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Decay spectroscopy of neutron-rich ¹²⁹Cd with the GRIFFIN spectrometer Saito, Yukiya 2018

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Decay spectroscopy of neutron-rich 129Cd with theGRIFFIN spectrometerbyYukiya SaitoB.Sc., Physics, The University of Tokyo, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)August 2018c© Yukiya Saito, 2018	 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, a thesis entitled:  Decay Spectroscopy of Neutron-rich 129Cd with the GRIFFIN Spectrometer  submitted by Yukiya Saito  in partial fulfillment of the requirements for the degree of Master of Science in Physics  Examining Committee: Dr. Reiner Krücken, Department of Physics and Astronomy Supervisor  Dr. Iris Dillmann, Physical Sciences Division, TRIUMF Supervisory Committee Member   Supervisory Committee Member  Additional Examiner     Additional Supervisory Committee Members:  Supervisory Committee Member  Supervisory Committee Member  AbstractNuclei around doubly magic 132Sn are of particular interest in nuclear structureas well as nuclear astrophysics. Their properties provide important input for ther-process as waiting-point nuclei. For example, their shell structure and half-livesaffect the shape of the second r-process abundance peak at A∼130. In terms ofnuclear structure, the evolution of single-particle levels near shell closures is idealfor testing the current nuclear models far from stability.There have been two studies on the decay of 129Cd, however, the level schemesof 129In have large discrepancies. Also, many of the spins of the excited statesremain unclear. Therefore, the main purpose of the present study is to resolve thedisagreements in the reported level schemes and to determine the properties of theenergy states.The experiment was performed at the ISAC facility of TRIUMF, Canada. A480 MeV proton beam, which was accelerated by the main cyclotron at TRIUMF,was impinged on an uranium carbide target to produce radioactive isotopes. 129Cdwas extracted using the Ion Guide Laser Ion Source (IG-LIS). γ-rays following thedecays of 129Cd were detected with the GRIFFIN spectrometer comprising of 16high-purity germanium (HPGe) clover type detectors, along with the β -particlesdetected with SCEPTAR. The high statistics and the high sensitivity of the detec-tors allowed us to perform detailed and precise spectroscopy.A theoretical calculation was conducted using the shell model code NuShellX@MSU, employing the realistic residual interaction model jj45pna.The results of the analysis, including 29 new transitions and 5 new excitedstates, will be discussed and compared to the theoretical calculations.iiiLay SummaryOne of the ultimate goals in this work is to understand how the building blocks ofthe visible matter, namely protons and neutrons, form various nuclear species. Dueto the complexity of the nuclear interactions, theoretical predictions are still imper-fect in many aspects. Therefore, experimental studies are essential to approach thisproblem of the origin of matter. Furthermore, this study is also related to how theheavy elements, such as gold, uranium, and so on, are synthesized in the universethrough explosive events in the cosmos. Understanding the physics of nuclear mat-ter will help us pin down the exact location and scenario of the production of suchelements around us.The method of this study is to produce radioactive nuclear species which donot exist in nature and observe how they decay using various radiation detectors.ivPrefaceThis study is part of the experimental campaign of the GRIFFIN collaboration,in order to investigate the properties of neutron-rich cadmium isotopes 128−132Cd,proposed by N. Bernier (PhD student, UBC), Dr. I. Dillmann (Research Scientist,TRIUMF), Dr. R. Kru¨cken (Professor of Physics, UBC). The GRIFFIN spectrom-eter was commissioned in 2014, led by Dr. A. Garnsworthy (Research Scientist,TRIUMF). The experiment was conducted in August, 2015 and the author, Y. Saito,did not take part in the data collection.The analysis of the data was solely done by the author, with the help from N.Bernier, Dr. M. Bowry (Postdoctoral Fellow, TRIUMF), Dr. R. Caballero-Folch(Postdoctoral Fellow, TRIUMF) and the author’s supervisors Dr. R. Kru¨cken andDr. I. Dillmann. The analysis tool used in this work, GRSISort, was developedby the nuclear physics group at the University of Guelph, Canada and the Gamma-Ray Spectroscopy at ISAC (GRSI) group at TRIUMF.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Independent Particle Model . . . . . . . . . . . . . . . . 21.1.2 Interacting Shell Model . . . . . . . . . . . . . . . . . . . 41.2 Rapid Neutron Capture Process (r-process) . . . . . . . . . . . . 111.3 Previous Investigation on 129Cd and 129In . . . . . . . . . . . . . 152 Decay Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 Radioactive Decay . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.1 β -decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.2 γ-decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Radiation Detection . . . . . . . . . . . . . . . . . . . . . . . . . 32vi2.2.1 Radiation Interaction with Matter . . . . . . . . . . . . . 323 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1 Isotope Production . . . . . . . . . . . . . . . . . . . . . . . . . 373.1.1 TRIUMF ISAC Facility . . . . . . . . . . . . . . . . . . 373.1.2 Ion Source: IG-LIS . . . . . . . . . . . . . . . . . . . . . 383.1.3 Radioactive Beam . . . . . . . . . . . . . . . . . . . . . 393.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.1 The GRIFFIN Spectrometer and Ancillary Detectors . . . 413.2.2 HPGe Clover Detectors Array . . . . . . . . . . . . . . . 413.2.3 Ancillary Detectors . . . . . . . . . . . . . . . . . . . . . 413.2.4 Data Acquisition (DAQ) System . . . . . . . . . . . . . . 434 Data Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Event Construction . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.1 Data Sorting and Event Construction . . . . . . . . . . . 454.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.1 HPGe GRIFFIN Array Energy Calibration . . . . . . . . 464.2.2 HPGe GRIFFIN Array Efficiency Calibration . . . . . . . 464.2.3 SCEPTAR Energy Calibration . . . . . . . . . . . . . . . 514.3 Coincidence Analysis . . . . . . . . . . . . . . . . . . . . . . . . 524.3.1 Timing Gates . . . . . . . . . . . . . . . . . . . . . . . . 524.3.2 γ-γ Coincidence Matrix . . . . . . . . . . . . . . . . . . 544.4 Level Scheme of 129In . . . . . . . . . . . . . . . . . . . . . . . 554.4.1 11/2+, 13/2+, and 17/2− states . . . . . . . . . . . . . . 564.4.2 Excited States Feeding the 1/2− Isomeric State . . . . . . 584.4.3 Half-Life of the 17/2− Isomeric State . . . . . . . . . . . 624.5 Determination of Relative γ-ray Intensities . . . . . . . . . . . . . 624.6 Determination of β -feeding Intensities and log f t Values . . . . . 704.6.1 Fit of the Beam Implantation Cycle . . . . . . . . . . . . 704.6.2 The Decay 129Cd→ 129In . . . . . . . . . . . . . . . . . 73vii5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1 Shell Model Calculation . . . . . . . . . . . . . . . . . . . . . . 805.2 Decay Properties of 129Cd . . . . . . . . . . . . . . . . . . . . . 835.2.1 Gamow-Teller (GT) Decays . . . . . . . . . . . . . . . . 835.2.2 First-Forbidden (ff) Decays . . . . . . . . . . . . . . . . 846 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Appendix A Production of Radioactivity and Series of Decays . . . . . 95A.1 Bateman Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 95A.2 Bateman Equations for Production of Radioactivity . . . . . . . . 96viiiList of TablesTable 2.1 Selection Rules for β -Decays . . . . . . . . . . . . . . . . . . 29Table 2.2 Type of β -Decays and Typical log f t Values [29] . . . . . . . . 29Table 4.1 Isotopes and transitions used for γ-ray efficiency calibration.The intensities were taken from Ref.[28]. . . . . . . . . . . . . 49Table 4.2 Relative γ-ray Intensities of 129In. The star symbols denote tran-sitions that are identified as 129In but could not be placed in thelevel scheme. The Ilitγ values were taken from Ref. [22]. . . . . 65Table 4.3 Types of tape cycle . . . . . . . . . . . . . . . . . . . . . . . 70Table 4.4 β -decay feeding intensities to each excited state and their log f tvalues. “lit” are the values from Ref.[22] to compare to thecurrent analysis. The third and fourth columns are the total ob-served β -feeding intensities, the fifth and sixth columns (theseventh and eighth columns) from the left are the β -feeding in-tensities in the β -decay of 3/2+(11/2−) state in 129Cd. Out ofall the observed β -decays of 129Cd, 59(4)% is from the 11/2−state and 41(4)% is from the 3/2+ state (see Section 4.6.2). Thelog f t values are calculated based on the β -feeding intensitiesfor each β -decaying state. . . . . . . . . . . . . . . . . . . . . 77ixList of FiguresFigure 1.1 Comparison of single-particle energies based on the (modified)simple harmonic oscillator (S.H.O.) potential. The energy lev-els at the left are the harmonic oscillator potential without anymodification, the ones in the middle are with an l2 term, andthe ones at the right are with an l2 and spin-orbit (l · s) terms.The figure was taken from Ref.[1] . . . . . . . . . . . . . . . 5Figure 1.2 Nuclear chart which shows the classical shell closures and onepossible r-process path. The figure is taken from Ref.[6]. . . . 11Figure 1.3 r-process abundance pattern. The blue points show the abun-dance pattern where there are contributions from r-process,whereas the red points show the abundance pattern where onlyr-process has contributions. Figure courtesy of I. Dillmann,based on Ref.[11] . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 1.4 Possible configurations of the single-particle levels of proton(left) and neutron (right) in 12948Cd81. . . . . . . . . . . . . . . 16Figure 1.5 Level scheme of 129In populated by the β -decays of 129Cd,based on Ref.[21]. The figure was taken from ENSDF in Ref.[28]. 17Figure 1.6 Level scheme of 129In populated by the β -decays of the 11/2−state in 129Cd, based on Ref.[22]. The figure was taken fromENSDF in Ref.[28]. . . . . . . . . . . . . . . . . . . . . . . 18Figure 1.7 Part of the level scheme of 129In populated by the β -decaysof the 3/2+ state in 129Cd, based on Ref.[22]. The figure wastaken from ENSDF in Ref.[28]. . . . . . . . . . . . . . . . . 19xFigure 1.8 The other part of the level scheme of 129In populated by theβ -decays of the 3/2+ state in 129Cd, based on Ref.[22]. Thefigure was taken from ENSDF in Ref.[28]. . . . . . . . . . . . 20Figure 2.1 Sketch of Compton scattering process. An incident γ-ray withthe energy of hν is scattered by an electron and deflected throughan angle θ with respect to its original direction. θ can be anyangle. The figure was taken from Ref.[33]. . . . . . . . . . . 34Figure 2.2 Linear attenuation coefficient of NaI as a function of incidentγ-ray energy. The contributions from photoelectric absorption,Compton scattering, and pair production are also shown. Thefigure is taken from Ref.[34]. . . . . . . . . . . . . . . . . . . 36Figure 3.1 The schematics of the TRIUMF ISAC facility. The GRIFFINspectrometer has replaced the 8pi spectrometer and is now lo-cated at the neighbouring beam line (not shown in this picture).The figure was taken from Ref. [35]. . . . . . . . . . . . . . . 38Figure 3.2 The schematics of the Ion Guide - Laser Ion Source (IG-LIS).See section 3.1.2 for details. The figure was taken from Ref.[36]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 3.3 Element-selective multi-step laser excitation scheme for cad-mium. Figure courtesy of J. Lassen. . . . . . . . . . . . . . . 40Figure 3.4 3D model of a GRIFFIN HPGe clover. The figure was takenfrom Ref.[38] . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 3.5 The upstream hemisphere of SCEPTAR and the east hemi-sphere of the GRIFFIN HPGe array without the BGO shields.The figure was taken from Ref.[37]. . . . . . . . . . . . . . . 43Figure 4.1 Example of the gain matching of HPGe crystals with calibra-tion sources 152Eu and 56Co. Only the first 8 crystals are shownfor example. The x-axis shows the bin number of the uncali-brated energy histogram and the y-axis shows the calibratedenergy. The “+” marker of each point in the plot does notcorrespond to the size of the error. . . . . . . . . . . . . . . . 47xiFigure 4.2 Example of the gain matching of HPGe crystals with knownpeaks from the decay of 129Sn in the experimental data. Onlythe first 8 crystals are shown for example. The x-axis showsthe energy with the source calibration (E) and the y-axis showsthe calibrated energy (E ′). The “+” marker of each point in theplot does not correspond to the size of the error. From the lin-ear fitting functions shown in the figure, it can be observed thatsome of the offsets of the energy gain have non-zero deviationfrom the original energy calibration. . . . . . . . . . . . . . . 48Figure 4.3 Time difference of two consecutive hits in the same crystal.The x-axis is the time difference and the y-axis corresponds toeach crystal. Large statistics around 0 on the x-axis is assumedto be pile-up events due to the strong source. Except for thechannel 31, the minimum time difference is 7.5 µs. . . . . . 51Figure 4.4 GRIFFIN HPGe array γ-ray detection efficiency curves. Theblue and red lines show the efficiency curves for 62 HPGe crys-tals and 63 HPGe crystals, respectively. . . . . . . . . . . . . 52Figure 4.5 The x-axis shows the channel number of SCEPTAR and they-axis shows the gain-matched energy in arbitrary unit. Sincethe data collected by SCEPTAR is only used for β -tagging ofthe γ-rays, the energy of the β -particles has to be gain-matchedbut does not have to be accurate. . . . . . . . . . . . . . . . . 53Figure 4.6 The figure shows the projection of Fig.4.5 on to the energyaxis. The dashed line shows the threshold which distinguishesthe β -particles following the β -decays from the backgroundelectrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 4.7 Timing difference between γ-ray hits within the same event.The γ-γ coincidence timing gate and background (time ran-dom) timing gate are shown in the figure. . . . . . . . . . . . 55Figure 4.8 Timing difference between β and γ-ray hits within the sameevent. The β -γ timing gate is defined as shown in the figure. . 56xiiFigure 4.9 β -gated γ-γ coincidence matrix. The axes of the original figureextends up to 6 MeV but in this figure it only shows up to 2MeV for visibility of the vertical, horizontal, and diagonal lines. 57Figure 4.10 This shows a example of a projection of the 994.8 keV tran-sition on the β -gated γ-γ coincidence matrix. The region en-closed by the dashed line represents the energy gate on the994.8 keV transition and the region enclosed by the dotted linerepresents the Compton background which is to be scaled andsubtracted from the projection. . . . . . . . . . . . . . . . . . 58Figure 4.11 β -gated γ-γ coincidence matrix processed using TSpectrum2class. The background is smoothed and the peaks are deconvo-luted. The red triangles show the coincidence peaks detectedby the TSpectrum2 algorithm. . . . . . . . . . . . . . . . . 59Figure 4.12 β -gated single γ-ray spectrum with 129Sn transitions subtracted(for details see Section 4.5). . . . . . . . . . . . . . . . . . . 60Figure 4.13 Level Scheme of 129In. Red arrows and lines indicate newlyobserved transitions and excited states. Blue lines indicate iso-meric states. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 4.14 Difference in timestamp of β -particles and γ-rays within the50 µs event construction time window. The y-axis goes onlyup to 15 µs for visibility. The lines extending along the y-axis indicate the existence of isomers. The unlabelled verticallines in the figure are due to isomers of the daughter nucleus129Sn. The prominent line at y = 0 is due to prompt γ-rays andCompton background. . . . . . . . . . . . . . . . . . . . . . 63Figure 4.15 The half-life of the 17/2− isomer fitted to the time differenceof the timestamps of β -particles and γ-rays gated on 334.0,358.9, 994.8, and 1354.2 keV transitions. . . . . . . . . . . . 64Figure 4.16 Comparison between a raw β -gated single γ-ray spectrum (blue)and the 129Sn transitions subtracted β -gated single γ-ray spec-trum (red). The raw spectrum (blue) is scaled down to the129Sn transitions subtracted spectrum (red) for comparison. . . 65xiiiFigure 4.17 Decay chain model starting from 129Cd. The excited stateswhich do not decay via β -decay have been omitted. This modelallows one state to populate both β -decaying states in the daugh-ter nucleus. This is because in reality, the excited states popu-lated by the β -decays can populate both β -decaying states. . . 71Figure 4.18 Fit of the number of β -particles detected by SCEPTAR in thewhole beam cycle (top) and the quality of the fit (bottom).The dashed blue and red lines correspond to the ground state(T1/2 = 611 ms) and the 1/2− isomeric state (T1/2 = 1.23 s)in 129In, respectively. The dash-dotted lines show the twoβ -decaying states in 129Sn with averaged half-lives based onRef.[26] (see text). . . . . . . . . . . . . . . . . . . . . . . . 72Figure 4.19 Correlation between the 129Cd half-life and the number of de-cays. The yellow band corresponds to the systematic error onthe number of decays. . . . . . . . . . . . . . . . . . . . . . 