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A study of the b-delayed particle decay of 9C Gete, Ermias A. 2000

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A S T U D Y O F T H E / 3 - D E L A Y E D P A R T I C L E D E C A Y O F 9 C By Ermias A. Gete M.Sc, The University of British Columbia, 1994 B.Sc, The University of British Columbia, 1992 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F P H Y S I C S A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRIT ISH C O L U M B I A January 2000 © Ermias A. Gete, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date Q * ~ - SJ*, > r y f r DE-6 (2/88) A b s t r a c t The /3-decay of 9 C (ti / 2 =126.5 ms) to the particle-unstable nucleus 9 B has been studied at the T I S O L facility at T R I U M F with an intent to measure the /3-decay scheme and to study the structure of 9 B . A l l of the states in 9 B populated by the (3+ decay of 9 C are unbound and decay ultimately into 2a+p. It is therefore necessary to measure the decay particles in coincidence in order to measure the excitation energy of 9 B . Two coincidence experiments with different detector geometries were performed. In the first experiment, four A E - E particle telescopes arranged in face-to-face pairs measured singles and double coincidence particle spectra. For the second experiment, a pair of position sensitive Double-Sided Silicon Strip Detectors were used in conjunction wi th particle telescopes to record three-fold coincidence events. The coincidence measurements allowed us to kinematically identify events from 5 L i +a and 8 B e + p , the two major decay modes of the states of 9 B . Several states in 9 B have been observed and /3-branching ratios to these states, and a and proton decay branches from these states have been measured. Because most of the 9 B levels are broad, the state shapes fed by the f3 decay of 9 C are distorted by the /3-decay phase-space factor. In order to describe the states of 9 B , and the secondary compound states such as 5 L i into which the 9 B states decay, a simplified one-level .R-matrix description of these states was used, and the spectra from both decay modes were fitted. /3-decay branching ratios as well as Gamow-Teller strengths for the observed states of 9 B were extracted from these fits. Four strongly populated states of 9 B have been observed; the ground state (47±5) % and excited states at 2.34 (35±7) %, 2.8 (6 .7±0.7) % and 12.16 (6 .8±0 .7 )% M e V . The strong branch to the 12.16 M e V state corresponds to a large Gamow-Teller matrix element wi th B G T = 2 . 5 ± 0 . 2 5 . i i T a b l e o f C o n t e n t s A b s t r a c t i i L i s t o f T a b l e s v i i i L i s t o f F i g u r e s x i A c k n o w l e d g m e n t s x x i 1 I n t r o d u c t i o n 1 1.1 General 1 1.2 Objectives and summary of the thesis . • 5 2 R e v i e w o f P r e v i o u s W o r k 8 2.1 (5 decay studies • 10 2.2 The mirror pair 9 B - 9 Be • 12 2.2.1 The ground state 12 2.2.2 The = 1/2+ state . 13 2.2.3 The = 5/2" state . 16 2.2.4 The = 1/2" state 17 2.2.5 The £ x =12 .06 MeV state 18 2.2.6 The J71" = 3/2~ isobaric analogue state 18 3 E x p e r i m e n t 20 3.1 Introduction 20 iii 3.2 9 C production 20 3.3 Experimental setups 23 3.3.1 Detection system 23 3.3.2 Experimental setup, Expt. 1 24 3.3.3 Experimental setup, Expt. 2 26 3.4 Electronics and data acquisition 27 3.4.1 Electronics, Expt. 1 30 3.4.2 Electronics, Expt. 2 31 3.4.3 Data acquisition 32 3.5 Calibration runs 34 3.5.1 1 7Ne Calibration run 34 3.5.2 1 8 N Calibration Run 37 3.6 Energy calibration 38 3.6.1 Telescope calibration 39 3.6.2 SSD Calibration . . . 41 3.7 Timing spectra 46 4 Monte Carlo Simulation and Kinematics 48 4.1 Kinematic considerations 48 4.2 Monte Carlo simulation 51 4.3 Decay through broad states 53 4.4 A study of breakup channels from Monte Carlo simulations 56 4.4.1 9 B decay to 8Be( 0+) and p 57 4.4.2 9 B decay to 5Li(3/2~) + a 57 4.5 Determination of coincidence efficiencies for the breakup channels . . . . 60 4.6 Angular correlation 62 iv 4.7 Recoil broadening due to /3^-particle correlations 64 5 D a t a A n a l y s i s 66 5.1 Introduction 66 5.2 Da ta analysis for Expt . 1 67 5.2.1 Singles spectra 67 5.2.2 Double coincidence between the opposing telescopes 70 5.3 Data analysis for Experiment 2 76 5.3.1 Particle-/? coincidence 76 5.3.2 Triple coincidence • 79 5.3.3 S S D R double hit, T L triple coincidence events 91 5.3.4 SSD-Telescope-/3- coincidences 95 6 R e s u l t s a n d d i s c u s s i o n 98 6.1 Introduction 98 6.2 F i t t ing Procedures • 99 6.3 Branching ratio determination 102 6.3.1 Relative branches for the 9 B —» 8 B e 9 . s . + p break-up (Expt. 1) . . 102 6.3.2 Relative branches for the 5 L i g s . + a break-up (Expt . 2) 104 6.3.3 Normalizing the spectrum from the 5 L i + a channel to the 8 B e + p channel 106 6.3.4 Determination of the ground-state branching ratio from the (3-particle spectrum 106 6.4 F ina l branching ratios and BQT values 113 6.5 Reconstruction of the singles spectra 116 6.6 Error Analysis 117 6.6.1 Error in the 8 B e 3 . s . + p spectrum 119 v 6.6.2 5 L i + a spectrum (SSDL, S S D R , T L ) 120 6.6.3 Error in the ground state branch 121 7 Conclusion 123 A 0 decay 127 A . l Theory of (5 decay 127 A.2 Fermi and Gamow-Teller transitions 134 A . 3 /3-decay and nuclear structure 137 A.3.1 Fermi transition 137 A.3.2 Gamow-Teller transition 138 A.3.3 Giant resonances in nuclear /3-decay . 140 A.3.4 The (p,n) reaction and /?-decay strengths 141 A.3.5 The Ikeda sum rule for Gamow-Teller transitions 142 A.3.6 Quenching of the G T strength 142 A . 3.7 Mir ror asymmetry 143 B i?-matrix theory 144 B . l Scattering of a nucleon by a central potential 145 B.2 Reactions involving more than one channel 148 B . 2.1 Tota l reaction cross sections 150 B.3 Appl icat ion to f3 decay through broad states 152 B.3.1 Decay through secondary broad states 154 C R-matrix parameterization of the spectra of 9 B from the decay of 9 C 157 C l F i t t ing the spectra of 9 B from the j3 decay of 9 C 159 D Models describing the A=9, T=l/2 system 162 v i DM The shell model 162 D. 2 The cluster model 163 D.2.1 The a-N-a system as two-center molecular state 164 E Kinematics 167 E. l Kinematics 167 F Angular correlations 170 F. l Angular correlations 170 F . l . l Decay of 9 B —> 5 Li(3/2") + a 171 F. l .2 Decay of 9 B —> 8Be(2+)+p 173 G The E682 Collaboration 177 Bibliography 178 vii List of Tables 2.1 Previously known 9 B states below 15.476 M e V , the energy available in 9 C decay. Information on the properties of 9 B is taken from Ref. [2]. The states that are expected to be strongly populated by the (3 decay of 9 C are of (Jn = l / 2 ~ , 3/2~, 5/2~) corresponding to an allowed Gamow-Teller transition 10 2.2 Experimental measurements for the first excited state of 9 B 15 2.3 Various theoretical calculations of the first excited state of 9 B 16 3.1 Dimensions and location of the various silicon detectors used in Expt . 1, 6 and <fi are as indicated in Figure 3.3 26 3.2 Identification of the major a and proton peaks in the /3-delayed particle decay of 1 7 N e decay 36 3.3 A list of the 7-ray lines emitted by 2 2 N a and 1 3 7 C s . The Compton edges seen in the spectra from the /3-detector are indicated. 46 4.1 Angular correlations for the breakup of 9 B states into an intermediate (daughter) nucleus; only the lowest allowed angular momenta were in-cluded in the computation. Note that for the decay to 8 B e ( 2 + ) , there are two possible distributions for the different spin states of the proton. . . 63 5.1 T D C conditions applied to S2-S3 combination in sorting the coincidences from the singles data. Events that satisfy any of the four were selected. . 71 5.2 T D C conditions applied to the S S D L , S S D R and T L coincidence events. 81 v i i i 6.1 State energies, widths and relative feeding factors obtained from the fit to the 8 B e ( 0 + ) + p spectrum. The 9 B excitation energies (Ex) are obtained from Ex = Ecm — 0.185 M e V . In the last column the relative contribution of each state is given normalized to the intensity of the 2.8 M e V state. . 104 6.2 State energies, widths and relative feeding factors obtained from the fit to the 5 L i + a ; spectrum. The 9 B excitation energies (Ex) are obtained from Ex = E s u m — 0.277 M e V . In the last column the relative contribution of each state is given normalized to the intensity of the 2.34 M e V state. . . 105 6.3 Relative 9 C /?-decay branches to the excited states of 9 B obtained after normalizing the 8 B e ( 0 + ) + p spectrum to the the 5 Li (3 /2~)+a ; spectrum. The second column gives the relative intensity of each state wi th respect to the two separate spectra. The relative intensity wi th respect to the combined data is given in the third column. I\ is normalized to the intensity of the 9 B ( 2 . 3 4 ) ^ 5 L i ( 3 / 2 ~ ) + Q ; transition 107 6.4 Relative contributions of the 9 B states decaying to the 5 L i ( 3 / 2 ~ ) + a chan-nel in the particle-/3-coincidence spectrum. The number of simulated events (column 5) is determined according to I'x (column 4) I l l 6.5 Relative contributions of the 9 B states decaying to the 8 B e ( 0 + ) + p channel in the particle-/3-coincidence spectrum. The ground state branch has been arbitrarily assigned the same branching ratio as the 2.34 M e V state of 9 B . 112 ix 6.6 Level energies and branching ratios to states populated in the /3-delayed particle-decay of 9 C . Brp and Bra denote branching ratios to 8 B e (ground state and E x = 3 M e V state) and the s L i ground state relative to the total number of /^-decays. We use "background" to denote the fraction of the branching ratio which is due to states above Ex=15 M e V or states too broad to allow a unique identification to be made. BQT is calculated using E q . 6.16, l o g ( / i ) employing E q . 6.17. 114 6.7 Relative errors for the states observed in the summed ratio cut spectrum. SE, SQ and Ss are errors due to energy calibration, beamspot location and statistical errors respectively 120 6.8 Relative errors for the states observed in the triple energy sum spectrum S S D L , S S D R and T L . SE, SQ and Ss are errors due to energy calibration, beamspot location and statistical errors respectively 122 C l Allowed angular momenta for Jn = 1/2", 3 /2" , 5 /2" states in 9 B decaying to 8 B e ( 0 + ) , 5 L i ( 3 / 2 " ) or 8Be(2+) 157 C.2 A list of the possible decay channels for the states of 9 B fed by j3 decay of 9 C 158 C.3 A list of the possible decay channels for the J 7 1 - state of 9 B 159 C.4 A list of the reduced width parameters in the spectrum of 9 B for J 7 1 ' = 1/2", 3/2~, 5 /2-s ta tes 160 F . l Angular correlation parameters for the 9 B —» 5 L i ( 3 / 2 ~ ) + a breakup. . . 171 F.2 Angular correlation parameters for the 9 B —» 8 B e ( 2 + ) + p breakup. . . . 174 x List of Figures 1.1 Chart of the nuclides showing the different regions of nuclear stability in the (TV, Z) diagram. The stable nuclei are represented by the dark squares. The zig zag lines on both sides of the stable nuclei indicate the present limits of the experimentally observed nuclei, while the outer most lines represent the drip-lines. . 2 1.2 An illustration of /^ -delayed particle emission 3 2.1 Level diagram of 9 B with respect to the decay of 9 C , also showing possible decay channels of 9 B . The level properties are adopted from Ref. [2]. . . . 9 2.2 Low-lying states in 9 B - 9Be mirror pair. The thresholds for nucleon and a emission are indicated 13 3.1 A schematic diagram of the TISOL isotope separator facility 22 3.2 Setup for Expt. 1. The dimensions are given in Table 3.1. The axes of S2, S3 and the annular detectors all lie in the same horizontal plane. The collection foil is rotated by 45° about the vertical axis 24 3.3 A schematic diagram of the setup for Expt. 1, showing the location of the detectors with respect to the co-ordinate axes. 6 and <fi, given in Table 3.1, are indicated for S4 25 xi 3.4 Setup for Expt. 2. The axes of all the detectors and the collector foil lie in the horizontal plane, and the angle between the beam axis and the axes of the detectors is 45° for the strip detectors, 46°m for T R and 49° for T L . The distance of each detector from the center of the foil is 4.5, 4.7, 4.2, 4.1 and 5.8 cm for T L , TR, SSDL, SSDR, and the (3 detector respectively. 28 3.5 A schematic diagram of the Silicon Strip Detector and the readout system. The signals from the adjacent strips were paired together due to lack of enough electronics components such as amplifiers and ADCs 29 3.6 A schematic diagram of the essential features of the electronics used in Expt. 1 30 3.7 A schematic diagram of the electronics for the silicon strip detectors. . . 33 3.8 Particle spectrum (mainly protons) from the 1 7 Ne run recorded by the back detector of S2 in Expt. 1. Note that the particles are detected by B2 after passing through the front detector F2 35 3.9 a particle spectrum from the 1 8 N run recorded by the front detector of SI in Expt. 1 37 3.10 A fit to the 1081 keV 1 8 N peak in the front detecor of S3 in Expt. 1. The experimental energy resolution obtained after subtracting the recoil broadening is 27 keV. The recoil broadening is 39 keV (Section 3.5.2). . . 40 3.11 Fits to the front and back portions of the pulse height of the 7031 keV peak from the 1 7 Ne run in the front detector of T L in experiment 2. . . 42 3.12 Raw A D C spectra for the 8 front (vertical) strips of SSDL from the 1 7 Ne run. Note that the central strips see more events than the peripheral strips due to the back-to-back nature of the decay of 1 7 F . Also see Figure 3.13. 43 3.13 Hit pattern of SSDR for events in coincidence with T L in 1 7 Ne run. Only the central strips that are opposite the telescope have events 44 xii 3.14 Typical 7-ray spectra obtained from the (3 detector using: (a) 1 3 7 C s source (b) 2 2 N a sources. The Compton edges from the 7-rays are indicated in the spectra 45 3.15 T D C spectra for F l and F4 for a subset of the data from the 9 C run. The time period is 0.244 ns per channel 47 4.1 A n illustration of the kinematics of the breakup of 9B—> 5Li-f a 49 4.2 Kinematic loci calculated for the breakup of 9B(12 MeV)—>• aap, where the decay particles are observed in the laboratory in coincidence at an angle of 170°. The two points indicated in the figure correspond to decays that proceed through intermediate states of definite energy, as indicated in the figure 50 4.3 A n illustration of the decay of a 9 B state with distribution W(E) decaying to the ground state of 5 L i of state shape w(E'). The two different distri-bution of 5 L i , resulting from the decay of 9 B of excitation energies E\ and E2 are shown 54 4.4 A plot of the shape of the ground state of 5 L i (shaded area) as populated by the decay of a 9 B state at Ex = 2.36 MeV. The unmodified °Li f l. s. distribution is shown by the dotted line. The x-axis is the breakup energy [=2.36 - Ex(5Li)} MeV. (Inspired by Figure 19 in Ref. [51].) 56 4.5 Scatter diagram of simulated double coincidence spectra for detector tele-scopes SI and S4, the more distant opposite detector pair in experiment 1 geometry. The simulated events include: (i) 9B(12.2)—>8Be(gs)+p, (ii) 9 B(12.2)^ 5 Li(gs)+a, (iii) 9B(12.2)^ 8Be(3.0)+p and (iv) 9B(2.8)^ 8Be(gs)+p. The number of simulated events are: (i) 2 x 106, and (ii)-(iv) 4 x 106. . . 58 xiii 4.6 Simulated a-a coincidence events from 9 B —> 5 L i 9 . s . + a. detected by the back-to-back detectors (TL vs SSDR) in coincidence with the protons detected in SSDL. 2 x 106 events were simulated for each energy from 2 MeV to 14 MeV in steps of 1 MeV, and same detector thresholds were applied as in the experimental data 59 4.7 Simulated kinematic efficiency of Telescope SI for detection of protons in coincidence with one or two a particles in the 8Be(gs)+p channel as a function of proton energy. The crosses with the statistical error represent the calculated points. The smooth curve is a fourth order polynomial fit. Each point was simulated from 2 x 106 events. The minimum a energy accepted is 160 keV 61 4.8 A n illustration of cascade transition, a, b and c are the spins of the three nuclei, and S\ and S2 are the spins of the emitted particles 63 5.1 Singles spectra for: (a) Front detector F l (30 thickness) (b) Front detector F3 (13 /j,m thickness), showing the breakthrough features that are characteristic of the thicknesses of the detectors used and the identity of the particle being measured 68 5.2 E - A E spectrum for telescope 2 (F2=10 (im): A reliable particle identi-fication can be made for proton energies above about 1 MeV and for a energies above about 3 MeV 69 5.3 Singles particle spectrum measured by Telescope 1. The feature resulting from the different decay modes of the 12.2 MeV state are indicated. The proton peak near 160 keV and the plateau in the 1-1.5 MeV region are as indicated in Figure 5.1 70 x iv 5.4 Two dimensional plot for the opposing telescopes S1-S4 pair. The p — a band from 8 B e 5 . s +p and the a —a band from 5 L i g s . +a decay are indicated. 72 5.5 Time difference spectra for the detectors in the S1-S4 telescope pair: (a) TF1-TF4, (b) TF1-TB4. A coincidence peak centered around 0 is apparent in both spectra. The peaks near channel ±2000 in both spectra are from one of the two events being random events and causing overflow in the T D C . 73 5.6 A diagram showing the cuts applied on the subset of the data in S2-S3 coincidence to select the events from 8 Be(0 + )+p events in the case where the protons are detected by S2 and the a-particles are measured by the opposing front detector F3. Similar cuts were applied to the other telescope spectra 74 5.7 Proton spectra measured by S1-S4 (a-d) obtained after applying the ratio-cut in order to select events from the 9 B—> 8 Be 3 . s . + p decay mode . . . 75 5.8 Simulated kinematic efficiencies of Telescopes S1-S4 (a-d) for detection of protons in coincidence with one or two a particles in the 8Be(gs)+p channel as a function of proton energy, for a energy thresholds of F4=250 keV, F3=300 keV, F2=240 keV, and Fl=240 keV. The crosses with the statistical error represent the calculated points. The smooth curves are a fourth order polynomial fit. Each point was simulated from 2 x 106 events. 77 5.9 The summed (after efficiency correction) ratio cut spectrum from the decay of 9 B — • 8 Be 9 . s . + p for the four telescopes S1-S4. The apparent peaks that are associated with the excited states of 9 B are indicated in the figure. 78 5.10 Particle spectra detected by the 50 fxm front telescope (FL) : (a) singles spectrum (b) /3-particle coincidence spectrum with the (3 energy threshold of 4 MeV 78 xv 5.11 A Gaussian fit with linear background to the 164 keV peak measured by F L in coincidence with a 0 particle in the plastic scintillator. The width is from recoil broadening discussed in the text as well as from experimental resolution 79 5.12 Triple energy sum spectrum for the coincidences between SSDL, SSDR and T L , with no timing cut applied . 80 5.13 Time difference spectra of the triple coincidence events between SSDL, SSDR and T L plotted in Figure 5.12 for the three detector combination SSDL-SSDR, SSDR-TL and SSDL-TL (a-c). In (d-f) are plotted events that are selected by making the cut in TSSDL-TSSDR shown in (a). . . 82 5.14 Triple sum spectrum for the coincidences between SSDL,SSDR and T L , after timing cuts were applied 83 5.15 Triple sum spectra for SSDL, SSDR and T L as sorted out by the kinematic analysis program. These events are identified as: (a) paa coincidences (b) aap coincidences, (c) apa coincidence (d) random coincidence events. . 86 5.16 Angle difference spectra for the events which are paa type shown in Fig-ure 5.15a 88 5.17 A two-dimensional plot for the paa coincidence events for which the triple energy sum spectrum is plotted in Figure 5.15a 89 5.18 Scatter diagram of energy-energy correlations for different detector combi-nations for the 12.2 MeV 9 B state: a comparison between the experimental (a-c) and Monte Carlo (d-f) data 90 xvi 5.19 Simulated triple-coincidence efficiency of SSDL, SSDR and T L for the 9 B break-up through the 5 Li(3/2~)+a channel as a function of the breakup energy. The squares with statistical errors represent the calculated points. The smooth curve is a fourth order polynomial fit. Each point was sim-ulated from 2 x 106 events. In the region below 4.0 MeV, angular distri-butions for the 5/2~ 9 B state were used. For the rest of the points, an isotropic distribution was assumed corresponding to J 7 r =3/2 _ 91 5.20 Triple sum spectra for the SSDR-TL coincidence events with a double hit on the strip detector: (a) Raw spectrum with no timing cut, (b) Spectrum after timing cut 92 5.21 Two-dimensional plot for the 12.2 MeV state for the SSDR-TL events shown in the window in Figure 5.20. The a-a coincidence events are for the decay through the 8Be+p channel, and the p-a events are for the decay through the 5 L i + a channel. Particle identification was made using the techniques described in Equations 5.2-5.5 93 5.22 Centre-of-mass angular correlation between the first a particle and the particles from the secondary break-up in the decay of the 12.2 MeV 9 B state through the 5 L i + a channel. The solid line histogram is a Monte Carlo simulation that assumes a spin of J 7 r=3/2~ for this state, with an isotropic angular distribution. The dashed line histogram is a simulation for a spin of J 7 r =l/2~ for this state (Section 4.6). (a) shows the aT-p angular correlation and (b) the al-a2 correlation 94 5.23 Two dimensional plot of events detected in the face-to-face detectors SSDR-T L in coincidence with a (5 particle in the plastic scintillator. Only single hit events in the SSD were accepted 95 xvii 5.24 A plot of the Q 2 value for SSDR-TL coincidence events calculated us-ing Equation 5.6. The peak near 100 keV is due to events decaying to 8 B e 9 . s +p, the plot in the inset is for those events that are selected under this peak 97 5.25 Proton spectra measured by the two strip detectors in coincidence with the respective opposing telescopes after selecting the 8 Be g . s . +p events by the "Q 2 cut" shown in Figure 5.24: (a) SSDL, (b) SSDR 97 6.1 The shape of a 9 B state , 1(E), (£ x =12 .16 MeV, T=450 keV) decaying to 8 B e 9 . s + p. The shape of this state observed in the B decay of 9 C , 1(E)fp(E), is given by the broken line 102 6.2 Fit to the summed-energy telescope spectrum from Expt. 1, corrected for efficiency. The contribution from each 9 B state to the total spectrum obtained from the fit is shown. The error bars on the data points represent only statistical errors. The component labelled background is a continuum arising from an assumed higher energy state 103 6.3 Fit to the triple-coincidence energy-sum spectrum for SSDL, SSDR and T L from Expt. 2. The contribution of each 9 B state to the total-spectrum obtained from the fit is shown. The error bars represent only statistical errors 105 6.4 Simulated particle energy spectrum in detector T L in coincidence with a j3-particle (Ep > 4 MeV) detected by the plastic scintillator. This spectrum was compared with the experimental spectrum (Fig. 5.10) in calculating the ground-state branching ratio in the 9 C /3-decay 108 xviii 6.5 Monte Carlo-simulated efficiency of detection for /3-particles with Ep > 4 MeV in the scintillation detector used in Expt. 2, as as function of 9 B excitation energy 110 6.6 A comparison between w\(E) and w'x(E) for the 2.34 MeV and the 12.2 MeV state I l l 6.7 /3-decay, scheme of 9 C measured from this work. The /3-decay branching ratios are indicated 115 6.8 A comparison of the singles "reconstructed" spectrum with the experimen-tal data (Telescope TL) in the low energy region. The experimental data are shown by the open circle points, and the Monte Carlo simulations by the histogram 116 6.9 A comparison of the singles "reconstructed" spectrum with the experimen-tal data in the high energy region. The experimental data are shown by the open circle points, and the Monte Carlo simulations by the histogram. 117 6.10 A comparison of the protons "reconstructed" spectrum with the experi-mental data detected in S2. The experimental data are shown by the open circle points, and the Monte Carlo simulations by the histogram 118 6.11 Efficiency of detection of protons for S3 in coincidence with a-particles in the 8 Be(0 + )+ p channel, for the different beamspot positions assumed. The efficiency curve used in the fitting is for a Gaussian source density of circular symmetry with FWHM=0.25 cm 119 6.12 Triple coincidence efficiency of SSDL, SSDR and T L for the decay through the 5 L i + a for different energy thresholds in the SSDs, the threshold on T L is 200 keV 121 xix B . l An illustration of the decay of a broad state in 9 B through the ground state of 5 L i . The primed quantities refer to the daughter nucleus 156 E . l A n illustration of the kinematics of sequential decay for the breakup of 9 B - ^ 5 L i + a 168 xx Acknowledgments I would like to express my gratitude and appreciation to my supervisor Prof. David F. Measday for his constant guidance, advice and encouragement throughout this work. I am thankful to Dr. L. Buchmann who acted as the spokesperson of the experiment as well as my unofficial second supervisor. Special thanks is given to Prof. E . Vogt, who has been an acting supervisor while my supervisor is away on administrative leave. Thanks are also due to Prof. R .E Azuma and Dr. K .P . Jackson. Their suggestions and discussions at the latest stage of this work are well appreciated. Members of the T R I U M F E682 collaboration need to be thanked for their help during the experiments. I would also like to thank Professors R.R. Johnson and M . McMillan for their advice and reading of the manuscript. I wish to express my appreciation to my fellow graduate students with whom I have interacted over the years. Thanks are to Andre Wong, Belal Moftah, Jimmy Chow, Makoto Fujiwara, Michael Saliba and Trevor Stocki. Finally, I would like to thank my family and friends at home for their constant support and encouragement throughout my studies. xxi Chapter 1 Introduction 1.1 General Since the discovery of the structure of the atom by E. Rutherford in 1911, the proper-ties of the nucleus have been a subject of intense investigation, both theoretically and experimentally. A nucleus with any arbitrary combination of nucleons is not found in nature, and there are stringent limits imposed on the stability of a system of nucleons. set by the different forces that govern the interaction between the nucleons inside the nucleus. This is illustrated in Figure 1.1 that shows the chart of the nuclides indicating the different regions of nuclear mass in the (N,Z) diagram. The features shown in this diagram are determined by the intricate interplay of the strong nuclear force between nucleons, the repulsive Coulomb force between protons as well as by the weak force that results in the /3-decay of nuclei. The dark squares in Figure 1.1 represent the stable or very long lived nuclei found in nature. By adding more protons or neutrons to these stable nuclei, a departure from the region of stability is reached resulting in the loss of binding energy. This region of /3-instability extends to the outer most lines found both in the proton and the neutron rich side of the stable nuclei. These lines, called drip lines define the boundary between nuclei that are stable against nucleon emission and nuclei that are unstable with respect to nucleon emission. The weakly bound systems that are found at the border of stability against nucleon emission are called drip-line nuclei. In the past two decades, a great deal of effort has been spent in studying the properties 1 Chapter 1. Introduction 2 N u c l e a r L a n d s c a p e 126 terra incognita Figure 1.1: Chart of the nuclides showing the different regions of nuclear stability in the (TV, Z) diagram. The stable nuclei are represented by the dark squares. The zig zag lines on both sides of the stable nuclei indicate the present limits of the experimentally observed nuclei, while the outer most lines represent the drip-lines. of nuclei that are far away from the valley of /3 stability using highly improved techniques of production, separation, detection and measurement [1]. As a result, an impressive array of nuclei have been identified and their properties have been studied. This is illustrated by the grey region of Figure 1.1 defined by the two zig zag lines found both in the proton and neutron rich side of the region of stable nuclei. The region defined by these two lines indicates unstable nuclei that have been synthesized in the laboratory up to now. From the view point of nuclear physics, studies of these nuclei are performed not only to identify and characterize new isotopes, but also to gain more insight into the theoretical understanding of nuclear structure and decay. Moreover, a knowledge of the properties of nuclei far away from stability is very crucial for an understanding of astrophysical processes such as the evolution of stellar systems and supernovae, and the synthesis of heavy elements. Chapter 1. Introduction 3 (Z,N) X = 100 ms particle Q„ = 10 MeV particle (Z-1.N+1) Figure 1.2: A n illustration of /3-delayed particle emission. The weakly bound nuclei near the border of stability exhibit certain interesting prop-erties. These nuclei, having extreme neutron to proton ratios are fertile testing grounds for the different nuclear theories that describe the nucleus. As the drip line is approached, the mass difference between the neighboring isobars increases rapidly, resulting in a rel-atively large Q-value for the /3-decay. Owing to this large decay energy, the /3-decay of these nuclei often leaves the daughter nucleus in a highly excited state that decays by particle emission. This decay mode is called /3-delayed particle emission since the emission of the particle comes after the /3-decay, even though it is very fast (snlO - 2 2 s) compared to the /3-decay life time (~ Is) of the parent nucleus. The phenomenon of /3-delayed particle emission is illustrated in Figure 1.2. Because of the large amount of decay energy (on the order of 10 MeV) available in these decays, the decay populates a wide range of states in the daughter nucleus. This allows for spectroscopy to be performed in the daughter nucleus over a broad range of excitation energies. Measurements of the /3-decay branching ratios for the different levels in the daughter nucleus make it possible to determine the decay strength functions over an energy region that are usually not accessible in the /3-decay of nuclei closer to the Chapter 1. Introduction 4 valley of stability. The focus of this thesis is the study of the /3-delayed particle decay of an extremely proton reach isotope of carbon 9 C ( £ i / 2 = 126.5 ms), with = 2, a nucleus with the highest proton to neutron ratio of all bound nuclei above 3 He. The decay of 9 C liberates an energy of 15.5 MeV that is shared between the three decay products ( 9 B, (3+, ve) as shown in Equation 1.1. The daughter nucleus, 9 B can thus be left in a highly excited state. 