will alsoincrease the systematic error in the calculated wavefunctions as the excitation energy increases.The substantial differences between experiment and theory indicate that more comprehensivelarge scale shell models are required for this nucleus.The spectroscopic factors of the 1229 and 1465 keV 0+ states are combined into a singlespectroscopic factor in figure 6.2, which is drawn at an excitation energy of 1347 keV. Thisrepresents the total 2s12 strength in the excited 0+ states which is associated with a sphericalconfiguration. As was discussed in section 1.3, the excited 0+ states would be closer together inenergy in the absence of mixing between the spherical and deformed 0+ configurations. Giventhat the excited 0+ states have been found to contain about equal mixtures of spherical anddeformed configurations, the unperturbed energies of both configurations would be nearly de-generate and halfway between the experimental energies (which is 1347 keV). The deformedconfiguration in the excited 0+ states would not be predicted in the shell model, and so this is1346.2. Discussion of 95Sr(d,p) results0+Exp.2+0+2+2+4+1+ 2+1+ 2+ 4+2+1+ 2+3+x 0.54+1+ 2+glek b©0+1+1+2+2+2+2+3+3+4+4+0+glek c©0+0+1+1+2+2+2+2+3+3+3+3+3+4+4+4+4+2+E [MeV]00.511.522.533.50.1SF Scale1.0 1.02s121d320g 72Figure 6.2: Selected low-lying 96Sr states compared to shell model calculations, where the lengthof each line represents the spectroscopic factor. The experimental ground state spectroscopicfactor is normalized to one, and all experimental excited state spectroscopic factors are drawnrelative to the ground state. The calculated spectroscopic factors are drawn to scale (moredetails in the text).1356.2. Discussion of 95Sr(d,p) resultsnot included in figure 6.2.The spectroscopic factors for the 0+ states are significantly different to the experimental results.The addition of the proton 0g 92 orbital in model space c© lowered the energy of the first excited0+ state to well below the experimental value, which may indicate that the SPE of the 0g 92 istoo low. Table 6.3 compares the spectroscopic factors for the 96Sr 0+ states to the shell modelcalculations.Jpi SF [This Work] SF [glek] : b© SF [glek] : c©0+1 1.00 (0.114) 1.00 (1.575) 1.00 (1.455)0+2 2.98±0.79 (0.34) 0.06 (0.098) 0.07 (0.105)Table 6.3: Comparison of experimental to calculated spectroscopic factors for 0+ states in96Sr, relative to the ground state spectroscopic factor. Values in parentheses are absolutespectroscopic factors. b© and c© denote the different model spaces that were calculated withthe glek interaction, as discussed in section 1.4.The ground state of 96Sr was calculated to contain approximately 90% of the 2s12 spectroscopicstrength. This is not in agreement with the experimental result, which indicates that the groundstate contains at most ∼25% of the 2s12 spectroscopic strength. This large discrepancy can beaddressed further by comparing the underlying orbital occupancies for each of the 0+ states in96Sr to the 95Sr ground state. These occupation numbers are given in table 6.4, where orbitalcan contain a maximum of 2j + 1 nucleons. It can be seen that the calculated occupationnumbers for the ground state of 95Sr are very similar to the ground state of 96Sr for both ofthe models. Additionally, both wavefunctions are calculated to contain almost pure seniority(ν) zero configurations, where all nucleon pairs are coupled to J = 0. This makes a single steptransfer between the two states possible, as there is minimal re-coupling required to transitionbetween the states.On the other hand, there is a significant difference between the calculated occupation numbersfor the ground state of 95Sr and the excited 0+ states in 96Sr. The underlying single particleconfigurations for these states are therefore less similar to those in the 95Sr ground state. Both1366.2. Discussion of 95Sr(d,p) resultsJpi pi1p32 pi1p12 pi0g92 ν1d52 ν2s12 ν1d32 ν0g72 Seniority, ν = 0 [%]b© 95Sr 12+3.90 0.10 0.00 5.54 1.03 0.37 0.05 94.5b© 96Sr 0+1 3.92 0.08 0.00 5.47 1.87 0.55 0.10 95.7b© 96Sr 0+2 3.76 0.24 0.00 4.48 1.48 1.38 0.67 83.4b© 96Sr 0+3 3.64 0.36 0.00 4.11 1.28 1.14 1.48 73.4c© 95Sr 12+3.43 0.10 0.47 5.48 1.03 0.38 0.11 94.1c© 96Sr 0+1 3.36 0.09 0.56 5.19 1.82 0.62 0.37 90.6c© 96Sr 0+2 2.21 0.09 1.70 3.55 1.57 0.99 1.89 42.0c© 96Sr 0+3 2.38 0.17 1.45 3.46 1.28 1.23 2.02 49.7Table 6.4: Occupation numbers for the calculated ground state of 95Sr and the 0+ states in96Sr.models predict that the excited 0+ states contain a substantial component of 1d52 two-particletwo-hole (2p− 2h) excitations into the 1d32 and 0g 72 orbitals. Model space c© also predicts thatthe [0g 72 ]2 configurations are accompanied by proton [0g 92 ]2 configurations, which indicates thatthere is substantial correlation energy associated with these orbitals. This is consistent withthe deformation-driving mechanism in this mass region that was put forward by Federman andPitel [FP79], where the strong proton-neutron residual interaction between spin-orbital partnerorbitals lowers the energy of deformed configurations. The occupation of the proton 0g 92 andneutron 0g 72 spin-orbit partner orbitals indicate that this indeed is a low energy configuration.The excited 0+ states in 96Sr are also calculated to have significant ν 6= 0 components intheir wavefunctions. As a result of this, the ground state of 95Sr would require additional nu-cleon pairs to be broken in order to populate the excited 0+ states in 96Sr through the transferof a neutron, and so this process cannot be described as a simple single step process. Thedissimilar configurations between the calculated 95Sr ground state and excited 0+ states in 96Srgreatly reduce the spectroscopic factor, as can be seen in figure 6.2.1376.2. Discussion of 95Sr(d,p) resultsComparison to Other Shell Model WorkThe occupation numbers from this work can also be compared to shell model calculations for97,98Zr from [SNL+09] and [HEHJ+00], as is shown in table 6.5. It is expected that the neutronconfigurations for Sr and Zr will be similar. The occupation numbers for the ground state of97Zr are presented alongside the 0+ states in 98Zr so that their similarities can be