74Figure 4.20 Correlation between the 129Cd half-life and the β -decay branch-ing ratio of the 1/2− state in 129In. The yellow band corre-sponds to the systematic error on the β -decay branching ratioof the 1/2− state in 129In. . . . . . . . . . . . . . . . . . . . . 75Figure 5.1 The comparison between the experimentally observed excitedstates and the Shell-model calculations of 129In. “jj45pna” isthe current calculation, “Wang et al” is from Ref.[20], “Taproggeet al” is from Ref.[22], and “Genevey et al.” is from Ref.[24](see text for detail). The experimental values for the excitedstates at 1630 keV and 1911 keV are also taken from Ref.[24].Some of the excitation energies in “Wang et al.” were not pre-sented in Ref.[20], therefore they were read from the figure. . 82xivAcknowledgmentsFirst and foremost, I would like to acknowledge my supervisors Dr. Reiner Kru¨ckenand Dr. Iris Dillmann for continuous support and instruction throughout my M.Sc.program. I feel very excited for the upcoming PhD project.My M.Sc. program has been supported by the NSERC CREATE programIsoSiM (Isotopes for Science and Medicine) and I am grateful for a number ofopportunities the program has been providing me.I would also like to thank the members of the Gamma-Ray Spectroscopy atISAC (GRSI) group, especially N. Bernier, M. Bowry and R. Caballero-Folch forhelping me with the analysis.I feel very privileged to be able to pursue my graduate degrees here at UBC inCanada and without the tremendous amount of support from my parents, it wouldhave not been possible. I would like to thank you very much.All my friends in Canada, Japan, and other parts of the world have been ab-solutely essential for my life and I would have not made it here without thesewonderful people.Last but not least, I would like to acknowledge my cat April for being there allthe time (or coming back home even when you escape).xvChapter 1IntroductionThe nuclei around doubly magic 132Sn have been a topic of investigation in order tostudy the validity and limits of the nuclear shell model, since 132Sn is the heaviestnucleus far from stability which has closed shells for both protons and neutrons.Therefore, the nuclei in this region are expected to provide us with valuable infor-mation on the evolution of the shell structure. Furthermore, these shell closures areresponsible for the second abundance peak of the astrophysical rapid neutron cap-ture process (r-process) at A∼ 130. The experimental information of these nucleican have large impact on the r-process abundance calculations.1.1 The Shell ModelThe characteristics of the nuclear force is that, aside from a very short-range re-pulsive core, it is principally attractive, rather short range, saturates, and charge-independent [1]. The nucleus is a quantum-mechanical many-body system and inorder to describe a nucleus using nucleon (proton and neutron) degrees of freedom,the wave functions of the nucleus need to be expressed in terms of those for indi-vidual nucleons [2]. Using such single-particle basis, the many-body eigenvalueproblem to be solved is:HΨα(r1,r2, · · · ,rA) = EαΨα(r1,r2, · · · ,rA), (1.1)1where Eα is the energy of the state with wave function Ψα(r1,r2, · · · ,rA). TheHamiltonian consists of a sum of kinetic energy of each nucleon and two-bodyinteraction:H = T +V =A∑i=1p2i2mi+A∑i>k=1Vik(r i− rk). (1.2)The second term can be expressed in terms of a central potential and a residualinteraction:H =A∑i=1p2i2mi+A∑i=1Ui(r i)+{A∑i>k=1Vik(r i− rk)−A∑i=1Ui(r i)}(1.3)= T +U(r)+Hresid . (1.4)1.1.1 Independent Particle ModelThe “independent particle model” can be applied when the residual interaction isignored. In this approximation, the nuclear Hamiltonian Eq. 1.3 is a sum of single-particle terms:H =A∑i=1p2i2mi+A∑i=1Ui(r i). (1.5)Harmonic Oscillator PotentialOne of the simple attractive potential models is the harmonic oscillator potential:U(r) =12mω2|r|2 = 12mω(x2+ y2+ z2). (1.6)This is a central potential and creates bound states of nucleons. The schro¨dingerequation for the Hamiltonian with this harmonic oscillator potential can be solvedby separating the wave function into the radial part and angular part:Ψ= R(r) ·χ(θ ,φ), (1.7)2and the radial wave function is [3]Rnrl(r) =√2l−nr+2(2l+2nr +1)!!√pi(nr)!b2l+3[(2l+1)!!]2rl+1e−r2/2b2×nr∑k=0(−1)k2knr!(2l+1)!!k!(nr− k)!(2l+2k+1)!!(r/b)2k, (1.8)where nr = 0,1,2, · · · is the radial quantum number, which indicates the numberof times the radial wave functions cross the r axis, l is the index for sphericalharmonic functions Yl,m(θ ,φ) with −l ≤ m≤ l, andb =√h¯mω. (1.9)The resulting energy eigenvalues areE = h¯ω(N+32)where N = 2nr + l, (1.10)This harmonic oscillator potential leads to a shell structure of energy state char-acterized by N, the major-shell harmonic oscillator quantum number. The shellgaps appear when the total number of states are 2,8,20,40,70, · · · . However,these shell closures differ from the empirical nuclear magic numbers, which are2,8,20,28,50, · · · . This is due to the fact that the nucleons in the nuclear interiorshould experience interactions in all directions, hence no net force. Therefore thecentral part of the nuclear potential should be approximately constant.A possible correction to this harmonic oscillator potential is to add an attractiveterm l2. This term increases with the orbital angular momentum of the particle andhigh angular momentum particles, whose wave functions are localized at largerradii, feel a stronger attractive interaction that lowers their energies [1]. A Woods-Saxon potential:U(r) =U01+ er−R0A1/3a, (1.11)has a flatter bottom than the harmonic oscillator and also produces effects similarto an l2 term. For this Wood-Saxon potential, only numerical solutions are avail-3able. Nevertheless, with such corrections, the empirical magic numbers are stillnot reproduced correctly.Spin-Orbit CouplingAlthough the harmonic oscillator potential is a reasonable starting point for repro-ducing the structure of single-particle states in nuclei, since the first few shell clo-sures are reproduced with this potential, further corrections have to be introduced.Mayer [4] and Haxel, Jensen, and Suess [5] suggested that the magic numbers maybe explained by introducing a spin-orbit force, therefore the potential term of thesingle-particle Hamiltonian now takes on the form:U(r) =12mω2r2+as · l , (1.12)where the parameter a depends on the nucleon number A. Here the l2 term isomitted for simplicity, but it is necessary to reproduce the empirical nuclear magicnumbers. With this spin-orbit coupling term, the single-particle energy (Eq.1.10)becomesENl j =(N+32)h¯ω+12 al−12 a(l+1)for j = l+12for j = l− 12. (1.13)With a spin-orbit component, the force felt by a given particle differs accordingto whether the single-particle state has j = j> ≡ l + 12 or j = j< ≡ l− 12 . Sincethe parallel alignment of an intrinsic spin and an orbital angular momentum isfavoured, a < 0 and a single-particle state with j> is lowered in energy. With thiscorrection and the l2 term, as shown in Fig. 1.1, the magic numbers are correctlyreproduced using the harmonic oscillator potential.1.1.2 Interacting Shell ModelThe discussion in this section is mainly based on Ref. [2].Although the independent particle model can account for nuclear propertiessuch as shell structures, for more precise information, it is necessary to includethe residual interaction, which is ignored in the independent particle model. Since45 2 Shell Model and Residual InteractionsFig. 3.2. Single-particle energies for a simple harmonic oscillator (S.H.O.), a modified harmonicoscillator with / 2  term, and a realistic shell model potential with / 2  and spin orbit (/ • s) terms.It is this grouping of levels that provides the shell structure required of anycentral potential useful for real nuclei. If we recall that each energy  level has2(21+1) degenerate m states, then, by  the Pauli principle, each nl level can con-tain 2(21 + 1) particles. Therefore, if we imagine filling such a poten-tial wellwith fermions, each group or shell can contain, at most, the specific numbers ofparticles indicated in the figure. Hence, such a potential automatically  gives ashell structure rather than, say , a uniform distribution of levels.Unfortunately , except for the lowest few, these shells do not correspond tothe empirical magic numbers. Therefore, while the harmonic oscillator poten-Figure 1.1: Co parison of single-particle energies based on the (modified)simple harmonic oscillator (S.H.O.) potential. The energy levels at theleft are the harmonic oscillator potential without any modification, theones in the middle are with an l2 term, and the ones at the right are withan l2 and spin-orbit (l · s) terms. The figure was taken from Ref.[1]5the independent particle states are reasonably good approximations, the residualinteraction may be viewed as introducing configuration mixing among such states.Therefore the wave functions are made of linear combinations of these independentparticle states.Since the many-particle space which consists of different products of the single-particle states is infinite in dimension, for the purpose of practical calculation, theHilbert space has to be truncated to a finite one by using techniques such as theHartree-Fock approach. Furthermore, selecting the active space based on the shellstructure of the single-particle states, calculations may be carried out in a relativelysmall part of the complete space. This approach is called interacting shell model.There are three steps to be carried out in order to perform the calculations: thechoice of a single-particle basis, the selection of an active space, and the derivationof an effective iteration.Selection of the Shell-Model SpaceIn the spherical shell model, each nucleon has an intrinsic spin s and occupies astate of definite orbital angular momentum l . When A nucleons are put into single-particle state, the many-body basis states are formed by coupling the single-particlestates together to form states with definite total angular momentum J and isospinT .In the LS-coupling scheme, the orbital angular momentum l i and the intrinsicspin si of each nucleon is first coupled separately to total orbital angular momentumL and total intrinsic spin S:L =A∑i=1l i, S =A∑i=1si. (1.14)Therefore the total angular momentum J isJ = L+S. (1.15)Alternatively, the orbital angular momentum and the intrinsic spin of each nucleon6may be coupled together first to form the nucleon spin j ij i = l i+ si, (1.16)then the nuclear spin isJ =A∑i=1j i. (1.17)This is called the j j-coupling scheme.In a spherical basis, the Hamiltonian is invariant under a rotation and J is agood quantum number. Isospin T is also a constant of motion as long as symmetry-breaking effects due to electromagnetic interaction are ignored. Hamiltonian ma-trix elements between different J and T values vanish. Furthermore, the Hamil-tonian matrix in the complete shell-model space appears in a block-diagonal formaccording to their (J,T ) values. Therefore, the calculation can be carried out sep-arately within a specific (J,T ) subspace and this reduces the size of the Hilbertspace of the calculation.In order to truncate the Hilbert space, the nucleons are divided into two groups,core nucleons and valence nucleons. Also, the single-particle states are separatedinto core states, active states, and empty states. For investigation of low-lyingstates, only nucleons near the Fermi surface are directly involved. The rest of thenucleons can be assumed to form an inert core, which is not excited. Contributionsfrom the core nucleons cannot be ignored but can be accounted for in the definitionof single-particle energies of the valence nucleons.There are single-particle states much above the Fermi energy and if the inter-est is confined to the low-lying states, such high energy single-particle states arealways expected to be empty.Effective HamiltonianIn order for the shell model Hamiltonian to be in a manageable size for a calcu-lation, we need to find an effective Hamiltonian such that the effect of the single-particle states which are ignored in the calculation may be accounted for in anefficient manner. Mathematically this means a transformation from the infinite-dimensional space, specified by all the Hartree-Fock single-particle states, for ex-7ample, to a finite, truncated shell-model space. The effective Hamiltonian may bewritten in the form ofHeff = H0+Veff, (1.18)where H0 is the one-body Hamiltonian and Veff is the effective two-body interaction.However, there is no reason to rule out three-body and higher order terms. In fact,three-body interactions have to be taken into account and their influence is a topicof current investigations [6–8]. However, in real shell model calculations they areaccounted for by emulating their effects by modifications to the two-body matrixelements.Let P be an operator which projects out the active part of the space from thecomplete many-body space and be Q an operator which projects out the rest of theHilbert space. Then P and Q satisfyP+Q = 1, (1.19)P2 = P, Q2 = Q, (1.20)andPQ = QP = 0. (1.21)By operating P and Q we obtainP{H0+V +V1E−QH QV}PΨ= EPΨ. (1.22)The derivation is shown in detail in Ref. [2]. Since the Hamiltonian in the completeHilbert space is H =H0+V , where H0 is a one-body operator and V is a two-bodypotential, we identify thatVeff =V +V1E−QH QV =V +V Q1E−H0−QV QV. (1.23)Since the expectation value of the residual interaction V are smaller than thosefor H0 and also expected to be smaller than E −H0, by expanding the operator8(E−H0+QV )−1 in powers of (E−H0)−1QV ,V Q1E−H0−QV QV =V Q∞∑n=1(1E−H0 QV)n. (1.24)Therefore,Veff =V +V Q∞∑n=1(1E−H0 QV)n. (1.25)Although there is no known proof that the series is actually convergent, the effec-tive interaction to roughly second order has been shown to give shell-model resultsthat are in good agreement with various experimental data [2]. This procedure tofind the effective interaction in a shell-model space is known as a renormalizationprocedure.Two-Body Matrix Element (TBME) and Shell Model CalculationOnce the residual interaction is described by an effective interaction, which is smallenough to be treated in perturbation theory, the matrix elements of the interactioncan be expressed by a linear combination of two-body matrix elements (TBME)[9]:〈ψA∣∣Veff∣∣ψ ′A′〉= ∑i, j,k,lJ,TCi jk,JTA,A′ 〈i jJT |Veff|klJT 〉 , (1.26)(i≤ j, k ≤ l, i≤ k, and j ≤ l when i = k)where i, j,k, l are the labels of single-particle orbits and |i jJT 〉 and |klJT 〉 are anti-symmetrized normalized two-body states coupled to the total angular momentum Jand isospin T . By evaluating these TBME of Veff between the single-particle eigen-states of H0 (Eq.1.18), the eigenvalues and eigenvectors of this matrix expressedby Eq.1.26 are obtained.9For example, if the shell model Hamiltonian is given byH =Nsps∑i=1εini+Nsps∑i jklvi j,kla†i a†jalak (1.27)=Nsps∑i=1εini+Nsps∑i jkl〈i jJT |Veff|klJT 〉a†i a†jalak, (1.28)where Nsps corresponds to the model space, εi is the single-particle energies deter-mined from the experimental values, a† and a are creation operators and annihila-tion operators, respectively. The Hamiltonian is expressed asH =〈Φ1|H|Φ1〉 〈Φ1|H|Φ2〉 〈Φ1|H|Φ3〉 · · ·〈Φ2|H|Φ1〉 〈Φ2|H|Φ2〉 〈Φ2|H|Φ3〉〈Φ3|H|Φ1〉 〈Φ3|H|Φ2〉 〈Φ3|H|Φ3〉.... . . , (1.29)where |Φ1〉 , |Φ2〉 , |Φ3〉 , · · · are the Slater determinants. Solving this eigenvalueproblem:HΨ= EΨ, (1.30)is equivalent to diagonalizing the Hamiltonian matrix(1.29):〈Φ1|H|Φ1〉 〈Φ1|H|Φ2〉 〈Φ1|H|Φ3〉 · · ·〈Φ2|H|Φ1〉 〈Φ2|H|Φ2〉 〈Φ2|H|Φ3〉〈Φ3|H|Φ1〉 〈Φ3|H|Φ2〉 〈Φ3|H|Φ3〉.... . .c1c2c3...= Ec1c2c3... , (1.31)and the eigen wave function |Ψ〉 is|Ψ〉= c1 |Φ1〉+ c2 |Φ2〉+ c3 |Φ3〉+ · · · , (1.32)where c1,c2,c3, · · · are the probability amplitudes [10].10Figure 1.2: Nuclear chart which shows the classical shell closures and onepossible r-process path. The figure is taken from Ref.[6].1.2 Rapid Neutron Capture Process (r-process)In this section, the synthesis of heavy elements, especially the rapid neutron cap-ture process (r-process), which is one of the main processes responsible for suchelements, is discussed. The topic of the current study, 129Cd is one of the isotopesrelated to the formation of the r-process second abundance peak, which is closelyrelated to the shell closures at Z = 50 and N = 82 (Fig. 1.2 and 1.3). Since anyastrophysical consideration is beyond the scope of the current thesis, only a generaloverview of the r-process is given.11Figure 1.3: r-process abundance pattern. The blue points show the abun-dance pattern where there are contributions from r-process, whereas thered points show the abundance pattern where only r-process has contri-butions. Figure courtesy of I. Dillmann, based on Ref.