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Decay spectroscopy of europium-160 Yates, Daniel 2019

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Decay Spectroscopy of Europium-160byDaniel YatesB.Sc., Pacific University, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2019c© Daniel Yates 2019The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Decay Spectroscopy of Europium-160submitted by Daniel Yates in partial fulfillment of the requirements for thedegree of Master of Science in Physics.Examining Committee:Dr. Reiner Kru¨cken, Department of Physics and AstronomySupervisorDr. Iris Dillmann, Physical Science Division, TRIUMFSupervisory Committee MemberiiAbstractNuclei far from the traditional proton and neutron shell closures with “magicnumbers” often deviate from spherical shapes and are deformed. The excited-state structure of these nuclei is built on the interplay of collective motionswith single-particle degrees of freedom. Neutron-rich europium (Z = 63)nuclei around N = 104 are located midshell in both proton and neutronnumber. These nuclei exist in a region of large deformation, and the excited-state structure of these nuclei has not been studied in-depth yet. β-decaydata was taken for a number of neutron-rich europium isotopes, including160Eu, at TRIUMF-ISAC using the GRIFFIN spectrometer. For this thesis,a comprehensive β-decay study was carried out on 160Eu and the daughter160Gd. 10 new excited states and 41 new transitions have been identified inthe excited-state level scheme of 160Gd, and for the first time, a thoroughanalysis on the β-decay of 160Eu was completed.iiiLay SummaryThe atomic nucleus, commonly thought of as simply a static cluster of protonsand neutrons at the centre of an atom, is actually a complex and dynamicsystem that is still not thoroughly understood. Nuclei that contain more (orless) neutrons than the stable nuclei that are known to most people undergoradioactive decay and thus are difficult to produce and study. High-poweredaccelerators and state-of-the-art detector arrays, like those used at TRIUMF,allow scientists to study these radioactive nuclei in order to advance our un-derstanding of the complex system that is the atomic nucleus. This workpresents partial results of an experiment examining the radioactive decay ofunstable, neutron-rich europium isotopes at the TRIUMF-ISAC radioactivebeam facility.ivPrefaceThis work is part of a larger experimental campaign undertaken by the GRIF-FIN collaboration examining the nuclear structure of 160−166Eu. This cam-paign is lead by Dr. I. Dillmann (TRIUMF) and Prof. P.E. Garrett (U. ofGuelph), and the GRIFFIN spectrometer is overseen by Dr. A.B. Garnswor-thy (TRIUMF) and Prof. C.E. Svensson (U. of Guelph). This experimentwas conducted during June 2017. The author, D. Yates, was not a memberof the GRIFFIN collaboration at that time and did not participate in datacollection.The analysis presented in this thesis was done solely by the author, withhelp from Dr. N. Bernier (UBC/TRIUMF), Dr. M. Bowry (TRIUMF), Dr.B. Olaizola (TRIUMF), and Y. Saito (UBC/TRIUMF), as well as the author’ssupervisors, Prof. R. Kru¨cken and Dr. I. Dillmann. GRSISort, the analy-sis package used, was developed by the Gamma-Ray Spectroscopy at ISAC(GRSI) group at TRIUMF and the nuclear physics group at the Universityof Guelph.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Symbols and Acronyms . . . . . . . . . . . . . . . . . . . . xviAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Theory and Motivation . . . . . . . . . . . . . . . . . . . . . . . 22.1 Nuclear Structure . . . . . . . . . . . . . . . . . . . . . . . . . 22.1.1 Shell Model . . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Nuclear Deformation . . . . . . . . . . . . . . . . . . . 82.1.3 Collective Motion . . . . . . . . . . . . . . . . . . . . . 112.1.4 Two-state Mixing . . . . . . . . . . . . . . . . . . . . . 152.2 Radioactive Decay . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 Nuclear Decay Law . . . . . . . . . . . . . . . . . . . . 172.2.2 β-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.3 γ-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.4 Conversion Electrons . . . . . . . . . . . . . . . . . . . 273 Review of Literature . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Reaction Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Previous β-decay Studies . . . . . . . . . . . . . . . . . . . . . 28vi3.2.1 Study by D’Auria et al. [1] . . . . . . . . . . . . . . . . 283.2.2 Study by Morcos et al. [2] . . . . . . . . . . . . . . . . 303.2.3 Comparison of Previous Work . . . . . . . . . . . . . . 303.3 Recent β-decay Study by Hartley et al. [3] . . . . . . . . . . . 334 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1 Radioactive Beam Production . . . . . . . . . . . . . . . . . . 354.2 GRIFFIN Spectrometer . . . . . . . . . . . . . . . . . . . . . . 374.2.1 γ-ray Detectors . . . . . . . . . . . . . . . . . . . . . . 404.2.2 Other GRIFFIN Ancillary Detectors . . . . . . . . . . 425 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.1 Data Processing and Corrections . . . . . . . . . . . . . . . . . 445.1.1 Summing Corrections . . . . . . . . . . . . . . . . . . . 445.1.2 Addback Mode . . . . . . . . . . . . . . . . . . . . . . . 465.2 β-gated γ-singles . . . . . . . . . . . . . . . . . . . . . . . . . . 465.3 β-gated γ − γ Coincidence . . . . . . . . . . . . . . . . . . . . 495.4 Relative Intensities . . . . . . . . . . . . . . . . . . . . . . . . 545.4.1 “Gating From Above” Method . . . . . . . . . . . . . 545.4.2 “Gating From Below” Method . . . . . . . . . . . . . . 555.5 Absolute Intensities . . . . . . . . . . . . . . . . . . . . . . . . 575.6 β-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.6.1 β-feeding Intensities . . . . . . . . . . . . . . . . . . . . 575.6.2 log(ft) Values . . . . . . . . . . . . . . . . . . . . . . . 595.6.3 Spin-parity assignments . . . . . . . . . . . . . . . . . . 595.7 Half-lives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.8 Fast-timing Techniques . . . . . . . . . . . . . . . . . . . . . . 616 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.1 β − γ − γ Coincidence Analysis . . . . . . . . . . . . . . . . . 636.1.1 Doublets . . . . . . . . . . . . . . . . . . . . . . . . . . 636.1.2 Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.1.3 Low-energy Transitions . . . . . . . . . . . . . . . . . . 706.2 Level Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.3 log(ft) Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.4 Spin-parity Assignments . . . . . . . . . . . . . . . . . . . . . 866.4.1 Existing States . . . . . . . . . . . . . . . . . . . . . . . 876.4.2 New States . . . . . . . . . . . . . . . . . . . . . . . . . 896.5 Half-lives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.6 Half-life of 1071 keV State . . . . . . . . . . . . . . . . . . . . 966.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101vii6.7.1 Comparison to Hartley et al. [3] . . . . . . . . . . . . . 1016.7.2 Mixing of γ- and Kpi = 4+-bands . . . . . . . . . . . . 1017 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . 112Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114AppendicesA Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118B Data Analysis Techniques . . . . . . . . . . . . . . . . . . . . . 123viiiList of Tables2.1 Selection rules for β-decay. Parentheses indicate a transitionthat is not possible if either Ii or If is 0. . . . . . . . . . . . . . 212.2 Weisskopf estimates for a number of electric and magnetic tran-sitions. The energy E is the energy of the emitted γ-ray in MeVand A the number of nucleons. . . . . . . . . . . . . . . . . . . 242.3 γ-ray multipolarity for the lowest order angular momentumand parity of an emitted γ-ray. . . . . . . . . . . . . . . . . . . 256.1 Identified excited levels and associated depopulating γ-rays.Intensities are normalized out of each level. Results are com-pared to the ENSDF evaluation [9] and more recent data ifdifferent from the evaluation. . . . . . . . . . . . . . . . . . . . 766.2 γ-rays associated with the decay of 160Gd. Intensities are takenrelative to the 173 keV γ-ray. For absolute γ-ray intensities,multiply by 0.21(4). For absolute transition intensities, multi-ply the absolute γ-ray intensity by (1 + α). . . . . . . . . . . . 816.3 β-intensities and log(ft) values for states populated by low-spin(left) and high-spin (centre) β-decay. States that could not beassigned to the isomeric or ground state β-decay are given inthe right column and log(ft) values for the isomeric/groundstate decays were calculated. . . . . . . . . . . . . . . . . . . . 856.4 New spin-parity assignments and log(ft) values for excitedstates in 160Gd. Values are compared to the ENSDF evalu-ation [9]. Recent data that conflicts or updates the evaluationis noted as well. States that could not be definitively associatedwith a parent state have log(ft) values for both the isomeric/-ground state decays given. . . . . . . . . . . . . . . . . . . . . . 866.4 Continued from previous page. . . . . . . . . . . . . . . . . . . 876.5 Extracted half-lives for the isomeric state and ground state β-decay of 160Eu. Results are compared to those published inRef. [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92ixList of Figures2.1 Example of energy levels associated with solutions of theScho¨dinger equation for the 3D infinite well and harmonic os-cillator potentials. Circled numbers correspond to the numberof nucleons at each shell closure. Picture taken from Ref. [4]. . 42.2 Woods-Saxon Potential (orange) compared to the square well(green) and harmonic oscillator (blue dashed) potentials. Fig-ure adapted from R. Kru¨cken. . . . . . . . . . . . . . . . . . . . 52.3 Energy levels for the Woods-Saxon Potential of Eq. 2.5 (“In-termediate form”) and for the Woods-Saxon Potential with aspin-orbit term (“Intermediate form with spin orbit”). Themagic numbers (circled) are reproduced with inclusion of thespin-orbit term. Figure taken from Ref. [4]. . . . . . . . . . . . 62.4 Example energy splitting of different K substates in a prolatedeformed nucleus. Adapted from Ref. [6]. . . . . . . . . . . . . 82.5 Nilsson orbitals for the Z =50-82 region. The deformationparameter  is nearly the same as β and represents the defor-mation of the nucleus [6]. . . . . . . . . . . . . . . . . . . . . . 92.6 Nilsson orbitals for the N=82-126 region. The deformationparameter  is nearly the same as β and represents the defor-mation of the nucleus [6]. . . . . . . . . . . . . . . . . . . . . . 102.7 Measured ground state quadrupole deformation for isotopesnear 170Dy. Plot courtesy of I. Dillmann. Data taken fromENSDF [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.8 Example of two-state mixing. The pure states at energies E1and E2 are perturbed by an energy ∆ES , resulting in the mixedstates at energies EI and EII . Picture taken from Ref. [6]. . . . 162.9 Example of two-state mixing affecting transition rates. Theunmixed 2+2 state, which normally has forbidden (F) transitionsto the other states, may have allowed (A) transitions to thestates when mixed with the 2+1 state. Picture taken from Ref. [6]. 17x2.10 Relative importance of the three major γ-ray interaction types.The lines show where neighboring effects are of equal impor-tance. Germanium detectors have Z = 32. Picture taken fromRef. [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1 Adopted band structure of 160Gd. From Ref. [9]. . . . . . . . . 293.2 Published excited states and transitions associated with the β-decay of 160Eu from Ref. [1]. Dashed lines indicate tentativetransitions. The parentheses on transitions indicate the relativeγ-ray intensities (in percentage). Level scheme taken from Ref. [1] 313.3 Published excited states and transitions associated with theβ-decay of 160Eu from Ref. [2]. Level scheme taken from Ref. [2]. 323.4 Partial level scheme for excited states and transitions associ-ated with the 5− ground state β-decay of 160Eu. New tran-sitions and levels are shown in red, with blue indicating re-arranged transitions and black indicating previously publishedlevels and transitions. Level scheme taken from Ref. [3]. . . . . 344.1 Schematic of the IG-LIS. Surface-ionized species (red) are re-pelled by the electrostatic barrier. Neutral atoms of the ele-ment of interest are selectively ionized by the laser-ionizationprocess and then extracted through the exit electrode [31]. . . . 364.2 Laser scheme used to ionize europium atoms. The two-steplaser ionization leaves the electron in an auto-ionizing (AI)state. Laser scheme courtesy of J. Lassen. . . . . . . . . . . . . 374.3 GRIFFIN detector array. Radioactive beam is delivered fromthe far side of the array. The tape station box can be seen inthe foreground. . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4 Detected β-particles as a function of tape cycle time. Beamdelivery begins at approximately 10 seconds and proceeds for380 seconds. At t = 390 s, beam delivery is halted, and theimplanted sample is allowed to decay. . . . . . . . . . . . . . . 394.5 Render of GRIFFIN HPGe clover detector. The four individ-ual germanium clovers are shown as different colours. Thecryogenic dewar is seen in the background. Measurements areshown for the aluminum housing of the clover. Picture takenfrom Ref. [34]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.6 A section of the GRIFFIN detector array. A LaBr3 detec-tor (circular face) can be seen between three HPGe detectors(square faces). Picture taken from Ref. [33]. . . . . . . . . . . . 42xi5.1 Example of a γ-ray cascades where summing corrections mustbe taken into account. . . . . . . . . . . . . . . . . . . . . . . . 455.2 (a) Reduction of background at low energies with Addbackmode. (b) At high energies, the photopeak efficiency inAddback mode is significantly increased compared to single-crystal mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.3 Plot of the time difference between β-particles and γ-ray de-tection. The two-dimensional gate to select prompt γ-rays isshown in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.4 Comparison of γ-ray singles and β-gated γ-ray singles spectra.The suppression of the contaminant 570 keV γ-ray from 207Bican be seen in the inset. . . . . . . . . . . . . . . . . . . . . . . 495.5 Plot of the time difference between β-gated γ-rays versus theenergy of the first detected γ-ray. The two-dimensional timecut to select prompt γ − γ coincidences is overlaid in red. . . . 505.6 β-gated γ − γ coincidence matrix for (a) the overall spectrumand (b) a zoomed-in region showing coincidences between the409/417 and 925 keV γ-rays and the 409/413/417 and 995 keVγ-rays. Areas of increased statistics correspond to coincidencesbetween γ-rays. . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.7 (a) γ − γ coincidence matrix. A gate on the 310 keV γ rayis shown in red. Background gates on either side of the gateare shown in black. (b) Projection of the gated spectrum from(a). Background subtraction has been applied, resulting in thelabeled scatter peaks. . . . . . . . . . . . . . . . . . . . . . . . 535.8 Example level scheme for the “gating from above” and “gatingfrom below” methods. A gate on γf is used to determine thebranching ratio between γa and γc. If γa is weak, gating frombelow on γb can allow the determination of the intensity of γa. 565.9 Number of detected β-particles as a function of tape cycle time.The fit is that of Eq. 2.20 between 10 and 390 seconds and anexponential decay afterwards; both fits also include a constantbackground term. Approximately 1.32(7)·107 β-particles weredetected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.10 Time distribution and fit of γ-rays associated with the groundstate β-decay of 160Eu. The x-axis is the time elapsed sincebeam delivery to GRIFFIN halted, corresponding to t = 390 sin the overall tape cycle. . . . . . . . . . . . . . . . . . . . . . . 60xii5.11 Chop analysis performed on the ground state half-life. Thevariation in half-life with the first included bin in the fit isshown. The extracted half-life (solid line) and associated un-certainty (dashed lines) are overlaid. . . . . . . . . . . . . . . . 616.1 Gate on 925 keV γ-ray. The 264 and 826 keV components ofthe 266 and 822 keV singles doublets can be seen. . . . . . . . . 646.2 Comparison of the 1144 keV peaks in three different gates. Thepeak centroid in the 1042 and 1215 keV gates, which depop-ulate the same level, is different from the centroid location inthe 606 keV gate. . . . . . . . . . . . . . . . . . . . . . . . . . . 656.3 Comparison of 1302 and 1305 keV gates from the doublet peakat 1304 keV. Different peaks are clearly seen in the differentgates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.4 Gate on the 384 keV γ-ray. Coincidences with the 1150 and1225 keV transitions are clearly seen. No coincidence with the899 keV γ-ray is observed. . . . . . . . . . . . . . . . . . . . . . 686.5 Gate on 1188 keV γ-ray. Coincidences with only the groundstate band, as well as the feeding 1028 keV γ-ray (inset), canbe seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.6 Gate on the 914 keV (a) and the 989 keV (b) γ-rays. The 82keV γ-ray can be seen in both spectra. . . . . . . . . . . . . . . 726.7 Gate on the 809 keV (a) and the 983 keV (b) γ-rays. The 116keV γ-ray can be seen in both spectra. . . . . . . . . . . . . . . 736.8 Level scheme of 160Gd resulting from the β-decay of 160Eu.Levels, transitions, and spin-parity assignments that have notbeen previously published are shown in red. Arrow widthsindicate relative γ-ray intensities. . . . . . . . . . . . . . . . . . 756.9 Summed background-subtracted time spectra for the (a) 5−ground state and (b) 0− isomeric state β-decays of 160Eu. . . . 936.10 Chop analysis of the 5− ground state half-life. The change infitted half-life is observed when the first (a) and last (b) binsof the fit are changed. The extracted half-life and associateduncertainty are overlaid as horizontal lines in both plots. Thehalf-life and uncertainty reported in Ref. [3] are overlaid in redin both plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94xiii6.11 Chop analysis of the 0− isomeric-state half-life. The change infitted half-life is observed when the first (a) and last (b) binsof the fit are changed. The extracted half-life and associateduncertainty are overlaid as horizontal lines in both plots. Thehalf-life and uncertainty reported in Ref. [3] are overlaid in redin both plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.12 Half-life (a) and chop analysis (b-c) of the ground state decayusing only the γ-rays identified in Ref. [3]. The extracted half-life and associated uncertainty (black) and the half-life pub-lished in Ref. [3] (red) are overlaid in (b-c). . . . . . . . . . . . 976.13 Half-life (a) and chop analysis (b-c) of the isomeric state decayusing only the γ-rays identified in Ref. [3]. The extracted half-life and associated uncertainty (black) and the half-life pub-lished in Ref. [3] (red) are overlaid in (b-c). . . . . . . . . . . . 986.14 (a) Energy spectrum in first LaBr3(Ce) detector in an event.The gate on the 413 keV γ-ray to start the timing signal isshown. (b) Energy spectrum in the second LaBr3(Ce) detectorafter the gate in (a) is applied. The stop signal for the TAC isfrom hits in either the 822 or 995 keV gates (shown overlaid). . 996.15 Half-life of the 1071 keV state. The half-life is measured bythe time difference between the feeding 413 keV transition andthe depopulating 822 and 995 keV transitions using LaBr3(Ce)detectors and a TAC. The Gaussian-exponential convolutionfit is overlaid in red, and the Gaussian component in black. . . 1006.16 Comparison of Single Crystal and Addback modes for the (a)928 keV, (b) 1408 keV, and (c) 1750 keV peaks. The significantincrease in photopeak area in Addback mode is characteristicof summed peaks. . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.17 (Left) The unmixed 4+1 and 4+γ states are shown at unknownenergies X and Y. (Right) The two states after mixing, nowlocated at 1071 and 1148 keV. The not-yet observed 159 keV4+II −→ 2+γ is tentatively shown. . . . . . . . . . . . . . . . . . . 1036.18 Summary of results from the two-state mixing between the 4+γand 4+1 states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.19 Configurations in states in 160Gd. The hexadecapole configu-rations in the 1071 keV state are calculations for the Kpi = 4+1state in 166Er, and the 1071 keV state in 160Gd is assumed tohave the same hexadecapole configurations. × denotes cou-pling of single-particle orbitals, and⊗indicates the couplingof quasi-particle configurations. Data taken from Refs. [3, 47, 48].110xivA.1 Energy calibration for GRIFFIN HPGe crystals 0-7. . . . . . . 121A.2 Absolute efficiency of entire GRIFFIN HPGe array. . . . . . . . 122xvList of Symbols andAcronymsa Mixing ratio parameterA Total nucleon numberAγ Photopeak area of γ-rayα Internal conversion coefficientb Mixing ratio parameterβ Deformation parameterβ2 Quadrupole deformation parameterBRγ γ-ray branching ratio Deformation parameter (similar to β)γ Detection efficiency of γ-rayγ Axial deformation parameter (when not a γ-ray)HPGe High-purity Germanium (detector)I Angular momentum of nucleusIβ β-feeding intensityIγ Intensity of γ-rayj Angular momentum of a nucleonK Projection of total angular momentum onto symmetry axis` Orbital angular momentumL Total orbital angular momentum of systemλ Decay constantm mass (of nucleon)mfi Nuclear matrix elementN Neutron numberpi Parityφ Wavefunction (unmixed)ψ Wavefunction (mixed)Qβ− Q-value of β-decayR0 Nuclear radius scaling constant (≈ 1.2 fm)σ Electromagnetic type (Electric or Magnetic)s SpinxviS Total spin of systemt1/2 Half-lifeτ LifetimeV (r) Nuclear potentialZ Proton numberB(σL) Reduced transition rateDAQ Data acquisition SystemENSDF Evaluated Nuclear Structure Data FilesGRIFFIN Gamma-Ray Infrastructure For Fundamental Investigations of NucleiHPGe High purity germaniumIG-LIS Ion Guide-Laser Ion SourceISAC Isotope Separator and AcceleratorPACES Pentagonal Array for Conversion Electron SpectroscopyRIB Radioactive Isotope BeamRILIS Resonant ionization laser ion sourceZDS Zero-Degree ScintillatorxviiAcknowledgementsThank YouMy sincere thanks to my supervisors, Dr. Reiner Kru¨cken and Dr.Iris Dillmann, for their continued support, knowledge, and guidancethroughout the past 2 years.I would like to thank the members of the GRSI group for their supportand assistance. I would especially like to thank Dr. Nikita Bernierfor her continued assistance and mentorship throughout my time atTRIUMF and UBC.Thank you to all the friends I’ve had throughout my life, and thank youto the many friends I’ve made in Vancouver. Without you, I would notbe the person I am today.I would not have gotten to where I am today without the love, care, andsupport of my parents. I cannot thank them enough for all that theyhave done for me over the years.Financial Support fromThe University of British Columbia: Department of Physics andAstronomyTRIUMF: Canada’s particle accelerator centreIsoSiM: Isotopes for Science and Medicine (NSERC CREATE program)xviiiChapter 1IntroductionOf the four fundamental forces that govern nature, the strong nuclear force isless understood than the well-known electromagnetic and gravitational forcesand the lesser-known weak nuclear force. While great strides on understand-ing the strong nuclear force have been made in the past century, our currentunderstanding and ability to accurately model this force still leaves muchto be desired. This holds especially true when one attempts to predict andmodel heavy nuclei. While theory derived from first principles can fairlyaccurately describe light nuclei, attempting to describe heavy nuclei with asimilar approach is near impossible as of now. Thus, further information onthe structure of heavy nuclei is needed to improve the predicting power ofnuclear models and better our understanding of nuclear structure in thesenuclei.Though 160Eu exists relatively close to stability, its β-decay to 160Gd hasnot been thoroughly studied; until recently, clean neutron-rich lanthanidebeams were difficult to produce. Thus, at the time of this experiment, onlytwo publications [1, 2] from the 1970’s had supplied information on this β-decay. Last year, more information was published by Ref. [3], but the levelof information about 160Gd from β-decay is minor when compared to whathas been determined through reaction studies. An intensive study of the β-decay of 160Eu will provide important nuclear structure information on theneutron-rich, heavy, deformed nucleus 160Gd.This thesis presents detailed decay spectroscopy of 160Eu collected by theGRIFFIN spectrometer at TRIUMF-ISAC in 2017. The motivation and rel-evant nuclear structure theory is presented in Chapter 2, followed by a lit-erature review of the isotope and the experimental setup in Chapters 3 and4. The analysis techniques used for this analysis are presented in Chapter 5,and finally Chapter 6 gives the results determined from the data using theanalysis techniques of Chapter 5.1Chapter 2Theory and Motivation2.1 Nuclear Structure2.1.1 Shell ModelThe nuclear shell model is a powerful tool used to describe the single