discussedqualitatively.Jpi pi0f 52 pi1p32 pi1p12 pi0g92 ν1d52 ν2s12 ν1d32 ν0g72 ν0h112[SNL+09] 97Zr 12+5.66 3.78 1.84 0.71 5.54 0.96 0.17 0.15 0.1698Zr 0+1 5.52 3.76 1.77 0.93 5.37 1.57 0.33 0.40 0.3298Zr 0+2 5.41 3.60 1.53 1.44 5.07 0.71 0.54 1.26 0.40[HEHJ+00] 97Zr 12+6.00 4.00 1.91 0.09 5.73 0.99 0.15 0.03 0.1198Zr 0+1 6.00 4.00 1.91 0.09 5.76 1.87 0.17 0.06 0.1598Zr 0+2 6.00 4.00 1.56 0.44 5.50 0.26 1.26 0.72 0.2798Zr 0+3 6.00 4.00 0.59 1.41 5.49 0.58 0.98 1.46 0.19Table 6.5: Occupation numbers for the calculated ground state of 97Zr and the 0+ states in98Zr from references [SNL+09] and [HEHJ+00].The results in table 6.5 are in qualitative agreement with the neutron occupation numbers pre-sented in table 6.4. The ground states of 97,98Zr are predicted to have very similar occupationnumbers to each other, which was also found in table 6.4. Additionally, the excited 98Zr 0+states contain 1d52 2p − 2h excitations into the 1d32 and 0g 72 orbitals, which would result inmuch smaller spectroscopic factors compared to ground state transfer.To summarize, the present experimental findings for 95Sr(d,p) show that there is approxi-mately three times more spectroscopic strength for populating the excited 0+ state in 96Sr thanthe 0+ ground state. This indicates that there is a larger overlap between the wavefunctionsof the ground state of 95Sr and the excited 0+ 96Sr configuration than the 96Sr ground state.This inversion of spectroscopic strength was not correctly predicted by shell model calculationsthat were carried out using several different, although limited, model spaces. Large scale shell1386.3. Mixing Between the Excited 0+ States in 96Srmodel calculations with optimized SPEs and TBMEs in addition to a less restricted modelspace would therefore be an important next step to investigating this result further, althoughthese calculations are beyond the scope of this thesis.6.3 Mixing Between the Excited 0+ States in 96SrAs was discussed in section 1.3, the very large ρ2(E0) value between the excited 0+ states in96Sr is an indicator of shape coexistence. This means that there are two 0+ configurations ateffectively the same energy which have different shapes and significant mixing.It is expected that one of the unmixed states is a strongly deformed configuration that wouldnot be populated directly through the 95Sr(d,p) reaction, while the other unmixed state is anearly spherical configuration which would be populated through 95Sr(d,p). A consequence ofthe mixing between the excited 0+ states is that they will both contain some component of thespherical configuration, and so both can be populated through direct one-nucleon transfer. Theγ-ray analysis of the 1465 keV 96Sr state in section 5.3 indicates that the two excited 0+ stateswere populated with approximately equal strengths through 95Sr(d,p).The measured relative population strength S3S2 = 1.1(4) that was given in table 5.6 is con-sistent with a mixing strength of b2 = 1− a2 = 0.52(19) or a2 = 0.48(17). From this result thedegree of quadrupole deformation, β, can be determined by rearranging equation 1.3β = 4√16pi29Z2ρ2(E0)a2(1− a2) (6.1)and using the previously measured monopole strength ρ2(E0) = 0.185(50) from [Jun80] withthe experimental value of a2 = 0.48(17) from this work. The uncertainty of the β value iscalculated using the standard error propagation formulaδβ =√(∂β∂x)2x2 +(∂β∂y)2δy2 (6.2)1396.3. Mixing Between the Excited 0+ States in 96Srwhere x = ρ2(E0) and y = a2.This gives a quadrupole deformation of β = 0.31+0.03−0.02. The small relative error in the calculateddeformation arises because the partial derivative ∂β∂y is effectively zero for y = a2 ∼ 0.5. Figure6.3 shows a plot of equation 6.1, illustrating the uncertainties of a2, ρ2(E0) and β.2 Mixing Strength, a0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9β Quadrupole Deformation, 00.10.20.30.40.50.60.7Quadrupole Deformation as a Function of Mixing StrengthFigure 6.3: Quadrupole deformation as a function of mixing strength for a monopole transitionstrength ρ2(E0) = 0.185(50). The drawn bands represent 68% confidence intervals.The mixing strength that was determined in this work can also be discussed in a broader context.Taking the two excited 0+ states in 96Sr as having equal components of spherical and deformedconfigurations, one would expect approximately equal populations of these states through otherreactions and also through decays from higher-lying states. There are four higher-lying stateswhich are known to feed the 1229 keV state. These states, with their branching ratios, aresummarized in table 6.6.Given a mixing ratio of a2 ∼ 0.5, the matrix element for the γ-ray transition between a higher-lying state and each of the excited 0+ states would be the same. For this reason, the relativebranching ratios for decay to the 1229 keV state and the 1465 keV state would be related bythe energy dependence of the transition, which follows E2`+1 for a transition with multipolar-ity `. At this time, none of the decay branches to the 1465 keV state have been measured.1406.3. Mixing Between the Excited 0+ States in 96SrState Energy [keV] B1229exp B1465calc. [M1] B1465calc. [E2]1628 5.3(4) 0.36(3) 0.06(1)1995 5.0(4) 1.66(1) 0.79(6)2084 51(4) 19.4(15) 10.1(8)2217 8.3(11) 3.66(5) 2.12(3)Table 6.6: Branching ratios for states which are known to feed the 1229 keV 96Sr state, expressedrelative to the strongest decay branch. Estimated branching ratios to the 1465 keV state arealso given, assuming a mixing strength of a2 = 0.5, for a pure M1 and a pure E2 transition (aslabelled).Nevertheless, for many of the states in table 6.6 the calculated branching ratios can be seen tobe very small and so may be beyond sensitivities of previous experiments. The exception tothis is the rather strong 2084 keV→1229 keV transition, which would have been measured inthe work of Jung [Jun80]. Given that the spin of the 2084 keV state is not firmly established,both M1 and E2 multipolarities could contribute to the transition to the 1465 keV state, whichwould predict a branching ratio of at least 20% of the 2084 keV→1229 keV strength. If weinstead assume a mixing strength of 1 − a2 = 0.3, which is the