Β-Delayed Neutron Emission Studies of Neutron-Rich Palladium and Silver Isotopes
β-DELAYED NEUTRON EMISSION STUDIES OF NEUTRON-RICH PALLADIUM AND SILVER ISOTOPES
A Dissertation
Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Karl Isaac Smith
Michael C.F. Wiescher, Director
Graduate Program in Physics Notre Dame, Indiana April 2014 © Copyright by Karl Isaac Smith 2014 All Rights Reserved β-DELAYED NEUTRON EMISSION STUDIES OF NEUTRON-RICH PALLADIUM AND SILVER ISOTOPES
Abstract by Karl Isaac Smith
The rapid-neutron capture process (r-process) is attributed as the source of nearly half the elements heavier than iron in the solar system. The astrophysical scenario responsible is still unknown, making the specific astrophysical conditions uncertain. To gain insight into the r-process nucleosynthesis, uncertainties in the nuclear prop- erties of the involved isotopes must be minimized. The r-process path traverses nuclei far from stability through many isotopes which have not yet been produced in the laboratory. This forces astrophysical models to rely heavily upon theoretical models of the involved nuclear physics. To help constrain parameters used in both astro- physical and nuclear models, an experiment was performed to measure properties of neutron-rich nuclei just below the N = 82 shell closure. These nuclei are believed to be responsible for production of the A = 130 peak in the solar r-process abundance pattern. The first measurements of β-decay half-lives and neutron branching ratios,
Pn values, were performed for neutron-rich isotopes of palladium and silver. To Kristen. I’d be nowhere without you.
ii CONTENTS
FIGURES ...... v
TABLES ...... vii
ACKNOWLEDGMENTS ...... viii
CHAPTER 1: INTRODUCTION ...... 1 1.1 Nuclear Astrophysics Overview ...... 1 1.2 Big Bang Nucleosynthesis ...... 3 1.3 Stellar Fusion ...... 3 1.3.1 Hydrogen Burning ...... 4 1.3.1.1 p-p Chain ...... 4 1.3.1.2 CNO Cycle ...... 4 1.3.2 Helium Burning ...... 5 1.3.3 Heavy Element Fusion ...... 7 1.4 Neutron Capture Processes ...... 8 1.4.1 Slow Neutron Capture Process (s-Process) ...... 9 1.4.2 Rapid Neutron Capture Process (r-Process) ...... 15 1.4.2.1 Classical r-Process ...... 16 1.4.2.2 Astrophysical Site of the r-Process ...... 21 1.4.2.3 Nuclear Parameters Relevent to the r-Process . . . . 25 1.5 Other Processes ...... 28 1.5.1 p-Process ...... 28 1.5.2 Rapid Proton Capture Process ...... 29
CHAPTER 2: INVESTIGATION OF β-DELAYED NEUTRON EMISSION PROBABILITIES AND HALF-LIVES OF PD AND AG ISOTOPES NEAR N=82 ...... 31 2.1 Introduction ...... 31
iii 2.2 Experimental Setup ...... 32 2.2.1 FRagment Separator (FRS) ...... 33 2.2.1.1 Atomic Charge Determination ...... 37 2.2.1.2 Mass Determination ...... 43 2.2.1.3 Particle Identification ...... 48 2.2.1.4 Isomer Tagging ...... 53 2.2.2 Silicon IMplantation Beta Absorber (SIMBA) ...... 56 2.2.3 BEta-deLayEd Neutron (BELEN) Detector ...... 64 2.2.4 Triggering Scheme ...... 69
CHAPTER 3: ANALYSIS ...... 73 3.1 Event Identification ...... 73 3.2 Implant-Decay Correlation ...... 76 3.2.1 β-Decay Neutron Emission Correlation ...... 78 3.3 Background Estimation ...... 79 3.4 Kernel Density Estimation ...... 82 3.5 Differential Decay Solution ...... 86 3.6 Half Life Analysis ...... 91 3.6.1 124Pd Half-Life ...... 103 3.6.2 129Ag Half-Life ...... 106 3.7 Detection Limits ...... 106 3.8 Determination of Neutron Branching Probabilities ...... 108
CHAPTER 4: R-PROCESS IMPLICATIONS ...... 118 4.1 High Entropy Wind Model ...... 118 4.2 Conclusion ...... 125
BIBLIOGRAPHY ...... 127
iv FIGURES
1.1 Solar Abundances ...... 2 1.2 Nuclei Binding Energy ...... 7 1.3 s-Process Path ...... 10 1.4 Neutron Separation Energies as Function of Neutron Number . . . . . 11 1.5 r-Process Path ...... 20 1.6 Effect of Shell Quenching on r-Process Abundances ...... 26 1.7 Calculated Half-Life as a Function of Quadrupole Deformation . . . . 27
2.1 Layout of the Helmholtzzentrum für Schwerionenforschung (GSI) . . 33 2.2 Fragment Separator (FRS) Schematic ...... 35 2.3 Energy Loss as a Function of Fragment Velocity ...... 39 2.4 Resolution of the Atomic Charge Determination ...... 41 2.5 Energy Loss after Final Focal Plane Degrader ...... 42 2.6 Time Projection Chamber Calibration ...... 44 2.7 Time of Flight Calibration ...... 47 2.8 Particle ID for 126Pd Setting ...... 50 2.9 Particle ID for 127Pd Setting ...... 51 2.10 Particle ID for 128Ag Setting ...... 52 2.11 Isomer TAGging (ITAG) System Schematic ...... 54 2.12 Isomer Tagger Particle ID ...... 55 2.13 SIMBA Schematic ...... 57 2.14 SIMBA Tracking Energy Dependence Calibration ...... 59 2.15 Linear-Logarithmic Pre-amplifier Gain Matching ...... 61
v 2.16 Tracking and DSSD Comparison ...... 62 2.17 Particle ID of Implanted Ions ...... 64 2.18 BELEN-30 Schematic ...... 65 2.19 Neutron Moderation Time Distribution ...... 67 2.20 Wall Effect ...... 68 2.21 BELEN-30 Neutron Detection Efficiency ...... 69 2.22 Triggering Scheme ...... 70
3.1 DSSD Event Energy ...... 75 3.2 Correlation Probability ...... 77 3.3 Variation in Background Estimation ...... 81 3.4 Kernel Density Estimator ...... 83 123 125 3.5 Decay Curve Fits for − Pd...... 93 126 128 3.6 Decay Curve Fits for − Pd...... 94 124 126 3.7 Decay Curve Fits for − Ag...... 95 127 129 3.8 Decay Curve Fits for − Ag...... 96 3.9 β-Detection Efficiency ...... 97 3.10 Half-Life Literature Comparison ...... 100 3.11 Even-Even Excitation Energies in the Pd Region ...... 104 3.12 Level Scheme of 126Sn and 128Sn...... 105 3.13 Histogram of Deposited Energy within Correlation Volume for various Q-values ...... 112 3.14 Palladium Neutron Emission Probability Correction Factor ...... 113 3.15 Silver Neutron Emission Probability Correction Factor ...... 114 3.16 Neutron Branching Ratio Literature Comparison ...... 117
4.1 Comparison of a High Entropy Wind Model to Solar r-Process Abun- dances ...... 120 4.2 Impact of New Half-Life Values on r-Process Abundances ...... 122 4.3 Impact of 128Pd Half-Life Upper Limit on r-Process Abundances . . 123
4.4 Impact of Pn Values on r-Process Abundances ...... 124
vi TABLES
2.1 MUSIC VELOCITY CORRECTION PARAMETERS ...... 40 2.2 FRS SETTING PARAMETERS AND DURATION ...... 49 2.3 SIMBA TRACKING ENERGY DEPENDANCE CALIBRATION PA- RAMETERS ...... 60 2.4 SIMBA TRACKING AND DSSD POSITION COMPARISON . . . . 63
3.1 COMPARISON OF MEASURED HALF-LIVES TO PREVIOUS EX- PERIMENTAL VALUES ...... 99 3.2 COMPARISON OF MEASURED HALF-LIVES TO THEORETI- CAL CALCULATIONS ...... 102 3.3 NUMBER OF DETECTED EVENTS AND RESULTING NEUTRON BRANCHING RATIOS ...... 115 3.4 RESULTING NEUTRON BRANCHING RATIOS ...... 116
vii ACKNOWLEDGMENTS
I would like to thank the many people who were involved in my educational career. First, Fernando Montes who was willing to entertain our long distance discussions and encouraged me throughout the difficult steps in the analysis. My advisors: Hendrik Schatz who continued to offer support throughout my educational career and provided many opportunities to continue my career, and Michael Wiescher, who facilitated the final steps in the the long analysis process of this work, your advice was always useful and appreciated. I’d like to thank the members of both Hendrik Schatz’s group, including Zach Meisel, for the many discussions around the world, Matt Amthor, for the guid- ance early in my career, and Giuseppe Lorusso, for the interesting situations the he would get us into. I’d also like to thank those participating in the JINA journal club for the unique conversations about recent publications. Also, those in Michael Wiescher’s group including James deBoer for the many lunch time discussions and Ethan Uberseder for spending time talking about technical aspects of our research. As well as the those at Notre Dame who were willing to lend a hand around the lab. I’d like to thank the members of my committee for the occasional guidance and, Ani Aprahiman, Xiao-Dong Tang, Jonathan Sapirstein, and Manoel Couder who was willing to sit on my committee on short notice.
viii I’d like to thank my collaborators involved with the experimental efforts at GSI, Roger Caballero Folch for the many days spent slaving away during those hot days in Germany, Chiara Nociforo for allowing me to shadow her around the lab and showing me the ropes of the FRS, Alfredo Estrade for the assistance during the experiment, as well as the weekend adventures around Germany, and César Domingo Pardo and Iris Dillmann. Finally, I’d like to thank my parents who provided an environment which encour- aged my curiosity and allowed me to continue my studies. Last, but not least I would like to thank my wife, Kristen, who has been supportive and patient through it all.
ix CHAPTER 1
INTRODUCTION
1.1 Nuclear Astrophysics Overview
Nuclear astrophysics seeks to explain the formation of the elements observed in the universe. Specifically, the abundance distribution observed in our solar system. The study of the content of our sun has a long history starting with the quantum mechanical description of stellar fusion by Atkinson & Houtermans [1] in 1929 and the introduction of nucleosynthesis within the stellar evolution by Hoyle [2] in 1946. The knowledge of the period was then summarized in the seminal papers by Burbidge et al. [3] and Cameron [4] in 1957. These two papers outlined the possible stellar processes and provides the modern basis for current research in the field of nuclear astrophysics. The general trends of the solar abundance distribution, as shown in Figure 1.1, can be explained by a number of cosmic processes. During the Big Bang, the large fraction of observed hydrogen and helium was produced. Beryllium and boron were produced through cosmic spallation, while the heavier metals were produced through stellar processes. The first and primary source of energy generation in stars is from hydrogen and helium burning generating heavier material. In massive stars this is followed by burning of heavier elements through neon, oxygen, and silicon burning.