particlestructure of the nucleus. The following sections will describe the differentmodels, starting with the basic spherical shell model and advancing into morecomplex models that account for deformation and collective motion in thenucleus.Spherical Shell ModelThe starting point to nuclear structure is understanding the spherical shellmodel, commonly referred to simply as the “shell model.” This model assumesthat each nucleon moves in a central potential produced by all other nucleons.One begins by examining two different potentials: the infinite square well andthe harmonic oscillator. The following derivations follow the description inRef. [4].One looks to solve the three-dimensional time-independent Schro¨dingerEquation for each scenario:− ~22m(∂2ψ∂x2+∂2ψ∂y2+∂2ψ∂z2)+ V (x, y, z)ψ(x, y, z) = Eψ(x, y, z) (2.1)where ψ denotes the wavefunction, V (x, y, z) the potential of the system, mthe mass of the particle, and E the energy of the system.For the three-dimensional square well, each dimension has a length a,creating a box to confine the particle in. The Schro¨dinger equation takes theform− ~22m(∂2ψ∂x2+∂2ψ∂y2+∂2ψ∂z2)= Eψ(x, y, z). (2.2)2Solutions to Eq. 2.2 are found using the common method of solving lineardifferential equations: separation of variables. The resulting energy solutionisEnxnynz =~2pi22ma2(n2x + n2y + n2z)(2.3)where nx, ny, and nz are independent natural numbers. The ground stateenergy for this system has energy E111 = 3~2pi22ma2, while the first excited statehas energy E211 = 6~2pi22ma2and is triply-degenerate. Clear gaps in energy canbe seen in the energy spectrum of the solutions (Fig. 2.1), hinting at theexistence of shells within the nucleus.For the harmonic oscillator potential, the problem is cast in sphericalcoordinates. The associated potential of the harmonic oscillator is given asV (r) = 12kr2, where k denotes the oscillator constant and r the distance fromequilibrium. Solution of the Schro¨dinger equation with this potential yieldsenergy eigenvalues given byEn = ~ω0(n+32)(2.4)where n is a whole number corresponding to the oscillator shell. Energy levelsassociated with Eq. 2.4 are shown in Fig. 2.1. The angular momentum ` ofthe system is constrained to be a whole number less than n and must havethe same parity as n.As seen in Fig. 2.1, the shell closure numbers above 20 do not agreebetween the two potentials, and neither potential accurately predicts theexperimentally-determined shell closure numbers: 2, 8, 20, 28, 50, 82, 126,and 184 [5]. As both the 3D square well and 3D harmonic oscillator were ini-tial estimates of the potential, clearly a better potential is needed to describethe nuclear shell structure.A more realistic potential–known as the Woods-Saxon potential–gives abetter approximation of the nuclear potential:V (r) =V01 + er−R0A1/3a(2.5)where V0 ≈ −50 MeV, R0 ≈ 1.2 fm, A is the number of nucleons (and thusR0A1/3 gives the approximate spherical radius of a nucleus with A nucleons),and a ≈ 0.524 fm [4]. The Woods-Saxon potential can be seen overlaid withthe square well and harmonic oscillator potentials in Fig. 2.2. The Woods-Saxon Potential better describes the potential within the nucleus because itmore accurately follows the density distribution of nucleons. This can be3Figure 2.1: Example of energy levels associated with solutions of theScho¨dinger equation for the 3D infinite well and harmonic oscillator poten-tials. Circled numbers correspond to the number of nucleons at each shellclosure. Picture taken from Ref. [4].approximated as an incompressible centre, followed by a decrease in densityas one moves out from the centre of the nucleus, and finally no nucleon densitypast a certain radius.The associated energies for the Woods-Saxon potential, which can onlybe solved numerically, are shown in Fig. 2.3. One sees that the Woods-Saxon potential itself still does not reproduce the known magic numbers. Theaddition of a spin-orbit term of the form Vso(r)~` · ~s [5] to Eq. 2.5 reproducesthe shell closure numbers well, as can be seen in Fig. 2.3. Energy levels withinthe spherical nucleus have degeneracy 2j + 1 due to the spherical symmetry,4V(r)[MeV]Figure 2.2: Woods-Saxon Potential (orange) compared to the square well(green) and harmonic oscillator (blue dashed) potentials. Figure adaptedfrom R. Kru¨cken.where the total angular momentum j of a state is sum of the spin (s) andorbital angular momentum (`) of that state (j = ` + s). A more thoroughtreatment of the Woods-Saxon potential with the spin-orbit term can be foundin Ref. [4].Nilsson ModelAlthough the nuclear shell model does well to describe energy levels of nucleinear shell closures, it fails when moving away from the magic numbers tomore exotic nuclei in both proton and neutron number. The assumption ofa spherical potential is not valid far from shell closures, as nuclei are seento exhibit deformation in these regions. When that spherical symmetry isbroken, so is the 2j + 1 degeneracy of states, and the nuclear states must bedescribed in terms of other quantities.5Figure 2.3: Energy levels for the Woods-Saxon Potential of Eq. 2.5 (“Inter-mediate form”) and for the Woods-Saxon Potential with a spin-orbit term(“Intermediate form with spin orbit”). The magic numbers (circled) are re-produced with inclusion of the spin-orbit term. Figure taken from Ref. [4].Instead, the energy levels of the nucleons depend on the component of jprojected onto the symmetry axis of the nucleus in its rest frame. A nucleonin a g9/2 orbital, for example, can have a value Ω = proj(j) between −9/2and +9/2, with Ω changing by a value of ±1. Since the negative and positiveprojections have the same energy, the projection Ω is said to have a valueof 1/2, 3/2,...j and degeneracy 2 [4]. It should be noted that Ω refers tothe single-particle angular momentum; K is also used with these deformedorbitals, and it refers to the total angular momentum projection onto the6symmetry axis. Since rotational angular momentum in low-lying states ofaxially-symmetric nuclei is often perpendicular to the symmetry axis, K isequal to Ω in most cases, and the two are often used interchangeably [6].In the Nilsson Model [7], the 2j + 1 particle degeneracy is broken due tothe projection of j. The corresponding Nilsson orbitals are each filled with upto two nucleons (since the positive and negative projections of j each carrythe same energy). There are then j + 1/2 Nilsson orbitals in place of thecorresponding spherical shell model orbital, with each orbital containing asmany as two nucleons.To construct the diagram of Nilsson orbitals, one determines if the nucleusof interest is prolate (β2 > 0) or oblate (β2 < 0), where β2 is the quadrupoledeformation (see Sec. 2.1.2). For a prolate nucleus, orbitals of low K see adecrease in their energies, while higher K orbitals increase in energy. Foroblate nuclei, the opposite is true: higher K see a lowering of energy, and anincrease for lower K. It is important to note that the energy splitting is notuniform: orbitals that decrease in energy have smaller energy gaps betweenthem, whereas orbitals that have increased in energy see larger energy spacing.An example of the splitting for the i13/2 orbital in a prolate deformed nucleusis shown in Fig. 2.4.To achieve a description of all Nilsson orbitals in a region, one extendsthe energy splitting described above to all of the shell model orbitals withinthe region. The Nilsson orbital diagrams for the Z = 50 − 82 and N =82 − 126 regions, which are relevant for 160Gd, are shown in Figs. 2.5 and2.6. When drawing a Nilsson diagram, one must remember a basic fact ofquantum mechanics: two fermions with the same quantum numbers may notoccupy the same state. Thus, orbitals with the same K and parity cannotcross in the diagram. If two states with the same Kpi approach each other,they will repel in order to avoid crossing.Nilsson orbital energy eigenvalues are denoted asKpi[NnzΛ]where N is the major oscillator shell, nz the oscillator quanta projection, andΛ is the projection of the orbital angular momentum along the symmetry axis.Often, the parity pi is omitted, as it can be determined easily if the originalshell model orbital is known. A more comprehensive and complete derivationof Nilsson orbitals can be found in Ref. [4, 6].7250 Collectivity, Phase Transitions, DeformationPig. 7.3. Variation of single-particle energies of i^ orbits with different projections K (orientations6) as a function of deformation (/} > 0, prolate, to the right).Recalling a fundamental rule of quantum mechanics that no two levels withthe same quantum numbers may cross (an infinitesimal interaction will causethem to repel when they get sufficiently close) and noting that the onlyremaining good quantum number for these orbits is K, it then follows that notwo lines in the Nilsson diagram corresponding to the same K value (andparity) cross. As two such lines approach each other they must repel (see Fig.1.9). Thus, it is now possible to incorporate several; values into the Nilssondiagram and to extend it to realistic deformations where the energies ofdifferent orbits intermingle. This is shown in Fig. 7.4, which gives the Nilssondiagram for two different regions. Each line, representing a Nilsson state,starts out straight and is downward or upward sloping according to the angle ofthe orbit relative to the main mass of the nucleus. It only starts to curve whenit approaches another level with the same K and parity. The entire structure ofthe diagram relies thus on only three factors: K splitting (resulting from theeffects of a short-range nuclear interaction in a deformed field), level-levelrepulsion, and the input single-particle shell model energies.The Nilsson wave functions are equally easy to deduce qualitatively, eventhough they are a complex result of a multistate diagonalization. (We ignorehere the phases of the various terms in these wave functions, most of which canonly be obtained by explicit diagonalization.) The interaction that leads toconfiguration mixing in Nilsson model is of quadrupole form. (We will discussother deformed shapes later.) For very small quadrupole deformations j3, thenuclear wave functions must be nearly pure in;'; as the deformation increases,the configuration mixing will increase.The nondiagonal mixing matrix elements of the quadrupole interaction, not13/211/25/27/29/23/21/2i 13/2E!!=0K22Figure 2.4: Example energ splitting of different K substates in a prolatedeformed nucleus. Adapted from Ref. [6].2.1.2 Nucl ar DeformatioIn regions far from shell cl sures, ncluding the region containing 160Gd, thenuclear ground state shape is often not spherical, but instead is deformed.The most common and lowest order deformation is a quadrupole deforma-tion, wh re the nucleus appears either s r tched (like an American football)or squished (like a frisbee). The former is referred to as prolate, and thelatter as oblate. The quadrupole-deformed nuclear radius is described by theexpressionR = Rs[1 +∑µαµ(t)Y2µ(θ, φ)](2.6)with Rs being the theoretical spherical radius of the nucleus, αµ(t) (possiblytime-dependent) expansion coefficients, and Y2µ(θ, φ) the 2nd order spherical8The Deformed Shell Model or Nilsson Model 251surprisingly, tend to mix configurations that differ by two units in angularmomentum and in which the nucleon spin orientation is not changed. Forexample, in the 50-82 shell, the dM and s1/2 orbits have large matrix elementsand mix substantially, even though they are slightly further separated than thed5/2 and dM orbits. The g7/2 and d^ mix more than g7/2 and ds/2do. Likewise, inthe 82-126 shell, the quadrupole matrix element between the pM and f7/2 orbitsis strong. However, the closeness of the energies of the i,a and h9/2 orbits leadsto substantial mixing, even though the matrix element is not favored. There-fore, combining a regard for the energy separations of different shell modelorbits and the most important quadrupole mixing matrix elements, one canestimate the Nilsson wave functions, that is, the composition of the wavefunctions in terms of amplitudes for different; subshells.Consider the example of the 82-126 neutron shell shown in Fig. 7.4. As /?increases, the £,„ and hm orbits begin to mix. We recall that the angle of theorbital orientation depends primarily on the ratio Klj (8 ~ sin~J KJj ~ Klj forFig. 7.4. (a) Nilsson diagram for the Z = 50-82 regions. The abscissa is the deformation parameter£, which is nearly the same as /3. (Gustafson, 1967).Figure 2.5: Nilsson orbitals for the Z =50-82 region. The deformation param-eter  is nearly the same as β and represents the deformation of the nucleus[6].harmonics.For a static quadrupole deformation, conventionally one sets α1 = α−1 =0, as these coefficients represent the motion of the center of mass of thenucleus. Additionally, the other coefficients are redefined as α0 = β2·cosγand α2 = α−2 = β2·sinγ. β2 and γ are commonly used to characterize the9252 Collectivity, Phase Transitions, DeformationFig. 7.4. (b) Nilsson diagram for the N - 82-126 regions. The abscissa is the deformationparameter e, which is nearly the same as ft. (Gustafson, 1967).small K). Small angles can occur either because K is low, or for given K,because; is high. Thus, the energies of the K -1/2,3/2, and 5/2 orbits from theh9/2 shell decrease in energy faster with deformation than those from the f7/2orbit. This difference in rate of decrease of the Nilsson energies with deforma-tion can overcome the small spherical f7/2-h9/2 energy separation. The lowKtia and h9/2 orbits therefore approach each other, mixing more and more.However, the two orbits cannot cross and so repel each other, leading to aninflection point at the value of j8 where they would have crossed. This effect isvery clear for the K = 5/2 and K = 7/2, f^ and h9/2 orbits in Fig. 7.4.An interesting feature of the Nilsson diagram is apparent if one looks at theenergies past the "pseudo crossing." Starting at large deformations andtracing back toward /? = 0 the energy of the lowest K = 5/2 orbit is drawn as ifit stems from the f7/2 shell. However, one sees that it actually points directlyback to the h9/2 spherical energy. This reflects the fact that this orbit, for largedeformations, is actually the continuation of the hW2 shell. In effect, while theFigure 2.6: Nilsson orbitals for the =82-126 region. The deformation pa-rameter  is nearly the same as β and represents the deformation of the nucleus[6].quadrupole deformation of a nucleus: β2 gives the amount of quadrupole de-formation, with β2 > 0 being prolate and β2 < 0 oblate, and γ gives thedegree of axial sy metry present. Though β is often used to describe thenuclear deformation, with β2 being specifically the quadrupole deformation,other deformation parameters are used.  is also used as a deformation param-eter, which derives from the harmonic oscillator potential model for deformed10nuclei, though  is nearly equal to β. A more detailed description of thesedeformation parameters may be found in Ref. [6].The ground state quadrupole deformation β2 is often approximated fromthe strength of the 2+ −→ 0+ transition in the ground state band:β2 ≈ 4pi√53ZR20√B(E2; 2+g.s −→ 0+g.s.)/e2 (2.7)where B(E2; 2+g.s. −→ 0+g.s.) is the reduced transition strength or rate. A moredetailed discussion on reduced transition strengths follows in Sec. 2.2.3.The reduced transition strength for an electric quadrupole transition (E2)can be calculated from the half-life of the initial state Ipii by measuring thepartial half-life τP of the excited stateB(E2; Ipii → Ipif)=564τPE5γe2fm4MeV5ps (2.8)where the energy Eγ in MeV and the partial half-life is given in ps and is re-lated to the lifetime of the state via the branching ratio (BR) of the measuredtransition: τP = t1/2/BR. One (of many) methods to measure the half-life ofstates is possible with electronic timing techniques using LaBr3(Ce) detectors(see upcoming Sec. 4.2.1).The gadolinium (Z = 64) isotopes around N = 100 are of interest fortheir strong ground state quadrupole deformation. A naive model, wheredeformation is driven by proton-neutron correlations, would expect maximumdeformation for this region in 170Dy since it is midshell in both proton (Z =66) and neutron (N = 104) number. However, data shows that Gd isotopeshave a larger quadrupole deformation than Dy isotopes in this region aroundN = 100 (Fig. 2.7). Thus, isotopes around 170Dy, including the gadoliniumisotopes, are of interest to study.2.1.3 Collective MotionAlthough the Nilsson model advances on the shell model for deformed nuclei,it still has its limitations; namely, the Nilsson model, like the shell model,only describes single particle energy spectra. Oftentimes, nuclear excitationsare not simply due to single particles transitioning between orbitals, but dueto the collective motion of many nucleons within the nucleus.Evidence for collective excitations can be seen in medium- and heavy-mass nuclei. Many light and even-even nuclei near shell closures exhibit firstexcited 2+ states in the 1-2 MeV range. This first excited state correspondsto the breaking of a nucleon pair and excitation of one of the (now) unpaired11Figure 2.7: Measured ground state quadrupole deformation for isotopes near170Dy. Plot courtesy of I. Dillmann. Data taken from ENSDF [8].nucleons to another orbital. However, as one advances away from closed shellsin both proton and neutron number for heavier nuclei, the 2+ first excitedstate of even-even nuclei drops in energy significantly; some nuclei have firstexcited states at energies less than 100 keV above the ground state. Withpair breaking of nucleons requiring on the order of 1 MeV of energy, thesefirst excited states cannot come from single particle excitations.Instead, excitations of heavy mid-shell nuclei are built on collective ex-citations, where the nucleus excites as a collective unit as opposed to theindividual nature of excitations for nuclei near shell closures. The two typesof collective motion relevant for these deformed mid-shell nuclei (like 160Gd)are rotations and vibrations.Nuclear RotationsNuclear rotation is prevalent in heavy, even-even deformed nuclei away fromshell closures. These nuclei exhibit a ground state rotational band corre-sponding to the nucleus increasing its angular momentum from the Ipi = 0+12ground state. The Hamiltonian of the quantum-mechanical rotor is similar tothe classical rigid-rotor Hamiltonian; however, the eigenvalues of the Hamil-tonian are different in the quantum-mechanic situation. The energy of theexcited states in the rotational ground state band is given byE =~22II(I + 1) (2.9)where I denotes the moment of inertia of the nucleus. Eq. 2.9 does wellto predict the energy of the rotational band built on the ground state of160Gd. Using the ENSDF evaluation [9], with the 2+ member of the ground-state band at 75.3 keV, one solves for ~22I = 12.55 keV. The next expectedstates are calculated at 251 keV, 527 keV, and 904 keV for the 4+, 6+, and 8+states, respectively. The current evaluation places these levels at 249 keV, 515keV, and 868 keV, respectively. One sees that the simple formula of Eq. 2.9produces good estimates for the energy of the rotational excited states.Nuclear rotation is not restricted to being built on the ground state, andin fact rotational bands are often built on other excitations, including vibra-tion. When this rotation is build on a state with total angular momentumprojection K, the energies of the rotational states areE =~22I[I(I + 1)−K(K + 1)]. (2.10)It is important to remember that the expected energies of Eq. 2.10 are energiesrelative to the base energy of the excitation–the so-called “bandhead” of therotational band.If again, one compares the expected rotational energies to those of theexperimental levels [9], one sees that Eq. 2.10 does well to predict rotationalenergies. Taking the Kpi = 2+ γ-vibrational band built on the 989 keV statein 160Gd, and using a similar procedure to that described above, one calculatesthe rotational states built on the γ-vibration at 1148, 1261, 1397, and 1556keV for the 4+, 5+, 6+, and 7+ members of the band. respectively. Thecorresponding experimental levels for this band are at energies of 1148, 1261,1393, and 1549 keV. One can see that these simple models for rotationalenergies of excited states are a powerful tool to aide in predicting excitedstates in rotational nuclei.Characterization of which type of collective excitation is built on theground state is often done via the ratio E4+1/E2+1. With the 160Gd nucleus,one calculates this ratio as E4+1/E2+1= 248.5 keV/75.3 keV = 3.30–in excellentagreement with the theoretical value of E4+1/E2+1= 3.33 [4] for a rotationalnucleus.13Nuclear VibrationsMuch like the rotational excitations described above, deformed nuclei can alsoexhibit nuclear vibrations as a collective excitation. Just as the most commonnuclear deformation is a quadrupole deformation, the most common low-lyingvibrational excitation is also a quadrupole in nature. Because of this, theycarry two units of angular momentum, and thus are states with either K = 0or K = 2 [6]. Though vibrational states are found in all types of nuclei, thisdiscussion will focus only on vibrations in deformed nuclei which are relevantto 160Gd.The K = 0 scenario corresponds to the so-called “β-vibration.” This vi-bration is aligned with the symmetry axis. The β-vibration is called suchbecause it corresponds to changes in the quadrupole deformation parameterβ2. When the vibration is from a state with K = 2, the vibrational mode isknown as a “γ-vibration.” As one would guess, the γ-vibration is associatedwith changes in the γ deformation parameter. γ-vibrations see the nucleuschange the degree of axial symmetry in the nucleus. Both β- and γ-vibrationsare present in nearly all heavy deformed nuclei around 1 MeV of excitationenergy [6].Vibrational energy is carried by the quasiparticle “phonon.” Much likehow rotational momentum added to a nucleus increases the energy to thenext excited state, vibration of the nucleus also causes the nucleus to be ina higher-lying excited state. As with the ground state band, the additionof a quadrupole phonon carrying two units of angular momentum excites aneven-even nucleus from a 0+ ground state to a 2+ excited state. While anumber of spherical nuclei exhibit multi-phonon excitations, for the majorityof deformed nuclei, multi-phonon vibrational excitations like the ββ-, βγ-,and γγ-vibrations have not yet been identified.Of course, the quadrupole phonon is not the only phonon available forexcitation. Octupole phonons, carrying three units of angular momentum,lead to octupole vibrations. Hexadecapole vibrations (caused by hexadecapolephonons carrying four units of angular momentum) are also thought to occurin even-even deformed nuclei such as 160Gd [10]. Rotational excitations arethen built on these single-phonon vibrational states. A more comprehensiveand complete description of rotational and vibrational states in deformednuclei can be found in Refs. [4, 6].Characterization of vibrational excitation comes again from the E4+1/E2+1ratio. Vibrational excitations are often found with a ratio of E4+1/E2+1=2 − 2.2, as the addition of phonons adds approximately the same quanta ofenergy to the system.142.1.4 Two-state MixingOftentimes, observed nuclear states are not due to pure nuclear states, butrather an admixture of two (or more) pure states. While a multi-state mix-ture involves diagonalization of a large Hamiltonian, the two-state scenario issomewhat simple. The scenario, depicted in Fig. 2.8, starts with the unper-turbed, pure states |φ1〉 and |φ2〉 at energies E1 and E2, respectively. Due tomixing of the states, the perturbed wavefunctions |ψI〉 and |ψII〉 are given asa mixture of both unmixed wavefunctions:|ψI〉 = a |φ1〉+ b |φ2〉|ψII〉 = −b |φ1〉+ a |φ2〉(2.11)where the normalization condition a2 + b2 = 1 is satisfied. For states that areperturbed by an energy ∆ES compared to their unperturbed energy difference∆EUNP , the ratio of the perturbation to the unmixed energy splitting is givenas|∆ES |∆EUNP=12[√1 +4R2− 1](2.12)where R = ∆EUNPV is the ratio of the unperturbed energy splitting to theinteraction strength V . A derivation of Eq. 2.12 can be found in Ref. [6].The ratio R also relates to the mixing amplitude b and can be expressedasb =(1 +[R/2 +√1 +R2/4]2)−1/2. (2.13)Thus, if one knows the ratio R, it is possible to compute the mixing amplitudesa and b, and vice versa.Transition RatesWhen two states mix, there is an effect on the transition rates due to themixture. A conventionally “forbidden” transition between two unmixed statesacquires some transition probability due to the mixture of the states. For thesimplified case in Fig. 2.9, the pure 2+2 state has forbidden transitions to theunmixed 2+1 and 0+1 states, whereas the 2+1 state has an allowed transition to1518 IntroductionINTERACTION-. vFig. 1.6. Two-state mixing: definitions and notation.of the unperturbed energy spacing to the strength of the matrix element. Thenthe perturbed energies arewhere the + sign is for En and the - sign for Er It follows that the final energydifference isor, in units of the unperturbed splitting AEu, the final separation is given by thesimple resultA more useful result is the amount, AEs , by which each energy is shifted bythe interaction. A/^ I is given byor, again, in units of A£u, one