lower limit from this work,the ratio of matrix elements between the 1465 keV state and 1229 keV would drop to 0.43 andso the calculated branching ratio to the 1465 keV state would drop by more than a factor of two.In the recent work of Cle´ment et al. [CZP+16b], a measurement was made of the shape ofthe 1229 keV excited 0+ state, while there was no observation of the 1465 keV state. It is notknown at this time whether the experiment had sufficient sensitivity to measure both states, orwhether the long lifetime of the 1465 keV state prevented a γ-ray analysis from being carriedout.Taken together, these experimental results favour a smaller mixing strength, where the 1229keV state contains more of the spherical configuration and the 1465 keV state is dominated bythe deformed configuration.One can also use the measured mixing ratio to estimate the interaction strength, V, between1416.3. Mixing Between the Excited 0+ States in 96Srthe 0+ configurations. In section 1.3, the coexisting 0+ configurations were described using atwo-level mixing model. The experimental result of effectively 50:50 mixing between the con-figurations simplifies this discussion considerably as it indicates that the unmixed states weredegenerate in energy. As a result, the energy spacing between the 1229 keV state and the1465 keV state is entirely due to the interaction between them, or ∆Ep = 2V . This gives aninteraction strength of approximately 118 keV.We can now use this result to discuss the observed transition from a spherical to a deformedground state at N = 60 in the Sr isotopic chain. The deformed 0+ configuration in 96Sr is lessstrongly deformed than the ground state of 98Sr, which has β = 0.43 [Par12]. The differencein deformation between these structures in 96Sr and 98Sr can be explained in terms of the twoadditional neutrons in 98Sr. It is expected that the additional neutrons in 98Sr will increasethe occupancy of the neutron 0g 72 orbital, which in turn causes increases the occupancy of theproton 0g 92 spin-orbit partner orbital through the strong isoscalar residual p − n interaction.It has been argued by Federman and Pittel [FP79] that this residual interaction is the drivingmechanism behind the emergence of low energy deformed structures in these nuclei, and so theadded neutrons will lower the energy of the deformed configuration further. Indeed, in 98Sr thedeformed structure becomes the lowest energy configuration available and so it is the groundstate.However, the mixing coefficient b2 = 1 − a2 for the coexisting 0+ states in 98Sr was mea-sured to be b2 = 0.0146 [Par12], which is much lower than has been observed in 96Sr. Themixing coefficient is defined asb =1√1 +[R2 +√1 + R24]2 (6.3)where R = ∆EuV , V is the interaction strength between the unmixed configurations and ∆Eu isthe energy spacing between the unmixed states, as before. In the weak mixing limit (R  1),the energy shift due to the mixing interaction V is very small, and so the perturbed energies1426.3. Mixing Between the Excited 0+ States in 96Srare approximately the same as the unperturbed energies (∆Ep ∼ ∆Eu) which givesb ∼ V∆Eu∼ V∆Ep(6.4)From this, the interaction strength can be determined using the experimental energy spacingbetween the coexisting 0+ states in 98Sr. Given that the excited 0+ state is at 215 keV, theinteraction strength V ∼ 26 keV. Such an interaction strength predicts that the difference inthe energies of the unmixed configurations is ∆Eu ∼ 209 keV.From this analysis, the strong ρ2(E0) values between the low-lying 0+ states in 96Sr and 98Srappear to originate from different scenarios. In the case of 96Sr, the unmixed configurationshave a substantial deformation difference of β ∼ 0.31 and are degenerate in energy. There is astrong interaction between the configurations (V ∼ 118 keV) which causes an energy splittingof 235 keV and maximally mixed wavefunctions. Contrastingly, there is weak mixing in 98Srbetween 0+ configurations that have a very large deformation difference of β ∼ 0.43. The weakmixing is caused by an unperturbed energy difference of ∆Eu ∼ 209 keV and a weak interactionstrength (V ∼ 26 keV) between the coexisting 0+ states.From this, a picture of competing structures in the Z∼40, N∼60 region begins to emerge.There are both deformed and spherical structures which are available to the nucleus at lowenergy cost, which are a result of the rearrangement of a few nucleons across important valenceorbitals (namely the pi0g 92 and ν0g72 orbitals). There is a delicate interplay between the attrac-tive pairing interaction which couples like nucleons to J = 0 and favours sphericity, and theattractive residual p−n interaction which has a tendency to produce non-spherical shapes. Asthe number of neutrons increases in this mass region, the energy of the deformed structure isgradually lowered and the degree of deformation increases. At N = 58, it becomes degeneratein energy with low-lying excited spherical 0+ configurations which causes very large mixing andcorrespondingly large E0 transition rates. At N = 60, a critical point is reached where thedeformed structure becomes the lowest energy configuration available, and this new bindingenergy minimum stabilizes until the neutron drip line.143Chapter 7Summary and OutlookIn this work, successful measurements were made of the spins, parities and single particle charac-ter of states in 95Sr and 96Sr through 94Sr(d,p) and 95Sr(d,p) in inverse kinematics, respectively.For the first time, a direct measurement has been made of the [2s12 ]2 component of low-lying 0+states in 96Sr through 95Sr(d,p), which indicates that there is substantially more spectroscopicstrength in the excited 0+ state configuration than in the ground state. Based on a γ-ray in-tensity analysis, it was found that the 1229 keV and 1465 keV excited 0+ states were populatedwith approximately equal intensity, and this measurement allows an experimental constraint tobe placed upon the mixing strength between the underlying spherical and deformed configura-tions of a2 = 0.48(17).Firm spin assignments have been made for the first time and spectroscopic factors have beenextracted for the low-lying 352 keV, 556 keV