1 This process of fusion eventually terminates in what is referred to as the iron-peak. Elements heavier than iron are primarily produced through neutron capture pro- cesses. There are two dominant processes a slow and a rapid neutron capture process, known respectively as the s-process and r-process. In addition, a small fraction of material is produced by a proton process, known as the p-process. [5] The following section expands on the various processes with a significant focus on the r-process.
Big Bang 1 H H, He Burning 10-1 He α-Process O 10-2 C Iron Peak Ne 10-3 Si Fe Neutron capture (s,r-process)
Mass10 Fraction -4 Ca 10-5 10-6 r-process 10-7 s-process Xe -8 Ba Pb 10 Pt 10-9 10-10 Li 10-11 0 50 100 150 200 Mass Number
Figure 1.1. Solar abundances highlighting the processes responsible for their production [6].
2 1.2 Big Bang Nucleosynthesis
The universe formed during a massive expansion phase called the Big Bang. Dur- ing this phase the process of Big Bang Nucleosynthesis (BBN) [7,8] was responsible for the formation of the early universe, which was composed of mainly hydrogen and helium, as well as a small amount of deuterium and lithium. Although there is some discrepancy in the 7Li abundance [9] the mass fraction of hydrogen, 74%, and helium, 25%, is considered to be well understood. This early material condensed to form the first stars.
1.3 Stellar Fusion
The formation of the first stars marked the end of the cosmic dark ages, in which no visible or infrared sources of light existed. These early stars, called population III, were composed almost entirely of the hydrogen and helium ejected from the Big Bang. These stars initially burned the hydrogen fuel until it was exhausted. This was followed by helium burning of material from the star’s formation and that which was produced during hydrogen burning. The helium burning phase enriched the early stars with metals. After the exhaustion of helium, these metals continued to produce energy through fusion until reaching the most bound nuclei at which point energy generation by fusion is no longer favorable. For a detailed review of stellar evolution see Carroll & Ostlie, 2006 [10] or Iliadis, 2007 [11].
3 1.3.1 Hydrogen Burning
The majorty of a star’s life is spent in the hydrogen burning phase. This is the principal energy generation process during a star’s lifetime. There are two main reaction sequences responsible for the destruction of hydrogen, the p-p chain and the CNO cycle.
1.3.1.1 p-p Chain
The p-p chain converts protons into helium and releases 26.22 MeV of energy. The primary reaction branch, pp I , undergoes the following reactions:
p + p 2H + e+ + ν → e p + 2H 3He + γ → 3He + 3He 4He + 2p →
This process is responsible for a majority of the energy production in the pp-chain. There are additional reaction chains that release various amounts of energy. These include the pp-II, pp-III and the pep chains [12, 13]. The pp-chains are the dominent form of energy generation in low mass stars.
1.3.1.2 CNO Cycle
The second process responsible for energy generation by hydrogen burning is the CNO cycle. This process requires that the stellar material has already been enriched with material heavier than hydrogen and helium. Due to the requirement of enrichment this process cannot occur in population III stars until the onset of
4 helium burning. The material used in the formation of population II stars has been enriched by the prior generation’s stellar nucleosynthesis and this process is accessible prior to the helium burning phase. Massive star energy generation is dominated by the CNO cycle over the previously described pp-chain. During the CNO cycle four protons are consumed and a helium nucleus is pro- duced generating a net energy release of 26.73 MeV. The process undergoes the following reactions: p + 12C 13N + γ → 13N 13C + e+ + ν → e p + 13C 14N + γ → p + 14N 15O + γ → 15O 15N + e+ + ν → e p + 15N 12C + 4He → The limiting reaction is the 14N(p,γ)15O reaction. This reaction sets the time-scale of the cycle and causes an increased abundance of 14N. These reactions describe the main CNO cycle although there exist smaller branches into other similar cycles. Under high temperature conditions, this cycle may be modified into the Hot CNO cycle. These temperatures are achieved in more extreme cases such as novae and x-ray bursts [14].
1.3.2 Helium Burning
After the hydrogen fuel has been depleted in the stellar core, the core begins to collapse until the increasing temperatures become high enough that helium burning
5 may proceed. Helium burning proceeds initially through the triple-α process in which three helium nuclei, α particles, are fused to form 12C and releases 7.273 MeV of energy. This process involves a two-step reaction:
α + α 8Be → α + 8Be 12C →
This reaction chain is not significant until high stellar temperatures are reached. To successfully produce 12C, the fusion of another α particle with the beryllium nucleus must occur immediately following the first reaction, prior to the rapid decay of the unstable 8Be nucleus [15]. This process is followed by subsequent α captures producing 16O. Fusion beyond this point is inhibited as it becomes an endothermic process due to decreasing binding energies per nucleon. Figure 1.2 shows the measured nuclear binding energies as a function of nucleon number, A. The trend rises from hydrogen until reaching its maximum at A = 56.
6 9
8
7
6
5
Binding Energy / Nucleon4 [MeV]
3
2
1
0 0 20 40 60 80 100 120 140 160 180 Nucleon Number (A)
Figure 1.2. Binding energy per nucleon as a function of nucleon number. Each point represents a single isotope. Fusion continues increasing nucleon number until reaching the maximum at 56Ni. Only measured binding energies are shown. Data taken from the Atomic Mass Evaluation 2012 [16].
1.3.3 Heavy Element Fusion
Once the helium material is exhausted, nucleosynthesis continues through fusion of the heavier metals in the star. As with the onset of helium burning, the stellar core will collapse until temperatures are sufficiently high for fusion to overcome the increasing Coulomb barrier. The heavy element fusion processes occur by fusion of nuclei in the following order: carbon, neon, oxygen, and silicon burning. In the silicon
7 burning phase the fusion of two 28Si nuclei is highly unlikely and the process instead proceeds by dissociation of lighter nuclei. The dissociated neutrons, protons and alpha particles are then captured forming more tightly bound nuclei. This process continues eventually terminating near the most bound nucleus, 56Ni, forming what is referred to as the iron-peak in the solar abundance pattern. At this point, stars with sufficient mass will undergo a supernova explosion where the surface of the star is ejected into space enriching the interstellar medium for future generations of stars.
1.4 Neutron Capture Processes
To achieve the production of heavier masses, a different mechanism is necessary as fusion becomes prohibitive. Charged particle reactions are inhibited due to increas- ing Coulomb barriers that can only be overcome at temperatures greater than that which can be achieved in most astrophysical scenarios. The neutron capture pro- cesses provide a mechanism to move beyond the iron-peak. Nearly all the elements heavier than iron were formed through neutron capture reactions processes. There are two major types of these processes: a slow-neutron capture process, s-process, and a rapid-neutron capture process, r-process. Each of these processes account for nearly half of the abundance distribution of the heavy elements. In addition to these processes, a small fraction of the material is generated by the p-process and rp-process.
8 1.4.1 Slow Neutron Capture Process (s-Process)
When heavy seed nuclei such as 56Fe are exposed to a flux of neutrons, neutron capture reactions occur. In this process, the flux of neutrons is small such that the time-scale for β-decay is much shorter than that of the neutron capture. In this regime the β-decays can be considered nearly instantaneous. This suggests the name slow neutron capture process (s-process). The process starts with a seed nuclei (Z, N) that will capture a neutron changing it to (Z, N + 1). If the resulting nuclei is stable then it will undergo another neutron capture. Unstable nuclei will be transformed under β−-decay from (Z, N) to (Z +1, N 1). This process tends to follow the valley − of β stability. In general, the process cannot reach neutron-deficient stable nuclei if they are separated by one that is unstable to β−-decay. These unstable nuclei may be overcome when the decay rate is of similar order to the neutron-capture rate. The nuclei located where the process splits following two paths are known as branching points. A diagram of the s-process path is shown in Figure 1.3. As the neutron capture reactions are the limiting factor, the abundances as a function of mass number, A, involved in the process can be determined by the dif- ferential decay chain equation. Where production of the species A is given by the neutron capture on species A 1 reaction rate, σν A 1, and destruction is given by − h i − the capture reaction rate of species A, σν . h iA
dYA = Nn σν A YA + Nn σν A 1 YA 1 (1.1) dt − h i h i − −
This steady-state flow approximation has shown to correctly reproduce the s-process
9 2
1
Proton Number (Z) 0
0 1 2 3 Neutron Number (N)
Figure 1.3. Diagram of the s-process path. Stable isotopes are shown in grey. The path originates with the seed nucleus (Z = 0, N = 0). The nucleus (Z = 0, N = 1) is a branching point with the dashed line representing the new path due to the branching. pattern [17]. The time-scale of the s-process is set by the neutron capture rates with typical lifetimes of τnγ & 10 yr. The isotopes with small capture cross-sections will tend to pile up with increasing abundance. Those with a large cross-section will quickly be destroyed and processed into heavier nuclei. This pile up and destruction leads to peaks and troughs, respectively, in the s-process abundance. Figure 1.1 shows the solar abundance distribution with two significant, narrow peaks attributed to the s-process centered at A 138 and 208 corresponding to the closed shell numbers ≈ N = 82 and 126. The addition of a neutron to a nucleus with a closed shell requires substantial energy to overcome the large difference in neutron separation energy as shown in Figure 1.4. This reduces the available excitation energy thus forcing the reaction to occur where the level density is small, resulting in a reduction in the cross
10 section. The s-process continues along the valley of stability until reaching its termination point. The last stable nucleus along the path is 209Bi, at which point further neutron captures will produce isotopes unstable to α emission [18].