obtains a result independent of the initial spacing:Figure 2.8: Example of two-state mixing. The pure states at energies E1 andE2 are perturbed by an energy ∆ES , resulting in the mixed states at energiesEI and EII . Picture taken from Ref. [6].the 0+1 state. If the 2+1 and 2+2 state mix, the mixed states 2+I and 2+II aregiven by|2+I 〉 = a |2+1 〉+ b |2+2 〉|2+II〉 = −b |2+1 〉+ a |2+2 〉(2.14)Now, one calculates the matrix element for the 2+II −→ 0+1 transition as〈2+II |E2 |0+1 〉 = 〈−b2+1 + a2+2 |E2 |0+1 〉= −b 〈2+1 |E2 |0+1 〉+ a 〈2+2 |E2 |0+1 〉= −b 〈2+1 |E2 |0+1 〉(2.15)since the 2+2 −→ 0+1 is forbidden, and thus 〈2+2 |E2 |0+1 〉 = 0. With mixing,there is now a finite probability that the 2+II −→ 0+1 will occur. The matrixelement 〈ψf |σL |ψi〉 is related to the reduced transition rate B(σL) that holds16Introduction 25Finally, note that in all the multistate mixing cases considered, all of thecomponents of the lowest lying wave function have the same sign. Though thisresult depends on the phase conventions chosen, if consistent conventions areused for both wave functions and operators, then matrix elements (observ-ables) will contain coherent, in-phase sums, and can be extremely large. Thewave function has coherence, and such multistate mixing can lead to collectivityas reflected in enhanced transition rates, cross sections, and the like. Also,note that the sum of the initial and final energies is the same, as, of course, itmust be. Since these energies appear on the diagonal of the matrix to bediagonalized, this is equivalent to the formal statement that the trace isconserved.The importance and usefulness of the results in this section cannot beoveremphasized. With them, and an understanding or the basically attractivenature of the nuclear force, and of the effects of the Pauli principle and ofantisymmetrization, it is possible to understand nearly all of the detailedresults of most nuclear model calculations in an extremely simple, intuitiveway that illustrates the underlying physics that is often lost in complex formal-isms and computations.1.6 Two-State Mixing and Transition RatesOne application of the concept of two-state mixing that is worth discussing,even though it invokes concepts and excitation modes that will be introducedlater, is the effect of certain types of mixing on transition rates. Consider thesimple level scheme in Fig. 1.12 with 2+ levels from different intrinsic excita-tions (say, belonging to two bands of a deformed nucleus). Suppose that,according to some model, one 2f level has an allowed (A) ground state transi-tion and the other has forbidden (F) transitions to both 0^ and 25+ states. Oneoccasionally encounters statements of the following kind: "While the 22*—> 0,+Fig. 1.12. tiffed of mix ing on allowed (A) and forbidden (I-') /-ray transit ions.Figure 2.9: Example of two-state mixing affecting transition rates. The un-mixed 2+2 state, which normally has forbidden (F) transitions to the otherstates, may have allowed (A) transitions to the states when mixed with the2+1 state. Picture taken from Ref. [6].information on the nuclear structure and is discussed in more details in theupcoming Section 2.2.3.By a similar argument, the transition rate of the 2+II −→ 2+I transition isshown to be〈2+II |E2 |2+I 〉 = ab[ 〈2+2 |E2 |2+2 〉 − 〈2+1 |E2 |2+1 〉 ] (2.16)where again terms that are associated with the forbidden, unmixed transi-tions are eliminated. For this transition, the matrix element 〈2+II |E2 |2+I 〉 isproportional to the difference in quadrupole moments (〈ψ|E2 |ψ〉) of the un-mixed states. Of course, in the limit of no mixing (b = 0), the matrix elementis zero, corresponding to the forbidden transition assumed at the start of thisexample. More information on two-state mixing and associated transitionrates can be found in Ref. [6].2.2 Radioactive Decay2.2.1 Nuclear Decay LawRadioactive decay is a random process whereby the probability of decay in atime interval dt is constant and independent of the time since the last decay17occurred; therefore, radioactive decay is a Poissonian process.Within a given time frame dt, the number of nuclei dN that decay is givenbydNdt= −λN(t) (2.17)with N(t) being the number of nuclei available to decay at time t and λ thedecay constant for the nucleus. The decay constant λ is related to the half-lifet1/2 by λ =ln(2)/t1/2.The first order differential equation of Eq. 2.17 is readily solved to yieldthe radioactive decay law for a single decaying nuclear species:N(t) = N0 e−λt, (2.18)where N0 is the number of nuclei at an established time t = 0. The activityof the sample A(t) is given byA(t) = λN(t) = λN0 e−λt. (2.19)When a radioisotope is being produced at a rate R, Eq. 2.17 becomes:dNdt= R− λN. (2.20)The solution to the differential equation in Eq. 2.20 gives the number of nucleiand activity of the sample asN(t) =Rλ(1− e−λt) (2.21a)A(t) = R(1− e−λt) (2.21b)with the assumption that initially no radioisotope existed (N(0) = 0).2.2.2 β-decayMedium-mass neutron-rich nuclei away from stability will generally decreasetheir energy via the radioactive process known as β−-decay (β-decay forshort). In this decay, the nucleus converts a neutron into a proton, electron,and antineutrino:n −→ p+ + e− + νe. (2.22)18For an isotope of element X with Z protons, N neutrons, and A = N + Znucleons, the process of β-decay will convert a neutron into a proton andoften (but not always) leaves an isotope of element Y in an excited state:AZXN −→AZ+1 Y ∗N−1 + e− + νe. (2.23)This process can only occur if the decay is energetically favorable for thenucleus. For this to occur, the rest mass-energy of the parent nucleus mustbe greater than that of the daughter. This value is known as the Qβ−-valueand is defined in terms of the mass of the parent mp and mass of the daughtermd:Qβ− = mpc2 −mdc2. (2.24)If Qβ− > 0, the nucleus will undergo a β−-decay and release a total energyequal to Qβ− . Thus, the Qβ−-value not only determines if a β−-decay ispossible, but also puts a limiting value on the energy that the emitted electronand antineutrino can possess.Selection RulesThe four particles involved in β−-decay (neutron, proton, electron, and anti-(electron) neutrino) are all spin-1/2 fermion particles. The orbital angularmomentum Lβ and spin Sβ of the β−-decay are defined in terms of the angularmomenta and spins of the emitted particles:Lβ = `e + `ν (2.25a)Sβ = Se + Sν (2.25b)with the total angular momentum of the β-decay defined as Jβ = Lβ+Sβ. Lβmay be any natural number, while Sβ can only be equal to 0 or 1 dependenton if the spins of the electron and anti-neutrino are anti-parallel or parallel;the former case is known as a Fermi decay, and the latter a Gamow-Teller(GT) decay.The angular momentum of the nucleus (“nuclear angular momentum” or“nuclear spin”) is represented as I, with Ii and If corresponding to the nuclearspins of the initial and final states for a decay, respectively. During a β-decay,any change in nuclear spin must be due to the angular momentum of theparticles, as given byIi = If + Jβ = If + Lβ + Sβ (2.26a)19∆I = |If − Ii|. (2.26b)The case where Lβ = 0 is known as an allowed transition: the orbitalangular momentum of the emitted electron and neutrino are zero. An allowedFermi decay must then have no change in the nuclear spin, as Jβ = 0 bydefinition. Since a GT transition has the spin vector Sβ = 1, the change innuclear spin ∆I can be 0,±1, with 0+ −→ 0+ transitions not possible (for anallowed GT transition). The change in parity for a nuclear transition is givenaspii = pif · (−1)Lβ (2.27)where pii and pif are the parity of the initial and final state, respectively. FromEq. 2.27, it is clear that there is no change in parity for an allowed transition.Transitions with Lβ 6= 0 are known as “forbidden” transitions. Althoughcalled like this, these transitions are not truly forbidden, but rather suppressedrelative to allowed transitions. A first forbidden transition, correspondingto Lβ = 1, has a change in parity as required by Eq. 2.27. A Fermi firstforbidden transition (Sβ = 0) thus has a possible ∆I of 0,±1. For a firstforbidden GT transition, the possible ∆I values are 0,±1,±2. As with theallowed transitions, 0+ −→ 0+ are not possible for the forbidden transitions aswell. Higher Lβ values correspond to higher-order forbidden transitions, withthe selection rules for the transitions given by Equations 2.26b and 2.27. Asummary of the selection rules for β-decay is shown in Table 2.1.Because Fermi transitions have Sβ = 0, these transitions populate isobaricanalog states: the decaying neutron decays into a proton in the same orbitalshell, with the same ` and J . For isotopes far from the N = Z line–including160Eu–the corresponding valence neutron orbital is much higher than the va-lence proton orbital. Thus, it is highly unlikely for such nuclei that the decay-ing neutron will populate an isobaric analogue state in the daughter. Fermidecays are therefore not expected to contribute significantly in exotic nucleilike 160Eu.Comparative Half-livesThe nuclear matrix element mfi is the quantity of interest, as it carries thenuclear structure information. The transition probabilities for Fermi and GTtransitions are the quantities that can be measured and used to extract therelevant matrix elements. The transition strengths B are defined for the Fermi20Fermi GTType L ∆I ∆pi ∆I ∆pi log(ft)Allowed 0 0 No (0),1 No ∼4–7First Forbidden 1 (0),1 Yes 0,1,2 Yes ∼6–9Second Forbidden 2 (1),2 No 2,3 No ∼10–13Third Forbidden 3 (2),3 Yes 3,4 Yes ∼14–20Fourth Forbidden 4 (3),4 No 4,5 No ∼23Table 2.1: Selection rules for β-decay. Parentheses indicate a transition thatis not possible if either Ii or If is 0.and GT transitions asB(F ) = | 〈ψ∗f | τ |ψi〉 |2 (2.28a)B(GT ) = | 〈ψ∗f |στ |ψi〉 |2 (2.28b)where |ψi > and < ψf | represent the initial and final wavefunctions, respec-tively, σ represents the well-known Pauli spin matrices, and τ the isospinladder matrices. The transition probabilities are connected to the ft valuesof the decay via the equationft =kg2VB(F ) + g2AB(GT )(2.29)where k = 2ln(2)pi3~7m5ec4 , and gV and gA are the vector and axial-vector couplingconstants for the weak interaction, respectively [11].The ft value can also be calculated once information on the decay of theparent nucleus is known. The ft value is given byft = f(Z,Q,Ef ) · t1/2 (2.30)where f(Z,Q,Ef ) is the Fermi integral, which depends on quantities such asthe Qβ−-value of the reaction and the intensity and energy of the populatedstate. Since the ft values vary by many orders of magnitude across differ-ent nuclei, convention is to compare them as log(ft) values. Values for theFermi integral are readily available, as the integral is tabulated and log(ft)calculators are available online (Ref. [12]).Once log(ft) values have been calculated, the transition strengths B(GT)and B(F) can be computed using Eq. 2.29 and values for the matrix elementsdeduced. The log(ft) values also provide insight into the type of transition21that has occurred; allowed transitions generally have log(ft) < 6.0, with theforbidden transitions having larger and larger log(ft) values [11]. Ranges forlog(ft) values for different transition types are shown in Table 2.1. A com-prehensive study of log(ft) distributions in β-decay may be found in Ref. [13]2.2.3 γ-decayOnce a nucleus undergoes β-decay, the daughter nucleus is often left in anexcited state rather than its ground state. This excited nucleus then releasesenergy in the form of high-energy photons known as γ-rays as it de-excites tothe ground state–akin to electrons emitting X-rays as they de-excite from ahigher-energy electron orbital. The de-excitation of the nucleus is not limitedto a single γ-ray emission, but often includes the emission of a number ofsubsequent γ-rays in cascade. The energy of these γ-rays ranges from lessthan 100 keV to upwards of 10 MeV.Unlike β-decay, γ-decay does not change the number of protons and neu-trons within the nucleus, but rather just the energy of the nucleus itself. Theenergy Eγ of the emitted γ-ray is given by Eγ = Ei−Ef , with Ei and Ef beingthe energies of the initial and final states, respectively. A thorough analysisof the γ-rays emitted from a nucleus can yield the excited-state structure ofthe nucleus of interest.Electromagnetic RadiationFrom classical electromagnetism, the radiated power from a electric or mag-netic multipole is of the formP (σL) =2(L+ 1)c0L[(2L+ 1)!!]2(ωc )2L+2[m(σL)]2 (2.31)where σ is either E or M to represent electric or magnetic radiation, m(σL)represents the time-varying amplitude of the electric or magnetic moment,and the double factorial represents a factorial over odd terms only [4]. Theangular momentum term L defines the multipole order, with L = 1 being adipole, L = 2 a quadrupole, and subsequent higher-order terms defined in asimilar way.Transitioning to quantum mechanics, the multipole moment m(σL) isreplaced by the multipole operator σL that changes the nucleus from an initialstate ψi to a final state ψf . Much like with β-decay, the matrix element mfidictates the transition probability between the initial and finals states:mfi(σL) = 〈ψf |σL |ψi〉 . (2.32)22With the energy of a photon given as ~ω and the substitution of the multi-pole operator of Eq. 2.32, the decay constant for the photon emission can beexpressed in terms of the power P (σL) radiated (Eq. 2.31):λ(σL) =P (σL)~ω=2(L+ 1)0~L[(2L+ 1)!!]2(Eγ~c )2L+1B(σL; Ii −→ If ). (2.33)Transition Rates and Weisskopf EstimatesThe reduced matrix element | 〈ψf ||σL||ψi〉 |2 is proportional to the commonly-used reduced transition rate B(σL; Ii −→ If ) (often shortened to B(σL)) ofEq. 2.33. The matrix element and reduced transition rate are related via theequationB(σL; Ii −→ If ) = | 〈ψf ||σL||ψi〉 |22Ii + 1. (2.34)The reduced transition rate factors out the energy dependence of the transi-tion so that information on the initial and final wavefunctions can be obtained.Thus, the reduced transition rate includes aspects of the nuclear structure andis one of the quantities of focus for nuclear structure studies.The matrix element | 〈ψf ||σL||ψi〉 |2 must be evaluated before more infor-mation can be gained from Eq. 2.33. Under the assumption that the γ-rayemission is due to the transition of a single particle between two shell-modelorbitals (among other assumptions), the transition probability for an electrictransition EL is given asλW (EL) ∼= 8pi(L+ 1)L[(2L+ 1)!!]2e24pi0~c(Eγ~c)2L+1( 3L+ 3)2cR2L (2.35)and a magnetic transition probability ML asλW (ML) ∼= 8pi(L+ 1)L[(2L+ 1)!!]2(µp − 1L+ 1)2( ~mpc)2( e24pi0~c)×(E~c)2L+1( 3L+ 2)2cR2L−2(2.36)where the nuclear radius R is often approximated as R = R0A1/3 (with R0 ≈1.2 fm), and the energy E is in MeV. A full treatment of the derivation ofEqs. 2.35 and 2.36 is given in Ref. [4].23σL BW [e2fm2L] λW [s−1]E1 6.45× 10−2A2/3 1.0× 1014A2/3E3E2 5.94× 10−2A4/3 7.3× 107A4/3E5E3 5.94× 10−2A2 34A2E7E4 6.29× 10−2A8/3 1.1× 10−5A8/3E9M1 1.79 5.6× 1013E3M2 1.65A2/3 3.5× 107A2/3E5M3 1.65A4/3 16A4/3E7M4 1.75 A2 4.5× 10−6A2E9Table 2.2: Weisskopf estimates for a number of electric and magnetic tran-sitions. The energy E is the energy of the emitted γ-ray in MeV and A thenumber of nucleons.One may also estimate the reduced transition rates B(σL) for single-particle transition. Traditionally, the B(σL) single-particle estimates aregiven asBW (EL) =14pi( 3L+ 3)2e2R2L (2.37a)BW (ML) =10pi( 3L+ 3)2µ2NR2(L−1). (2.37b)Comparing to Eq. 2.33, the estimates in Eqs. 2.37 correspond to the factorssubstituted for | 〈ψf |σL |ψi〉 |2, with other minor factors added. A thoroughderivation and justification for these estimates can be found in Ref. [4, 14, 15].These estimates are known as the Weisskopf estimates, corresponding tocrude estimates of single-particle transition rates within a nucleus. The Weis-skopf estimate, though independent of the wavefunctions of the initial andfinal states, can give insight into the transition. Experimental decay ratesorders of magnitude larger than the Weisskopf estimate might suggest thatmore than one nucleon contributes to the decay.By setting the term [µp − 1/L + 1]2 equal to 10 in Eq. 2.36, Eqs. 2.35and 2.36 can be evaluated at different angular momenta L to get energy- andnucleus-dependent Weisskopf estimates [4]. Selected lower-order electric andmagnetic Weisskopf estimates are shown in Table 2.2.Selection RulesMuch like for β-decay, γ-ray transitions must obey conservation of angularmomentum. The initial and final angular momentum Ii and If are related to24∆I = |Ii − If | 0∗ 1 2 3 4 5piipif = −1 E1 E1 M2 E3 M4 E5piipif = +1 M1 M1 E2 M3 E4 M5*Not 0 −→ 0 transitions.Table 2.3: γ-ray multipolarity for the lowest order angular momentum andparity of an emitted γ-ray.the angular momentum Lγ of the γ-ray via the equationIi = If + Lγ (2.38)and the possible values that Lγ may have are obtained by the coupling of thethree vectors:|Ii − If | ≤ Lγ ≤ Ii + If . (2.39)It should be noted that, because photons must carry at least one unit of an-gular momentum, a γ-ray transition with L = 0 is not possible; furthermore,a 0 −→ 0 transition is also not allowed to occur via γ-ray emission. Instead,0 −→ 0 transitions proceed via conversion electron emission (see Sec. 2.2.4).Since γ-ray transitions are electromagnetic, they must also conserve parity.The parity of the electric and magnetic transitions is given by the classicalparity of the radiation fields:pi(EL) = (−1)Lpi(ML) = (−1)L+1. (2.40)The parity of electric and magnetic transitions is opposite for a given angularmomentum. The angular momentum coupling and parity of the transitionthus give the selection rules for γ-decay. The character of γ-rays are oftenreferred to by the lowest order multipole of the emission. Table 2.3 givescommon designations for low angular momentum transitions.γ-ray InteractionsThere are three mechanisms by which γ-rays interact with matter: photo-electric absorption, Compton scattering, and pair production. Photoelectricabsorption occurs via the photoelectric effect, where a γ-ray is absorbed by anatom and a photoelectron is emitted. The energy of a photoelectron emittedby the absorption of a γ-ray of frequency f is given byEe− = hf − Eb (2.41)2552 Chapter 2 Radiation Interactions so0 Figure 2.19 A polar plot of the number of photons (incident from the left) Compton scattered into a unit solid angle at the scattering angle 0. The curves are shown for the indicated initial energies. after slowing down in the absorbing medium, two annihilation photons are normally produced as secondary products of the interaction. The subsequent fate of this annihilation radiation has an important effect on the response of gamma-ray detectors, as described in Chapter 10. No simple expression exists for the probability of pair production per nucleus, but its magnitude varies approximately as the square of the absorber atomic number.l The impor- tance of pair production rises sharply with energy, as indicated in Fig. 2.18. The relative importance of the three processes described above for different absorber materials and gamma-ray energies is conveniently illustrated in Fig. 2.20.The line at the left represents the energy at which photoelectric absorption and Compton scattering are equally probable as a function of the absorber atomic number. The line at the right repre- sents the energy at which Compton scattering and pair production are equally probable. Three areas are thus defined on the plot within which photoelectric absorption, Compton scattering, and pair production each predominate. L -4 100 - - - Photoelectric effect Pair production - dominant - - 0.01 0.05 0.1 0.5 1 5 10 50 100 hv in MeV Figure 2.20 The relative importance of the three major types of gamma-ray inter- action. The lines show the values of Z and hv for which the two neighboring effects are just equal. (From The Atomic Nucleus by R. D. Evans. Copyright 1955 by the McGraw-Hill Book Company. Used with permission.) Figure 2.10: Relative importance of the three major γ-ray interaction types.The lines show where neighboring effects are of equal importance. Germaniumdetectors have Z = 32. Picture taken from Ref. [16].with Eb being the binding energy of the photoeletron prior to its emissionfrom the atom. Emitted photoelectrons interact with the electrons on then-type germanium semiconductor. Electron/hole pairs drift to the p− andn-contacts, respectively, where charge is then deposited and collected. Pho-toelectric absorption dominates γ-ray interactions with germanium detectors(like those used in this experiment) for low-energy γ-rays (/ 200 keV) [16].γ-rays of medium energy (200 keV / Eγ / 5 MeV for germanium) canCompton scatter within the detector crystal. Rather than depositing theirentire energy in the crystal, Compton-scattered γ-rays result in incompletecharge collection of the γ-ray due to scattering. The energy E′γ of a Compton-scattered γ-ray is given byE′γ =Eγ1 +Eγmec2(1− cosθ)(2.42)where Eγ is the initial γ-ray energy, mec2 is the rest-mass energy of an elec-tron, and θ is the scattering angle of the γ-ray. Because scattering at allangles is possible, energy deposition due to Compton scattering is a continu-ous distribution; as an example, 1 MeV γ-rays can deposit up to ≈ 800 keVdue to scattering, and 4 MeV γ-rays can deposit as much as ≈ 3.75 MeV ofenergy while still scattering and escaping the germanium detector.When the energy of a γ-ray exceeds 1.022 MeV, pair production is en-26ergetically possible. An incoming γ-ray may produce an electron-positronpair–each with a rest mass of 0.511 MeV/c2. Any energy in excess of 1.022MeV will be carried away as kinetic energy by the electron and positron. Thecreated positron will annihilate with a nearby electron and create two 511keV γ-rays. If one of these γ-rays escapes the crystal, it can be detected byneighboring crystals and appear as a 511 keV peak in the data. Pair pro-duction can only occur above 1.022 MeV γ-ray energy, and is the dominantinteraction process for energies above ≈5 MeV (for germanium detectors) [16].A plot showing the relative importance of each interaction type is shown inFig. Conversion ElectronsSometimes, the nucleus will undergo internal conversion and release an elec-tron instead of a γ-ray. Internal conversion occurs when the electromagneticfield couples to an atomic electron and ejects it from the nucleus. The ejectedelectron is referred to as a “conversion electron.” The energy of the ejectedelectron is equal to the energy between the excited states that the nucleusis transitioning between (i.e. the energy of the theoretically emitted γ-ray)minus the binding energy of the atomic electron. The conversion electron nor-mally comes from the K or L electron orbital, giving the conversion electronmultiple possible energies.Conversion electrons occur when the emission of a γ-ray is suppressed.In some even-even nuclei, the first excited state has spin-parity of 0+. Be-cause the ground state of even-even nuclei is also 0+, a single γ-ray transitionbetween the two states cannot occur, and the nucleus de-excites instead byan E0 internal conversion transition. Additionally, transitions between ex-cited states of close energy can occur via internal conversion in addition toγ-ray emission. From Eqs. 2.35 and 2.36, the transition rate is proportionalto the energy of the emitted γ-ray: λ ∝ E2L+1. Low energy γ-rays withlow multipolarity have small transition rates, and the transition can often behighly converted–indicating there is a large probability of internal conversioncompeting with γ-ray emission.The character α denotes the conversion coefficient, which is the numberof conversion electrons emitted for every one γ-ray emission: a conversioncoefficient of α = 9 indicates that nine out of ten transitions are conversionelectrons, and one out of ten is a γ-ray emission, for example. Theoreticalconversion electron coefficients can be attained from online calculation toolssuch as BrIcc [17].27Chapter 3Review of Literature3.1 Reaction Studies160Gd, being a stable nucleus, has been studied extensively, including via(n, n′γ), (p, p′), and Coulomb excitation. The ENSDF evaluation [9] estab-lishes over 90 excited states, as well as the corresponding transitions, up to anenergy of 3.5 MeV. The ground state rotational band, γ-vibrational band, andoctupole-vibrational band have been established. Other bands, including theKpi = 4+ and Kpi = 0+ bands, have been partially identified as of the mostrecent evaluation (2005). The evaluated band-structure of 160Gd is shown inFig. 