3C7 B + (3+ + ue, ti / 2=126.5 ms, Q j 9 = 15.476 MeV (1.1) A l l levels in 9 B (including the ground state) are unbound to particle emission, and they can either decay to 8 Be by proton emission or to 5 L i by emission of an a particle. Both 8 Be and 5 L i are also unbound and their subsequent decay results in a proton and two a particles in the final state (Equation 1.2). g.s. V 8Be*+p 5Li* + a 2a + p . (1.2) Due to the complex nature of the decay scheme of the /^-delayed particle decay of 9 C , the (3+ decay branches to the various states in 9 B are not well established. In addition, the decay branches of the states in 9 B are poorly known. A review of the previous theoretical and experimental studies of /3-decay of 9 C , and of the structure of the 9 B - 9 Be mirror pair is given in Chapter 2. In this review, we shall use the compendium of Ajzenberg-Selove [2] to define the properties of the nuclei 5 < A < 10. Before proceeding to Chapter 2, a summary of the work performed in this thesis together with a discussion of the objectives of this work is given in the following Section. Chapter 1. Introduction 5 1.2 Objectives and summary of the thesis The major objectives of this work are the the measurement of the /3-decay branching ratios of 9 C to the states in 9 B , and the identification of the excited states of 9 B along with the determination of their properties (e.g. energies, widths, spins and parities, and particle decay modes). In Ref. [3] it was pointed out, correctly, that to achieve these objectives the acquisition of single-particle spectra is not sufficient, as kinematic broadening severely distorts the spectra and leads to ambiguous interpretations. As the 9 C /3-decay occurs at rest in the laboratory, for each j3 + v energy, the sum of the energies of the two a particles and the proton from the decay of the 9 B daughter nucleus is independent of the successive decay mode when the full solid angle is covered (ignoring small 9 B recoil from the /3-decay). This was indeed achieved in Ref. [3], but, at the price of losing all kinematic information in the sum spectrum. Such kinematic information is crucial for any theoretical understanding of the spectra, because the spectra which are eventually obtained cannot be properly interpreted without knowledge of the separate decay channels. For example, it is easy to see that the penetrabilities for decay channels leading to states in 8 Be and 5 L i , used in such calculations, are quite different from one another. To accomplish the objectives of this work, two experiments with different detector geometries were performed at the TISOL facility at T R I U M F , which provide previously unattainable yields of 9 C , and data were taken and recorded in an event-by-event mode to allow full kinematic reconstruction and analysis. This also allowed the simultaneous collection of singles, and double and triple coincidence spectra. The experimental set-ups for these two experiments (henceforth identified as Expt . l and Expt.2) are described in Chapter 3. After a short description of the TISOL facility and the production of radioactive beams (Section 3.2), a description of the detection systems (Section 3.3), Chapter 1. Introduction 6 electronics and data acquisition systems (Section 3.4) are presented. Energy calibrations are discussed in Section 3.5 and 3.6. Chapter 4 contains a detailed discussion of the Monte Carlo simulations performed for the different breakup modes of the states of 9 B . Kinematics of sequential decay is discussed in Section 4.1. The procedures followed in the simulations are detailed in Sec-tion 4.2. The model adopted for describing breakups through broad states is discussed in Section 4.3. Simulation results are presented in Section 4.4. Calculations of coinci-dence efficiencies are presented in Section 4.5, and angular correlations are discussed in Section 4.6. Data analyses for both experiments are described in Chapter 5. It will be seen that there are two dominant modes of break-up of the states of 9 B , i.e. decay through the ground states of either 8 Be or 5 L i . It is shown that the geometry of Expt . l can be exploited to select kinematically only the break-ups through the ground state of 8 Be, while that of Expt.2 can be used to select preferentially break-ups through the ground state of 5 L i . The data analyses for these two experiments are discussed in Sections 5.2 and 5.3, respectively, with the results of Expt . l shown in Fig. 5.9, Section 5.2.2, and those of Expt.2 in Fig. 5.14, Section 5.3.2. The results presented in these two figures account for more than 40% of the /3-decays of 9 C . In addition to these modes of decay, it was known from previous work that /3-decay to the ground state of 9 B could account for about 50% of the total /3-decays. This is corroborated in the data described in Section 5.3.1 and shown in Fig. 5.10b. Angular correlation results for the E;E=12.2 MeV state are presented in Section 5.3.3. The three sets of results, Figs. 5.9, 5.14 and Fig. 5.10b, are combined in Chapter 6 to give the final /3-decay branching ratios as well as the experimental energies, widths, and particle-decay branching ratios for the states of 9 B . These results were extracted from fits to the experimental spectra, where each level was assumed to be an isolated Chapter 1. Introduction 7 resonance that is represented by the one-level approximation in the .R-matrix theory. The theoretical basis for this model are formulated in Appendix B and C. Conclusions are presented in Chapter 7. Chapter 2 Review of Previous Work The nucleus 9 C is the first in the series of Tz = — §, A = 4n+1, /3-delayed proton emitters that have been investigated experimentally up to 6 1 Ge [4, 5]. For most of the nuclides in this series, the particle spectra following /3-decay are composed of several narrow resonance peaks that provide a wealth of spectroscopic data in the daughter nucleus. In the case of 9 C , however, the singles particle spectrum is dominated by a continuum of a and proton energies in the range of 100 keV to about 14 MeV, with only a few proton peaks. This is because, except for the decay through the 8 Be ground state, the decay of 9 B proceeds through broad states that result in the emission of particles with energies that are not related to the excitation energy in 9 B in any simple way. For this reason, inadequate experimental data exist on the branching ratios of the decay of 9 C and the state properties of the daughter nucleus 9 B . The previously known states of 9 B that can be populated by the j3 decay of 9 C are listed in Table 2.1, and the decay scheme of 9 B following the (3 decay of 9 C is given in Figure 2.1. This level scheme is adopted from Ref. [2]. In the following section, we shall review the previous experiments performed to mea-sure the decay scheme of the /3-delayed particle decay of 9 C . Experimental and theoretical studies of the 9 B - 9 Be mirror pair are discussed in more detail in Section 2.2. 8 Chapter 2. Review of Previous Work 9 Figure 2.1: Level diagram of 9 B with respect to the decay of 9 C , also showing possible decay channels of 9 B . The level properties are adopted from Ref. [2]. Chapter 2. Review of Previous Work 10 Table 2.1: Previously known 9 B states below 15.476 MeV, the energy available in 9 C decay. Information on the properties of 9 B is taken from Ref. [2]. The states that are expected to be strongly populated by the (3 decay of 9 C are of (Jn = l / 2 ~ , 3/2~, 5/2~) corresponding to an allowed Gamow-Teller transition. E x (MeV) T(keV) Jn,T Decay mode g.s. 0.54±0.21 3 - . l 2 ' 2 P (1.6) «700 1 + . 1 2 .' 2 (P.a) 2.361(5) 81±5 5- . 1 2 ' 2 a 2.788(30) 550 ± 40 (3 5\+ . 1 V.2'2'' ' 2 P 4.8 (1) 1200 ±200 a 6.97 (6) 2000±200 7 - . 1 2 ' 2 ft-) • I V2 > ' 2 P 11.70(7) 800 ±50 P 12.06(6) 800±200 . 1 ' 2 P 14.01(7) 390 ±110 . 1 ' 2 14.6550 (2.5) 0.395±0.042 3 - 7 3 2 ' 2 7.P 14.70 (18) 1350±200 ( 5 - ) • I V2 / ' 2 15.29 (4) . 1 ' 2 2.1 (3 decay studies The first observation of 9 C was made in 1956 by Swami, Schneps and Fry [6] in a nuclear emulsion. They were able to observe a connected double star in a photographic emulsion exposed to 300 MeV protons. They interpreted the secondary star as coming from the track of an electron, a proton and two a particles from the decay of 9 C . In 1971, Mosher, Kavanagh, and Tombrello [7] determined the mass and the half life of 9 C by detecting the delayed protons from the reaction 7Be(3He, n) 9 C. In 1965, Hardy et al. [8] performed the first experimental measurement of delayed protons following j3 decay of 9 C . They produced 9 C from 12C(p,d2n)9C, 1 0 B(p, 2ra)9C and nB(p, 3n) 9 C reactions, and measured the singles particle spectra. They observed two proton peaks on the high energy side of the spectrum. However they were not able to observe transitions that result in low Chapter 2. Review of Previous Work 11 energy protons and a-particles, due to the /3 pile-up in the low energy region. Esterl et al. also measured the /3-delayed particle spectrum over the range of 12 MeV [9], but in the presence of an intense (5 tail. They also observed a number of peaks which they identified with states in 9 B . More recent work was performed by Mikolas et al. [3]. They produced 9 C by bom-barding a thick Ni target with a 1 2 C ion beam using the cyclotron at the National Superconducting Laboratory in Michigan. They implanted 104 9 C ions in a pair of A E - E detector telescopes and measured the 9 B energy from the energy of the decay particles whenever all the decay particles were contained in the detector. In their spectra, they were clearly able to observe peaks corresponding to transitions to the ground state, and to the states at 2.36, 2.79, and 12.06 MeV. From this measurement, they deduced the branching ratios for the transitions to the ground state (60 ± 10 % ), and the states at 2.36 and 2.78 MeV (17 ± 6 % ) and (11 ± 5% ) respectively. They also set a lower limit (1.8 ± 0.6 % ) on the branching ratio to the 12.06 MeV state. No identification of the decay channels of the states of 9 B was made from this experiment. /3-decay of 9 L i Like its mirror counterpart, the /^-delayed particle decay of 9 L i is also complicated since all the excited states of the daughter nucleus 9 Be are unbound and their decay results in 2a particles and a neutron, involving several possible intermediate decay channels. Chen et al. [10], and Macefield et al. [11] measured the neutron time of flight spectrum from the /3-delayed particle decay of 9 L i . Chen et ai, determined the ground state branch to be (65 ± 3)% . In another /3-neutron coincidence experiment by Bjornstand et al. [12], and Langevin et al. [13], they measured the ground state branching ratio and found (50 ± 3) % . A more detailed study was recently performed by Nyman et al. [14] at the ISOLDE Chapter 2. Review of Previous Work 12 facility at C E R N . They measured the neutron time of flight and the a energy spectra from the decay of 9 Be separately and determined the branching ratios for the decay of 9 L i to five excited states in 9 Be. For the ground state transition, they adopted the value from Ref. [12] and [13]. They used one-level i?-matrix formalism to fit their spectra, and to determine the Gamow-Teller strengths for transitions to these levels. In particular they measured a value of B G T of 5.6 for the transition to a level at 11.81 MeV. In a recent theoretical calculation of B G T for (3 decay of 9 L i performed in Ref. [15], a strong resonance is predicted in the region £' x=8.5-11.5 MeV. When the B G T values for the 9 L i decay in Ref. [14] are compared with the B G T values for the mirror transitions in the /3-decay of 9 C measured in Ref. [3], discrepancies exist. The source of these discrepancies could be due to the different interpretations of the spectra in the two experiments. 2.2 The mirror pair 9 B - 9 B e Over the past four decades, numerous transfer reaction experiments have been performed to populate and study the properties of the levels of 9 B . A compilation of the experimental and the theoretical works is found in Ref. [2]. The interpretation of the experimental data from the reaction studies is often times complicated by the four-body final state that can sometimes result in misidentification of a peak or the break-up mode. In the following sections, we shall discuss the low lying states of 9 B - 9 Be (Figure 2.2) in more detail. 2.2.1 The ground state The ground state of 9 B is unbound to proton emission by 185 keV, and has a width of only 0.54 keV and decays through the 8 Be ground state. Its relatively small width is due to the small amount of energy available for its decay. This state has been observed in Chapter 2. Review of Previous Work 13 Be + p Figure 2.2: Low-lying states in 9 B - 9 Be mirror pair. The thresholds for nucleon and a emission are indicated. several reaction experiments [2]. According to the shell model configuration discussed in Appendix D, this state is expected to have a spin parity assignment of 3/2~, and this has been confirmed experimentally [2]. The ground state of 9 B is strongly populated by the /3-decay of 9 C . The mirror state in 9 Be is bound. 2.2.2 T h e Jn = 1/2+ s ta te The first excited state of 9 B is a | + state and is predicted to lie in the region Ex = 1 — 2 MeV. The analog state in the mirror nucleus 9 Be is very well known, yet despite many attempts to measure this state in 9 B , its existence has not been confirmed unambiguously, and different theoretical calculations predict different level widths and locations. This state should not be strongly populated in the p+ decay of 9 C since the P transition to this state is a first forbidden transition. However, owing to the large Q-value for the /3-decay, a population in the range of 1% is possible. Chapter 2. Review of Previous Work 14 This level is expected to have a shift from the level of the mirror nucleus. This expected shift, also known as the Thomas-Ehrman shift is mainly observed in unbound states. In the context of the i?-matrix theory, this shift results from the discrepancy in matching the external and the internal wave functions at the surface for the case of 8 Be + p and 8 Be + n. The reason for this shift is the distortion of the Coulomb wave function of the outer proton when compared with the wave function of the corresponding neutron in the region beyond the channel radii. The shift is larger when the long range behavior of the wave functions is very different as is the case in mirror pairs with a bound neutron and a corresponding unbound proton. In the case of the first excited states of 9 B and 9 Be for example, the 9 Be first excited state is unbound by only 20 keV while the corresponding mirror state in 9 B is predicted to be unbound by more than 1 MeV. This shift is well established for the first excited states of the mirror pairs of 1 3 N and 1 3 C [16, 17]. Several transfer reaction experiments have been performed to locate the energy and width of the first excited state of 9 B , however contradictory results have been obtained. Difficulty in locating this state stems from the fact that it is expected to be a broad state (T ~ 1 MeV). Another difficulty encountered in a majority of these experiments is due to the large background caused by the strongly populated nearby states such as the 2.36 MeV state. A summary of the experimental data available for this state is given in Table 2.2. The results obtained from these experiments are not in agreement. Several theoretical calculations have also been performed to predict the location and the width of this state. Barker [24], calculated the shift for the first excited state of 9 B with respect to the location of the analogue state in 9 Be using the i?-matrix method. His calculation for the shift from the mirror level include contributions from: • Internal Coulomb interaction. Chapter 2. Review of Previous Work -~ 15 Table 2.2: Experimental measurements for the first excited state of 9 B . Experiment E e x (MeV) T (MeV) Reference" 1 0 B( 3 He,a) 1.5 «0 .7 [18] 9Be( 3He,t) 1.16±0.05 1.08±0.05 [19] 10B(3He,oO 1.8±0.2 0.8±0.3 [20] 9 Be( 6 Li, 6 He) 1.32±0.08 0.86±0.26 [21]1 6 Li ( 6 Li , t ) 1.6±0.1 0.77 [22] xThe findings in Ref. [21] were not later confirmed by a later, more precise, measurement performed by Catford et al. [23]. • Electromagnetic spin-orbit interaction. • The different external wave functions in 9 Be and 9 B . and found E e x=1.8 MeV and T=l-2 MeV. Descouvemont [25] used a microscopic cluster model with a 8 Be core and a nucleon in the calculation of the low-lying states of 9 Be and 9 B . He performed a phase-shift calculation of 8 Be+n and 8 Be+p and found E x=1.34 MeV and T=1.3 MeV for the | + state in 9 B . This model, however, predicts a bound state for the first excited state of 9 Be. Sherr and Bertsch [26] used a single particle model to calculate the Coulomb dis-placement energy between the mirror levels and found E x=0.93 MeV and T=1.4 MeV. Recently, Fortune [27] systematically studied the energy differences in the splitting be-tween the 2s i and the 2ds levels in a number of light nuclei in the A = l l - 1 7 region. These 2 2 differences were interpreted as due to change in TZ which was assumed to be proportional to a quantity called A that was common to all the pairs. He found a nearly constant value of A for all the pairs. He then used this value to calculate the difference in the splitting in the 9 B e - 9 B pair, from which the first excited state of 9 B was deduced. In his recent paper, Barker [28] showed that, when applied to the mass-9 system, the value of A departs significantly from the constant value assumed by Fortune. Despite all the Chapter 2. Review of Previous Work 16 Table 2.3: Various theoretical calculations of the first excited state of 9 B . Model E e x (MeV) T (MeV) Reference 9 Be + Coul. Displ. 2.0 1-2 [24] Microscopic Three Cluster 1.16 1.3 [25] Single Particle 0.93 1.4 [26] One-Body Potential 1.13 - [27] effort spent in studying this state, its existence is still a subject of major controversy. To illustrate the spread in the theoretically predicted values of the width and the location of this state, the results are summarized in Table 2.3. 2.2.3 The Jn = 5/2" state The lowest \~ state in 9 B is located at 2.36 MeV above the ground state and has a width of 81 keV. It almost always decays to the ground state of 5 L i . The first detailed measurement of the decay mode of this state was performed by Wilkinson et al. [29]. They populated this state by the 10B(3He, a)9B reaction and measured the outgoing a particles in coincidence with a proton from the decay of 9 B . This coincidence measurement allowed them to establish the decay mode. They saw no evidence for any decay through the 8 Be channel and set an upper limit of less than 1% for this decay. This state has also been observed in other transfer reaction experiments [19, 21]. Wilkinson et al. studied the decay mode of this level to measure the admixture of the p-shell configuration, i.e. ( l s ) 4 ( lp ) 5 with other high-lying configurations for this state. If this level has a pure p-shell configuration, it cannot decay through the p+ 8 Be(0 + ) channel, since this requires the proton to have / = 3. On the other hand, an observation of such a decay mode would indicate that this state is an admixture of ( l s ) 4 ( l p ) 4 ( l / ) 1 with the p-shell configuration. Chapter 2. Review of Previous Work 17 A theoretical investigation of the mirror state (2.43 MeV state in 9 Be) by Henley and Kunz [30] found that between 5 and 20 % of the decay of this state proceeds through the n+ 8 Be(0 + ) channel. This prediction is supported by /?-decay experiments performed by Chen et al. [10] and in (3He,p) transfer reaction experiments performed by Christensen et al. [31]. This discrepancy in the decay modes for these two mirror states is proba-bly due to the difference in the energy available for the decay through the a channel (Figure 2.2). The restricted phase space for the decay of 9 Be through the a channel, makes the n+ 8 Be(0 + ) more likely. This restricted phase space is also reflected in the large difference in the widths of these states. The width of this state in 9 Be is ~ 1 keV [32], whereas the mirror state in 9 B has a width of 81 keV. 2.2.4 T h e J* = 1/2" s ta te The lowest \~ state in 9 Be is located at 2.78 MeV, and has a width of 1080 keV. The mirror state in 9 B , on the other hand is not very well-known. In Ref. [25], Descouvemont's calculation predicts that the mirror state in 9 B should lie near Ex=2.5 MeV, with a width of about 1.1 MeV, decaying mainly to the ground state of 8 Be. Descouvemont also pointed out that this state had not been observed experimentally. His model also requires a nearby | + state, and a relatively narrow state has been observed at Ex=2.78 MeV, T= 550 keV in transfer reaction experiments [2] that is assumed to be «/* = | + or | + . Evidence for a | state was first seen in (p,n) reactions by Clough et al. [33], Anderson et al. [34] and lastly by Byrd et al. [35]. Better evidence was found by Fazely et al. in a (p,n) reaction experiment [36] at IUCF which was analyzed in detail by Pugh [37]. From a fit to the neutron spectrum from the (p, n) reaction experiment in Ref. [37], Pugh assumed that two states were needed in addition to the 2.36 MeV level viz:- Ex = 2.71(1) MeV and T = 0.7(1) MeV and another at Ex = 2.75(3) MeV and T = 3.1(2) MeV. This analysis was never published but is highly likely that there are two levels one | + , one Chapter 2. Review of Previous Work 18 \ , very close together near 2.8 MeV. The calculations of Descouvemont require both, and all shell model calculations demand a low-lying | state, but do not include positive parity states in their space. The best measurement of the \ state comes from recent studies by Tiede et al. in 6Li(6Li,t)9B reaction [22]. They made a fit to their spectrum by including both the \ and the | + states and by fixing the width and the location of the | + state. They found the best x2 from their two-state fit for _E'X=2.91, T=3.05 MeV for the | state. This state has also been observed in the /3-decay of 9 C by Mikolas et al. [3], who found Ex = 2.9 MeV and T = 1.6 MeV. However these values do not appear to have been extracted from a fit. 2.2.5 The £ x = 1 2 . 0 6 MeV state The state at 12.06 MeV has a width of about 400 keV and it decays both by proton and a emission. The existence of this state has been observed in the /3-delayed particle decay of 9 C . Estrel et al. [9] observed a peak in the /3-delayed proton spectrum that they identified with this state. Mikolas et al. [3] also observed this state decaying by a emission in their spectrum. The spin and parity of this state have not been determined definitively. However, because of the large strength of /3 transition to this state, it is possible to conclude that the Jw of this state should be either ( l /2~, 3/2~ or 5/2 _ ) . In the /3-decay study of 9 L i to 9 Be, Nyman et al. reported observation of two states at 11.28 and 11.81 MeV with a rather strong BGT value of 5.6 ± 1.2 for the 11.81 MeV state. This state is probably the analogue state to the 12.06 MeV state in 9 B . 2.2.6 The Jn = 3/2~ isobaric analogue state The isobaric analogue state has been identified through reaction experiments and it is located at 14.65 MeV. Because particle decay for this state is isospin forbidden, it has a Chapter 2. Review of Previous Work 19 very small natural width (T = 0.4 keV). Due to the small energy difference between this state and the ground state of 9 C , the /3-decay branch to this state is expected to be very small. Chapter 3 Experiment 3.1 Introduction The work of this thesis is based on two separate experiments carried out at the T R I U M F isotope separator facility (TISOL). The first experiment was performed during a course of a one week run in September 1995. Another experiment which ran for a period of 5 days was performed in August 1997. These two experiments will be discussed in this Chapter; with emphasis on the beam production, experimental set-ups and energy calibrations for these two experiments. Both experiments used the same production methods for the 9 C beam, but different detector configurations and electronics were utilized. The beam production method is discussed in Section 3.2. In Section 3.3, the experimental set-ups for both experiments are described. The electronics and data acquisition systems are explained in Section 3.4, and energy calibrations are discussed in Sections 3.5 and 3.6. Finally in Section 3.7, we discuss the method of establishing coincidences from the timing information measured. 3.2 9 C production 9 C (< 104 s _ 1 ) was produced at the T R I U M F isotope separator facility (TISOL) [39] using a 13 g/cm 2 Zeolite production target and the TISOL E C R ion source [40]. A layout of the TISOL facility is shown in Figure 3.1. A 500 MeV proton beam of up to 1.5 / iA from the T R I U M F cyclotron was used to irradiate the Zeolite target in the form 20 Chapter 3. Experiment 21 of small pellets. The target was continuously heated under vacuum to about 600 °C and the molecules that diffuse out of the target were transported to the ion source via a short transfer tube. After ionization, a 12 keV ion beam was extracted, and magnetic mass separation was used to select the desired nuclide. In this experiment the separator was operated at mass 25 in order to select the 9 C isotope that is produced in the form of a molecular ion ( 9 C 1 6 0 + ) 1 . Following mass analysis, the ion beam was directed to the collection chamber where the beam is stopped at a 10 pg/cm2 C catcher foil, and the decay products are observed by the multi-detector set up, consisting of several silicon semiconductor detectors. A liquid nitrogen trap and a collimator were mounted upstream close to the detection chamber to improve the vacuum and to minimize carbon deposits on the beam catcher foil. A Faraday cup downstream of the experimental set-up assisted the focusing of the ion beams. With the 12 keV mass A=25 ( 9 C 1 6 0 + ) beam from the TISOL isotope separator, the 10 pg/cm2 C foils did not show any sign of deterioration in the course of the experiment as there was no significant stable ion beam at this mass position (7. < 1 nA). However, strong yields of the short-lived radioactive isotopes 1 1 C (ti / 2=20.4 min, n C 1 4 N + ) and 1 3 N (t 1 / 2=10.0 min, 1 2 C 1 3N+) were observed. These two isotopes caused considerable /3-background in the low energy regions of detection. During Expt. 1, the ion beam was deflected out periodically in the beam transport system, for a small fraction of the run time in order to test for the presence of longer-lived contaminant beams, and no charged particles from long-lived nuclide decays were identified. ^^ No 9 C was observed at the A=9 position. Figure 3.1: A schematic diagram of the TISOL isotope separator facility. Chapter 3. Experiment 2 3 3.3 Experimental setups 3.3.1 Detection system The measurement of the /3-delayed particle decay of 9 C requires a detection system that is capable of detecting protons and a particles over the range of 100 keV to 14 MeV with good energy resolution and coincidence efficiency in a high background environment caused by the (3 particle flux. Silicon detectors with their good energy resolution were the natural detector of choice in these experiments. It is also preferable if the detection system can distinguish between the different particles being detected. This was partly achieved by using a pair of thin and thick surface-barrier detectors (also called A E - E telescopes). The rate of energy loss per path length for charged particles, as they traverse through matter, is dependent on their mass and charge. A comparison between the energy loss in the front detector and the energy deposited in the back detector can help identify the particle (ft, p, a) in this experiment. For particles that are not energetic enough to break through the front detector, however, it is not possible to make particle identification using this method. Moreover, identification of the /3-particles with the A E - E telescopes is not always possible, since there is a small probability that the /3-particles can scatter nearly parallel to the surface of the detector, depositing a large fraction of their energy in the detector as a result. This interaction produces a high-energy exponential tail in the particle spectra observed by these detectors. The exponential tail intensity is a function of the thickness of the detector. Hence the A E - E detection system cannot altogether remove all the (3 background, although appropriate cuts remove a fair fraction. As discussed in Chapter 1, due to the three-body nature of the 9 B break-up, the information obtained from the singles spectra following the break-up of 9 B is very limited. We thus employed a coincidence technique, whereby events detected by each detectors Chapter 3. Experiment 2-1 Carbon Collimator Figure 3.2: Setup for Expt. 1. The dimensions are given in Table 3.1. The axes of S2, S3 and the annular detectors all lie in the same horizontal plane. The collection foil is rotated by 45° about the vertical axis. are recorded simultaneously and a software coincidence could be established during the analysis. In order to have full kinematic information on the coincident particles, position-sensitive detectors were used in the second experiment. We now discuss the experimental setups for the two experiments. 3.3.2 Experimental setup, Expt. 1 Schematics of the experimental setup for Expt. 1 is given in Figure 3.2. A multi-detector setup was built around a thin (10 //g/cm 2) carbon collector foil in which the ion beam was stopped. The setup consists of a movable target ladder (not shown in the figure) holding several carbon foils and an a calibration source, and 4 A E - E telescopes arranged Chapter 3. Experiment 25 \ 1 7 7 /\> / v \ / 2 / 7 / - O S -, / t - B - B - . / 3 Figure 3.3: A schematic diagram of the setup for Expt. 1, showing the location of the detectors with respect to the co-ordinate axes. 6 and </>, given in Table 3.1, are indicated for S4. in face-to-face pairs as well as a pair of annular detectors positioned on the beam axis also in face-to-face geometry2. Protons and high energy a-particles were detected in the A E -E telescopes and the annular detectors. The A E - E telescopes and the annular detectors are Si surface-barrier detectors of various thicknesses and dimensions (Table 3.1). The A E (front) detectors were either 10 or 30 pm thick, while the E-counters were 1500 or 2000 /xm thick to exceed the full range of the protons. The annular detectors were 300 pm thick. The dimensions of the detectors together with the locations of the center of the front detectors with respect to the beam spot center are given in Table 3.1. With this arrangement, a coverage of about 8% of 4-7T was obtained. It will be later shown in Chapter 5 that the coincidence spectra between the opposing telescopes (S1-S4, and S2-S3) were the most useful in this experiment. 