and 681 keV excited states in 95Sr, and a con-straint has been made on the spin of the higher-lying 1666 keV excited state in 95Sr. Theseresults have been compared to shell model calculations, and the agreement between calculatedand experimentally determined spectroscopic factors for 95Sr was found to be satisfactory. Sim-ilarly, spectroscopic factors have been extracted for 14 states in 96Sr from 95Sr(d,p), and newexperimental constraints have been assigned to the spins and parities of 8 states in 96Sr. Ad-ditionally, two new states have been added to the 96Sr level scheme as of this work. Whilea detailed comparison of the experimental and theoretical 96Sr spectroscopic factors was notpossible at this time, it is my hope that large scale shell model calculations will be carriedout so that the experimental results from this work can be further interpreted within the shellmodel. In particular, it remains a challenge to reproduce the experimentally observed [2s12 ]2component of low-lying 0+ states in 96Sr within the shell model.144Chapter 7. Summary and OutlookDuring the 2013 and 2014 experiments, a substantial amount of 94,95Sr(d,t) data was alsotaken, which is currently under analysis by one of the spokespeople for this experiment. In94,95Sr(d,t) reactions, a neutron is stripped from the Sr nuclei and so these reactions probethe overlap of the 94,95Sr ground state wave function plus one neutron hole configurations withstates in 93,94Sr. These reactions are an excellent tool to study single particle structure andare very similar to (d,p) reactions, however they are suited to the study of occupied valenceorbitals as opposed to empty orbitals in (d,p). Taken together, (d,p) and (d,t) experimentsprovide complimentary measurements of the configurations of valence neutrons across the va-lence orbitals. An analysis of the 94,95Sr(d,t) data will also provide constraints on the spins andparities of states in 93,94Sr, which are mostly unknown.In addition, a 94Sr(t,p) experiment has been approved at TRIUMF and will be undertakensome time in 2017. In 94Sr(t,p), two neutrons are added to 94Sr to populate states in 96Sr. Thecross section for this reaction favours the population of 0+ states with configurations such as[1d52 ]2, [2s12 ]2, [1d32 ]2 and [0g 72 ]2 where the two neutrons are added as an S = 0 pair in a singlestep. For this reason the 94Sr(t,p) experiment will provide a very useful probe of the 0+ statesin 96Sr. Furthermore, it is expected that this data will allow the experimental constraint of themixing strength between the excited 0+ states in 96Sr to be further refined.In conclusion, there are many exciting prospects for furthering our understanding of this region.The results from this thesis and combined with the (d,t) and (t,p) studies will constrain the spinsand parities and elucidate the single particle configurations underlying states in 93,94,95,96Sr. Asystematic analysis of the underlying configurations of states across these nuclei will also offera broad perspective of the role of evolving single particle structure in this transitional region.In turn, these experimental data will provide a benchmark for theoretical models which aimto describe shape coexistence in the N∼60, Z∼40 region, and we hope that this will furtherencourage the vibrant discussion that is taking place.145Bibliography[Ago03] S. Agostinelli. GEANT4 - A Simulation Toolkit. Nuclear Instruments and Methodsin Physics Research Section A, 506(3):250–303, 2003.[BBD+] P. C. Bender, V. Bildstein, R. Dunlop, et al. GRSISort - A Lean, Mean SortingMachine (https://github.com/GRIFFINCollaboration/GRSISort) June 2017.[BF73] C. R. Bingham and G. T. Fabian. Neutron Shell Structure in 93Zr, 95Zr, and 97Zrby (d, p) and (n, He) Reactions. Physical Review C, 7(4), 1973.[BG69] F. D. Becchetti and G. W. Greenlees. Nucleon-Nucleus Optical-Model Parameters,A>40, E<50 MeV. Physical Review, 182(1190), 1969.[BHK16] G. C. Ball, G. Hackman, and R. Kru¨cken. The TRIUMF-ISAC facility: TwoDecades of Discovery with Rare Isotope Beams. Physica Scripta, 91(9), 2016.[BR97] R. Brun and F. Rademakers. ROOT - An Object Oriented Data Analysis Frame-work. Nuclear Instruments and Methods in Physics Research Section A, 387, 1997.[BR14] B. A. Brown and W. D. M. Rae. The Shell-Model Code NuShellX@MSU. NuclearData Sheets, 120, 2014.[Cas00] R. F. Casten. Nuclear Structure from a Simple Perspective. Oxford Science Pub-lications, 2 edition, 2000.[Cat14] Wilton N. Catford. The Euroschool on Exotic Beams, Vol. IV, volume 879 ofLecture Notes in Physics. Springer Berlin Heidelberg, 2014.[CZP+16a] E. Cle´ment, M. Zielin´ska, S. Pe´ru, et al. Low-Energy Coulomb Excitation of 96,98SrBeams. Physical Review C, 94(5), 2016.146Bibliography[CZP+16b] E. Cle´ment, M. Zielin´ska, S. Pe´ru, et al. Spectroscopic Quadrupole Moments in96,98Sr : Evidence for Shape Coexistence in Neutron-Rich Strontium Isotopes atN=60. Physical Review Letters, 116(2):1–6, 2016.[DCV80] W. W. Daehnick, J. D. Childs, and Z. Vrcelj. Global Optical Model Potential forElastic Deuteron Scattering from 12 to 90 MeV. Physical Review C, 21, 1980.[DFS+11] C. Aa. Diget, S. P. Fox, A. Smith, et al. SHARC: Silicon Highly-segmented Arrayfor Reactions and Coulex used in Conjunction with the TIGRESS γ-ray Spectrom-eter. Journal of Instrumentation, 6(02):P02005–P02005, 2011.[DKM14] J. Dilling, R. Kru¨cken, and L. Merminga, editors. ISAC and ARIEL: The TRIUMFRadioactive Beam Facilities and the Scientific Program. Springer Netherlands, 1edition, 2014.[EHJH+93] T. Engeland, M. Hjorth-Jensen, A. Holt, et al. Structure of Neutron Deficient SnIsotopes. Physical Review C, 48(2), 1993.[EMS88] J. Eberth, R. A. Meyer, and K. Sistemich. Nuclear Structure of the ZirconiumRegion, volume 1. Springer-Verlag, 1988.[Fes92] H. Feshbach. Theoretical Nuclear Physics: Nuclear Reactions, volume 2. JohnWiley and Sons, 1992.[FP79] P. Federman and S. Pittel. Unified Shell-Model Description of Nuclear Deforma-tion. Physical Review C, 20(2), 1979.[FPE84] P. Federman, S. Pittel, and A. Etchegoyen. Quenching of the 2p12 − 2p32 ProtonSpin-Orbit Splitting in the Sr-Zr Region. Physics Letters, 140B, 1984.