30
25
20 ) [MeV] n 2
S 15
10 Energy (
Two Neutron Separation 5
6
) [MeV] 5 +2 n 2 4 S −
n 3 2 S 2 1 Energy ( 0 Difference in Two Neutron Separation 20 40 60 80 100 120 140 Neutron Number (N)
Figure 1.4. Upper panel show the two neutron separation energies for even N nuclei. Bottom panel shows the difference in two neutron separation energy between nuclei with neutron number N and N +2. Red dashed lines indicate the neutron magic numbers at N = 28, 50, 82, and 126.
11 At least two different components of the s-process are required to correctly re- produce the solar s-process abundance distribution [19]. The main component is responsible for production of isotopes from A = 90 to 205. The weak component which contributes far less to overall production is responsible for the majority of the abundance of isotopes with A < 90. Both components, the main and weak s-process, have different astrophysical sites. The main component of the s-process is believed to originate from the late ther- mal pulsing of low mass (1.5 3 M ) Asymptotic Giant Branch (AGB) stars [20–22]. − In this scenario the AGB star alternates between hydrogen and helium burning. As the hydrogen burning produces more helium, the helium layer mass increases until it ignites. This new source of energy expands the start causing the hydrogen layer temperature to decrease extinguishing the hydrogen burning. This expansion phase is referred to as a thermal pulse. During the thermal pulse the helium intershell between the hydrogen and helium layers becomes completely convective mixing car- bon products from helium burning with protons from the hydrogen layer. Between thermal pulses the carbon in the helium intershell undergoes the following reactions:
p + 12C 13N + γ → 13N 13C + e+ + ν → e p + 13C 14N + γ →
These reactions form a pocket of 13C and 14N. Once the temperature is high enough, the lifetime of the neutron producing reaction 13C(α,n)16O is shorter than the time between pulses, a sufficient neutron flux is generated. Produced neutrons are then
12 captured by seed nuclei fueling the s-process. The neutron density during this period
7 3 is relatively low, N 10 cm− , such that most branches are not activated. n ≈ As the hydrogen shell burning phase ends, the star contracts, increasing the temperature, until the helium shell is again ignited forming a thermal pulse. Higher temperatures are achieved and the 14N is converted to 22Ne through the reactions:
α + 14N 18F + γ → 18F 18O + e+ + ν → e α + 18O 22Ne + γ →
The neutron source 22Ne(α,n)25Mg is activated leading to higher neutron densities. This phase is shorter and does not contribute substantially to production. The higher neutron density activates the branches thus changing the overall shape of the distribution. As the helium layer becomes inactive and contracts the hydrogen shell will ignite again repeating the process. The 13C(α,n)16O provides a neutron source of density,
7 3 3 22 25 N 10 cm− , active for 20 10 yr and the Ne(α,n) Mg source provides n ≈ ∼ × 10 3 densities of N 10 cm− for a few years. n ≈ The weak component of the s-process occurs in core helium burning of stars with mass greater than 13 M [23, 24]. The more abundant 14N, produced during the
13 CNO cycle, is converted to 22Ne through the following reactions:
α + 14N 18F + γ → 18F 18O + e+ + ν → e α + 18O 22Ne + γ →
Neutrons are then provided by the reaction 22Ne(α,n)25Mg towards the end of the helium burning phase, once the temperature is sufficient to overcome the Coulomb barrier. In addition, 22Ne(α,n) competes with 22Ne(α,γ) and is dominated by the later at lower temperatures. More massive stars will produce more neutrons as stellar temperatures increase as a function of mass. The s-process may be impeded by reactions that act as neutron sinks or poisons. In fact, the neutron source reaction 22Ne(α,n)25Mg is self-poisoning since the product 25Mg has a high neutron-capture cross-section. Other reactions may also act as poisons, such as 12C(n,γ)13C and 16O(n,γ)17O. In the case of neutron capture on 12C, the neutrons are eventually recycled by 13C(α,n)16O thus reducing its effectivity as a neutron poison. In the 16O case, the recycling reaction 17O(α,n)20Ne competes with the α-capture reaction 17O(α,γ)21Ne. It has been shown that the ratio of the 17O(α,n) to 17O(α,γ) reactions is such that the neutron capture of 16O is an efficient neutron poison [25, 26]. Astrophysical modeling of these two components has been very successful and generally reproduce the solar s-process well. The s-process is responsible for nearly half the production of elements heavier than iron. Once the s-process abundances have been determined they can be subtracted from the solar abundance pattern and
14 the remainder will require another process to explain their production.
1.4.2 Rapid Neutron Capture Process (r-Process)
The well understood s-process does not provide a mechanism for the production of some neutron-rich nuclei that are shielded by isotopes with very short β-decay half- lives. To produce these, a mechanism with a much higher neutron flux is required. In this regime, the time scale for neutron-capture is much smaller than that of the β- decay. This leads to a successive and rapid capture of neutrons giving the mechanism the name rapid neutron capture, r-process [27]. The astrophysical site providing this high neutron flux is still unknown, two favored scenarios will be described in Section 1.4.2.2. The initial discussion of the r-process mechanism will assume we have conditions with a high neutron flux that last for some length of time. This process is responsible for production of the remaining half of the solar abundance heavier than iron, including the peaks in the solar abundance pattern, Figure 1.1, at A = 80, 130 and 195. The solar r-process pattern is determined by subtracting the s-process pattern from the total solar abundance pattern. The remainder of the pattern must be generated by the r-process. Recent models of the s-process have provided an in- creased predictive confidence providing a reliable r-process abundance pattern. Any uncertainties in the s-process abundances will directly impact the r-process abun- dances. The s-process pattern can be confirmed by comparison of stable nuclei that are shielded from the r-process by stable neutron-rich nuclei. These shielded nuclei can only be produced by the s-process and are called s-only nuclei. The shielding
15 isotope is typically separated from the s-process path by an unstable nucleus such that only the r-process is able to produce that isotopes. These shielding isotopes are referred to as r-only nuclei. The r-only nuclei provide an unambiguous confirmation of the solar r-process abundance pattern.
1.4.2.1 Classical r-Process
The path of the r-process is much further from the valley of stability than that of the s-process, passing through very neutron-rich isotopes. In the simple classical model of the r-process, the nuclei are exposed to a temperature of T 1 GK and ≥ 21 3 neutron density of N 10 cm− . In these conditions, both the neutron capture, n ≥ (n, γ), and the photo-disintegration, (γ, n), reactions have time-scales much smaller than the β-decay rates. With sufficient temperature and neutron flux, an equilibrium along an isotopic chain is established such that the rate of abundance change for a fixed Z will be constant. The abundance of an isotope along this chain, Y (Z, N), can then be determined by a ratio to its neighbor using the Saha equation.
2 !3/2 Gnorm Y (Z, N + 1) h MZ,N+1 gZ,N+1 Z,N+1 Qnγ /kT = Nn norm e (1.2) Y (Z, N) 2πkT MZ,AMn gZ,N gn GZ,N where g = (2j + 1) and M describe the spin and mass of the involved particles,
norm GZ,A is the normalized partition function of nucleus Z, A and Qnγ is the Q-value, or neutron separation energy of the neutron capture reaction. Repeatedly applying this equation m times one can determine the abundance of species m relative to the
16 initial species m = 0.