3.1.Since the most recent evaluation (over 14 years ago), there have been anumber of (n′, n′γ) [18, 19, 20, 21] studies on 160Gd that have improved theknowledge of the nuclear structure of this nucleus. New levels associatedwith both new and existing bands have been added to the structure of 160Gd.The following literature review and subsequent analysis will focus mainly onnuclear structure that can be determined from β-decay; it will compare withall literature when appropriate.3.2 Previous β-decay StudiesAlthough the structure of 160Gd has been extensively studied via reactions,there have been few studies on the β-decay of 160Gd. Complimentary β-decayspectroscopy on the 160Eu−→160Gd decay will give new, detailed informationon the structure of 160Gd. Prior to the proposal of this experiment, only twoβ-decay studies had been published on the decay of 160Eu to 160Gd [1, 2].3.2.1 Study by D’Auria et al. [1]The first study [1] used the bombardment of 14.8 MeV neutrons on samples ofnatural (≈22% 160Gd) and enriched 160Gd (≈91%) to produce the β-decaying160Eu via (n, p) reactions. Gamma-ray spectroscopy was carried out usingGe(Li) detectors after the bombardment of the sample. A NaI(Tl) detector2816064Gd96-17 From ENSDF16064Gd96-17Band(A): Kπ=0+ground-state rotationalbandBand(B): Kπ=2+γ-vibrational bandBand(C): Kπ=4+ bandBand(D): Kπ=0−octupole-vibrationalbandBand(E): Tentative firstexcited Kπ=0+ band Band(F): Second excitedKπ=0+ band (possibleβ−vibration ?)0.00+75.262+248.524+514.756+867.98+1300.710+1806.312+2377.314+3008.116+75173266353433506571631988.402+1057.543+1147.784+1261.075+1392.806+1548.61(7+)1717(8+)2118(10+)2582(12+)2032452883254014641070.424+1192.695+1331.116+122138 1224.281−1290.103−1427.865−1643(7−)1941(9−)2313(11−)138 1325.73(0+)1377.06(2+)1537.414+1601379.560+1435.992+16064 Gd96Adopted Levels, Gammas17Figure 3.1: Adopted band structure of 160Gd. From Ref. [9].29was utilized to search for γ-rays above 2 MeV, and β-spectroscopy was carriedout with a solid anthracene scintillator.16 γ-rays associated with the 160Gd daughter were published [1]. Coinci-dence analysis, much like that described later in this thesis, was completedin order to build a level scheme of the excited states in 160Gd. A tentativelevel scheme, including 10 of 16 transitions, was published. The remainingsix γ-rays could not be definitively placed. The corresponding partial levelscheme can be seen in Fig. 3.2. The dominant depopulating branch occursvia the 412/514/822 keV cascade to the 4+ member of the ground state band.A Qβ− value of 3.9(3) MeV for the decay was determined via a Fermi-Kurieanalysis from the β-endpoint [22], and a half-life of 50(10) s was extracted.3.2.2 Study by Morcos et al. [2]The second study [2] used the same production method as Ref. [1]: bom-bardment of an enriched 160Gd (≈96%) sample with 14.7 MeV neutrons tocreate (n, p) reactions. Again, a Ge(Li) detector was used for detection ofγ-rays, and a NaI(Tl) and a plastic scintillator was used for collecting β − γand β − γ − γ coincidences. 15 transitions were placed in the published levelscheme shown in Fig. 3.3. The dominant 413/515 keV de-excitation cascadeis placed differently than in Ref. [1]. The measured Qβ− value of 4.2(2) MeV(again via Fermi-Kurie analysis) and associated half-life of 52.8(10) s are inagreement with that of Ref. [1].3.2.3 Comparison of Previous WorkA comparison of the results from both works show a number of discrepancies.Firstly, the published level schemes shown in Figures 3.2 and 3.3 are clearlynot in agreement. Placement of a number of γ-ray transitions, including thedominant 413/515 keV cascade, are conflicting between the two level schemes;the associated excited states are also not in agreement. Six excited states (at515, 986, 1173, 1289, 1688, and 2101 keV) published in Ref. [2] are not seenin Ref. [1]; similarly, 2 states (1585 and 1997 keV) published in Ref. [1] areabsent from Ref. [2].Furthermore, the level scheme in Fig. 3.2 is missing many spin-parity as-signments. The assigned 3− spin-parity of the 1070 keV state [1] conflictswith the tentative 2+ assignment given in Ref. [2]. While both acknowledgethe previously determined spin-parity of 2+ for that state [23], the new assign-ment of 3− in Ref. [1] comes from a comparison of the transition probabilitiesout of the 1070 keV state to predictions from a symmetric core model of thenucleus.30CAN.  J .  PHYS. VOL. 51, 1973 TABLE 6. Reduced transition probabilities -- -- -   - - - - .- -A. - Assumed B(EL)l/B(EL)z Level energy ETl(K,Ii -> KrIr) rnultipolarity (keV) E.,z(KJi + KrIr) (EL) Theorya Experimental - 'Pred~cted accord~ng to  the symmetric core model of Bohr and Motlelson, descr~bed in Nathan and Nllsson (1965). FIG. 7. The proposed decay scheme for the @ decay of lf iOEu to the levels of 160Gd. The numbers in paren- theses are relative intensities of the y rays and do not include internal conversion. possibility. The ratios of the transition preba- bilities have been calculated using the observed y-ray intensities feeding the ground state, assum- ing E2 transitions and El transitions. These are displayed in Table 6. Reasonable agreement is obtained with predictions of the symmetric core - - - -model of Boh'r-and Mottelson as mentioned in Nathan and Nilsson (1965), if an assignment of 13-11 is used for the 1070 keV level. Considerable efforts were made to clarify the origin of the observed second component exhibiting a half-life of 8 f I min. The associ- ated gamma rays could not be correlated with any levels in the nuclides 1 5 9 ~ b ,  lS8Gd, and 160Gd, eliminating assignment to the beta decay of an isomeric level in Is9Gd, L 5 8 E ~ ,  and respectively. Further a definite coincidence was noted between the 198 keV y-ray and a 2.4 MeV beta ray. In addition these unassigned radiations were not observed in the studies of Gujrathi and D'Auria (1971) on the decay of 164Tb, elimina- ting the possibility of an isomeric level in 161Gd. Since the only remaining reaction of reasonable cross section is the (n,a) reaction, the possibility of an assignment to the decay of lS7Sm is quite likely. Correlating this with the recent work of Morcos et al. (1972) indicates that isomerism may exist in the levels of Is7Sm. Such a possibility could arise from the presence of the 1112 - [505T] level in this region of deformation. Summary The decay of 160Eu was studied using fast neutrons on enriched samples of gadolinium. A half-life of 50 f 10 s, associated with gamma rays definitely arising from transitions in I6OGd, is assigned to the decay of 1 6 0 ~ u .  A Qg of 3.9 + 0.3 MeV has been measured with a characteriza- tion of [l-11 for the decaying level in 160Eu. A tentative, although incomplete decay scheme is proposed including 10 y rays. A second component of 8 f 1 min is observed in these studies associated with several y rays. The most likely assignment appears to be the decay of lS7Sm, but this could not be determined unambiguously. NOTE ADDED I N  PROOF: A recent study, com- pleted at the same time as the present paper, by N. A. Morcos, W. D. James, D. E. Adan~s, and P. K.  Kuroda (private communication) has indi- cated similar results to those presented here. BLOCK, R., ELBEK, B., and TJOM, P. 0 .  1967. Nucl. Phys. A, 91, 576. BROWN, R. A.. ROULSTON, K .  I., EWAN,  G. T., and ANDERSON. G. 1. 1969. Can. J .  Phys. 47, 1017. D'AURIA, J .  M., OSTROM, D., and GUJRATHI, S. C. 1971. Nucl. Phys. A, 178, 172. Can. J. Phys. Downloaded from www.nrcresearchpress.com by UBC Central Serials (FNHL) on 01/08/18For personal use only. Figure 3.2: Published excited states and transitions associated with the β-decay of 160Eu from Ref. [1]. Dashed lines indicate tentative transitions. Theparentheses on transitions indicate the relative γ-ray intensities (in percent-age). Level scheme taken from Ref. [1]313666 N.A. MORCOS, W. D. JAMES, D. E. ADAMS and P. K. KURODA Figure 7 shows the tentatively proposed decay scheme for 16°Gd as constructed from Tables 2, 3 and 4. The levels at 75.4, 248.9 and 515-5 keV are rotational members of the K = 0 ground state of t6°Gd and have been well characterized[3-6]. The higher .energy levels at 986.4, 1070.6, 1224, 1289, 1462 and 1688 keV have been ob- served by Block e t  a/.[5] and the levels at 1173.4 and 2101 keV were established from coincidence results and energy considerations. From the results of the singles B- spectrum, a branching ratio of 48 and 52 per cent was observed for the 2-1 and 4.2 MeV B-components, respectively. No B-ray was observed to decay to the 2 + (75.4 keV) level in the ground state rotational band of 16°Gd. Since the ground state of  16°Eu is expected to be a I ~ = 1- level, an allowed B-transition should occur at least to a small extent. A log ft of 5.0 and 6.3 was calculated for the 2.1 and 4.2 MeV B-components, respectively. The ground-state spin-parity of :6°Gd can be deduced from the coupling rules given by Gallagher and Muszkowski[9]. For 16°Eu the 63rd odd proton and the 97th odd neutron can be assigned to the 5/2+ [413] and 5 / 2 -  [523] Nilsson orbitals[10], respectively. From the coupling of these two nucleons, a spin-parity of 1 - (the i - member of the K = 0) band is assigned. The energetically favorable high 9"1" 52*10 Isec) ~l • 4'2*0.2 (MeV),52 (%),6"3 (log ft) ~= • 2"1 *0"2 (MeV),48 (%), 5'0 (log ft) 2101 0 1688 OJ~, 4.2"0.2 (MeV) 3" 1462 3- o ,  ,- i ~ ~ ,,,1269 ~1 ~ 1224 - =-.~-- ,~ ~ . 7 ~  o ~  . _ ® ® o ~ , o : , ~  o~ 986.4 , 0-6 ° . - . L - -  - -  - - ~ - - ~ 5 1 5 . 5  J p. Q'~* - T ' ~  2 4 8 " 9 .  ~ 0.2' '7~P4 0,0" | o K.=',r =6°Gd S ' f o b l e  Fig. 7. Proposal decay scheme of ~°Eu. 9. C. J. Gallagher, Jr. and S. A. Moszkowski, Phys. Rev. l l l ,  1282 0958). 10. O. Prior, F. Bochm and S. G. Nilsson, Nucl. Phys. All0, 25"7 0968). Figure 3.3: Published excited states and transitions associated with the β-decay of 160Eu from Ref. [2]. Level scheme taken from Ref. [2].32While both the Qβ−-value and β-decay half-life are in agreement withinuncertainty, large uncertainties on both (especially for the half-life) leave spacefor improvement.3.3 Recent β-decay Study by Hartley et al. [3]While this analysis was carried out, another independent study of the β-decayof 160Eu was published [3]. Using the ATLAS facility at Argonne NationalLaboratory, 160Eu was produced via the fission of 252Cf through the Cali-fornium Rare Isotope Breeder Upgrade (CARIBU) facility [24]. The X-array,consisting of four germanium clover detectors and a low-energy photon spec-trometer, was used for the detection of γ-rays. The SATURN detector, con-sisting of four plastic scintillators and a moving tape system, was used forβ-particle detection [25].Mass measurements using the Canadian Penning Trap [26] at the ATLASfacility indicated the presence of an isomeric state in the 160Eu parent atan excitation energy of 93.0(12) keV that had not been previously observed.Calculations suggest a particle configuration of pi5/2[413] × ν5/2[523]; theseNilsson orbitals are derived from the 1g7/2 proton and 2f7/2 neutron sphericalorbitals, respectively. These particle configuration lead to spin assignmentsof Kpi = 5− for the ground state of 160Eu and Kpi = 0− for the isomeric state[3].Furthermore, β-decay half-lives of the ground and isomeric states weremeasured via γ-rays associated with the respective decays. Half-lives of42.6(5) s for the 5− ground state decay and 30.8(5) s for the 0− isomericstate were reported. These new half-lives are in stark contrast with the previ-ously identified half-life of 38(4) s, which is the ENSDF [9] weighted average ofa number of half-life measurements published on 160Eu ranging from 31s [27]to 53 s [2]. Identification of a β-decaying isomeric state in the parent isotopeexplains half-life measurements that have varied greatly over the years.An updated partial level scheme with excited states up to 2 MeV waspublished in Ref. [3]. The level scheme(Fig. 3.4) shows states associated withthe 5− ground state β-decay of 160Eu. Many improvements on the publishedlevel schemes from Ref. [1, 2], including the rearrangement and placement ofthe 413/515 keV cascade and identification of new levels and transitions, canbe seen.The 5− ground state preferentially decays to the 1999 keV state in 160Gd,which then depopulates via many cascades, including the strongest branchthrough the 515/413 keV γ-ray cascade previously identified by Refs. [1, 2].A tentative spin of Kpi = (5−) was assigned to the 1999 keV state, indicating33Figure 3.4: Partial level scheme for excited states and transitions associatedwith the 5− ground state β-decay of 160Eu. New transitions and levels areshown in red, with blue indicating rearranged transitions and black indicatingpreviously published levels and transitions. Level scheme taken from Ref. [3].an allowed β-decay from the 5− ground state of the parent 160Eu.The ENSDF evaluation for 160Gd [9] does not list an excited state at1999 keV; however, there is a state at 1996.67(9) keV which was identifiedby (n, n′γ) reactions, as well as in the β-decay study of Ref. [1]. This stateincludes a strong 412.56(10) keV γ-ray to a 1584.03(10) keV excited state,followed by a strong 513.6(2) keV transition to the 1070 keV state. Boththe 1996 and 1584 keV states are given a spin assignment of Kpi = 2+. Theenergy levels and spin assignments of the 1582 and 1999 keV states in Ref. [3]clearly conflict with the currently adopted levels in ENSDF [9]; either newexcited states of close energy have been discovered, or there is a disagreementbetween the established levels and the newly published levels in Ref. [3].Additionally, a new spin assignment of 4+ for the 1071 keV state updatesthe contrasting assignments of 3− and 2+ from Ref. [1, 2]. This state, listedat 1070.42(9) keV in the ENSDF evaluation, has previously been determinedto be the band head of the first Kpi = 4+ band and had spin assignmentsfrom (n, n′γ) [28, 29] and (d,d’) [23] reactions. The Kpi = (6+) assignment tothe 1296 keV state disagrees with the ENSDF adopted (4+, 5+) assignmentsfor a 1295.57(15) keV state, as well.Since there exists conflict between some published excited states in Ref. [3]and the established levels from ENSDF [9], a more thorough study of theexcited levels of 160Gd is needed to reconcile this newly published level scheme.34Chapter 4ExperimentIn June 2017, nine days of beam were delivered to the GRIFFIN spectrometerin the low-energy ISAC-I experimental hall at TRIUMF. Data was collectedfor the six isotopes 160−165Eu. Details of the experiment are discussed in thefollowing.4.1 Radioactive Beam ProductionThe TRIUMF 520 MeV cyclotron is able to produce beams of protons thatare used to produce intense beams of radioactive nuclei for the experimentalfacilities at the Isotope Separator and Accelerator (ISAC) facility at TRIUMF[30]. The ISAC facility utilizes the isotope separator on-line (ISOL) methodfor beam production; a 9.8-µA beam of 480 MeV protons is delivered to theISAC target station and impinges on a uranium carbide (UCx) target in orderto produce the beams of neutron-rich europium nuclei for this experiment.Production of clean, high-intensity beams for comprehensive studies ofthe lanthanides is challenging because of contamination due to other surface-ionized species. Although laser ionization is used to select the element ofinterest, the very similar and low first ionization potential of many lanthanidesallows them to diffuse out of the target already ionized and contaminate thebeam. High transmission mass separator magnets–like those used at ISAC–have relatively limited resolution (ISAC HRS: m/δm ≤ 3000) and thus cannoteffectively separate out isobaric beam contaminants.Implementation of the ion guide laser ion source (IG-LIS) [31] at TRIUMFhas allowed suppression of surface-ionized species and clean beam productionof nuclei. To suppress the surface-ionized contaminants, an electrostatic bar-rier is placed between the target transfer line and the laser ionization step.Surface-ionized species diffusing out of the target are repelled by the electro-static repeller, and only neutral atoms–including the isotope of interest–passto the laser ionization step. A simplified view of the IG-LIS is shown inFig. 4.1. More detailed information on the IG-LIS can be found in Ref. [31].Element-selectivity for atoms that pass the electrostatic repelling barrierof IG-LIS is achieved via laser ionization using the resonant ionization laser35Figure 4.1: Schematic of the IG-LIS. Surface-ionized species (red) are repelledby the electrostatic barrier. Neutral atoms of the element of interest areselectively ionized by the laser-ionization process and then extracted throughthe exit electrode [31].ion source (RILIS) [32]. After passing into the ionizer tube, a Z-specificmulti-step laser excitation allows selection of the element of interest whilefurther suppressing other contaminant elements. For europium, which hasa first ionization energy of 5.67 eV, a two-step laser ionization scheme wasdeveloped. The scheme uses a 460 nm laser to excite the atom initially, anda 415 nm laser excites the atom into an auto-ionizing state, which is a moreefficient ionizing procedure than simply exciting the atomic electron past theionization potential non-resonantly. The laser scheme used for europium isshown in Fig. 4.2.Mass-selectivity occurs via the high resolution mass separator, which al-lows selection of a certain mass-to-charge ratio. The radioactive beam isdelivered to the GRIFFIN facility via the low energy beam transport sys-tem, which consists of electrostatic bending and focusing elements to steerthe beam. An electrostatic barrier (kicker) allows the GRIFFIN facility toreceive the beam in a cyclic nature that is needed for decay spectroscopy.This allows control of the beam delivery to set-up isotope-specific cycle struc-tures for collection and decay measurements, which allows extraction of decayhalf-lives as well.36Figure 4.2: Laser scheme used to ionize europium atoms. The two-step laserionization leaves the electron in an auto-ionizing (AI) state. Laser schemecourtesy of J. Lassen.4.2 GRIFFIN SpectrometerThe Gamma-Ray Infrastructure For Fundamental Investigations of Nuclei(GRIFFIN) spectrometer consists of a number of different components thatallow for precise decay spectroscopy of exotic nuclei. The GRIFFIN array canbe seen in Fig. 4.3.Moving Tape CollectorThe GRIFFIN array utilizes a moving tape system in order to minimize con-taminant data from decaying daughter nuclei during an experiment. Radioac-tive beam is implanted on a Mylar tape at the centre of the GRIFFIN detectorarray. The tape can be moved out of the array and behind a lead shieldingwall to block γ-rays from long-lived daughter nuclei decays. A typical tapecycle consists of a tape move and background reading, followed by implan-tation of radioactive beam. The kicker is then used to prevent further beamdelivery and allow decay data collection.For the 160Eu decay experiment, the tape cycle consisted of 10 s for thetape move and background collection, 380 s of beam implantation, and 114s of beam-blocked data collection. With an adopted half-life of 38 s [9], thiscorresponds to an implant time of 10·t1/2 and a decay time of 3·t1/2. Thetape cycle can be seen via the number of detected β-particles relative to thetape cycle time in Fig. 4.4. Since 160Eu decays to stable 160Gd, and becausethe beam delivered for this experiment contained only minimal contaminants,subtraction of the subsequent daughter decays by using the tape cycle wasnot required.37Figure 4.3: GRIFFIN detector array. Radioactive beam is delivered from thefar side of the array. The tape station box can be seen in the foreground.38Figure 4.4: Detected β-particles as a function of tape cycle time. Beamdelivery begins at approximately 10 seconds and proceeds for 380 seconds.At t = 390 s, beam delivery is halted, and the implanted sample is allowed todecay.394.2.1 γ-ray DetectorsHigh-Purity Germanium DetectorsGRIFFIN is composed of 16 high-purity germanium (HPGe) clover detectorsin a close-packed array. The GRIFFIN structure surrounds the implantationpoint in a rhombicuboctahedral shape, with 16 of the 18 faces covered by theHPGe detectors; the two remaining faces service the beamline and tape system[33]. The GRIFFIN HPGe detectors consist of four germanium crystals in aclosely-packed clover arrangement. The crystals have a diameter of 60 mm,a length of 90 mm, and are tapered at 22.5◦ over the first 30 mm lengthon the outer edges to enable close-packing with neighboring clovers aroundthe implantation point. GRIFFIN HPGe crystals are reverse biased in orderto extend the depletion region across the entire volume of the crystal. Thecrystals are cooled with liquid nitrogen via a single cryostat [34]. A schematicof a HPGe clover detector is shown in Fig. 4.5. For this experiment, 15HPGe clover detectors were used; one clover was removed to accommodatethe conversion electron spectrometer PACES (see Sec. 4.2.2).HPGe detectors are used for γ-ray spectroscopy due to their excellentenergy resolution. The 64 HPGe crystals, when initially commissioned forGRIFFIN, all had energy resolutions better than σ = 0.54 keV and σ = 0.86keV for a 122.0 keV and 1332.5 keV source peaks, respectively. Each crystalalso had a relative efficiency (relative to a 3”×3” NaI scintillator) equal to orgreater than 37.9% at a distance of 25 cm from the source. The average valuefor energy resolution and efficiency for all 64 crystals was σ = 0.80(3) keVand 41(1)% for the 1332.5 keV source peak, respectively. Timing resolutionfor all GRIFFIN HPGe crystals was better than 10 ns for detected γ-raysabove 100 keV. More details on the GRIFFIN HPGe detectors can be foundin Ref. [34].Lanthanum Bromide DetectorsWithin the GRIFFIN rhombicuboctahedral geometry, there are eight trian-gular faces in addition to the 18 square faces that house the HPGe detectors,beamline, and tape. A host of other detectors may be used in these ancillarylocations, including cerium-doped (≈ 5%) lanthanum bromide (LaBr3(Ce))detectors. While the HPGe detectors provide great energy resolution, theirnanosecond timing resolution isn’t fast enough for a thorough fast-timinganalysis of excited states. The LaBr3(Ce) detectors, on the other hand, havebeen shown to determine lifetimes to the precision of ≈ 10 ps. However, theabsolute efficiency of the eight LaBr3(Ce) detectors for GRIFFIN at 1 MeV40Figure 4.5: Render of GRIFFIN HPGe clover detector. The four individualgermanium clovers are shown as different colours. The cryogenic dewar isseen in the background. Measurements are shown for the aluminum housingof the clover. Picture taken from Ref. [34].41Figure 4.6: A section of the GRIFFIN detector array. A LaBr3 detector (cir-cular face) can be seen between three HPGe detectors (square faces). Picturetaken from Ref. [33].is only 1.8(1)% [33].Determination of the half-lives of excited states allows a better under-standing of the nucleus by characteristics such as transition strengths, whichcan be determined once the excited state half-lives are known. More details onthe LaBr3(Ce) detectors and the analysis to extract half-lives can be found inRef. [33]. A cylindrical LaBr3(Ce) detector can be seen between three HPGedetectors in Fig. Other GRIFFIN Ancillary DetectorsIn addition to the HPGe and LaBr3(Ce) γ-ray detectors, GRIFFIN has anumber of ancillary detectors.Zero-Degree ScintillatorMounted directly behind the implantation position within the vacuum cham-ber of the beamline is a fast β scintillator. At 0◦ to the beam axis, the so-called42Zero-Degree Scintillator (ZDS) allows for detection of β-particles emitted dur-ing the β-decay of the nucleus of interest. The ZDS is a circular fast plasticscintillation disk of 25 mm diameter and 1 mm thickness and is mounted ona movable rod within the vacuum chamber so that its position within thechamber can be modified without breaking the vacuum. When mounted atthe closest position–a few millimeters from the tape–the ZDS gives a solidangle coverage of ≈ 25% of 4pi. The ZDS may be retracted up to a distanceof 5 cm from the tape in order to reduce the detection rate without reducingthe amount of delivered beam [33]. For this experiment, the ZDS was located≈1 mm behind the tape implant position.Conversion Electron SpectrometerLocated upstream of the implantation chamber is the conversion electron spec-trometer, PACES (Pentagonal Array for Conversion Electron Spectroscopy).PACES consists of five cryogenically-cooled lithium-drifted silicon detectors.The cylindrical silicon detectors have a circular area of 200 mm2 and a thick-ness of 5 mm. Tilted at 24◦ relative to the implantation point, the fivedetectors are spaced evenly in a circle surrounding the beamline. Locatedat a distance of 31.5 mm from the implantation point, the overall solid anglecoverage of PACES is 7.4% of 4pi [33]. PACES is used to detect the conversionelectrons emitted from the decay of the nucleus of interest.PACES was used for this experiment due to the low-lying first excitedstates associated with deformed nuclei. The highly-deformed gadolinium iso-topes resulting from the β-decay are known to have first excited states atless than 100 keV above the ground state. Since the transitions between thefirst excited states and the ground states of these nuclei are low-energy, theyare expected to be highly converted. Thus, PACES was used for electronspectroscopy on these nuclei.43Chapter 5Data AnalysisThis chapter discusses the data analysis procedure used for the analysis ofthe 160Eu decay data; the described analysis follows the general GRIFFINanalysis procedure for β-decay data.5.1 Data Processing and CorrectionsData is collected by the GRIFFIN data acquisition system (DAQ) [35], whichis integrated into MIDAS [36]. Data is unpacked and sorted via GRSISort[37]–a custom-made analysis framework built in the ROOT [38] environment–into fragment and analysis trees. These trees are readily sorted into his-tograms via GRSISort for further analysis.Though not discussed here, energy and efficiency calibrations were com-pleted before data analysis started. The basic procedure used for calibrationsof the HPGe detectors is outlined in Appendix A.5.1.1 