[HEHJ+00] A. Holt, T. Engeland, M. Hjorth-Jensen, et al. Application of Realistic EffectiveInteractions to the Structure of the Zr Isotopes. Physical Review C, 61(August1999):1–11, 2000.[Hey90] K. Heyde. The Nuclear Shell Model. Springer-Verlag, 1 edition, 1990.147Bibliography[HJS49] O. Haxel, J. H. D. Jensen, and H. E. Suess. On the ”Magic Numbers” in NuclearStructure. Physical Review, 75, 1949.[Hod71] P. E. Hodgson. Nuclear Reactions and Nuclear Structure, volume 1. ClarendenPress, Oxford, 1971.[HRH+04] J. K. Hwang, A. V. Ramayya, J. H. Hamilton, et al. High spin states in 95Sr.Physical Review C, 69(067302), 2004.[HW11] K. Heyde and J. Wood. Shape Coexistence in Atomic Nuclei. Reviews of ModernPhysics, 83(December), 2011.[Jam] F. James. MINUIT Function Minimization and Error Analysis(https://inspirehep.net/record/1258343?ln=en) June 2017.[Jon13] K. L. Jones. Transfer Reaction Experiments with Radioactive Beams: From Halosto the R-Process. Physica Scripta, T152:014020, 2013.[JPA+80] G. Jung, B. Pfeiffer, L. J. Alquist, et al. Gamma-Gamma Angular Correlations ofTransitions in 94Sr and 96Sr. Physical Review C, 22(1), 1980.[Jun80] G. Jung. Nuclear Spectroscopy on Neutron Rich Rubidium With Even Mass Num-bers. PhD thesis, Justus Liebig-Universitat, Giessen, 1980.[KAW77] G. E. Knoll, A. Arbor, and M. J. Wiley. Radiation Detection and Measurement.Wiley, 4th edition, 1977.[Kra88] K. S. Krane. Introductory Nuclear Physics. John Wiley and Sons, 2 edition, 1988.[Lab] Brookhaven National Laboratory. Evaluated Nuclear Structure Data File(http://www.nndc.bnl.gov/ensdf/) June 2017.[LPK+94] G. Lhersonneau, B. Pfeiffer, K. L. Kratz, et al. Evolution of Deformation in theNeutron-Rich Zr Region from Excited Intruder State to the Ground State. PhysicalReview C, 49(3), 1994.[Mat] A. Matta. NPTool, A ROOT/Geant4 Based Framework for Nuclear Physics(http://nptool.org/) June 2017.148Bibliography[May49] M. G. Mayer. On Closed Shells in Nuclei. II. Physical Review, 75, 1949.[MM69] J. B. Moorhead and R. A. Moyer. Nuclear-Structure Studies in Mo and Nb Isotopesvia Stripping Reactions at 12 MeV. Physical Review, 184(4), 1969.[MSI+16] P. Mo¨ller, A. J. Sierk, T. Ichikawa, et al. Nuclear Ground-State Masses and Defor-mations : FRDM ( 2012 ). Atomic Data and Nuclear Data Tables, 109-110:1–204,2016.[MWG+90] H. Mach, E. K. Warburton, R. L. Gill, et al. Meson-Exchange Enhancement of theFirst-Forbidden 96Y(0−) to 96Zr(0+) Beta Transition: Beta Decay of the Low-SpinIsomer of 96Y. Physical Review C, 41(1), 1990.[MXY+12] H. Mei, J. Xiang, J. M. Yao, et al. Rapid Structural Change in Low-Lying Statesof Neutron-Rich Sr and Zr Isotopes. Physical Review C, 85(3):1–10, 2012.[Nob13] C. Nobs. Simulating and Testing the TRIUMF Bragg Ionisation Chamber. Master’sthesis, University of Surrey, England, 2013.[Par12] J. Park. Study of Shape Coexistence in 98Sr, 2012.[PP76] C. M. Perey and F. G. Perey. Compilation of Optical Model Parameters. AtomicData and Nuclear Data Tables, 17(1-101), 1976.[Rad] D. Radford. RadWare Software Package (http://radware.phy.ornl.gov/) June 2017.[RGSR+10] R. Rodrguez-Guzman, P. Sarriguren, L. M. Robledo, et al. Charge Radii and Struc-tural Evolution in Sr, Zr, and Mo Isotopes. Physics Letters, Section B, 691(4):202–207, 2010.[RUSU+09] T. Rza¸ca-Urban, K. Sieja, W. Urban, et al. (h112 ,g72)9− Neutron excitation in92,94,96Sr. Physical Review C, 79(2):2–11, 2009.[Sat83] G. R. Satchler. Direct Nuclear Reactions. Oxford University Press, Oxford, 1983.[SG14] C. E. Svensson and A. B. Garnsworthy. The GRIFFIN spectrometer. HyperfineInteractions, 225(127-132), 2014.149[SNL+09] K. Sieja, F. Nowacki, K. Langanke, et al. Shell Model Description of ZirconiumIsotopes. Physical Review C, 79(6):1–9, 2009.[SPH+05] H. C. Scraggs, C. J. Pearson, G. Hackman, et al. TIGRESS Highly-SegmentedHigh-Purity Germanium Clover Detector. Nuclear Instruments and Methods inPhysics Research Section A, 543(2-3):431–440, 2005.[Tho88] I. Thompson. Coupled Reaction Channels Calculations in Nuclear Physics. Com-puter Physics Reports, 7(167-212), 1988.[TN09] I. Thompson and F. M. Nunes. Nuclear Reactions for Astrophysics. CambridgeUniversity Press, 2009.[TTO+16] T. Togashi, Y. Tsunoda, T. Otsuka, et al. Quantum Phase Transition in the Shapeof Zr isotopes. Physical Review Letters, pages 1–5, 2016.[V+91] R. L. Varner et al. A Global Nucleon Optical Model Potential. Physics Reports,201(57), 1991.[W+12] G. L. Wilson et al. Towards 26Na via (d,p) with SHARC and TIGRESS and aNovel Zero-Degree Detector. Journal of Physics: Conference Series, 381(1), 2012.[WHC+04] C. Y. Wu, H. Hua, D. Cline, et al. Multifaceted Yrast Structure and the Onset ofDeformation in 96,97Sr and 98,99Zr. Physical Review C, 70(6):1–9, 2004.[WZD+99] J. L. Wood, E. F. Zganjar, C. De Coster, et al. Electric Monopole Transitions fromLow Energy Excitations in Nuclei. Nuclear Physics A, 651(4):323–368, 1999.[ZZB10] J. F. Ziegler, M. D. Ziegler, and J. P. Biersack. SRIM - The Stopping Power andRange of Ions in Matter. Nuclear Instruments and Methods in Physics ResearchSection B, 268(1818-1823), 2010.150Appendix AFRESCOFigure A.1 gives an example FRESCO input file, which was used to calculate the DWBAcross section for 95Sr(d,p) to the 96Sr ground state assuming a pure∣∣∣95Sr; 12 +g.s.〉 ⊗ ∣∣ν2s12〉configuration.95Sr(d,p)96Sr 0.000 MeV state l=0 total spin j=0 @ E = 5.378 MeV/u NAMELIST &FRESCO hcm=0.1 rmatch=30.0 hnl=0.1 rnl=3.2 centre=-0.1 nnu=36 jtmax=30thmin=1.00 thmax=180.00 iter=1 chans=1 xstabl=1 elab=510.9 / &PARTITION namep='95Sr' massp=94.9194 zp=38 namet='d'    masst=2.0141   zt=1 qval=0.000 pwf=F nex=1  / &STATES jp=0.5 bandp=1 jt=1.0 bandt=1 cpot = 1 fexch=F /&PARTITION namep='96Sr' massp=95.9217 zp=38 namet='p'    masst=1.008   zt=1 qval=3.654 pwf=F nex=1  /&STATES jp=0.0 bandp=1 ep=0.000 jt=0.5 bandt=1 cpot=2 fexch=F/&partition /&POT kp=1 ap=95 rc=1.300  / INCOMING CHANNEL: DETUERON + 95SR.. &POT kp=1 type=1 shape=0 p1=130.000 p2=0.970 p3=0.860 p4=0.000 p5=0.000 p5=0.000   / REAL VOLUME [WS]&POT kp=1 type=2 shape=0 p1=0.000 p2=0.000 p3=0.000 p4=12.000 p5=1.250 p6=0.771   / IMAG SURFACE [WS]&POT kp=1 type=3 shape=0 p1=7.000 p2=0.750 p3=0.500 p4=0.000 p5=0.000 p6=0.000   / SPIN ORBIT [WS] &POT kp=2 ap=96 rc=1.250  / OUTGOING