2 !3m/2 3/2 norm " m 1 # m h Mm gm Gm 1 X− Ym = Y0Nn m m norm exp Qi (1.3) 2πkT M0Mn g0gn G0 kT i=0
One can see from this equation that the determination of the most abundant species is strongly dependent on neutron separation energies. The process can be thought of as successive neutron captures continuing until the neutron separation energy of the next nucleus is less than the a particular constant value established for a given astrophysical condition. As the nuclear properties are fixed, the only varia- tion in the r-process is due to the neutron density and temperature. An increase in neutron density will shift the abundance maximum toward more neutron-rich, smaller neutron separation energies. An increase temperature will increase the rate of photo-disintegration pushing the maximum toward the less neutron-rich nuclei, larger neutron separation energies. Material will typically build up in one or two isotopes in the isotopic chain. Due to pairing effects, these isotopes are typically those with even number of neutrons, even–N. Even–N nuclei have large separation energies while odd–N nuclei have small separation energies. The isotopic chain will continue capturing neutrons until reaching the equilibrium separation energy, which typically occurs between an even–N and odd–N isotopes. The isotopes with the largest abundance in an isotopic chain then represent waiting-points at which the process may not proceed until a β-decay occurs. The β-decay transfers material between isotopic chains and controls the ratio of abun- P dances between chains, YZ = N Y (Z, N). Two isotopic chains are related by the
17 differential decay chain:
dYZ = λZ YZ + λZ 1YZ 1 (1.4) dt − − − where λZ is the decay constant for the isotopic chain given by
X Y (Z, N) λZ = λβ(Z, N) (1.5) N YZ
where λβ(Z, N) is the β-decay constant for nucleus (Z, N). The solution to this type of differential equation is described in Section 3.5. In the steady flow approximation the rate of abundance change between isotopic chains is constant giving the following relation between chains:
dY Z = 0 dt (1.6)
λZ YZ = λZ 1YZ 1 = const − −
The solar r-process abundance pattern peaks at A = 80, 130 and 195 can be attributed to the neutron magic numbers N = 50, 82 and 126. These peaks are shifted to lower mass than the corresponding peaks in the s-process. As the r- process path reaches a nucleus with a magic number of neutrons, the separation energies drop significantly. This leads to a build-up of abundance in the nucleus with a magic number creating a waiting point. The path then continues after a β- decay, where the situation is again repeated in the next isotopic chain. The path will follow the magic numbers to increasing Z. Meanwhile, as the process continues along the waiting points at the magic numbers the β-decay half-lives become increasingly long, as well as increase in neutron separation energy. Eventually, the process is able
18 to continue by neutron capture once the neutron separation energy is high enough. The waiting point with the largest β-decay half-life will have the largest abundance along the magic numbers. These peaks correspond to the classical r-process waiting point nuclei of 80Zn, 130Cd and 195Tm. A schematic view of the r-process is shown in Figure 1.5. The figure demonstrates the successive neutron capture and subsequent β-decay proceeding along a path of constant neutron separation energy with typical values between 2-3 MeV [28]. The figure also shows the successive waiting points along a magic number. When the neutron source is exhausted the unstable nuclei will undergo succes- sive β-decays along constant isobars toward stability. The final abundances will be influenced by the duration, τ, of the neutron flux determined by the astrophysical scenario. At the time of freeze-out, when the neutron flux is exhausted, the abun- dance pattern has a strong odd-even effect due to variation in neutron separation energy as explained above. This odd-even effect is much stronger than observed in the solar abundance pattern. It has been shown that this pattern is smoothed by the process of β-delayed neutron emission [28]. Many of the β-decays have Q-values larger than the neutron separation energy. This allows those nuclei to emit a neutron immediately following the β-decay shifting the abundance pattern from isobar A to A 1. The neutron emission branching ratio describes the probability that a β-decay − leads to a neutron emission and is often labeled as a Pn value. In addition, heavier nuclei may also undergo β-delayed fission. This fission re- cycling may be an additional source of neutrons that could provide a short rejuve- nation to the r-process. Finally, the heaviest isobars will reach β-stable nuclei that
19 52
51
50 Proton Number (Z)
49
48
47
46
78 79 80 81 82 83 84 85 Neutron Number (N)
Figure 1.5. The r-process path follows successive neutron captures (right arrows) until a waiting-point is reached. The waiting-points are determined by a specific neutron separation energy (red contour), dependent on the astrophysical properties. The path may then only proceed following a β- decay (diagonal arrows). The path tends to have numerous waiting-points along neutron shell closures, such as N = 82, (black rectangle) where the separation energy drops significantly. Separation energies (blue contours) are shown in 1 MeV steps, data from the Atomic Mass Evaluation 2012 [16] and the Finite Range Droplet Model 1995 [29]. The nearest stable nuclei 130Tc (gray box) is shown.
20 are unstable to α emission. These long lived α emitters may be used as nuclear chronometers [30]. The combination of the waiting point approximation and the steady flow approx- imation provides a simple model that demonstrates the importance of astrophysical and nuclear properties on the r-process. This describes the classical r-process, a first approximation, a more dynamic calculation would follow the subsequent neu- tron freeze-out, neutron production via β-delayed neutron emission, and subsequent neutron capture period. The process depends on the astrophysical scenario which sets the neutron flux of density, Nn, temperature, T , and duration, τ, as well as the nuclear data providing the neutron separation energies, Sn, the β-decay rates, λβ, and the neutron branching ratios, Pn values. Properties of nuclei along the r-process path are difficult to determine experimentally as the nuclei are very neutron-rich and thus hard to produce. Many nuclei along the path have not been produced in a laboratory setting and current astrophysical models rely heavily on theoretical calculations. Precision determination of the nuclear data will reduce the ambiguity in the astrophysical model as well as assist in constraining nuclear models. This will allow more precise studies of the astrophysical scenario without ambiguity due to the properties of the involved nuclei.
1.4.2.2 Astrophysical Site of the r-Process
The astrophysical site of the r-process is still unknown [31, 32]. Many parametric studies of the process have been performed and have shown that a single constant neutron density and temperature is insufficient to reproduce the r-process pattern. A
21 superposition of at least three components is required [28] indicating multiple sites or a dynamic process in which the neutron flux and temperature vary. Although many sites have been proposed [33], there are currently two preferred sites: type II supernovae and neutron star-mergers [5]. Both sites offer a dynamic environment, sufficient neutron densities and temperatures that are not so large as to increase the dissociation rate. The first of the preferred r-process sites is the type II supernova scenario which occurs at the end of a massive star’s life [34, 35]. Once the star has completed the silicon burning phase the star’s structure is composed of concentric spheres of various material, with heavier material toward the center. The outer core comes into nuclear statistical equilibrium, in which all the reaction are balanced by their counterpart. The central core is formed by the tightly bound 56Fe nuclei which no longer has a nuclear source of energy generation. The stellar core becomes electron degenerate as its mass continues to increase. The electron degeneracy pressure maintains the growing core which continues to increase in temperature. Once the Chandrasekhar limit (1.4 M ) has been reached the electron degeneracy is not sufficient to prevent its eventual collapse. In addition, at these temperature the photo-dissociation of iron occurs. This produces helium and neutrons, the former of which is also dissociated into protons and neutrons. This proceeds through the following reactions:
56Fe + γ 13 4He + 4n → (1.7) 4He + γ 2p + 2n →
22 With a large abundances of protons, electron captures on protons, p + e− n + ν , → e occur removing the electron degeneracy pressure and accelerating the collapse of the core. This reaction produces a substantial number of neutrinos, which then stream out of the core toward the outer layers. As the core collapses, the density will continue to increase until reaching the nuclear density, at which point the core stiffens due to the repulsion caused by the strong force. This stiffening will cause the core to bounce and create an outgoing shock wave. The shock wave will propagate outward through the star, as it encounters more inward falling material the temperature is increased and leads to further dissociation robbing the shock wave of energy and forcing the shock to stall. This stalled shock wave is dense enough that the outgoing neutrinos interact with the material depositing energy and thus providing a heating source. The heating causes the shock to continue outwards leading to a delayed explosion. This phenomenon is called a neutrino-driven wind. The outer core, which was in nuclear statistical equilibrium at high temperatures, will be composed of mostly protons and neutrons with a large neutron excess. As the material expands and cools, α-particles are formed and eventually become the dominant particle. At this point, the equilibrium collapses and the large excess of α-particles assisted by the neutrons produce material beyond 56Ni into the A = 100 region by (α, n) and (n, γ) reactions. As the temperature continues to drop, the alpha-induced reactions slow and if the neutron excess is large enough an r-process begins using the A = 100 material as seeds. At this point, the neutron density will still be large and the temperature of the material will have dropped sufficiently to allow the process to run far from stability. At the termination of the r-process the
23 enhanced material will be ejected along the outgoing shock wave. The second promising site is the merging of two neutron stars [36–39]. This envi- ronment provides a low temperature, low entropy, high neutron density environment that has been shown to reproduce the solar r-process abundances with A > 140. Neutron stars that become gravitationally bound will continue to spiral and in-fall until merging to form a single body. During this merging phase, the stellar material is shocked enough that temperatures increase such that nuclear statistical equilib- rium (NSE) is formed. As with the core-collapse scenario the NSE material expands and cools and alpha particle fusion produces seed material. As the electron fraction in this scenario is lower there will be a larger fraction of available neutrons. As temperature drops more quickly the environment is considered relatively cold. The lower temperatures and increased neutron density will cause the process to run fur- ther from stability producing waiting-point nuclei with short half-lives. This allows the process to continue more quickly to the heavier mass regions. The r-process enhanced material is then ejected by tidal forces. There are some drawbacks to these scenarios. The neutrino-driven wind models have not been successful in one-dimensional models. The shock stalls and does not become dense enough to allow for sufficient heating. Recent explorations of multidimensional models have shown that a shock revival at later times is possible [40]. The neutron star merger scenario suffers from an unknown occurrence rate that may be too low to produce sufficient enrichment of the interstellar environment [41].
24 1.4.2.3 Nuclear Parameters Relevent to the r-Process
Three nuclear properties define the r-process path and eventual decay to stability, these are the neutron separation energies, β-decay half-lives, and β-delayed neutron emission branching ratios. The involved nuclei are neutron-rich and thus far from stability making experiments difficult due to low production rates. Typically, the half-lives and branching ratios are accessible by experimental measurements prior to measurements of nuclear masses and thus separation energies. Since these nuclei have been difficult to access in the laboratory, typical astrophys- ical models rely almost entirely on theoretical mass models. Commonly used mass models include the Finite Range Droplet Model (FRDM) of Möller et al. [29], the Ex- tended Thomas-Fermi plus Strutinsky Integral (ETFSI-1) of Aboussir et al. [42], and a Hartree-Fock approach with Skyrme forces and Bardeen-Cooper-Schrieffer parining (HFBCS-1) of Goriely et al. [43]. These models include strong shell effects, in other words a large change in separation energies across magic numbers, that continue to neutron-rich nuclei. There has been some question whether these shell closures are quenched and thus weakened as nuclei become more neutron rich. An extension of ETFSI-1 by Pearson et al. [44] has produced a mass model (ETFSI-Q) including a reduction in shell strength, called shell quenching. It has been shown that the addi- tion of shell quenching can improve reproduction of the solar abundances at masses just prior to the abundance peaks at A = 130 and 195. Figure 1.6 shows a compar- ison between the two ETFSI models. The ETFSI-1 model with strong shell effects shows a considerable under production prior to the the abundances peaks that is less prominent when including shell quenching with the ETFSI-Q model.