Summing CorrectionsBecause the GRIFFIN HPGe crystals are large-volume, there exists a non-zero chance that multiple γ-rays will enter a HPGe detector during signalprocessing of an initial γ-ray. When this happens, the energies of these mul-tiple γ-rays will sum together and appear only as a single γ-ray. Correctionsmust be made to photopeak areas to account for this summing.Two summing scenarios exist: “summing-in” and “summing-out.” Bothscenarios occur in a situation like that of Fig. 5.1, where a two-γ cascade runsparallel to a single emitted γ-ray. In the summing-in scenario, both γa andγb enter the same detector, and their full energies are detected and recordedas an event for γc–and thus are missing in the photopeaks for γa and γb. Thisartificially increases the count rate of γc.With summing-out, again both γa and γb enter the same detector; how-ever, one (or both) Compton-scatter out of the detector and do not depositthe full energy associated with the γ-rays. This scenario artificially lowers44γaγbγcFigure 5.1: Example of a γ-ray cascades where summing corrections must betaken into account.the count rate of γa and γb, as this summing-out event exists in the Comptoncontinuum of the spectrum.For γ-ray emission, the probability that two γ-rays are emitted from thesame nucleus with an angle of 0◦ is roughly equal to the probability of anangle of 180◦ between them. Since the scenario with an angle of 0◦ createsthe summing effects, examining coincidences between detectors at 180◦ givesan estimate of the summing effects present in the data.To account for summing-in, one gates on either γa or γb (in the 180◦coincidence matrix) and measures the photopeak area of the other γ-ray.This gives an estimate of the summing-in effects, and the number of coincidentcounts in the gated photopeak must be subtracted from the singles photopeakarea of γc. For summing-out, one projects the 180◦ coincidence matrix ontoan axis and measures the photopeak area of the γ-ray, γc. This value mustbe added to the singles photopeak area of γc to account for the summing-outeffects. After correcting the singles photopeak area of γc for summing-in andsumming-out effects, one can then calculate the efficiency-corrected area andsubsequent intensities of interest.455.1.2 Addback ModeThe efficiency of GRIFFIN HPGe clovers can be increased by using Addbackmode. In Addback, for a given event, the energy extracted from GRSISortfor the entire clover is combined into the energy of the event, as opposedto only the energy in the crystal that detected the γ-ray. With Addbackmode, the Compton background in the singles spectra is reduced, as Compton-scattered events are recombined to recover the initial energy of the γ-raybefore it scattered. Addback mode increases the overall efficiency of the HPGedetectors. The improvement in efficiency is especially significant at higherenergies, as high-energy γ-rays have an increased probability of Compton-scattering and not depositing their full energy in the HPGe detector. Theeffects of background suppression and increased efficiency in Addback modecan be seen in Fig. 5.2.Cross-talk CorrectionIn the GRIFFIN HPGe clovers, when a γ-ray is deposited in one crystal, asignal is also induced in the neighboring crystals. If another γ-ray event isdetected in the neighboring crystal at the same time, the apparent energydetected by the crystal will be different because of the cross-talk effects. InAddback mode, where neighboring crystals are added together, cross-talksignificantly affects the shape of the photopeak and must be addressed.To address cross-talk effects, one has to look at multiplicity-2 events withina clover. By plotting the Compton-scattered events within a clover, correc-tions to cross-talk can be made to improve the energy resolution of Addbackpeaks. Automated GRSISort scripts calculate the cross-talk based on the pro-cedure outlined in Appendix A. Further information on cross-talk correctionscan be found in Ref. [33].This analysis focused on using single crystal data as opposed to Addbackspectra. There are few high-energy γ-rays–where the increased efficiency ofAddback mode is useful–in this data set. The summing effects for Addbackmode are also more substantial than for single-crystal mode. Enough statisticswere collected in single-crystal mode to not require Addback for this analysis.All spectra and matrices will be solely single crystal spectra unless otherwisenoted.5.2 β-gated γ-singlesThe data was initially collected in singles mode, where every event detected ina detector was written to disk. Because of this, a histogram of the γ-ray singles46 Energy [keV]200 250 300 350 400 450 500Counts410510610 Single CrystalAddback(a) Energy [keV]2420 2440 2460 2480 2500Counts0500100015002000250030003500 Single CrystalAddback(b)Figure 5.2: (a) Reduction of background at low energies with Addback mode.(b) At high energies, the photopeak efficiency in Addback mode is significantlyincreased compared to single-crystal mode.47events includes γ-rays associated with room background and contaminantisotopes within the beam. To counteract this, a gate requiring both a β-particle and γ-ray detection is used in order to remove most contaminantsnot associated with the isotope of interest.As seen in Fig. 5.3, the majority of γ-ray events fall within a 200 ns timewindow following the detection of a β-particle. The low-energy tail seen inFig. 5.3 results from slow charge collection within the germanium crystalsdue to smaller electron production for low-energy γ-rays. Horizontal linescorrespond to events with no β−γ time correlation–likely due to backgroundor contaminant γ-rays. By requiring only prompt γ-rays in close time coin-cidence (.200 ns, extending longer for low-energy γ-rays) with a β-particledetected by the ZDS, contaminants in the spectrum can be eliminated to en-sure only γ-rays associated with 160Eu are analyzed. The prompt gate usedis shown overlaid in red in Fig. 5.3.Figure 5.3: Plot of the time difference between β-particles and γ-ray detection.The two-dimensional gate to select prompt γ-rays is shown in red.The effect of the β-gate on γ-ray singles data can be seen in Fig. 5.4.A significant reduction in statistics is seen when applying the β-gate. Theβ-gate also eliminates γ-rays not associated with the 160Eu decay. The 78848and 1435 keV γ-rays associated with the decay of 138La from the LaBr3(Ce)detectors are eliminated with the β-gate, as is the 1460 keV γ-ray associatedwith the decay of 40K, which is abundant in the experimental hall due to itspresence in concrete. Suppression of the 570 keV γ-ray associated with 207Bidecay can be seen in the inset of Fig. 5.4. All histograms and matrices usedfor analysis will be β-gated unless otherwise stated.Figure 5.4: Comparison of γ-ray singles and β-gated γ-ray singles spectra.The suppression of the contaminant 570 keV γ-ray from 207Bi can be seen inthe inset.5.3 β-gated γ − γ CoincidenceSimilar to the matrix in Fig. 5.3, one can construct a β-gated γ − γ timedifference matrix. Shown in Fig. 5.5, similar features to that of the β−γ timedifference matrix can be observed, including the horizontal “time-random”lines of the random coincidences and the low-energy tail. Again, selection ofthe prompt γ-rays located within the time cut of Fig. 5.5 allows removal ofcontaminant data and ensures most (if not all) of the gated data is from thedecay of interest.49Figure 5.5: Plot of the time difference between β-gated γ-rays versus theenergy of the first detected γ-ray. The two-dimensional time cut to selectprompt γ − γ coincidences is overlaid in red.50Once only prompt β − γ − γ events have been selected, a coincidencematrix of these events is constructed. One plots the energy of the first γ-rayon the x-axis and the energy of the second γ-ray on the y-axis; the first γ-rayis then plotted on the y-axis and the second on the x-axis to ensure symmetryof the coincidence matrix. By examining the matrix, one can begin to noticecoincidences between γ-rays.The β − γ − γ coincidence matrix for this decay is shown in Fig. 5.6. Thetotal spectrum is shown in Fig. 5.6a, and a zoomed-in portion in Fig. 5.6b.Horizontal and vertical lines in the matrix correspond to coincidences withthe Compton continuum. Diagonal lines are the Compton-scattered γ-raysthat have been detected as a coincidence event. Spots of increased statisticscorrespond to coincidences between the γ-rays. From the coincidence matrix,one can quickly tell whether or not two γ-rays are in coincidence.Gating on a γ-ray from the coincidence matrix creates a spectrum showingγ-rays in coincidence with the gated γ-ray. Fig. 5.7 shows an example gateapplied on the 310 keV γ-ray. The red box in Fig. 5.7a represents the gateapplied, and the black boxes are background gates taken near the γ-ray ofinterest to allow for background-subtraction of the gated spectrum. The re-sultant background-subtracted gated spectrum is shown in Fig. 5.7b. Becausethe diagonal Compton-scatter lines appear at different energies within the twobackground gates, so-called “scatter peaks” are present in the background-subtracted gated spectra. These scatter peaks can give the appearance of”false” coincidences within the spectrum and must be identified. Two promi-nent scatter peaks in the 310 keV gate are labeled in Fig. 5.7b.After recording coincidences between γ-rays, the transitions are placedin a level scheme for the nucleus. Coincident transitions are identified usingthe analysis rules outlined in Appendix B. New excited states are establishedbased on coincident (and non-coincident) transitions as needed.51(a)(b)Figure 5.6: β-gated γ − γ coincidence matrix for (a) the overall spectrumand (b) a zoomed-in region showing coincidences between the 409/417 and925 keV γ-rays and the 409/413/417 and 995 keV γ-rays. Areas of increasedstatistics correspond to coincidences between γ-rays.52(a)(b)Figure 5.7: (a) γ − γ coincidence matrix. A gate on the 310 keV γ ray isshown in red. Background gates on either side of the gate are shown in black.(b) Projection of the gated spectrum from (a). Background subtraction hasbeen applied, resulting in the labeled scatter peaks.535.4 Relative IntensitiesIntensities are taken relative to the highest intensity γ-ray; in this case, this isthe 173 keV γ-ray. The 173 keV γ-ray corresponds to the 4+ −→ 2+ transitionin the ground state band, and while the 75 keV 2+ −→ 0+ transition followingit should be more intense, the high conversion electron rate (α ≈ 7.3) of the75 keV transition makes the 75 keV γ-ray less intense than the 173 keV γ-ray(α ≈ 0.4).The relative intensity of a γ-ray is calculated by taking the ratio of theefficiency-corrected singles area of the photopeak of interest to the efficiency-corrected photopeak area of the 173 keV transition:Iγ1 =Aγ1/γ1A173/173(5.1)where Aγ is the (summing-corrected) singles photopeak area of the γ-ray ofinterest and γ is the singles efficiency of the γ-ray.5.4.1 “Gating From Above” Methodγ-ray branching ratios may be determined via the so-called “gating fromabove” method. In this, a gate is taken on a strong transition populatingthe excited state of interest. Efficiency-corrected areas of the γ-rays depopu-lating this level can be compared to determine the branching ratio out of thatlevel. For a strong transition γa populating a state, the branching ratio BRbof a transition depopulating that state is given asBRγb =Aγb(a)/γb∑nAγn(a)/γn. (5.2)In Eq. 5.2, Aγn(a) denotes the the photopeak area of a γ-ray γn in the spec-trum gated on γa, and γn denotes the singles efficiency of γn. It must beremembered that the sum over n in Eq. 5.2 only sums over γ-rays depopu-lating the state of interest–not all γ-rays seen in the gate. In Fig. 5.8, a gateis taken on the feeding γ-ray γf , and fits on the photopeaks of γa and γc areused to determine the branching ratio between γa and γc.Using the “gating from above” method, one can determine branching ra-tios for γ-rays that are too weak to be seen in the singles spectrum. If a strongdepopulating transition can be readily fit in singles, then the branching ratiosof the strong transition and weak transition can be used to determine theefficiency-corrected “singles” area of the weak transition.545.4.2 “Gating From Below” MethodWhen no strong feeding transition can be gated on, the intensity of a weaktransition may be determined via the “gating from below” method. By gatingon a strong transition depopulating the state that the weak transition feeds,one can determine the intensity of the weak transition from the gated spec-trum. As an example, consider Fig. 5.8. If γa cannot be fit in singles, andthere exists no strong transition feeding it, one can use the method of “gatingfrom below” to determine the intensity of γa.When examining a coincidence spectrum between two γ-rays in cascade,with a feeding transition γa and depopulating transition γb, calculating thesingles photopeak area from the gated spectrum is not trivial. As stated inRef. [39], the area of the coincidence photopeak Aab between γa and γb isgiven byAab = NIγaγaBRγbγbγaγbη(θγaγb). (5.3)In Eq. 5.3, Iγa represents the intensity of the feeding γ-ray (γa), γa and γbthe singles efficiencies of γa and γb, respectively, BRγb is the branching ratio ofγb, γaγb is the coincidence efficiency between the two γ-rays, η(θγaγb) the an-gular correlation attenuation factor between the γ-rays, and N a coincidencenormalization constant that is specific to the isotope. With the use of a lib-eral coincidence timing gate, γaγb ≈ 1, the overall efficiency of coincidencesis simply the product of the two individual efficiencies.With the assumption that γaγb ≈ 1, rearranging Eq. 5.3 to obtain IγayieldsIγa =AabNγaBRγbγbη(θγaγb). (5.4)Because γa , γb , and BRγb are known, a redefinition of I′γa =AababBRγbsim-plifies Eq. 5.4. Since N and η(θγaγb) are still unknown, one cannot directlycalculate the intensity of γa. However, by gating from below on all transitionsdepopulating the level of interest, one can use a similar approach to that ofSec. 5.4.1 to calculate the associated branching ratio of the γ-ray of interest:BRγa =I′abNη(θγaγb )∑nmI′nmNη(θγnγm )(5.5)where the sum over nm runs over the cascades depopulating parallel to thebranch of interest: the cascades γa − γb and γc − γd in Fig. 5.8, for example.From Eq. 5.5, one can see that the normalization factor N–which is energyindependent–cancels out. For highly-symmetric arrays, such as the 8pi array55γaγbγcγdγfFigure 5.8: Example level scheme for the “gating from above” and “gatingfrom below” methods. A gate on γf is used to determine the branching ratiobetween γa and γc. If γa is weak, gating from below on γb can allow thedetermination of the intensity of γa.used in Ref. [39], the angular correlation attenuation factor η(θγaγb) can beapproximated as 1.Unfortunately, because GRIFFIN is not a highly-symmetric array, η(θγaγb)cannot be approximated as 1. Furthermore, because one GRIFFIN HPGeclover was removed to accommodate PACES during this experiment, the an-gular correlation attenuation factors become even trickier to calculate. Be-56cause of this, the “gating from below” method was not used to determineintensities and branching ratios for this analysis. However, the “gating frombelow” method can be used to determine the presence of weak transitions,as well as energies of peaks contained within a singles multiplet. Thus, the“gating from below” method still has use aside from the determination ofbranching ratios.5.5 Absolute IntensitiesThe absolute intensity of a transition corresponds to the number of times thetransition occurs per 100 β-decays. To calculate the absolute γ intensity of atransition, one takes the efficiency-corrected counts from the β-gated γ-singlesphotopeak of interest and divides by the number of detected β-particles.The number of β-particles is obtained by fitting the histogram of thenumber of detected β-particles as a function of the tape cycle time. Thefit, shown in Fig. 5.9, consists of fitting the beam-on portion of the cycle(10 < t < 390 s) with Eq. 2.21a and the beam-off portion with an exponentialdecay (Eq. 2.18); both fits also include a constant background term.5.6 β-decay5.6.1 β-feeding IntensitiesTo calculate the β-feeding intensities, one must first correct for conversionelectron transitions, as these compete with γ-decay. The conversion coefficientα denotes the ratio of conversion electrons to γ-decays for a given transition.Conversion coefficients were calculated using the BrIcc online calculator tool[17]. Corrections for conversion electrons are especially important for the 75keV 2+ −→ 0+ transition in the ground state band of 160Gd. This transitionis highly converted, as the calculated conversion coefficient for this transitionis α ≈ 7.3. Conversion corrections were applied to transitions where theconversion coefficient is greater than α = 0.05, indicating a conversion rate ofmore than 5%.With conversion electrons taken into account, one can then determinethe β-feeding intensities to each excited state in the 160Gd daughter nucleus.For each state, the β-feeding intensity is given by the difference between thesum of the number of transitions (efficiency-corrected γ-rays and conversionelectrons) feeding and depopulating the state of interest. Any “extra” decaysthat are seen depopulating, but not feeding, the state of interest indicate β-decay feeding to this excited state. The intensity of a transition is normalized57Cycle time [s]50 100 150 200 250 300 350 400 450 500Counts05001000150020002500300035004000'sβ 71.32(7)x10Figure 5.9: Number of detected β-particles as a function of tape cycle time.The fit is that of Eq. 2.20 between 10 and 390 seconds and an exponentialdecay afterwards; both fits also include a constant background term. Approx-imately 1.32(7)·107 β-particles were detected.58to the number of detected β-particles, which is determined via a fit to thenumber of detected β-particles (1.32(7)·107), as described in Sec. 5.5.The sum of the β-feeding intensities to all excited states gives an indicationof missing states and transitions: the total β-feeding to the daughter shouldequal 100%. Any missing β-intensity puts an upper limit on the amount ofdirect ground state feeding in the decay. The actual value of the ground statefeeding intensity may be less than the limit; unobserved weak transitions andthe Pandemonium Effect [40] reduce the feeding to the ground state, but sincethese are unable to be observed, they cannot be quantified.5.6.2 log(ft) ValuesAs described in Sec. 2.2.2, calculation of log(ft) values allows comparison ofthe transition rates to each state during β-decay. log(ft) values are calculatedusing the NNDC log(ft) calculator application [12]. An updated Qβ− valueof 4.4483(16) MeV, calculated from the mass excess of 160Gd from Ref. [41]and the newly-measured mass excess of 160Eu from Ref. [3], was used for thecalculations. The half-life used in the calculation depends on if the populatedexcited state is associated with the decay of the 5− ground state or 0− isomericstate. Determination of the half-life for each decay is discussed in Sec. 5.7, andthe determined half-lives from this work were used for the log(ft) calculations.5.6.3 Spin-parity assignmentsThe log(ft) value can be used to determine possible spin-parity assignmentsto a particular excited state. By examining the log(ft) value for a state andcomparing it to other log(ft) values from nearby similar nuclei, and with therough guidelines of Table 2.1, one can determine the likely change of spinand parity from the β-decaying state to the excited state of interest. Withknowledge of the parent state’s spin-parity, possible spin-parity assignmentsof the daughter’s excited state can be determined.One can further constrain the spin-parity of a level by examining whatstates transition to and from the excited state of interest. Because electric andmagnetic transitions of low angular momentum are preferred, the examinationof the multipolarity of a given transition or cascade with different spin-parityassignments can constrain the spin-parity of a level. An E1-E2 cascade wouldbe preferred over an M2-M3 cascade, for instance, so this gives an indicationas to which spin-parity assignments are more likely.While log(ft) values and cascade multipolarities provide information onlikely spin-parity assignments, sometimes they cannot constrain the assign-ments completely. In this case, γ − γ angular correlation analyses, like those59described in Ref. [42], can be used to further constrain the spin-parity assign-ment of a level. γ−γ angular correlation analysis has not yet been completedfor this thesis but will be performed for the forthcoming publication.5.7 Half-livesOne can determine the half-life of the radioactive decay by fitting the decayportion of the tape cycle with Eq. 2.18 and adding a constant backgroundterm. When multiple β-decaying states exist in an isotope, such as with160Eu, one examines the time distribution of γ-rays to extract the differenthalf-lives. The time distribution of γ-rays associated with the β-decay of aparticular parent state are fit to extract the β-decaying half-life of that state.By summing the (background-subtracted) time distribution of multiple γ-rays associated with the decay of a single parent state, one can obtain a moreaccurate half-life measurement of the β-decay. An example of the fit of γ-raysassociated with the ground state β-decay of 160Eu is shown in Fig. 5.10.Time [s]0 20 40 60 80 100Counts100200300400500Ground State Decay = 42.7(7) s1/2tFigure 5.10: Time distribution and fit of γ-rays associated with the groundstate β-decay of 160Eu. The x-axis is the time elapsed since beam delivery toGRIFFIN halted, corresponding to t = 390 s in the overall tape cycle.Systematic uncertainties of the half-life fit are examined via performinga “chop analysis” [43]. By plotting the half-life as a function of fit region,60First Bin Included [s]390 395 400 405 410Half-life [s]40414243444546Half-life vs First Bin IncludedFigure 5.11: Chop analysis performed on the ground state half-life. Thevariation in half-life with the first included bin in the fit is shown. Theextracted half-life (solid line) and associated uncertainty (dashed lines) areoverlaid.one can determine the extent to which the fit region influences the extractedhalf-life. Rate-dependent effects on the half-life will be reflected in a “chopanalysis” of the first bin of the fit. Long-lived contaminants will be seen in the“chop analysis” of the ending bin of the fit. Because the different data pointsin a “chop analysis” are not independent of each other–each data point in afirst bin “chop analysis” contains all of the data held in each point to the rightof it–the data points are not statistically clustered around the mean half-lifevalue. The extracted half-life is obtained by fitting the largest region wherethere is no significant change in half-life when a “chop analysis” is performed.A plot showing the effect of the first bin of the fit for the ground state half-lifeis shown in Fig. Fast-timing TechniquesOne can use the LaBr3(Ce) detectors as the ancillary detectors for GRIFFINin order to determine the half-life of states within the excited nucleus. Half-lives of excited states are determined by computing the time difference be-61tween γ-rays in the LaBr3(Ce) detectors. The DAQ uses a time-to-amplitudeconverter (TAC), among other electronics, to process the signal. ROOT treesare built when there are events in two LaBr3(Ce) detectors and a TAC, as wellas an optional HPGe hit. Gates can then be placed on transitions in the twoLaBr3(Ce) detectors and the TAC spectrum plotted to show the difference intime between the LaBr3(Ce) events.When the half-life of the state is less than ≈100 ps, the time-difference sig-nal will resemble a Gaussian distribution. When longer half-lives are present,the signal will appear as a Gaussian with an exponential tail. For the long-lived half-lives, a fit of a Gaussian convoluted with an exponential decay allowsextraction of the decay constant for the state, which can easily be translatedinto the half-life for the state.While analysis of LaBr3(Ce) data was not a focus for this thesis, onepreliminary half-life was extracted from the LaBr3(Ce) data to aide in thediscussion of results obtained from the HPGe analysis. Much more detailedinformation on LaBr3(Ce) fast-timing techniques and analysis is available in[33, 44].62Chapter 6ResultsApproximately 3.8 hours of 160Eu decay data was collected using the GRIF-FIN setup. This chapter will discuss the results of the analysis carried out inChapter 5.6.1 β − γ − γ Coincidence Analysis6.1.1 Doublets264/266 and 822/826 keV DoubletsApplying a gate on the 925 keV γ-ray allows identification of two doubletpeaks that are difficult to spot in the singles data. In the singles spectra, a266 and 822 keV γ-ray can be observed; however, since the 925 keV γ-ray isnot in coincidence with either of them, a gate on the 925 keV reveals thatthose peaks are actually doublets. The 266 keV singles peak also contains a264 keV peak, and the 822 keV singles peak also contains a 826 keV peak, asseen in Fig. 6.1. The ≈2 keV separation between each doublet peak shouldbe resolved in singles. However, since both the 266 and 822 keV γ-rays arevery intense, their photopeak areas encompass the adjacent doublet