CHANNEL: PROTON + 96SR.. &POT kp=2 type=1 shape=0 p1=58.725 p2=1.250 p3=0.650 p4=0.000 p5=0.000 p5=0.000  / REAL VOLUME [WS]&POT kp=2 type=2 shape=0 p4=0.000 p5=0.000 p6=0.000 p4=13.500 p5=1.250 p6=0.470  / IMAG SURFACE [WS]&POT kp=2 type=3 shape=0 p1=7.500 p2=1.250 p3=0.470 p4=0.000 p5=0.000 p6=0.000  / SPIN ORBIT [WS] &POT kp=3 ap=95 rc=1.3  / BINDING OF NEUTRON TO 95SR &POT kp=3 type=1 shape=0 p1=50  p2=1.3  p3=0.66 / REAL VOLUME [WS]&POT kp=4 at=1 rc=1.0  / BINDING OF NEUTRON TO PROTON &POT kp=4 type=1 shape=2 p1=72.15  p3=1.538  p7=1.0 / REAL VOLUME [GAUS]&POT kp=5 ap=96 rc=1.250  / INTERACTION BETWEEN CORES: PROTON TO 95SR [WS] &POT kp=5 type=1 shape=0 p1=58.725 p2=1.250 p3=0.650 p4=0.000 p5=0.000 p5=0.000  / REAL VOLUME [WS]&POT kp=5 type=2 shape=0 p4=0.000 p5=0.000 p6=0.000 p4=13.500 p5=1.250 p6=0.470  / IMAG SURFACE [WS]&POT kp=5 type=3 shape=0 p1=7.500 p2=1.250 p3=0.470 p4=0.000 p5=0.000 p6=0.000  / SPIN ORBIT [WS] &pot /  &OVERLAP kn1=1 ic1=1 ic2=2 in=2 kbpot=4 isc=1 nn=1 l=0 sn=0.5 j=0.5 be=2.225 / N + P &OVERLAP  kn1=11 ic1=2 ic2=1 in=1 kbpot=3 isc=1 nn=3 l=0 sn=0.5 j=0.5 be=5.879  / N + 95Sr binding in 95Sr 0.000 MeV STATE  &overlap  / DESCRIBES THE BOUND STATE OF THE TRANSFERRED PARTICLE IN BOTH THE INITIAL AND FINAL STATE &COUPLING icto=2 icfrom=1 kind=7 ip1=0 ip2=-1 ip3=5 / &cfp  in=2 ib=1 ia=1 kn=1 a=1.0   / [DEUTERON]  &cfp  in=1 ib=1 ia=1 kn=11 a=1.0  / [96Sr 0.000 MeV STATE] &cfp /  &coupling / Figure A.1: Example FRESCO input file for 95Sr(d,p) to the 96Sr ground state.151Appendix BSHARC Solid AnglesA solid angle describes the two-dimensional angular coverage of an object in space, and ismeasured in steradians (sr). This is effectively an angular area. In spherical co-ordinates aninfinitesimal solid angle dΩ is simply defined asdΩ = sin θdθdφ (B.1)and the total solid angle Ω is the closed surface integral over the angular range of the objectΩ =∮sin θdθdφ (B.2)which has a total integral over all possible angles of 4pi. This is the maximum possible solid angleand physically reflects an object which is completely enclosed. For a non-spherical geometry,the general co-ordinate system-independent expression for the solid angle is defined asΩ =∮rˆ · dAr2(B.3)where rˆ is the unit vector from the origin and dA is an inifitesimal normal vector of the surface.The dot product rˆ · dA effectively maps this arbitrarily oriented surface element onto the unitsphere. The solid angle subtended by an object of fixed size decreases with its distance from theorigin, which is accounted for by the 1r2term. This integral, computed over the entire surfaceA, gives the total solid angle of any object from a given point in space. This expression can beused to calculate the total solid angle coverage of an array of separate objects, such as elementsof a detector array.152Appendix B. SHARC Solid AnglesThe total acceptance of SHARC is the solid angle sum over all active detector regions. Themost useful way to evaluate this sum is to divide the detector into a large set of small pixelsand add up the solid angle of each of these. That way, individual strips can be easily excludedfrom the calculation if they are removed from the analysis.SHARC has two principal sections, each with a symmetry which can be used to simplify thesolid angle calculation. The QQQ detectors have cylindrical symmetry with respect to thebeam axis and so the solid angle of each pixel in a QQQ can easily be calculated by simplyusing the θ and φ range subtended by the pixel as limits in equation B.2. The solid angle ofpixels within the planar DSSD detectors are most readily calculated by converting to cartesianco-ordinates. For a surface in the x-z plane, equation B.3 simplifies toΩ =∫ x2x1∫ z2z1y(√x2 + y2 + z2)3dxdz =[[arctan(xzy√x2 + y2 + z2)]x2x1]z2z1(B.4)where the integral limits can be chosen to calculate either an individual pixel or an entire strip.The calculated solid angle for all pixels of a DSSD detector and a QQQ detector are shown infigures B.1a and B.1b. The solid angles (in milli-steradian) are plotted as a function of strip(a) (b)Figure B.1: Solid angle of each individual pixel in (a) a DSSD detector and (b) a QQQ detector.number and not geometrical position, which is why the quadrant cylindrical geometry of a QQQ153Appendix B. SHARC Solid Anglesdetector is not shown in B.1b. As the number of counts in a given detector region is dependenton the solid angle of that detector element, these plots also describe the intensity pattern ofa calibration source placed at the center of the array. The strips that were excluded from theanalysis of the 2013 94Sr data set and the 2014 95Sr data set are listed in tables B.1 and B.1,respectively.Detector Front Strips Excluded Back Strips Excluded5 0, 23 476 0-23 0-477 0, 23 23, 39, 478 0, 23 41, 44, 479 1 2, 3, 21, 36, 46, 4710 4, 12 ,19 21, 33, 35, 3711 0, 1, 12, 16 11, 3712 1, 2, 12 18, 28, 29, 32, 3613 0-15 0-2314 0-15 0-2315 12 0, 5, 10, 1116 6, 8 1, 16, 23Table B.1: Summary table of SHARC the 204 strips that were excluded from the 2013 94Sranalysis.Detector Front Strips Excluded Back Strips Excluded5 0, 2, 3, 10, 13, 22, 23 24, 26 476 0, 1, 2, 3, 6, 23 0, 1, 2, 3, 3, 5, 37, 477 0, 2, 3, 5, 6, 7, 11 5, 6, 26, 418 0, 4, 6, 10, 13, 14, 17, 23 2, 10, 20, 24, 28, 29, 35, 38, 39, 40, 42, 44, 469 2, 6, 14 18, 36, 4510 3, 7, 15 4, 5, 35, 4711 2, 10 18, 25, 26, 29, 31, 3312 2, 6 8, 35, 4513 6, 7, 8, 9, 10 4, 16, 18, 19, 2314 0 0, 2315 0 1, 20, 2316 5, 7, 9, 12 8, 16, 23Table B.2: Summary table of SHARC the 107 strips that were excluded from the 2013 95Sranalysis.154Appendix CAnalysis CodesThe analysis for this project was carried out using the ROOT framework [BR97], which wasadapted to suit the specific needs of this experiment. The data was stored as ROOT trees, andwas sorted using the GRSISort package [BBD+].GRSISortGRSISort [BBD+] was founded and developed by P. C. Bender circa 2013, with ongoing de-velopment by the GRIFFIN collaboration. GRSISort is based on the ROOT framework, andprovides additional detector classes which are designed to efficiently store the tree data and todescribe in detail the geometry of detector systems such as SHARC and TIGRESS. The primaryfunctionality