25 Figure 1.6. A comparison of two r-process calculations with differing mass models compared to the solar r-process abundances (black). The first mass model, ETFSI-1 [45], has a strong shell effect, no shell quenching (blue) and shows large deviations prior to the abundance peaks. The second mass model, ETFSI-Q [44], extends the previous model with a reduced shell effect, including shell quenching (red) shows a closer agreement to solar values.
The unmeasured β-decay rates are determined using either semiempirical global models or a Quasiparticle Random Phase Approximation (QRPA) calculations [46– 48]. The QRPA calculations make use of the same mass models described above. Comparison of measured half-lives with calculated values can help constrain these models to more accurately predict β-decay rates as well as improve mass models. For example, the quadrupole deformation term, describing the nucleus shape, plays a significant role in the determination of the decay half-life from these QRPA mod- els [49, 50]. A comparison of experimental and theoretical values as a function of quadrupole deformation, Figure 1.7, can provide an indication of the required nu-
26 cleus shape in that region. In addition, if the shape is already known, the half-life can constrain the Q-value through this type of calculation.
Figure 1.7. An example of a constraint on mass models from measured β- decay half-lives. Measured half life of 39Si (dashed line) compared to QRPA calculation (solid line) performed at various quadrupole deformations. To reproduce the measured half-life a large prolate or oblate deformation is required. Shaded region corresponds to range of QRPA calculations given the uncertainties in Qβ. Figure adapted from [49].
Finally, β-delayed neutron emission probabilities play an important role in the final stage of the process. These branching ratios are responsible for the relative odd-even smoothness of the final abundance distribution. These parameters are also calculated using QRPA models. An under prediction of these values tends toward a more staggered odd-even effect in the final abundances. Uncertainties in these
27 parameters can make it difficult to compare calculated final abundances to r-process abundances determined from the solar system. Providing measurement in this neutron-rich region of the chart reduce uncertain- ties in the specific measured nuclei as well providing information for nuclear models. Variations in r-process models due to nuclear uncertainties can be significant. If fur- ther astrophysical constraints on the scenario are to be extracted from these models the uncertainties in the input parameters must be reduced.
1.5 Other Processes
A small deviation exists between the solar abundances and the sum of the s- and r-process solar abundances. These deviations occur in nuclei on the proton rich side of stability, which are shielded from both of the neutron capture processes by stable nuclei. These nuclei are referred to as p-nuclei as they are proton-rich. The abundance of the p-nuclei is nearly two orders of magnitude smaller than that produced by the s- and r- process. Two possible processes are discussed below.
1.5.1 p-Process
The p-process takes place in stellar environments where high temperatures can be achieved. The initial seed material is taken from the products of the s- and r-process and is photo-disintegrated to the proton-rich side of the valley of stability via (γ, n) reactions. The process follows successive (γ, n) reactions moving the abundance toward neutron-deficient nuclei until the neutron separation energies become large. Meanwhile, the proton and alpha separation energies decrease and open the (γ, p)
28 and (γ, α) reactions [51]. These reactions shift the abundance to different isotopic chains. This process produces the p-nuclei as well as contribute to the s- and sr- nuclei, nuclei produced by both the s- and r-process. This process was initially thought to take place in hydrogen rich layers of a star undergoing a supernova [3]. As the shock passes through these layers, the temperature and density would become sufficient to initiate the process. In addition, proton-capture reactions may also occur at these temperatures sufficient to overcome the Coulomb barrier. Achieving these temperatures and pressures was shown to be unreasonable and the current accepted scenario occurs in hydrogen poor regions such as the O-Ne rich layers [52, 53]. In these regions there is insufficient hydrogen to support (p, γ) reactions. The process is limited at closed shell nuclei where the alpha and proton thresholds become high preventing further dissociation. Once the increased temperatures fall, the unstable material under goes β+-decay toward stability.
1.5.2 Rapid Proton Capture Process
Another process with a path along the proton-rich nuclei follows a series of rapid proton captures, giving it the name the name rapid proton capture process or rp- process. This process unlike the r-process occurs with some β+-decay time scales on the order of the proton capture reactions. The proton capture then competes with the photo-disintegration and the β+-decays and even proton emission as the path follows nuclei along the proton drip-line. The process is inhibited by waiting points which have small or negative proton separation energies and relatively long
29 β+ half-lives. At the exhaustion of the hydrogen fuel, the material β+-decays back to stability and may undergo β-delayed proton emission. This process has been shown to occur on the surface of neutron stars in binary systems [54]. In the latter case, the neutron star accretes hydrogen rich material from its neighboring companion. As the material falls onto the surface of the neutron star its temperature increases such that a break-out of the CNO cycle occurs, resulting in a type I X-ray burst. The breakout initially leads to the αp-process, a series of (α, p) and (p, γ) reactions that bypass many of the slow β+-decays. The process then continues through the rp-process as described above. These environments have large gravitational potential and it is unlikely that the produced nuclei can be ejected into the interstellar medium.
30 CHAPTER 2
INVESTIGATION OF β-DELAYED NEUTRON EMISSION PROBABILITIES AND HALF-LIVES OF PD AND AG ISOTOPES NEAR N=82
2.1 Introduction
Astrophysical models of the r-process are sensitive to the astrophysical site, as well as nuclear data inputs. To understand the constraints of the astrophysical scenario the other uncertainties must be minimized. In an effort to provide precise nuclear data many experimental campaigns measuring relevant nuclear properties have been performed. These campaigns are pushing the current boundaries of rare isotope production. For very neutron-rich nuclei, current production rates are very low not permitting many types of experiments. The measurement of half-lives and β- delayed neutron emission branching ratios of neutron-rich nuclei along the r-process path is possible even with very small numbers of produced nuclei. These values are important themselves as nuclear data inputs for the r-process, but the also may provide constraints on theoretical mass models which are currently the only available source for mass information in much of the r-process region. Production methods for nuclei in this region also produce a large number of contaminants which must be separated. These low production rates coupled with contamination require a
31 number of experimental detector systems to provide clear identification of the type of produced nucleus as well as measure its decay properties.
2.2 Experimental Setup
An experiment to investigate half-lives and β-delayed neutron emission branching ratios of neutron-rich nuclei was performed at the Helmholtzzentrum für Schwerio- nenforschung (GSI), located in Wixhausen, Germany. See Figure 2.1 for a schematic
1 238 layout of the facility. A 900 MeV u− , U beam was delivered by two stages of ac- ∼ celeration. The first stage of acceleration was accomplished by the linear accelerator,
1 UNILAC [55], delivering ions at energies up to 20 MeV u− . The beam is then in- jected into the second acceleration stage, the SIS-18 [56] synchrotron. After multiple revolutions and accelerations, the accelerated and bunched beam was then delivered
2 to a 2.5 g cm− thick lead target placed at the entrance to the FRagment Separa- tor (FRS). The primary beam intensity was measured by the SEcondary Electron TRAnsmission Monitor (SEETRAM) [57] at up to 2 109 ions/spill. After in-flight × separation and identification, the fission fragments were implanted into the Silicon IMplantation Beta Absorber (SIMBA). This detector system was capable of time and position correlation of ions and their decays providing a method to determine the β-decay half-lives. Neutrons released from nuclei undergoing β-delayed neutron emission were detected by the surrounding BEta-deLayEd Neutron (BELEN) Detec- tor providing a neutron branching ratio, Pn value, determination.
32 Figure 2.1. Layout of the Helmholtzzentrum für Schwerionenforschung (GSI). Beam is accelerated from the ion sources via the UNILAC and SIS18 accelerators and then delivered to the Fragment Separator (FRS).
2.2.1 FRagment Separator (FRS)
The FRagment Separator (FRS)[58] is a high resolution spectrometer consisting of a series of bending sections (magnetic dipoles) and focusing sections (quadrupole doublets and triplets). It allows in-flight physical separation and identification of reaction products produced by projectile fragmentation or fission. The resulting beam can be studied at the final focal plane or transported to another experimental station. A schematic drawing of the FRS depicting the separation and focusing
33 elements as well as the various detector systems is shown in Figure 2.2. The system has two focal planes. The first, called the intermediate focal plane comes after the first separation stage, consisting of the first two dipoles. At this point additional systems can allow for a second separation stage, the final two dipoles. The final separated beam is then delivered to the final focal plane where the experimental end station was placed. In the FRS, the primary beam first impinges upon a fragmentation or fission target. At the high energies provided by the SIS18 often both channels are avail- able. In this experiment, we use the FRS to select the fission fragments, producing a secondary cocktail beam. This secondary beam contains a large number of species many of which are not particular interesting to any given experiment. The remain- ing primary beam and secondary contaminates are separated using the Bρ–∆E–Bρ method. Isotopes are first separated by making use of the Lorentz Force Law:
d~p F~ = = q(E~ + ~v B~ ) (2.1) dt ×
The FRS makes use of magnetic dipoles providing a magnetic field tangential to the ion’s velocity. We can simplify the Lorentz force law for this specific application in which there is only a magnetic field and equate it with the centripetal force with radius ρ. vp γmv2 F = qvB = = (2.2) ρ ρ
34 Target Quadrupole Scintillator MUSIC BELEN Dipole Degrader TPC SIMBA
Intermediate Focal Plane
Final Focal Plane
Figure 2.2. Schematic view of the FRS. The beam direction is from left to right, impinging first on the fission target, intermediate focal plane, and final focal plane. The intermediate focal plane consisted of a scintillator, a degrader to assist in separation, and two TPCs used in conjuction with the scintillator for particle identification. The final focal plane consisted of another TPC, two MUSICs, another TPC, a scintillator, an additional degrader, another MUSIC and the final implantation end station consisting of SIMBA, a silicon stack for implant and β-decay detection, surrounded by BELEN-30, a neutron detector system to detect β-delayed neutron emission. The TPCs, scintillator and MUSICs allowed a for particle identification. The variable degrader was used to slow ions such that they were implanted into SIMBA.