2The arrangement of the detectors is mostly in face-to-face pairs. This is because two of the three particles from the break-up of 9 B are emitted nearly back-to-back. Chapter 3. Experiment 26 Table 3.1: Dimensions and location of the various silicon detectors used in Expt. 1, 9 and <j) are as indicated in Figure 3.3. Detector Thickness (pm) Active area (mm2) distance (mm) 9 (degrees) 4> (degrees) si I F 1 b i i B l 30 1500 100 100 33±2 45 225 S 2 J F 2 b Z \ B2 10.6 2000 50 100 22±2 90 " 270 S 3 { B3 12.9 2000 50 100 17±2 90 90 « { BI 30 1500 100 100 38±2 135 45 F5 300 830 48 ± 2 90 0 F6 300 830 60±2 90 180 3.3.3 Experimental setup, Expt. 2 The main features of the second experiment were the use of position sensitive detectors and the additional detection of (3 particles with a thick plastic scintillator. A pair of A E - E telescopes as well as two position sensitive double-sided silicon strip detectors (SSDs) 3 with 5 cmx 5 cm x 300 pm were used for detection of protons and a-particles. A schematic of the experimental setup for Expt.2 is given in Fig. 3.4. The detectors are referred to with the labels shown in this figure (SSDL, SSDR, T L , TR) . Each SSD was placed face-to-face with respect to a A E - E telescope. Each A E - E telescope has an active area of 100 mm 2 . The thickness of the front and back detectors F L and F R are 50 and 30 pm respectively, while the back detectors are 2000 pm thick. The collector foil was located near the mid-point between the two detectors and oriented perpendicular to the beam direction. With this arrangement, a solid angle coverage of about 15% of 47r was obtained for proton and a particle detection. This setup allowed a measurement of the 3Manufactured by Micron Inc. U . K . . Chapter 3. Experiment 27 angles between the coincident particles being detected in the back-to-back detectors with an angular resolution of about 11°. A thick plastic scintillation detector (</> = 6 cm, I = 9 cm) was placed directly along the beam direction for the detection of (3 particles. Triple coincidences were selected to provide most of the information for this run, and this is discussed in detail in Chapter 5 where it is shown that this setup preferentially selects decays through the ground state of 5 L i . The Silicon Strip Detectors The SSDs are made by etching several individual electrodes onto the surface of a rectan-gular silicon wafer both on its front and back. Dividing the electrodes into several pieces results in a corresponding geometric division of the charge-collecting electric field. The interaction of a particle in the detector generates a charge pulse in one of the p-contact strips (columns) and in one of the n-contact strips (rows). The position of interaction is then localized as having occurred in the detector pixel where the electrode strips cross. At room temperature the strip detectors posed some problems because of high leakage currents, especially when some stable ion beam from the separator was present. Therefore a cooling system was designed that kept the SSDs at around 4 °C. 3.4 Electronics and data acquisition The main goal of the electronics and the data acquisition system in this experiment was to collect and process data from all the detection components whenever a valid event is registered in any of the detectors. In doing so, all the kinematic variables of the decay particles could simultaneously be recorded for coincident events. The relative timing between the trigger and an event detected in any of the detectors was also measured. This timing information is later used to discriminate against random coincidence events Chapter 3. Experiment 28 Double sided Scintillator Figure 3.4: Setup for Expt. 2. The axes of all the detectors and the collector foil lie in the horizontal plane, and the angle between the beam axis and the axes of the detectors is 45° for the strip detectors, 46°m for T R and 49° for T L . The distance of each detector from the center of the foil is 4.5, 4.7, 4.2, 4.1 and 5.8 cm for T L , TR, SSDL, SSDR, and the (3 detector respectively. Chapter 3. Experiment 29 Double-sided SSD Front strips -mm Pre-amp Motherboard A N A L O G U E O U T P U T Shaping Amplifier and Discriminator Board L O G I C O U T P U T A N A L O G U E O U T P U T T O A D C , T D C Figure 3.5: A schematic diagram of the Silicon Strip Detector and the readout system. The signals from the adjacent strips were paired together due to lack of enough electronics components such as amplifiers and ADCs. Chapter 3. Experiment 30 (128 ns) STOP TFA CFD Com. START Computer Busy 1 Det. PA TR1G1 VETO Coin. Unit MASTER TRIGGER GATE 1- ° J ADC SA Figure 3.6: A schematic diagram of the essential features of the electronics used in Expt. 1. mainly caused by the high (3 flux environment. 3.4.1 Electronics, Expt. 1 The major features of the electronics system are shown in Figure 3.6. Each detector is connected using a co-axial cable to a pre-amplifier (OR142B PA), located outside the vacuum chamber. In order to reduce the noise pickup, the detectors were electrically isolated from the support structure. A precision pulser signal was fed to the test input of each pre-amplifier in order to check the electronic resolution and to monitor the stability of the electronics. Each pre-amplifier provided two output signals: energy and time signals. The energy signal from each pre-amplifier was sent to a spectroscopy amplifier (TC241 SA) in order for the signals to be shaped and amplified. The outputs from the spectroscopic amplifiers were then sent to the high resolution voltage A D C (Lecroy AD811), where their energy-dependent amplitudes were digitized and recorded. The timing signals obtained from the pre-amplifier outputs were sent to timing filter amplifiers (OR474 TFAs) for shaping and amplification. Each T F A output signal was then fed to a Constant Fraction Discriminator (OR934 CFD) . The discriminators were set to trigger on any pulse above a certain energy threshold which is set above the noise level. For Chapter 3. Experiment 31 input signals above the set threshold level, the discriminators generate logic pulses as their output. One of the C F D outputs from each detector were combined into a single Master Trigger to start the data acquisition. This signal also opened a gate for all the A D C channels, and in addition was used as the T D C common start signal. Another C F D output from each detector was sent to T D C (Lecroy 2228A) stop after a 128 ns delay. The T D C information was useful in establishing software coincidences between the various detector pairs during the analysis (Section 3.7). 3.4.2 E l e c t r o n i c s , E x p t . 2 Te le scope e l e c t r o n i c s a n d t r i g g e r l og i c The electronics for the telescope detectors is similar to that employed for Expt . l , except that in addition to the detector telescope signals, the signal from the (3 detector was also included in the trigger logic. The SSDs were not included in the trigger due to the high count rate observed by these detectors. S S D e l e c t r o n i c s A schematics of the readout system of the SSD signals has been shown in Figure 3.5. The multiple signals from each SSD are processed with a modular amplifier system developed at the Rutherford Appleton Laboratory [41]. The complete circuit diagram is shown in Figure 3.7. Each SSD provides 32 signals (16 from the back strips and 16 from the front strips). Due to the lack of electronic components such as amplifiers and ADCs to process these signals, the signals from adjacent strips for both SSDs were combined4 and routed to the Charge Sensitive Pre-Amplifiers (CSPAs) that are mounted as close as possible to the SSDs outside the vacuum chamber in order to minimize the noise. The amplified 4Not shown in Figure 3 .7 but shown in Figure 3 .5 . Chapter 3. Experiment 32 output signals from the CSPAs were fed to the shaping amplifiers (RAL108 8-channel SAs) via twisted ribbon cables that connect the CSPA motherboard and the SAs. For each input signal that is received from the CSPA motherboard, the Shaping Amplifier (SA) produces a shaped and amplified signal. It also generates a logic pulse for each input signal that is above a certain threshold using a leading-edge discriminator. The shaped and amplified signals from the shaping amplifiers are then sent to a peak-sensing A D C . The logic outputs are sent to the 32-channel T D C (Lecroy 2277) after passing through the Lecroy 4532 Majority Logic Unit ( M A L U ) . The Analog Majority Output (AMO) from the M A L U for each SSD detector was also sent to the Lecroy 2228A T D C . The M A L U has 16 input channels. In the A M O output channel, the M A L U provides a NIM signal, if it receives one or more input signal in the 16 channels. The minimum number of input signals needed to create an output can be adjusted to be between 1 and 16. In this case the requirement was that a single signal was registered in one of the 16 channels so that any of the 16 signals from the front and the back strips that were above threshold were registered by the T D C . 3 . 4 . 3 Data acquisition The data acquisition for both experiments was controlled by the T R I U M F V D A C S sys-tem [42] and it consists of a single-crate C A M A C acquisition system, a V A X workstation and a PDP-11 front-end processor housed in the C A M A C crate also known as the Star-burst. The V A X computer was used for online monitoring, data storage and down-loading the user-defined T W O T R A N program to the Starburst. The T W O T R A N program [43] instructs the Starburst to collect the relevant data from the C A M A C modules, once the Starburst is activated by the Master Trigger. For each trigger event, the energy and timing signals from each detector were digitized and recorded on an 8 mm V C R tape. The Chapter 3. Experiment 33 m in o 73 in in o 34—way Vacuum Feed—through RAL 108 Preamp Motherboard Figure 3.7: A schematic diagram of the electronics for the silicon strip detectors. Chapter 3. Experiment 34 event bit pattern, which is used to identify the detector signal that caused the trigger was also recorded. On-line monitoring was performed using the TRIUMF-developed analysis program NOVA [44], which allows the user to provide the definition of various spectra as well as conditions or cuts to be imposed on the data to select events for inclusion into specific histograms. This analysis software was also used later for the off-line analysis (Chapter 5). 3.5 C a l i b r a t i o n r u n s Besides the standard a calibration source runs ( 2 4 1 A m and 1 4 8 G d ) , 1 7 Ne and 1 8 N cali-bration runs were performed. These calibration runs and identification of the calibration peaks from these sources are discussed in this Section. 3.5.1 1 7 N e C a l i b r a t i o n r u n The /3-delayed particle spectrum of 1 7 Ne consists of several mono-energetic proton as well as a peaks from the decay of the daughter nucleus 1 7 F [45]. This was an ideal calibration source for this experiment, since it constitutes several well-known proton peaks ranging from 0.8-10 MeV as well as a few a peaks, and because of the large yield of this isotope obtained from TISOL. The peaks of interest for the calibration in this work are given in Table 3.2. Because this spectrum is complicated with many closely spaced peaks, proper identi-fication of the peaks of interest is essential. As given in Table 3.2, the two decay modes for 1 7 F following /3-decay of 1 7 Ne are as follows: 17F*—>P+l60 (3.1) 17F* —• a +13 N Chapter 3. Experiment 35 Figure 3.8: Particle spectrum (mainly protons) from the 1 7 N e run recorded by the back detector of S2 in Expt . 1. Note that the particles are detected by B2 after passing through the front detector F2. Chapter 3. Experiment 36 Table 3.2: Identification of the major a and proton peaks in the /3-delayed particle decay of 1 7 Ne decay. Energy (keV) particle transition 864 P i 1 7F(8.44) -* 1 6 0 (6.92) 1341 p2 1 7F(8.07) • -> 1 6 0 (6.05) 1680 p3 1 7F(8.44) -» 1 6 0 (6.05) 1725 al 1 7F(8.07) -> 1 3 N (g.s.) 2301 a2 1 7F(11.19) 1 3 N (g.s.) 4598 P 4 1 7F(5.49) - 1 6 0 (g.s.) 5114 p5 1 7F(6.04) -> 1 6 0 (g.s.) 7031 p6 1 7F(8.07) - 1 6 0 (g.s.) 7371 p7 1 7F(8.44) -> 1 6 0 (g.s.) 9965 P 8 1 7F(11.19) - 1 6 0 (g.s.) Neglecting the small recoil of 1 7 F * from the (3 decay, the break-ups of 1 7 F given in Equation 3.1 occur while the 1 7 F is at rest in the lab frame. Hence, the available energy from the decay of 1 7 F is shared between the two decay products according to their masses. The ratio of the energy of the protons to the 1 6 0 energy will be 16:1 and the a to 1 3 N energy ratio will be 13:4. These ratios can be used to discriminate between the proton and the a peaks using a 2-dimensional plot of the energy of the coincident events. For break-ups of 1 7 F with low Q-values, the 1 6 O events from the 1 7 F —>16 O + p decay will have too low an energy to be detected and only the 1 3N-o; events are observed in the coincidence spectrum. In both experiments, the separator was switched to A=17 at the beginning and end of the 9 C runs to perform 1 7 Ne calibration run. A particle spectrum from the decay of 1 7 Ne recorded by the back detector of S2 in Expt. 1 is shown in Figure 3.8, showing a number of proton peaks corresponding to those that are listed in Table 3.2. Chapter 3. Experiment 37 120 3 O 100 300 500 700 E F 1 (Channels) 900 1100 Figure 3.9: a particle spectrum from the 1 8 N run recorded by the front detector of SI in Expt. 1. 3.5.2 1 8 N Calibration Run For the 1 8 N run, the separator was operated at the A=32 position ( 1 8 N 1 4 N + ) . In contrast to the 1 7 Ne spectrum, the 1 8 N spectrum is quite simple, having only two major narrow a peaks at 1081.0 and 1409.0 keV [46]. These are very suitable a calibration peaks for this experiment as most of the a particles from 9 C decay are in this energy range, and because these two peaks are well-known. These peaks have a very small natural width. However, they are kinematically broadened by the (3 - v recoil effects discussed in Section 4.7, and the calculated broadening for these two lines are 39 and 43 keV respectively [47]. The a spectrum from the 1 8 N run recorded by the front detector of S i in Expt. 1 is given in Figure 3.9. Similar spectra are obtained for the front detectors of the telescopes in Expt. 2. However, it was not possible to operate the SSDs for the 1 8 N run due to their high Chapter 3. Experiment 38 leakage current for this run. The high leakage current observed in these detectors was found to be correlated with the intensity of the stable beam current that is being stopped in the collector foil. Unfortunately at mass 32, the stable beam current (mainly 1602~) intensity was as high as 8 pA, and it was not possible to operate the detectors under this condition. Source runs For the a energy calibration, we also used 2 4 1 A m and 1 4 8 G d sources which emit 5485.7 and 3182.8 keV a particles respectively. 3.6 Energy calibration Accurate determination of the energies of the particles being measured is a very important aspect of these experiments as the state energy of 9 B , as well as its breakup mode, are determined from the energy and the direction measurements of the decay particles. One of the difficulties encountered in the energy calibration procedure is the pulse height defect, i.e, different ionizing particles of the same energy do not produce a signal of the same amplitude [48]. Consequently, the detector calibration is different for different particle types. The pulse height defect is caused by various effects, and the three different contributions to the total pulse height defect are given in Equation 3.2. A = Aw + An + A r (3.2) In the above equation, A ^ is known as the window defect. It is caused by the difference in the energy lost by different ionizing particles as they travel through the thin detector window (also called the dead layer). An is the nuclear-stopping defect that results from a net ionization defect due to the nuclear stopping process, and A r results from recombina-tion of electron-hole pairs in the plasma produced along the ionized track. The window Chapter 3. Experiment 39 defect is more serious for detectors with a thick dead layer. For the detectors used in these experiments, the SSDs had thicker dead layer (~ 0.7pm of Al) than the telescopes (~ 0.15/im of Al) , and thus the pulse height defect was more pronounced for the SSDs. Because the decay of 9 B produces protons as well as a-particles, separate energy cali-bration for the a-particles and the protons had to be be performed. Accurate calibration hence requires a knowledge of the particle identification a priori. For the triple coinci-dence events, a preliminary calibration was made by assuming two of the three particles to be a particles and one of them to be a proton. Using this calibration, each event was kinematically analyzed (Chapter 5) and particle identification was made from this analy-sis. The events were then recalibrated using the correct particle identification. Whenever a particle identifcation was not possible, an a calibration was used for energies less than 6 MeV and a proton calibration was used for energies above 6 MeV. In the following, the a and the proton calibration procedures followed in the two experiments are discussed. 3.6.1 Telescope calibration The a-particle calibration for the front detectors and the annular detectors was performed using the 1 8 N peaks and the peaks from 2 4 1 A m and 1 4 8 G d . The peak locations were found from a gaussian fit to these peaks. A fit to the 1081 keV 1 8 N peak detected in the front detecor of S3 in Expt. 1 is given in Figure 3.10. The resolution of the detector at this energy has a FWHM=25 keV, and is found by subtracting (in quadrature) the contribution from recoil broadening (see Section 4.7). The detector resolutions typically varied between 20-30 keV. Once the peak locations were found from the fits, a linear calibration was performed using these peaks. For the protons that break through the front detector, a combined calibration for both the front and the back detectors was performed from the high energy proton peaks in the 1 7 Ne run. This was done using the combined pulse height spectra of the front and Chapter 3. Experiment 40 centroid : 320.89884 250 300 350 400 450 E F 3 (Channe ls ) Figure 3.10: A fit to the 1081 keV 1 8 N peak in the front detecor of S3 in Expt. 1. The experimental energy resolution obtained after subtracting the recoil broadening is 27 keV. The recoil broadening is 39 keV (Section 3.5.2). Chapter 3. Experiment 41 the back detectors as shown in Figure 3.11. A gate was set on the high energy proton peaks in the back detectors as shown Figure 3.11a for detector B L to select the events associated with this peak in the front detector, and these events measured by F L are plotted in Figure 3.11b. These two peaks are then fitted to a gaussian to locate the pulse height positions x and y for the front and the back detector respectively. The pulse height positions were then fitted with: E — a*x + b*y + c. (3.3) Where x and y are the pulse height positions recorded by the front and the back detectors and a, b and c are parameters of the fit which are just the gains of the front and the back detectors and the combined offset respectively. The gain of the front detector was fixed from the front detector calibration. Five of the proton peaks shown in Figure 3.8 (p4-p8) were included in the fit. 3.6.2 SSD Calibration The A D C spectra from 1 7 Ne run recorded by the vertical strips 1-8 of SSDL are shown in Figure 3.12. Proton calibration for these strips was performed with the strong pro-ton peaks p3, p4, and p5 (Table 3.2), also labeled in Figure 3.12d. Note that the 1680 keV proton peak appears at a higher channel position than the 1725 keV a peak (Fig-ure 3.12d). This is due to the pulse height defect discussed earlier5. As discussed in Section 3.3, it was not possible to operate the SSDs during the 1 8 N run due to their high leakage current, thus it was not possible to have the a-calibration peaks from the 1 8 N run. For the central strips where it was possible to detect the 1725 keV a-peak from the 1 7 Ne run, a-calibration was performed using this peak and the 5 T h e identification of the two peaks (p3 and a l ) was made from the coincidence plot with the associated heavy recoils detected in the telescope (Section 3.5.1). Chapter 3. Experiment 42 s u m : 1 6 8 5 8 1 0 0 0 1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 EBL (Channels) If) - g 8 0 0 -3 o U 6 0 0 -4 0 0 -2 0 0 - \ s u m : c e n t r o i d : f w h m ; red. x2 : f i t l i m i t s 1 9 4 8 9 1 0 6 . 3 3 9 0 5 9 1 5 . 1 8 3 4 8 7 2 2 . 5 1 3 2 5 4 8 8 0 : 1 3 1 (b) — i 1 r — i r 1 — r 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 • EFL (Channels) 1 6 0 Figure 3.11: Fits to the front and back portions of the pulse height of the 7031 keV peak from the 1 7 Ne run in the front detector of T L in experiment 2. Chapter 3. Experiment 43 (<=) s t r i p #3 a : 200 GSD ezo izao 1040 2000 EsrL= (Channels ) (e) . s t r i p #5 aoo Deo 020 izac 16*0 2000 ESTLS (Channels ) (9) -St r ip #7 u : (d) p4 St r ip #4 \ ^ -. V" A J p5 -r=^—: 1——--=—1 • 1 i — r ZOO 560 9Z0 12B0 1B40 2000 Esrw ( C n a n n e l s ) (f) s t r i p #6 ^ J J L *J\J EDO D60 020 I2B0 1B40 2000 S m (Channels) (h) s t r i p #8 mill .nJSnAi - iti-^VV u 690 1120 1280 18*0 E (Channels) 200 &60 12B0 I EM E m , (Channels) Figure 3.12: Raw A D C spectra for the 8 front (vertical) strips of SSDL from the 1 7 Ne run. Note that the central strips see more events than the peripheral strips due to the back-to-back nature of the decay of 1 7 F . Also see Figure 3.13. Chapter 3. Experiment 44 1 2 3 4 5 6 7 Ver t i c a l s t r ip number Figure 3.13: Hit pattern of SSDR for events in coincidence with T L in 1 7 Ne run. Only the central strips that are opposite the telescope have events. 2 4 1 A m peak. Because of lack of a calibration peaks for the outer strips (only the 2 4 1 A r a peak), the a-calibrations for the outer strips were deduced from the calibration of the inner strips where the a calibration was possible. This was done by evaluating the ratio of the proton calibration gain to the a calibration gain for the inner strips and by using this ratio to deduce the a calibration gain for the outer strips from the proton calibration gain. /3 -detector c a l i b r a t i o n Calibration for the (3 detector was performed using 2 2 N a and 1 3 7 C s 7-ray sources. The 7-ray spectra recorded by the plastic scintillator from these runs are given in Figure 3.14. The Compton edges produced by the 7 rays from these sources are also indicated in this figure. Since the plastic scintillator has a very low photo-electric absorption cross-section Chapter 3. Experiment 45 10000 16000 12000 A 8000 4000 50 140 80 120 160 E (channels) 230 320 E (channels) 200 A 511 \ / (b) 1274 / 410 500 Figure 3.14: Typical 7-ray spectra obtained from the j3 detector using: (a) 1 3 7 C s source (b) 2 2 N a sources. The Compton edges from the 7-rays are indicated in the spectra. due to the low Z of the elements it is made of, it is very improbable for a 7 to deposit its full energy in the detector. Hence only the Compton edges can be observed in these spectra. The Compton edges can be calculated from the full energy of the 7-ray using: ECE 2 £ 2 me + 2E1 The energy of the 7-rays emitted by these sources is given in Table 3.3. (3.4) Chapter 3. Experiment 46 Table 3.3: A list of the 7-ray lines emitted by 2 2 N a and 1 3 7 C s . The Compton edges seen in the spectra from the /3-detector are indicated. Source photon energy (keV) Compton edge (keV) A D C channel 2 2 N a 511.0 340.7 89 2 2 N a 1274.0 1061.7 250 1 3 7 C s 661.6 477.3 118 3.7 Timing spectra The T D C spectra are very useful in defining true coincidence conditions, and in discrim-inating against random background. The T D C spectra for the Front detectors of the two telescopes SI and S4 are shown in Fig 3.15. The sharp peak at the beginning of the spectra is the peak corresponding to events where the same detector caused the trigger. This peak is sharp because the T D C STOP is caused by the same signal that is input to the T D C START, but with a fixed delay. So it is only affected by the resolution of the T D C . The "coincidence peak" is due to events where another detector caused the START. As a result, this peak is broadened by the time resolution of the detector and the electronics. Chapter 3. Experiment 47 0 4 0 0 8 0 0 1200 1600 2000 0 4 0 0 8 0 0 1200 1600 2 0 0 0 T (Channels) T (Channels) Figure 3.15: T D C spectra for F l and F4 for a subset of the data from the 9 C run. The time period is 0.244 ns per channel. Chapter 4 Monte Carlo Simulation and Kinematics 4.1 Kinematic considerations Because the initial beam ions are stopped completely in the catcher foil, the laboratory and center-of-mass system are identical in the breakup of 9 C . After the /3-decay of the 9 C nucleus, the 9 B nucleus has a small recoil momentum which has been ignored in the kinematics calculations. Then the breakup of a recoiling fragment of the 9 B decay is viewed in its own moving reference frame. The kinematic variables for the second breakup are then transformed into the laboratory frame with the application of momentum and energy conservation. Following the derivation in Appendix E, the following expression is obtained for the relation between the energy of the particle from the first breakup (E\) and the energy of one of the two particles from the second breakup (E2)\ m2(m2 + m3) (m 2 + m 3) 2 m2 m2 + m 3 y m2 with the index 1 for the stable fragment of the first breakup and the index 2 for one of the particles of the second breakup (Figure 4.1). Q2 is the total energy of the second breakup, 0\2 is the laboratory angle between particles 1 and 2. In the case where the breakup of 9 B leaves the recoiling fragment in a sharp state, Q2 is well defined. Then for a given excitation energy in 9 B and a given value of #12, the energies of particles 1 and 2 are uniquely defined and would be represented by a single point in a two-dimensional plot of El vs E2. On the other hand, if the second breakup is through a broad state, Q2 can have a range of values determined by the width of the 48 Chapter 4. Monte Carlo Simulation and Kinematics 49 P (2) 0 ) a 9| B o i l l a (3) Figure 4.1: An illustration of the kinematics of the breakup of 9B—>5Li+a:. state, and the energies of the breakup particles would lie on an extended locus in the E1-E2 plot. In general three different kinematic loci exist for the three different particle combination. In the case of the decay of 9 B , however, there are only two possible loci since two of the decay products are identical. For the purpose of demonstration, the two kinematic loci calculated for the breakup of 9B(EX = 12 MeV) —> aap and 0 1 2 = 170° are plotted in Figure 4.2. In the general case where there is no restriction for the secondary breakup energy, Q 2 , the coincidence events are found along the loci labelled ct-p and a-a. For the case where the decay proceeds through an intermediate state of a definite energy, the correlation between the energies of the two particles appear as a point in the El vs E2 plot (Figure 4.2). In an experiment, the distributions of particles along the possible loci are modified by the shape-functions of the secondary states (Sec. 4.3), as well as by finite solid angles of the detectors and the finite size of the source. As noted above, in addition to the possibility of decays through secondary broad states, the experiment is further complicated by the fact that /3-decay does not in general Chapter 4. Monte Carlo Simulation and Kinematics 50 E2 (MeV) Figure 4.2: Kinematic loci calculated for the breakup of 9 B ( 1 2 MeV)—>• aap, where the decay particles are observed in the laboratory in coincidence at an angle of 170°. The two points indicated in the figure correspond to decays that proceed through intermediate states of definite energy, as indicated in the figure. Chapter 4. Monte Carlo Simulation and Kinematics 51 leave 9 B in sharp states but rather with a combination of sharp and broad states along wi th a continuum of excitation in 9 B . The interpretation and analysis of coincidence spectra hence becomes extremely complex. The only way that is possible to surmount these difficulties is to perform extensive Monte Carlo calculations which simulate the decay of 9 B through al l the possible channels for the appropriate choice of excited states in 9 B . 4.2 Monte Carlo simulation A computer program was written to simulate the breakup of 9 B via each of the several decay channels available, wi th the experimental geometry for both runs incorporated in the simulations. The simulations were done mainly for three inter-related reasons. Firstly, the coincidence detection efficiency is a function of the excitation energy in 9 B , the particle breakup mode, the energy available in the decays and the experimental geometry. Monte Carlo simulation was the only way that is possible to determine these efficiencies. Secondly, Monte Carlo simulations of the various decay modes including the calculated efficiencies were used to determine the /3-decay branching ratios to states in 9 B and the particle decay branches of these states by adjusting the input parameters of the Monte Carlo simulations until they reproduced the experimental spectra. Finally, simulations were used to determine the angular correlations of the decay modes of some 9 B states by making comparison between the experimental and simulated singles as well as coincidence spectra. These angular correlation data were useful for l imit ing spins and parities of the 9 B states (see Section 5.3.2). The Monte Carlo programme B 9 D E C A Y was written in F O R T R A N , and the simu-lation was carried out using the T R I U M F U N I X Workstations A L P H 0 3 and A L P H 0 4 . In a typical simulation, several mil l ion events were calculated. Particles from the first Chapter 4. Monte Carlo Simulation and Kinematics 52 decay were assumed to be emitted isotropically in space. Where necessary, both angu-lar distributions and widths of states were sampled for the subsequent decay. For the widths, the probabilities were sampled according to the distributions given in Sec. 4.3, and the angular distribution probabilities are given in Sec. 4.6. After all events were calculated, the geometric constraints of the detectors and electronic detection thresholds were applied to make the final selection of events for each decay channel. The Monte Carlo efficiencies and detector spectra were determined from these selected events ( Sec-tion 4.5). For decay via 8Be(0 +) in Expt.l, neither angular correlations nor state shape functions were required and the process was straight forward. However, for Expt.2 where decay is via 5Li(3/2~) the situation is more complicated, and this is discussed in more detail in Sec. 4.3. First, the steps followed in the Monte Carlo simulation are outlined in the following: 1. Select an excitation energy of 9 B for the state of interest by sampling from a Breit-Wigner distribution, modified by the Fermi phase-space factor1. 2. Select the excitation energy of the daughter nucleus according to the 9 B excitation energy selected in step 1 (see discussion in Section 4.3) , and calculate the energies of the first breakup particle and the recoiling nucleus2. 