peak ineach case.The 826 keV γ-ray was placed transitioning between the 1999 and 1173keV states, as it is seen in coincidence with γ-rays depopulating the 1173keV state. By gating on transitions from higher-energy states than the 1173keV state, one can look for coincidences to reaffirm this placement. A lackof coincidences between the 826 keV transition and transitions depopulatingstates above 1173 keV reaffirms this placement. However, since the 822 and826 keV transitions are in cascade, one cannot gate from above on the 826 keVtransition to further confirm its placement, and since the 822 keV transitionis so much more intense than the 826 keV transition, one cannot gate directlyon the 826 keV γ-ray to observe all coincidences with it.The 266 keV transition has been previously established as the 6+ −→ 4+transition in the ground state band. Placement of the 264 keV γ-ray of thisdoublet is in the discussion of the 1437/1438 keV states in Sec. [keV]0 100 200 300 400 500 600 700 800 900Counts010002000300040005000600070008000Gate: 925 keVx-rays173.375.4264.4 309.9 417.0408.5515.6560.7825.8763.0Figure 6.1: Gate on 925 keV γ-ray. The 264 and 826 keV components of the266 and 822 keV singles doublets can be seen.1144 keV DoubletThe singles peak at ≈1144 keV is also a doublet. Gating on the 606 keV γ-ray, this peak has a centroid location of 1144.87(6) keV (statistical uncertaintyonly). However, when one gates on the 1042 keV transition, this γ-ray has anenergy of 1143.33(8) keV. Gating on the 1215 keV transition, which depop-ulates the same level as the 1042 keV transition, this peak has an energy of1143.32(6) keV. While the systematic energy uncertainty is surely larger thanthe statistical uncertainty on these fits, any systematic uncertainty shouldaffect these peaks almost identically, since they are at (essentially) the sameenergy. The fact that they are seen at different energies that are significantlylarger than the statistical uncertainty indicates that these are indeed differentpeaks. The different peaks at 1144 keV can be seen overlaid in Fig. 6.2.The 1145 keV transition is placed depopulating the 1393 keV state due toits coincidence with the feeding 606 keV γ-ray, as well as coincidences withthe ground state band. The 1143 keV transition is placed as a transition64between the 2434 and 1291 keV states due to its coincidences with the 1042and 1215 keV transitions which depopulate the 1291 keV level. Energy [keV]1130 1135 1140 1145 1150 1155 1160 1165 Counts020406080100120140Gate: 606 keVGate: 1042 keVGate: 1215 keVFigure 6.2: Comparison of the 1144 keV peaks in three different gates. Thepeak centroid in the 1042 and 1215 keV gates, which depopulate the samelevel, is different from the centroid location in the 606 keV gate.1302/1305 keV DoubletThe singles photopeak around 1304 keV is also a doublet. With a distortedpeak shape in the singles spectrum, this doublet is easier to identify and placethan many of the others. One can create tight gates on each edge of the peakto identify coincidences with each γ-ray that constitutes the doublet peak.Fig. 6.3 shows the two gated spectra. There are clearly different peaks ineach gate corresponding to the different coincidences between γ-rays.The 1302 keV transition is placed depopulating the 1377 keV state due tocoincidences with the feeding 1088 keV transition and a lack of coincidencewith the 173 keV transition in the ground state band. The 1305 keV γ-ray isplaced depopulating the 2363 keV state due to its coincidences with the 808and 983 keV transitions which depopulate the 1058 keV state, in additionto no coincidences with transitions depopulating states of higher energy thanthe 1058 keV state.65Figure 6.3: Comparison of 1302 and 1305 keV gates from the doublet peakat 1304 keV. Different peaks are clearly seen in the different gates.666.1.2 Levels1148 keV 4+γ StateOf the known low-spin members of the γ-band in 160Gd, only the 4+ statewas not observed in this analysis. The 2+, 3+, 5+, 6+, and 7+ states wereall observed and confirmed in this analysis. The 4+γ state is placed at 1148keV from the ENSDF evaluation [9], and recent publications confirm thisplacement [19, 20, 21].Three γ-rays are placed as depopulating this state: 633, 899, and 1073 keV[9]. Only the 899 keV transition is observed in this analysis. A gate takenon the 899 keV γ-ray is unhelpful, as there are a number of scatter peaksthat clutter the spectrum. The 384 keV γ-ray was placed by Refs. [19, 20]as populating the 1148 keV 4+γ state. The 384 keV transition is clearly incoincidence with the 1150 and 1225 keV transitions depopulating the 1225keV state. Furthermore, the 899 keV transition is not seen in coincidencewith the 384 keV transition, as seen in Fig. 6.4. The placement of this γ-rayby Refs. [19, 20] is clearly incorrect.Refs. [19, 20] also placed a 543 keV γ-ray as populating the 4+γ state. Thisγ-ray is not seen in this analysis. Finally, Ref. [20] places a 288 keV γ-raypopulating the 4+γ state. The proximity of a 286 keV γ-ray makes the 288 keVγ-ray difficult to confirm. However, when gating on the 899 keV transition,there is no coincidence with a 288 keV transition. Thus, the 1148 keV 4+γstate could not be confidently placed in the decay scheme for 160Gd based onthis analysis.1437/1438 keV StatesIdentification of two closely-spaced excited states at 1437 and 1438 keV comesfrom coincidences and gates between multiple γ-rays. Applying a gate on the1188 keV γ-ray (Fig. 6.5) shows only coincidences with the ground state bandand the 1028 keV γ-ray–though very faint. A gated spectrum for the 1361keV γ-ray shows similar coincidences, indicating the presence of an excitedstate at approximately 1437 keV and allowing the 1028 keV γ-ray to decayfrom the newly-established 2465 keV state. The 1437 keV state has previouslybeen identified as a state in 160Gd, appearing in the ENSDF evaluation [9] asa 2+ state at 1435.99(11) keV identified via 160Gd(n, n′γ) reactions.As previously shown in Fig. 6.1, a coincidence exists between the 264 and925 keV γ-rays. Gates on the 1052 and 1122 keV γ-rays also show coincidenceswith the 264 and 925 keV γ-rays (among others depopulating the 1173 keVstate). The 1052 and 1122 keV γ-rays also do not show coincidences with other67 Energy [keV]900 950 1000 1050 1100 1150 1200 1250 Counts020406080100120140Gate: 384 keV11501225No 899Figure 6.4: Gate on the 384 keV γ-ray. Coincidences with the 1150 and 1225keV transitions are clearly seen. No coincidence with the 899 keV γ-ray isobserved.68Figure 6.5: Gate on 1188 keV γ-ray. Coincidences with only the ground stateband, as well as the feeding 1028 keV γ-ray (inset), can be seen.69γ-rays depopulating the identified 2490 and 2560 keV states, respectively.Thus, another excited state exists at approximately 1438 keV; this state ispopulated by the 1052 and 1122 keV transitions and depopulates via the 264keV γ-ray to the previously-established 1173 keV state. This state has onlybeen published once before in [21]. In that paper, it was assigned as the 7+member of the Kpi = 4+ band and given at an energy of 1437.2 keV. There isno published information on transitions to and from this level, so this analysisprovides the first knowledge on the transitions related to this excited state.6.1.3 Low-energy TransitionsThere exist a number of low-energy, low-intensity transitions seen in thisnucleus that hint towards a new understanding of the structure of this nucleus.82 keV TransitionThe 82 keV transition from the 1071 keV state to the 989 keV state is anextremely weak transition that has not been before observed. Located in theCompton continuum, the transition is not seen in singles, nor is it seen inthe majority of gated spectra. The only indication of the 82 keV transitioncomes when gating from below on the transitions depopulating the 989 keVstate. The 989 keV state has previously been identified as the bandhead ofthe Kpi = 2+ γ-vibration band [9] and is depopulated by the 914 and 989 keVγ-rays. Gates on either of the depopulating transitions show an 82 keV γ-rayin coincidence. Furthermore, the gated spectra show coincidences with the516/413 γ-ray cascade that feeds the 1071 keV state; a transition between the1071 and 989 keV states is further supported by these coincidences. Gatedspectra showing the 82 keV transition are shown in Fig. 6.6.The intensity of the 82 keV transition is difficult to determine. Since itcannot be fit in singles and when gating from above, one must use the “gatingfrom below” method to determine the intensity of the transition. However,as stated in Sec. 5.4.2, the angular correlation attenuation factors neededfor GRIFFIN are nontrivial. If the approximation holds that the angularcorrelation attenuation factors are negligible (η(θγaγb) ≈ 1), then the “gatingfrom below” method yields a relative (to the 173 keV γ-ray) γ-ray intensityof Iγ ≈ 0.06% for this transition.One can also put a limit on the intensity by examining the intensity ofnearby low-intensity transitions that can be observed in the singles spectrum.The 99 keV transition has a γ-ray intensity of 0.9% and can clearly be seenand fit in the singles spectrum. Since the singles efficiencies of those γ-raysis similar, and since the 82 keV transition cannot be seen in singles, the 8270keV transition must have an intensity that is much less than 0.9%. This isconsistent with the approximated intensity of 0.06% from the “gating frombelow” method. Since other γ-rays with intensities as small as 0.1% wereobserved in this data, an intensity limit of Iγ < 0.1% has been assigned forthe intensity of this transition.116 keV TransitionMuch like the weak 82 keV transition, there also exists a weak 116 keV tran-sition that can only be seen when gating from below. The 116 keV transitionis placed as a transition from the 1173 keV state to the 1058 keV state, as it isonly seen when gating on the two transitions depopulating the 1058 keV state(809 and 983 keV γ-rays). In the gated spectra, there are also coincidenceswith the intense 409, 417, and 516 keV transitions, which all feed (either di-rectly or indirectly) the 1173 keV state, indicating that there is a transitionbetween the 1173 and 1058 keV states. Again, the intensity of this transitionis difficult to determine, so a limit of Iγ < 0.1% is placed for this transitionalso. The gated spectra showing the 116 keV transition are shown in Fig. 6.7.71(a)(b)Figure 6.6: Gate on the 914 keV (a) and the 989 keV (b) γ-rays. The 82 keVγ-ray can be seen in both spectra. 72(a)(b)Figure 6.7: Gate on the 809 keV (a) and the 983 keV (b) γ-rays. The 116keV γ-ray can be seen in both spectra. 736.2 Level SchemeThe identified γ-rays associated with the de-excitation of 160Gd were placedin the level scheme in Fig. 6.8. Previously published levels, transitions, andspin-parity assignments are shown in black, with those not (yet) published orupdated shown in red.Identified excited states and the associated depopulating transitions andintensities are shown in Table 6.1. Results are compared to the ENSDFevaluation [9] and recent literature when different from the evaluation. Thevalue of the excited state energy is the average of all possible γ-ray cascadesleading to that level. Transition intensities are normalized out of each excitedstate for both this work and the ENSDF evaluation.Energy uncertainties were calculated by extracting photopeak energies ofstandard sources and comparing the energies to those of ENSDF for eachisotope. The standard deviation of the residuals for 27 known γ-rays fromfour different source isotopes mentioned in Appendix A was σ = 0.154 keV.Thus, an uncertainty of 0.2 keV is assigned to each γ-ray unless the statisticaluncertainty is greater than 0.2 keV, in which case that value is assigned.The uncertainty in energy level was calculated by the quadrature sum ofthe uncertainties of transitions summing to an excited state energy. Thestandard deviation of energies from all possible cascades contributing to theaverage energy of each excited state was also calculated. The uncertainty onexcited state energy values is whichever is greater between the uncertaintypropagation and the standard deviation of possible excited state values.Table 6.2 shows the energies of γ-rays associated with the de-excitationof 160Gd following the 160Eu β-decay. γ-rays are ordered by energy, and theintensity is the γ-ray intensity relative to the 173 keV transition. If the transi-tion is placed in the level scheme, the associated initial and final state energiesare included. γ-rays that appear in Table 6.2 without an associated initial andfinal level are thought to be associated with the 160Gd de-excitation, but havenot yet been placed in the level scheme–likely due to uncertain or insufficientcoincidence measurements with known γ-rays.Conversion coefficients are included in Table 6.2 for γ-rays below 600keV that have α > 0.05, corresponding to a conversion electron contribu-tion greater than 5% of the γ-ray intensity. Absolute γ-ray intensities weredetermined by a fit of the β-particle time-distribution histogram, as describedin Sec. 5.6.1. The total number of β-particles detected was 1.32(7) · 107. Therelative-to-absolute γ-ray intensity coefficient was determined to be 0.21(4)and was calculated by dividing the efficiency-corrected photopeak area of theintense 173 keV γ-ray by the number of β-particle detected.7464160 Gd960+ 02+ 75.44+ 248.66+ 515.08+ 868.32+ 989.03+ 1057.84+ 1070.85+ 1173.41- 1224.85+ 1261.63- 1290.5(4+,5+,6+) 1295.7(1,2+) 1351.7(2) 1377.36+ 1393.52+ 1436.9(4+,5+,6+,7+) 1437.84+ 1483.47+ 1548.7(5+) 1582.0(0+,1+,2+) 1608.92+ 1932.4(5-) 1999.12328.2(4,5,6) 2345.02362.9(1,2+) 2433.6(1-) 2465.0(5-) 2489.9(0+,1,2+) 2511.7(5-) 2560.03090.33115.675.4173.2266.4353.4913.7988.9809.1982.581.6555.7822.1995.5102.7115.6658.3925.01149.51224.8203.9746.41012.91041.71215.3224.9780.81047.31276.41351.61301.9878.41144.91188.31361.5264.4187.6309.9412.6968.61235.0680.21033.898.6286.2408.5384.0874.8943.31683.71856.91932.1417.0450.5515.6605.7737.8825.81484.11270.41339.2763.01171.81138.61304.92287.61056.51143.31208.92357.9855.81028.31088.01113.52389.62464.8490.71006.81052.61194.41316.81159.41286.92436.0560.81076.81122.61264.51386.82841.72044.8Figure 6.8: Level scheme of 160Gd resulting from the β-decay of 160Eu. Levels, transitions, and spin-parity assign-ments that have not been previously published are shown in red. Arrow widths indicate relative γ-ray intensities.75Table 6.1: Identified excited levels and associated depopulating γ-rays. Intensities are normalized out of each level. Results arecompared to the ENSDF evaluation [9] and more recent data if different from the evaluation.This work ENSDF Recent workElevel[keV] Eγ [keV] Iγ [%] Elevel[keV] Eγ [keV] Iγ [%] Elevel Eγ75.4(2) 75.4(2) 100 74.26(1) 75.26(1) 100248.6(2) 173.3(2) 100 248.52(6) 173.19(9) 100515.0(3) 266.4(2) 100 514.75(8) 266.31(8) 100868.3(3) 353.4(2) 100 867.9 352.9 100989.0(1) 988.40(8) 740.02(20) 3.4(4)913.7(2) 59(16) 913.25(16) 100.0(11)988.9(2) 100(19) 988.52(15) 76.7(22)1057.8(2) 809.1(2) 20(6) 1057.54(9) 809.01(15) 38.7(11)982.5(2) 100(17) 982.13(15) 100.0(12)1070.8(2) 81.6(2) <0.3 1070.42(9)555.7(2) 1.7(4) 556.4(5) 2.8(13)822.1(2) 100(14) 822.04(18) 100(6)995.5(2) 64(13) 995.13(15) 61.1(10)1173.4(1) 102.7(2) 17(3) 1173.11(4)1, 1173.112, 11733115.6(2) <2.3658.3(2) 17(3) 658.20(12)1, 6583925.0(2) 100(21) 924.59(3)1, 92531224.8(1) 1224.28(9) 235.5(3) 4.1(11)1149.5(2) 100(17) 1149.18(15) 100.0(22)1224.8(2) 57(14) 1224.14(15) 65.2(15)1261.6(2) 203.9(2) 2.4(5) 1261.07(12) 203.0(4) 11(6)746.4(2) 21(4) 746.30(20) 23(2)1012.9(2) 100(15) 1012.47(15) 100(4)1 from Ref.[20], 2 from Ref.[21], 3 from Ref.[3], 4 from Ref.[19]76Continued from previous page.This work ENSDF Recent workElevel[keV] Eγ [keV] Iγ [%] Elevel[keV] Eγ [keV] Iγ [%] Elevel Eγ1290.5(2) 1290.10(9) 301.9(4) 2.3(8)1041.7(2) 45(10) 1041.42(16) 52(2)1215.3(2) 100(16) 1214.92(15) 100(2)1295.7(1) 1295.57(15) 1223148.2(9) 8(7)224.9(2) 100(10) 225.0(3) 34(4)238.5(10) 22(18)780.8(2) 37(7) 781.71(30) 23(4)1047.3(2) 25(5) 1046.72(18) 100(6)1351.7(1) 1351.09(10) 126.7(2) 9.4(18)203.0(4) 8(4)1276.4(2) 100(17) 1275.90(15) 100(3)1351.6(2) 33(8) 1351.0(2) 29.7(21)1377.3(2) 1377.06(13) 305.7(7) 1.6(9)319.0(4) 2.6(4)1129.0(6) 5.2(15)1301.9(2) 100 1301.46(3)1, 1301.46(5)41393.5(2) 1392.80(11) 244.9(5) 91(38)878.4(2) 100(15) 878.10(8) 100(9)1144.9(2) 59(13) 1142.9(5) 40(13)1436.9(2) 1435.99(11) 288.0(3) 11(2)1188.3(2) 100(16) 1187.7(2) 100(6)1361.5(2) 37(9) 1360.69(17) 30(3)1435.8(2) 20(1)1437.8(1) 264.4(2) 100 1437.221 from Ref.[20], 2 from Ref.[21], 3 from Ref.[3], 4 from Ref.[19]77Continued from previous page.This work ENSDF Recent workElevel[keV] Eγ [keV] Iγ [%] Elevel[keV] Eγ [keV] Iγ [%] Elevel Eγ1483.4(1) 187.6(2) 3.0(5) 14833 1883309.9(2) 7(1) 3103412.6(2) 100(12) 4133968.6(2) 5(1) 96831235.0(2) 10(2) 12353140831548.7(2) 1548.61(23) 288.0(3) 100(20)680.2(2) 12(3)1033.8(2) 100(15) 1033.4(3) 100(25)1582.0(1) 98.6(2) 10(1) 15823 993286.2(2) 6(1)408.5(2) 100(12) 40931608.9(2) 384.0(3) 1001932.4(2) 874.8(2) 53(14) 1931.89(13)1, 1931.96(7)4 874.42(28)1, 874.51(6)4943.3(2) 95(23)1683.7(2) 49(13) 1683.22(21)1, 1683.45(7)41856.9(2) 100(18) 1856.63(13)1, 1856.65(6)41932.1(2) 30(8) 1931.94(29)1,1932.00(7)41999.1(2) 417.0(2) 30(5) 19993450.5(2) 7(1)515.6(2) 100(12)605.7(2) 3.9(8)737.8(2) 14(3)825.8(2) 5(1)1484.1(2) 0.8(2)1 from Ref.[20], 2 from Ref.[21], 3 from Ref.[3], 4 from Ref.[19]78Continued from previous page.This work ENSDF Recent workElevel[keV] Eγ [keV] Iγ [%] Elevel[keV] Eγ [keV] Iγ [%] Elevel Eγ175132328.2(2) 1270.4(2) 75(20)1339.2(2) 100(17)2345.0(2) 763.0(2) 16(3)1171.8(2) 100 2346.5(3)2362.9(4) 1138.6(2) 26(6) 2361.91(16)11304.9(2) 100(17)2113.40(33)12287.6(2) 98(25) 2286.51(26)12433.6(2) 1056.5(2) 36(23)1143.3(2) 100(16)1208.9(2) 46(11)2357.9(2) 41(10)2465.0(3) 855.8(3) 5(1)1028.3(3) <0.11088.0(2) 27(7)1113.5(2) 18(5)2389.6(2) 6(2)2464.8(2) 100(20)2489.9(3) 490.7(2) 100(12) 49131006.8(2) 59(12)1052.6(2) 24(5)1194.4(2) 39(8)1316.8(2) 57(12)2511.7(3) 1159.4(4) 10(6)1 from Ref.[20], 2 from Ref.[21], 3 from Ref.[3], 4 from Ref.[19]79Continued from previous page.This work ENSDF Recent workElevel[keV] Eγ [keV] Iγ [%] Elevel[keV] Eγ [keV] Iγ [%] Elevel Eγ1286.9(2) 100(16)2436.0(2) 73(19)2560.0(3) 560.8(2) 100(13) 56131076.8(2) 43(9)1122.6(2) 17(3)1264.5(2) 22(5)1386.8(2) 40(8)3090.3(3) 2841.7(2) 1003115.6(2) 2044.8(2) 1001 from Ref.[20], 2 from Ref.[21], 3 from Ref.[3], 4 from Ref.[19]80The 82, 116, and 1028 keV γ-rays were too weak to be fit in singles. Inlieu of the “gating from below” method, upper bounds were placed on theintensities of these weak transitions. Intensities of nearby weak transitionsthat can be seen and fit in singles were assigned as upper bounds on theintensities of these weak transitions that could otherwise only be determinedvia gating from below.Table 6.2: γ-rays associated with the decay of 160Gd. Intensities are takenrelative to the 173 keV γ-ray. For absolute γ-ray intensities, multiply by0.21(4). For absolute transition intensities, multiply the absolute γ-ray inten-sity by (1 + α).Eγ [keV] Iγ [%] α Ei[keV] Ef [keV]75.4(2) 29(4) 7.28 75 081.6(2) <0.1 5.28 1071 98998.6(2) 0.9(1) 2.08 1582 1483102.7(2) 4.1(5) 1.86 1173 1071115.6(2) <0.1 1.51 1173 1058122.2(2) 0.7(3)173.3(2) 100 0.361 249 75187.6(2) 4.4(6) 0.273 1483 1296203.9(2) 3.8(1) 0.211 1262 1058224.9(2) 6.7(9) 0.150 1296 1071264.4(2) 3.0(4) 0.0899 1438 1173266.3(2) 16(2) 0.0876 515 249286.2(2) 4.7(7) 0.110 1582 1296300.2(3) 1.5(2)309.9(2) 9(1) 0.0876 1483 1173353.4(2) 1.7(3) 868 515367.3(2) 2.4(4)384.0(3) 1.8(3) 1609 1225408.5(2) 19(3) 1582 1173412.6(2) 81(12) 1483 1071417.0(2) 17(2) 1999 1582432.8(2) 2.9(4)450.5(2) 3.4(5) 0.110 1999 1549490.7(2) 5.4(8) 2490 1999515.6(2) 84(13) 1999 1483555.7(2) 1.1(2) 1071 515560.7(2) 8(1) 2560 1999605.7(2) 2.7(4) 1999 139381Continued from previous page.Eγ [keV] Iγ [%] α Ei[keV] Ef [keV]658.3(2) 4.1(7) 1173 515680.2(2) 1.6(3) 1549 868703.5(3) 1.7(3)732.7(2) 1.2(2)737.8(2) 12(2) 1999 1262746.4(2) 1.5(3) 1262 515763.0(2) 3.7(6) 2345 1582769.7(2) 3.3(6)780.8(2) 2.6(4) 1296 515809.1(2) 0.9(2) 1058 249822.1(2) 61(10) 1071 249825.8(2) 1.7(3) 1999 1173855.8(3) 0.6(1) 2465 1609874.8(2) 0.6(2) 1932 1058878.4(2) 1.2(2) 1393 515899.7(2) 1.1(2)913.7(2) 2.7(5) 989 75924.9(2) 19(3) 1173 249943.3(3) 0.8(1) 1932 989968.6(2) 6(1) 1483 515977.8(2) 0.39(8)982.5(2) 2.9(5) 1058 75988.9(2) 2.1(4) 989 0995.5(2) 39(7) 1071 751006.8(1) 3.7(7) 2490 14831012.9(1) 12(2) 1262 2491028.3(3) < 0.1 2465 14371033.8(2) 2.4(4) 1549 5151041.7(2) 1.5(3) 1291 2491047.3(2) 2.1(4) 1296 2491052.5(2) 1.4(3) 2490 14381056.5(2) 0.5(1) 2434 13771076.8(2) 3.3(6) 2560 14831088.0(2) 1.5(3) 2465 13771113.5(2) 1.1(2) 2465 13521122.6(2) 0.9(2) 2560 14381138.6(2) 0.8(2) 2363 122582Continued from previous page.Eγ [keV] Iγ [%] α Ei[keV] Ef [keV]1143.3(2) 1.5(3) 2434 12911145.0(2) 0.7(1) 1393 2491149.5(2) 4.6(8) 1225 751159.4(4) 0.22(7) 2512 13521161.8(2) 0.3(1)1165.9(2) 0.20(6)1171.8(2) 0.41(8) 2345 11731184.0(2) 1.0(2)1188.3(2) 3.2(6) 1437 2491194.4(2) 2.4(4) 2490 12961208.9(2) 0.6(1) 2434 12251215.3(2) 3.0(6) 1291 751224.7(2) 3.2(6) 1225 01227.6(2) 0.23(8)1235.0(2) 8(2) 1483 2491240.5(2) 2.4(4)1264.5(2) 1.5(3) 2560 12961270.4(2) 0.7(1) 2328 10581276.4(2) 4.4(8) 1352 751286.9(2) 1.1(2) 2512 12251301.9(2) 4.9(9) 1377 751304.9(2) 3.2(6) 2363 10581316.8(2) 3.3(6) 2490 11731339.2(2) 0.9(2) 2328 9891351.6(2) 0.9(2) 1352 01361.5(2) 1.2(2) 1437 751386.8(2) 2.9(6) 2560 11731389.6(2) 0.5(1)1442.3(2) 0.23(5)1484.1(2) 0.22(5) 1999 5151683.7(2) 0.5(1) 1932 2491811.8(2) 1.9(4)1856.9(2) 0.9(2) 1932 751932.1(2) 0.36(8) 1932 02044.8(2) 0.19(5) 3116 10712202.7(2) 1.1(2)2242.8(3) 0.33(7)83Continued from previous page.Eγ [keV] Iγ [%] α Ei[keV] Ef [keV]2277.7(2) 1.3(3)2287.6(2) 3.0(6) 2363 752333.6(2) 2.5(5)2357.9(2) 0.6(1) 2343 752389.6(2) 0.4(1) 2465 752396.2(4) 0.22(8)2436.0(2) 0.7(2) 2512 752454.9(2) 0.33(7)2464.8(2) 6(1) 2465 02471.3(2) 0.18(5)2841.7(2) 0.13(3) 3090 2496.3 log(ft) Valuesβ-feeding intensities were calculated to each excited state in the 160Gd daugh-ter nucleus. The absolute γ-ray intensities of Table 6.2 were corrected forconversion electrons with the conversion coefficients in Table 6.2. Conversioncorrections were also only applied to transitions with α > 0.05 since γ-rayintensity uncertainties are generally on the order of 10% or greater in thisanalysis.log(ft) values were calculated using the determined β-feeding intensitiesand the half-lives determined in Sec. 6.5. The Qβ− of 4.4483(16) MeV [3, 41]was used. The log(ft) values were calculated using the NNDC log(ft) onlinecalculator tool [12]. β-feeding intensities and the corresponding calculatedlog(ft) values for each excited state are shown in Table 6.3.A number of states could not be unambiguously assigned to the 0− iso-meric state or the 5− ground state; those states are listed in the right columnof Table 6.3, and both log(ft) values corresponding to the isomeric and groundstate decays were calculated.84Table 6.3: β-intensities and log(ft) values for states populated by low-spin (left) and high-spin (centre) β-decay.States that could not be assigned to the isomeric or ground state β-decay are given in the right column and log(ft)values for the isomeric/ground state decays were calculated.0− isomer 5− ground state Unknown originState (keV) Iβ− [%] log(ft) State (keV) Iβ− [%] log(ft) State (keV) Iβ− [%] log(ft)0 <43 >6.1 248 2(3) 7.5(5) 1291 0.6(1) 7.33(8)/7.52(8)75 9(5) 6.7(3) 515 0 - 1933 0.69(6) 6.88(4)/7.05(4)989 0.5(1) 7.57(9) 868 0 - 2328 1.1(2) 6.39(8)/6.55(8)1058 0 - 1071 0 - 2363 1.5(2) 6.22(6)/6.39(6)1225 0.7(2) 7.3(1) 1173 0 - 3090 0.03(1) 7.2(1)/7.4(1)1352 0.8(2) 7.2(1) 1262 1.3(6) 7.2(2) 3116 0.04(1) 7.1(1)/7.2(1)1377 0.7(2) 7.2(1) 1296 0 -1437 0.9(1) 7.07(5) 1392 0 -1609 0.24(5) 7.5(1) 1438 0.2(1) 7.9(2)2434 1.2(1) 6.26(4) 1483 4(3) 6.6(4)2465 2.1(3) 6.00(7) 1549 0 -2512 0.44(5) 6.64(6) 1582 1.4(7) 7.0(2)1999 23(2) 5.48(5)2345 0.8(1) 6.67(6)2490 3.4(3) 5.92(4)2560 3.5(3) 5.85(4)85As seen in Table 6.3, the 0− isomeric state preferentially decays to the 2433and 2465 keV excited states, as well as into the 0+ and 2+ members of theground state band. The 5− high-spin ground state of the parent preferentiallydecays to the 1999 keV state. Lesser β-feeding occurs to the 2490 and 2560keV states, which feed the 1999 keV state, as well as to the 1483 keV state.The sum of β-feeding intensities to all identified excited state totals57(8)%, leaving approximately 43% of β-decays unaccounted for–either indirect ground state feeding or undetermined excited states. Thus, we assignIβ < 43% for the ground state feeding. However, direct ground state transi-tions from the 0− isomeric state are first forbidden transitions. It is highlyunlikely that a first forbidden transition has 43% feeding, so a significant frac-tion of the 43% unaccounted β-feeding is likely due to states and transitionsthat have not yet been observed in this analysis.6.4 Spin-parity AssignmentsExcited states that are new or have conflicting spins from previous assign-ments were assigned spin and parity based upon log(ft) values on transitionmultipolarities, as described in Sec. 5.6.3. States that are well-established anddo not have conflicting spin assignments did not undergo extensive scrutinyin this analysis. Table 6.4 summarizes the existing and proposed spin assign-ments for states identified in this analysis.Table 6.4: New spin-parity assignments and log(ft) values for excited statesin 160Gd. Values are compared to the ENSDF evaluation [9]. Recent datathat conflicts or updates the evaluation is noted as well. States that couldnot be definitively associated with a parent state have log(ft) values for boththe isomeric/ground state decays given.This work Previous workState (keV) Ipi log(ft) Ipi log(ft)0 > 6.1 0+ 6.275 6.7(3) 2+248 7.5(5) 4+515 - 6+868 - 8+989 7.79(9) 2+1058 - 3+1071 - 4+1173 - 5+ 1,2, (5+)31 from Ref. [20], 2 from Ref. [21], 3 from Ref. [3], 4 from Ref. [19]86Table 6.4: Continued from previous page.This work Previous workState (keV) Ipi log(ft) Ipi log(ft)1225 7.3(1) 1−1262 7.2(2) 5+1291 7.33(8)/7.52(8) 3−1296 - (4+, 5+), 6+ 1,2, (6+)31352 7.2(1) 1, 2+, 1− 1,2,41377 7.2(1) (2+), 2− 1,2,41393 - 6+1437 7.07(5) 2+1438 (4+, 5+, 6+, 7+) 7.9(2) 7+ 21483 6.6(4) 4+ 31549 - 7+1582 7.0(2) (5+) 31609 (0+, 1+, 2+) 7.5(1)1933 6.88(4)/7.05(4) 2+ 1,41999 5.48(5) (5−) 3 5.1 32328 6.39(8)/6.55(8)2345 (4, 5, 6) 6.67(6)2363 6.22(6)/6.39(6)2434 (1, 2+) 6.26(4)2465 (1−) 6.00(7)2490 (5−) 5.92(4)2512 (0+, 1, 2+) 6.64(6)2560 (5−) 5.85(4)3090 7.2(1)/7.4(1)3116 7.1(1)/7.2(1)1 from Ref. [20], 2 from Ref. [21], 3 from Ref. [3], 4 from Ref. [19]6.4.1 Existing States1296 keV StateThe discrepancy in the range of spins for the 1296 keV state cannot be im-proved on. The evaluated data [9] assigns this state as (4+, 5+), but recentpublications [3, 20, 21] place an assignment of 6+ for the state. No β-feedingis seen to this state, so log(ft) values cannot be used to assist in restrictingthe spin. γ-ray multipolarities also do not help constrain the spin further87than what is already seen in Table 6.4. A spin assignment for this state fromthe 160Eu β-decay cannot be made without further γ − γ angular correlationanalysis.1352 keV StateNo further determination is made on the 1352 keV state. The state hadpreviously been assigned spins of (1, 2−) from the ENSDF evaluation [9].Recent studies ([19, 20, 21]) have updated the spin assignment to 1−. Thelog(ft) value from this study is 7.2(1), indicating an allowed or first forbiddentransition. The direct ground state γ-ray rules out spin of zero for the state,leaving option of spin 1 or 2−. The assignment of 1− from previous paperscannot be improved upon without angular correlation analysis.1377 keV StateNo determination is made for the spin of the 1377 keV state. log(ft) valuespoint to an allowed or first forbidden transition, giving a possible spin of(0+, 1, 2+). Only one feeding and one depopulating transition are observedfor this state, so one cannot easily narrow down spins based on transitionmultipolarities. Furthermore, if recent spin assignments of Ipi = 2− [19, 20, 21]are correct, then this state would result from a second forbidden GT transitionof the 0− isomeric state. A log(ft) value of 7.2(1) does not immediately pointto a second forbidden transition. Second forbidden transitions generally havelog(ft) values above 10, and there is not sufficient evidence to suggest thatisotopes in this region of the nuclear chart have log(ft)≈7 correspond to asecond forbidden transition. A γ − γ angular correlation analysis performedin Ref. [20] ruled out spins of 0+, 1, 3−, and 4+ for this state. Based on thisanalysis and the analysis of Ref. [20], a spin of 2+ is the only possibility forthis state. This directly conflicts with the previous spin assignments of 2−,however, so further analysis is required. Thus, no spin assignment was madefor this state.1438 keV StateThe 1438 keV state is given an assignment of (4+, 5+, 6+, 7+). The log(ft)value of the β-feeding to this state is 7.9(2). This indicates a first forbiddenGT transition, so there is a change in parity and a spin change of 0, ±1, or ±2.