Solving for the magnetic rigidity, Bρ, we find:
p γβm c Bρ = = 0 (2.3) q q where we have introduced γ, the Lorentz factor, and β as the ratio of the velocity of ion to the speed of light, c. Finally, m0 is the rest mass and q is the charge state of the ion. The values of ρ are fixed by the acceptance of the FRS. The magnetic dipoles
35 have a central radius of ρ0 = 11.2641 m. Using the first two dipole sections, which deliver beam to the intermediate focal plane, we can chose a value of the magnetic field and make a selection in the mass to charge ratio, A/Q. The chosen A/Q is physically centered while contaminant ions will be horizontally offset and can then be stopped with slits. The second Bρ stage will then refocus the beam at the final focal plane. To further separate the selected A/Q, a degrader is placed at the intermediate focal plane. As the beam passes through the degrader, the momentum of the particles is changed as a function of atomic charge, Z. The beam then passes through the final two dipole section to the final focal plane. Making use of the degrader, the second Bρ stage, with a fixed Bρ, now separates these ions providing a physical separation as a function Z. The ion of interest is again centered, now at the final focal plane, and offset contaminants are again stopped using slits. In addition, the degrader can be offset from a tangential alignment so that there is an angle between the beam and degrader. This allows for delivery of the ions of interest at the final focal plane to be either focused in position or mono-energetic. The FRS is capable of transmitting a number of distinct isotopic species depend- ing on the chosen acceptance. This requires an unambiguous particle identification (PID) of the fragments on an event by event basis. This is achieved at the FRS by measurement of the energy loss, time of flight and position of the ion as determined by ionization chambers, plastic scintillators and time projection chambers, respec- tively. The combination of these values provides an identification via atomic number, Z, and mass to charge ratio, A/Q, of each fragment. A description of each of these
36 methods and corresponding detector systems follow.
2.2.1.1 Atomic Charge Determination
To identify the atomic charge of the ions and thus their elemental type, multiple ion chambers were employed. Three MUltiple Sampling Ionization Chambers (MU- SICs) [59] were placed at the final focal plane to determine the fragments atomic charge. The MUSIC chambers are operated at room pressure and temperature and
filled with P10 gas (Ar + 10% CH4). The chamber is operated under an electric field between the cathode plate and eight anode strips. As the beam passes through the gas, charged particles are released and drift toward the anodes. The deposited charge signal is then readout using a series of amplifiers and a peak-sensing ADC for each anode. These chambers made use of the fact that the energy deposition is proportional to the atomic charge squared, described by the Bethe formula:
dE 4πe4Z2 " 2m c2β2 ! # N Z e β2 β2 = 2 2 mat mat ln ln(1 ) (2.4) dx mec β I − − −
where Nmat and Zmat are the number density and atomic charge of the stopping material, me and e are the rest mass of the electron and the electronic charge, and I is the mean excitation energy. Z is the atomic charge of the material and β is the velocity of the ion relative to the speed of light, c. The energy loss in the MUSICs is determined by the geometric mean of the eight
37 anode segments: 8 !1/8 Y ∆E = ∆Ei (2.5) n=1 The ion chambers were calibrated by measuring the energy deposition as a function of fragment velocity, determined in Section 2.2.1.2, of a known reference fragment. The energy loss of another fragment was determined by scaling the calibration by Z2. This approximation can be written as:
Z 2 ∆E = f(β) Z0 (2.6) f(β) = a + bβ + cβ2
where Z0 is the atomic charge of our calibration ion and f(β) is determined from
fitting ∆E as a function of β where Z = Z0 as shown in Figure 2.3. The resulting parameters are shown in Table 2.1. The atomic charge, Z can then be found by:
s ∆E Z = Z (2.7) 0 f(β)
This method provides a clear separation between elements and provides a dis- tinct identification of the elemental type of the fragments. The Z identification for MUSIC 1 is shown in Figure 2.4. The second MUSIC chamber was not necessary for this experiment. Another experiment in our campaign implanted heavy ions, A > 200, that may carry a charge state through the fragment separator. This affects the resolution of the particle identification. The second chamber permitted a high enough sensitivity to separate
38 1110 Second Order Polynomial
1100 MUSIC 1 ) [arb. units]
E MUSIC 2 ∆ 1090
1080 Energy Loss ( 1070
1060
1050
1040 0.79 0.795 0.8 0.805 0.81 0.815 Relative Velocity (β)
Figure 2.3. Energy loss for Silver reference fragments shown as a function fragment velocity with respect to the speed of light. Multiple silver iso- topes were used from multiple settings to increase the range of β values. Two ionization chambers are shown, in order of interaction with beam, MU- SIC 1 (blue) and MUSIC 2 (red).
these cases. See Benlliure et al. [60] for further details. The final MUSIC chamber, MUSIC 3, was placed after the degrader material at the final focal plane. As fragments pass through the degrader there is a probability that another fragmentation reaction may occur. This is due to the high energies of
2 the fragments and the substantial amount of material, 9 g cm− , necessary to slow ∼ the ions for implantation. This MUSIC chamber allowed identification of fragments
39 TABLE 2.1
MUSIC VELOCITY CORRECTION PARAMETERS
Parameter MUSIC 1 Value MUSIC 2 Value
a 6289.19 7476.40
b 10807.1 13950.3 − − c 5364.21 7421.96 that underwent an additional fragmentation event in which the atomic charge was changed. A proper velocity calibration of this detector was not possible due to the spread in velocities of the primary beam after impinging on the detector. Without this calibration the resolution of the detector is significantly reduced and only a cut on whether a charge changing reaction occurred is available, shown in Figure 2.5. Events undergoing this additional fragmentation are removed from further analysis as there exact properties are unknown. This method is not sensitive to fragmentation reactions in which only neutrons are removed. These neutron fragmentation reactions will contaminate our PID as they change species after the PID was performed.
40 0.025 Z 0.02
0.020
0.015 Fraction of Events / 0.010
0.005
0.000 44 45 46 47 48 49 50 Atomic Charge (Z)
Figure 2.4. Histogram of measured atomic charges, Z. There is a clear separation between the various atomic charges allowing for an unambiguous determination of Z.
41 3500
3000
2500
2000
1500 Energy Loss in MUSIC 3 [arb. unit]
1000
500
0 43 44 45 46 47 48 49 50 51 52 Atomic Charge (Z)
Figure 2.5. Energy loss in MUSIC 3, following the final focal plane degrader. Ions undergoing an additional fragmentation reaction will have lower atomic charge and will deposit less energy in the MUSIC chamber (red). Those events which have not undergone an additional fragmentation will deposit the most energy (black). Events undergoing another fragmentation are cut from the final analysis as there exact isotopic information is unavailable.
42 2.2.1.2 Mass Determination
Mass Determination is done by identifying the magnetic rigidity and the time of flight of each ion during the second half of the FRS. The horizontal positions and angles of each fragment are determined at the intermediate and final focal planes by four Time Projection Chambers (TPCs) [61]. The time-of-flight was determined by the scintillators that measured the time difference of the ion between the two focal planes. The TPC is composed of a vertical drift chamber between a plate cathode and a shielding grid. Four proportional counters with C-pads were placed just below the shielding grid. Use of the C-pads allowed use in higher ion intensities. The drift volume was filled either with Ar+10% CH4 (P10) at room temperature and pressure. The gas mixture was held under a uniform electrical field with voltages of
1 up to 400 V cm− . Fragment interaction with the drift volume lead to the ionization of the gas. Produced electrons and corresponding ions drift toward the anode and cathode respectively. The four 20 µm anode wires were surrounded by the C-pad cathodes, which were each connected to two independent integrated delay line chips. The vertical position was determined by the electron drift time to each anode wire, providing four measurements. The horizontal position was determined by the difference between the left and right side of each delay line, providing two mea- surements. Calibration of the TPC was performed by coincidence with a series of scintillator fibers 1 mm thick. The fibers were arranged into a grid with a horizontal separation of 2 cm and vertical separation of 1 cm. The relative offset was then de- termined by closing the slits to 1 mm. An example of this calibration is shown in ±
43 Figure 2.6.
105 m
µ × 3 2 1
Counts / 250 20
15
10
5
0
-5 Vertical Position [mm] -10
-15
-20 105 -40 -30 -20 -10 0 10 20 30 40 2 4 6 × Horizontal Position [mm] Counts / 250 µm
Figure 2.6. Time Projection Chamber (TPC) position Calibration. A set of fiber scintillators are inserted and coincident events are shown. The x-y positions (black, dots) clearly show the separation between the fiber grid. Histograms of projections of the horizontal (red upper pane) and vertical position (blue right pane).
44 With the known positions of the fragments at the intermediate focal plane, xF 2 and final focal plane xF 4, the magnetic rigidity can be determined via the following equation: ! xF 4 fmagxF 2 Bρ = Bρ0 1 + − (2.8) fdisp where Bρ0 is the central magnetic rigidity and the magnification, fmag, and dis- persion, fdisp are properties of the chosen ion optics between the intermediate and final focal planes. The magnification describes the change in image size of the beam between the intermediate and final focal planes. The slope from linear fit of the hor- izontal positions at both focal planes provides the magnification as fmag = 1.115225. The dispersion can be found found by varying the magnetic fields by a known scal- ing factor and measuring the resulting change in the final focal plane position. The slope from a linear fit of the position as a function of scaling factor dispersion of fdisp = 7.2202681 mm/%. The time-of-flight was determined by a pair of scintillators at the intermediate and final focal planes. Each of these scintillators had photo-multiplier tubes at- tached to the left and right ends of the scintillators. Due to the high rate of ions at the intermediate focal plane, it is beneficial to allow fragments reaching the final scintillator to provide a start condition for the Time-to-Analog Converter (TAC). Many of the contaminant ions at the intermediate focal plane do not reach the final focal plane. The stop signal is provided by a delayed signal from the scintillator at the intermediate focal plane. The velocity of the particle is then determined by the
45 following relationship with the time-of-flight, tflight:
1 l β = eff (2.9) c tflight
The path length, leff , is nearly constant for all fragments, with some deviations due to varying flight paths. The effective flight path can be determined as a function of a fragment’s horizontal positions at the intermediate, xS2, and final, xS4, focal planes as well as the angle at the final focal plane, aS4:
leff = l0 + cxS2 xS2 + cxS4 xS4 + caS4 aS4 (2.10)
where cxS2 , cxS4 , and caS4 are constant for each setting of the FRS. The central path length, l0, is the path of the particle for which the FRS system tune was chosen.