[11]Nucleosynthesis up to IronCosmological nucleosynthesis starts shortly after Big Bang, in which hydrogen,helium, and lithium isotopes are produced. After the formation of stars, up tonuclei with intermediate mass up to iron are formed through fusion reactions in thecore burning processes inside the stars. The main source of iron peak nuclei aroundA ≈ 56 are explosive events such as Type IA supernovae, where fusion reactionsalong N = Z occur up to 56Ni which decays into 56Fe.Nuclei beyond iron cannot be formed through nuclear fusion since the nuclearbinding energy peaks at iron. Heavier nuclei are formed through neutron captureand successive β -decays, about half each in a slow (s) and rapid (r) neutron captureprocess. While most of the s-process and its astrophysical sites are understood, the12astrophysical site of the r-process is still unknown.Mechanism of r-processWith the existence of seed nuclei such as iron in a very hot and neutron-rich envi-ronment (T & 109K, Nn & 1020cm−3) [12], the nuclei start to capture neutrons andto equilibrate the reactions:(A,Z)+n (A+1,Z)+ γ, (1.33)on a very fast timescale. Such situations are achieved in massive stars (M > 8M)close to the forming neutron star during the core collapse supernova, or in neutronstar mergers [12, 13]. This shifts the nuclei away from the valley of β -stability tothe neutron rich side. However, there is a limit of the number of neutrons that canbe attached to a certain nucleus, which is defined as the neutron separation energySn:Sn = B(Z,A+1)−B(Z,A), (1.34)where B(Z,A) is the binding energy of the nucleus (Z,A). When Sn is zero (thedashed line on the neutron-rich side in Fig.1.2) or negative, the neutron is notbound. The addition of neutrons could stop even before the neutron-drip line(Sn = 0) is reached due to the balance of neutron capture and photodisintegrationreactions under the given astrophysical conditions (temperature and neutron den-sity). After capturing neutrons, the nuclei have to wait for the β -decays to occur(t1/2 ' 10−1− 10−2 s) (for β -decay see Section 2.1.1). This “waiting point” canbe approximately described by the canonical r-process (CAR) model. This modelrelies on the assumptions that the neutron density Nn remains constant over thewhole timescale τ , and Nn is high enough so that the (n,γ) reaction on the neutron-rich nuclei happens faster than their β -decays (Nn & 1020cm−3). In addition, thephoto-disintegrations (γ,n) are also expected to happen faster than the β -decays.This means that the temperature of the environment T is high (T & 109K). Underthese conditions, starting from pure 56Fe, the evolution of the abundances can be13expressed in the form:dN(Z,A)dt= λ Z,A+1γ,n N(Z,A+1)−〈σv〉Z,A N(Z,A)Nn, (1.35)where N(Z,A) is the number density of nucleus (Z,A), λγ,n is the rate of photo-disintegration, and 〈σv〉 is the cross section of neutron capture averaged overMaxwell-Boltzmann velocity distribution. If the equilibrium (1.33) holds for thewhole timescale τ for all the isotopes heavier than iron, the following equationholds:N(Z,A+1)N(Z,A)=〈σv〉Z,Aλ Z,A+1γ,nNn (1.36)=G∗(Z,A+1)2G∗(Z,A)(2pi h¯2NAmkT)3/2Nn exp[Sn(Z,A+1)kT], (1.37)where NA is the Avogadro number, and the reduced mass m is approximated by thenucleon mass mn for the heavy nuclei. The partition function G∗ is defined asG∗ = (2J0+1)G, (1.38)where J0 is the ground state nuclear spin of the nucleus of interest, and G representsthe temperature-dependent normalized partition function:G = GI(T ) =∑µ(2JµI +1)(2J0I +1)exp(− εµIkT). (1.39)For a given Nn and T , Eq.(1.37) indicates the abundance of a element Z concen-trated on its isotope with a neutron separation energy Sn(Z,A) approaching thevalueS0a[MeV] =(34.075− logNn[cm−3]+ 32 logT9)T95.04, (1.40)where T9 is the temperature in the unit of 109 K [14]. Finally, this process iseither stopped by the lack of free neutrons or by spontaneous or neutron-inducedfission of the synthesized heavy nuclei. The process is called the rapid neutroncapture process (r-process) because it is assumed that during the build-up of heavy14elements, the neutron capture rates are much greater than the β−-decay rates [12].Possible Sites of r-processThe astrophysical sites for the r-process are still a topic of investigations. Accord-ing to the thermodynamical conditions, they can be classified in low-entropy andhigh-entropy sites. Low-entropy sites include the decompression of cold neutronstar material, prompt explosion of ONeMg cores and jets from accretion disks.The neutrino-driven wind from the nascent neutron star in a core-collapse super-nova is classified as a high-entropy environment [15]. In addition to this, themulti-messenger observations prompted by the detection of the binary neutronstar merger event GW170817[13] suggest that the observed kilonova/macronovais powered by the radioactive decay of r-process nuclei synthesized in the ejecta ofthe binary neutron star merger [16].1.3 Previous Investigation on 129Cd and 129Inβ -decaying States in 129CdThere are two known β -decaying states in 129Cd, which are the 11/2− state andthe 3/2+ state. These spin assignment was confirmed by Yordanov et al. with laserspectroscopy [17]. Although Ref.[18–20] suggested 11/2− as the ground state andan excited 3/2+ β -decaying isomer, there has been no experimental evidence tosupport this assignment. The configuration of the neutron single-particle levelsand the effect of the proton-neutron interaction decide the order of these states(Fig.1.4).Previously, three experiments for the half-lives of the two β -decaying statesin 129Cd have been reported. Arndt et al.[21] proposed the half-lives of 104(6)ms for the 11/2− state and 242(8) ms for the 3/2+ state, respectively, by mea-suring the β -delayed neutrons. These values are based on the Diploma thesis ofO. Arndt, which, however, has not been published. The next measurement wasdone by Taprogge et al.[22] and the reported values were 155(3) ms for the 11/2−and 148(8) ms 3/2+, respectively, which do not agree with the previous measure-ment within the uncertainties. Finally, another measurement was done by Dun-15Figure 1.4: Possible configurations of the single-particle levels of proton(left) and neutron (right) in 12948Cd81.lop et al. with the GRIFFIN spectrometer [23] using the same dataset as the cur-rent study, which resulted in the half-lives of 147(3) ms for the 11/2− state and157(8) ms for the 3/2+ state. The measurements reported in Ref.[22, 23] used theβ -γ-gating method. These values are in agreement with the results by Taproggeet al.[22] within 2σ for the 11/2− and 1σ for the 3/2+ state, respectively, whichhave surperseded the half-lives reported by Arndt et al..Decay Spectroscopy of 129CdSo far, there have been two studies on the decay of 129Cd, one of which was the ex-periment performed at the CERN On-Line Isotope Mass Separator (ISOLDE) facil-ity, reported in Ref.[21]. In this study, more than 50 γ-ray transitions following theβ -decay of 129Cd were observed, confirming the placement of the 17/2− isomericstate with a half-life of 8.5(5) µs at 1687 keV reported by Genevey et al.[24] andthe β -decaying 1/2− isomer reported in Ref.[25–27]. The level scheme is shownin Fig.1.5. This 1/2− isomeric state was also confirmed by a new mass measure-ment with the TITAN facility at TRIUMF-ISAC, reporting the excitation energyof 444(15) keV, which is in agreement with the previous result [27]. The mostrecent decay spectroscopy of 129Cd before the current study was carried out byTaprogge et al.[22] at the Radioactive Isotope Beam Factory (RIBF) at the RIKENNishina Center (Japan). This study expanded the level scheme of 129In reported in16Ref.[21] with 31 newly observed transitions, resulting in establishing 27 new ex-cited states (Figs.1.6 - 1.8). At the same time, out of 53 transitions,19 placementsof them, and 14 excited states reported in Ref.[21], 12 transitions, 11 placements,and 11 excited states, respectively, were not confirmed in Ref.[22]. In addition, theβ -feeding intensities and log f t values were reported in Ref.[22] for the first time.12949 In80-4 From ENSDF12949 In80-4(9/2+) 0 611 ms 5(1/2−) 459 1.23 s 3(5/2) 858.8(11/2+) 995.1(5/2) 1020.5(3/2−) 1091.0(13/2+) 1354.0(5/2+) 1422.8(5/2) 1562.0(9/2+) 1585.7(5/2−) 1632.8(17/2−) 1687.8 8.7 µs 7(13/2−) 2419.1(5/2) 2918.9(13/2−) 3150.13183.9(5/2−) 4578.94119.92.03487.81.03184.13.01760.919.02155.19.01796.1321462.211.02918.51.02460.26.01065.28.0731.18.5333.5(M2)13.0541.811.01585.712.01561.55.01103.45.01422.6201354.121.0358.850631.9301020.38.5561.78.0995.0100858.13.3400.57.012948 Cd8103/2+ 242 ms 8Q=9330 SY%β−=100.00+x11/2− 104 ms 6Q=9330 SY%β−=100.0Intensities: Relative IγDecay Scheme129Cd β− decay:mixed 2009Ar0412949 In80γ Decay (Uncertain)Iγ > 10%×ImaxγIγ < 10%×ImaxγIγ < 2%×ImaxγLegend4Figure 1.5: Level scheme of 129In populated by the β -decays of 129Cd, basedon Ref.[21]. The figure was taken from ENSDF in Ref.[28].1712949 In80-3 From XUNDL12949 In80-39/2+ 0<28 >5.211/2+ 995.1213 5.413/2+ 1354.088 5.517/2− 1688.16<1 >6.31693.33<2 >6.02015.192 5.92085.05<1 >6.22551.21<1 >6.12561.21<1 >6.12588.46<2 >5.7(13/2−) 3150.3652 4.2 2155.66.41796.526.41462.312.21457.01.41135.26.81065.48.5599.22.8589.13.2561.88.51234.67.7895.01.4873.03.6863.13.8731.06.2391.71.41020.110.1339.16.5334.119.91354.220.5358.957.4995.193.712948 Cd810.011/2− 154 ms 2Q−=9330 SY %β−=100Iβ− Log f tIntensities: Relative IγDecay Scheme129Cd β− decay:T1/2:XUNDL-4 2015Ta1312949 In80Iγ > 10%×ImaxγIγ < 10%×ImaxγIγ < 2%×ImaxγLegend3Figure 1.6: Level scheme of 129In populated by the β -decays of the 11/2−state in 129Cd, based on Ref.[22]. The figure was taken from ENSDF inRef.[28].1812949 In80-4 From XUNDL12949 In80-49/2+ 0 611 ms 5(5/2−) 1219.846 5.61620.582 6.01761.75<4 >5.72060.193 5.72088.23<1 >6.22143.982216.953 5.72432.68<2 >5.82445.993 5.63286.1<2 >5.5(5/2+) 3347.7017 4.63487.8<2 >5.43702.013 5.23888.742 5.33913.92 5.33966.524 5.03970.11.5 5.44081.52.5 5.24100.12.8 5.14118.072 5.2 4118.31.12356.01.02880.22.82460.92.51524.11.53966.43.32204.80.43913.81.83888.40.72127.60.91744.40.83701.93.11940.30.43487.70.93348.02.01586.25.41287.34.61259.60.91203.50.91130.62.2915.00.93286.10.812948 Cd810+x3/2+ 146 ms 8Q−=9330 SY %β−=100Iβ− Log f tIntensities: Iγ per 100 parent decaysDecay Scheme129Cd β− decay:T1/2:XUNDL-5 2015Ta1312949 In80Iγ > 10%×ImaxγIγ < 10%×ImaxγIγ < 2%×ImaxγLegend4Figure 1.7: Part of the level scheme of 129In populated by the β -decays ofthe 3/2+ state in 129Cd, based on Ref.[22]. The figure was taken fromENSDF in Ref.[28].1912949 In80-5 From XUNDL12949 In80-59/2+ 0 611 ms 51/2− 450.72 1.23 s 3<18 >5.311/2+ 995.13(3/2−) 1082.717 5.6(5/2−) 1219.846 5.61554.82<2 >6.01620.582 6.01761.75<4 >5.72060.193 5.72088.23<1 >6.22143.982216.953 5.72432.68<2 >5.82445.993 5.6(5/2+) 3184.5826 4.4 3184.42.61422.911.01124.11.61096.04.41040.61.5967.64.0752.00.91363.60.71227.32.3891.00.92433.43.42216.84.91221.64.12143.40.92088.05.3840.27.0439.71.81761.610.7542.09.4537.91.8400.84.11555.20.8471.91.0769.228137.2[M1]2.5631.913.7995.14.112948 Cd810+x3/2+ 146 ms 8Q−=9330 SY %β−=100Iβ− Log f tIntensities: Iγ per 100 parent decaysDecay Scheme (continued)129Cd β− decay:T1/2:XUNDL-5 2015Ta1312949 In80Iγ > 10%×ImaxγIγ < 10%×ImaxγIγ < 2%×ImaxγLegend5Figure 1.8: The other part of the level scheme of 129In populated by the β -decays of the 3/2+ state in 129Cd, based on Ref.[22]. The figure wastaken from ENSDF in Ref.[28].20Chapter 2Decay SpectroscopyIn this chapter, the basic of decay spectroscopy and the relevant experimental tech-niques are discussed.2.1 Radioactive DecayA nuclear (many-body) system can go through different decay processes includingβ -decay, γ-decay, α-decay, spontaneous fission, and so on, depending on the num-ber of protons and neutrons. In this section, the decay processes relevant to thecurrent study, namely β -decay and γ-decay, are discussed.2.1.1 β -decayIn this section, the theory of β -decay is discussed based on Ref. [29].β -decay consists of the following three decay processes:AZXN → AZ+1XN−1+ e−+νe (β−-decay), (2.1)AZXN → AZ−1XN+1+ e++νe (β+-decay), (2.2)e−+ AZXN → AZ−1XN+1+νe (electron capture). (2.3)Here Z and N denote the number of protons and neutrons, respectively, and A =Z+N. These decays are driven by the weak interaction and they occur if the decayis energetically allowed. In the case of the β -decay of 12948Cd to12949In, this is β−-21decay.In oder to describe β -decays, let us first introduce the Pauli spin matrices andthe ladder operators:σ x ≡ σ 1 =(0 11 0), (2.4)σ y ≡ σ 2 =(0 −ii 0), (2.5)σ z ≡ σ 3 =(1 00 −1), (2.6)σ± =12(σ 1± iσ 2). (2.7)Similarly, we define the isospin matrices and the isospin ladder operators, whichare numerically identical to the Pauli spin matrices:τ x ≡ τ 1 =(0 11 0), (2.8)τ y ≡ τ 2 =(0 −ii 0), (2.9)τ z ≡ τ 3 =(1 00 −1), (2.10)τ± =12(τ 1± iτ 2). (2.11)Allowed nuclear β -decayThere are two types of allowed β -decays, classified by the coupling of the spinsof the leptons which are emitted from the β -decay. When the spins of the leptons(e− and νe for β−-decay, and e+ and νe for β+-decay) are coupled to total spin 0(singlet state), this is called Fermi decay. If the spins of the lepton are coupled tototal spin 1 (triplet state), this is known as Gamow-Teller (GT) decay. In allowednuclear β -decays, the leptons do not carry any orbital angular momentum and theconversion of a proton (neutron) into a neutron (proton) can be expressed using the22isospin ladder operators:τ− |n〉= |p〉 , (2.12)τ+ |n〉= 0, (2.13)τ− |p〉= 0, (2.14)τ+ |p〉= |n〉 . (2.15)These hold true since the u and d quarks can be represented as the following vectors[29]:|u〉=(01), (2.16)|d〉=(10). (2.17)Therefore, the operators for the two allowed β−-decay can be written as:A∑i=1τ−(i)≡ T − for Fermi decay, (2.18)A∑i=1−→σ (i)τ−(i)≡ Y − for Gamow-Teller decay, (2.19)where A denotes the mass number of the nucleus. The summation runs over all thenucleons. The operators for the β+-decay are obtained by replacing τ− with τ+,T − with T +, and Y − with Y +.Decay Rates for Allowed TransitionsIn order to calculate the decay rates for the allowed transitions, let us introducethe probability of transition Pf i from an initial state |i〉 to a final state | f 〉. Pf iis calculated via the scattering matrix S f i, defined in time-dependent perturbationtheory. The perturbation series for the time-evolution operator is defined as [30]:23Uˆ(t, t0) =∞∑n=01n!(−i)n∫ tt0dt1 · · ·∫ tt0dtnT (Hˆ1(t1) · · · Hˆ1(tn)), (2.20)where Hˆ1 is the perturbation part of the Hamiltonian of the system andT (Hˆ1(t1) · · · Hˆ1(tn)) is the time-ordered product of Hˆ1(ti1), Hˆ1(ti2), · · · , Hˆ1(tin). Withthis time-evolution operator, S f i is expressed asS f i = limt2→+∞limt1→−∞〈 f |Uˆ(t2, t1) |i〉 (2.21)=∑n(−i)nn!〈 f |∫ +∞−∞d4x1d4x2 · · ·d4xnT (Hˆ(x1),Hˆ(x2), · · · ,Hˆ(xn)) |i〉=δ f i− i〈 f |∫ +∞−∞d4xHˆ(x) |i〉− 12〈 f |∫ +∞−∞∫ +∞−∞d4x1d4x2T (Hˆ(x1),Hˆ(x2)) |i〉+ · · · , (2.22)where | f 〉 and |i〉 are the final state and the initial state, individually, and Hˆ is theHamiltonian density. Let |Ψ(t)〉 denote the time-dependent state vector and therelation between |Ψ(t)〉 and |i〉 islimt→−∞ |Ψ(t)〉= |i〉 . (2.23)With this S f i, the probability of transition Pf i is written asPf i = S∗f iS f i. (2.24)Since four-momentum is conserved for all processes,(2pi)4δ 4(∑p f −∑pi)≡ (2pi)3δ 3 (∑−→p f −∑−→p i)2piδ (E f −Ei), (2.25)where p f and pi denote energy-momentum four-vectors of the particles involved.By using this, if we define the T matrix asS f i = δ f i+(2pi)4δ 4(∑p f −∑pi)iTf i. (2.26)24If we compare this with the series expansion of the S matrix, in first order pertur-bation theory, we have(2pi)4δ 4(∑p f −∑pi)Tf i =−〈 f |∫d4xH(x) |i〉 (2.27)=−(2pi)4δ 4 (∑p f −∑pi)M f i.We see that in first order perturbation theory, the matrix element M f i of the Hamil-tonian operator in momentum space and Tf i differ only in their signs.Let V be the interaction volume and t be the duration of the interaction, weobtain the following result [29]:Pf i = (2pi)4δ 4(∑p f −∑pi)Vt|Tf i|2. (2.28)By dividing this by V and t, we obtain a transition rate dWf i/dt per particle in theinitial state:dWf idt= (2pi)4δ 4(∑p f −∑pi) |Tf i|2. (2.29)Since all appropriate final states consistent with four-momentum conservation haveto be taken into account and every particle of the final state contributes a phase-space factor d3 p/(2pi)3, by integrating over the momentum−→p , we obtain the decayratedWdt= (2pi)4∑f∫δ 4(∑p f −∑pi)∏fd3 p f(2pi)3|Tf i|2. (2.30)If the matrix element Tf i is independent of the kinematics, it may be removed fromthe integral and Eq. 2.30 becomesdWdt= ρ · |T |2 = ρ · |M|2, (2.31)where T and M are the spin-averaged matrix elements and ρ isρ = (2pi)4 ∑spins∫δ 4(∑p f −pi)∏fd3 p f(2pi)3. (2.32)The assumption that the T matrix is independent of the kinematics is more or lesssatisfied in nuclear β -decay [29]. This assumption leads to the allowed transitions25and we have ∣∣T ∣∣2 = G2β [BF( f )+BGT ( f )], (2.33)ρ =1(2pi)5∫d3 p f d3 ped3 pνδ 3(−→p f +−→pe+−→pν)δ (Ei−E f −Ekinf −Ee−Eν), (2.34)where Gβ is the interaction constantGβ = 1.008 ·10−5m−2p (mp is the mass of the proton), (2.35)and B±F and B±GT are the reduced transition probability for Fermi decay and Gamow-Teller decay, individually:B±F =∣∣〈N f ∣∣|T±|∣∣Ni〉∣∣22Ji+1, (2.36)B±GT =c2A∣∣〈N f ∣∣|Y±|∣∣Ni〉∣∣22Ji+1. (2.37)(The factor cA is the renormalization of the weak interaction in the GT decay)If we define the decay energy ∆ f = Ei−E f , neglect the kinetic energy of the finalnucleus Ekinf , and integrate over−→p f , the quantity ρ can be calculated and we obtain:ρ =∫dρ =1(2pi)3∫ ∆ fmepeEe(∆ f −Ee)2dEe, (2.38)dρ =1(2pi)3peEe(∆ f −Ee)2dEe. (2.39)Therefore, for the total decay rate dWf /dt to the final state f , we havedWfdt=G2β2pi3[BF( f )+BGT ( f )]∫ ∆ fmepeEe(∆ f −Ee)2dEe, (2.40)ordWfdtdEe=G2β2pi3peEe(∆ f −Ee)2[BF( f )+BGT ( f )]. (2.41)However, this calculation does not generate realistic results since we have ne-glected the Coulomb interaction between nuclei and electrons. In order to correct26this, the following correction factor is introduced:F(Z,Ee) = |ψ(0)with/ψ(0)without|2, (2.42)whereψ(0)with is the electron wave function evaluated at the position of the nucleustaking into account Coulomb interaction for an extended nucleus, and ψ(0)withoutis the corresponding value when the Coulomb interaction is not taken into account.This correction factor F(Z,E) is known as the Fermi function. For non-relativisticelectrons in the field of a pointlike nucleus, it has an analytic form:FNR(Z,E) =2piη1− e−2piη , (2.43)whereη =±Ze2vefor β∓-decay, (2.44)and ve is the velocity of the emitted electron (positron) at infinity.For heavy nuclei (large Z), F(Z,E) must be calculated by solving the rela-tivistic Dirac equation with the Coulomb potential for an extended nucleus. dρ ismodified to take into account the Fermi function and we have:dρ =12pi3F(Z,E)(∆ f −Ee)2 peEedEe. (2.45)Thus we obtain the decay rate of allowed transitions per electron-energy intervaldWdtdEe=G2β2pi3F(Z,E)peEe(∆ f −Ee)2[BF +BGT ], (2.46)and by integrating over the electron energy:dWdt=∫ ∆ fmEdWdtdEedEe =G2βm5e2pi3f [BF +BGT ], (2.47)wheref ≡ 1m5e∫ ∆ fmeF(Z,E)peEe(∆ f −Ee)2dEe. (2.48)This Fermi integral f is tabulated by Gove and Martin [31]. The relation between27the decay rate and the partial half-life t1/2 is given byt1/2 = [dW/dt]−1 ln2. (2.49)Therefore we havef t1/2 =2pi3 ln2G2βm5e1[BF +BGT ]≡ 4piD[BF +BGT ], (2.50)where t1/2 is the partial half-life for a transition to a given level E f of the daughternucleus. Eq.