of GRSISort is to produce lean trees which contain essential event informationsuch as measured charges, calibrated energies, timestamps and addresses of detector elements.A number of analysis tools have also been developed for GRSISort, several of which weremy contributions. These are summarized as follows;• TReaction - A relativistic two-body reaction kinematics class. The main feature of thistool is to calculate the kinematics of a general two-body reaction by using Lorentz trans-formations to boost between the laboratory (lab) frame and center-of-mass (CM) frame.This class was used to convert from measured lab frame kinetic energy to CM excitationenergy, and to convert between lab and CM angles. Both of these functionalities wereessential to the angular distribution analysis presented in this thesis. It was also used todetermine the velocity of the recoil nucleus which was vital for the Doppler reconstructionof γ-rays. TReaction is also a useful tool for plotting kinematics curves, and was used topredict the angular coverage of SHARC for each of the reaction products.155GRSISort• TNucleus - A nuclear data storage structure. TNucleus parses a mass file from [Lab] tolook up the mass, Z and N of a given nucleus. By creating multiple instances of TNucleus,TReaction is able to determine the Q-value of reactions.• TSRIM - An energy loss calculation tool. TSRIM parses the stopping power tables whichare produced by the software package TRIM [ZZB10] to calculate the energy lost by ionsin the target and in SHARC as a function of distance and energy. TSRIM was usedextensively throughout this analysis to determine the reconstructed energy, as defined inequation 4.4.A number of additional tools were also developed in order to make the analysis flexible androbust and to facilitate extensive consistency checks. The following short sections describe themain programs that were developed by myself for this thesis work.TSharcAnalysisTSharcAnalysis is a suite of functions that is designed to facilitate calibrations and analysis ofSHARC experiments. The main functionalities are summarized as follows;• Calculates the effective target and detector thicknesses as a function of angle (using for-mulae such as equations 4.2 and 4.3).• Calculates the energy deposited by reaction products in SHARC by calculating energylosses in the target, detector dead layers and sensitive detector material (using TSRIM andTReaction). This functionality also works in reverse, and so can calculate the kinematicenergy of the reaction products based on measured energies.• Calculates the solid angle of each element within SHARC and uses this information toproduce an acceptance curve (in both the lab frame and CM frame).• Parses an optional bad strips file, which removes selected strips from the acceptance curve.• Allows a position offset to be applied to the target within SHARC. An uncertainty canalso be assigned to the target position, which is then included in the acceptance curve.156GRSISort• Carries out simulated angular distribution measurements and particle spectra using DWBAcurve outputs from FRESCO.This code is freely available and can be found at https://github.com/steffencruz/SharcAnalysis.TTigressAnalysisTTigressAnalysis is a suite of functions for analysis of TIGRESS (plus SHARC) experiments.The primary analysis functionalities are summarized as follows;• A separate program, MakeEasyMats, loops over the data trees to produce numerous 1D,2D and 3D analysis histograms of excitation energy, γ-ray energy, and center-of-massangle. These histograms are used as an input to TTigressAnalysis.• TTigressAnalysis carries out projections on the 2D and 3D histograms from MakeEasy-Mats to create γ-gated and particle energy/angle gated histograms for analysis (such asthose in figure 4.18).• Calculates the TIGRESS absolute efficiency curve from raw data files and calculates theefficiency at any energy up to 4 MeV.• Fits 1D γ-ray spectra to carry out background subtraction and to extract photo peakcounts.• Fits 2D γ-ray versus TIGRESS angle matrix as a series of 1D γ-ray energy spectra andfitting a user-specified photo-peak. In this way, the Doppler correction was examined byproducing plots such as figure 4.11b. Angular correlations can also be calculated in asimilar way.In addition to these data analysis tools, a γ-ray simulation suite is also included in TTigress-Analysis. The main functions of this suite are summarized as follows;• A text file containing the excitation energies, transition energies and transition intensitiesbetween states in a nucleus is parsed and us used to determine the photo-peak energiesand areas of a theoretical γ-ray spectrum. Compton scattering effects are not included.157GRSISort• The experimental efficiency and resolution of TIGRESS can be applied to the theoreticalγ-ray spectrum to make realistic photo-peak spectra, which can also be gated on anexcitation energy range.• By adjusting the population strength of different states, a realistic γ-ray spectrum canbe produced which reproduces the observed experimental γ-ray spectrum. These statestrengths can also be saved and read for use across multiple sessions.• Theoretical γ-gated γ-ray spectra and γ−γ matrices can also be produced, and comparedto data.This code is freely available and can be found at https://github.com/steffencruz/TigressAnalysis.TFrescoAnalysisTFrescoAnalysis is a program that was developed to carry out DWBA analyses on the experi-mental angular distributions. The main functionalities are summarized as follows;• Automatically constructs FRESCO files from various templates using a simple interactiveuser interface (within the GRSISort environment).• Carries out DWBA calculations by running FRESCO in session with the generated inputfiles.• Fits data to calculated DWBA cross sections to extract normalization constant and spec-troscopic factors.• Enables multiple optical model parameters to be fitted in-session and parameter sets tobe saved and read for use across multiple sessions.