The time-of-flight offset, toffset, is due to the delay in transmitting the signal from one scintillator to the other. Both l0 and toffset can be determined using well known velocities of the 238U primary beam as calibration. The measured value of the time of flight deviates by some offset such that:
t = t t (2.11) flight meas − offset
Using this relation and the fact that the primary beam should follow the central path, rearranging Equation 2.9 we find the following relationship:
l t β = t β + 0 (2.12) meas offset c
46 We have selected a β value and tuned the FRS to transmit the primary beam with this velocity along the central path. Since the scintillators has two photo-multiplier tubes we can determine the time of flight from the time difference between the left or right sides. We chose to use an average between these times. A simple linear fit using the above equation can then extract the central path length and the offset time.
Figure 2.7 shows the linear fit, we find toffset = 201.7632 ns and l0/c = 122.2968 ns. [ns] 45 meas β t 40
35
30
25
20
15
10
0.65 0.7 0.75 0.8 0.85 β
Figure 2.7. Time of flight calibration points (black dots) fitted with a linear function, Equation 2.12, (red line) to determine the time of flight offset and central path length.
47 We can rewrite Equation 2.3 to solve for the atomic mass to charge ratio, where we replace the particle mass, m0, as the atomic mass, A, of our fragments:
A 1 Bρ = (2.13) Q f γβc
2 Where f is the conversion factor between amu and MeV c− . This allows us to determine the mass to charge ratio using a particle’s magnetic rigidity determined from Equation 2.8 and its velocity determination from Equation 2.9.
2.2.1.3 Particle Identification
Using the atomic charge, Z, determined in Section 2.2.1.1 and the atomic mass to charge ratio, A/Q, determined in Section 2.2.1.2 we can make a particle identifi- cation. Plotting Z vs A/Q there is a clear separation between different fragments. Two dimensional graphical cuts can be placed around each isotope to separate each case for an independent analysis. During the experiment three settings were used: centered on 126Pd, 127Pd, and 128Ag. The tune parameters for each setting are shown in Table 2.2. The maximum thickness of the final degrader provided at the final focal plane did not provide enough material to sufficiently reduce the beam energy such that ions of interest would be implanted in SIMBA. An additional aluminum plate
2 of thickness 15.4 mm corresponding to an aerial density 4.158 g cm− was added. A particle ID of ions reaching the final focal plane prior to the final variable degrader is shown for each setting in Figure 2.8, Figure 2.9, and Figure 2.10. The amount of beam time spent on each setting varied.
48 TABLE 2.2
TUNING PARAMETERS OF THE FRS 49
Magnetic Field B [Tesla] Intermediate Focal Plane Degrader Final Focal Plane Degrader Centered Ion Beam Time [hr] 2 2 Dipole 1 Dipole 2 Dipole 3 Dipole 4 Thickness [g cm− ] Angle [mrad] Thickness [g cm− ]
126Pd 1.2275 1.2275 1.0131 1.0128 5.88071 7 9.35928 44.25
127Pd 1.2407 1.2407 1.0259 1.0256 5.88071 7 9.70962 36.13
128Ag 1.2407 1.2397 0.9890 0.9887 5.88071 10 8.66180 3.53 50
49 102
Atomic Charge (Z) 48
47
46 10 45
44
43 1 2.55 2.6 2.65 2.7 2.75 2.8 Atomic Mass to Charge Ratio (A/Z)
Figure 2.8. Particle Identification for the setting centered on 126Pd. A clear separation between distinct groupings is visible. 126Pd is indicated as a reference (red circle).
50 50
49 10
Atomic Charge (Z) 48
47
46
45
44
43 1 2.55 2.6 2.65 2.7 2.75 2.8 Atomic Mass to Charge Ratio (A/Z)
Figure 2.9. Particle Identification for the setting centered on 127Pd. A clear separation between distinct groupings is visible. 126Pd is indicated as a reference (red circle).
51 50 102 49
Atomic Charge (Z) 48
47
46 10
45
44
43 1 2.55 2.6 2.65 2.7 2.75 2.8 Atomic Mass to Charge Ratio (A/Z)
Figure 2.10. Particle Identification for the setting centered on 128Ag. A clear separation between distinct groupings is visible. 126Pd is indicated as a reference (red circle).
52 2.2.1.4 Isomer Tagging
The measured Particle ID (PID) was confirmed by studying microsecond isomers produced in the fission target, by using the Isomer TAGging (ITAG) [62] system used at the FRS. This technique was first demonstrated by Grzywacz et al. [63]. This system consists of two plastic scintillators, a passive stopper and two high purity germanium detectors as shown in Figure 2.11. This system is placed at the final focal plane of the FRS to allow for particle ID explained above. During the isomer datat collection, one of the germanium detectors was replaced with a series of LaBr3 detectors. Degrading material was added upstream of the focal plane scin- tillator such that the ions would be implanted in the stopper. A second scintillator was placed down stream of the stopper and was used to veto ions that were not im- planted. This second scintillator is not part of the productions setup and is removed after confirmation of the PID. Adjacent to this scintillator, the γ sensitive detectors were placed. Lead shielding was placed around these detectors to block beam in- duced radiation upstream of the stopper. The γ-rays emitted by the transition from the isomeric state to the ground state were measured and correlated with the particle identification, providing an unambiguous identification. The ITAG was used to confirm our Particle ID by examining the 1497 keV, 7.9 µs isomeric state of 205Bi. By selecting γ-decays from this specific state, only 205Bi appears on the PID. Making use of the long fission fragment tails and overlaying multiple settings we can confirm the PID to the region of interest by simply counting the isotopes from identified species. This confirmation is shown in Figure 2.12.
53 Figure 2.11. Schematic of Isomer TAGging (ITAG) system used at the FRS. The system consists of two high purity germanium detectors, two plastic scintillators and a passive stopper. Beam is implanted into the stopper and characteristic isomeric decays were detected with the germanium detectors. Figure used with permission from Farinon [62].
54 90 205 88 83Bi122 86 202Bi 84 83 119 82 192 80Hg 80 112 196 182 78 80Hg 77Ir105 116
Atomic Atomic Charge 76 186 77Ir109 74 170 73Ta97 72 174 73Ta101 70 68 148 66 65Tb83 154 64 65Tb89 62 60 58 130 55 Ba 56 56 74 135 54 Sb 126 51 84 52 Sn 133 50 76 Sn 50 50 83 114 48 Pd 106Ru 46 68 46 44 62 118 127 44 46Pd72 Ag 110 47 80 42 44Ru66 40 2.20 2.22 2.24 2.26 2.28 2.30 2.32 2.34 2.36 2.38 2.40 2.42 2.44 2.46 2.48 2.50 2.52 2.54 2.56 2.58 2.60 2.62 2.64 2.66 2.68 2.70 2.72 2.74 Atomic Mass to Charge Ratio
Figure 2.12. Particle Identification confirmed with ITAG. 205Bi was unam- bigoulsy identified through a correlation with a detected γ-ray from the decay of 1497 keV isomeric state. Multiple FRS setting are overlayed and a confir- mation of the region of interest was performed by identifying various species by counting from the known 205Bi. 2.2.2 Silicon IMplantation Beta Absorber (SIMBA)
In order to study the decay properties of neutron-rich nuclei, an efficient detec- tor system capable of detecting charged particles and neutrons was required. The charged particles were implanted and resulting decays were measured within a series of silicon detectors placed at the final focal plane of the FRS. Following the particle identification stages explained above, the beam interacted with a degrader with cho- sen thickness, see Table 2.2, such that ions would be stopped in these silicon layers. The highly-segmented silicon detectors that were utilized were placed in a closely packed stack to maximize geometric efficiency. The stack consisted of two tracking detectors, two Single-Sided Silicon Detectors (SSSDs), three Double-Sided Silicon Detectors (DSSDs) and two more SSSDs as shown in Figure 2.13. The two tracking detectors were SSSDs with an active area of 60 60 mm2, × 300 µm thick with 60 strips each. The tracking detectors were oriented in the hor- izontal and vertical direction. Each half of the tracking detector was read out via resistor chain. By performing a center of mass calculation on the charge read on ei- ther side of the detector, one can determine the location on the detector that the ion traversed. The following equation is used to determine the raw position in arbitrary units: E E u = a − b (2.14) Ea + Eb where u is the raw position, and Ea / Eb are either the left / right or up / down signals respectively. The positions have a higher order position dependence which can be corrected by fitting the ratio of the calculated position to its maximum value
56 Figure 2.13. Schematic view of SIMBA silicon layers. Beam direction is from the left to right, encountering following layers from front to back: vertical tracking, horizontal tracking, front SSSDs, DSSDs, and rear SSSDs. The separation between detectors has been expanded for ease of viewing. as a function of energy deposited, E. The following functional form was used for the calibration function, f(E):
u c4 f(E) = = c1 + c2(E c3) (2.15) umax(E) −
where the constant c4 is less than 1, providing an inverse power law. umax(E) is the position calculated at large energy deposition. As each resistor chain of the detector is independent, a separate calibration function was used for each side. Where the constants, c1 4 are dependent on the side, a or b. The corrected positions at any −
57 energy can then be computed by the following relationship:
1 Ea Eb ucorr = − (2.16) fa(Ea)fb(Eb) Ea + Eb
Figure 2.14 demonstrates the higher order energy dependence correction. The maxi- mum value of the tracking position in the detector, umax, is determined at the largest energy deposition, at roughly 3500 in Figure 2.14. This position is chosen as the true position value for the entire strip and f(E) describes the correction fraction needed to remove the energy dependence. Prior to the correction, the resolution of the outer strips position was poor, providing an implant position with a large uncertainty. Af- ter applying the correction each strip was well resolved with a 1σ uncertainty in both the horizontal and vertical direction of 0.05 mm. Parameters used in the calibration function are given in Table 2.3
The absolute scaling for the tracking position was then determined by perform- ing a polynomial fit between the central value of the energy corrected positions for each strip and the correct strip positions. The central values were computed as the centroid of a fitted gaussian. After the tracking detectors there were two SSSDs of dimension 60 40 mm2, × 1000 µm thick with seven strips each in the horizontal direction. These detectors allowed identification of decay-like background events, see Section 3.1. Following the first set of SSSDs was the implantation region comprised of three DSSDs, each being 60 40 mm2, 700 µm thick with 1 mm pitch between each strip. × The vertical strips were read out by a system with a linear high-gain amplification
58 3500
3000
2500
2000
1500 Energy Deposited [arb. unit]
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
3000
2500
2000
1500 Energy Deposited [arb. unit] 1000 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Position [arb. unit]
Figure 2.14. The energy deposition as a function of raw position determined by center of mass calculation (upper, black) for the horizontal position detec- tor. A single inverse power law calibration function (upper, red) is overlayed. The calibration is scaled for each strip for comparison. Energy dependence is removed by applying the calibration (lower, black). (The large groupings of positions correspond to various Z values.)