3. Generate a random direction for the first decay particle. This direction was sampled from a randomized azimuthal ((f)) direction. The polar angle (6) was determined by choosing an isotropic distribution for the first break up. 4. In the CM frame of the recoiling nucleus, calculate the energies of the two particles from the second breakup. JFor the simulation performed in the efficiency calculation, discrete set of 9 B excitation energies were chosen (Section 4.5). 2For decay to 8Be s . s .+p, the excitation energy of the daughter nucleus is a constant. For decay to 5 L i 9 . s . + a, see discussion in Section 4.3. Chapter 4. Monte Carlo Simulation and Kinematics 53 5. Generate random directions for the particles from the second breakup in the C M frame of the recoiling nucleus. This direction is sampled from a random azimuthal direction (</>cm). The polar angle (9cm) was determined by sampling from an angular distribution w(8). This angular distribution depends on the spins and parities of the parent and daughter nuclear states, as well as the angular momenta involved in the transitions. (See discussion in Section 4.6.) 6. Convert the C M energies and angles of the particles from the secondary breakup to lab energies and angles, and write the kinematic variables of each of the three paritcles (Ei, di, 4>i) to an event file. The generated events are then further analyzed by incorporating the experimental geometries, energy thresholds, energy resolutions and coincidence conditions, same as the ones set for experimental data. Once the accepted events were written to a data file, the T R I U M F developed software, P H Y S I C A [50] was used to generate spectra and make the appropriate energy threshold cuts in the selected event. 4.3 Decay through broad states For the decay of 9 B through broad resonances, the distribution of the daughter state is modified by the penetrability of the particle emitted by 9 B . The spectral shape of the daughter nucleus is therefore dependent on the initial 9 B excitation energy. A n example of this dependence is illustrated in Figure 4.3, showing the decay of a broad state in 9 B to the ground state of 5 L i , and the two resulting distributions of 5 L i from the decay of 9 B at two different excitation energies. The spectral shape of a broad state such as the ground state of 5 L i represented by the one-level i?-matrix approximation is discussed in Appendix B. The distribution of this state observed in the decay of 9 B is further modified by the penetrability of the a-particle Chapter 4. Monte Carlo Simulation and Kinematics 54 [E'-E,. - A(E')]2 + [rx,(E')];4 Figure 4.3: A n illustration of the decay of a 9 B state wi th distribution W(E) decaying to the ground state of 5 L i of state shape w(E'). The two different distribution of 5 L i , resulting from the decay of 9 B of excitation energies E\ and E2 are shown. Chapter 4. Monte Carlo Simulation and Kinematics 55 in the decay of 9 B , and is given by: I(E, E') = P(E-E')w(E') (4.2) where w(E') = (4.3) \E' - Ex. - Ay(E')}2 + [Yx,{E')f/A is the one-level i?-matrix expression for the ground state of 5 L i . The primed quantities in Equation 4.3 and in Figure 4.3 refer to the the p-a system in the 5 L i nucleus. The energy dependencies of the width Ty(E') and of the shift function Ay(E') in the one-level .R-matrix expression, are given in Appendix B: In the above Equations 4.4 and 4.5, 7 2 is the reduced width for the decay, P is the penetration factor, and S is the shift function. Because of this energy dependence of the 5 L i spectrum, for each excitation energy of 9 B selected, the 5 L i spectrum has to be calculated using Equation 4.2. This requires an excessive computational time, because of the computation involved in the evaluation of the Coulomb wave functions, needed in the calculation of the penetrability and the shift function (Appendix B). In order to surmount this difficulty, it was necessary to pre-calculate the 5 L i spectra resulting from decays of discrete excitation energies in 9 B . Once a specific 9 B state was selected, a 5 L i spectrum from which the energy of the 5 L i is sampled is chosen according to the selected excitation energy of 9 B . The interplay between the penetrability of the a-particle from 9B(EX = 2.36 MeV) —»• 5 L i + a and the 5 L i ground state distribution is illustrated in Figure 4.4. The 5 L i 9 s distribution populated by this decay is shown by the shaded area. The penetrability of TX,{E') = 212P{E') (4.4) AX,(E') =-^[SiE') ~ S(Ey)} (4.5) Chapter 4. Monte Carlo Simulation and Kinematics 56 Eb r e a k u p ( 9 B ( 2 - 3 6 ) - 5 L l + «) ( M e V ) Figure 4.4: A plot of the shape of the ground state of 5 L i (shaded area) as populated by the decay of a 9 B state at Ex = 2.36 MeV. The unmodified 5 L i 5 . s . distribution is shown by the dotted line. The x-axis is the breakup energy [=2.36 — Ex(°Li)] MeV. (Inspired by Figure 19 in Ref. [51].) the oj-particle is calculated for I = 2 and channel radius, a c = 4.62 fm. The effect of the penetrability is more significant at energies below the Coulomb barrier, and this can be seen by the the drastic drop of the penetrability at low a energies. The ground state distribution of 5 L i is calculated for the channel radius ar — 3.62 fm and reduced width 7 2 = 3.33 MeV [52]. 4.4 A study of breakup channels from Monte Carlo simulations In this section, the simulation of the breakup of the states of 9 B for the different breakup modes will be discussed. Chapter 4. Monte Carlo Simulation and Kinematics 57 4.4.1 9 B decay to 8Be( 0+) and p As noted in Chapter 3, the two sets of opposite detector pairs S1-S4 and S2-S3 were used to select uniquely the decays of 9 B through the ground state of 8 Be. Events for this decay channel were simulated with the Monte Carlo program for a Gaussian source density (FWHM=0.25 cm) and the geometry of Fig. 3.2. For the purposes of demonstration, it was assumed that the excitation in 9 B consisted of broad levels at 2.8 and 12.2 MeV. The results of the S1-S4 simulation are shown in Fig. 4.5. In this figure, the coordinates of each point are the energies of the coincident pair. The data for this channel are found exclusively in the nearly horizontal (slope~ 1/10) and nearly vertical (slopes 10/1) bands. This can be understood qualitatively by noting that the ground state of 8 Be is very narrow, only a very small energy (Q2=0.092 MeV) is available for the breakup into two a particles, and the coincidences are obtained in back-to-back geometry. The protons are the higher energy partners of the coincident pair. The sharp component of the bands corresponds to the detection of both a particles in the same detector, and the broader bands closer to the axes are from events where only one a particle is detected. For each pair of face-to-face detectors two proton spectra can therefore be obtained by projection of the events selected by the ratio cuts onto their respective energy axes. Because the ground state of 8 Be is very narrow (T = 6.8 eV), these proton spectra are directly related to the 9 B excitation energy [EX(9B) = (Ep -.0.164)9/8] MeV. 4.4.2 9 B decay to 5Li(3/2") + a Also in Fig. 4.5 is shown a simulation of the decay of 9B(12.2) —> 5 L i g s . + a. This simulation shows a densely populated region on the diagonal ( £ 1 — EA) at approximately 6 MeV accompanied by two low-energy crescents, which correspond to a-a coincidences. The two wings on both side of the cluster around El = EA = 6 MeV are proton-a events Chapter 4. Monte Carlo Simulation and Kinematics 58 0.0 3.5 7.0 10.5 14.0 E l (MeV) Figure 4.5: Scatter diagram of simulated double coincidence spectra for detector tele-scopes S i and S4, the more distant opposite detector pair in experiment 1 geometry. The simulated events include: (i) 9B(12.2)^ 8Be(gs)+p, (ii) 9 B(12.2)^ 5 Li(gs)+a, (iii) 9B(12.2)^ 8Be(3.0)+p and (iv) 9B(2.8)^ 8Be(gs)+p. The number of simulated events are: (i) 2 x 106, and (ii)-(iv) 4 x 106. Chapter 4. Monte Carlo Simulation and Kinematics 59 Figure 4.6: Simulated a-a coincidence events from 9 B —» 5 L i 9 . s . + a. detected by the back-to-back detectors (TL vs SSDR) in coincidence with the protons detected in SSDL. 2 x 106 events were simulated for each energy from 2 MeV to 14 MeV in steps of 1 MeV, and same detector thresholds were applied as in the experimental data. mainly from the decay of 9 B —> 8 B e 3 . 0 + p, with a weak contribution from the decay of 9 B to the tail of the ground state of 5 L i . The approximate back-to-back geometry is also kinematically favored for decays through the 5 L i channel, because in the subsequent breakup of the 5 L i recoil nucleus the a particle receives only 1/5 of the breakup energy and has a very low velocity. Con-sequently, the vector addition of this velocity with the recoil velocity causes only a little deviation from back-to-back coincidences. The Monte Carlo simulations have been re-produced for a set of excitation energies in 9 B from 1 MeV to 14 MeV for decays through both the ground state of 5 L i and the first excited state of 8 Be. The former produces a continuous set of mainly diagonal events at lower energies due to the decay of states of lower excitation energy in 9 B . The latter mode [8Be(3 MeV)] also appears as events Chapter 4. Monte Carlo Simulation and Kinematics 60 mainly on the diagonal but at a somewhat lower energy for the same excitation in 9 B . As it will be evident in Chapter 5, limited kinematic information can be obtained from the double coincidence data (in the geometry of this experiment) for decays that proceed through broad intermediate states such as 5 L i g s . . Hence only from the triple coincident data, a reliable information on the excitation of 9 B and its decay mode can be extracted. Monte Carlo simulations for the triple coincidence between the two strip detectors, and one of the telescopes were carried out for the breakup of 9 B —> 5 L i g s . + a for 9 B excitation energies in steps of 1 MeV for Ex — 2 — 14 MeV. A density plot of the triple coincident events detected by SSDL, SSDR and T L for the face-to-face detectors (SSDR, TL) is shown in Figure 4.6. The energy thresholds applied to the experimental data are also applied to the simulated events. These two-energy correlations in the triple coincidence events were useful in determining the breakup modes of the states of 9 B (Section 5.3.2) 4.5 Determination of coincidence efficiencies for the breakup channels The coincidence efficiencies for the 9B—>8Beg s +p decay mode detected in the face-to-face detectors in Expt. 1, and for the triple coincidence between the two SSDS and one of the telescopes in Expt. 2 for the 9BS.S.—>-5Li9.s.-r-a: decay mode were calculated from Monte Carlo simulations. For both decay modes, it is evident from kinematics considerations that the coinci-dence efficiency for a given detector combination depends on the excitation energy in 9 B . In the case of 9B—» 8Be 9. s. -f-p decay mode, Monte Carlo calculations were carried out to determine the coincident efficiency as a function of proton energy for each of the telescopes. Because the ground state of 8 Be has J=0, angular correlations effects are not present in these calculations. The efficiency curve for detection of protons in SI in Chapter 4. Monte Carlo Simulation and Kinematics 61 0 . 0 0 3 5 0 Z 4 6 8 10 13 1 4 E p (MeV) Figure 4.7: Simulated kinematic efficiency of Telescope SI for detection of protons in coincidence with one or two a particles in the 8Be(gs)+p channel as a function of proton energy. The crosses with the statistical error represent the calculated points. The smooth curve is a fourth order polynomial fit. Each point was simulated from 2 x 106 events. The minimum a energy accepted is 160 keV. coincidence with a particles (Ea > 160 keV) in the opposing telescope S4 for the decay mode 9F3—>8Beg.s. +p is shown in Fig. 4.7. In addition, the Monte Carlo coincidence efficiency as a function of the excitation energy in 9 B for the decay of 9B—> 5Li g s + a was calculated for the SSDL, SSDR, and T L combination of detectors following a similar procedure. Also threshold effects, as set experimentally and in the analysis, were simulated. Unlike in the 9 B - ^ 8 B e f f S . + p breakup, the angular correlation probabilities for each energy have to be included in the Monte Carlo calculation for the 9 B—> 5 Li 9 s + a breakup. More discussion on the calculation of the efficiency curve for this decay mode is given in Section 5.3.2. Chapter 4. Monte Carlo Simulation and Kinematics 62 4.6 A n g u l a r c o r r e l a t i o n When a particle is emitted from a definite nuclear state, its probability of emission generally depends on the angle between the nuclear spin axis and the direction of emission. This results in an anisotropic distribution of the emitted radiation. In order to observe this anisotropy, the spin of the decaying nuclei should be oriented in some preferred direction, and this can be achieved by polarizing the nuclei. Another situation where this anisotropy can be observed is by measuring the angular correlation of two particles that are emitted successively in a cascade-decay process. The observation of the first decay particle in a predefined direction (ki) selects those nuclei that have an anisotropic distribution of spin orientation. The second particle then shows an angular correlation with respect to this predefined direction, k\. A comprehensive treatment of the theory of angular correlations is found in a paper by Devons and Goldfarb [53]. These correlations can be complicated if several angular momenta for the initial breakup are allowed, because of the mixing between the different angular distributions. The mixing is dependent both on the spins and angular momenta involved as well as nuclear structure. If each of the decays is pure with no mixing, the angular correlation is given by: WClcM = ( 4 7 r ) 2 ( - l ) a + c - 2 t ( 2 6 + l ) E {-l)'L'~L2Cko{C1Cl)ClQ{C2C2)W{bbLlL1-ka)W{bbL2L2; kc)Pk(cos 6) k=2i (4.6) In the above equation, 6 is the angle between the first and the second breakup. The variables a, b and c are the spins of the states of the three nuclei involved in the cascade transition (Figure 4.8). L\ and L 2 are the total angular momenta of the emitted particles. Hence, Lj = ti + s~l where Zj and s, are the angular momenta and the intrinsic spins Chapter 4. Monte Carlo Simulation and Kinematics 63 Table 4.1: Angular correlations for the breakup of 9 B states into an intermediate (daugh-ter) nucleus; only the lowest allowed angular momenta were included in the computation. Note that for the decay to 8 Be(2 + ) , there are two possible distributions for the different spin states of the proton. J* of 9 B Daughter nucleus Angular distribution I -2 5 Li(§") 1 + P2(cos8) 3~ 2 5 L i ( | - ) 1 5 -2 5 L i ( | - ) 1 - 0.7141P2(cosfl) 1 + 2 5Li(fr) 1 + P2(cos8) 1-2 8 Be(2 + ) 1 + P2{cos9) 3-2 8Be(2+) 1 5~ 2 8Be(2+) 1, 1 - 0.7141P2(cosfl) 1+' 2 8 Be(2 + ) 1 + P2{cos6), 1 + 1.1427P2(cos0) + 0.8568P4 (cos9) \ r \ s2 r Figure 4.8: A n illustration of cascade transition, a, b and c are the spins of the three nuclei, and si and S2 are the spins of the emitted particles. Chapter 4. Monte Carlo Simulation and Kinematics 64 of the emitted particles respectively, and C denotes the set of quantum numbers L and I. Furthermore W(bbL\Li\ ka) etc., are the Racah coefficients, and the coefficients Cfco(£, £ ) are given as: 2L 4-1 Ck0(C, C) = -—^-(-1)<(Z0, Z0|fc0) (4.7) 4TT for a particles. And Ck0(C,C) = 2^{-l)L-ll2+\L\,L-\\kO) (4.8) for protons. Pk(cos6) are the usual Legendre polynomials of cos#. The selection rule for the sum over k requires that k be even and the maximum is given by kmax = mAn{2Lu2eu 2 L 2 , 2£2, 2b) (4.9) Because J=0 for the 8 Be ground state, i.e. 6 = 0, all decays from this state will decay isotropically. Decays through the 5 L i ground state and the 8 Be(2 + ) state can show an angular distribution of the pattern: W{6) = a0 + a 2 cos2 # + a 4 cos4 # (4.10) In Table 4.1, the angular correlations for the different J77 states in 9 B decaying through various channels are given. Sample calculations of the angular correlation coefficients are given in Appendix F. 4.7 Recoil broadening due to /^-particle correlations The energy spectrum of the particle emitted following /3-decay is distorted by the mo-mentum emparted to the recoilling nucleus from the /3-decay. Although the recoil energy of the 9 B nucleus from emission of /3 and v is typically on the order of a keV, the energy of the particle emitted from the recoiling nucleus in flight can deviate significantly from Chapter 4. Monte Carlo Simulation and Kinematics 65 the energy of the particle if it were emitted from the nucleus at rest. The magnitude of this energy shift is dependent on the angle between the (5 particle and the neutrino. Maximum recoil is obtained when the B and the neutrino are emitted in the same di-rection, and minimum recoil is obtained when the two are emitted in opposite direction. Experimentally this is observed as a broadening of the energy spectrum of the particle being emitted. This broadening of the a-peak has been observed in /3-delayed a emitters in Refs. [54, 55, 56]. The magnitude and shape of this broadening depends upon the Q-values for the B decay and the Q-value for the subsequent breakup, as well as the B — v angular correlation. These angular correlations depend on the spin and parities of the parent and daughter nuclei in the B decay as well as the nature of the decay, i.e. whether the decay is Fermi or Gamow-Teller transition. A calculation of these correlation coefficients is given in Ref. [57]. In the case of the /?-decay of 9 C , this recoil-broadening can be observed in the proton peaks from 9B—>• 8 Be g . s . + p. In this work, a broadening of the proton peak from the decay of the ground state of 9 B has been observed. This recoil effect also causes the broadening of the narrow 1 8 N peaks that were discussed in Section 3.5.2. C h a p t e r 5 D a t a A n a l y s i s 5.1 I n t r o d u c t i o n The primary goal of this experiment is to measure the /3-decay branches of 9 C , and to study the various levels of 9 B that are fed by the /3-decay of 9 C . As discussed in Chapter 1, except for the ground state, all the states of 9 B are either broad or decay through a broad intermediate state. Hence, complete kinematic information is often needed in order to deduce the 9 B excitation and its decay mode from the measurement of the decay particles. In both experiments, data were recorded in singles mode so that the singles information could be retained, while double and triple coincidences could be established in software and analyzed. The results from the analysis of the singles as well as the coincidence spectra are important in finally reconstructing the full /3-delayed particle spectrum of 9 C and in establishing the subsequent decay schemes of the levels of 9 B . In this chapter, the analysis of the experimental data and comparisons with Monte Carlo simulations will be discussed in full detail. As mentioned in Section 3.4, preliminary analysis of the data was carried out using the analysis software NOVA. In addition, NOVA was used to sort the coincident events and output these events to an event file for further analysis. In order to insure that any good coincidence event is not rejected in the primary sorting, "loose" coincidence conditions were established in NOVA and events that satisfy these conditions were selected and written out. User-defined codes were written in order to output the signals for these 66 Chapter 5. Data Analysis 67 selected events, such as A D C , T D C channels in the desired format. Further cuts and kinematic analyses were performed on the selected coincidence events, using kinematic analysis codes written for this analysis. P H Y S I C A was used as a graphic interface for putting gates on the two-dimensional plots and in plotting the spectra. In the following two sections, the analyses of the data from the two experiments are described. In Section 5.2, the data analysis for Expt. 1 is discussed. As noted in Chapter 3, the back-to-back coincidence data from Expt. 1 was mainly used to select the decay through the ground state of 8FJe, and the steps followed in selecting these events are discussed in detail. The analysis for Expt. 2 is discussed in Section 5.3. Also, as noted in Chapter 3, the triple coincidence spectra from Expt. 2 were useful in the analysis of the decay through the ground state of 5 L i , and this analysis of the triple coincidence data from Expt. 2 is discussed in Section 5.3. In addition, analysis of the particle-f3 coincidence spectra, as well as double coincidences between the SSDs and the opposing telescopes are discussed in the same Section. 5.2 D a t a ana l ys i s for E x p t . 1 5.2.1 S ing les s p e c t r a The singles spectra recorded by the front detectors F l (30 pm) and F3 (13 pm) are shown in Figure 5.1. The narrow peaks that are observed at the low energy sides in both spectra are from the decay of the ground state of 9 B to 8Be g . s .+p, and the plateau in the region 1-1.5 MeV is mainly due to a-particles from the decay of the 2.34 MeV state in 9 B to 5 L i g s +ot. From these two spectra, it is possible to see that these two states of 9 B are strongly populated. It will be shown later in Chapter 6 that the decay of 9 C to these two states constitutes more than 80% of the decay strength. The decay of the ground state of 9 B to 8Beg.s+ p results in a proton of energy 164 keV, Chapter 5. Data Analysis 68 0 1200 2 4 0 0 3 6 0 0 4 8 0 0 6 0 0 0 0 1200 2 4 0 0 3 6 0 0 4 B 0 0 6 0 0 0 EF1 (keV) EF3 (keV) Figure 5.1: Singles spectra for: (a) Front detector F l (30 pm thickness) (b) Front detector F3 (13 fim thickness), showing the breakthrough features that are characteristic of the thicknesses of the detectors used and the identity of the particle being measured. and two a particles with mean energy of about 55 keV. The energies of these a-particles are too low to be measured with our method of detection and hence, the ground state can only be observed by detecting the protons from this decay Because the protons from this decay mode are of such low energy, the intense (3 response tail in this energy region caused difficulty in the analysis of this region of detection. In order to suppress this /3-induced background, simultaneous detection of the (3 particles was adopted in Expt. 2, and will be discussed in Section 5.3.1. In addition to the two features discussed above, the break-through features in both spectra in Figure 5.1 are apparent. Since the full energy range of the protons as well as the a-particles cannot be stopped in the front detectors, break-through edges appear in the spectra at energies where the particles have enough energy to break through the front detectors. These edges are characteristic of the thicknesses of the detector used and the type of particle detected. The edge near 1500 keV seen in Figure 5.1a for F l (thickness of 30/im) is due to protons starting to break through the front detector. Moreover, the edges near 900 keV and 3000 keV in Figure 5.1b are due to protons and a-particles Chapter 5. Data Analysis 69 14000 10500 -\ 0 > P —'7000 -\ 3500 -i 0 1000 2000 3000 4000 E F 2 (keV) Figure 5.2: E - A E spectrum for telescope 2 (F2=10 pm): A reliable particle identification can be made for proton energies above about 1 MeV and for a energies above about 3 starting to break through F3 (thickness of 13 ^m). For particles that have enough energy to break through the front detector, particle identification is possible from the plot of E vs A E . Such an E - A E plot measured by S2 is given in Figure 5.2. From this plot, it can be seen that the protons and the a particles can reliably be separated above an energy of about 1 MeV and 3 MeV respectively. Furthermore, a significant fraction of the (3 particles can be identified from this plot. After removing the (3 events by the " A E - E cut" in the two dimensional plots, the calibrated events in both the front and back detectors were added. Such a calibrated particle spectrum measured by telescope S i is shown in Figure 5.3. In the low energy region of the spectrum, the peak at 164 keV and the plateau in the region 1-1.5 MeV super-imposed on the exponential-like 0 response tail are again apparent. The edge at about 6 MeV is due to a-particles from the decay of the 12.2 MeV state to 5 Li f l . s .+ a, and almost all the events located above this edge are protons with the broad peak at about 8 MeV MeV. Chapter 5. Data Analysis 70 0 3000 6000 9000 12000 15000 E l (keV) Figure 5.3: Singles particle spectrum measured by Telescope 1. The feature resulting from the different decay modes of the 12.2 MeV state are indicated. The proton peak near 160 keV and the plateau in the 1-1.5 MeV region are as indicated in Figure 5.1 caused by protons from the decay of the 12.2 MeV state to 8Be3.o+ p, and the proton peak at about 10.5 MeV from the decay of the 12.2 MeV state to 8 B e g s.+ p. 5.2.2 Double coincidence between the opposing telescopes As noted in Section 4.4.1, the two sets of data from the face-to-face coincident pairs S1-S4 and S2-S3 were useful in separating the two major decay modes of the states of 9 B , and in particular in selecting uniquely the decays of 9 B through the ground state of 8 Be. These two sets of coincident events were sorted from the singles data, and the selected events were written to an event file for further analysis. A loose coincidence condition was applied in selecting these events. This coincidence requirement was that two of the four detector elements that are in back-to-back geometry register an event in the T D C spectra. For (S2-S3) coincidence for example, the coincidence requirement was either of Chapter 5. Data Analysis 71 the conditions listed in Table 5.1. A D C and T D C signals for events satisfying one of Table 5.1: T D C conditions applied to S2-S3 combination in sorting the coincidences from the singles data. Events that satisfy any of the four were selected. Condition Possible coincidence (0<TF2<2047).AND. (0<TB3<2047) a in S2 and P in S3 (0<TF2<2047).AND. (0<TF3< 2047) a in S2 and a in S3 (0<TF3<2047).AND. (0<TB2< 2047) p in S2 and a in S3 (0<TB2<2047).AND. (0<TB3< 2047) a in S2 and a in S3 these requirements were then written to an event file for further analysis. A coincidence plot of events measured by SI and S4 is given in Figure 5.4. One of the most visible features in this plot are coincidence events that are found in the nearly horizontal (slope ~ 1/10) and nearly vertical (slope ~ 10/1). As explained in Section 4.4.1, these are p-a and a-p events from the decay of 9 B to 8 B e 9 s + p, and are to be compared with the Monte Carlo simulation results discussed in the same section. Clearly an energy ratio cut incorporating the two bands will select those events associated with this channel. One can already see in this figure that 9 B states at about 3 and 12 MeV are strongly populated in the /3-decay of 9 C . Another feature is the intense band along the diagonal. This band is due to a-a coincidence events with roughly equal energies that are mainly from the decay through the 5 L i 9 S + a channel (Section 4.4.2). Also, the crescents due to the decay of the 12.2 MeV state to 5 L i 9 s + a shown in Figure 4.5 are clearly visible in Figure 5.4. Also shown are events along a straight line of negative slope, to be compared with the simulated cluster of events from the decay of the 12.2 MeV state to 8 B e 3 0 +p shown in Figure 4.5. From Figure 5.4, it is also possible to see that the region near (1,1) MeV is densely populated. These events are mainly a-a coincidences from the decay of the 2.34 MeV Chapter 5. Data Analysis 72 0.0 3.5 7.0 10.5 14.0 E l (MeV) Figure 5.4: Two dimensional plot for the opposing telescopes S1-S4 pair. The p — a band from 8 B e s . s +p and the a — a band from 5 L i 9 . s . +a decay are indicated. state corresponding to the plateau in the singles spectrum in Figure 5.1a. 9 B d e c a y to 8 B e 5 5 + p As discussed in Section 4.4.1, there is a one-to-one correspondence between the proton energy and the 9 B excitation energy since the ground state of 8 Be has a very small natural width of only 6.8 eV. Thus it is possible to study the excited states of 9 B that decay by this mode by selecting events decaying through this mode and by studying the protons spectra. Events from the decay through the 8 Be(0 + ) + p channel were selected using a cut on the two-dimensional plot as shown in Figure 5.61. This selection was applied to both S1-S4 and S2-S3 coincidences. After the ratio cut was applied to these data sets, E-A E cuts on the protons were applied. Further timing cuts on the time difference spectra remove the random coincidence events. The time difference was established between the 1 We refer to this cut as "ratio cut" throughout this thesis since events selected by this cut have energy ratio of as 1/10. Chapter 5. Data Analysis 73 3 o u o u T - 1 3 3 3 - 6 6 7 0 6 6 7 1333 T F 1 - T F 4 ( C h a n n e l s ) - 1 2 0 0 - 4 0 0 4 0 0 1200 T F 1 - T B 4 ( C h a n n e l s ) Figure 5.5: Time difference spectra for the detectors in the S1-S4 telescope pair: (a) TF1-TF4, (b) TF1-TB4. A coincidence peak centered around 0 is apparent in both spectra. The peaks near channel ±2000 in both spectra are from one of the two events being random events and causing overflow in the T D C . a-proton coincidences. For example, in order to select a — p coincidences between F l and S4, a gate was set on the two time difference spectra (TF1-TF4), and (TF1-TB4), while the former insures that low energy protons that do not deposit enough energy in the back detector are included, the later includes high energy protons that deposit high enough energy in the back detector to trigger the discriminator that provides the T D C stop signal for the back detector. These time difference spectra are shown in Figure 5.5, and a coincidence peak in both the spectra are apparent. For the telescopes with thin front detectors (S2, S3), only the front-back time difference conditions were important (for example, TF3-TB2 for detection of protons in S2). For these telescopes, all the protons above about 1.5 MeV have energies above the discriminator thresholds to provide the T D C stop signals for the back detectors. Note that all the a particles from this decay mode are stopped in the front detectors, and all the protons above about 1.5 MeV break through all of the front detectors. The sequence of cuts applied to a subset of the data to select the protons detected by S2 in coincidence with an a particle in F3 are shown Chapter 5. Data Analysis 71 T „ - T _ (Channels) " * E2 (keV) Figure 5.6: A diagram showing the cuts applied on the subset of the data in S2-S3 coincidence to select the events from 8 Be(0 + )+p events in the case where the protons are detected by S2 and the a-particles are measured by the opposing front detector F3. Similar cuts were applied to the other telescope spectra. Chapter 5. Data Analysis 75 c 3 O O E l (keV) 2000 4000 6000 8000 10000 12000 14000 E3 (keV) a o o 6000 8000 10000 E2 (keV) 12000 14000 C D o U E4 (keV) 10000 12000 14000 Figure 5.7: Proton spectra measured by S1-S4 (a-d) obtained after applying the ratio-cut in order to select events from the 9 B—> 8 Be f f s + p decay mode . in Figure 5.6. These cuts were applied to the four detector telescopes in order to select the protons from the decay to 8 Be s . s . + p. These proton spectra observed by the four detectors after these cuts were applied are given in Figure 5.7. The variation of the coincidence efficiency with the 9 B excitation energy and the Monte Carlo simulations performed to calculate this coincidence efficiency for each co-incidence pair were discussed in Section 4.5. A plot of these coincidence efficiencies for all the four telescopes is given in Figure 5.8. The low energy cutoffs for the a-particles detected in the opposing front detector are given in the caption. Efficiency corrections Chapter 5. Data Analysis 76 for all the four telescope spectra in Figure 5.7 were applied and the resulting spectra were summed. This summed, efficiency corrected spectrum is given in Figure 5.9. The apparent peaks observed in this spectrum are indicated. It will be shown later in Chap-ter 6 that this spectrum is used in the determination of the 9 C branching ratios to the states of 9 B . 