(2.50) is the so-called f t value. The total half-life T1/2 for allowed β -decay to the daughter nucleus is given by summing over all final states E f involvedin the β -decayT−11/2 =∑ f fBF(E f )+BGT (E f )4piD. (2.51)Forbidden TransitionsIn the decay of an extended object such as an atomic nucleus, transitions in whichone or both leptons carry orbital angular momentum can occur. Such transitionsare termed forbidden. The term forbidden expresses the fact that transitions withtransfer of orbital angular momentum have a strongly reduced decay rate. Thisreduction in the decay rate is due to the fact that a transition with lepton orbitalangular momentum l corresponds to the lth order in a multipole expansion of thelepton wave function with expansion parameter Rq, where R is the radius of thenucleus and q is the momentum transferred between the nucleus and the leptons[29].For unique forbidden transitions, where only a single multipole component andonly one transition operator contribute, there is a correspondence between fnt1/2and the reduced transition strength Bn:fnt1/2 =2pi3 ln2G2βm5eBn. (2.52)28Table 2.1: Selection Rules for β -DecaysFermi GTType L ∆J ∆J ∆piAllowed 0 0 (0),1 NoFirst Forbidden 1 (0),1 0,1,2 YesSecond Forbidden 2 (1),2 2,3 NoThird Forbidden 3 (2),3 3,4 YesTable 2.2: Type of β -Decays and Typical log f t Values [29]Type ∆J log f t (typical)Superallowed (Fermi + Gamow-Teller) 0 ≈ 3Allowed (Gamow-Teller) 0,1 (not 0+→ 0+) ≈ 4−6First Forbidden 0,1,2 ≈ 6−9Second Forbidden 2,3 ≈ 11−13Third Forbidden 3,4 ≈ 18Selection Rules and log f t ValuesFor allowed β -decays and forbidden β -decays discussed above, the selection rulesare shown in Table 2.1 and typical log f t values are shown in Table 2.2. The valuesare taken from Ref.[29].2.1.2 γ-decayMost decays from one nucleus to another as well as nuclear reactions leave thefinal nucleus in an excited state. These excited states typically decay rapidly tothe ground state through the emission of one or more γ rays, which are photons ofelectromagnetic radiation [32].29Reduced Transition probabilities for γ-decayThe interaction of the electromagnetic field with the nucleus can be expressedthrough the following operator [3]:O=∑λ ,µ[O(Eλ )µ +O(Mλ )µ], (2.53)where O(Eλ )µ and O(Mλ )µ are the electric and magnetic multipole operators,respectively, with tensor rank λ :O(Eλ ) = rλYλµ(rˆ)etze, (2.54)O(Mλ ) =[−→l2gltzλ +1+−→s gstz]−→∇[rλYλµ(rˆ)]µN , (2.55)where Yλµ are the spherical harmonics and etz are the electric charges for the protonand neutron in units of e. For the free-nucleon charge, ep = 1 and en = 0 for theproton and neutron, respectively.For a given E or M operator of rank λ , the electromagnetic transition rateWMi,M f ,µ is given byWMi,M f ,µ =(8pi(λ +1)λ [(2λ +1)!!]2)(k2λ+1h¯)∣∣〈J f M f ∣∣O(λ )µ ∣∣JiMi〉∣∣2 , (2.56)where k is the wave-number for the electromagnetic transition of energy Eγ givenby:k =Eγh¯c=Eγ197 MeV fm(2.57)By averaging over the Mi states and summing over M f and µ , the total rate for aspecific set of states and a given operator is obtained:Wi, f ,λ =1(2Ji+1)∑Mi,M f ,µWMi,M f ,µ (2.58)=(8pi(λ +1)λ [(2λ +1)!!]2)(k2λ+1h¯)|〈J f ∣∣|O(λ )|∣∣Ji〉 |2(2Ji+1). (2.59)30The last factor in Eq.2.59 is known as a reduced transition probability B:B(i→ f ) =∣∣〈J f ∣∣|O(λ )|∣∣Ji〉∣∣2(2Ji+1). (2.60)Weisskopf Units for γ DecayIn order to judge the relative strength of a transition, the reduced transition prob-ability is often given in Weisskopf units. The Weisskopf unit is an oversimplifiedestimate of the reduced transition probability for a single-particle and dependenceupon mass. By convention it is defined by [3]:BW (Eλ ) =(14pi)[3(3+λ )]2(1.2A1/3)2λ e2fm2λ , (2.61)BW (Mλ ) =(10pi)[3(3+λ )]2(1.2A1/3)2λ−2µN2fm2λ−2. (2.62)Angular Momentum and Parity Selection RulesAn electromagnetic transition between an initial nuclear state i and a final nuclearstate f can take place only if the emitted γ ray carries away an angular momentum−→l such that −→J f =−→Ji +−→l . (2.63)Therefore|Ji− J f | ≤ l ≤ JI + J f (where J = |−→J |). (2.64)Since the photon has an intrinsic spin of 1, l = 0 γ transitions are forbidden.The electromagnetic interaction conserves parity, and the elements of the op-erators for Eλ (Eq. 2.54) and Mλ (Eq. 2.55) can be classified according to theirtransformation under parity change [3]:POP−1 = piOO, (2.65)31wherepiO = (−1)λpiO =−1piO =+1for Yλ ,for vectors −→r ,−→∇ ,and −→p ,for pseudo vectors−→l =−→r ×−→p and −→σ .For a given matrix element〈Ψ f∣∣O∣∣Ψi〉= 〈Ψ f ∣∣P−1POP−1P∣∣Ψi〉= piipi fpiO 〈Ψ f ∣∣O∣∣Ψi〉 , (2.66)and the matrix element vanishes unless piipi fpiO =+1. Thereforepiipi f =+1 for M1,E2,M3,E4, · · · , (2.67)piipi f =−1 for E1,M2,E3,M4, · · · . (2.68)(2.69)2.2 Radiation DetectionIn this section, the principles of radiation detection are discussed. There are varioustypes of radiation in nature such as α-radiation, β -radiation, γ-radiation, and neu-trons. In this study, however, the crucial types of radiation are β−-particles emittedfollowing β−-decays and γ-rays emitted from the decay of the excited states of anucleus. Therefore, the focus on this section is the two types of radiation and theirdetection. The discussion in this section is based on Ref.[33].2.2.1 Radiation Interaction with MatterIn order to detect radiation, they have to interact with the matter, namely the ma-terial of the detectors. Different types of radiation interact with matter in differentways.Interaction of β -particlesβ -particles, namely electron and positrons, are charged particles and they interactwith matter through Coulomb forces. An incident electron can lose its energy byinteracting with the orbital electrons of atoms as well as with the positively charged32nucleus. In addition to this, a positron annihilates with one of the orbital electronsat the end of its track, resulting in an emission of two 511 keV γ-rays in oppositedirections.The specific energy loss due to ionization and excitation for fast electrons maybe described by the Bethe formula:−(dEdx)c=2pie4NZm0v2(lnm0v2E2I2(1−β 2) − (ln2)(2√1−β 2−1+β 2)+(1−β 2)+ 18(1−√1−β 2)2), (2.70)where v is the velocity of the primary particle, N and Z are the number densityand atomic number of the absorber atoms, m0 is the electron rest mass, e is theelectronic charge, and β ≡ v/c. The parameter I represents the average excitationand ionization potential of the absorber. The subscript c on the left hand sidedenotes that this is a collisional process. An electron can also lose its energy byradiative processes called bremsstrahlung, whose specific energy loss is−(dEdx)r=N ·E ·Z(Z+1)e4137m20c4(4ln2Em0c2− 43)(2.71)where the subscript r denotes that this is a radiative process. This happens when acharged particle decelerates (accelerates) and corresponds to the deflections of theelectron by interactions with the material of the detector.Interaction of γ-raysA γ-ray is electromagnetic radiation arising from the transitions between statesin atomic nuclei. In measurements of γ-ray, three types of interaction play animportant role: photoelectric absorption, Compton scattering, and pair production.They lead to the partial or complete transfer of the γ-ray photon energy to electronenergy.In the photoelectric absorption process, a photon interacts with an absorberatom and an energetic photoelectron is ejected by the atom from one of its bound33Chapter 2 Interaction of Gamma Rays 51all angles of scattering are possible, the energy transferred to the electron can vary fromzero to a large fraction of the gamma-ray energy.The expression that relates the energy transfer and the scattering angle for any giveninteraction can simply be derived by writing simultaneous equations for the conservationof energy and momentum. Using the symbols defined in the sketch belowon(energy = ks)we can show’ thathv’=hv (2.17)1+ 2(1—cos0)m0cwhere m0c2 is the rest-mass energy of the electron (0.511 MeV). For small scatteringangles 0, very little energy is transferred. Some of the original energy is always retained bythe incident photon, even in the extreme of 0 = 7t. Equations (10.3) through (10.6) describesome properties of the energy transfer for limiting cases. A plot of the scattered photonenergy predicted from Eq. (2.17) is also shown in Fig. 10.7.tThe probability of Compton scattering per atom of the absorber depends on the number of electrons available as scattering targets and therefore increases linearly with Z. Thedependence on gamma-ray energy is illustrated in Fig. 2.18 for the case of sodium iodideand generally falls off gradually with increasing energy.The angular distribution of scattered gamma rays is predicted by the Ktein—Nishinaformula for the differential scattering cross section dcr/d:do- 1 2 1 + cos2 0 a2(1 — cos 0)2= Zr(+ a(1 — cos 0)) ( 2 ) ( + i + cos2 0)[1 + a(1 — cos 0)1) (2.18)where a = hv/m0c2 and r0 is the classical electron radius. The distribution is shown graphically in Fig. 2.19 and illustrates the strong tendency for forward scattering at high valuesof the gamma-ray energy.3. PAIR PRODUCTIONIf the gamma-ray energy exceeds twice the rest-mass energy of an electron (1.02 MeV), theprocess of pair production is energetically possible. As a practical matter, the probability ofthis interaction remains very low until the gamma-ray energy approaches several MeV andtherefore pair production is predominantly confined to high-energy gamma rays. In theinteraction (which must take place in the coulomb field of a nucleus), the gamma-ray photon disappears and is replaced by an electron—positron pair. All the excess energy carriedin by the photon above the 1.02 MeV required to create the pair goes into kinetic energyshared by the positron and the electron. Because the positron will subsequently annihilatetThe simple analysis here neglects the atomic binding of the electron and assumes that the gamma-ray photoninteracts with a free electron. If the small binding energy is taken into account, the unique energy of the scatteredphoton at a fixed angle predicted by Eq. 2.17 is spread into a narrow distribution centered about that energy (seeFig. 13.9).Figure 2.1: Sketch of Compton scattering process. An incident γ-ray with theenergy of hν is scattered by an electron and deflected through an angleθ with respect to its original direction. θ can be any angle. The figurewas taken from Ref.[33].shells. The energy of the photoelectron Ee− is given byEe− = hν−Eb, (2.72)where Eb is the binding energy of the photoelectron in its initial shell. The p ot -electric process is the predominant mode of interaction in the lower energy regioncompared to the other two processes. The probability of the photoelectric interac-tion of the photon with the material of the detector is approximately proportionalto:τ ∝ZnE3.5γ, (2.73)where τ is the photoelectric mass attenuation coefficient, Z is the element numberof the material, and Eγ is the energy of the incident γ-ray. n varies between 4 and5 depending on the energy region of the γ-ray.Compton scattering is a scattering process between the incident γ-ray photonand an electron in the absorbing material. In this process, the incoming γ-rayphoton is deflected through the scattering with an electron and transfers a portionof its energy to the electron. The schematics of the process is shown in Fig. 2.1.34The relation between the energy transfer and scattering angle is given byhν−hν ′ = hν1− 11+hνm0c2(1− cosθ) , (2.74)where m0c2 is the rest-mass of an electron.If the energy of the γ-ray is larger than twice the rest-mass of an electron (511keV×2), the pair production process is energetically possible. In practice, the prob-ability of this process is very low until the energy of the γ-ray approaches severalMeV (see Fig. 2.2). Therefore this process is predominant in the high energy re-gion above 6 MeV. In this process, the γ-ray photon disappears and is replaced byan electron-positron pair. If the energy of the γ-ray is more than required to createa electron-positron pair, the excess energy goes into the kinetic energy of electronand positron. Since the positron annihilates in the material after slowing down, thesecondary products of this process are two annihilation photons with the energy of511 keV emitted in the opposite direction.The sum of the probabilities of occurrence of these three processes per unitpath length in the absorber is called the “linear attenuation coefficient”. Fig. 2.2shows the linear attenuation coefficient for a NaI crystal as a function of the γ-rayenergy.35Gamma-Ray Interactions with Matter 29a much lower fraction of incident gamma radiation than does a similar thicknessof aluminum or steel. The attenuation coefficient in Equation 2-1 is called the linearattenuation coefficient. Figure 2.3 shows the linear attenuation of solid sodium iodide,a common material used in gamma-ray detectors.Alpha and beta particles have a well-defined range or stopping distance; however,as Figure 2.2 shows, gamma rays do not have a unique range. The reciprocal of theattenuation coefficient 1/p/ has units of length and is often called the mean free path.The mean free’path is the average distance a gamma ray travels in the absorber beforeinteracting; it is also the absorber thickness that produces a transmission of l/e, or0.37.103 , ,,, : 1!, , 1E L@3ag 10-1zd Photoelectric zPair production10-2 I 1!! 11,11 1 1 1 1 11111 ()-2 10-1 1.0 101Photon Energy (MeV)Fig. 2.3 Linear attenuation coeflcient of NaI showingcontributions from photoelectric absorption,Common scatterirw, and ~air production..,. .Figure 2.2: Linear attenuation coefficient of NaI as a function of incident γ-ray energy. The contributions from photoelectric absorption, Comptonscattering, and pair production are also shown. The figure is taken fromRef.[34].36Chapter 3Experiment3.1 Isotope ProductionIn this section, a method of isotope production relevant to this study, which isthe Isotope Separation On-Line (ISOL) technique, is discussed. There are twomajor unstable isotope production methods, which are the in-flight fragmentationtechnique and the ISOL technique, and TRIUMF’s ISAC facility employs the ISOLtechnique. The in-flight fragmentation technique is used at facilities such as GSI inGermany, NSCL in the United States, and RIBF at RIKEN in Japan. On the otherhand, facilities such as ISOLDE at CERN and TRIUMF in Canada use the ISOLtechnique for isotope production.3.1.1 TRIUMF ISAC FacilityThe TRIUMF ISAC facility is one of the world class facilities for low energy nu-clear physics using radioactive beams. The primary beam for isotope production isthe 480 MeV proton beam accelerated by the world largest cyclotron. This protonbeam impinges on a production target, in this experiment a uranium carbide (UCx)target was used, and through secondary reactions including fission, spallation andfragmentation, various isotopes are produced. Different ionization methods can beused to extract and transport the isotope. The ionized isotopes are reacceleratedusing high-voltage electrodes and after going through the mass separator, the ra-dioactive beams are delivered to the experimental facilities. The schematics of the37Figure 3.1: The schematics of the TRIUMF ISAC facility. The GRIFFINspectrometer has replaced the 8pi spectrometer and is now located atthe neighbouring beam line (not shown in this picture). The figure wastaken from Ref. [35].ISAC facility is shown in Fig. 3.13.1.2 Ion Source: IG-LISIn general, the products of the reactions in the target which are caused by the pri-mary proton beam result in the production of various isotopes. However, most ofthe nuclear physics experiments require a pure beam which consists of an isotope ofinterest. The first step of the purification is done at the time of the extraction of iso-topes using an ion source. There are different types of ion sources available at theISAC facility, such as surface ion sources, the Force Electron Beam Induced ArcDischarge (FEBIAD) ion source, and the resonance-ionization laser ion-sources(TRILIS and IG-LIS). In the current study, Ion Guide - Laser Ion Source (IG-38Figure 3.2: The schematics of the Ion Guide - Laser Ion Source (IG-LIS). Seesection 3.1.2 for details. The figure was taken from Ref. [36].LIS) [36] was used for extraction and purification of the beam. In the IG-LIS,after the initial production and diffusion of isotopes from the target, the surface-ionized species are repelled with an electrostatic electrode. Only