γ-ray transitions that feed this state come from proposed (5−) states. Thestate also has a transition to a 5+ state, so this state is associated with the 5−ground state β-decay. Possible spins of the state are then (3+, 4+, 5+, 6+, 7+).88Spins (4+, 5+, 6+) are preferred, as these lead to E1−M1 cascades throughthe state. Spin assignments of (3+, 7+) give M2 − E2 cascades, which aresuppressed compared to an E1 −M1 cascade. Ref. [21] assigned this stateIpi = 7+ by an analysis of inertial parameters associated with states in a givenband. Because of this previous assignment of 7+, this spin has not been ruledout here. A more thorough analysis, such as with γ − γ angular correlations,is needed to refine the spin assignment for this state.2363 keV StateNo determination on the spin of the 2363 keV state has been made. Transi-tions to low-spin states from this level indicate association with the low-spinisomeric parent state, as this state has γ-rays to states of spin 1−, 2+, and 3+.However, Ref. [20] reported transitions to 2+ and 4+ states. If the transitionto a 4+ state is true, then one would assume that the 2363 keV state has spinof at least 2, otherwise this transition would be E3 or higher, which is unlikelyto occur compared to the E1/M1/E2 transitions to the low-spin states seenhere. This would indicate a large change in spin during the β-decay; however,a log(ft) of ≈6.3 does not indicate a highly forbidden β-decay, but rather anallowed or first forbidden transition. More analysis is needed before a spincan be proposed to this state.6.4.2 New States1609 keV StateThe 1609 keV excited state is given an assignment of (0+, 1+, 2+). A log(ft)value of 7.5(1) indicates a first forbidden GT transition. The state transitionsto a 1− state and is fed by a proposed 1− spin state as well, indicating that thestate is fed by the 0− isomeric state β-decay. Because only a single feeding anddepopulating γ-ray have been identified for this state, one cannot constrainthe spins further based on cascade multipolarities. The state is left with aspin assignment of (0+, 1+, 2+) until further analyses can be undertaken.2328 keV StateNo spin assignment is given to the 2328 keV state. Only two transitions wereidentified as depopulating this state: the 1270 keV γ-ray to a 3+ state andthe 1339 keV transition to a 2+ state. With log(ft)≈6.5, an allowed or firstforbidden transition is suggested to this state, but no further assignments canbe made without identification of more transitions from this state.892345 keV StateThe 2345 keV state is assigned a spin of (4, 5, 6). The log(ft) value of the stateis 6.67(6), indicating an allowed or first forbidden transition. Only two de-populating transitions have been identified for this state, and both transitionto a (5+) state. Spin assignments of (4, 5, 6) give transition multipolarities ofE1 or M1 (depending on the parity of the 2345 keV state). Spin assignmentsof (3+, 7+) yield E2 transitions, which are suppressed compared with the E1and M1 transitions. Nothing more can be ruled out without γ − γ angularcorrelation analysis.2434 keV StatePossible spins of (1, 2+) are assigned to the 2434 keV state. The log(ft) valueof 6.26(4) indicates an allowed or first forbidden transition, and the state hasγ-rays to 1−, 2+, and 3− states. The 0+ assignment is ruled out based onunfavorable transition multipolarities to these states. No further definitivespin assignment can be made without more information for this state.2465 keV StateA spin assignment of (1−) is given to the 2465 keV state. The β-decay tothis state is an allowed or first forbidden transition due to the log(ft) valueof 6.00(7). Since the state decays to low-spin states, it is associated with the0− isomeric state decay, and possible spin assignments due to the selectionrules of Table 2.1 are (1, 2+). The 2465 keV state cannot have spin of 0,since the parent state has a spin 0, and since it decays to the 0+ ground statevia γ-ray emission. The large intensity of the 2465 keV γ-ray to the groundstate indicates that this is a favorable transition. A spin assignment of (1−)makes this an E1 transition, which is enhanced compared to the M1 and E2transitions that come from a spin assignment of (1+) and (2+), respectively.Other transitions from this level feed into 2+ states, which again give E1transitions if the 2465 keV state has spin 1−.2490 and 2560 keV StatesThe 2490 and 2560 keV states are given spin assignments of (5−). Bothhave log(ft) values of ≈5.9 (5.92(4) and 5.85(4), respectively), which indicateallowed GT transitions. γ-rays to high-spin states, including a strong feedingto the (5−) 1999 keV state, mean that these states are associated with the5− ground state of 160Eu. Thus, likely spins are (4−, 5−, 6−). Both statesalso have transitions to state with spins 4+, 5+, 6+. A spin assignment of 5−90gives an E1 transition to the three positive-parity states, whereas an spin of4− (or 6−) gives an M1 transition to the 1999 keV state and a γ-ray of M2to the 6+ (4+) states. Since the M1 and M2 multipolarities are suppressedcompared to the E1 transition, the 5− spin assignment to the 2490 and 2560keV states is preferred.2512 keV StateThe 2512 keV state has a spin assignment of (0+1, 2+). A deduced log(ft)value of 6.64(6) prefers an allowed or first forbidden GT β-decay. Decays tolow-spin states indicate that this state is associated with the isomeric stateβ-decay, so spins of (0+, 1, 2+) are possible. The 0− assignment is ruled outvia β-decay selection rules. γ-rays to states of spin 1 and 2+ do not help torule out possible spins of the 2512 keV state. The lack of a 2512 keV γ-raymeans a spin of 0+ cannot be ruled out. Further γ− γ analysis must be doneto constrain the spin of the 2512 keV state.3090 keV StateNo spin assignment is made for the 3090 keV state. Only one γ-ray wasidentified as depopulating this state: the 2842 keV transition to a 4+ state.The log(ft) value of ≈ 7.3 indicates a likely first forbidden transition, butfurther information is needed to assign a spin to this state.3116 keV StateThe 3116 keV state is not assigned a spin. The 2045 keV γ-ray to a 4+ stateis the only transition identified as depopulating the 3116 keV state. A log(ft)value of ≈7.15 hints at a first forbidden transition to this level, but no spinassignment is attempted for this state. More information is required for aspin assignment.6.5 Half-livesAs first noted in Ref. [3], there exist not just one, but (at least) two β-decayingstates in the parent 160Eu nucleus. This can be seen from the population of alarge range of low and high spins in the daughter 160Gd nucleus. The identifiedβ-decaying states were assigned spins of 5− and 0− for the ground state andisomeric state, respectively [3].The time-distribution histogram for the ground state decay was con-structed from 24 γ-rays that were identified as being associated with the91high-spin ground state β-decay, including the intense 413, 516, 822, and 995keV γ-rays. The background-subtracted time spectra were summed to cre-ate the time histogram in Fig. 6.9a. The time in the histogram correspondsto time elapsed since beam delivery halted to GRIFFIN; t = 0 in Fig. 6.9corresponds to t = 390 s in the overall tape cycle of Fig. 4.4. The extractedhalf-life for the ground state decay is t1/2 = 42.7(7) s. This half-life is in greatagreement with the published half-life of t1/2 = 42.6(5) s from [3].The time-distribution histogram of γ-rays for the 0− isomeric state decayis shown in Fig. 6.9b. 15 γ-rays associated with the isomeric state decay,including the 1276, 1352, 2288, and 2465 keV transitions identified in [3],were summed to produce the histogram. The fitted isomeric state half-lifewas determined to be t1/2 = 24.7(9) s. This is significantly shorter than thereported half-life of 30.8(5) s from [3]. A summary of the half-life results from[3] and this work is shown in Table 6.5.Table 6.5: Extracted half-lives for the isomeric state and ground state β-decayof 160Eu. Results are compared to those published in Ref. [3].This work Previous work [3]t1/2 [s] t1/2 [s]5− ground state 42.7(7) 42.6(5)0− isomeric state 24.7(9) 30.8(5)Systematic effects of the half-life fit region were investigated using the“chop analysis” method described in Sec. 5.7. The first and last bins ofthe fit region were changed in 1 s increments, and the resulting half-life andassociated uncertainty were plotted against the first and last bin numbers.The ground state (Fig. 6.10) and isomeric state (Fig. 6.11) half-lives show nosignificant change when removing the first ≈10 s at the beginning and end ofthe fit region. The resulting half-life is indicated by horizontal lines in bothfigures and is the half-life extracted from the largest fit region available, asdiscussed in Sec. 5.7. The half-lives reported in Ref. [3] are overlaid in redin Figs. 6.10 and 6.11. The ground state time-distribution histogram wasbinned in 100 ms bins, and the isomeric state was binned with 500 ms binsdue to lower statistics. No significant change is seen in either half-life whenincreasing the bin width of the histograms.The time-distribution histograms used in this analysis were the sum ofa large number of γ-rays associated with each β-decaying state. Time-distribution histograms for the isomeric and ground state decay were alsocreated using only the γ-rays identified and used in Ref. [3]. Time-distributionand chop analysis plots of the ground and isomeric state decays are shown92Time [s]0 20 40 60 80 100Counts100200300400500Ground State Decay = 42.7(7) s1/2t(a)Time [s]0 20 40 60 80 100Counts020406080100120Isomeric State Decay = 24.7(9) s1/2t(b)Figure 6.9: Summed background-subtracted time spectra for the (a) 5−ground state and (b) 0− isomeric state β-decays of 160Eu.93First Bin Included [s]390 395 400 405 410Half-life [s]4142434445Half-life vs First Bin Included(a)Last Bin Included [s]482 484 486 488 490 492 494 496 498 500 502 504Half-life [s]4142434445Half-life vs Last Bin Included(b)Figure 6.10: Chop analysis of the 5− ground state half-life. The change infitted half-life is observed when the first (a) and last (b) bins of the fit arechanged. The extracted half-life and associated uncertainty are overlaid ashorizontal lines in both plots. The half-life and uncertainty reported in Ref. [3]are overlaid in red in both plots.94First Bin Included [s]390 395 400 405 410Half-life [s]22242628303234Half-life vs First Bin Included(a)Last Bin Included [s]484 486 488 490 492 494 496 498 500 502 504Half-life [s]2224262830Half-life vs Last Bin Included(b)Figure 6.11: Chop analysis of the 0− isomeric-state half-life. The changein fitted half-life is observed when the first (a) and last (b) bins of the fitare changed. The extracted half-life and associated uncertainty are overlaidas horizontal lines in both plots. The half-life and uncertainty reported inRef. [3] are overlaid in red in both plots.95in Figs. 6.12 and 6.13. The half-lives for the ground and isomeric state de-cays using only the γ-rays identified in Ref. [3] are t1/2 = 43.5(10) s andt1/2 = 23.1(11) s, respectively. These values agree within uncertainty withthe half-lives obtained by using a larger number of γ-rays. The half-life forthe ground state decay using the limited number of γ-rays also agrees withthe value published in Ref. [3]. Using only the γ-rays identified in Ref. [3],the half-life of the isomeric state decay is still significantly lower, so at thediscrepancy between Ref. [3] and this work cannot yet be resolved.6.6 Half-life of 1071 keV StateThe observation of the low-energy transitions identified in Sec. 6.1.3 makesthe excited states associated with these transitions of interest. Because ofthis, the half-life of the 1071 keV state was determined via the LaBr3(Ce)detectors. To determine the half-life of the state, a gate was taken on thefeeding 413 keV transition in the first LaBr3(Ce) detector of the event, and agate on the depopulating 822 or 995 keV transitions in the second LaBr3(Ce)detector. The two LaBr3(Ce) detector spectra and associated gates are shownin Fig. 6.14.As is obvious from the spectra in Fig. 6.14, the low energy resolution onthe LaBr3(Ce) yields wide peaks. One cannot resolve the 409/413/417 keVtriplet in the LaBr3(Ce) spectrum, for instance, as this triplet appears justas a single peak. Since the three transitions used to determine the half-lifeof the 1071 keV state (413, 822, and 995 keV) are all very intense comparedto neighboring peaks, it is assumed that the contribution to the half-life fromthe neighboring peaks is negligible.The resulting time-difference spectrum is shown in Fig. 6.15. The distri-bution clearly shows an exponential tail, corresponding to a half-life that islarger than ≈100 ps. A fit with the Gaussian-exponential convolution func-tion yields a half-life for the 1071 keV state of 1.3(1) ns. The convolutedfunction is shown in red in Fig. 6.15, and the Gaussian component is shownin black.This half-life is particularly interesting, as it is significantly longer thanexpected for a nuclear state. Other states of similar energy in this nucleus havebeen measured through the Doppler-shift attenuation method [45]. Thoseresults, published in Ref. [19], give half-lives on the order of femtoseconds topicoseconds: 14(1) fs for the 1225 keV state and 0.7(5) ps for the 1148 keVstate). The significance of the nanosecond half-life of the 1071 keV state isdiscussed in Sec. [s]0 20 40 60 80 100Counts50100150200250300Ground State Decay = 43.5(10) s1/2t(a)First Bin Included [s]390 395 400 405 410Half-life [s]4041424344454647Half-life vs First Bin Included(b)Last Bin Included [s]484 486 488 490 492 494 496 498 500 502 504Half-life [s]4042444648Half-life vs Last Bin Included(c)Figure 6.12: Half-life (a) and chop analysis (b-c) of the ground state decayusing only the γ-rays identified in Ref. [3]. The extracted half-life and as-sociated uncertainty (black) and the half-life published in Ref. [3] (red) areoverlaid in (b-c). 97Time [s]0 20 40 60 80 100Counts0102030405060Isomeric State Decay = 23.1(11) s1/2t(a)First Bin Included [s]390 395 400 405 410Half-life [s]1820222426283032Half-life vs First Bin Included(b)Last Bin Included [s]484 486 488 490 492 494 496 498 500 502 504Half-life [s]1820222426283032Half-life vs Last Bin Included(c)Figure 6.13: Half-life (a) and chop analysis (b-c) of the isomeric state decayusing only the γ-rays identified in Ref. [3]. The extracted half-life and as-sociated uncertainty (black) and the half-life published in Ref. [3] (red) areoverlaid in (b-c). 98Energy [keV]0 200 400 600 800 1000Counts210310(Ce) Detector 13LaBrGate: 413 keV (a)Energy [keV]0 200 400 600 800 1000Counts110210(Ce) Detector 23LaBrGate: 822 keV Gate: 995 keV(b)Figure 6.14: (a) Energy spectrum in first LaBr3(Ce) detector in an event. Thegate on the 413 keV γ-ray to start the timing signal is shown. (b) Energyspectrum in the second LaBr3(Ce) detector after the gate in (a) is applied.The stop signal for the TAC is from hits in either the 822 or 995 keV gates(shown overlaid). 99Time [ps]15000 20000 25000 30000 35000 40000Counts110 = 1.3(1) ns1/2tFigure 6.15: Half-life of the 1071 keV state. The half-life is measured bythe time difference between the feeding 413 keV transition and the depop-ulating 822 and 995 keV transitions using LaBr3(Ce) detectors and a TAC.The Gaussian-exponential convolution fit is overlaid in red, and the Gaussiancomponent in black.1006.7 Discussion6.7.1 Comparison to Hartley et al. [3]The results from this analysis overall agree with those published in Ref. [3].This analysis agrees with the placement of all nine published excited states.Spin-parity assignments from here do not conflict with those proposed inRef. [3].Of the 27 placed transitions from Ref. [3], this analysis agrees with theplacement of 23 of them. Of the four transitions that are not in agreement,three (928, 1408, and 1750 keV) are identified as likely being summed peaks.While these peaks exist as weak transitions in single crystal spectra, theirphotopeak area is drastically increased in Addback mode–much more thanexpected and than other nearby peaks. This characteristic is associated withsummed peaks, as Addback mode adds multiple crystals from the same de-tector together, and these summed peaks are amplified due to this. A com-parison of single crystal and Addback mode for each of these peaks is shownin Fig. 6.16.The 122 keV transition between the 1296 and 1173 keV states was omittedfrom the level scheme of Fig. 6.8 and the corresponding analysis. When gatingon the 122 keV peak, no coincidence is seen with the 658 keV transition. Theplacement of the 122 keV γ-ray in Ref. [3] should clearly make the 122 keVγ-ray in coincidence with the 658 keV γ-ray. Due to this lack of a coincidencethat should be present, the 122 keV could not be confidently placed in thelevel scheme as of now.Since Ref. [3] does not present comprehensive results on the β-decay of160Eu, it is nearly impossible to compare the results here to that of Ref. [3].A log(ft) value of 5.1 is given for the β-decay to the 1999 keV state in 160Gd,however. This value is smaller than the determined value of 5.48(5) from thisstudy.As discussed in Sec. 6.5, the determined half-life for the low-spin isomericstate decay of 160Eu is significantly shorter than the value published in Ref. [3],while the ground state decay’s half-life is in excellent agreement with thepreviously published value.6.7.2 Mixing of γ- and Kpi = 4+-bandsThe low-energy 82 keV transition between the 1071 and 989 keV states dis-cussed in Sec. 6.1.3 gives an indication to two-state mixing occurring in thisnucleus. The 82 keV transition competes with the very intense 822 and 995keV transitions from the same state. Based on Eq. 2.33, the transition proba-101 Energy [keV]900 920 940 960 980 1000 Counts050100150200250Gate: 995 keVSingle CrystalAddback928(a) Energy [keV]1360 1380 1400 1420 1440 1460Counts5001000150020002500 Single CrystalAddback1408(b) Energy [keV]1680 1700 1720 1740 1760 1780Counts100150200250300350400450 Single CrystalAddback1750(c)Figure 6.16: Comparison of Single Crystal and Addback modes for the (a)928 keV, (b) 1408 keV, and (c) 1750 keV peaks. The significant increase inphotopeak area in Addback mode is characteristic of summed peaks. 102bility for the E2 82 and 995 keV γ-rays is proportional to E5γ . Assuming thatthe matrix elements for the two transitions are the same, then the ratio oftransition probabilities is equal to the ratio of the energies to the fifth power:λ(E2; 995 keV)λ(E2; 82 keV)=9955825≈ 2.63 · 105. (6.1)Thus, the 995 keV γ-ray is over 105 times more likely to occur than the82 keV transition; this does not factor in the M1/E2 822 keV transitionfrom this state as well. In order for the 82 keV transition to compete, thistransition must be a collective transition, and the B(E2) value of the 995keV transition has to be small. The Kpi = 4+ band, of which the 1071 keV4+ state is the bandhead, is not expected to have collective transitions tothe γ-band containing the 989 keV state. Because of the observation of ahighly-collective transition between these two bands, it is natural to assumethat this transition is a result of configuration mixing between the Kpi = 4+bandhead (4+1 ) and the 4+ member of the γ-band (4+γ ).64160 Gd960+ 02+ 754+ 2492γ+98941+X4γ+Y64160 Gd960+ 02+ 754+ 2492γ+9894I+10714II+1148(159)82995Figure 6.17: (Left) The unmixed 4+1 and 4+γ states are shown at unknownenergies X and Y. (Right) The two states after mixing, now located at 1071and 1148 keV. The not-yet observed 159 keV 4+II −→ 2+γ is tentatively shown.103Characterization of MixingObservation of this 82 keV transition may be enhanced if the 4+ 1071 keVstate is mixed with another nearby state–in this case, the 4+ member of theγ-vibrational band. The 4+ member of the γ-band is quoted as being at 1148keV [9, 19, 20, 21], although it has not yet been confirmed in this analysis(see Sec. 6.1.2).If the 1071 and 1148 keV 4+ states are indeed mixed, then one can con-strain the level of mixing based on the data obtained in this work and withother known data. First, the mixed wavefunction for the 1071 keV state(|4+I 〉) and the 1148 keV state (|4+II〉) are written (from Eq. 2.11) as a linearcombination of the unperturbed wavefunctions |4+1 〉 and |4+γ 〉:|4+I 〉 = a |4+1 〉+ b |4+γ 〉 (1071 keV)|4+II〉 = b |4+1 〉 − a |4+γ 〉 . (1148 keV)Simplified level schemes showing the unmixed and mixed scenarios are shownin Fig. 6.17. It is assumed that the 2+γ state is not mixed, as there is no 2+member of the Kpi = 4+ band for it to mix with.The B(E2) value for the 82 keV transition from the 4+I −→ 2+γ is given asa function of the unmixed B(E2) values:B(E2; 4+I −→ 2+γ ) = a2B(E2; 4+1 −→ 2+γ ) + b2B(E2; 4+γ −→ 2+γ )±2ab√B(E2; 4+1 −→ 2+γ ) ·B(E2; 4+γ −→ 2+γ )(6.3)where again B(E2) ∝ | 〈ψf |E2|ψi〉 |2. If the transition between the unmixed|4+1 〉 state and the |2+γ 〉 is assumed to be forbidden (B(E2; 4+1 → 2+γ ) = 0),then the B(E2) value of the 4+I to 2+γ is proportional to the B(E2) value ofthe 4+γ to 2+γ transition:B(E2; 4+I −→ 2+γ ) = b2B(E2; 4+γ −→ 2+γ ). (6.4)Since the 4+γ −→ 2+γ is unobserved, one can estimate the B(E2; 4+γ −→ 2+γ )by connecting them to the B(E2) values from the ground state band viathe Alaga rules [6]. The Alaga rules state that the ratio of B(E2) values fortransitions connecting the same intrinsic initial and final states is proportionalto the square of their Clebsch-Gordon coefficients:B(E2; Ii −→ If )B(E2; Ii −→ I ′f )=| 〈IiKi2∆K|IfK〉 |2| 〈IiKi2∆K|I ′fK〉 |2. (6.5)104where again I represents the nuclear spin of the state and K is the angularmomentum projection onto the symmetry axis. While not quite connectingthe same initial and final intrinsic states, one can use the Alaga rules toestimate the strength of the 4+γ −→ 2+γ transition based on the strength of the4+g.s. −→ 2+g.s. transition in the ground state band. The Alaga ratio isB(E2; 4+γ −→ 2+γ )B(E2; 4+g.s. −→ 2+g.s.)