5.3 Data analysis for Experiment 2 5.3.1 Particle-/? coincidence As discussed in Section 5.2.1, in both the front detectors and the telescope spectra, it was not possible to remove the P background altogether by a A E - E cut alone, and this /3-induced background caused difficulty in the detection of the lower energy portion of the particle spectrum, especially the protons from the decay of the ground state of 9 B . In order to suppress this /3-induced background, a coincidence with a P particle detected in the plastic scintillator was demanded. A comparison between the singles spectrum and the spectrum after demanding a coincidence with a signal in the P detector is shown in Figures 5.10a-b. It can be seen in Figure 5.10b that the P response tail that was superimposed on the 164 keV peak, and that was clearly visible in the singles spectrum, is essentially removed in the particle-/? coincidence spectrum. It will be shown in Section 6.3.4 that this particle-/? coincidence spectrum, together with the branching ratio information from the coincidence spectra is used in the determination of the ground state branch. The 164 keV peak shown in Figure 5.10b was further analyzed by fitting it to a Gaussian as shown in Figure 5.11. This peak has a very small natural width of 0.54 keV [2]. The width of this peak observed in this spectrum is 30 keV (after subtracting the experimental resolution of 22 keV). This is due to the recoil broadening discussed in Chapter 5. Data Analysis 77 Figure 5.8: Simulated kinematic efficiencies of Telescopes S1-S4 (a-d) for detection of protons in coincidence with one or two a particles in the 8Be(gs)+p channel as a function of proton energy, for a energy thresholds of F4=250 keV, F3=300 keV, F2=240 keV, and Fl=240 keV. The crosses with the statistical error represent the calculated points. The smooth curves are a fourth order polynomial fit. Each point was simulated from 2 x 106 events. Chapter 5. Data Analysis 78 2 4 6 8 10 12 1 4 E (MeV) Figure 5.9: The summed (after efficiency correction) ratio cut spectrum from the decay of 9B—> sBegs, + p for the four telescopes S1-S4. The apparent peaks that are associated with the excited states of 9 B are indicated in the figure. 0 1000 Z 0 0 0 3 0 0 0 4 0 0 0 0 1000 2 0 0 0 3000 4 0 0 0 EFL (keV) EFL (keV) Figure 5.10: Particle spectra detected by the 50 pm front telescope (FL) : (a) singles spectrum (b) /3-particle coincidence spectrum with the j3 energy threshold of 4 MeV. Chapter 5. Data Analysis 79 1000 • 800 H Area : centroid fwhm : X 2/dof : bin size : fit limits 3406.6599 160.694004 37.0956631 1.15765244 9,99617871 1.12 : 212 c O O 600 • 400 • 200 • 100 150 E F L (keV; 200 Figure 5.11: A Gaussian fit with linear background to the 164 keV peak measured by F L in coincidence with a (3 particle in the plastic scintillator. The width is from recoil broadening discussed in the text as well as from experimental resolution. Section 4.7. 5.3.2 Triple coincidence The triple coincidence spectra with significant statistics were from those that consist of events detected in the two strip detectors in conjunction with an event in one of the telescopes i.e., (SSDL, SSDR, TL) and (SSDL, SSDR, TR) , as well as double-hit events in the strip detector together with another event in the opposing telescope (2SSDR, TL) and (2SSDL, TR) . In a similar manner to the analysis of coincident events described in Section 5.2.2, NOVA was used to create coincidence conditions, and for sorting the coincidence events. Once an event that satisfies the prescribed condition is selected, the A D C and T D C signals from the telescope detectors, as well as signals from the front and the back elements of the SSDs were written out to an event file. In addition, the strip Chapter 5. Data Analysis 80 9000 ( k e V ) 12000 15000 Figure 5.12: Triple energy sum spectrum for the coincidences between SSDL, SSDR and T L , with no timing cut applied. addresses that provided these signals were written out. These addresses provided the position informations used in the kinematic analysis. In the following, the analysis of the triple coincidence spectra for (SSDL, SSDR, TL) detector combination will be explained. SSDL, SSDR, T L coincidence events The triple sum spectrum obtained from events detected in coincidence by SSDL, SSDR and T L is given in Figure 5.12. The two peaks from the decay of the 2.34 and 12.2 MeV state are evident in this spectrum. Moreover, the shoulder below the 2.34 MeV peak consists of mainly random coincidence events with at least one of the detectors not observing a particle from the breakup of 9 B . In order to remove these random coincidence events, further timing cuts were applied. Chapter 5. Data Analysis 81 1. Timing C u t s The time difference spectra of the three coincidence pairs (TSSDL-TSSDR, T S S D L - T F L and TSSDR-TFL) for the raw triple coincidence between SSDL, SSDR and T L are shown in Figures 5.13a-c. In all the three spectra, "coincidence peaks" near 0 are apparent. Note that these peaks are broad and asymmetric due to the "time walk" of the leading edge discriminators used for the SSDs. The spectra plotted in Figures 5.13d-f are for those events that are selected after making the cut shown in Figures 5.13a. Already after applying this cut, it is possible to see that a majority of the events located outside the coincidence peaks in Figures 5.13b-c are removed (Figures 5.13e-f). Similar cuts were applied to the other two spectra in finally selecting good coincidence events. In order to understand the systematic effects of these cuts, different regions and sequences of cuts were applied to the three spectra, and the final region of cuts and the associated energy thresholds are given in Table 5.2. The triple sum spectrum obtained after these cuts were applied is given in Figure 5.14. Table 5.2: T D C conditions applied to the SSDL, SSDR and T L coincidence events. Time difference variable Region of cut Energy threshold (keV) TSSDL-TSSDR (-850,1060) ESSDL^400 T S S D R - T F L (-850,1060) ESSDR^850 T S S D L - T F L (-330,-10) E T L ^ 7 0 In addition to suppressing background events from false coincidences, the time differ-ence cuts also reject some good events that have at least one of the three energies below the discriminator thresholds to generate a T D C stop signal. This can also be seen from the suppression of the peak corresponding to 2.34 MeV excitation in Figure 5.14. The energy thresholds imposed on the spectra due to these time difference cuts are also given Chapter 5. Data Analysis 82 -4000 -3000 T S S D L - T S S D R (Channels ) -1000 o 1000 2000 T S S D R - T F L (Channe l s ) - r -1000 D 1000 T S S D L - T F L (Channe ls ) -4000 -3000 -2000 -1000 0 1000 2OO0 3O00 4OO0 T S S D L - T S S D R (Channe l s ) -4000 -3000 -2000 -1000 0 1000 T S S D R - T F L (Channe l s ) -2000 -1000 o 1000 T S S D L - T F L (Channe l s ) Figure 5.13: Time difference spectra of the triple coincidence events between SSDL, SSDR and T L plotted in Figure 5.12 for the three detector combination SSDL-SSDR, SSDR-TL and SSDL-TL (a-c). In (d-f) are plotted events that are selected by making the cut in TSSDL-TSSDR shown in (a). Chapter 5. Data Analysis 83 800 600 - \ K) o u 400 H 200 - \ 0 0 3000 6000 9000 12000 15000 E sum ( k e V ) Figure 5.14: Triple sum spectrum for the coincidences between SSDL,SSDR and T L , after timing cuts were applied. in Table 5.2. A further kinematic analysis has been carried out on these selected events after the timing cuts and the analysis procedure is discussed in the following. K inema t i c analysis A kinematic analysis code K I N A N A L was written in order to identify each of the three particles detected by the detectors on an event by event basis. This identification is useful as a check on the suppression of the random coincidences. In addition, using these identifications, a detailed comparison of the experimental data with Monte Carlo simu-lations was performed in order to understand the break-up modes observed in the triple coincidence spectra. Kinematic analysis of the triple coincidence events was performed by comparing the measured angles between the particles with the angles calculated from the measured energies of the particle pairs using conservation of momentum. Chapter 5. Data Analysis 84 The lab angles between each particle pair were calculated from the location of the detectors which observe the particles. For the strip detectors, the center of the pixel where the hit occurred is considered as the position of the hit. The pixel location is determined from the horizontal and vertical strip addresses. For the telescopes, the geometric center of the detector was taken as the position of the hit. In the calculation of the angles between each particle pair from the measured energies, conservation of momentum was used. Neglecting the small recoil from the j3 decay and assuming that the 9 B decay occurs at rest requires the vector sum of the momenta of the three particles to be zero. The relationship between the momenta of any two of the three decay pair is then given by: Pi +Pl -pl = -2pip 2cos(9i2 . (5.1) Where pi, p2 and p3 are the momenta of the particles and #12 is the angle between the decay pair. In calculating the momenta from the measured energies, a certain particle identification should be assumed. Since true coincidence events should always contain two a particles and a proton, only three possible combinations exist, namely aap, apa, and paa. The angles between the three particle pairs are then calculated for the three different possibilities and a comparison with the measured angles is made. This comparison is made by subtracting the measured angle from the calculated angle. If the assumed set of particle identification is incorrect, or if one of the particles happens to be a (3 - particle, two possible situations arise in the calculated cosine of angles between the decay pair: 1. The cosine of the angle gives an unphysical result, i.e. it does not fall between the range of -1 and 1. 2. The calculated angles will be significantly different from the experimentally mea-sured angle. Chapter 5. Data Analysis 85 This procedure is repeated for the three possible identifications and the differences be-tween the calculated and measured angles are compared for the three possible combina-tions. Thus for each possible particle identification, the difference between the calculated and measured angles for all three angle pairs is obtained. = {Oc - Om)m (5.2) #13i = {0C — 0m)m (5.3) ^23* = {Oc ~ 0m)23l . (5.4) In the above equation, 9C and 6m are the calculated and the measured angles between the three decay pair. The label i denotes the particle identification assumed, i = 1,2,3 for paa, apa,aap respectively. Following the above determination, the sum of the absolute values of the above three quantities (Aj) was evaluated : ^ = \612i\'+ |<Ji3i| + |<*23i| • (5-5) From the three different possibilities for the particle identifications, the combination which gives the smallest value of Aj is taken as the combination with the correct identifi-cation for the three particles. If the value of one of the cosine of the calculated angle for all the three different possibilities does not fall within the range between -1 and 1, the event will be considered as a random coincidence. These events are then sorted according to the particle identifications namely paa, apa, aap and random coincidence candidate events. Figure 5.15a-d show the triple coincidence energy sum spectra obtained after per-forming the sorting using the kinematic analysis procedure. Referring to Figures 5.15a-d, more than 96% of the events are sorted as paa events. The "angle difference" spectra between the calculated and the measured values from the three detector combination for the paa events are given in Figure 5.16. In all the Chapter 5. Data Analysis 86 Figure 5.15: Triple sum spectra for SSDL, SSDR and T L as sorted out by the kine-matic analysis program. These events are identified as: (a) pact: coincidences (b) aap coincidences, (c) apa coincidence (d) random coincidence events. Chapter 5. Data Analysis 87 three combinations, a peak centered near zero is evident. These peaks are broadened by the angular and energy resolutions of the detectors, as well as by the recoil broadenings from (5 decay of 9 C , that has not been taken into account in the calculatons. A two-dimensional plot of SSDR-TL coincidences for the paa events displayed in Figure 5.15a is shown in Figure 5.17. A l l the events lie on the diagonal band, and this feature is consistent with the decay through the 5 L i 3 s . + a, to be compared with the Monte Carlo simulation shown in Figure 4.6 in Section 4.4.2. In order to verify that these events are indeed associated with decays through the ground state of 5 L i , the peak at 12.2 MeV observed in the triple energy sum spectrum was studied in more detail by selecting the events under this peak. The correlation between the energies of the individual particles were compared with Monte Carlo simulations. The Ex= 12.2 M e V state The triple energy sum peak near 12.4 MeV from the decay of the 12.2 MeV state has been studied by selecting the events under this peak. A Monte Carlo simulation has been performed for this state and a comparison between the Monte Carlo and experimental spectra has been made. The experimental data were selected from those of Figure 5.15a with an energy gate set on the 12.2 MeV state of 9 B . The two dimensional coincidence distributions for each pair of detectors are compared with those from the Monte Carlo simulations, and show good qualitative agreement (Figure 5.18). Coincidence efficiency As discussed in Section 4.2, the coincidence efficiency for a given energy and breakup mode is a function of the breakup energy. This coincidence efficiency was calculated from Monte Carlo simulations for SSDL, SSDR and T L combinations for the 9B—> 5 L i 3 . s . + a breakup for excitations in 9 B from 2-14 MeV. Also threshold effects, as set experimentally Chapter 5. Data Analysis 88 Figure 5.16: Angle difference spectra for the events which are paa type shown in Fig-ure 5.15a. Chapter 5. Data Analysis 8 9 6 H > CD 2 4 2H E (MeV) SSDR v y Figure 5.17: A two-dimensional plot for the paa coincidence events for which the triple energy sum spectrum is plotted in Figure 5.15a. and in the analysis, were incorporated in the simulations. Also as noted in Section 4.2, the angular correlation probabilities for each energy have to be included in the Monte Carlo calculation for this decay mode. This requires the knowledge of, or an a priori assumption about, the spins of the 9 B states and the relative contributions of these states at the energy in question for the spectrum, to which the efficiency curve is applied. The main contributions in this spectrum are the J^=5/2: -, E x=2.34 MeV and E x=12.2 MeV states, and the continuum between them. Consequently, in the region up to 4.0 MeV the angular correlation for Jn = 5/2~ with only the lowest allowed /-value was assumed. Since the angular distribution for the 12.2 MeV state is shown to be consistent with isotropy (see section 5.3.3), an isotropic correlation has been taken for the entire region above 4.0 MeV. Low energy cuts were incorporated for each detector which reflected those imposed by the timing discriminators used for the experimental data. The efficiency curve is Chapter 5. Data Analysis 90 0 2 4 8 B 0 2 4 6 8 Figure 5.18: Scatter diagram of energy-energy correlations for different detector combi-nations for the 12.2 MeV 9 B state: a comparison between the experimental (a-c) and Monte Carlo (d-f) data. Chapter 5. Data Analysis 91 Figure 5.19: Simulated triple-coincidence efficiency of SSDL, SSDR and T L for the 9 B break-up through the 5 Li(3/2 _ )+o; channel as a function of the breakup energy. The squares with statistical errors represent the calculated points. The smooth curve is a fourth order polynomial fit. Each point was simulated from 2 x 106 events. In the region below 4.0 MeV, angular distributions for the 5/2~ 9 B state were used. For the rest of the points, an isotropic distribution was assumed corresponding to Jn=3/2~. shown in Figure 5.19. 5.3.3 S S D R d o u b l e h i t , T L t r i p l e c o i n c i d e n c e even ts As discussed in the introduction of this Section, the triple coincidence events detected by the face-to-face detectors, [i,e 2(SSDR)-TL and 2(SSDL)-TR] with two of the three events detected by the SSD, also had significant statistics. The energy sum spectrum for the triple coincidence between SSDR-TL is given in Figure 5.20a, and the coincidence events obtained after timing cuts are shown in Figure 5.20b. The peak at 12.2 MeV in Figure 5.20a is broadened and shifted due to the energy calibration used. Since particle identification has not been assigned in this spectrum, Chapter 5. Data Analysis 92 (keV) 12000 14000 (keV) 12000 14000 Figure 5.20: Triple sum spectra for the SSDR-TL coincidence events with a double hit on the strip detector: (a) Raw spectrum with no timing cut, (b) Spectrum after timing cut. proton calibration is assumed for all the three particles. After selecting the events under this peak, a two-dimensional plot between the two energies observed in the strip detector is shown in Figure 5.21. From this plot the p — a coincidences from 9 B ;L2 . 2 —> 5 Lfg.s . + a. and the a — a coincidence from 9 B i 2 . 2 —> 8 B e g s . + p are displayed. The p — a events in Figure 5.21 were selected to study the angular correlation between the decay particles, for the breakup of this state to 5 L i 5 s . + a. Angular correlation for the 1 2 . 2 M e V state After selecting those events decaying through the 5Li+a channel from Fig 5.21, and after making a kinematic determination of the particle identification, the center of mass angles between the first a particle and the secondary breakup particles were determined for each event. A plot of these angular distributions and the comparisons with the Monte Carlo simulations assuming a spin of either 3/2 or 1/2 are given in Figures 5.22. It can be seen from this plot, that the observed angular correlation is consistent with isotropic distribution. Although based on the information from this plot we cannot assign with Chapter 5. Data Analysis 93 7000 • 5250 -> CD 3500 • p —a 1750 • 1 I I 1750 3500 5250 7000 Figure 5.21: Two-dimensional plot for the 12.2 MeV state for the SSDR-TL events shown in the window in Figure 5.20. The a-a coincidence events are for the decay through the 8 Be+p channel, and the p-o; events are for the decay through the 5 L i + a channel. Particle identification was made using the techniques described in Equations 5.2-5.5. Chapter 5. Data Analysis 94 E x p e r i m e n t 0 24 48 72 96 120 0 a i P (degrees) 100 116 132 148 164 180 8 (degrees) ala2 \ => ' Figure 5.22: Centre-of-mass angular correlation between the first a particle and the particles from the secondary break-up in the decay of the 12.2 MeV 9 B state through the 5 L i - | - Q ! channel. The solid line histogram is a Monte Carlo simulation that assumes a spin of Jn=3/2~ for this state, with an isotropic angular distribution. The dashed line histogram is a simulation for a spin of Jn—l/2~ for this state (Section 4.6). (a) shows the a l -p angular correlation and (b) the al-a2 correlation. Chapter 5. Data Analysis 95 6000 4500 -\ Figure 5.23: Two dimensional plot of events detected in the face-to-face detectors SSDR-TL in coincidence with a (3 particle in the plastic scintillator. Only single hit events in the SSD were accepted. certainty a spin of 3/2 for this state, the J = 1/2 possibility can be ruled out. 5 . 3 . 4 SSD-Telescope-/3- coincidences The two dimensional spectra between SSDR, T L and SSDL and T R recorded in coinci-dence with a 0 particle were useful in the determination of the normalization of the 8 Be+ p and the 5 L i + a spectra. The additional coincidence requirement with the P detector suppressed random coincidence events. A density plot of SSDR-TL coincidence is given in Figure 5.23. The densely populated region around (1,1) MeV is again from the decay of 9B2.34 —• 5 L i s . s . + a and the nearly horizontal band is mainly from 9B2.s -+ 8 Be 3 . s .+ p. The absence of the vertical band from the decay to 8 Be s . s .+ p is due to the low energy cutoff in SSDR. Chapter 5. Data Analysis 96 In order to select the p-a events decaying through the ground state of Be , the sec-ondary breakup energy (Q 2) for this channel is calculated using Equation 4.1 in Chapter 4. Substituting the appropriate masses for the 8 Be+p decay mode in Equation 4.1, Q2 is given by: Q2{8Be) = + 2Ea + ^EpEacos9pa . (5.6) A plot of the spectrum of the Q 2 value is given in Figure 5.24, and a peak around 100 keV in this plot is apparent. This peak is due to events decaying to 8 B e g s . + p 2 . By applying a cut on this peak, as shown in Figure 5.24, events from the decay of 9 B —» 8 Be 5 . s . +p were sorted. Shown in the inset of this figure is the two-dimensional plot of the events selected under this peak. The plot in this inset shows that events selected by the "Q 2 cut" have the energy ratio that is expected from decay through the 8 Be channel. The proton energy spectra observed by SSDL and SSDR are given in Figure 5.25. The 1/2 - state at 2.8 MeV peak is prominent in this spectrum. There is also an excess of counts around 1 MeV in both spectra that could be due to the decay of the l / 2 + state in 9 B , but definitive determination requires a good understanding of the background in this region, such as possible contributions from the strongly populated 2.34 MeV state. The quality of the data can be improved if a better position resolution is achieved. Since with better position resolution the broadening of the "Q 2 peak" can be reduced, and this would allow us to put a tighter cut on this peak. Because this region is very weakly populated, better statistics are also needed. 2Note that Q2(8Beg.s.) = 92 keV, with a width of only 6.8 eV Chapter 5. Data Analysis 97 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Q2(8Be) (keV) Figure 5.24: A plot of the Q 2 value for SSDR-TL coincidence events calculated using Equation 5.6. The peak near 100 keV is due to events decaying to 8Be 3 . s .+p, the plot in the inset is for those events that are selected under this peak. Figure 5.25: Proton spectra measured by the two strip detectors in coincidence with the respective opposing telescopes after selecting the 8 Be 9 . s . +p events by the "Q 2 cut" shown in Figure 5.24: (a) SSDL, (b) SSDR. Chapter 6 Results and discussion 6.1 Introduction In this Chapter, the results obtained from the work of this thesis are presented. These results mainly consist of the /3-decay branching ratios of 9 C to states in 9 B , and the decay branches for the subsequent breakup of these 9 B states. Because all the excited states of 9 B are broad, and because there is, in most cases, more than one decay channel available for the breakup, the procedure used in determining the decay branches involves analyzing the spectra for the different decay modes separately and relating the information obtained on the decay branches from one data set to another. These steps will be discussed in the following sections, and the discussion is organized as follows. We begin with a discussion of the fitting procedure followed in this work in Section 6.2. The results for the fits to the summed ratio-cut spectrum from 9B—> 8 Be 5 . s . + p break-up are presented in Section 6.3.1. In Section 6.3.2, the fitting results for the triples spectrum from 9 B - ^ 5 L i s . s . + a break-up are given. The steps followed in the normalization between the spectra for the two decay channels and in the subsequent relative branch determination for the excited states of 9 B are discussed in Section 6.3.3. In Section 6.3.4, the determination of the relative strength of the ground-state transition to the sum of the strengths of transitions to all of the excited states is discussed. In Section 6.4, the final /3-decay branching ratio results are presented, and the BGT values are determined. A consistency check was performed (Section 6.5) on the branching ratio results by simulating the singles particle spectra from the breakup 98 Chapter 6. Results and discussion 99 of the 9 B states using the branching ratios and the state properties determined from the fits, and by comparing the simulated results with the measured spectra. Finally in Section 6.6, the error analysis on the branching ratio results is discussed. 6.2 F i t t i n g P r o c e d u r e s Each 9 B state resulting from the (3 decay of 9 C is considered to be an isolated resonance that can be represented by the line shape that follows the one-level approximation to the i?-matrix theory discussed in Section B.3. Following these formalisms, the shape of the spectrum of 9 B , wc(E) from /3-decay of 9 C decaying through a broad state such as 5 L i 9 s . is given by, : r ( n f V AXc r Pc(E-E')Px,(E')ri w ^ - ^ 2 r { E x + A x _ E y + i _ r l { E ) J { E x , + AX> - E>y + \n,(E>)d±J • ( b - i j Here AXc is an arbitrary feeding factor for each state, and the remaining variables are as defined in Equation B.57, Section B.3. A n approximation was made by setting the shift function for the 9 B state, AX(E) to zero. In addition, the total width of the 9 B state TX(E) was assumed to be not dependent on energy. Making these approximations, the /5-delayed 9 B spectrum decaying through a broad intermediate state such as 5 L i 5 . 5 . +a is given by: • W c { E ) (EX - Ef + I « / 4 J (Ex> + A A , ( £ ' ) - E>Y + T\,{E<)I± ' ^ ] For the decay to 8Be 9 . s .+p the intermediate state is sufficiently narrow and E1 can be taken as a constant thereby eliminating the integral in Equation 6.2: w ^ = f ^ { E x - E y + n/4- ( 6 - 3 ) Where PP(E — E') is now the proton penetrability in the decay of 9 B to 8 B e g s . + p. Chapter 6. Results and discussion 100 The fittings for the spectra from 8 Be+p and 5 L i + a breakup channels were carried out using Equations 6.2 and 6.3 respectively. Only states that correspond to allowed Gamow-Teller decay (J 7 r = l /2 - , 3 /2~ ,5 /2~) are considered in the fit, and only the lowest allowed angular momenta between the decay pair are included. The fitting parameters to be adjusted until the x2 w a s minimized are the feeding factors A\, the resonance energies E\ and the level widths Fx- The x 2 was calculated from: Nbin ( A T _ AT , \2 ^2 _ ^ expt lscalc) ^ i=l a 2 where Nexpt is the experimental value, Ncaic is the theoretical value convoluted with the energy resolution of the detectors and o is the statistical error. The fittings were carried out using the CERN-written minimization package MINUET [58]. Instead of calculating the Fermi phase space factor fp, and the penetrabilities Pi every time wc(E) is calculated, these functions were calculated beforehand for each energy bin in the spectra to be used in the calculations of the fitted spectra. 1. The Fermi phase space factor The Fermi phase space factor was evaluated using the parameterization equation given in Ref. [59], and with the parameters provided in the same reference. This function is parameterized with: / = Sfz=0, (6.5) where fz=o is the phase-space factor for a nucleus with Z = 0 and is evaluated analyti-cally, fz=o = ^(2VFo 4 - 9VF02 - 8)p0 + \w0\n(W0 +p). (6.6) Chapter 6. Results and discussion 101 W0 in Equation 6.6 is the total electron end-point energy in units of mec2 and po = (W0 2 — l ) 1 / 2 . S is given by the following parameterization equation: l n S = ^ > n ( l n £ 0 ) n (6.7) n=0 The coefficients an are evaluated numerically in Ref. [59] by using electron wavefunc-tions for an atom with a uniform spherical coulomb field of the nucleus, and also by including first order radiative corrections. EQ is the end-point energy of the electrons in units of MeV. The coefficients an are tabulated in Ref. [59] for nuclei with atomic mass A = 3 — 257, both for (3+ and (3~ decay. The Fermi function, / , calculated from these parameterization is plotted in Figure 6.1. 2. The penetrabilities The penetrabilities are calculated from the regular and irregular Coulomb Functions using the code C O U L L from the C E R N library with channel radii, ac=3.62, 4.2, 4.44 and 4.62 fm for the a+p, 8Be+p, ct+ct and 5 Li - | - a systems respectively. 3. The 9 B state shape observed in the /3-decay of 9 C The shape of a broad state in 9 B observed in the /3-decay of 9 C is distorted by the Fermi-phase space factor. In order to demonstrate this distortion, the distribution of a 9 B state (Ex=12.16 MeV, T=450 keV decaying to 8 Be 9 . s . ) is plotted by the solid curve in Figure 6.1. The same state observed in the (3 decay of 9 C is plotted by the broken line. Chapter 6. Results and discussion 102 10= 1(E) 10' 0 2 4 6 8 10 12 14 16 E x ( 9 B ) ( M e V ) Figure 6.1: The shape of a 9 B state , 1(E), (EX=12.16 MeV, T=450 keV) decaying to 8 Be 9 . s . + p. The shape of this state observed in the (3 decay of 9 C , 1(E) fp(E), is given by the broken line. 6.3 Branching ratio determination 6.3.1 Relative branches for the 9 B —> 8 B e 9 s + p break-up (Expt. 1) The summed, efficiency corrected proton spectrum from the break-up of 9B—>8Beg s +p has been discussed in Section 5.2.2, and a plot of the spectrum was shown in Figure 5.9. The spectrum is mainly composed of a weak narrow peak near 2.3 MeV from the 5/2~ state, a broad peak around 3 MeV from the l / 2 ~ state as well as peaks near 12.2, 13.3 and 14.0 MeV. Included in the fit were these five states as well as a background state that is represented by a broad level at a higher excitation energy in 9 B . The contribution from each level to the fit is shown in Fig. 6.2. The fitting parameters and the spin and parity assignments used in the fit are given in Table 6.1. The intensity of each state normalized to the intensity of the 2.8 MeV state is given in the last column of this table. In Table 6.1, the parametrs E\, F\ and Ax are determined from the fit and the relative intensities are determined by calculating the peak shape using these parameters and integrating the Chapter 6. Results and discussion 103 2 4 6 8 10 12 14 16 E (MeV) cm. ' Figure 6.2: Fit to the summed-energy telescope spectrum from Expt. 1, corrected for efficiency. The contribution from each 9 B state to the total spectrum obtained from the fit is shown. The error bars on the data points represent only statistical errors. The component labelled background is a continuum arising from an assumed higher energy state. Chapter 6. Results and discussion 104 Table 6.1: State energies, widths and relative feeding factors obtained from the fit to the 8 Be(0 + )+p spectrum. The 9 B excitation energies (Ex) are obtained from Ex — Ecm — 0.185 MeV. In the last column the relative contribution of each state is given normalized to the intensity of the 2.8 MeV state. Ex (MeV) in 9 B T (MeV) Ax relative intensity 2.34 0.12 5/2- 4.0 0.030±0.002 2.8 2.50 1/2- 86.9 1 12.16 0.45 3/2" 61.4 0.095±0.007 13.3 0.03 - 0.047 0.00033±0.00002 14.0 0.60 - 358.7 0.031±0.002 Background _ - 181.6 0.09±0.01 peak over an energy interval from 0 to 15.475 MeV, the /3-decay Q-value. 