neutral atoms canpass the repeller electrode, and the isotope of interest is ionized by the element-selective multi-step laser excitation. The ionized isotopes are confined and trans-ported towards the high-voltage extraction field by a radio frequency quadrupole.The schematics of the system is shown in Fig. 3.2. The wavelengths of the laserexcitation are characteristic to the respective element. Therefore, the isotope ofinterest can be ionized with highly selectivity. The laser excitation scheme forcadmium is shown in Fig. 3.33.1.3 Radioactive BeamThe isotope of interest 129Cd was produced with the method described in Section3.1. The beam of 129Cd, which consists of both the ground state and the isomericstate was delivered to the GRIFFIN spectrometer with a intensity of∼120 pps. Thedata was collected approximately for 13 hours. For a few runs, the laser in IG-LIS(see Section 3.1.2) was blocked in order to facilitate the identification of the peaksin the γ-ray spectra that originate only from surface-ionized species such as In and390 cm-1 4d10    5s20 IPCd 72540 cm-1 excitation scheme cadmium I   online 12/2013 with  ig-LIS       J. Lassen 43692 cm-1  4d10   5s5p1 65135 cm-1  4d10   5s6d2 λ = 466.3 nm λ = 1064 nm 74533 cm-1 λ = 228.87 nm Aki = 5.3e8 Figure 3.3: Element-selective multi-step laser excitation scheme for cad-mium. Figure courtesy of J. Lassen.Cs. In the “laser on” spectra, also the peaks of from the decay of 129Cd becomevisible.3.2 Experimental SetupIn this experiment, the beam of 129Cd was implanted on the moving tape collector,which was surrounded by the β -tagger SCEPTAR. SCEPTAR was located at thecentre of the HPGe detectors array GRIFFIN. In order to suppress high-energeticbremsstrahlung, SCEPTAR was surrounded by a 20 mm thick Delrin absorber.This setup enables us to conduct a β -γ correlation analysis, which is discussed40further in Chapter 4.3.2.1 The GRIFFIN Spectrometer and Ancillary DetectorsThe Gamma-Ray Infrastructure For Fundamental Investigations of Nuclei (GRIF-FIN) Spectrometer is a high efficiency γ-ray spectrometer at the TRIUMF’s ISACfacility, newly commissioned in 2014. This spectrometer is primarily designed fordecay spectroscopy of unstable nuclei [37].3.2.2 HPGe Clover Detectors ArrayThe heart of the GRIFFIN spectrometer is the array of 16 high volume high-purityGe (HPGe) clover detectors. They cover almost all the 4pi solid angle and achievehigh γ-ray detection efficiency. Each clover detector is segmented into four crystalsof HPGe and its fine angular resolution is suitable for γ-γ angular correlation anal-ysis. These four n-type HPGe crystals are housed together in a cryostat, forming aclose-packed “clover” arrangement (Fig. 3.4).The average energy resolution at 122.0 keV and 1332.5 keV for all 64 crystal is1.12(6) keV and 1.89(6) keV, respectively [38]. The dominant interaction processfor γ-rays with the energy of a few hundreds keV up ∼7 MeV in germanium isCompton scattering (see Section 2.2.1). In a Compton scattering event, it is possi-ble that a γ-ray deposits part of its energy in several neighbouring crystals. Suchenergy depositions are time correlated, therefore the full γ-ray energy may be re-covered by summing these individual events. This procedure is called “add-back”and the clover detectors can be operated in the add-back mode in addition to thesingle crystal mode, where such summing is not done.In the most recent upgrade, the bismuth germanate (BGO) scintillator shieldswere installed to each HPGe detector in order to suppress Compton scattering sig-nals caused by the scattered γ-rays escaping from the detector.3.2.3 Ancillary DetectorsThe GRIFFIN array may be combined with various types of ancillary detectors fordifferent purpose, such as β -γ coincidence analysis, neutron detection, fast timing,and internal conversion electron detection. In this section, only those ancillary41signal into the front of the preamplifier as a direct alternative tocharge collection from the crystal.A SHV bulkhead connector is provided for each crystal to allowfor individual high voltage bias to be applied to each crystal.Operating bias voltages are specified by the manufacturer indivi-dually for each crystal in the Detector Specification Sheet [6]provided with each detector. The operating bias voltages havevalues of either þ3.5 or þ4 kV. During all measurements reportedhere the bias voltage applied to each crystal was the same as thatstated on the Detector Specification Sheet for that crystal. Eachclover is equipped with an internal temperature sensor andassociated circuitry which is used for high voltage bias shutdownin the instance of accidental warm up of the detector. Protectionfrom application of bias voltage during a thermal cycle to roomtemperature is essential as this can lead to electrical discharge thatcan damage the field-effect transistor.The four preamplifier chips and bias-shutdown alarm card arelocated on a common Printed-Circuit Board (PCB) and share acommon electrical ground. Preamplifier power at a voltage of712 V is provided by a 9-pin D-sub connector on the bulkhead.3. GRIFFIN clover detector propertiesThe performance of the GRIFFIN clover detectors was char-acterized at the Simon Fraser University Nuclear Science Labora-tories as part of the initial acceptance testing procedure and isdescribed in this section.3.1. Energy resolutionEnergy resolution measurements were performed by placing aseries of radioactive sources 25.00(5) cm from the center of theface of the clover detector.1 A 152Eu source was used for energyresolution measurements at 122.0 keV. A 60Co source was used forenergy resolution determination at 1332.5 keV. In all measure-ments the total counting rate of each crystal was less than 1 kHz.The preamplifier output signal was fed directly into an ORTECDSPEC jr 2.0 Multichannel Analyzer (MCA) via a 50Ω, 5 m coaxialcable. In separate measurements, a minimum of 105 counts in thebackground subtracted 122.0 keV and 1332.5 keV photopeakswere collected. The background subtraction was achieved throughthe automatic background subtraction routines in the Maestroapplication software [7]. Energy resolution was defined as the full-width at half-maximum (FWHM) of the photopeak, extractedusing gf3, a least square fitting program, from the RADWAREgamma-ray spectroscopy software package [8]. The full-width attenth-maximum (FW.1M) resolution was extracted from Maestro'speak fitting algorithm [7]. The FWHM of the photopeaks for allcrystals at 122.0 keV and 1332.5 keV were better than 1.26 keVand 2.02 keV, respectively, while the FW.1M was below 2.3 keVand 4.3 keV, respectively. The results for all crystals are shown inFigs. 3 and 4. The average energy resolution at 122.0 keV and1332.5 keV for all 64 crystals is 1.12(6) keV and 1.89(6) keV,respectively.3.2. EfficiencyAn activity-calibrated (72% at the 90% confidence level) 60Cosource was placed in the same source holder used in the energyresolution measurements. Data were collected for each crystalseparately without moving the source such that more than 105counts were accumulated in the 1332.5 keV background-subtracted photopeak. The total counting rate of each crystal waskept below 1 kHz during the data collection.The relative efficiency of each GRIFFIN crystal at 1332.5 keV at25.00(5) cm with respect to a 3 in:" 3 in: sodium iodide (NaI)scintillator [9] is presented in Fig. 5. The relative efficiency of allcrystals was greater than or equal to 37.9% at 25.00(5) cm source-to-detector distance.Fig. 6 summarizes the overall energy resolution and relativeefficiency performance at 1.3 MeV of the 64 crystals in the 16clovers that will form the GRIFFIN spectrometer. The performanceof all GRIFFIN detectors is excellent. The average values, indicatedby the square data point in Fig. 6, for energy resolution and rela-tive efficiency are 1.89(6) keV and 41(1)% at 1.33 MeV, respectively.3.3. Timing resolutionThe timing resolution of each HPGe crystal was measured withrespect to a barium fluoride (BaF2) scintillator for all gamma-rayinteractions above 100 keV using a 60Co source. Timing resolutionwas defined as the FWHM of the HPGe-BaF2 coincidence-timingpeak with a minimum of 103 counts. Standard NIM electronicsFig. 1. 3-D model of a GRIFFIN HPGe clover with the exterior dimensional toler-ances of the aluminum crystal housing indicated.Fig. 2. Block diagram showing the basic layout of the detector electronics.1 The detector manual states that the front of the crystals are 7 mm from thissurface.U. Rizwan et al. / Nuclear Instruments and Methods in Physics Research A 820 (2016) 126–131 127Figure 3.4: 3D model of a GRIFFIN HPGe clover. The figure was taken fromRef.[38]detectors that were relevant to the current study are explained.SCEPTARThe Scintillating Electron Positron Tagging ARray (SCEPTAR) is used to dete tβ -particles emitted from β -decay events. It was built as an ancillary detector forGRIFFIN’s predecessor, the 8pi spectrometer. The detector consists of 20 plasticscintillators and is located in the vacuum chamber at the centre of the GRIFFINHPGe detectors array (Fig.3.5). In order to suppress bremsstrahlung γ-rays, thedetectors are covered by Delrin absorber. The thickness of the absorber is 10mmor 20mm, depending on the energies of γ-rays of interest. For the 129Cd decayspectroscopy, a D lrin absorber of 20 mm thickness was used.42Table 2: Coordinates of the centre of each HPGe crystal inthe GRIFFIN array at source-to-detector distances of 11 and14.5 cm. The coordinate system is defined in Section 3.2.Array Crystal 11 cm 14.5 cmPosition Position ✓lab lab ✓lab lab(deg) (deg) (deg) (deg)1 1 36.5 83.4 37.9 80.12 55.1 79.0 53.2 77.23 55.1 56.0 53.2 57.84 36.5 51.6 37.9 54.92 1 36.5 173.4 37.9 170.12 55.1 169.0 53.2 167.23 55.1 146.0 53.2 147.84 36.5 141.6 37.9 144.93 1 36.5 263.4 37.9 260.12 55.1 259.0 53.2 257.23 55.1 236.0 53.2 237.84 36.5 231.6 37.9 234.94 1 36.5 353.4 37.9 350.12 55.1 349.0 53.2 347.23 55.1 326.0 53.2 327.84 36.5 321.6 37.9 324.95 1 80.6 32.0 82.3 30.32 99.4 32.0 97.7 30.33 99.4 13.0 97.7 14.74 80.6 13.0 82.3 14.76 1 80.6 77.0 82.3 75.32 99.4 77.0 97.7 75.33 99.4 58.0 97.7 59.74 80.6 58.0 82.3 59.77 1 80.6 122.0 82.3 120.32 99.4 122.0 97.7 120.33 99.4 103.0 97.7 104.74 80.6 103.0 82.3 104.78 1 80.6 167.0 82.3 165.32 99.4 167.0 97.7 165.33 99.4 148.0 97.7 149.74 80.6 148.0 82.3 149.79 1 80.6 212.0 82.3 210.32 99.4 212.0 97.7 210.33 99.4 193.0 97.7 194.74 80.6 193.0 82.3 194.710 1 80.6 257.0 82.3 255.32 99.4 257.0 97.7 255.33 99.4 238.0 97.7 239.74 80.6 238.0 82.3 239.711 1 80.6 302.0 82.3 300.32 99.4 302.0 97.7 300.33 99.4 283.0 97.7 284.74 80.6 283.0 82.3 284.712 1 80.6 347.0 82.3 345.32 99.4 347.0 97.7 345.33 99.4 328.0 97.7 329.74 80.6 328.0 82.3 329.713 1 124.9 79.0 126.8 77.22 143.5 83.4 142.1 80.13 143.5 51.6 142.1 54.94 124.9 56.0 126.8 57.814 1 124.9 169.0 126.8 167.22 143.5 173.4 142.1 170.13 143.5 141.6 142.1 144.94 124.9 146.0 126.8 147.815 1 124.9 259.0 126.8 257.22 143.5 263.4 142.1 260.13 143.5 231.6 142.1 234.94 124.9 236.0 126.8 237.816 1 124.9 349.0 126.8 347.22 143.5 353.4 142.1 350.13 143.5 321.6 142.1 324.94 124.9 326.0 126.8 327.8Figure 6: The upstream hemisphere of the SCEPTAR arrayof plastic scintillators for beta tagging and the east hemi-sphere of the GRIFFIN spectrometer.Figure 17) but the reduction in background is often526of more benefit in the study of isotopes populated527in a beta decay with a large Q value.5283.4. SCEPTAR529The primary ancillary detector system used in530the GRIFFIN facility is the Scintillating Electron531Positron Tagging ARray (SCEPTAR) consisting of53220 plastic scintillators located inside the vacuum533chamber [4]. These detectors subtend roughly 80%534of the solid angle and are used to tag on beta par-535ticles emitted in the decay of a parent nucleus.536The twenty trapezoidal paddles are arranged in537four pentagonal rings concentric with the beam538axis. Table 3 lists the coordinates of the cen-539tre of each SCEPTAR paddle with respect to the540beam axis. This geometry matches that of the 8⇡541spectrometer [4] and originally there was a one-to-542one correspondence between scintillator and HPGe543crystal for the purpose of a Bremsstrahlung veto544with the associated HPGe crystal. The two up-545stream rings are shown in Figure 6. Light produced546in the 1.5mm thick, BC404 trapezoidal-shaped547scintillator is collected by a 1.5mm thick Ultra-548Violet Transmitting (UVT) acrylic light guide that549is contoured and glued to a 1 cm diameter UVT550acrylic light guide rod. The light guide rod is inte-551grated with the vacuum chamber as an optical vac-552uum feedthrough. A ring of 13mm diameter Hama-553matsu H3165-10 Photo-Multiplier Tubes (PMTs)554sit around the outside of the beam pipe, coupled555to the light-guide rod on the atmospheric side of55610Figure 3.5: The upstream hemisphere of SCEPTAR and the east hemisphereof the GRIFFIN HPGe array without the BGO shields. The figure wastaken from Ref.[37].Moving Tape CollectorThe beam is implanted on the moving tape collector located at the centre of thedetector array. The tape is usually operated in a cycle mode (see Section 4.6.1 forthe cycle types in this experiment), which consists of background, implantation,and decay part. After a cycle the used part of the tape is transported behind a leadshield so that the following measurements are not affected by the decay products.3.2.4 Data Acquisition (DAQ) SystemThe GRIFFIN DAQ system consists of custom-designed hardware and firmwarewhich were designed to achieve high-rate data processing with high precision.Each HPGe crystal can operate at a count of rate up to 50 kHz and its reliability43is compatible for precision measurements such as Fermi super-allowed β -decayswhich requires a measurement of half-life and branching ratio to better than 0.05%[39]. The signals of the radiation interacting with the detectors are first amplifiedby the preamplifier provided for each crystal and then digitized by the GRIF-16module. The digitized signals are collected from each crystal by a GRIF-C mod-ule and after filtering, stored in the storage disk. The interface with the GRIFFINelectronics modules is based on the MIDAS (Maximum Integrated Data Acquisi-tion System) system developed at Paul Scherrer Institute (PSI), Switzerland andTRIUMF, which can write experimental data to disk efficiently in the MIDAS fileformat [40].44Chapter 4Data Analysis and ResultsIn this chapter, the data analysis procedure is explained in detail and the results arediscussed.4.1 Event Construction4.1.1 Data Sorting and Event ConstructionThe MIDAS file is sorted into a .root file using the analysis software packageGRSISort [41] based on ROOT. GRSISort produces two types of .root fileswhich are the FragmentTree and AnalysisTree. The FragmentTreeis a collection of time-ordered hits and it also contains basic information of thedetector system. On the other hand, AnalysisTree is a collection of hits sortedinto physics events. A physics event is defined as a group of hits within a certaincoincidence time window, and those hits are expected to originate from the samephysical process. The length of the time window is set to 2 µs by default but canbe modified if necessary.4.2 CalibrationThe first step of the analysis is various calibrations of the detectors. Through thisprocedure, the correct energy gain and efficiency of the HPGe clover detector arrayis obtained.454.2.1 HPGe GRIFFIN Array Energy CalibrationInitially the gain matching of the HPGe detectors was conducted crystal by crystalusing standard calibration sources such as 56Co, 60Co, 133Ba, and 152Eu. Howeverit is known that the energy gain of each crystal deviates as a function of time.Therefore, it is necessary to perform fine gain matching at reasonably short timeintervals. In this analysis the gain matching was done roughly on a day-by-daybasis using well-known γ-ray transitions [28] of 129Sn, which are present in theonline data.4.2.2 HPGe GRIFFIN Array Efficiency CalibrationIn order to determine γ-ray intensities, the energy dependence of the γ-ray detec-tion efficiency has to be determined. Unlike the energy gains, this does not deviateover time unless detector channels fail during the experiment, and therefore thesource runs at the beginning of the experiment were used for the efficiency calibra-tion. The γ-ray sources used for this calibration are listed in Table 4.1. In order toobtain the correct detection efficiency, several corrections have to be applied andthey are discussed below.Summing Effect CorrectionWhen multiple γ-rays are emitted from the same decay event, it is possible thatmore than one γ-ray deposits energy in the same crystal of the detector. If suchevent is detected as one hit, its energy is registered as the sum of the depositedenergy and it is impossible to reconstruct their original energies. The followingmethod is one of the ways to estimate such summing effects.It is known that the angular correlation of the γ-rays emitted in the same decaycascade is the same at 0◦ and 180◦ since the angular correlation W (θ) has thefollowing form:W (θ) = A0[1+a22P2(cosθ)+a44P4(cosθ)+ · · · ], (4.1)where θ is the angle between the two γ-rays, Pn(cosθ) is the nth order Legendrepolynomial, and A0, a22, a44, · · · are some constants that can be calculated. There-46Channel0 500 1000 1500 2000 2500 3000Channel0 500 1000 1500 2000 2500 3000Channel0 500 1000 1500 2000 2500 3000Channel0 500 1000 1500 2000 2500 300Energy [keV]050010001500200025003000Channel0 500 1000 1500 