≈ | 〈4220|22〉 |2| 〈4020|20〉 |2 . (6.6)The B(E2) value for the 4+g.s. −→ 2+g.s. has not yet been measured in thisnucleus; however, the 2+g.s. −→ 0+g.s. transition has been, so one can again usethe Alaga rules to determine the desired B(E2) value:B(E2; 4+g.s. −→ 2+g.s.)B(E2; 2+g.s. −→ 0+g.s.)=| 〈4020|20〉 |2| 〈2020|00〉 |2 (6.7)and thus an estimate for the B(E2; 4+γ −→ 2+γ ) is given asB(E2; 4+γ −→ 2+γ ) ≈| 〈4220|22〉 |2| 〈4020|20〉 |2| 〈4020|20〉 |2| 〈2020|00〉 |2B(E2; 2+g.s. −→ 0+g.s.). (6.8)The B(E2) for the 2+g.s. −→ 0+g.s. from the ENSDF evaluation [9] is 201.2(16)W.u., and the Clebsch-Gordon coefficients can be easily calculated. The re-sulting B(E2) estimate for the 4+γ −→ 2+γ is B(E2; 4+γ −→ 2+γ ) ≈ 120 W.u.B(E2; 4+γ −→ 2+γ ) values for some deformed even-even nuclei in this regionhave been measured, and their values (158Gd = 113+166−13 W.u.,166Er = 138(9)W.u., and 168Er = 92(20) W.u.) indicate that the approximation of 120 W.u.is reasonable for 160Gd [8].As discussed in Sec. 6.6, the preliminary half-life of the 1071 keV statewas determined to be 1.3(1) ns. The branching ratio of the depopulating 995keV γ-ray is 38(7)% (calculated from Table 6.1). From Eq. 2.8, the B(E2)value of the 995 keV γ-ray, which is the 4+I −→ 2+g.s. transition, is given asB(E2; 4+I −→ 2+g.s.) =564(E4+I −→2+g.s.)5 · t1/2BRγe2fm4MeV5psB(E2; 4+I −→ 2+g.s.) =564(0.995 MeV)5 · 1300 ps0.38e2fm4MeV5ps=0.17(3) e2fm4(6.9)105where the error in the B(E2) value comes from propagating the errors in en-ergy, half-life, and branching ratio in quadrature. Via the Weisskopf estimatesof Table 2.2, the B(E2) value for the 995 keV transition is 3.3(6) · 10−3 W.u.With the B(E2) value for the 995 keV transition calculated, one canapproximate the B(E2) value of the 82 keV transition. Similar to the Alagarules, the ratio of B(E2) values for transitions γ1 and γ2 depopulating thesame state is related to their intensities and energies:B(E2; γ1)B(E2; γ2)=Iγ1(1 + αγ1)/E5γ1Iγ2(1 + αγ2)/E5γ2(6.10)where again α denotes the conversion coefficient of the transition. The the-oretical E2 conversion coefficient for the 82 keV transition is 5.374, and theconversion coefficient for the 995 keV transition is 2.8×10−3 and thus neg-ligible [17]. With the intensities from Table 6.2, the B(E2) of the 82 keVtransition is calculated asB(E2; 82 keV) ≈ < 0.1(1 + 5.374)/82539/9955B(E2; 995 keV). 14.2 W.u.(6.11)Now, with estimates for the B(E2) values for both the unmixed and mixedtransitions, one can constrain the amount of mixing between the two 4+ states.Rearranging Eq. 6.4, the mixing parameter b2 is thusb2 ≈ B(E2; 4+I −→ 2+γ )B(E2; 4+γ −→ 2+γ ). 0.118 (6.12)corresponding to less than ≈ 12% mixing between the two 4+ states.One can then use the mixing ratio and approximate B(E2) value to esti-mate the strength of the 4+II −→ 2+γ transition. The strength of the unobserved159 keV 4+II −→ 2+γ γ-ray is related to the strength of the unmixed 4+γ −→ 2+γtransition byB(E2; 4+II −→ 2+γ ) = a2B(E2; 4+γ −→ 2+γ ). (6.13)In the limiting case of a2 ≈ 0.88 (corresponding to b2 ≈ 0.12 since themixed wavefunctions are normalized), and assuming that the estimate ofB(E2; 4+γ −→ 2+γ ) ≈ 120 W.u. is reasonable, then the strength of the 159106keV transition isB(E2; 4+II −→ 2+γ ) ≈ 0.88 · 120 W.u.≈ 106 W.u.≈ 5450 e2fm4.(6.14)Rearrangement of Eq. 2.8 also yields the branching ratio of the transition:BR =B(E2) · E5γ · t1/2564e−2fm−4MeV−5ps−1.(6.15)The lifetime of the parent 1148 keV state was reported by Ref. [19]as 1080+730−320 fs, corresponding to a half-life of 0.749+0.51−0.22 ps. Evaluation ofEq. 6.15 yields a branching ratio for the 159 keV 4+II −→ 2+γ transition asBR ≈ 7.4 · 10−4. With a theoretical conversion coefficient of α = 0.484 [17],the expected γ-ray branching ratio would be BRγ ≈ 5 · 10−4. Such a smallbranching ratio confirms why the 159 keV 4+II −→ 2+γ γ-ray transition has notyet been observed in this nucleus. A summary of these results is shown inFig. 6.18.The above discussion shows that by using a simple two-level mixing model,accompanied by reasonable assumptions for the B(E2) values for the groundstate and γ-bands, a consistent description of the data can be achieved.Nilsson ConfigurationsNow, one can consider what the structure of these levels are and how thatmight lead to the observed mixing scenario. Calculations performed in Ref. [3]using a deformed Woods-Saxon potential [46] show a likely single-particleconfiguration of pi2(5/2[413]× 5/2[532]) for the 1999 keV 5− state, as well asa proposed configuration of pi2(5/2[413] × 3/2[411]) for the 4+ state at 1483keV. The strong 515 keV transition is interpreted as an E1 transition betweenproton orbitals: pi5/2[532] −→ pi3/2[411].According to Ref. [47], the Kpi = 4+ band is formed by the orbital pairspi2(5/2[413] × 3/2[411]) and ν2(5/2[523] × 3/2[521]). The 4+ state at 1071keV then has the same single-particle proton configuration (pi2(5/2[413] ×3/2[411])) as the 4+ state at 1483 keV. One would expect that the 5− 1999keV state should then decay to both 4+ states; it is reasonable to expect thatthe decay to the 1071 keV state should have higher intensities due to the higher10764160 Gd960+ 02+ 754+ 2492γ+9894I+10714II+1148(159)82995 B(E2)=0.0033(6) W.u.B(E2)   14 W.u.≲B(E2)≈120 W.u.BR  ≈5×10 4 t     =1.3(1) ns1/2! -Figure 6.18: Summary of results from the two-state mixing between the 4+γand 4+1 states.108energy of the emitted γ-ray. Ref. [3] places a weak 928 keV γ between the1999 and 1071 keV states. As mentioned in Sec. 6.7.1, this analysis indicatesthat this transition is a sum peak and not a real transition. Either way, the1999 keV state preferentially decays to the 1483 keV state as opposed to the1071 keV state of the same proton configuration.The above single-particle orbital structure for the 1071 keV state is thusinsufficient to describe the state. Ref. [10] notes that the Kpi = 4+ bands indeformed even-even nuclei like 160Gd are predominantly hexadecapole vibra-tions as opposed to double-γ phonon states. While calculations for the hex-adecapole components in 160Gd are not readily available, they are available innearby nuclei. Calculations for configurations in 166Er indicate that the first4+ state–predominantly a hexadecapole vibration–has contributions from thepi2(7/2[523]⊗1/2[541]) and ν2(5/2[523]⊗3/2[521]) quasi-particle configu-rations [48]; here⊗denotes coupling of quasi-particle states as opposed tosingle-particle states. Examination of Nilsson orbitals near the Fermi-surfaceindicates that the hexadecapole configurations in 160Gd are likely based on asimilar configuration to those in 166Er from Ref. [48].The pi2(5/2[413]× 3/2[411]) single-particle configuration in the 1483 keVstate couples to the same configuration in the 1071 keV state, which allowsthe strong transition between the two states. However, since the 1999 keVstate does not decay (or decays only weakly) to the 1071 keV state, thisindicates that the pi2(5/2[413]× 3/2[411]) contribution to the 1071 keV stateis much smaller than the pi2(5/2[413] × 3/2[411]) contribution to the 1483keV state. From this, one can conclude that the hexadecapole configurationsare the dominant contributors to the structure of the 1071 keV state. Thepresence of the single-particle configurations in the Kpi = 4+γ 1071 keV stateare essential components of the state’s configuration, however. Without thesesingle-particle states, the strong transition from the 1483 keV state to the1071 keV 4+1 state and mixing of the 4+ states would not be possible.Ref. [49] used the random-phase approximation [50, 51] to calculate theNilsson orbital contributions to the 2+γ states in well-deformed nuclei, in-cluding 160Gd. Since the 4+γ state is a rotational state built on the 2+γlevel, information on the structure of the 2+γ level aides in understandingthe structure of the 4+γ state. The dominating proton correlations identifiedin Ref. [49] are the pi2(5/2[413]⊗1/2[411]) and the pi2(3/2[411]⊗1/2[411])orbitals. The dominant neutron correlations are the ν2(5/2[523]⊗1/2[521])and ν2(3/2[521]⊗1/2[521]) orbitals.The single-particle configurations that contribute to the 1071 keV 4+ stateinvolve some of the same orbitals that contribute to the γ-band. Mainly,the pi5/2[413], pi3/2[411], ν5/2[523], and ν3/2[521] orbitals contribute both10964160 Gd960+ 02γ+98941+10714γ+114842+14835- 1999!  (5/2[413]×5/2[532])!  (5/2[413]×3/2[411])!  (5/2[413]×3/2[411])"  (5/2[523]×3/2[521])!  (7/2[523]⨂1/2[541])"  (5/2[523]⨂3/2[521])!  (5/2[413]⨂1/2[411])!  (3/2[411]⨂1/2[411])"  (5/2[523]⨂1/2[521])"  (3/2[521]⨂1/2[521])single-particlehexadecapole2222222222Figure 6.19: Configurations in states in 160Gd. The hexadecapole configura-tions in the 1071 keV state are calculations for the Kpi = 4+1 state in166Er,and the 1071 keV state in 160Gd is assumed to have the same hexadecapoleconfigurations. × denotes coupling of single-particle orbitals, and⊗ indicatesthe coupling of quasi-particle configurations. Data taken from Refs. [3, 47, 48].110to the single-particle configurations in the Kpi = 4+ band, as well as to thecorrelations involved in the γ-band. It should come as no surprise that mixingbetween the two 4+ states occurs, as the underlying configurations of the twobands are similar. A summary of the configurations for the states involved isshown in Fig. 6.19Mixing between these bands is further supported by the observation ofthe weak 116 keV transition between the 5+ state at 1173 keV and the 3+member of the γ-band at 1058 keV. Again, this transition from the Kpi = 4+band to the γ-band would normally be expected to be forbidden. However, ifthe 1173 keV state mixes with the 5+ member of the γ-band–proposed to bethe 1261 keV state [9, 19, 20, 21]–then this transition would be allowed, muchlike the 82 keV transition is. Since these higher-spin states are just rotationalstates built on the underlying bandhead structure, mixing is possible betweenall of the higher-spin states between the γ- and Kpi = 4+ bands.111Chapter 7Conclusions and OutlookIn summary, the work presented here details the analysis of the decay of160Eu. Present in a region of large nuclear deformation, this nucleus is oneof many studied using the GRIFFIN spectrometer in order to increase theunderstanding of the complex nuclear structure of heavy, deformed nuclei.Detailed spectroscopy has been carried out on the 160Eu−→160Gd β−-decay.For the first time, a comprehensive study of the β-decay of 160Eu was com-pleted, and information on β-feeding intensities and log(ft) values of theβ−-decay to 160Gd has been extracted. Using the GRIFFIN spectrometer,knowledge of the excited-state nuclear structure of 160Gd has been improved.10 new excited states, as well as 41 new transitions, have been established.Associated relative and absolute intensities for the transitions were calculated,and tentative spin-parity assignments have been made for a number of thenewly-established excited states.The presence of the low-energy 82 and 116 keV transitions connecting theKpi = 4+ and γ-vibrational bands provides new information on the structureof this nucleus. A particularly long half-life of 1.3(1) ns has been determinedfor the 1071 keV 4+ excited state. Evidence is presented here for configura-tion mixing between the Kpi = 4+ and γ-bands, and with some reasonableassumptions on transition strengths, constraints of less than 12% mixing weredetermined for the configuration mixing between the 4+ members of thesebands.While the analysis presented here has greatly increased the understandingof the β-decay of 160Eu, there is still more data available to further improvethis knowledge. More fast-timing data from the LaBr3(Ce) detectors is avail-able for analysis, and this data may allow measurements of half-lives of excitedstates in 160Gd. Information on state half-lives will allow precise computationsof the reduced transition rates for transitions depopulating these levels, whichgives information on the degree of collectivity occurring in these transitionsand the nucleus.Conversion electron spectra are also available from the PACES detector.PACES data will allow the identification of conversion electrons, which canbe used to confirm the highly-converted nature of a number of transitions,112including the newly-identified low-energy transitions presented in this work.Additionally, more detailed analysis may be undertaken on the γ-ray datathat was analyzed in this work. A number of γ-rays could not yet be placed inthe level scheme, so further analysis on these transitions is needed, including amore in-depth γ−γ coincidence analysis. As discussed, many spin assignmentsto new levels could not be unambiguously determined. To further constrainthe spins of these states, γ−γ angular correlation analysis must be completed.γ − γ angular correlations generally require a singles photopeak area of onemillion or more counts for best results. Since this 160Eu β-decay data isnot super high statistics, only the intense cascades have enough statistics toperform a conclusive angular correlation analysis.113Bibliography[1] J.M. D’Auria, R.D. Guy, and S.C. Gujrathi. The Decay of 160Eu. Cana-dian Journal of Physics, 51(6):686, 1973.[2] N.A. Morcos et al. Decay of 160Eu. Journal of Inorganic and NuclearChemistry, 35(11):3659, 1973.[3] D.J. Hartley et al. Masses and β-Decay Spectroscopy of Neutron-RichOdd-Odd 160,162Eu Nuclei: Evidence for a Subshell Gap with Large De-formation at N = 98. Phys. Rev. Lett., 120:182502, 2018.[4] K.S. Krane. Introductory Nuclear Physics. John Wiley & Sons, 1987.[5] Maria Goeppert Mayer. On Closed Shells in Nuclei. II. Phys. Rev.,75:1969, 1949.[6] R.F. Casten. 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Elbek, and P.O. Tjm. Collective vibrational states in evenGd nuclei. Nuclear Physics A, 91(3):576, 1967.[24] G. Savard et al. Radioactive beams from gas catchers: The CARIBUfacility. Nucl. Instr. Meth. Phys. Res. B, 266(19):4086, 2008.[25] A. Mitchell et al. The X-Array and SATURN: A new decay-spectroscopystation for CARIBU. Nucl. Instr. Meth. Phys. Res. A, 763:232, 07 2014.[26] J.C. Wang et al. The Canadian Penning Trap mass spectrometer. Nu-clear Physics A, 746:651, 2004.[27] H. Mach et al. Identification of Four New Neutron-Rich Rare-EarthIsotopes. Phys. Rev. Lett., 56:1547, 1986.115[28] Y.Y. Berzin et al. Study of the Excited States of 160Gd by the (n, n’γ)Reaction. Lat.PSR, Zinat.Akad.Vestis:Fiz.Teh.Ser. No.5, 3, 1983.[29] Y.Y. Berzin et al. The Levels of 160Gd Nucleus. Izv.Akad.Nauk,SSSR:Ser.Fiz. 53, 901, 1989.[30] J. Dilling, R. Kru¨cken, and G. Ball. ISAC overview. Hyperfine Interac-tions, 225(1):1, 2014.[31] S. Raeder et al. An ion guide laser ion source for isobar-suppressed rareisotope beams. Review of Scientific Instruments, 85(3):033309, 2014.[32] P. Bricault et al. An overview on TRIUMFs developments on ionsource for radioactive beams (invited). Review of Scientific Instruments,79(2):02A908, 2008.[33] A.B. Garnsworthy et al. The GRIFFIN facility for Decay-Spectroscopystudies at TRIUMF-ISAC. Nucl. Instr. Meth. Phys. Res. A, 918:9, 2019.[34] U. Rizwan et al. Characteristics of GRIFFIN high-purity germaniumclover detectors. Nucl. Instr. Meth. Phys. Res. A, 820:126, 2016.[35] A.B. Garnsworthy et al. The GRIFFIN data acquisition system. Nucl.Instr. Meth. Phys. Res. A, 853:85, 2017.[36] S. Ritt and P.-A. Amaudruz. Midas – Maximum Integrated Data Ac-quisition System. Proceedings of the 10th IEEE Real Time Conference,Beaune, 309, 1997.[37] P.C. Bender et al. GRSISort- A lean, mean, sorting machine, 2019. URLhttps://github.com/GRIFFINCollaboration/GRSISort.[38] R. Brun and F. Rademakers. ROOT – An object oriented data analysisframework. Nucl. Instr. Meth. Phys. Res. A, 389(1):81, 1997.[39] W. D. Kulp et al. N = 90 region: The decays of 152Eum,g to 152Sm.Phys. Rev. C, 76:034319, 2007.[40] J.C. Hardy et al. The essential decay of pandemonium: A demonstrationof errors in complex beta-decay schemes. Phys. Lett. B, 71(2):307, 1977.[41] M. Wang et al. The AME2016 atomic mass evaluation (II). tables, graphsand references. Chinese Physics C, 41(3):030003, 2017.116[42] J.K. Smith et al. Gamma- gamma angular correlation analysis tech-niques with the GRIFFIN spectrometer. Nucl. Instr. Meth. Phys. Res.A, 922:47, 2019.[43] G. F. Grinyer et al. High precision measurements of 26Na β− decay.Phys. Rev. C, 71:044309, 2005.[44] B. Olaizola et al. Shape coexistence in the neutron-deficient Pb region: asystematic study of lifetimes of the even-even 188−200Hg with GRIFFIN.Phys. Rev. C, To be published, 2019.[45] T. Belgya, G. Molna´r, and S.W. Yates. Analysis of Doppler-shift attenu-ation measurements performed with accelerator-produced monoenergeticneutrons. Nuclear Physics A, 607(1):43, 1996.[46] S. Cwiok et al. Single-particle energies, wave functions, quadrupole mo-ments and g-factors in an axially deformed woods-saxon potential withapplications to the two-centre-type nuclear problems. Computer PhysicsCommunications, 46(3):379, 1987.[47] V. G. Soloviev, A. V. Sushkov, and N. Yu. Shirikova. Quadrupole andHexadecapole Vibrational Excitations in Deformed Nuclei. InternationalJournal of Modern Physics E, 06(03):437, 1997.[48] V. G. Soloviev, A. V. Sushkov, and N. Yu. Shirikova. Description oflow-lying vibrational and two-quasiparticle states in 166Er. Phys. Rev.C, 51:551, 1995.[49] Ch. Hinke et al. Evolution of 2+γ wave functions and gamma-stiffness inwell-deformed rare-earth nuclei. European Physical Journal A, 30:357,11 2006.[50] D. Bohm and D. Pines. A Collective Description of Electron Interactions:III. Coulomb Interactions in a Degenerate Electron Gas. Phys. Rev.,92:609, 1953.[51] P. Ring and P. Schuck. The Nuclear Many Body Problem. Springer, 1980.117Appendix ACalibrationsHPGe Energy Calibrations• Identify runs associated with each source– 152Eu, 133Ba, 56Co, and 60Co• Sort fragment and analysis trees into histograms and matrices– Create γ-singles spectrum, γ-ray singles spectrum for each individ-ual crystal, and matrix of detector channel/crystal number versusγ-ray singles energy– Initially sort with online calibration file to get rough gains for eachcrystal– Use GetCharge function to extract raw charge collected rather thanalready gain-matched data• For each HPGe crystal:– Identify intense transitions of well-known energy using histogramscreated with online calibration file from each source run– Identify and fit same peaks in histograms created with GetChargefunction– Plot known energy of peak versus raw centroid channel, fit with:Energy = a1 ∗ channel + a0 (A.1)• Compile fit parameters a1 and a0 into calibration file• Re-sort source runs using created calibration file• Fit known peaks, including peaks not used in original calibration, toexamine nonlinearities and systematic deviations• Example fits of Eq. A.1 for GRIFFIN HPGe crystal numbers 0-7 areshown in Fig. A.1118HPGe Efficiency Calibrations• For each source run (152Eu, 133Ba, 56Co, and 60Co):– Get number of counts Nγ in γ-singles for transitions with knownintensities Iγ– Correct for summing effects for each transitions (Sec. 5.1.1)– Determine source activity As at time of run and run time t for eachrun∗ Note: 56Co source is prepared in-house and does not have acertified activity– Determine deadtime d of DAQ by plotting time difference betweenevents versus detector channel number– Determine total number of events N recorded by examining γ-singles spectrum– Calculate absolute γ-ray detection efficiency γ for each γ-ray via:γ =NγIγ ·As · (t− d ·N) (A.2)where t− d gives the live-time of the detector• Plot γ for each γ-ray from 152Eu, 133Ba, and 60Co, fit with:γ(Eγ) = ep0+p1ln(Eγ)+p2ln2(Eγ)+p3ln3(Eγ)+p4ln4(Eγ) (A.3)• Manually scale γ from 56Co to match fit– Note: the same scaling factor should be used regardless of the γ-rayenergy for each γ value obtained• Re-fit efficiencies from all sources with Eq. A.3 to obtain absolute effi-ciency curve• The efficiency curve for the entire GRIFFIN array is shown in Fig. A.2Cross-talk Corrections• Using a 60Co source, create matrix of multiplicity-2 events within aclover• Construct γ − γ coincidence matrix for these events for each pair ofcrystals in clover119• Fit Compton-scattered events with linear fit to extract offset and slope• Create cross-talk correction matrix for clover• Apply corrections to calibration file120Channel0 0.5 1 1.5 2 2.5310×Energy [keV]00.511.522.533.544.55310×/ndf = 1.200/152χ = 0.114(0.223)0p = 1.42984(0.00019)1pGriffin Crystal 0Channel0 0.5 1 1.5 2 2.5310×Energy [keV]00.511.522.533.544.55310×/ndf = 12.789/152χ = -1.080(0.224)0p = 1.36429(0.00019)1pGriffin Crystal 1Channel0 0.5 1 1.5 2 2.5310×Energy [keV]00.511.522.533.544.55310×/ndf = 1.671/152χ = 0.205(0.223)0p = 1.41093(0.00019)1pGriffin Crystal 2Channel0 0.5 1 1.5 2 2.5310×Energy [keV]00.511.522.533.544.55310×/ndf = 19.461/152χ = 0.905(0.223)0p = 1.29442(0.00018)1pGriffin Crystal 3Channel0 0.5 1 1.5 2 2.5310×Energy [keV]00.511.522.533.544.55310×/ndf = 2.531/152χ = 0.541(0.223)0p = 1.32190(0.00018)1pGriffin Crystal 4Channel0 0.5 1 1.5 2 2.5310×Energy [keV]00.511.522.533.544.55310×/ndf = 11.622/152χ = 0.296(0.224)0p = 1.27862(0.00017)1pGriffin Crystal 5Channel0 0.5 1 1.5 2 2.5310×Energy [keV]00.511.522.533.544.55310×/ndf = 2.581/152χ = 0.744(0.223)0p = 1.39004(0.00019)1pGriffin Crystal 6Channel0 0.5 1 1.5 2 2.5310×Energy [keV]00.511.522.533.544.55310×/ndf = 16.834/152χ = 0.388(0.232)0p = 1.27024(0.00018)1pGriffin Crystal 7Figure A.1: Energy calibration for GRIFFIN HPGe crystals 0-7.121Energy (keV)210 310Efficiency0. / ndf 2χ  9.025 / 17Prob   0.9395p0        0.04603±42.05 − p1        0.01115± 26.59 p2        0.001647±6.298 − p3        0.0002235± 0.6473 p4       05− 2.322e±0.02493 − Absolute Efficiency, entire arrayFigure A.2: Absolute efficiency of entire GRIFFIN HPGe array.122Appendix BData Analysis TechniquesCoincidence Analysis• Look at singles, determine peak energies to gate on• Construct coincidence matrix for prompt coincidence events• Gate on γ1, gate on γ2 background on both sides of γ1• Create background-subtracted gated spectrum– Create histogram of events γ2 that have γ1 in the gate– Create (separate) background histograms of events γ2 that haveevents γ1 in the background gates– Normalize background gated spectra to width of γ1 gate width– Average background gated spectra– Subtract averaged background gated spectrum from γ1 gated spec-trum• Identify γ-rays in coincidence based on gated spectra– γ-rays in coincidence with 511 keV γ-ray are likely escape peaks(not real γ-rays)– Single (double) escape peaks are 511 (1022) keV less than real γ-rayenergy– Scatter peaks will appear at different energies in different gates andwill often have dips before and after due to background-subtraction– The peak being gated on should not be seen in the gated spectrumunless that peak is a doublet– Peaks seen in Addback and not in singles (Sec. 6.7.1, for example)are likely summed peaks and not real– Examining γ− γ matrix can help with clarifying coincidences thatare ambiguous in gated spectra123• Build decay scheme based on coincidences– γ-rays in coincidence are in cascade (either directly or throughintermediate γ-ray(s))– γ-rays not in coincidence run parallel to each otherIntensities and Branching Ratios• Get relative intensities– Determine photopeak area in β-gated γ-singles– Correct photopeak area for summing effects– Divide photopeak area by corresponding efficiency– Normalize efficiency-corrected photopeak area to the most intensetransition• Determine absolute intensities– Determine number of detected β-particles– Divide efficiency-corrected photopeak area by number of detectedβ-particles to achieve absolute intensity– Divide relative intensity by number of β-particles to achieve relative-to-absolute intensity factor• Calculate β branching ratios– Correct absolute γ-ray intensity for conversion electron effects– Calculate difference in intensity between all states feeding and de-populating a given excited state– Calculate associated log(ft) value based on β-decay branching ra-tioHalf-lives• Identify states likely populated by only one parent β-decaying state– Assume allowed transitions, so spin change should be 0, ±1∗ This ensures that there is no crossover from other β-decayingstates with different spins• Identify γ-rays associated with state124• Identify γ-rays in cascade that also associated with state– γ-rays should not come directly from states populated by otherβ-decaying states– γ-rays should not have transitions from states populated by otherβ-decaying states feeding them– Avoid using γ-rays from multiplets that contain γ-rays associatedwith different β-decaying states• Gate on identified transitions and background, create background-subtracted histogram of time-distribution of γ-rays in tape cycle• Sum background-subtracted time-distribution histograms of all identi-fied γ-rays associated with a give β-decaying parent state• Fit decay portion of summed histogram to extract half-life of β-decayingstate• Perform chop analysis (Sec. 5.7) to examine effect of fit region on half-life125


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