6.3.2 R e l a t i v e b ranches for t he 5 L i f f . s . + a b r e a k - u p ( E x p t . 2) The triple coincidence spectrum measured by SSDL, SSDR and T L in Expt. 2 has been discussed in Section 5.3.2. This spectrum consists mainly of break-ups through the 5 L i + a channel. A plot of the spectrum after efficiency corrections is given in Figure 6.3 with the x-axis corresponding to the sum energy (Esum = Ex + 0.277 MeV) of the three particles detected in coincidence. The dominant features in this spectrum are the two peaks from the states at 2.34 and 12.2 MeV. There is also a bump near 4 MeV which is interpreted here as the break-up of the J 7 r =l /2" state. These three peaks were included in the fitting routine. In addition, in order to account for the excess counts observed between the two peaks, a continuum ("background"), modeled by a broad state at higher excitation energy, was included in the fit. The fitting parameters and the spin and parity assignments used in the fitting are given in Table 6.3.2. The intensity of each state normalized to the intensity of the 2.34 MeV state is given in the last column of this table. Chapter 6. Results and discussion 105 Figure 6.3: Fit to the triple-coincidence energy-sum spectrum for SSDL, SSDR and T L from Expt. 2. The contribution of each 9 B state to the total spectrum obtained from the fit is shown. The error bars represent only statistical errors. Table 6.2: State energies, widths and relative feeding factors obtained from the fit to the 5 L i + a spectrum. The 9 B excitation energies (Ex) are obtained from Ex = Esum — 0.277 MeV. In the last column the relative contribution of each state is given normalized to the intensity of the 2.34 MeV state. Ex (MeV) in 9 B T (MeV) r Ax relative intensity 2.34 0.10 5/2". 17.4 1 (4.0)a 0.57 (1/2-) 6 0.24 0.020±0.003 12.16 0.45 3/2" 18.24 0.135±0.007 Background - (5/2-)" 80.7 0.106±0.006 a Assumed to correspond to the lowest J 7 r =l /2 state, see text. b Assumed in fit. Chapter 6. Results and discussion 106 6.3.3 Normalizing the spectrum from the 5 L i + a channel to the 8 Be+p channel Once the relative branches for both data sets are known, what remains in the determina-tion of the relative /3-feeding factors to the excited states of 9 B is to relate these relative branches by normalizing the 5 L i + a spectrum to the 8Be-fp spectrum. The 2.34 MeV and the 2.8 MeV states, decaying almost entirely to 5 Li4-a and 8Be+p, respectively, were used for the normalization. The ratio of the decay branches of these two state was deter-mined from the two dimensional (SSDR-TL) coincidence spectrum (Figure 5.23). After making corrections for the acceptances for these two channels, the ratio of the strengths of the two branches was obtained: •^ 2.8^ 8Be+p = 5.8 ± 0 . 6 (6.8) Using this ratio, the 8 B e ± p spectrum (Fig. 6.2) was normalized to the 5 L i + a spectrum (Figure 6.3). In the fourth column of Table 6.3, the relative intensities of each level in the total spectrum (Ix), calculated from the normalization factor and the relative yields determined in Sections 6.3.2 and 6.3.3, are listed. 6.3.4 Determination of the ground-state branching ratio from the /3-particle spectrum The analysis of the /5-particle coincidence spectrum and the decay of 9 C through the ground state of 9 B have been discussed in Section 5.3.1. This spectrum (Fig. 5.10a) is used to determine the ratio of the /3-feeding factor of the ground state to the sum of the feeding factors of all the excited states of 9 B . This was done by simulating the /^-particle coincidence spectrum resulting from the break-up of all states of 9 B and comparing the experimental spectrum with that generated by the Monte Carlo simulation. In order to simulate the contribution of each level to the particle spectrum recorded in coincidence Chapter 6. Results and discussion 107 Table 6.3: Relative 9 C /3-decay branches to the excited states of 9 B obtained after nor-malizing the 8 Be(0 + )+p spectrum to the the 5 Li (3 /2 - ) - | -a i spectrum. The second column gives the relative intensity of each state with respect to the two separate spectra. The relative intensity with respect to the combined data is given in the third column. I\ is normalized to the intensity of the 9B(2.34)—> 5 Li(3/2~)-r -a transition. Ex (MeV) in 9 B T T ( N2.8^a Be+p \ / A - / ( S £ e ( 0 + ) + P ) ^ 2 3 4 ^ 5 L I + J 8 Be(0 + )+p I(sBe(0+)+p) 2.34 0.030 0.0052±0.0006 2.8 1.0 0.170±0.018 12.16 0.095 0.016±0.002 13.3 0.00033 0.000057±0.000006 14.0 0.031 0.0053±0.0006 Background 0.09 0.015±0.002 12.16^ 8 Be(2 + )+p 0.034±0.004 5 Li(3 /2- )+a A5ii(3/2-)+a) = -f(5Li(3/2-)+a) 2.34 1.0 1.0±0.19 (4.0) 0.020 0.020±0.004 12.16 0.135 0.135±0.014 Background 0.106 o.io6±8:SI a Measured from the singles proton spectrum. Chapter 6. Results and discussion 108 0 1000 2000 3000 4000 E (keV) Figure 6.4: Simulated particle energy spectrum in detector T L in coincidence with a /3-particle (Ep > 4 MeV) detected by the plastic scintillator. This spectrum was com-pared with the experimental spectrum (Fig. 5.10) in calculating the ground-state branch-ing ratio in the 9 C /3-decay. Chapter 6. Results and discussion 109 with /3-particles in the /3-detector, it was necessary to include the distortion of the spectral shape of the 9 B states caused by the coincidence requirement. Due to the low energy cut-off of the /3-detector (4 MeV), the particle-/? coincidence efficiency is dependent on the 9 B energy into which the transition takes place. This energy-dependent efficiency was calculated for /3-spectra for a number of excitation energies in 9 B by Monte Carlo simulations. Simulation of the response of the @ detector A Monte Carlo program to simulate the response of the /3-detector was written using the CERN-developed program, G E A N T . It provides user-defined subroutines to track var-ious particles through a user-specified geometry, simulating the physical processes that take place during the interaction such as energy loss, multiple scattering, bremsstrahlung, pair production and Compton scattering. Incorporated in the simulation were the exper-imental setup and the theoretical /3-particle spectra. The desired output of the program, the energy deposited in the active volume of the detector, was recorded for each event. The calculated efficiency curve is shown in Fig. 6.5. This efficiency is the ratio of the number of detected events that are above a certain (5 energy (in this case 4 MeV) to the total number of the detected events (Equation 6.9) Distortion of the particle spectrum due to the (5 efficiency The spectral shape observed in the particle-/3-coincidence spectrum, w'x(E) is then given £ = N(Ep>AMeV) N-total (6.9) by w'x(E) = wx(E)e(E) , (6.10) Chapter 6. Results and discussion 110 7.0 E (9B) (MeV) Figure 6.5: Monte Carlo-simulated efficiency of detection for /3-particles with Ep > 4 MeV in the scintillation detector used in Expt. 2, as as function of 9 B excitation energy. where w\(E) is the spectral shape for a 9 B state A. A comparison between wx(E) and w'x(E) for the 2.34 MeV and the 12.2 MeV state is given in Figure 6.6. The mean value of the efficiency, < ex(E) > for each 9 B state used in the simulation is determined from Eq. 6.11, <ex(E) >--f0Q'e(E)wx(E)dE J?" wx(E)dE (6.11) The relative strength of each state in the particle-/^-spectrum, I'\(E), is therefore given by I'x = Ix<ex(E)> , (6.12) where Ix is the relative strength of the 9 B state determined in Section 6.3.3. The spec-tral shapes, w'x, and the relative intensities, I'x, were used in simulating the particle-/? spectrum. The entries in the fourth columns of Table 6.4 and 6.5 are the relative con-tributions of each state to the /^-particle spectrum, and are calculated from the relative branches (column 2) and the /3-detector efficiency (column 3). The number of simulated events is listed in the last column. 2.5 x 106 events were simulated for the 9B(2.34) —> 5 L i Chapter 6. Results and discussion 111 E y ( SB) (MeV) E x ( 9B) (MeV) Figure 6.6: A comparison between wx{E) and w'x(E) for the 2.34 MeV and the 12.2 MeV Table 6.4: Relative contributions of the 9 B states decaying to the 5Li(3/2~)+a: channel in the particle-/3-coincidence spectrum. The number of simulated events (column 5) is determined according to Ix (column 4). Ex (MeV) in 9 B h ' < 6 ^ E > >~ [wx(E)dE I'x = h< ex(E) > N 2.34 1.0 0.421 0.421 2.5 x 106 (4.0) 1.96 x IO" 2 0.373 7.15 x 10" 3 42,450 12.16 0.135 0.095 1.29 x IO" 2 76,300 Background 0.106 0.178 1.90 x 10" 2 112,225 Chapter 6. Results and discussion 112 Table 6.5: Relative contributions of the 9 B states decaying to the 8 Be(0 + )+p channel in the particle-/3-coincidence spectrum. The ground state branch has been arbitrarily assigned the same branching ratio as the 2.34 MeV state of 9 B . Je(E)wx(E)dE~ f wx(E)dE Ex (MeV) in 9 B < ex(E) >= I'x = h<ex[\E)> N 0.0 2.34 2.8 12.16 14.0 Background 5.08 x 10~3 0.17 1.6 x IO" 2 5.2 x IO" 3 1.5 x I O - 2 12.16(8Be(2+)+p) 3.4 x 10" 0.450 0.426 0.405 0.137 0.134 0.287 0.095 0.450 2.16 x 10" 3 6.96 x 10~2 2.24 x IO" 3 7.04 x 10~4 4.35 x IO" 3 3.23 x IO" 3 2.67 x 106 12,852 413,250 13,300 4,180 25,800 19,000 -f- a decay. The number of events for the remaining states were determined accordingly. The ground state branch, the only unknown variable in the simulation, was first arbi-trarily assigned to have the same branch as the 2.34 MeV state. The relation between the ground state branching ratio, B(gs) and the branching ratio to the 2.34 MeV state, 5(2.34), was determined by calculating the ratio of the counts under the ground state peak to the counts for the rest of the spectrum in both the experimental (ReXp) a n d the Monte Carlo (RmC) data and by comparing these ratios (Figures 5.10b and 6.4). The branching ratios for these two states are related by the following expressions: Rexv = (~) = 2.40 (6.13) \ 9s / exp and Rmc = ( ~ ) = 3.20 (6.14) Chapter 6. Results and discussion 113 C Rmc \ Rexp / B(gs) = 73(2.34) = (1.33 ± 0.13)73(2.34) . (6.15) 6.4 Final branching ratios and BQT values With the ratio of the strengths of the ground state and the 2.34 MeV determined from relation 6.15, the absolute branching ratio can be calculated from Table 6.3 and Equa-tion 6.15. These calculated values are shown in Table 6.6. The energies of the states were taken from the fits to the 8 Be(0 + )+p spectrum where a good energy calibration was available. Because most states extend over a large energy range, the value of BGT (except for the ground state transition) was calculated using an averaged inverse Fermi function folded over the distribution of each individual state, as discussed in Section A.2, and using Equation A.41 that is given below for a reminder. K Jc?" jfeW(E)dE < B G T >= r > — • (6.16) h/2 f^W(E)dE with 7^=6177 s and £1/2 the partial half-life for the decay to that state. The log (ft)' value is then given by, ft = , (6.17) < OGT > and shown in Table 6.61. The /3-decay branching ratios and the state energies of 9 B measured in this work are shown in the decay scheme in Figure 6.7. Clearly the state at 12.2 MeV shows a very large Gamow-Teller strength making it a significant part of ^ef. [14] uses = <e.u> for Gamow-Teller transitions with an unquenched ^- = 1.59 (<M and gy being the axial vector and vector 2 coupling constants in /3-decay, respectively). Following Ref. [3] a quenched value for ^ « 1 is taken 91 here. When values of BGT are compared with Ref. [14], the values here have to be divided by 1.58. Ref. [14] uses a similar averaging procedure for BGT to that described here. Chapter 6. Results and discussion 114 Table 6.6: Level energies and branching ratios to states populated in the /3-delayed particle-decay of 9 C . Brp and Bra denote branching ratios to 8 Be (ground state and E^^S MeV state) and the 5 L i ground state relative to the total number of /3-decays. We use "background" to denote the fraction of the branching ratio which is due to states above Ex=lb MeV or states too broad to allow a unique identification to be made. BQT is calculated using Eq. 6.16, log (ft) employing Eq. 6.17. E x (MeV) J w Brp [%} Brv [%} Bra [%} log/ t < BGT > 0.0 3-2 46.9 ± 5.0 46.9±5.0 a b 5.38 0.026±0.003 2.34±0.03 5-2 35.2±6.7 0.19±0.02a 35.0±6.7 5.01 0:061±0.013 2.8±0.2 1 -2 6.7±0.7 6.0±0.6 0.7±0.1 c 5.61 0.015±0.0018 12.16±0.10 6.8±0.7 1.8±0.2 e 5.0±0.5 3.39 2.50±0.25 13.3±0.1 . / 0.0020±0.0004 0.0020±0.0004 b 5.79 0.010±0.003 14.0±0.2 . / 0.19±0.02 0.19±0.02 b 4.17 0.42±0.05 background5 . / A 0+2.1 h ^•z-0.7 0.5±0.1 o 7+2.0 h °- '-0.6 4.17 u- 4 Z-0.07 a 8Be(gs) only. 6 No evidence found for a break-up; assumed small. c Break-up through both 5 L i and 8Be(3MeV) are possible. d 3n > l / 2 " , see text. e This is composed of 0.58±0.07% for 8Be(gs) and 1.2±0.2% for 8 Be(3MeV). Not known. 9 From states which cannot be identified uniquely, see text. h Error large and asymmetric due to unknown spin composition. the Gamow-Teller giant resonance. Errors were derived, as discussed in Section 6.6, by Monte Carlo-simulated variations of the efficiency curves, beam spot positions and other significant parameters (Section 6.6). The errors are mainly systematic; statistical errors are comparatively small. While the errors are highly correlated (because of the 100% sum rule and the different normalizations applied here), they are treated as independent of one another. Energies of states are taken from the fits. Chapter 6. Results and discussion 115 Figure 6.7: /3-decay scheme of 9 C measured from this work. The /3-decay branching ratios are indicated. Chapter 6. Results and discussion 116 0 1000 2000 3000 4000 E (keV) Figure 6.8: A comparison of the singles "reconstructed" spectrum with the experimental data (Telescope TL) in the low energy region. The experimental data are shown by the open circle points, and the Monte Carlo simulations by the histogram. 6.5 Reconstruction of the singles spectra Using the branching ratios and the state shapes determined from Section 6.4 the singles spectra measured in the telescopes were simulated and compared with the experimental singles spectra to check for consistency. For the low energy part, the experimental spec-trum used for comparison is the /3-particle coincidence spectrum measured by T L owing to the background due to /3-particles in the measured singles spectrum (Fig. 6.8). The spectrum obtained by Monte Carlo simulation has been normalized to the experimental spectrum by matching the counts under the proton peak at 164 keV. For the high energy part (above 4 MeV), the singles spectrum measured by the Telescope 2 in Expt. 1 was used. This spectrum is given in Fig. 6.9. The proton peak at 12.2 MeV was used in the normalization. Chapter 6. Results and discussion 117 10* o 10° 3500 5500 7500 9500 11500 13500 E (keV) Figure 6.9: A comparison of the singles "reconstructed" spectrum with the experimental data in the high energy region. The experimental data are shown by the open circle points, and the Monte Carlo simulations by the histogram. Also, the proton events were selected from the E - A E plots, and the proton spectra were compared to the simulated proton spectra. This comparison for the proton spectrum observed by S2 is shown in Fig. 6.10. Clearly in all the three spectra, the qualitative features of the singles spectra are well reproduced by the Monte Carlo simulation. Minor inconsistencies are attributed to (i) a not entirely correct state shape using the approxi-mations described above, and (ii) incomplete knowledge of angular momentum fractions, in particular, in the continuum region. 6.6 Error Analysis Error analysis was carried out in order to deduce the systematic errors associated with the branching ratios of the decay of 9 C to the states of 9 B . For the errors in the branching ratio to the excited states of 9 B , first the uncertainties are determined for the relative intensity of each state in the spectra for which the state is observed (i.e 8 Be+p and Chapter 6. Results and discussion 118 2000 4000 6000 BOOO 10000 12000 14000 E p (keV) Figure 6.10: A comparison of the protons "reconstructed" spectrum with the experimen-tal data detected in S2. The experimental data are shown by the open circle points, and the Monte Carlo simulations by the histogram. 5 L i + Q ! ) . These uncertainties are then propagated in order to determine the final errors in the total decay branches. The major source of the errors in the relative intensities are the experimental inputs into the Monte Carlo simulation used in the calculation of the coincidence efficiencies. These uncertainties in the experimental inputs are the uncertainties in the energy calibrations and the description of the geometry of the system. In order to deduce these efficiency-induced errors in the spectra that were fitted, several efficiency curves were calculated for possible deviations in the geometries and energy calibrations of the detectors based on reasonable assumptions of the possible uncertainties in source and threshold position. Once the efficiencies for one set of conditions are determined, the spectra are.efficiency-corrected (for this condition) and fitted in order to determine the relative intensities of each state in that spectrum. This procedure is repeated for several conditions and finally the maximum deviations from the values determined in Sections 6.3.1 and 6.3.2 are taken as the uncertainties of the the relative Chapter 6. Results and discussion 119 0 . 0 0 9 0 . 0 0 8 - \ 0 . 0 0 7 - \ CD 0 . 0 0 6 -«5 0 . 0 0 5 -^ CU 0 . 0 0 4 -CJ o < 0 . 0 0 3 -0 . 0 0 2 - \ ( 0 . 0 ) 0 . 0 0 1 - \ 0 . 0 0 0 2 4 6 8 10 12 E (MeV) Figure 6.11: Efficiency of detection of protons for S3 in coincidence with a-particles in the 8 Be(0 + )+ p channel, for the different beamspot positions assumed. The efficiency curve used in the fitting is for a Gaussian source density of circular symmetry with FWHM=0.25 cm. intensities of the peaks. These error analyses for the two fitted spectra are described in the following sections. 6.6.1 Error in the 8Be 9. s.+ P spectrum 1. Uncertainty in beamspot size and location In order to determine the error in the efficiency curve due to the uncertainty in beamspot location and size, several locations (within the foil) were simulated. The different ef-ficiency curves corresponding to the beamspot locations for S2 assumed are given in Figure 6.11. These errors for the different states, 5c are listed in Table 6.7. Chapter 6. Results and discussion 120 Table 6.7: Relative errors for the states observed in the summed ratio cut spectrum. SE, SQ and Ss are errors due to energy calibration, beamspot location and statistical errors respectively. EX (MeV) in 9 B SG (%) SE (%) Ss (%) ST = ^SG + 5% + S2S 2.34 5.9 4.2 1.2 7.3 2.8 1.1 3.0 0.4 3.2 12.16 5.3 4.8 0.51 7.2 14.1 6.3 2.9 0.73 7.0 Background 10.5 3.0 0.6 11.0 2. Energy threshold Two efficiency curves were calculated for each telescope for a energy thresholds in the opposing detector of EEXP± 20 keV, where EEXP is the threshold in the experimental data, in order to determine the error in the relative branches caused by the uncertainty in the energy calibration of the front detector telescopes. These efficiency curves were then used to correct the proton spectra in the same way as discussed above. The errors due to the thresholds, SE are listed in Table 6.7. 6.6.2 5 L i + a spectrum (SSDL, SSDR, TL) The systematic errors in the triple coincidence efficiency were determined in a manner similar to that described in Section 6.6.1. The error due to the uncertainty in the energy calibration was found to be the most significant in the triple coincidence spectrum. This is because of the high threshold associated with the two SSDs, (450 and 890 keV respectively for SSDL and SSDR). Two efficiency curves were determined for a threshold at EEXP ± 25 keV in both SSDL and SSDR, where EEXP are the experimental energy thresholds. These efficiency curves and the efficiency curve used in the actual fitting are shown in Figure 6.12, and the error associated with this uncertainty, SE are given in the second Chapter 6. Results and discussion 121 0.0005 0.0004 o.oooi -\ 0.0000 ESSEL-4&0,ESSDR -890 knV ESSDL-4S5, E5SDR=B65 KeV ESSDL=475, ESDR=915 keV 2000 4000 6000 8000 E b r e a k u p 10000 ( k e V ) 12000 14000 16000 Figure 6.12: Triple coincidence efficiency of SSDL, SSDR and T L for the decay through the 5 L i + a for different energy thresholds in the SSDs, the threshold on T L is 200 keV. column of Table 6.8. From simulations of the efficiency for different beamspot locations, the uncertainty in the geometry was found to be small compared with the case of the geometry of Expt . l and was estimated to be less than 5% . In addition, because of the lack of knowledge of the angular distribution of the decay particles in the continuum region, additional uncertainties are introduced in the branching ratio of the continuum region (the "background state"). Events in the continuum region were assumed to decay isotropically, and an estimate in the upper limit on the error in the branching ratio due to this effect was determined by inspection of the singles reconstructed spectra. 6.6.3 Error in the ground state branch The source of error in the branching ratio of the ground state that was considered was the one due to the uncertainty of the energy calibration of the (3 detector. In order to estimate the systematic error due to energy calibration of the (3 detector, the procedure described Chapter 6. Results and discussion 122 Table 6.8: Relative errors for the states observed in the triple energy sum spectrum SSDL,SSDR and T L . 5E, 8Q and 5s are errors due to energy calibration, beamspot location and statistical errors respectively. Ex (MeV) in 9 B SG (%) 5E (%) 5S (%) 5T = < si®; + + (%) 2.34 5 15.1 1.3 16.0 (4.0) 5 15.2 2.8 16.2 12.16 5 0.8 1.7 5.1 Background 5 2.9 1.6 6.0 in Section 6.3.4 was repeated by putting the threshold at Ep — 3.5 and Ep = 4.5 MeV in the experimental data and calculating Rexp for these cases. A deviation of 5% and 11% respectively was obtained from the value of Rexp = 2.40 determined in Equation 6.13 for Ep = 4 MeV. Hence an error of 11% was estimated for the ground state branch. Chapter 7. Conclusion In this thesis, a detailed study of the states of 9 B populated by the (3+ decay of 9 C has been performed. Coincidence detection of the particles from the breakup of 9 B was made feasible owing to the large yield of 9 C produced at TISOL. By performing kinematic analyses on these coincident events, and by making comparisons with Monte Carlo simulations, it was possible to determine the breakup modes of these states in 9 B , and to separately study the spectra from the different breakup modes. It was found that most of the decay of the states of 9 B proceed through 5 L i g s . + a and 8 Be s . s . + p. There has been no definitive evidence for three-body breakup of any of the 9 B states, although it cannot be completely excluded at a low branching level. In order to describe the states of 9 B , and the secondary compound states such as 5 L i into which 9 B decays, a simplified one-level P-matrix description of these states was adopted. The spectra from both decay modes were then fitted using this model, and the relative branching ratios were extracted from the fits. From these branching ratios, the Gamow-Teller strengths for the observed states of 9 B were determined. Further consis-tency check was performed by simulating the singles spectra using the state properties and relative strengths determined from these fits, and by comparing these simulated spectra with the experimentally measured spectra. A reasonably good agreement was found in most cases. For many of the states observed in this work, the two objectives of this work, i.e., 123 Chapter 7. Conclusion 124 measurement of the /3-decay branching ratios of 9 C , and the identification of the ex-cited states of 9 B along with the determination of their properties has been achieved, as outlined below: 1. A detailed study of the states of 9 B populated by the (3+ decay of 9 C has been performed. (a) proton and a decay branches of these states have been measured for the first time. (b) Locations and the widths of these states have been measured. (c) A new state in 9 B at 13.3 MeV has been found and studied. (d) The Ex = 14.0 MeV state has been observed for the first time in /?-decay studies. 2. The /?-decay branches and the transition strengths for the (5 decay of 9 C have been measured. (a) The (5 decay branches to the higher lying states (Ex = 12.16,13.3 and 14.0 MeV) have been measured for the first time, and improved measurements to that of Ref. [3] have been obtained for the (3 decay branches to the lower lying 9 B states (Ex = 0, 2.34 and 2.8 MeV). (b) A significant fraction of the transition strength (BQT) has been observed (3.45 ± 0.45) compared to the sum rule of 3(Z — N)=9 for the total value of BQT-(c) The large transition strength measured for the 12.2 MeV state BQT = 2.50 ± 0.25 is identified with the Gamow-Teller Giant Resonance. Chapter 7. Conclusion 125 Relatively poor agreement was found in the present work between the branching ratios of 9 B states with low excitation energy (EX = 0, 2.34 and 2.8 MeV) and those given by Mikolas et al. [3], the only data available on the /3-decay branches of 9 C . This can be attributed to (i) a different description of the shapes of the states here relative to Ref. [3] and (ii) some of the corrections and cuts applied in [3]. Good agreement is found with the range of Gamow-Teller elements BGT suggested in the shell model calculations of Ref. [3] for all states identified there. The strength of the 14 MeV 9 B state corresponds well with the one of the next higher J 7 r =l/2~ state above the 3/2~ state predicted in the Ref. [3]. It is also found that a large fraction of the breakup of the 12.2 MeV state proceeds through the 5 L i + Q ! channel, also an appreciable breakup branch for decay to 8 Be 9 . s . + p and 8Be3.o + p for this state has been measured. Furthermore, most of the observed Gamow-Teller strength is concentrated in this state. Having observed these properties, the | state in group (c) of Table IV of Ref. [3] predicted by the calculation of Brown is the most likely candidate for this state. A spin of | for this state is also in agreement with the observation of the angular correlation of the decay particles in this work. As discussed in Ref. [3] this state may then be regarded as the antianalog T = l / 2 state to the T=3/2 IAS state at 14.7 MeV. In relation to the /3 decay data in the mirror nucleus 9 L i , good agreement of the branching ratios with those of Nyman et al. [14] is found, if: (i) the branching ratio of the 2.8 MeV state and the background state in the present work are added and compared with that of the branching ratio for the l / 2~ state in Ref. [14]. (ii) one assumes that there is only one mirror state at 11 MeV in 9 Be instead of the two assumed in the decay of 9 L i , since there is only one apparent state in our data at 12 MeV. The Gamow-Teller matrix elements BGT are too sensitive to the state shape functions to allow a definitive comparison, unless 9 L i decay data of the same quality as the present 9 C data become Chapter 7. Conclusion 126 available, and are analyzed with the same method. The decay of the continuum in the region Ex ~ 4 — 10 MeV was also found to proceed mainly through the 5 Li+o; channel. Although in our fits we have assumed this continuum to arise from a higher-lying "ghost" state above 15 MeV, the nature of the origin for this continuum is not known. One other possibility for the continuum is that there exist one or two broad states in this region that cannot be resolved or defined. Such broad states have been predicted to exist in this region by the shell model calculations of Brown [3]. In conclusion, a consistent description of several 9 B levels has been achieved, and new information on their decay modes has been obtained. These measurements fit quite well into our general understanding of the mass-9 mirror pair that has been achieved via other reactions and by nuclear structure calculations. A n improved experiment with even more statistics, large solid angle coverage, improved particle identification and position resolution should allow one to measure the low excitation energy region in 9 B (1-2 MeV), and perhaps make an unambiguous determination of the location and width of the long-sought l / 2 + state in 9 B . It would also be desirable to perform a full i?