2000 2500 300Energy [keV]05001000150020002500300Channel0 500 1000 1500 2000 2500 300Energy [keV]05001000150020002500300Energy [keV]05001000150020002500300HPGe Crystal 0 HPGe Crystal 1HPGe Crystal 2 HPGe Crystal 3HPGe Crystal 4 HPGe Crystal 5HPGe Crystal 6 HPGe Crystal 7Bin No.Calibrated Energy [keV]E = 1.3056(3) Bin + 0.9(3)χ 2 / NDF = 3.4 / 9E = 1.3404(4) Bin − 1.1(3)χ 2 / NDF = 3.4 / 9E = 1.4402(3) Bin − 0.1(2)χ 2 / NDF = 2.7 / 9E = 1.1887(3) Bin − 0.1(3)χ 2 / NDF = 4.6 / 9E = 1.2863(3) Bin − 0.1(2)χ 2 / NDF = 2.9 / 9E = 1.1683(2) Bin − 0.5(2)χ 2 / NDF = 2.6 / 9E = 1.2624(3) Bin − 0.03(30)χ 2 / NDF = 3.7 / 9E = 1.1366(3) Bin − 0.1(2)χ 2 / NDF = 3.0 / 90500100015002000250005001000150020002500050010001500200025000500100015002000250025002000150010005000 25002000150010005000Figure 4.1: Example of the gain matching of HPGe crystals with calibrationsources 152Eu and 56Co. Only the first 8 crystals are shown for example.The x-axis shows the bin number of the uncalibrated energy histogramand the y-axis shows the calibrated energy. The “+” marker of eachpoint in the plot does not correspond to the size of the error.47Old Energy [keV]0 500 1000 1500 2000 2500 3000Old Energy [keV]0 500 1000 1500 2000 2500 3000Old Energy [keV]0 500 1000 1500 2000 2500 3000Old Energy [keV]Old Energy [keV]0 500 1000 1500 2000 2500 300Energy [keV]050010001500200025003000Old Energy [keV]0 500 1000 1500 2000 2500 300Energy [keV]05001000150020002500300Old Energy [keV]0 500 1000 1500 2000 2500 300Energy [keV]05001000150020002500300Energy [keV]05001000150020002500300HPGe Crystal 7HPGe Crystal 5HPGe Crystal 3HPGe Crystal 1HPGe Crystal 0HPGe Crystal 2HPGe Crystal 4HPGe Crystal 6Calibrated Energy [keV]E' = 1.001(3) E − 0.1(6)χ 2 / NDF = 1.2 / 3E' = 0.9984(4) E + 1.8(7)χ 2 / NDF = 1.7 / 3E' = 1.0003(1) E − 0.7(3)χ 2 / NDF = 0.3 / 3E' = 1.0044(3) E − 1.0(6)χ 2 / NDF = 1.4 / 3E' = 1.0003(4) E − 0.3(8)χ 2 / NDF = 2.3 / 3E' = 1.0001(2) E − 0.05(6)χ 2 / NDF = 0.7 / 3E' = 1.001(3) E − 0.1(6)χ 2 / NDF = 1.2 / 3E' = 1.0001(4) E − 0.3(7)χ 2 / NDF = 1.8 / 325002000150010005000 2500200015001000500005001000150020002500050010001500200025000500100015002000250005001000150020002500Figure 4.2: Example of the gain matching of HPGe crystals with knownpeaks from the decay of 129Sn in the experimental data. Only the first 8crystals are shown for example. The x-axis shows the energy with thesource calibration (E) and the y-axis shows the calibrated energy (E ′).The “+” marker of each point in the plot does not correspond to the sizeof the error. From the linear fitting functions shown in the figure, it canbe observed that some of the offsets of the energy gain have non-zerodeviation from the original energy calibration.48Table 4.1: Isotopes and transitions used for γ-ray efficiency calibration. Theintensities were taken from Ref.[28].Source Energy [keV] Intensity per 100 Decays [%]152Eu121.7 28.5(5)244.6 7.55(4)344.2 26.5(2)411.1 2.23(1)778.9 12.93(8)867.3 4.23(3)964.0 14.51(7)1085.8 10.11(5)1112.0 13.67(8)1408.0 20.87(9)133Ba53.1 2.14(3)80.9 32.9(3)276.3 7.16(5)302.8 18.3(1)383.8 8.94(6)56Co846.7 99.9399(1)1037.8 14.05(4)1238.2 66.4(1)1360.2 4.28(1)1771.2 15.41(6)2015.2 3.01(1)2034.7 16.97(4)2598.5 1.03(1)3009.6 3.20(1)3253.5 7.92(2)3203.7 1.875(2)fore, the probability of the two γ-rays entering the same crystal of the detectorshould also be the same as those γ-rays entering the crystals located at 180◦ withrespect to the decay point. The GRIFFIN array has 4× 16 = 64 crystals and thenumber of crystal pairs separated by 180◦ is also 64. By measuring the number ofcounts in coincidence with each γ-ray listed in Table 4.1 for each 180◦ crystal pair(with removal of time random coincidences), the summing effect can be estimated.49HPGe Deadtime CorrectionEvery time a detector detects a hit, the corresponding channel becomes inactivedue to the signal processing and it does not accept another hit for a certain amountof time. In the latest DAQ system, the length of the deadtime is programmed to be1.2 µs so that the deadtime can be uniform for any energy of γ-rays. However, thiswas not yet implemented at the time of this experiment. Therefore, the deadtimewas obtained by plotting the time interval of two consecutive γ-rays for each crystal(Fig. 4.3) and it was measured to be 7.5µs. In addition to this, it is possible that hitsget rejected for other reasons such as the memory buffer on the GRIF-16 modulebeing full. Such deadtime period is recorded in the 14-bit deadtime word of eachevent. Furthermore, even though hits are processed the data can be lost due tonetwork limitations, for example. The sum of all these deadtime and data loss isthe deadtime that has to be taken into account.Efficiency CurveOnce the corrections are applied, the γ-ray detection efficiency is obtained with thefollowing formula.Efficiency =Nγt ·A · Iγ , (4.2)where Nγ is the number of counts of each γ-ray above background with the sum-ming effect corrected, t is the livetime of the detector (Livetime = Runtime−Deadtime), A is the activity of the calibration source at the time of the efficiencymeasurement, and Iγ is the intensity of the γ-ray per 100 decays.The fitting function for the γ detection efficiency is [42]ln(εγ) =c0+ c1 ln(Eγ)+ c2(ln(Eγ))2+ c3(ln(Eγ))3+ (4.3)c4(ln(Eγ))4+ c5(ln(Eγ))5, (4.4)where εγ is the intrinsic γ-detection efficiency, Eγ is the energy of each γ-ray usedfor the efficiency calibration, and cn(n = 1, · · · ,5) are the fitting parameters. TheGRIFFIN array is equipped with 64 HPGe crystals (16 clover detectors). However,the experimental data was collected with one or two crystals disabled due to thetechnical issues during the run, and the efficiency curve was scaled to 62 and 63500 2 4 6 8 10 120102030405060110210310410Time Difference [μs]GRIFFIN HPGe ChannelFigure 4.3: Time difference of two consecutive hits in the same crystal. Thex-axis is the time difference and the y-axis corresponds to each crystal.Large statistics around 0 on the x-axis is assumed to be pile-up eventsdue to the strong source. Except for the channel 31, the minimum timedifference is 7.5 µs.crystals. The resulting efficiency curves are shown in Fig.4.4.4.2.3 SCEPTAR Energy CalibrationSCEPTAR does not have a sufficient energy resolution to conduct electron spec-troscopy, however, it works as a β -tagger and is important to discriminate β -particles following β -decays from background electrons. In order to achieve this,the energy gains of each channel of SCEPTAR were matched so that the energythreshold is the same for each scintillator (Fig. 4.5) and then the threshold was setat the lowest edge of the distribution of the β -particles correlated to the β -decays(Fig. 4.6).510 500 1000 1500 2000 2500 3000 3500 4000510152025Efficiency [%]Energy [keV]: 63 crystals: 62 crystalsFigure 4.4: GRIFFIN HPGe array γ-ray detection efficiency curves. The blueand red lines show the efficiency curves for 62 HPGe crystals and 63HPGe crystals, respectively.4.3 Coincidence AnalysisOne of the main purposes of the decay spectroscopy is to construct the level schemeof the isotope of interest. Unless isomeric states with sufficiently longer half-livesthan the time resolution of the detector (∼10 ns for HPGe detectors) exist, allthe γ-rays in a cascade are assumed to be emitted promptly and simultaneously,following the β -decay. Therefore, by observing which γ-rays are detected at thesame time, i.e. in coincidence, information on the level structure can be obtained.4.3.1 Timing GatesIn order to define “coincidence”, proper timing gates have to be determined. Thiscan be achieved by constructing a histogram that shows the time difference of twoconsecutive hits (Fig.4.7). γ-rays within this timing window of 1.8 µs around t = 0520 2 4 6 8 10 12 14 16020004000600080001000012000110210310410SCEPTAR Channel No.β Energy [arb. Unit]Figure 4.5: The x-axis shows the channel number of SCEPTAR and the y-axis shows the gain-matched energy in arbitrary unit. Since the datacollected by SCEPTAR is only used for β -tagging of the γ-rays, theenergy of the β -particles has to be gain-matched but does not have to beaccurate.are expected to be correlated. In 129In there is one µs-isomer known so far: thelifetime of the 17/2− isomer at 1688 keV was measured to be 10.7(1) µs (seeSection 4.4.3). In order to remove the time random coincidence events, γ-rays thatare outside of the time window are scaled and subtracted from the events inside thetime window.A similar timing gate is applied to the β -γ correlation and this can be usefulto discriminate the γ-rays originate from β -decay events from time-random back-ground γ-rays. As opposed to the γ-γ time difference, the β -γ time difference isnot symmetric around x = 0 as shown in Fig.4.8. This is because γ-rays that arecorrelated to β -particles are emitted following β -decays. By imposing such tim-ing constraint on β -γ timing, a strong suppression of background γ-rays can be530 1000 2000 3000 4000 5000 6000 7000 8000020406080100120140310×β Energy [arb. Unit]Countsβ Energy ThresholdFigure 4.6: The figure shows the projection of Fig.4.5 on to the energy axis.The dashed line shows the threshold which distinguishes the β -particlesfollowing the β -decays from the background electrons.achieved.4.3.2 γ-γ Coincidence MatrixWith proper timing gates as discussed before, one can construct matrices (2D-histograms) that show γ-rays in coincidence following β -decays. Such matricesare constructed by plotting the energy of one γ-ray on one axis and the energy ofother γ-rays on the other axis. γ-rays that are in coincidence can be observed as apeak along the z-axis on the matrix. The diagonal lines are scattered events, wherea γ-ray does not deposit all of its energy in one crystal and travels into an othercrystal. The matrices are symmetric with respect to the y= x line but built in a waythat there is no self-coincidence or double-counting.Typical usage of the matrices is to project a part of the matrix which corre-sponds to a certain γ-ray energy onto one axis with Compton background subtrac-54500− 400− 300− 200− 100− 0 100 200 300 400 500110210310410510610710Time Difference [10ns]CountPrompt Timing GateTimeRandomTimeRandomFigure 4.7: Timing difference between γ-ray hits within the same event. Theγ-γ coincidence timing gate and background (time random) timing gateare shown in the figure.tion, which are represented by the neighbouring part of the matrix and scaled tothe background component of the gate region. See Fig. 4.10 for an example. Inaddition to this, ROOT offers a library TSpectrum2which can detect coincidencepeaks directly from the 2D histogram without being affected by the time-randomcoincidences (Fig.4.11). However, such coincidences still need to be verified byother means.4.4 Level Scheme of 129InUsing the coincidence matrices explained in the section 4.3.2, as well as the β -gated single γ-ray spectrum with a subtraction of 129Sn transitions, which is ex-plained in detail in the following Section 4.5 and shown in Fig. 4.12, 93 transitionsincluding 29 new transitions and 5 new excited states were identified and placed55500− 400− 300− 200− 100− 0 100 200 300 400 500110210310410510610β-γTiming Gateβ-γ Time Difference [10ns]Count / 10nsFigure 4.8: Timing difference between β and γ-ray hits within the sameevent. The β -γ timing gate is defined as shown in the figure.in the level scheme of 129In, which is shown in Fig.4.13. For all of the analysisexcept for the 17/2− isomeric state, a 1.8 µs coincidence time window was used(Fig.4.7).4.4.1 11/2+, 13/2+, and 17/2− statesThe decay scheme of the 17/2− isomer at 1688 keV was established by Geneveyet al.[24] and is also confirmed in this experiment by coincidence analysis. There-fore the four γ-ray transitions following the decays of 17/2− isomer were used asa starting point of constructing the level scheme. This establishes excited states of11/2+ at 995 keV and 13/2+ at 1354 keV.By gating on γ-ray transitions, starting with the 994.8 keV transition from the11/2+ state to the 9/2+ ground state as well as γ-rays in coincidence with the994.8 keV transition, excited states at 1693, 2015, 2085, 2589 and 3151 keV are56Figure 4.9: β -gated γ-γ coincidence matrix. The axes of the original figureextends up to 6 MeV but in this figure it only shows up to 2 MeV forvisibility of the vertical, horizontal, and diagonal lines.established. This result agrees with the analysis by Taprogge et al.[22] and thedetails on how the levels are established is discussed in the literature.As for the excited states at 2277 and 2551 keV, although the 598.9 and 862.8keV, and the 588.7 and 872.9 keV transitions were observed in coincidence in thestudy by Taprogge et al.[22], the order of those transitions could not be determinedand the excited states were proposed at 2551 and 2561 keV tentatively. In the cur-rent study, the 922.4 keV transition in coincidence with the 872.9 keV transitionwas newly observed and the sum of the energies of the 334.0 and 588.7 keV tran-sitions is 922.4 keV. Therefore, this suggests a new excited state at 2277 keV. Atransition with 274.7 keV in coincidence with the 598.9 and 588.7 keV transitionsalso supports the placement of the new excited state.57Figure 4.10: This shows a example of a projection of the 994.8 keV transi-tion on the β -gated γ-γ coincidence matrix. The region enclosed bythe dashed line represents the energy gate on the 994.8 keV transi-tion and the region enclosed by the dotted line represents the Comptonbackground which is to be scaled and subtracted from the projection.A number of transitions in coincidence with the 994.8 keV transition such asthe 1515.5, 1546.0, 1610.4, 2155.5, 2918.6, and 2971.5 keV transitions are foundto be feeding the 11/2+ state at 995 keV. Correspondingly, ground state γ-ray tran-sitions with 2510.5, 2541.5, 2605.8, 3150.5, 3913.9 and 3966.2 keV are observedand this establishes three new excited states at 2510, 2541 and 2606 keV.4.4.2 Excited States Feeding the 1/2− Isomeric StateExcited states which have decay branches feeding the 1/2− isomeric state are con-firmed to agree with the analysis by Taprogge et al.[22], while one new excitedstate is established and a few transitions were in disagreement.The transition that connects the 3967 keV state and the 1762 keV state were re-ported in Ref.[22], however, we do not see any evidence for the transition. Also, the58900 920 940 960 980 1000 1020 1040 1060 1080 11003003203403603804004204404604805002−101−10110210310410γ1 Energy [keV]γ 2 Energy [keV]Figure 4.11: β -gated γ-γ coincidence matrix processed using TSpectrum2class. The background is smoothed and the peaks are deconvo-luted. The red triangles show the coincidence peaks detected by theTSpectrum2 algorithm.ground state transition with 1555.2 keV γ-ray which had been previously reported[22] was not placed in the level scheme in this analysis. Although a transition witha 1555.4 keV γ-ray was identified as one of the transitions in 129In, due to the lackof information on transitions in coincidence, the position of the transition could notbe determined.A new excited state was established at 2135 keV based on the 579.9, 1051.5,1684.2 keV γ-rays in coincidence with the 1835.5 keV transition. Due to the factthat the 1051.5 keV transition is in coincidence with the 631.7 keV γ-ray and notwith the 137.1 nor 768.8 keV γ-rays, this transition is placed at the top of the 631.7keV transition.5940005000700015600030002000100000 200 400 600 800 1000  600  80012001000  400  20001000 1200 1400 1600 1800 2000  600  80012001000  400  2000  100  80  60  40  20  03530252010502000 2200 2400 2600 2800 30003000 3200 3400 3600 3800 40004000 4200 4400 4600 4800 5000Energy [keV]Count / keV136.8298.5326.6333.7338.7358.8392.0 400.5439.7471.7 504.5537.4541.6561.5588.7598.9631.9730.4768.8840.2862.8872.9922.4967.0994.81020.01040.61065.01095.91103.71123.91130.61134.91170.11221.51226.81234.51287.31354.21396.51423.01458.01462.61500.41524.31555.41561.11586.41610.4 1690.01684.01761.51796.61835.81889.91940.21964.42001.02088.02127.72143.52155.52216.22267.8 2295.82352.2DE2357.12388.02415.72432.12460.72498.82510.3 2521.22527.02541.12605.7 2619.9SE DE2669.42880.72919.0DE 2999.63024.1DE 3150.93184.6SE3285.8 3347.83385.1SE SE SE3487.53701.83888.63913.93966.23977.34118.6Figure 4.12: β -gated single γ-ray spectrum with 129Sn transitions subtracted(for details see Section 4.5).60451137.1108312201/2- 0x = 451.017623185β-decaying isomeric state  T1/2  = 1.23 s    2088206033481620370238894119221715552446408224322144995135416932085201525893151168825102541260639149/2+11/2+ 13/2+ Isomeric State