• Performs χ2 + 1 searches to give good estimations of parameter uncertaintiesThis code is freely available and can be found at https://github.com/steffencruz/FrescoAnalysis.158Appendix DLow Energy Background in SHARCA large quantity of β-decay was measured in the UBOX and UQQQ sections of SHARC. Thiswas caused by the decay of radioactive beam-like nuclei in the downstream beam dump. FigureD.1a shows the overlaid ∆E energy spectrum for the UBOX and UQQQ sections in the 201495Sr experiment. It can be seen that the quantity of background data was substantially largerthan the transfer data, with up to 100 times more statistics at very low energies.Figure D.1b shows the γ-ray spectrum gated on all particles measured in the UBOX and UQQQsections of SHARC for a series of different energy thresholds. For very low thresholds such as400 keV and 600 keV, the γ-ray spectrum is dominated by the Doppler corrected β-decaytransition lines. The Compton continuum created by these background γ-rays made it difficultto identify transitions from the nucleus of interest. An energy threshold of 1 MeV was thereforeused in SHARC so that the γ-ray transitions of interest (those between states in 96Sr in thisexample) were much larger in size than those from the β-decay background.(a) (b)Figure D.1: (a) ∆E energy spectrum for upstream detectors. (b) γ-ray spectrum gated onparticles with different threshold ∆E energy in upstream detectors.159Appendix ECalculation of ElectromagneticTransition RatesElectromagnetic transition rates can be calculated using the so-called Weisskopf approximation,which is a commonly used for order-of-magnitude estimates. This model assumes that theradiation is produced by the transition of a single nucleon within a nucleus of mass A from aninitial state with energy Ei and spin Ii to a final state with Ef and If . The model assumesthat there is maximal overlap between the initial and final wavefunction so that the matrixelement for the transition is unity. In this way, the multipolarity and energy dependence ofelectromagnetic transitions can be studied [Cas00]. The emitted photon can carry angularmomentum L which is restricted by the electromagnetic selection rules|If − Ii| < L < If + Ii (E.1)Where L=0 is not allowed, since a photon has an intrinsic spin of s = 1~. At a given multipoleorder, electric and magnetic transitions induce opposite parity changes.Electric: ∆pi = (−1)L (E.2)Magnetic: ∆pi = (−1)L+1 (E.3)All transitions that do not violate the selection rules will in principle occur, however contribu-tions from higher multipole orders quickly become extremely small. In this way, only the firstfew multipole orders need to be considered when estimating electromagnetic transition rates.The total transition rate between two states is the sum of all possible multipole transitions.The transition rates for the first few multipole orders, λ(σL), are given in table E.1.160Appendix E. Calculation of Electromagnetic Transition RatesMultipole Order, L λ(EL) λ(ML)1 1.0 x 1014A23E3 5.6 x 1013E32 7.3 x 107A43E5 3.5 x 107A23E53 34 A2E7 16 A43E74 1.1 x 10−5A83E9 4.5 x 10−6A2E9Table E.1: Weisskopf estimates for first few electromagnetic rates. Energy is in units of MeV.In this work, positive parity to positive parity sates were studied, and so only M1, E2, M3 andE4 transitions need to be considered, which have relative strengths 1 : 1.4x10−3 : 2.1x10−10 :1.3x10−13. It is well known that E2 transition rates are frequently at least an order of mag-nitude larger than the Weisskopf estimates, suggesting that many nucleons participate in thetransition. This is strong evidence for collectivity in nuclei. For this reason E2 transition ratescan be comparable in magnitude to M1 rates. For an initial state that can decay to more thanone final state, the transition rate is the sum of each individual (or partial) transition rates.The branching ratio for a given transition, Bi, is the ratio of the partial decay rate to the totaldecay rate.Bi =λiΣkλk(E.4)The branching ratios are usually expressed as a relative quantity so that the strongest transitionis normalized to 100% strength. Table E.2 shows an example of calculated transition rates forthe 2084 keV 96Sr state using Weisskopf estimates. Each possible spin and parity assignmentfor the 2084 keV state leads to different prediction for the branching ratios to lower lying states.Table E.3 compares these values to the measured branching ratios. All branching ratios in tableE.3 are given relative to the strongest transition. The calculated Weisskopf estimates for thebranching ratios were compared to experimental values using a χ2 analysis.χ2 =∑ (Bexp −BW )2δBexp(E.5)The branching ratios for Jpi = 1+ were in better agreement with the data than Jpi = 2+ andJpi = 3+, indicating that this spin and parity assignment is the most likely. The significant161Appendix E. Calculation of Electromagnetic Transition RatesJpiinitial Efinal [MeV] Eγ [MeV] Jpifinal λ(M1) [s−1] λ(E2) [s−1] λ(M3) [s−1] λtotal [s−1]1+, 2+, 3+ 1.507 0.577 2 1.1x1013 2.1x109 1.5x102 1.1x10131+ 1.229 0.855 0 3.5x1013 1.5x1010 2.4x103 3.5x10132+ 1.229 0.855 0 - 1.5x1010 2.4x103 1.4x10103+ 1.229 0.855 0 - - 2.4x103 2.4x1031+, 2+, 3+ 0.815 1.269 2 1.1x1014 1.1x1011 3.7x104 1.2x10141+ 0 2.084 0 5.1x1014 1.3x1012 1.2x106 5.1x10142+ 0 2.084 0 - 1.3x1012 1.2x106 1.3x10123+ 0 2.084 0 - - 1.2x106 1.2x106Table E.2: Weisskopf estimates for the decay of the 2084 keV 96Sr state to states with establishedspin and parity, using measured branching ratios from [Lab].difference in calculated branching ratios between the 2084 keV to 1229 keV transition and 2084keV to ground state transition is purely a result of the energy dependence of the Weisskopfformula. The experimentally measured values of 51% and 100% for these transitions indicatesthat the matrix elements could be significantly different, and that there may be enhanced E2strengths due to collectivity in these transitions.Efinal [MeV] Eγ [MeV] Bexp [%] BW for Jpi = 1+ [%] BW , 2+ [%] BW , 3+ [%]1.507 0.577 15.6(21) 2.1 9.4 9.41.229 0.855 51(4) 6.9 0.0 0.00.815 1.269 27(3) 22.5 100.0 100.00 2.084 100(11) 100 1.1 0.0χ2/N 3.21 7.26 7.27Table E.3: Weisskopf estimates for the decay of the 2084 keV 96Sr state to states with establishedspin and parity, using known branching ratios.162

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