-matrix analysis to extract the level properties of these states such as resonance energies and reduced widths that can be compared with model calculations, for example, the shell model or the cluster model. A p p e n d i x A /? d e c a y In this Appendix, we give a review of the theory of nuclear /3 decay. We begin by outlining the most important features of nuclear /3-decay in Section A . l . Since the purpose of this work is to use /3-decay as a tool to probe the structure of the daughter nucleus following /3-decay, more emphasis is given on the discussion of the nuclear structure information that can be obtained by studying nuclear /3-decay. This is discussed in more detail in Sections A.2-A.3. A . l T h e o r y o f (5 decay In 1934 Fermi formulated a successful theory of /3 decay [60] that follows time dependent perturbation theory. Fermi's hypothesis is that the electron and the neutrino in nuclear /3-decay are created as a result of the transformation of a neutron state into a proton state, or vice versa, inside the nucleus, the same way as a photon is created with the change of state of a charged radiating system. The /? transition rate between initial state (i) and final state (/) can then be calculated using the same "golden rule" of time dependent perturbation theory: where is the number of final states in the energy interval dE, and \Hfi\ is the matrix element for the interaction Hamiltonian between the initial and the final states. A = dE (A.l) 127 Appendix A. (3 decay 128 T h e (3 i n t e r a c t i o n H a m i l t o n i a n The simplest form of nuclear /3-decay is the decay of the free neutron, n —> p + e~ + Ve . (A.2) Because all the particles involved in the /3-decay are spin | fermions, their wave functions are four component spinors in Dirac's relativistic theory. Based on this fact, Fermi proposed that the interaction responsible for /3-decay is proportional to a four-vector current. The interaction Hamiltonian can then be written as: H = g@p01>n)@eOrp,,) + h.c., (A.3) h.c. is the hermitian conjugate of the expression preceding it and accounts for the creation of the positron, and ijjpe = V>pe74> 9 *s a measure of the strength of the interaction, ipn and ipu are the wave functions of the annihilated particles1, vb* and ijj* are the hermitian conjugates of the wavefunctions of the created particles; O is the operator that causes the annihilation of the two particles to create the new particles, and it can be any operator that is constructed from Dirac matrices. Using Dirac matrices to construct O, it is possible to have sixteen different kinds of interactions. Of these, only those that satisfy the requirement of Lorentz invariance are accepted. This limits the possible forms of the interaction to five types, namely scalar (S), vector (V), tensor (T), axial vector (A) and pseudoscalar (P). Furthermore, the scalar, tensor and pseudoscalar interactions predict wrong angular correlations between the direction of the electron and the neutrino, and should be discarded. Thus only the vector and the axial vector interactions are possible, and Equation A.3 can be generalized as follows, H = g Y, CSpOM@eOi4>v) + h.c, (AA) i=V,A 1The j3 decay process n —> p + e~ + Ve can equivalently be written as n + ve —> p + e~ Appendix A. (5 decay 129 where Cy and CA are appropriate constants, and Oy and OA are operators mediating the vector and the axial vector interaction respectively, Oy = 7 M and OA = 7 ^ 7 5 - Note that 7 M has even parity whereas 7 ^ 7 5 has odd parity. Before Yang and Lee suggested the possibility of parity non-conservation in weak interactions [61], the even and odd interactions had been used as equivalent alternate ways in which one could formulate a theory of /3-decay. The possibility of their co-existence in any decay was never considered because it was believed that parity was conserved in all physical interactions. Immediately following Yang and Lee's suggestion that parity may be violated in weak interactions, C S . Wu et al. performed the classic experiment in which they measured the spatial distribution of (3 particles emitted from a polarized 6 0 C o source [62]. Their measurement proved that parity is not conserved in weak interactions, and hence both odd and even interactions can co-exist in a decay. Lee and Yang obtained a general form of parity violating Hamiltonian given by: H = 9 E ®POM®eOi(Ci + CMu) + h.c. (A.5) i=V,A Because time reversal invariance requires that CA, C'a, Cy and C'v be real, and be-cause the neutrinos are produced in a definite helicity, CA — C'A and Cy = C'v. The Hamiltonian in Equation A.5 can then be written as: H = g E CSvOM\<PeOl{l + 7 5 ) ^ ] + h.c. (A.6) i=V,A In order to calculate the matrix element for the two types of interactions, we make a sim-plification by using a non-relativistic approach for the nuclear part of the matrix element in Equation A.6. This approximation is well justified since the recoil energy imparted to the neutron is much smaller than its rest mass. The matrix element representing the Appendix A. (3 decay 130 vector interaction is then given by:-J % ^ n d \ = \MF\ (A.l) The (3 transitions mediated by the vector operator are called Fermi transitions since they were originally proposed by Fermi, and the matrix element for the transition is denoted by \MF\. The matrix element corresponding to the axial-vector interaction is The operator a is an axial vector given by the Pauli spin matrices. Transitions mediated by the axial vector operator are called Gamow-Teller transitions, after Gamow and Teller who first proposed this mode of transition, and the matrix element for this transition is denoted by | M G T | . More discussion on the properties of these transitions is given in Section A.2. In the following discussion, we use non-relativistic approach in order to calculate the /3-spectrum and decay rate in nuclear j3 decay. T h e (3 s p e c t r u m In order to calculate the (3 energy spectrum using Equation A . l , we first find an expression for the number of final states per unit energy. Ignoring the small recoil energy of the nucleus, the number of final states, dNf, is the product of the number of final states accessible to the electron and the neutrino, i.e., Where dNe and dNv are the number of final states available to the electron and the neutrino respectively. Although recent experiments indicate that electron neutrinos have mass, the value is extremely small (~ 10 _ 5 eV/c 2 ) , so we may set the mass of the neutrino (A.8) dNf = dNedNv . (A.9) Appendix A. (3 decay 131 to be zero. Then the following expression is obtained for the number of final states in the interval E0 and E0 + dE0: dN V2 lit = 4 ^ P < E ' { E ° ~ E ' ? d E ' • ( A 1 0 ) Where ED is the total disintegration energy, and Ee and pe are the (3 energy and momen-tum respectively. To calculate the j3 spectrum using Equation A . l , we make the following assumptions: 1. The initial wave function, tpi = describes the nucleus before decay. The final state wave function is given by tpf = w/"0eVv which is the product of the nuclear, the electron and the neutrino wave functions in the final state. Hence, the transition matrix element is given by, \Hfi\ = Ju}^:Huid3r . ( A . l l ) 2. The electron and neutrino wave functions are plane waves, and their wave functions normalized to the nuclear volume are given by: V>e = ( ^ ) 1 / 2 e x p ( ^ ) (A.12) ^ = ( ^ ) 1 / 2 e x p ( Y ) where p and q are the momenta of the electron and the neutrino respectively. 3. Since the decay energy available in nuclear (3 decay is typically on the order of a few MeV, the wave lengths of the emitted electron and neutrino are very long when compared to the size of the nucleus, hence, to a good approximation, their wave function can be assumed to be constant over the nuclear volume. This approximation results in restricting the orbital angular momenta of the electron and the neutrino to be 0. This approximation is valid for a majority of the (3 transitions observed in experiments, and it is called the allowed approximation. There is however a small probability that the transition can involve higher orbital angular momenta for the electron or the neutrino. Appendix A. (3 decay 132 Such transitions are called "forbidden transitions" since the transition matrix element is several orders of magnitude lower than in the case of allowed transitions, thus not truly forbidden. The electron and neutrino wave functions in Equation A. 13 can be expanded as follows: exp(ip.r/h) = 1 + ^ + • • • (A.13) ZQ.T exp(iq.f/h) = 1 + -I In the allowed approximation, only the first terms in the summations are retained. Hence, ipe = ybu = (^) 1 / / 2 . Making use of Equations A . l , A.10, and the above assumptions, the following expression is obtained for the energy distribution of the electrons, 1E = ¥ ^ ^ M " ^ - E ^ A - ( A 1 4 ) Expression for | M / j | for neutron decay is given in Equations A.7 and A.8. The transition matrix element for the (3 decay of a complex nucleus is discussed in Section A.2. Due to the Coulomb field of the nucleus, the wave function of the /3-particle is distorted from the plane wave given in Equation A.13. Because of this distortion, the electron density relative to unit density at infinity is not unity and depends on the charge and the radius of the nucleus as well as the electron energy. This dependence is given by the function, p(Z,R,E), and can be evaluated analytically, P(Z,R,E)= ^ z (A.15) p(Z, R, E) involves a complicated expression, and this expression is given in Ref. [63]. For a non-relativistic electron in the field of a point like nucleus, it is given by the following: 27T?7 p(Z,R = 0,E) = where 1 - e2nr> (A.16) Appendix A. B decay 133 However, for large nuclei, this approximation is not adequate and p(R, Z, E) must be calculated by solving the relativistic Dirac equation with the Coulomb potential for an extended nucleus. When the distortion due to the Coulomb field is included in Equa-tion A . 14, the B energy spectrum is given by, % = ]^^92p(Z,R,E)\Mfl\2(E0 - Ee)2PeEe . (A.18) We define a new dimensionless coupling constant G, G = ^ ( f ) 3 . (A.19) Furthermore, using the unit of energy of the electron normalized to the electron mass we obtain the following expression for the energy distribution of the electrons: ^^^S^'^^ 1^ 1^ 0-^ 2- 1)V2 " (A-20) From the above equation, it is evident that the shape of the (3 energy spectrum for allowed decay depends only on the statistical factor and p(Z,R,e), i.e, ^<xp{Z,R,e){eo-t)2e{<?-l)1/2- (A.21) The decay rate The total probability of the decay A is obtained by integrating Equation A.20, and is given by: mr2 (l2 A = ^ 7 ^ / ( ^ , e , e 0 ) | M / 2 | 2 (A.22) where f(Z, e, e0) = £° dep(Z, R, e)(e0 - e)2e(e2 - l)1'2 . (A.23) The above integral, f(Z,e,e0), is known as the integrated Fermi function and is evalu-ated numerically. In Ref. [59], Wilkinson and Macefield have tabulated the parameters necessary to evaluate the Fermi function for a range of nuclei, both for j3+ and (3~ decay. Appendix A. (3 decay 134 To have a qualitative understanding of the dependence of / on the decay energy, we ignore the Coulomb field and evaluate Equation A.23. It can easily be shown that the integral for zero Coulomb field of the nucleus is roughly proportional to the fifth power of e0. Hence, the decay rate is a strong function of the energy available in the decay. The dependence of the rate on the decay energy can be removed by considering the ratio / / A , a quantity that is known as the ft value. The expression for ft can then be obtained remembering that: In 2 , k S ti/2 = — , (A.24) and thus from Equation A.22 where, h 2TT3 In 2 _ C/g2 f t l / 2 ~ ^G^WfJ2 ~ Wtf ( 5) „ 27T 3(ln2)ft 7 / A x C = V , / . A.26 m 5 c 4 v ; Since / is a known function and t i / 2 is measured experimentally, it becomes possible to determine | M j , | 2 for each nuclear transition by measuring the partial half-life and the decay energy for each nuclear transition. A.2 Fermi and Gamow-Teller transitions The matrix element for a Fermi transition, \ M F \ , is given in Equation A.7. This inter-action just changes the neutron into a proton without affecting the spin or the space component of the wave function of the neutron. The selection rules for allowed Fermi transition are, therefore, AJ — 0, and no change in parity. The operator mediating this interaction is therefore similar to the isospin raising (lowering) operator, r±. The matrix element for Gamow-Teller transitions, \MGT\ is given in Equation A.8. The spin operator a in \MQT\ can cause a change As = 0 or 1 in the spin part of the Appendix A. (3 decay 135 nucleoli wave function but no change in the orbital angular momentum. Therefore, the allowed Gamow-Teller transitions are A J = 0 or 1 but no change in parity. However, transitions with J = 0 —> 0 are not allowed. The operator mediating the Gamow-Teller transition is o~r±. In general (except for the case of 0 + —> 0 + transitions) , both Fermi (r) and Gamow-Teller (ar) parts of the weak interaction can contribute to the transition matrix element, a prime example is free neutron decay | + —> | + . The total matrix element for nuclear (3-decay is calculated by summing over all the transition matrix elements for each individual nucleon of the decaying nucleus, \Mf,? = E I < f\ £r(A:)|i > |2 + - ^ f - E I < /I 5>(A:)a| i > | 2 (A.27) where Ji is the spin of the parent nucleus, and the sum over k is the sum over the different nucleons. The Pauli spin matrix has three components and a sum over all the three states has to be performed for the case of the Gamow-Teller transition, i.e, | < f\^r(k)a\z > f = E I < f\ T,T(kWk)\i > I2 • (A-28) fc j k Equation A.27 can be re-written as follows: \Mfi\2 = C2F\MF\2 + CGT\MGT\2 (A.29) where: and \MF\2 = -r1— E I < / IE T WI Z > I2 (A.30) \M, |2 1 GT = T E I < / I E ^ M * 0 I * > | 2 - (A-31) 2 Ji + ± f i k Substituting the value of \Mfi\2 obtained in Equation A.29 in Equation A.25, A / 2 = Cl\MF\?i9C2GT\MGT\2 ( A ' 3 2 ) Appendix A. f3 decay 136 We define the vector (gy) and the axial-vector (gA) coupling constants by: gCF = gv (A.33) gCGT = gA (A.34) The ft value can therefore be expressed as follows f h / 2 = 92y\MF\2 + g\\McT? = \ M f \ 2 1%\MGT\2 ^ ' ^ 9V Defining a new constant K — C/gv = 6177 ± 4 s. [68, 3], we arrive at the following expression for the ft value: /* i/2 ^ (A.36) | M F | 2 + ^ | M G r | 2 We define the strength of Fermi and Gamow-Teller transitions by BF and BQT respec-tively as follows: BF = \MF\2 (A.37) BGT = %MGT\2 (A.38) 9v Hence, fti/2 = p * (A.39) t5F + tsGT For transitions between different isospin states, the contribution from Fermi transition is negligible. Therefore, we set BF — 0, and the following expression is obtained for the value of BGT BGT = (A.40) 7*1/2 The above expression is valid for transitions to well-defined states in the daughter nucleus. For decay that proceeds through broad resonances having a distribution W(E) Appendix A. (3 decay 137 in the daughter nucleus, the /-factor varies across the level, hence the transition strength has to be calculated for each energy bin and integrated over the distribution of the state: Jo0" )W(E)dE < BGT >= K o (A-41) t1/2tf0 W(E)dE A.3 /3-decay and nuclear structure A.3.1 Fermi transition For the Fermi transition, the effect of the operator (r) in the transition matrix element Mp on the nucleon is to change a proton into a neutron and vice-versa without otherwise affecting the wave function of the nucleus. The effect of this operator on the nuclear wave function is therefore to convert each term in it into a series of terms in each of which, one of the neutrons is changed into a proton (or vice versa) in the same orbit. This is analogous to the isospin raising or lowering operator that connects states in isobars with the same spin and isospin, commonly referred to as isobaric analogue states. The operator increases (or decreases) the third component of the isospin (T 3) by one unit. The isospin selection rule for Fermi transition in (3 decay is therefore: A T = AJ = 0, (A.42) A T 3 = ± 1 . (A.43) In the absence of the electromagnetic interaction and assuming that the nuclear force is charge independent, both the parent and the daughter states for Fermi transitions are described by the same wave function. For this reason, the Fermi matrix element is independent of the details of the nuclear wave functions of the parent and the daughter nuclei. This results in a model independent expression for the matrix element for Fermi Appendix A. (3 decay 138 transition between isobaric analogue states2 given by: | M F | 2 = [T(T+ 1) - (T3)i(T3)f] . (A.44) There are two classes of Fermi transitions occurring between isobaric analogue states that are interesting. The first class is the transitions that occur between ground states of members of the isospin doublets, i.e transitions between mirror nuclei. The ground states of these mirror pairs differ only by the interchange of neutrons and protons. Using Equation A.44, it can be shown that the Fermi matrix element for such a transition is unity. The other class of Fermi transition that has been a subject of many studies is the transitions of type (J* = 0 + , T = 1) -» (Jn = 0+,T = 1). As discussed in Section A.2, the Gamow-Teller matrix element for such transition vanishes, and for this reason they are called pure Fermi transitions. Because these transitions are of pure Fermi type, the value oi CF for these transitions can be experimentally determined with a high accuracy. Very high precision measurements have been performed in measuring the ft values of such transitions since they provide critical information on the fundamental properties of the weak interaction [64, 65]. By making an accurate measurement of the transition rate to the IAS, and comparing with the theoretically expected value for a pure isospin state, the isospin purity of the IAS can be measured. In a majority of (3 transitions, the isobaric analogue states in the daughter state are not energetically accessible with the exception of (3+ decays of highly proton rich nuclei such as 1 7 Ne. A.3.2 Gamow-Teller transition In contrast with the Fermi matrix element, the Gamow-Teller matrix element is strongly dependent on details of the nuclear wave functions of the parent and the daughter nucleus. 2 Due to the Coulomb interaction, and because of charge dependence in the nuclear force, a weak mixing of this state occurs. Hence, the transition strength of the Fermi matrix element is concentrated on a single state (the IAS) with only weak contributions to other nearby states. Appendix A. (3 decay 139 Because nuclear forces are strongly spin-dependent, the Gamow-Teller operator (err) does not connect nuclear states with similar wave functions. As a result, the strength of the Gamow-Teller transition is spread across states in the daughter nucleus with different spin and isospin from that of the parent nucleus. A measurement of the Gamow-Teller strength can therefore serve as a probe of the structure of the nuclei involved in the transition. By comparing the experimentally measured strengths with theoretical calculations such as shell model calculations, the validity of a model can be tested. The coupling strength for the Gamow-Teller transition (CQT) for the free nucleon is determined from the ft value for the neutron decay. This is possible since the the matrix elements for both interaction are known, and because Cp is known from transitions involving pure Fermi transitions. The value of the ratio of these strengths for the free neutron decay is determined by Wilkinson [66]: ^ = 1.2606 ±0 .009 . (A.45) For complex nuclei, however, this value is found to be significantly smaller than the free nucleon value of 1.2606. In his pioneering work, Wilkinson [67] compared the experimen-tal data for A = 6 — 21 nuclei with shell model calculations available at the time and found that ^ F 1 = 1.13. Brown and Wildenthal [68] did a systematic study by comparing the experimentally measured G T transition rates in the A — 17 — 39 nuclei with shell model calculations, and found that best agreement between the calculated values and the experimental data is obtained for = 0.95. This discrepancy with the free nucleon value is referred to as the quenching of the coupling constant for the G T transition, and a discussion of the possible sources for this quenching is given in Section A.3.6. Appendix A. /3 decay 140 A.3.3 Giant resonances in nuclear /3-decay A giant resonance arises from the collective motion of many nucleons in the nucleus, and the best known giant resonance is the electromagnetic dipole resonance that is described by the oscillation of protons with respect to neutrons. The large (p,n) cross section observed for the excitation of the isobaric analogue of the target ground state in charge exchange reactions had been recognized as the signature of a giant resonance, in this case arising from the isospin-lowering operator. Since the Fermi operator is the same as the isospin operator, and because isospin in nuclei is a nearly conserved quantum number, the total Fermi giant resonance is an eigenstate of the nuclear Hamiltonian, so that the full transition strength is concentrated in the isobaric analog state. On the other hand, because the Gamow-Teller operator does not commute with either T 2 or H , the strength of the Gamow-Teller Giant resonance is spread across states with different spin and isospin. The Gamow-Teller strength is concentrated in an excitation region in the daughter nucleus in the vicinity of the isobaric analogue peak. This region is known as the Gamow-Teller Giant Resonance (GTGR), and is energetically inaccessible in P decay with N > Z. The only case in which it may be excited is in P+ decay of nuclei with N < Z. The G T G R may, however be excited by the (p, n) reaction (Section A.3.4). The source of the G T G R was explained in a series of papers by Fujita and Ikeda and Fujii [69, 70, 71, 72] as the coherent excitation of the nucleons resulting from the imbalance between the proton and the neutron number, i.e. these are proton-hole neutron (p _ 1-n) or neutron-hole proton (n _ 1-p) excitations. These excitations correspond to the coherent neutron-hole neutron and proton-hole proton excitation in the known E M giant resonance. Appendix A. (3 decay 141 A.3.4 The (p, n) reaction and /3-decay strengths The central isovector component of the effective nucleon-nucleon interaction that medi-ates low momentum transfer charge exchange reaction is given by V = VaT(ai.ffp)(Ti.Tp) + VTTi.Tp . (A.46) VaT describes the strength of the spin flip (S=l) transition and VT describes the strength of the non-spin flip (S=0) transition. The two terms in the above equation are analo-gous to the operators for Gamow-Teller and Fermi transitions described in Section A.2. Therefore, a linear relationship is expected between allowed /3-decay transition strengths and forward angle (p, n) scattering cross sections3 [73]. Such a relationship has been experimentally observed for a range of nuclei studied by Goodman et al. [74] who found that at forward angles, the G T transitions dominate for proton bombarding energies be-tween 100 MeV and 600 MeV. The review of Osterfeld [75], discusses the general topic of spin-isospin excitations in nuclei, and a more recent review by Alford and Spicer [76] focuses on (p, n) reactions at intermediate energies. For /3-decay measurements, the energy window for which the strength of the transition can be measured is limited by the Q-value of the decay, but the advantage of the (p. n) reaction is that, one can measure the forward angle scattering cross section for a wide range of energies, exploring the regions of the Fermi and the Gamow-Teller resonances. For the case of 9 B , for example, Fazely et al. [36] have measured cross sections for the 9Be(p, n)9B reaction. In Ref. [3], these (p,n) reaction cross sections are compared with the P-dec&y strengths from the decay of 9 C . 3Because (3 decay occurs with nearly no momentum transfer, it is necessary to measure the forward angle scattering cross section in order to make a direct comparison with the strength measured from /3 decay. Appendix A. j3 decay 142 A.3.5 The Ikeda sum rule for Gamow-Teller transitions As discussed in Section A.3.3, the matrix element for the Gamow-Teller transition de-pends on the details of the structure of both the parent and daughter nuclei, hence the calculation of the transition matrix element is model-dependent. On the other hand, the sum of the strength of the matrix elements for all the transitions, depends only on the properties of the parent and daughter nuclei, hence the total transition strength does not depend on the specific model used in the calculation. If 5_ and S+ are the summed G T strengths for /3~ and f3+ decay respectively, S - - S V = £ E I < f\°kT-\i > f - E E I < f'WkT+\i > | 2 . (A.47) k=l f k=l /' From the above equation, a simple expression for the sum is obtained in Ref. [77], 5_ - S+ = 3(A^ - Zi) , (A.48) where the subscript i refers to the parent nucleus. The calculated sum of the the G T strength using Equation A.48 has been found to be consistently larger than the measured value in (p, n) reactions, where only about 60% of the calculated strength is measured experimentally. A.3.6 Quenching of the G T strength As discussed in Section A.3.2, the measured G T strength in nuclei is smaller than calcu-lated from models using the free nucleon value of = 1.2606. This reduction in the G T strength has been observed in (p, n) reactions [78] as well as in j3 decay measurements. The possible sources of the the missing G T strength are discussed in [79, 80]. 1. A mechanism. i.,e the nucleon-hole excitation associated with the G T G R could mix with the A-hole excitation, shifting the G T strength to the region of the A resonance at about 300 MeV. Appendix A. (3 decay 143 2. Nucleonic mechanism. This is a nuclear configuration mixing whereby high lying two-particle-two-hole (2p2h) states mix with low-lying 1-particle-l-hole ( lplh) G T states and shift the G T strength into the energy region far beyond the G T G R . It is generally agreed that most of the quenching (probably 2/3 of it) comes from higher-order configuration mixing in the nuclear wavefunction. A.3.7 Mirror asymmetry If the Coulomb interactions are ignored and the charge dependent nuclear force is assumed to be negligible, nuclei with a mirror combination of neutrons and protons should have the same properties. For example, allowed (3 decays from mirror nuclei should have the same ft values. In nature this mirror symmetry is slightly broken due to the Coulomb interaction and due to the charge dependent nuclear force. Experimentally, an asymmetry 8 [81] has been observed indicating either a charge symmetry breaking in the nuclear force or a new weak-interaction effect such as a second class current. The asymmetry 8 is defined by: 8 = ^ ± - l . (A.49) Jt-The largest contribution to this asymmetry is in the binding energy difference between the proton that makes the (3 transition in the (3+ decay and and its mirror neutron in the (3~ decay. This contribution has been estimated by Wilkinson [82], and a comparison of 8 with the experimental measurement gives a residual asymmetry that is not accounted for by this effect, especially for odd nuclei. This asymmetry can be more reliably calculated for light nuclei. A p p e n d i x B i t ' - m a t r i x t h e o r y In this section, a brief discussion of the i?-matrix theory is given. The i?-matrix theory is a formalism that is used in describing low energy nuclear reactions. The first step in describing a reaction in .R-matrix theory is to divide the configuration space of all the nucleons into two regions: the interior region and the exterior region. In the interior region it is assumed that all the nucleons interact with each other by the short range nuclear force. For reactions that proceed via a resonance, this region includes the com-pound nucleus. The exterior region is defined by the region outside the nuclear volume where the strong nuclear force between the individual nucleons is negligible. This re-gion is further subdivided into all the pairs of nuclei whose nucleons sum to the total number of nucleons in the interior region, and these pair of nuclei are assumed to be far enough apart that only the Coulomb interaction between them is important. The minimum distance between the two reaction pair nuclei beyond which the strong nuclear force becomes negligible is called the channel radius, ac, and in practice this is defined by the sum of the radii of the two nuclei. The interior and the exterior regions are separated by the nuclear surface. The distinction between the interior and the exterior region is significant because in the exterior region, one can solve the radial wave equations for each channel. In the interior region, we want to solve the Schrodinger equation with ap-propriate boundary condition at the surface. These boundary conditions are determined by matching the wave function and the derivative of the internal region to those of the external region at the surface. 144 Appendix B. R-matrix theory 145 A comprehensive treatment of the i?-matrix theory is found in Ref. [84, 85]. We shall closely follow the formulations in the latter in the discussion in Appendix B . l where we discuss the simplest case of scattering of a spinless particle from a one dimensional square well potential, and in section B.2 where we extend this discussion to the general case of reactions involving more than one channel. In Section B.3, we discuss how this theory is used in describing /3-delayed particle spectra. A discussion of the description of the /3-delayed particle spectra of 9 C using .R-matrix theory is given in Appendix C. B. l Scattering of a nucleon by a central potential We first consider the simple case of elastic scattering of a spinless particle from a central potential V. The radial part of the wave function in the interior region, <f> satisfies the Schrodinger equation: Considering the standing wave character of the wave function cj) at certain energies, we make an explicit construction of a complete set of standing waves, X\ in terms of which we expand the wave function <fi. • **> + %{E-V)Xk = 0. (B.2) In addition to Equation B.2, we impose the following boundary'condition at the surface of the interior region: r-^r =bX* lr=« (B.3) b is a real number chosen so that a set of standing waves, X\ is obtained from Equa-tion B.2. Because X\ form a complete set of eigenfunctions in the interior region, we can Appendix B. R-matrix theory 146 expand (fi in terms of Xx: = £ A A X A , (B.4) A where the A\ are the coefficients defined by: A\ = fXX(fidr (B.5) Jo In order to solve for A\ in the above Equation B.5, we first multiply Equations B . l and B.2 by Xx and (fi respectively. We then integrate the difference between the resulting equations in the region r < a to obtain: h2 \ (,dx A Xx^) =(E- Ex) f cfiXxdr . (B.6) / r=a ® \2mJ V dr Substituting the integral in Equation B.6 by the expression for Ax given in Equation B.5, Ax = (Ex - E)-1 (^-) Xx(a)[<fi\a) - bcfi(a)} (B.7) where (fi' = r^. By substituting the value of Ax given above in Equation B.4, an expression for (fi in the interior region region is obtained in terms of its value and its derivative at the surface; (fi(r) = G(r,a){(fi'(a)-b(fi(a)}. (B.S) The Green's function G(r,a) gives (fi at any point r in terms of its derivatives and its value at r = a, and is given by The Green's function G(a, a) that relates the value of (fi on the surface to its derivative on the surface is called the i?