T1/2  = 10.7(1) μs       17/2-400.5537.7471.7840.2439.7298.51761.52088.02432.23184.61040.61203.73348.72143.52216.81363.81635.52510.32541.12605.73150.53913.9338.81020.1730.5392.01234.5895.1504.91134.91065.5561.51458.01226.8227725513285.832863487.53488994.81354.2358.9334.01796.6541.71103.7768.8631.74118.63888.63701.8890.939711524.32861.72460.7 (13/2- )2135579.9326.81051.81684.0872.9598.91462.6588.7922.4862.83966.239661743.8(5/2- )(3/2- ) (5/2+) (5/2+)2352.21940.32127.72357.12267.81835.52415.72669.42619.92059.62999.62527.01221.5861.7274.71423.01123.91095.9752.6967.01586.41287.41259.4914.71130.61964.41515.51546.11610.42155.52919.02971.7Figure 4.13: Level Scheme of 129In. Red arrows and lines indicate newlyobserved transitions and excited states. Blue lines indicate isomericstates.614.4.3 Half-Life of the 17/2− Isomeric StateThe presence of the 17/2− isomer was first reported by Genevey et al. to be 8.5(5)µs [24] and adopted in ENSDF [28]. The half-life has been measured in severalother studies to be 11(1) µs [43, 44], 2.2(3) µs [45], and 11.2(2) µs [46]. AlthoughRef. [43, 44] and Ref. [46] reported consistent values, it is worthwhile to inves-tigate the half-life in this study. In order to determine the half-life of the 17/2−isomer, the number of counts of β -gated γ-rays are used. The γ-rays of interestare the 334.0, 358.9, 994.8, and 1354.2 keV transitions following the decay of the17/2− isomer. The decay of these four γ-rays are visible in Fig. 4.14. The half-lifeof this isomer is expected to be around 10 µs, therefore the time window of theevent construction was set to be 50 µs (see Section 4.1.1 for the event constructionmethod). Fig. 4.15 shows histograms of the number of counts of the four γ-raysfollowing the β -decays. They are plotted by gating on each energy of the γ-rayswith time-random background subtraction on the 2D histogram that shows the β -γtime difference.By taking the weighted average of the half-lives obtained from the four γ-raysshown in Fig. 4.15, the half-life of the 17/2− isomeric state is determined to be10.7(1) µs. This value is consistent with the half-life reported in Ref. [43, 44],however, it is not consistent with the half-lives reported in Ref. [24, 45, 46] withintheir given uncertainties.4.5 Determination of Relative γ-ray IntensitiesIntensities of γ-rays relative to the 994.8 keV transition were determined for 115transitions, with 22 transitions not placed in the level scheme. For most of thetransitions, the β -gated single γ-ray spectrum with the subtraction of the granddaughter isotope (129Sn) transitions was used to determine the intensities. Thedetail of the subtraction is discussed in Section 4.5. For the transitions that are closeto each other in energy or too weak to fit in the single γ-ray spectrum, projectionsof the coincidence matrix were used to determine the intensities.For the 137.1 keV transition, the intensity was corrected for the internal con-version using the conversion coefficient calculator BrIcc v2.3S [47] assuming anM1 transition. The calculated conversion coefficient was αtot = 0.211(3), which62Figure 4.14: Difference in timestamp of β -particles and γ-rays within the 50µs event construction time window. The y-axis goes only up to 15 µsfor visibility. The lines extending along the y-axis indicate the exis-tence of isomers. The unlabelled vertical lines in the figure are due toisomers of the daughter nucleus 129Sn. The prominent line at y = 0 isdue to prompt γ-rays and Compton background.contributes to the γ-ray intensity with 1.19(1) %.Subtraction of the Grand-daughter Isotope 129Sn TransitionsAs explained in the section 3.2.3, the beam is implanted on the moving tape collec-tor in a cycle mode, which consists of background, implantation, and decay parts.Since the half-life of 129Cd (T1/2 ∼ 150 ms) is a few times smaller than that of129In (T1/2 ∼ 600 ms for the ground state and T1/2 ∼ 1 s for the 1/2− β -decayingisomeric state), it is expected that the most of the decays that happen in the decaypart of the beam implantation are dominated by the decay of 129In. This can alsobe seen in the fit of the cycle structure shown in Fig.4.18. Therefore, if the decaypart of the cycle is subtracted from the implantation part of the cycle with an ap-63105χ2/ndf = 230.1/245T1/2 = 10.8(1) μsEγ = 334.0 keVχ2/ndf = 275.9/245T1/2 = 10.5(1) μsEγ = 358.9 keVχ2/ndf = 117.2/96T1/2 = 10.3(4) μsEγ = 1354.2 keVχ2/ndf = 261.3/245T1/2 = 10.8(2) μsEγ = 994.8 keV104103102101041031021010 10 20 30 40 500 10 20 30 40 501010210310410510102103104Counts / 200 nsCounts / 200 nsCounts / 200 nsCounts / 500 nsβ-γ Time Difference [μs] β-γ Time Difference [μs]Figure 4.15: The half-life of the 17/2− isomer fitted to the time difference ofthe timestamps of β -particles and γ-rays gated on 334.0, 358.9, 994.8,and 1354.2 keV transitions.propriate amount of scaling, the contribution from the transitions in 129Sn can beremoved from the γ-ray spectrum.641400 1500 1600 1700 1800 1900 2000 2100 22000200400600800100012001400Energy [keV]Counts [arb. unit]129Sn TransitionsFigure 4.16: Comparison between a raw β -gated single γ-ray spectrum (blue)and the 129Sn transitions subtracted β -gated single γ-ray spectrum(red). The raw spectrum (blue) is scaled down to the 129Sn transitionssubtracted spectrum (red) for comparison.Table 4.2: Relative γ-ray Intensities of 129In. The star symbols denote tran-sitions that are identified as 129In but could not be placed in the levelscheme. The Ilitγ values were taken from Ref. [22].Eγ [keV] Iγ [%] Ilitγ [%] Ei [keV] E f [keV] Jpii Jpif136.8a 5.8(4) 4.7(6) 1220 1083 (5/2−) (3/2−)273.9 0.55(9) 2551 2277298.5 1.1(1) 2060 1762326.6 3.2(3) 2088 1762Continued on next page65Table 4.2 – Continued from previous pageEγ [keV] Iγ [%] Ilitγ [%] Ei [keV] E f [keV] Jpii Jpif333.7 17.2(8) 19.9(12) 1688 1354 17/2− 13/2+338.7 5.4(5) 6.5(7) 1693 1354 13/2+358.8 59.3(15) 57.4(30) 1354 995 13/2+ 11/2+392.0 1.5(2) 1.4(7) 2085 1693400.5 7.2(4) 6.3(6) 1620 1220 (5/2−)439.7 4.8(4) 2.7(3) 2060 1620471.7 4.7(3) 1.5(8) 1555 1083 (3/2−)504.5 1.6(2) 2.1(2) 2589 2085537.4 4.5(3) 2.8(8) 1620 1083 (3/2−)541.6 16.1(8) 14.5(10) 1762 1220 (5/2−)561.5 9.1(7) 8.5(8) 3151 2589 (13/2−)588.7 4.4(3) 3.2(4) 2277 1688 17/2−598.9 4.6(3) 2.8(4) 3151 2551 (13/2−)631.9 28.0(14) 21.0(12) 1083 451 (3/2−) 1/2−730.4 7.5(6) 6.2(7) 2085 1354 13/2+752.6 0.53(9) 1.4(12) 3185 2432768.8 38.5(20) 43.8(40) 1220 451 (5/2−) 1/2−840.2 5.4(3) 10.8(9) 2060 1220 5/2−861.7 0.6(1) 2217 1354 13/2+862.8 4.2(3) 3.8(4) 2551 1688 17/2−872.9 3.6(3) 3.6(5) 3151 2277 (13/2−)890.9 0.65(5) 1.4(12) 2447 1555914.7 0.6(1) 1.4(12) 3348 2432 (5/2+)922.4 2.8(2) 2277 1354 13/2+967.0 4.9(3) 6.2(7) 3185 2217 (5/2+)994.8 100.0(40) 100.0(51) 995 0 11/2+ 9/2+1020.0 10.3(6) 10.1(8) 2015 995 9/2+1040.6* 2.3(2)1040.8 0.6(1) 2.3(5) 3185 2144 (5/2+)Continued on next page66Table 4.2 – Continued from previous pageEγ [keV] Iγ [%] Ilitγ [%] Ei [keV] E f [keV] Jpii Jpif1051.8 0.6(2) 2135 1083 (3/2−)1065.0 7.3(4) 8.5(7) 3151 2085 (13/2−)1095.9 2.6(2) 6.7(7) 3185 2088 (5/2+)1103.7 2.9(2) 2.5(13) 1555 451 1/2−1123.9 2.0(2) 2.5(13) 3185 2060 (5/2+)1130.6 2.4(3) 3.4(9) 3348 2217 (5/2+)1134.9 6.0(4) 6.8(7) 3151 2015 (13/2−)1170.1* 1.1(1)1203.7 0.2(1) 1.4(12) 3185 2144 (5/2+)1221.5 2.5(2) 6.3(9) 2217 995 11/2+1226.8 1.4(2) 3.6(27) 2447 1220 (3/2−)1234.5 5.8(4) 7.7(11) 2589 1354 13/2+1259.7 0.7(1) 1.4(5) 3348 2088 (5/2+)1273.9* 0.8(1)1287.3 6.4(4) 7.0(7) 3348 2060 (5/2+)1354.2 17.0(6) 20.5(12) 1354 0 13/2+ 9/2+1363.8 1.3(2) 1.0(5) 2447 10831387.0* 0.8(1)1396.5* 3.2(4) 3.0(15)1423.0 17.4(10) 16.9(10) 3185 1762 (5/2+)1458.0 1.5(2) 1.4(1.2) 3151 1693 (13/2−)1462.6 8.6(5) 12.2(12) 3151 1688 (13/2−) 17/2−1500.4* 3.7(3) 4.4(5)1515.5 1.0(1) 2510 995 11/2+1524.3 2.6(2) 2.3(4) 3971 24471538.0* 0.4(1)1546.1 0.6(1) 2541 995 11/2+1555.4* 2.7(3) 1.3(4)1561.1* 5.3(4) 2.7(11)Continued on next page67Table 4.2 – Continued from previous pageEγ [keV] Iγ [%] Ilitγ [%] Ei [keV] E f [keV] Jpii Jpif1586.4 7.2(5) 8.3(3) 3348 1762 (5/2+)1610.4 0.9(1) 2606 995 11/2+1635.5 1.7(2) 4082 24471659.9* 0.7(1)1684.0 2.2(2) 2135 451 1/2−1690.0* 5.4(4) 4.5(5)1743.8 0.4(1) 1.2(6) 3889 21441761.5 16.0(9) 16.4(12) 1762 0 9/2+1796.6 28.5(17) 26.4(15) 3151 1354 (13/2−) 13/2+1835.8 1.6(1) 3971 21351889.9* 3.3(2) 5.0(6)1940.2 1.4(2) 0.6(4) 3702 17621964.4* 1.2(2)2001.0* 3.4(3) 5.3(12)2059.6 0.3(1) 2060 0 9/2+2088.0 5.9(4) 8.1(9) 2088 0 9/2+2127.7 1.7(2) 1.4(7) 3889 17622143.5 1.7(2) 1.4(7) 2144 0 9/2+2155.5 8.0(5) 6.4(8) 3151 995 (13/2−) 9/2+2216.2 7.4(5) 2217 0 9/2+2267.8 0.7(1) 3889 16202295.8* 1.7(1) 1.3(4)2352.2 1.1(2) 3348 995 (5/2+) 11/2+2357.1 1.0(2) 1.5(7) 4119 17622388.0* 0.8(1)2415.7 1.2(2) 3971 1220 (5/2−)2432.1 1.4(2) 5.3(6) 2432 0 9/2+2460.7 4.4(3) 3.8(8) 4082 16202498.8* 0.7(1)Continued on next page68Table 4.2 – Continued from previous pageEγ [keV] Iγ [%] Ilitγ [%] Ei [keV] E f [keV] Jpii Jpif2510.3 0.5(1) 2510 0 9/2+2521.2* 1.0(4)2527.0 0.7(3) 4082 15552541.1 1.0(1) 2154 0 9/2+2605.7 0.4(1) 2606 0 9/2+2619.9 1.4(1) 3702 1083 (3/2−)2669.4 1.1(3) 3889 1220 (5/2−)2880.7* 2.4(2) 4.3(10)2919.0 0.5(1) 3914 995 11/2+2971.7 0.8(1) 3967 995 11/2+2999.6 0.5(1) 4082 1083 (3/2−)3024.1* 0.4(1)3150.9 0.60(8) 3151 0 (13/2−) 9/2+3184.6 4.4(6) 3.3(10) 3185 0 (5/2+) 9/2+3285.8 1.0(1) 1.3(4) 3286 0 9/2+3347.8 2.0(2) 1.4(6) 3348 0 (5/2+) 9/2+3385.1* 0.7(1)3487.5 2.5(3) 1.4(6) 3488 0 9/2+3701.8 10.3(8) 4.7(17) 3702 0 9/2+3888.6 1.5(2) 1.1(7) 3889 0 9/2+3913.9 5.3(4) 2.7(5) 3914 0 9/2+3966.2 6.1(4) 5.0(12) 3966 0 9/2+3977.3* 0.9(1)4118.6 4.1(4) 1.7(5) 4119 0 9/2+a Intensity corrected for internal conversion using BrIcc v2.3S [47].69Table 4.3: Types of tape cycleTape CycleRun No. Tape Move [s] Background [s] Implant [s] Decay [s]04516 - 04521 Continuous04529 - 04541 1 0.5 0.6 0.7504652 - 04656 1 0 10 1.34.6 Determination of β -feeding Intensities and log f tValuesIn Section 4.5 the relative γ-ray transition intensities are determined. However,absolute γ-ray transition intensities are necessary to determine β -feeding intensi-ties and log f t values. In this section, the procedure to obtain the absolute γ-raytransition intensities and log f t values is explained.4.6.1 Fit of the Beam Implantation CycleIn order to determine the absolute γ-ray transition intensities, the number of ob-served β -particles from the decay of 129Cd has to be determined. As long as theγ-ray transition intensities come from the β -gated spectrum, the obtained absoluteγ-ray transition intensities are independent of the β detection efficiency. In thisexperiment, there are three types of tape cycle, which are shown in Table 4.3. Forthe fit, runs 04529 - 04541 were used with a tape cycle consisting of 1 s of tapemove, 0.5 s of background measurement, 0.6 s of beam implantation, and 0.75 s ofdecay part (no beam implantation).The fit was performed using a program developed by Jorge Agramunt Ros(IFIC, Valencia), which was originally intended to be used for half-life fits in BE-LEN experiments. The key parameters in this fit were the structure of the cycle,decay branch, its branching ratio, half-lives of the β -decaying states of each iso-tope, beam implantation rate, and background. The decay branch is shown in theschematics in Fig.4.17.In this fit, excited states which decay via γ-ray emission were not consideredand it is assumed that the β -decay directly populates β -decaying states in its daugh-70129Cdx [keV]~300 + x [keV]11/2−3/2+129In1/2−9/2+451 [keV]β−129Sn11/2−3/2+35.2 [keV]β−0 [keV]0 [keV]Figure 4.17: Decay chain model starting from 129Cd. The excited stateswhich do not decay via β -decay have been omitted. This model allowsone state to populate both β -decaying states in the daughter nucleus.This is because in reality, the excited states populated by the β -decayscan populate both β -decaying states.ter nucleus. This is because the excited states of each nucleus that are not shownin Fig. 4.17 decay by emitting γ-rays promptly or with negligible half-lives com-pared to the β -decay half-lives and populate the β -decaying states. Therefore thisassumption that the β -decays populate the β -decaying states of the daughter nu-cleus does not affect the results of the fit.129In and 129Sn each have two β -decaying states (ground state and first excitedstate). For 129In, the half-lives were fixed to the value taken from NNDC [28],which are T1/2 = 611 ms for the ground state and T1/2 = 1.23 s for the β -decayingisomeric state. For the half-lives of 129Sn, since the programs only allows the9/2+(1/2−) state in 129In to populate the 11/2− (3/2+) state in 129Sn, which is notthe case in reality, the weighted averages of the two half-lives were used accordingto the branching ratio to each β -decaying state reported by Gausemel et al.[26].The fit program uses Bateman equations in order to fit the β -decay chainswhich spans several succeeding generations. The details of the equations are dis-cussed in Appendix A.For the half-life of 129Cd, although the analysis of the same data set reported147(3) ms for the 11/2− state and 157(8) ms for the 3/2+ state in 129Cd [23], inthe current fit, the mixed half-life of 154(2) ms reported in Ref.[22] was used for71Entries = 25708400 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.21−10110210310Counts/msCycle Time [s]00.20.40.40.2(Counts - Fit) / Fit per ms129Sn129CdBackground129Inχ 2 / NDF = 3138.7 / 2248Figure 4.18: Fit of the number of β -particles detected by SCEPTAR in thewhole beam cycle (top) and the quality of the fit (bottom). The dashedblue and red lines correspond to the ground state (T1/2 = 611 ms) andthe 1/2− isomeric state (T1/2 = 1.23 s) in 129In, respectively. The dash-dotted lines show the two β -decaying states in 129Sn with averagedhalf-lives based on Ref.[26] (see text).72the following reason. This fit treats the two half-lives of the β -decaying states in129Cd as one half-life and the reported half-life of 154(2) ms was obtained in thesame manner.The fit resulted in 1.346(8)×106 of 129Cd decays and 77(4)% (23(4)%) of themend up in the 9/2+ ground state (1/2− β -decaying isomeric state) of 129In, whosehalf life is 611 ms (1.23 s). The correlation of the number of decays of 129Cd andthe β -decay branching ratio of 129In are shown in Fig.4.19 and Fig. 4.20, whichresult in a systematic uncertainty for the branching ratio. The overall uncertaintiesare calculated to be the square root of the squared sum of the systematic uncer-tainties and the statistical uncertainties from the fit. That is, the uncertainty of thenumber of β particles (σβ ) and the β -decay branching ratio (σBRβ ) areσβ =√σβ ,sys2+σβ ,stat2, (4.5)σBRβ =√σBRβ ,sys2+σBRβ ,stat2. (4.6)Based on this information, the absolute intensities of the γ-ray transitions areobtained viaIabs.γ =Nβ-γεγ ·Nβ, (4.7)where Nβ-γ is the number of the β -gated γ-ray transitions, εγ is the γ-ray detectionefficiency, and Nβ is the number of β -particles obtained from the fit. This way, theabsolute γ-ray intensity can be obtained independent of the β -particle detectionefficiency.4.6.2 The Decay 129Cd→ 129InAs discussed in Section 1.3, both the ground state and the first excited state of129Cd decay via β -decay, and the ordering of the 11/2− and 3/2+ states has notbeen experimentally confirmed. In addition to this, those two β -decaying stateshave similar half-lives as reported in the previous studies [22, 23]. Therefore thediscrimination of the origin of the β -decay is difficult. However, it is necessary toidentify which states are fed by the β -decays of the 11/2− and 3/2+ states in orderto determine the β -feeding intensities and log f t values. The β -feeding intensities730.152 0.154 0.156 0.158 0.1601.331.341.351.361.37-129Cd Half life [s]NumberofDecay[⨯106]Figure 4.19: Correlation between the 129Cd half-life and the number of de-cays. The yellow band corresponds to the systematic error on the num-ber of decays.and log f t values of each state are shown in the Table 4.4.Decay of the 11/2− State in 129CdExcept for the direct transition to the ground state, transitions from the 3151 keVstate go through the 13/2+ state at 1354 keV or the 11/2+ state at 995 keV. Con-sidering the strong β -feeding to the state at 3151 keV (Fig. 4.4) and the mostintense transition with a 1796.6 keV γ-ray to the 13/2+ state at 1354 keV, it can beinferred that the state at 3151 keV is populated by the β -decay of the 11/2− statein 129Cd. Based on the analysis by Taprogge et al. [22], nearly all of the 11/2−decays proceed via the 994.8 keV and 1354.2 keV transitions or directly populatethe 9/2+ ground state. It is likely that the weak ground state transition from thestate at 3151 keV, newly confirmed by the current analysis, is taken into account asa direct β -feeding to the 9/2+ ground-state in their analysis. Therefore, it is mostlikely that this 3151 keV state and other states populated by the decay of the 3151740.150 0.152 0.154 0.156 0.158 0.1601520253035129Cd Half-life [s]129 In1/2−BRβ[%]Figure 4.20: Correlation between the 129Cd half-life and the β -decay branch-ing ratio of the 1/2− state in 129In. The yellow band corresponds tothe systematic error on the β -decay branching ratio of the 1/2− statein 129In.keV state are populated only by the β -decay of the 11/2− state.For the states at 2510, 2541, and 2606 keV, both ground-state transitions andthe population of the state at 995 keV are observed. This suggests that these excitedstates are also populated by the β -decay of the 11/2− state.In the previous investigation by Taprogge et al. [22], the states at 3914 and 3967keV were assumed to be populated by the β -decay of the 3/2+ state. However, ouranalysis confirmed the population of the state at 995 keV from those states andthis results in the same decay pattern as observed for the states at 2510, 2541, and2606 keV. Moreover, the log f t values of ∼ 5 [22] suggest that the β -decays thatpopulate these states are either allowed or first-forbidden decay. Therefore, it isinferred that the states at 3914 and 3967 keV are populated by the allowed β -decayof the 11/2− state.75Decay of the 3/2+ State in 129CdTwo states at 3185 and 3348 keV that are strongly populated by the β -decay have avery similar decay pattern. A significant fraction of their decay branches populatethe 1/2− β -decaying states through the intermediate states. Since the β -decay ofthe 11/2− state in 129Cd either proceeds via the 994.8 and 1354.2 keV transitionsor directly populates the 9/2+ ground state of 129In, it is likely that these two statesare populated by the β -decays of 3/2+ state. Other states whose decays mostlybypass the 11/2+ state at 995 keV in a similar manner to the 3185 and 3348 keVstates, including high-energy direct ground-state transitions, can also be expectedto be populated by the β -decays of the 3/2+ state. The excited states at 3971and 4082 keV do not decay via ground-state transitions, however, considering thefact that all the transitions starting from 3971 and 4082 keV states feed 1/2− β -decaying state, it is likely that these states are populated by the 3/2+ state in 129Cd.β -decay to the 9/2+(1/2−) State and Pandemonium EffectSince the total number of β -decays of 129Cd, the number of decays ending up in the9/2+(1/2−) state in 129In, and the absolute γ-ray intensities are obtained, the upperlimits of the β -feeding intensity to the 9/2+(1/2−) state can be obtained. Thereare several reasons why only upper limits can be obtained. Firstly, it is possiblethat there are high-energy γ-ray transitions which the detectors cannot capture dueto the low efficiency in the high-energy region. Also, if the intensity of a transitionis too small to be distinguished from the background fluctuations, such transitionscan not be observed. This effect is known as “Pandemonium Effect”.The resulting upper limits of the β -feeding intensities to the 9/2+ ground stateand the 1/2− isomeric state are 28% and 7%, respectively.β -feeding Intensities and log f t ValuesBased on the previous discussion, the excited state at 3151 keV, the states populatedby the decays of the 3151 keV state, and the states at 3914 and 3966 keV arepopulated by the β -decays of the 11/2− state in 129Cd. All the other states areassumed to be populated by the β -decays of the 3/2+ state in 129Cd.The absolute intensities of the 994.9, 1354.2, 2510.5, 2541.5, 2605.8, 3150.5,763913.9, and 3966.2 keV transitions sum up to 36(1)%, and the β -feeding intensityto the 9/2+ ground state including the “Pandemonium Effect” is 24(4)%. The sumof these two intensities subtracted by the absolute intensities of the 861.7, 1221.5,and 2352.2 keV transitions, which populates the 995 and 1354 keV states, amountsto 59(4)%. This means that out of all the β -decay of 129Cd observed, 59(4)% isfrom the 11/2− state. Consequently, the β -decays of the 3/2+ state amounts to41(4)%.With this information, β -feeding intensities from each β -decaying state in129Cd can be determined. Once the β -feeding intensities are determined, log f tvalues of each state are obtained using the program LOGFT[48]. The Q-value ofthe β -decay Qβ = 9330(200) keV was taken from Ref.[49]. Also, according to thecalculation reported in Ref.[20], it was assumed that the 11/2− state is the groundstate of 129Cd and the 3/2+ state is placed around 300 keV above the ground state.The resulting β -feeding intensities and log f t values are shown in Table 4.4.Table 4.4: β -decay feeding intensities to each excited state and their log f tvalues. “lit” are the values from Ref.[22] to compare to the current anal-ysis. The third and fourth columns are the total observed β -feeding inten-sities, the fifth and sixth columns (the seventh and eighth columns) fromthe left are the β -feeding intensities in the β -decay of 3/2+(11/2−) statein 129Cd. Out of all the observed β -decays of 129Cd, 59(4)% is from the11/2− state and 41(4)% is from the 3/2+ state (see Section 4.6.2). Thelog f t values are calculated based on the β -feeding intensities for eachβ -decaying state.3/2+ 11/2−Ex Ipi Ilitβ− Iβ− Ilitβ− Iβ− Ilitβ− Iβ− log f tlit log f t(keV) (%) (%) (%) (%) (%) (%)0 9/2+ <14 <28 <28 <48 >5.3 >5.0451 1/2− <9 <7 <18 <19 >5.4 >5.4995 11/2+ 6.5(20) 5.3(14) 13(4) 8.9(24) 5.4(1) 5.7(1)1083 (3/2−) 3.6(6) 2.2(5) 7(1) 5(1) 5.7(1) 5.7(1)1220 (5/2−) 3.0(17) 1.1(3) 6(4) 2.8(10) 5.7(6) 6.0(1)Continued on next page77Table 4.4 – Continued from previous page3/2+ 11/2−Ex Ipi Ilitβ− Iβ− Ilitβ− Iβ− Ilitβ− Iβ− log f tlit log f t(keV) (%) (%) (%) (%) (%) (%)1354 13/2+ 3.2(12) 2.6(11) 8(3) 4.3(19) 5.5(2) 5.72(2)1555 <1.0 1.1(3) <2 2.9(8) >6.1 5.9(1)1621 1.2(4) 0.4(2) 2(1) 1.1(6) 6.1(2) 6.3(3)1688 17/2− <0.4 <0.3 <1 <0.6 >6.3 >6.41693 0.7(6) 0.2(2) 1(1) 0.43(39) 6.3(5) 6.7(5)1762 0.5(10) <0.1 2(2) <0.2 6.0(5) >6.82015 1.1(3) 1.1(2) 2(1) 1.9(4) 5.9(2) 5.8(1)2060 1.3(6) 0.9(2) 3(1) 2.2(6) 5.8(2) 5.9(1)2085 <0.1 <0.2 <1 <0.4 >6.2 >6.52088 <0.4 1.6(1) <1 4.0(6) >6.2 5.6(1)2135 0.5(1) 1.3(3) 6.1(1)2143 0.10(8) 0.26(22) 6.8(4)2217 1.4(6) 0.8(2) 3(1) 1.4(4) 5.7(2) 5.9(1)2277 0.8(1) 1.4(3) 5.9(1)2433 <1.1 <0.1 <2 <0.3 >5.9 >6.62446 1.3(11) 3(2) 5.7(3)2510 0.42(6) 0.7(1) 6.2(1)2541 0.45(7) 0.7(1) 6.1(1)2551 <0.5 <0.2 <1 <0.3 >6.1 >6.52589 <1.2 <0.2 <2 <0.3 >5.8 >6.42606 0.3(1) 0.6(1) 6.2(1)3151 (13/2−) 25.3(8) 21.7(6) 52(5) 36(3) 4.2(1) 4.30(8)3185 (5/2+) 13.3(8) 9.2(4) 26(3) 22(2) 4.5(1) 4.61(9)3286 0.4(1) 0.29(5) 1(1) 0.7(1) 5.9(5) 6.0(1)3348 (5/2+) 8.6(9) 5.9(2) 17(2) 14(1) 4.7(1) 4.75(9)3488 0.7(2) 0.62(8) 1(1) 1.5(2) 5.8(5) 5.6(1)3702 1.6(6) 3.6(2) 3(1) 9(1) 5.3(2) 4.8(1)Continued on next page78Table 4.4 – Continued from previous page3/2+ 11/2−Ex Ipi Ilitβ− Iβ− Ilitβ− Iβ− Ilitβ− Iβ− log f tlit log f t(keV) (%) (%) (%) (%) (%) (%)3889 1.2(4) 1.5(1) 2(1) 3.8(5) 5.4(2) 5.1(1)3914 0.9(2) 1.6(1) 2(1) 2.7(3) 5.4(2) 5.16(9)3966 1.9(4) 1.9(1) 4(1) 3.2(3) 5.1(1) 5.07(9)3971 0.7(1) 1.5(1) 3.8(5) 5.1(1)4082 2.3(13) 2.5(1) 6.3(8) 4.8(1)4119 1.0(3) 1.2(1) 2(1) 3.0(4) 5.3(2) 5.1(1)79Chapter 5Discussion5.1 Shell Model CalculationThere have been several shell-model (SM) calculations reported for 129In. In Ref.[24],a realistic effective interaction derived from the CD-Bonn nucleon-nucleon (NN)potential was employed. In this calculation, the doubly magic 132Sn was consid-ered to be a closed core. Neutron holes occupy the five levels of the 50-82 shell(1g7/2, 1d5/2, 2s1/2, 1d3/2, and 0h11/2) and the proton hole is assumed to occupythe four levels of the 28-50 shell (1p1/2, 1p3/2, 0 f5/2, and 0g9/2). Calculated exci-tation energies of 1295, 1300, and 1540 keV were obtained for the 11/2+, 13/2+,and 17/2− levels (Fig. 5.1). In Ref.[22] another calculation is shown, employingan empirically optimized two-body interaction based on the one used in Ref.[24]and 1p3/2 and 1p1/2 proton single-hole energies from Refs.[27, 50]. The multi-pole part of the interaction was modified for a constant description of 46≤ Z ≤ 50,N ≤ 82 nuclei [50, 51].The most recent SM calculation was done by Wang et al.[20], employingthe extended pairing-plus-quadrupole-quadrupole model combined with monopolecorrections (EPQQM) model, where the paring and quadrupole forces describe theshort and long range parts of the interaction. While the pairing and quadrupoleterms take care of the main smooth part of the structure properties, the monopoleterms play important roles for the shell evolution and often are responsible for ex-plaining animalous behaviours in spectra and transitions [52]. In this calculation,80the EPQQM model for nuclei having one or two protons and/or neutrons less than132Sn [53] was used, with two matrix elements for protons modified from zero to〈p1/2,g9/2,J = 4∣∣V ∣∣p1/2,g9/2,J = 4〉= 0.32, (5.1)〈p1/2,g9/2,J = 5∣∣V ∣∣p1/2,g9/2,J = 5〉=−0.22. (5.2)The two monopole correction termsM1 = kmc(νh11/2,ν f7/2) = 0.52 MeV, (5.3)M2 = kmc(pig9/2,νh11/2) =−0.40 MeV, (5.4)were introduced into the Hamiltonian. In 129Cd, this M2 term pushes down the11/2− state with the configuration of pig−29/2νh−111/2 due to the enhanced monopoleattraction between the pig9/2 and the νh11/2 orbits, resulting in predicting the 11/2−state to be the ground state.In the current study, we used the shell-model code NuShellX@MSU[54]. Theinteraction jj45pna was employed, in which the residual two-body interaction is de-rived from the CD-Bonn nucleon-nucleon interaction through the G matrix renor-malization method [55] with single-particle energies adjusted to the 132Sn region[56]. The model space is the same as in the first two calculations mentioned above,which are 1g7/2, 1d5/2, 2s1/2, 1d3/2, and 0h11/2 for the neutron holes and 1p1/2,1p3/2, 0 f5/2, and 0g9/2 for the proton hole.Fig. 5.1 shows a comparison between the experimentally observed excitedstates in 129In and several SM calculations. For the current calculation, if thereare levels with the same spin and parity, only the first ones are shown. Above theenergy shown in the figure, the level density rapidly increases. Therefore, it isdifficult to compare the results of the experiment and the calculations. The highspin isomers at 1630 keV with the tentative spin assignment of 23/2− and at 1911keV with the tentative spin assignment of 29/2+ are taken from Ref.[24].As shown in Fig.5.1, the current calculation does not reproduce several exper-imental properties of 129In. Firstly, the excitation energy of the first excited state1/2− is underestimated by ∼300 keV while the calculations by Wang et al. [20]and Taprogge et al. [22] are in a rather good agreement. Although the order of the81exp.9/2+1/2-11/2+(3/2-)13/2+(5/2-)(23/2-)17/2-(29/2+)jj45pna11/2+,129513/2+,13255/2+,14707/2-,154917/2-,160523/2-,166519/2-,18237/2+,19749/2-,202429/2+,20449/2+,01/2-,1743/2-,7165/2-,8369/2+,01/2-,42211/2+,99413/2+,11435/2+,12053/2-,153317/2-9/2+23/2-5/2-29/2+,2134Wang et al.9/2+,01/2-,40311/2+,10063/2-,10935/2-,119323/2-,149913/2+,129917/2-,160329/2+,1804Taprogge et al.11/2+,129513/2+,130017/2-,154023/2-,157329/2+,19629/2+,0Genevey et al.04519951083122013541630(56)16881911(56)129InFigure 5.1: The comparison between the experimentally observed excitedstates and the Shell-model calculations of 129In. “jj45pna” is the cur-rent calculation, “Wang et al” is from Ref.[20], “Taprogge et al” is fromRef.[22], and “Genevey et al.” is from Ref.[24] (see text for detail). Theexperimental values for the excited states at 1630 keV and 1911 keV arealso taken from Ref.[24]. Some of the excitation energies in “Wang etal.” were not presented in Ref.[20], therefore they were read from thefigure.3/2− and 5/2− states are reproduced correctly, the position of these states doesnot agree with the experiment. The energy difference of the 3/2− state betweenthe present calculation and the work presented by Wang et al. [20] is about 800keV, whereas Taprogge et al. [22] presented the excitation energies which are in agood agreement with the experimental values. The order of the 17/2− isomer and23/2− isomer [24] has also shown the dependency on the models of the residualinteraction.Our calculation suggests that the 9/2+ ground state has the main configurationof pig−19/2νh−211/2 mixed with ∼10% of the pig−19/2νd−23/2 configuration. The contri-bution of this configuration is about 10% lower than the value reported by Wanget al. [20]. The first excited 1/2− state at 451 keV was reported to have the main82configuration of pi p−11/2νh−211/2 [20], and our calculation shows that the main con-figuration is pi p−11/2νh−211/2, with a ∼ 15% mixture of the pi p−11/2νd−23/2 configuration.Wang et al.[20] suggested that the 3/2− and 5/2− states are single-particle likestates with the configurations pi p−13/2νh−211/2 and pid−15/2νh−211/2, respectively. How-ever, while our calculation agrees with the configuration for the 3/2− state, for the5/2− state it suggested ∼40% of the pig−19/2ν(d−13/2h−111/2) configuration.The two positive parity states 11/2+ and 13/2+ at 995 keV and 1354 keV,respectively, are reported to have the pig−19/2νh−211/2 configuration [22]. The currentshell-model calculation indicates that while this is the case for the 11/2+ state, the13/2+ state has non-negligible contributions from several other configurations.5.2 Decay Properties of 129Cd5.2.1 Gamow-Teller (GT) DecaysAs shown in Table.4.4, the level at 3151 keV with the tentative spin assignment of13/2− receives a strong β -decay feeding. The log f t value of 4.30(8) indicates thatthis is an allowed GT decay. It is likely that this state is populated by the β -decayof the 11/2− state in 129Cd, whose main configuration is pig−29/2νh−111/2 [20]. Wanget al.[20] suggested the configuration pi p−11/2νh−211/2 for this 13/2− state, however,since the only GT decay which populates excited states in 129In is the ν0g7/2 →pi0g9/2 GT single-particle transition, the configuration of pig−19/2ν(g−17/2h−111/2) shouldbe assigned to this state [22]. Our calculation shows the 13/2− state with thecorresponding configuration at 2390 keV.There are two other strongly β -populated excited states at 3185 keV and 3348keV with the tentative spin assignments of (5/2+). The corresponding log f t valuesare 4.61(9) and 4.75(9), respectively. It is expected that these states are populatedby the decay of the 3/2+ state in 129Cd, whose main configuration is pig−29/2νd−13/2.For the same reason as the (13/2−) state at 3150 keV, the ν0g7/2 → pi0g9/2 GTsingle-particle transition is expected to populate these states, and the main config-uration of pig−19/2ν(g−17/2d−13/2) may be assigned to these states to form 5/2+ states[22].835.2.2 First-Forbidden (ff) DecaysThe excited states at 995 keV and 1354 keV with the spin assignments of 11/2+and 13/2+, respectively, are expected to have large contribution from the configu-ration of pig−19/2νh−211/2 (see Section 5.1). The log f t values of these states are 5.7(1)and 5.72(2), respectively. This indicates that these states are likely populated viathe ff transition ν0h11/2→ pig9/2 from the 11/2− state in 129Cd. The 9/2+ groundstate (log f t > 5.0) is expected to receive β -feeding from the same type of ff tran-sition [22].For the 1/2− state at 451 keV, our study shows a log f t > 5.4, and it is likelythat this state is populated via the ff transition νd3/2 → pi p1/2. This is consistentwith the results of our SM calculation. Similarly, the (3/2−) state at 1083 keVwith the log f t value of 5.7(1) is expected to be populated via the ff transitionνd3/2→ pi p3/2. Taprogge et al. [22] suggested that the (5/2−) state at 1220 keVis populated through the ff transition νd3/2 → pi f5/2 and the decay pattern sup-ports this argument. However, the results of our SM calculation as well as Wanget al.[20] do not agree with this.84Chapter 6ConclusionIn this thesis, new results of the β -decay of 129Cd as well as the properties ofthe excited states in 129In were reported. The dominant fraction of the β -decayshappen via the GT decay ν0g7/2 → pi0g9/2 which populates the excited states at3151, 3185, and 3348 keV. This is in agreement with the previous study [22]. Inaddition to this, we observed non-negligible contributions from ff transitions suchas ν0h11/2→ pig9/2, νd3/2→ pi p1/2, and νd3/2→ pi p3/2. However, the mechanismwhich populates the (5/2−) state at 1220 keV remain inconclusive in our study, asopposed to the results discussed in Ref.[22], where it was suggested to be the fftransition νd3/2→ pi f5/2.In terms of the 129In levels, we have newly established 29 γ-ray transitions and5 excited states, with some corrections to the previously reported level scheme inRef. [22], demonstrating the great sensitivity of the GRIFFIN spectrometer. Ingeneral, the level scheme, as well as the γ-ray intensities and β -feeding intensitiesare in a good agreement with Ref.[22].A shell-model calculation was done with the shell-model code NuShellX@MSU using the realistic residual interaction model jj45pna and compared to theexperimental results. There has been no reported shell-model calculation for 129Inusing a residual interaction based on the CD-Bonn renormalized G-matrix. Ourcalculation falls short in reproducing the energy levels of the low-lying states in129In, which are better reproduced in the shell model calculations in Refs. [20, 22,24]. These previous calculations were, however, fine-tuned in their single-particle85energies and interactions to better reproduce the experimental results. 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URLhttps://link.aps.org/doi/10.1103/PhysRevC.71.044317. → page 8194Appendix AProduction of Radioactivity andSeries of DecaysThe discussion in this chapter is partially based on Chapter 6 in Ref.[32].A.1 Bateman EquationLet us label the generation of radioactive nuclei with 1,2,3, · · · where 1,2,3, · · ·denotes the parent nucleus, the daughter nucleus, the grand-daughter nucleus, andso on, respectively. With the initial condition on the number of nuclei:N1(t = 0) = N0 (A.1)Ni(t = 0) = 0 (i = 2,3,4, · · ·) (A.2)the differential equationsdN1dt=−λ1N1 (A.3)dNidt=−λi−1Ni−1−λiNi (i = 2,3,4, · · ·) (A.4)95where λi is the decay constant of i-th generation of radioactive nucleus, have thesolutionNn(t) =N0λnn∑i=1cie−λit (n = 1,2,3, · · ·) (A.5)wherecm =n∏i=1λin∏i=1,i 6=m(λi−λm). (A.6)Hence the activity of the n-th generation radioactive nucleus An(t) isAn(t) = λnNn(t)= N0n∑i=1cie−λit . (A.7)A.2 Bateman Equations for Production of RadioactivityIn order to fit the beam implantation (production) part of the decay curve, the pro-duction of radioactivity has to be taken into account. Here we assume that theproduction rate R is constant over time.First let us consider the differential equation for the parent nucleus. Since theproduction rate of this nucleus is R, the differential equation isdN1(t)dt= R−λ1N1(t) (A.8)and the solution isN1(t) =Rλ1(1− e−λ1t) (A.9)under the condition N1(t = 0) = 0.The differential equations for the i-th generation of nucleus is the same asEq.A.4 and by modifying the solution A.7, we obtain the solution for the activ-ity of the n-th generation nucleus A′n(t)96A′n(t) = R(1+n∑i=1c′ie−λit) (n = 1,2,3, · · ·) (A.10)wherec′m =n∏i=1,i 6=mλin∏i=1,i 6=m(λm−λi). (A.11)97

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