-function, R = G{a,a). (B.10) Appendix B. R-matrix theory 147 The value of cf> on the surface, 0(a) is thus given by: <f>(a) = R[<l>'(a) + b(l>{a)]. (B . l l ) Hence, (B.12) </>'(a) _ 1 0(a) _ R ' If we define the reduced width, 7^ belonging to the state A by: * = (Is) <B-13> then, ( B 1 4 ) In the exterior region, 0 is a linear combination of the incoming (/) and outgoing (O) waves. The incoming wave is chosen to have a unit flux I = (4ixv)~l/2e~lkr', then the outgoing wave is given by O = [Aixv)~xl2elkr, <j> = I-UO (B.15) where U is the collision function. The cross section is given by a = £ | l - t f | 2 . (B.16) The collision function, U is calculated by matching the internal and external wave func-tions at the channel radius ac. This is done by equating the logarithmic derivatives for the wave functions at the channel radius, ac, = I-MR- I'R = 2 l k a l - b R + zkR O - bOR -O'R 1-bR-ikR' { ' 1 The cross section is then given in terms of the reduced widths and the level energies by using Equations B.16 and B.17. If we assume that only a single level with level energy, Appendix B. R-matrix theory 148 E0, and reduced width, 7 2 , contributes, then R is given by: R = 7o2 (E0 - E) " The one-level collision function is then given by: (B.18) U = e -lika 1 + (B.19) (E0-E)-(b + ik)1l Substituting the one-level collision function in Equation B.16, the cross section in the one-level approximation is given by: 7T fc2 2sin/ca.e + i f c a (E0-E)-(b + ikYfl (B.20) B.2 Reactions involving more than one channel If more than one channel is available in the reaction, all the functions derived in Sec-tion B . l become matrices with the rows containing the incoming channel information and the columns the outgoing channel information. The R- function then becomes the R- matrix with the elements given by: Red = l\el\e' Ex-E (B.21) A channel is defined by a set of quantum numbers (a, j, m, I, s) where a labels the pairs of particles and their relative energy, j is the total angular momentum, I is the orbital angular momentum between the reaction pairs, and 5 is the channel spin composed of the intrinsic spins S i and S 2 of the interacting particles. In analogy to the single channel formalism, The R matrix is related to the collision matrix U by matching the value and the derivative of the internal and the external wave functions at r = ac. The total wave function in the exterior region is given by: (B.22) Appendix B. R-matrix theory 149 Where <f>c is the wave function of the relative motion between the pair and ipc is the wave function that describes the internal states of the channel pairs, and can be expressed as follows: A = —K {lsmims\J'mj)ilYirniXsms • (B.23) '"c mi,ms Where s is the sum of the spins of the channel pair, the label a refers to a certain channel, I is the orbital angular momentum of the pair, rc is the channel radius, iYimi are the spherical harmonics associated with an angular momentum I, and Xsms is the wave function of the spins of the pair. For open channels, d>c is expressed as a superposition of incoming (Ic) and outgoing (Oc) waves: 0C = v:l'2{AcIc - BcOc) . (B.24) Ac and Bc are the amplitude coefficients. (Ic) and (Oc) are expressed with the regular Coulomb function Fc and the irregular Coulomb function Gc. For open channels, they are given by: Ic = {Gc ~ iFc)exp{iLuc) , (B.25) and Oc = (Gc + iFc)exp(-iuc) (B.26) where uic is the Coulomb phase shift given in terms of the orbital angular quantum number / and the Coulomb parameter 77, uic = ^2 arctan(77 c/n) . (B.27) n=l Following similar steps used to match the wave function and its derivative in the one dimensional case, an expression for the collision matrix is obtained, and it is given by the following: Appendix B. R-matrix theory 150 Ucd = (Pc)1/20;\1 - RL)-\l - RL*)Ic,(Pc>r1/2 • (B.28) A l l the matrices in Equation B.28 are diagonal matrices except the R-matrix. Then, Lc = 0'cOZ1 -bc = S c - B c + iPc (B.29) and pc = kcrc. The penetration factor, Pc and the shift factor, Sc for open channels are given by: and pc(FcF'c + GCG'C) . , Sc = F 2 + G 2 • (B.31, Referring to Equation B.28, the problem of calculating the cross section involves calcu-lating the inverse of (1-RL). For the case where the number of channels involved in the reaction exceed the number of levels, it is easier to express the collision matrix U in terms of the level matrix A , in calculating reaction cross sections. This relation is given by: U c c , = e l ^ + J V ) i^cc, + 2 i p i/2 g 7 A c 7 / , c A v P c V 2 j . (B.32) The matrix elements for the level matrix are given by its inverse as follows: = [Ex - E)SX^ -YslxdASc ~BC + iPc) (B.33) c and Qc = UJC - arctan(F c/G c) (B.34) B.2.1 Total reaction cross sections The total cross section for the reaction, A + a—>B—>D + d, (B.35) Appendix B. R-matrix theory 151 proceeding through the formation of states in the compound nucleus B of spin and parity J11 is derived in Ref [84, 86]. In terms of the level matrix, A , the cross section is given by, 47T& 3Pd (B.36) where gj is the spin statistical factor, Pd and Pa are the penetrabilities of the incoming and outgoing channels. We next consider a reaction in which the compound nucleus B in Equation B.35 is formed together with a stable particle b resulting in three particles in the final state, and we are interested in calculating the cross section for the formation of particle d (oa-,d)-A + a B + b D + d + b (B.37) The expression for oa^d is derived by Barker [86, 87]. In order to derive an expression for oa^d, Barker modified Equation B.36 by replacing 4 7 r f{ P a 7 2 by G\ab, where G\ab is a feeding factor that is a slowly varying function of E. |2 On oc G Xab "yxdAx^ X/j, Pr d • (B.38) Note that the most important distinction between the expressions given in Equations B.36 and B.38 is that the cross section given in Equation B.38 does not depend directly on the penetrability of the incoming channel. However it depends on the penetrability of particle d in the compound nucleus B. Because the energy of d is directly related to the excitation energy of B , this cross section also describes the spectral shape of the compound nucleus B. In the analysis of particle spectra from (3 decay to particle unbound states,we have similar problem of measuring the excitation function of the compound nucleus. In the case of j3 decay, however, the probability of formation of the compound nucleus is determined Appendix B. R-matrix theory 152 by the the Fermi phase space factor, and the strength of the transition matrix element. For example in the (3 decay of 9 C to states in 9 B that subsequently decay to 8 Be(0 + )+p, the probability of formation of the compound nucleus 9 B * is dependent only on the (3 decay matrix element for that particular transition and on the Fermi phase-space factor. 9C —>9 B* + (/?+ + u) —> s Be+p. (B.39) In Ref. [88], Barker and Warburton modify Equation B.38 to derive an expression for the spectrum of an unstable nucleus formed by (3 decay by incorporating the phase space factor fp and the transition matrix element, g\x, in this equation. Following the above argument, the following equation is obtained for /3-delayed particle spectra [88]:-w. •{E) = C2fpPcY, Xfj. 2 (B.40) where x = F or G corresponding to Fermi and Gamow-Teller transitions. B.3 Application to f3 decay through broad states As discussed in Section A.2, one of the physics goals in nuclear (3 decay experiments is to extract the strength of the transition matrix elements Bp and BQT from ft values determined in experiment. For decays involving stable states in the daughter nucleus, these matrix elements can readily be calculated using Equation A.25. For (3 decays to states involving unresolved broad resonances such as 9 B , it is not possible to apply Equation A.25 directly in calculating Bp and BGT because of the interference between these states. Moreover, since the Fermi phase-space factor fp varies across the broad level it is not possible to calculate fp corresponding to a level. A correct description of the spectra is therefore necessary in order to disentangle the contribution of each state to the total spectrum and to subsequently extract the strength of the transition from the spectrum. Appendix B. R-matrix theory 153 In the following discussion, we shall discuss the .R-matrix formulation of Barker and Warburton [88, 89], used to describe spectra from /3-decay to states consisting of broad resonances. We define the /3 decay probability per unit excitation energy E by w(E), then the transition probability is given by: w = = -= f w{E)dE (B.41) ti/2 r J where w(E) is the total excitation energy spectrum summed over all the decay channels, w(E) = 1£wc(E) (B.42) c and wc(E) is given in Equation B.40. The probability per unit energy interval, w(E) leads to a spectrum in the daughter nucleus N(E), and the total number of decays, N, is obtained by integrating over the energy spectrum: N = J N{E)dE (B.43) where N(E) = (NT)W(E) = £ NC(E) . (B.44) c NC(E) is derived from Equation B.40 and is given by, Nc(E) = f0PcY^ X where 2 I (B.45) BXx = CiNrfl'gxr. • (B.46) Equation B.45 is the general description of the excitation spectrum of a particle-unstable nucleus produced by /3 decay in the .R-matrix theory. In the general case where the spectrum consists of several levels decaying through many channels, Equation B.45 can be quite complicated with too many unknown parameters to be determined from fits to Appendix B. R-matrix theory 154 measured spectra. It is therefore necessary to make some approximations, when possible, so as to simplify the problem. First, we assume that the contribution from the Fermi transition is negligible and set <?AF=0. The justification for this approximation is based on the discussion given in Section A.3.1. With this approximation, Equation B.45 is given by: NC(E) = fpPc 2 (B.47) ^Bxcl^cA X/i Further simplification is made by considering two special cases. The first case involves a spectrum consisting of many levels, all decaying via a single channel. For this case, only the diagonal elements in the level matrix exist, and it is possible to show that Equation B.47 reduces to the following expression, N(E) = fPP\ \^BXG1X/{EX-E)\2 In the second case, we consider a spectrum consisting of only one level, but with more than one channel available for its decay. In this case, Equation B.47 reduces to the familiar Breit-Wigner formula given by )^-/^ (^ +fr-^ +in- (B49) Here, AA and T\ are given by the usual expression for the one-level approximation. AA = E-(&-£cbL. (B.50) c and r A = £ 2 P c 7 L . (B.51) c B.3 .1 D e c a y t h r o u g h s e c o n d a r y b r o a d s ta tes In the discussion given in Section B.3, both fragments resulting from the decay of the compound nucleus are assumed to be stable. In this section we consider the case where Appendix B. R-matrix theory 155 one of the fragments is unstable to particle decay, which is true for the decay of 9 B . Since the intermediate state into which the first decay occurs is unbound, the intermediate state can generally be a broad resonance. Such is the case of the decay of 9 B where, all the intermediate states are broad except for the ground state of 8 Be. The simplest case of a single level having more than one channel for its decay, and decaying through broad intermediate states is discussed in Sections XII and XIII.2 of Ref. [84], as well as in Ref. [14]. Following the discussion in Ref. [84], the particle spectrum for a given energy in the intermediate state E' is given by: AT(F - f V B \ G P c ( E - E ' h l c ( E ' ) , R r<V> N{E> E ) ~ f ^ (Ex + Ax-Er + \rr ( B - 5 2 ) where the energy-dependent reduced width is given by, ] = ^ (Ey + AX, - + \ri(E') ( 5 3 ) Note that the primed quantities in the above two equations refer to the intermediate state (Figure B . l ) . The shift function and the total width are given by, AX(E) = - W dE'(Sc(E - E') - B^KE') . (B.54) c J TX(E) = 2 W dE'Pc(E - E'h2Xc(E') . (B.55) c J Combining Equations B.52 and B.53 gives; N { E > c (EX+AX- Ey+in (Ex/+Ax> - E>y+\n,(E>) ( R 5 6 ) Because the energy of the intermediate state, E', is not observed in an experiment, the above equation is integrated over all values of E' to obtain N(E): Appendix B. R-matrix theory 156 Figure B . l : A n illustration of the decay of a broad state in 9 B through the ground state of 5 L i . The primed quantities refer to the daughter nucleus. Inspection of Equation B.57 shows that the integral in this equation is just the average of the penetrability of the first particle, Pc weighted by the state shape of the intermediate nucleus, and we define the mean penetrability VC(E) by this integral, PC(E - E')Py(E')ri Therefore, Equation B.57 is given by: V c { E ) = J (Ex, + Ax,-E>y + lri(E>)dE • ( R 5 8 ) ^)-f {^ESt-^\n (B.59) Appendix C R-matrix parameterization of the spectra of 9 B from the decay of 9 C The allowed (5+ decays of 9C(J7T = 3/2") can populate 1/2", 3/2~ and 5/2" states in 9 B . Each of these Jn states can decay by proton emission or a emission, populating the ground and first excited states in 8 Be and 5 L i . However, from the experimental data, the evidence is that the decays occur only through 8 Be(0 + ) , 8 Be(2 + ) and 5 Li(3/2~) and in the analysis given in this section, only these decay modes are considered. In Table C l , we list the allowed values of the angular momenta between the decay pair for each J w state in 9 B decaying through the three possible decay modes. For some of the decay modes, more than one angular momentum is allowed. Table C l : Allowed angular momenta for Jn = 1/2 ,3/2 , 5/2 states in 9 B decaying to 8Be(0+), 5 Li(3 /2- ) or 8Be(2+). Decay mode r(9B) allowed / 8Be(0+)-f-p 8Be(0+)+p 8 Be(0 + )+p i -2 3-2 1 1 3 5 Li(3/2~)+a 5 Li(3/2")+a 5 Li(3 /2- )+a I -3-2 5-2 2 0,2 2,4 8Be(2+)+p 8Be(2+)+p 8 Be(2 + )+p 1-2 3-2 1,3 1,3 1,3,5 As discussed in Section B.2, a channel is defined by the set of quantum numbers (a, j, m, /, s) where a labels the pairs of particles and their relative energy, j. is the total 157 Appendix C. R-matrix parameterization of the spectra of9B from the decay of 9 C 158 angular momentum, / is the orbital angular momentum between the reaction pairs, and s is the channel spin composed of the intrinsic spins S i and s 2 of the interacting particles. Using all the possible angular momentum states listed in Column 3 of Table C l , we identify and list all the decay channels of the three Jn catagories of states of 9 B in Table C.2. Table C.2: A list of the possible decay channels for the states of 9 B fed by (3 decay of 9 C . Decay channel Channel label 8Be(0+)+p, I = 1 8 Be(0 + )+p, / = 3 1 2 5 Li(3 /2- )+a , I = 0 5 Li(3 /2- )+a , I = 2 5Li(3/2-)+o;, I = 4 3 4 5 8Be(2+)+p, I = 1 8Be(2+)+p, I = 3 8 Be(2 + )+p, / = 5 6 7 8 Note that for the decay to the 8 Be(2 + ) state, for each I value, the proton and the 8 Be spins can couple in two ways, resulting in two channel spins for each /. Since the measured spectrum of 9 B is not sensitive to the polarization of any of the particles involved in the decay process, these different channel spins have been ignored. In fitting the spectrum of 9 B , the contribution of each Jn state to the total spectrum is calculated from Equation B.40, and the contribution from these different states are summed to be compared with the experimental data. In order to investigate the available decay channels for the different states, we tabulate the three types of Jn states in 9 B and their corresponding decay channels in Table C.3. From this table, it is possible to see that there are 4,5, and 6 decay channels for the 1/2", 3 / 2 - , 5/2~ states respectively. Appendix C. R-matrix parameterization of the spectra of9B from the decay of9C 159 Table C.3: A list of the possible decay channels for the Jn state of 9 B . J^( 9 B) Decay channel Channel label to to to to 1 1 1 1 8Be(0+)+p, I = 1 5 Li(3/2")+a, I = 2 8Be(2+)+p, / = 1 8Be(2+)+p, / = 3 1 4 6 7 3/2" 8 Be(0 + )+p, 1 = 1 1 3/2- 5Li(3/2-)+o;, I = 0 3 3/2" 5 Li(3 /2- )+a , 1 = 2 4 3/2" 8Be(2+)+p, 1 = 1 6 3/2" 8Be(2+)+p, I = 3 7 5/2" 8Be(0+)+p, / = 3 5 Li(3 /2- )+a , I = 2 5 Li(3 /2- )+a , / = 4 2 5/2- J 4 5/2" 5 5/2- 8 Be(2 + )+p, 1 = 1 6 5/2" 8Be(2+)+p, I = 3 7 5/2" 8 Be(2 + )+p, / = 5 8 C l Fitting the spectra of 9 B from the P decay of 9 C The experimentally measured spectra consist of the triples sum spectrum from 5 L i ( 3 / 2 " ) + Q ; and the ratio cut spectrum from 8 Be(0 + )+p (Chapter 5). In fitting these two spectra we assume that the spectra consist of only the 1/2", 3/2" and 5/2" states in 9 B . In the two-level approximation, we assume that only two levels contribute to each category of J71" state fed by the p decay From Table C.3, it is possible to see that a total of 15 decay channel exist. In the two-level approximation we are considering, there are 30 reduced width amplitudes 7 ^ , 6 state energies and 6 feeding factors, making the number of free parameters in the fit 42. Attempting to fit the spectrum with such a large numer of free parameters is obviously not practical, and it is desirable to reduce the number of param-eters by making some approximations. We make the first approximation by ignoring the decay of the 1/2" and 5/2" levels to the 8 Be(2 + ) state. This approximation is based on Appendix C. R-matrix parameterization of the spectra of9B from the decay of9C 160 the experimental evidence that only the decay of the 9B(12.2) state to the 8 Be(2 + ) state is observed. Another simplification is made by considering only the decay chanels with the lowest angular momenta between the decay pair. With these approximations, we list the parameters needed to be determined from the fit in Table C.4: Table C.4: A list of the reduced width parameters in the spectrum of 9 B for r = l / 2 - , 3 / 2 - , 5 / 2 - states. J 7 r ( 9 B) Decay channel levels Red. width Number of parameters 1/2- 1,4 2 7A4, 7AI 4 3/2" 1,3,6 2 7AI, 7A3, 7A6 6 5/2- 2,4 2 7A2, 7A4 4 Referring to Equation B.47 the /3-delayed spectrum in the two-level approximation can be written as: wc(E) = f0Pc\B1G(^lcAn + 7 2 c A i 2 ) + B 2 G ( 7 i c ^ 2 i + 72^22) | 2 • ( C l ) The first step in the fitting involves construction of the inverse of the level matrix, A-1, given by: ( A - % = (Ex - E)SX^ - X > / A c 7 ^ C • (C2) c Where LC = SC-BC + iPc . (C.3) The shift functions Sc and the penetrabilities Pc can be evaluated for each decay channel. The reduced widths jXc, and the level energies Ex, on the other hand, are the parameters to be extracted from the fits. Assuming only two levels contribute, A"1 is given by the matrix in Equation C.4. Where the index c in each matrix element represents the summation over all the channels. Appendix C. R-matrix parameterization of the spectra of9B from the decay of 9C 161 A = ( \ (Ei - £ ) - £ L c 7 2 c £ £ c 7 i c 7 2 c c c £ Ld2clic (E2 - £ ) - £ L c 7 | c (C.4) Each element of the inverse of the level matrix given in the above equation is re-written in Equation C.5: An = ( £ ! - £ ) - £ (Sc -Bc + iPc)llc c A \ 2 =2\2(Sc-Bc + i P c ) 7 i c 7 2 C c A2i = £ (Sc - Bc + i P c ) 7 2 C 7 i c c A2i = (E2-E)-YJ (Sc -Bc + iPch22 (C.5) rl Appendix D Models describing the A=9, T = l / 2 system The structure of light nuclei can be described by the single-particle shell model as well as by the cluster model. Some aspects of the properties of light nuclei in particular can be better understood by the cluster model. A brief discussion of these models for light nuclei is presented in this Appendix. D . l The shell model The nuclei in the mass number range of 5 < A < 16 are called the "lp" shell nuclei since the lowest-lying configurations for these nuclei are described by (ls)4(lp)A~4. Shell-model calculations using this configuration have been successful in predicting the locations, widths, spins, parities and /3-decay matrix elements for the low lying states for these nuclei. In addition to the "lp" shell configuration, higher lying configurations should also be included in order to describe the general wave function, ip(E), ip(E) = a0ME) + £ aME) + E ^{E) + • • •, (D.l) ip0{E) represents the lowest configuration (ls)4(lp)A~4, and 4>i(E) represents all one-nucleon excitations such as (ls)4(lp)A~5(lf), and 4>2(E) represents all the two-nucleon excitations such as ( l s ) 4 ( lp) ' 4 ~ 6 ( ld) 2 . In general, the low-lying states are not expected to have higher-order configurations because of the large amount of energy required to transfer a single nucleon into a higher orbital. However if the nucleus is deformed, higher 162 Appendix D. Models describing the A=9, T—l/2 system 163 excitations do become important. Omitting this complexity for the moment, the low-lying excited states are then obtained by re-coupling the nucleons in the p-shell, resulting in different angular momenta. In this model the lowest lying states in the A=9 system are of configuration ( l s ) 4 ( lp) 5 . Calculations for the low-lying states in light nuclei including 9 B have been made by several authors [90]-[95]. However it is important to realize that all calculations have restricted themselves to normal parity states, i.e. negative in 9 B , thus l / 2 + , 5 / 2 + states are not studied. D.2 The cluster model The cluster model has been successful in describing some aspects of the properties of light nuclei. Perhaps the most well known examples of clustering in nuclei are the en-cluster structures that are found in the ground-state wave functions of nuclei such as 8 Be, 1 2 C and 2 0 Ne. In addition, cluster structures in light nuclei such as 6 L i and 7 L i have been observed experimentally. In this model, the nuclear wave function ip of a nucleus consisting of n nucleons may be represented as the sum of all possible two-cluster wave functions ip2(E), three-cluster wave functions ^(E), etc. Hence, at a separation distance between clusters greater than the range of the nuclear force, the total wave function is given by iP(E) = E 0 2 A - M - E ) + X > A < M £ ) + • • • (D.2) A A Where 6i\ are expansion coefficients representing amplitudes of the wave function of a particular cluster configuration. Early work in the cluster model [96] suggested that, at low excitation energies^ a few MeV), the two-body cluster configurations are the most dominant, and the nucleus can adequately be described by the sum of the two-cluster wave functions iP(E) « 5>2A-02A(£) = ^OxME) (D.3) A A Appendix D. Models describing the A=9, T=l/2 system 164 where 6\ is the probability of finding the state in configuration A, also called the spectro-scopic factor. For well defined resonant states in the parent nucleus, and in the Wigner limit of a uniform nucleus [97], 0\ can be approximated by Where fi\ is the reduced mass of the cluster pair, and 7 2 and rc are the reduced width amplitude and the channel radius respectively, discussed in Section B. The spectroscopic factor for a cluster pair can therefore be calculated using the cluster model and this value can be compared with experimentally measured widths using the relation in Equa-tion D.4. In Ref. [98] for example, the a-particle spectroscopic amplitudes for states in 1 2 C , 1 6 0 and 2 0 Ne were calculated using the SU(3) cluster for the a particles. A similar calculation was performed in Ref [3] for the 9 B states decaying to 5 L i + a. D.2.1 The a-N-a system as two-center molecular state In Ref. [99, 100], it is pointed out that the 9 Be ground state and the first excited state can be described as bound states of resonances of a three-body system composed of 2 a particles and a neutron. In this model, the presence of the neutron is responsible for the existence of bound state in 9 Be, by creating an effective potential between the two a particles, which when summed with the two-body a-a interaction, gives enough attraction to allow a bound state to exist. Such systems are often called Borromean after the similar effect observed with the gravitational force [101]. From calculations performed in Ref. [100], a qualitative agreement is found for the location of the low-lying negative parity states in 9 Be. These initial calculations were then followed by that of Descouvement [25] who took a microscopic three-cluster model, using the generator co-ordinate method. He included the 8Be(/ = 2) component (i.e the first excited state of 3h2 (DA) Appendix D. Models describing the A=9, T=l/2 system 165 8 Be at 3.04 MeV), but did not include the 5 L i + a structure so some states were badly described. However this remains one of the best overall calculations and it does include some positive parity states. Recent calculations have been performed by treating the bound states of 9 Be in a similar manner to the H 2 + molecule, thus the 9 Be nucleus has been described as a two center state [102, 103], where the binding between the two 4 He nuclei is provided by the valence neutron. These calculations reproduce the observed level sequence of the low-lying states of 9 Be. In Ref. [103], it was suggested that other isotopes of Be can exhibit a-xn-a structure. In a recent experiment at G A N I L by M . Freer et al. [104], rotational states in 1 2 B e were observed, that they identified with the cluster decay of a molecular structure in 1 2 B e of the form a-An-a. Appendix E Kinematics E . l Kinematics In this appendix, we discuss some of the important features of the kinematics of sequential decay that lead to three-body final states, using non-relativistic kinematics. One example of sequential decay is illustrated in Figure E . l . First decay: We consider a case where the first breakup occurs at rest, therefore the laboratory frame is also the center of mass frame for the first breakup. Applying conservation of energy and momentum: Ei + ER — Qi (E.l) and P i + P R = 0 (E.2) Substituting p± = \j2rriiEi in Equation E.2; E\m\ (E.3) m2 + m 3 Second decay: From conservation of energy: E2 + E3 = Q2 + ER (EA) substituting Equation E.3 in Equation E.4 to eliminate ER, E2 + E% = Q2 + EiTTli (E.5) m2 + m3 166 Appendix E. Kinematics 167 (a) (b) 3? a (1) 5 Li (R) P (2) a (3) Figure E . l : A n illustration of the kinematics of sequential decay for the breakup of 9 B - + 5 L i + a. From conservation of momentum in both decays; P i + P2 + P3 = 0 resulting in: Pi + v\ - vl = - 2 p i P 2 cos 0i2 substituting = ^JlmEi in Equation E.7 gives, miEi + m2E2 - m3E3 = - 2 y m1m2E1E2 cos 6i2 Using Equation E.5, we can eliminate E3 in Equation E.8 resulting in: 2 m 3 m i m 2 ( m 2 + m 3 ) ( m 2 + m 3 ) : -E\ E2 m2 m2 + m3y m2 —E1E2 cos 912 (E.6) (E.7) (E.8) (E.9) The above equation is a quadratic equation in \[E~2 and solving for \fE~2 gives two solutions: E2 = x ± Jx2 + y (E.IO) Appendix E. Kinematics 168 where: and ^mlm2El x = cos6>i2 (E- l l ) m2 + m3 m 3 mxm2 V = Q2 Ex- (E.12) m2 + m3 (m2 + m3)z From Equations E.IO- E.12, it is possible to see that E2 is a function of the three variables (Ei,Q2,6i2). Hence, for a given set of directions for particles 1 and 2 (#12), all the coincidence events between 1 and 2 are restricted to a kinematic curve in the two dimensional energy spectrum. This requirement is independent of the decay process involved. If Q 2 is well-defined, then E2 can have a maximum of two possible values for given (Eu912). A p p e n d i x F A n g u l a r c o r r e l a t i o n s F . l A n g u l a r c o r r e l a t i o n s In this Appendix, we list the values of the parameters a, b, c, l\,l2, Lx and L2, needed in the calculation of the angular distributions W(9) discussed in Section 4.6, for the decays of 9 B —* 5 L i + a and 9 B —> 8 Be + p. We also make sample calculations for each decay mode. The angular correlation for pure radiations was given in Section 4.6, and is re-written here in Equation F . l . WClcM = (47r ) 2 ( - i r + c - 2 6 (26+l) k=2i (F.l) The coefficients Cko(C,C) are given by: o r _ i _ i Ck0(C, C) = —^—(-1)1(10:10\k0) (F.2) for a particles, and Ck0(C,jr) = 2J^(-l)L-1/2+k(L\, L - \\k0) (F.3) for protons. The selection rule for the sum over k requires that k be even and the maximum of k is given by kmax = min(2Lu 2£l,2L2, 2£2, 2b) . (F.4) 169 Appendix F. Angular correlations 170 Table F . l : Angular correlation parameters for the 9F3 —• 5 Li(3/2 ) + a breakup. a b c h Li £2 k 1/2- 1/2 3/2 0 2 1 2 3/2 min(4,4,3,2,3) 3/2" 3/2 3/2 0 0 1 0 3/2 min(0,0,3,2,3) 3/2" 3/2 3/2 0 2 1 2 3/2 min(4,4,3,2,3) 5/2- 5/2 3/2 0 2 1 2 3/2 min(4,4,3,2,3) 5/2" 5/2 3/2 0 4 1 4 3/2 min(8,8,3,2,3) 1/2+ 1/2 3/2 0 1 1 1 3/2 min(2,3,3,2,3) Note that, because J 7 r =0 + for the 8 Be ground state, i.e. 6 = 0, all angular correlations through this state will be isotropic due to the /c-selection rule. Decays through the 5 Li(3/2~) and the 8 Be(2 + ) states can show an angular distribution of the pattern: W(9) = a0 + a2cos29 + --- (F.5) These two decays are discussed below. F . l . l Decay of 9 B —> 5 L i ( 3 / 2 " ) + a In Table F . l , we list all the parameters necessary in the calculation of W{6) for this decay mode for states in 9 B of J 7" = 1/2", 3/2", 5/2" and l / 2 + . For this breakup mode, 6 = 3/2, c = 0. Substituting these values in Equation F . l , WClcM = ( 4 7 r ) 2 ( - i r 3 ( 4 ) £ ( - l ) - ^ - i 2 C f c 0 ( / : i / : i ) C f e * 0 ( £ 2 £ 2 ) k=2i 3 3 3 3 W(--L1Ll] ka)W(--L2L2; fcO)Pfc(cos 9) . (F.6) Appendix F. Angular correlations 171 Decay of 9 B ( l / 2 " ) —> 5 Li(3 /2~) + a From Table F . l , for the above break-up, a — 1/2, b — 3/2 c = 0, Zi = 2, Z2 — 1,-f-a = 2 , L 2 = 3/2 and k = 0, 2. Substituting the value of a, L x and L 2 in equation F.6, we obtain W W ? ) = (47r) 2(-l)1/2-3(4) E ( - l ) - 2 - 3 / 2 ^ o ( A , A ) ^ 0 ( £ 2 , £ 2 ) fc=0,2 3 3 1 3 3 3 3 W ( | | 2 2 ; A : - ) W ( - - - - ; f c O ) P f c ( c o s f l ) . (F.7) The coefficients Cko are then given by: C f c 0 ( A , A ) = ^(20,20|/cO), C M ( £ 2 , £ 2 ) = A H « { 2 I , I _ I | M ) . (i) A: = 0 Coo(£ i ,A) = ^ ( 2 0 , 20|00), - l i _ ~ 4^75' c„o (£ 2 , £ 2 )= -£ (^y4 ioo ) . 2 47T (ii) A; = 2 C 2 0 ( A , A ) = 7-(20,20|20), 47T C 2 0 ( £ 2 , £2) — — _ _5_ J L _ 31 3 _ 1_ 4 ^ 2 2 ' 2 ~ 2 |20). 2 4TT Appendix F. Angular correlations 172 The Racah coefficients are: (i) k = 0 W{~22^\) = -v 22 ' 2J _,_,3333 (ii) k = 2 Therefore, ^iii§:20>—I-+ ( - s ) i / f x L s x ( - i / S » x - i f t ( C 0 " > ] - ( R 8 ) = l + P2(cos#) . F . l . 2 Decay of 9 B —> 8Be(2+)+p In Table F.2, we list all the parameters necessary in the calculation of W(6) for this decay mode for states in 9 B of Jn = 1/2", 3 /2 _ , 5/2" and l / 2 + . For this breakup mode, b = 3/2, c = 0. Substituing these values in Equation F . l , WClCa(B) = ( 4 7 r ) 2 ( - l ) a - 4 ( 5 ) £ ( - l ) - ^ - ^ C f c 0 ( A ! / : i ) ^ 0 ( / : 2 , £ 2 ) k=2i W(22L1L1; ko)W(22L2L2; kc)Pk{cos6) . (F.9) Appendix F. Angular correlations 173 Table F.2: Angular correlation parameters for the 9 B —> 8 Be(2 + ) -f- p breakup. J*(9B) a b c h k L2 1/2- 1/2 2 0 1 2 3/2 2 min(3,2,4,4,4) 1/2- 1/2 2 0 3 2 5/2 2 min(5,6,4,4,4) 3/2" 3/2 2 0 1 2 1/2 2 min(l,2,4,4,4) 3/2- 3/2 2 0 1 2 3/2 2 min(3,2,4,4,4) 3/2" 3/2 2 0 3 2 5/2 2 min(5,6,4,4,4) 3/2- 3/2 2 0 3 2 7/2 2 min(7,6,4,4,4) 5/2- 5/2 2 0 1 2 1/2 2 min(l,2,4,4,4) 5/2- 5/2 2 0 1 2 3/2 2 min(3,2,4,4,4) 5/2" 5/2 2 0 3 2 5/2 2 min(5,6,4,4,4) 5/2- 5/2 2 0 3 2 7/2 2 min(7,6,4,4,4) 1/2+ 1/2+ 1/2 2 0 2 2 3/2 2 min(3,4,4,4,4) 1/2 2 0 2 2 5/2 2 min(5,4,4,4,4) Decay of 9 B(l/2") —>8Be(2+)+p Considering the lowest allowed angular momentum li gives a = 1/2, b = 2, c = 0, l\ l,l2 = 2,1/!* = 3/2 and L2 = 2. Substituting these values in Equation F.9, W £ l j C a (0 ) = (4TT)2(-1)1/2-4(5) E ( - 1 ) " 1 / 2 " 2 ^ O ( A , A ) ^ 0 ( £ 2 , £ 2 ) fc=0,2 3 3 1 W(22--;/c-)Fi/(2222;A;O)P f c(cos0) . (F.10) The coefficients Cko are then given by: C M ( A , A ) = s ( - ) ' + 1 ( ^ , ~ l * 0 ) , C l o ( £ 2 , £ 2 ) = ~(-)2(20,20|*:0). 47T (i) k = 0 Appendix F. Angular correlations 174 _ _ 4 1 ~ 4^r2 C00(C2,C2) = ^(2020|00) _ ~ 5 1 (ii) k = 2 ^ , £ 0 = ^ ( ^ , ^ 1 2 0 ) , 4 1 ~ 4^2 ' C2Q{C2,C2) = -^-(20,20|20)., _ _5 2_ ~~ 4T7n The Racah coefficients are: (i) k = 0 iy(2222;00) = -5 (ii) k = 2 VF(2222;20) = \ . Therefore, W(9) = 5(4vr)2(-)-7/2(-)-5/2[(-A)l x ( __5_ } x (_ / l } x (- 1 )lpo(cos0) Appendix F. Angular correlations 175 4 1 5 2 H x = 4 7 T 2 4 7 T x/U = l + P2(cosfl) . - \ ™ ) * ( - e ) P * ( < X » ° ) ] - ( F . l l ) A p p e n d i x G The E682 Col labora t ion Name Institution D. Anthony Simon Fraser University R.E. Azuma University of Toronto N . Bateman T R I U M F L. Buchmann T R I U M F J. Chow University of Toronto J . M . D'Auria Simon Fraser University M . Dombsky T R I U M F E. Gete University of British Columbia U. Giesen T R I U M F C. Iliadis T R I U M F K.P . Jackson T R I U M F J. D. King University of Toronto D.F. Measday University of British Columbia A . C . 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