to clearly identify decay events, depositing energies less than 1 MeV. Heavy-ions stopping or traversing the silicon, depositing energies up to 1 GeV, caused the ∼ linear amplification to overflow a number of neighboring strips. For this reason, we used the tracking detector previously described to identify the location of the
59 TABLE 2.3
SIMBA TRACKING ENERGY DEPENDANCE CALIBRATION PARAMETERS
Horizontal Tracking Vertical Tracking Parameter Left Right Upper Lower
c1 0.04602 0.04602 -0.5059 -0.3014
c2 0.6645 0.6645 1.074 0.8132
c3 1024 1024 605.0 704.2
c4 0.04646 0.04646 0.04367 0.05917 implanted ion. The DSSDs horizontal strips, as well as the SSSDs, made use of a linear-logarithmic preamplifier that allowed sufficient resolution for both heavy ions and the subsequent decays, as well as discrimination between the two types of events. The Mesytec MPR-16 LOG pre-amplifiers provided a linear gain up to 10 MeV and ∼ beyond that a logarithmic gain was provided. Each horizontal strip was gain matched to provide a rough energy calibration. The calibration function made use of a linear and exponential term as shown in Figure 2.15. To reduce the amount of amplification and digitization electronics, the first and last six horizontal strips were grouped into four groups of three strips each so that the 40 strips of one DSSD could be read by a module with 32 channels. This left 28 strips in the middle section which were connected to independent amplification and digitization channels. Since each DSSD has 1920 pixels, the effective pixelation of
60 120
12 100
Energy [MeV] 10
8 80 Energy [MeV] 6
4 60 2
0 40 0 200 400 600 800 1000 1200 1400 ADC Channel
20
0 0 500 1000 1500 2000 2500 ADC Channel
Figure 2.15. Each strip of each DSSD was gain matched using a pulser input on the linear-logarithmic pre-amplifiers. The manufacturer provided 1 the capacitance 0.78(8) pF giving a conversion factor of 17 MeV V− . The pulser voltage was varied and the corresponding ADC channel was deter- mined. The calibration function (red) was composed of a linear (black) and an exponential (blue) term. The linear region has been magnified in the inset. the full implantation region was 5760 pixels. After all corrections and calibrations have been applied for the tracking detectors, a comparison to the redundant position measurement provided from the DSSDs was performed. Figure 2.16 shows a nearly one to one correlation between the vertical / horizontal DSSD and the vertical / horizontal tracking detectors. The linear fit parameters are given in Table 2.4.
61 60 60 60 50 50 50 40 40 40 30 30 30 20 20 20 10 10 10 0 0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Horz. Tracking Pos. [mm] Horz. Tracking Pos. [mm] Horz. Tracking Pos. [mm] DSSD-0 Horz. Strip Pos. [mm] DSSD-1 Horz. Strip Pos. [mm] DSSD-2 Horz. Strip Pos. [mm]
30 30 30 25 25 25 20 20 20 15 15 15 10 10 10 5 5 5 0 0 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
DSSD-0 Vert. Strip Pos. [mm] Vert. Tracking Pos. [mm] DSSD-1 Vert. Strip Pos. [mm] Vert. Tracking Pos. [mm] DSSD-2 Vert. Strip Pos. [mm] Vert. Tracking Pos. [mm]
Figure 2.16. Comparison of measured positions between the tracking layer and the three DSSD layers: DSSD-0 (left), DSSD-1 (middle), and DSSD-2 (right). The horizontal positions (upper panes) have a poor resolution in the DSSD due to overflows in linear pre-amplifiers attached to the vertical strips. The horizontal strips made use of logarithmic pre-amplifiers allowing single strip resolution for the strips vertical positions (lower panes). The first two and last two DSSD vertical strip positions correspond to three physical strips as they were connected. Linear fits (red) show a near one to one correlation between the DSSD and Tracking positions.
Two more layers of SSSDs of the same type as previous followed the DSSDs. These layers provided a rejection for non-implanted events. An event that did not
62 TABLE 2.4
SIMBA TRACKING AND DSSD POSITION COMPARISON
Tracking Detector Parameter DSSD-1 DSSD-2 DSSD-3
Slope 0.9810 0.9805 0.9800 Horizontal Offset 0.5860 0.4938 1.963
Slope 0.9870 1.003 0.9933 Vertical Offset -4.630 -5.180 -5.120 stop in the DSSD would subsequently deposit energy in these SSSD layers provide a veto condition. In addition, these detectors were used to reject β-like background events, see Section 3.1. The energy thresholds of each DSSD and SSSD strip were set independently such that each threshold was just above the noise peak in that strip. As we were only interested in fragments that impinged on SIMBA, we added a requirement of a detection above threshold in the SIMBA tracking detectors as another gating condition to the Particle ID. In addition, fragments depositing energy in the rear SSSDs are rejected. Figure 2.17 shows a particle identification of implanted events over the entire experiment using these additional constraints.
63 102 ) Z 48.5
48 Atomic Charge ( 47.5
47 10
46.5
46
45.5
1 2.6 2.62 2.64 2.66 2.68 2.7 2.72 2.74 2.76 2.78 2.8 Atomic Mass to Charge Ratio (A/Q)
Figure 2.17. Particle Identification of implanted ions.
2.2.3 BEta-deLayEd Neutron (BELEN) Detector
BELEN-30 is a specific implementation of the BELEN detector system[64]. In this configuration, 30 3He proportional counters were arranged in a polyethylene matrix in two concentric rings surrounding a large central bore. The inner ring was composed of 10 counters at a pressure of 10 atm and the outer ring of 20 counters at a pressure of 20 atm. The large central bore had a diameter of 223 mm to accommodate the SIMBA detector. An additional 20 cm of material was added on each side of the central matrix to shield from external neutron sources. A schematic drawing of this
64 setup is shown in Figure 2.18.
Figure 2.18. BELEN-30 is composed of two concentric rings of 3He propor- tional counters embedded in a polyethylene matrix. The inner ring (teal) contained 10 counters at a pressure of 10 atm and the outer ring (red) con- tained 20 counters at a pressure of 20 atm. A large central bore (black) al- lows SIMBA to be placed inside. Additional polyethylene shielding is placed around the central matrix.
65 The polyethylene matrix acted as a moderating material reducing the neutron energy, by successive scatterings. After randomly scattering through the matrix, an encounter with a proportional tube may lead to a neutron capture reaction inside the proportional counters. The reduction of neutron energy increased the detection efficiency in the proportional counter as the 3He neutron capture cross-section in- creases as neutron energy is decreased. The typical moderation time distribution, Figure 2.19, follows an exponential decay with a time constant of 110(15) µs. The proportional counters were maintained with a 1500 V potential between the central anode and the concentric cathode forming the counter’s exterior. The cap- ture reaction, 3He(n,p)3H, releases two charged particles with a Q-value of 764 keV. Assuming the neutrons have been completely thermalized and thus do not contribute additional kinetic energy, the outgoing particles are released with characteristic ener- gies, the proton energy, Ep = 573 keV, and the tritium energy, E3H = 191 keV. These charged particles then ionize the gas. Electrons are attracted toward the anode and create an electron avalanche thus multiplying the output signal. The output signal is proportional to the energy deposited in the gas. The reaction products have significant ranges in the gas and they may escape the counter without depositing their full energy. This phenomenon is called the wall effect and produces a stair step pattern in the detected energy as shown in Figure 2.20. This leads to a large range of energy depositions for a neutron detection from the smallest energy of a single product, E3H = 191 keV, up to the full Q-value, Q = 764 keV. Due to the moderation, the incident neutron energy is not detected. The arrangement of the two concentric rings of proportional counters was chosen
66 s µ 4
103 Counts /
102
100 200 300 400 500 600 700 800 900 1000 Time After β-Decay [µs]
Figure 2.19. Emitted neutron are moderated in BELEN’s polyethylene ma- trix until becoming thermalized and detected in a proportional tube. This moderation distribution (grey) has been fit (red) with an exponential de- cay (green) with a time constant of 110(15) µs and a constant term (blue) accounting for uncorrelated background neutrons. so that the detection efficiency would be flat with respect to neutron energy. A series of Geant4 [65] simulations were performed to optimize this configuration. The final orientation gave a relatively flat distribution in the detection efficiency from 10 keV to 1 MeV. The simulation efficiencies are shown in Figure 2.21. These simulations have shown a good agreement with similar systems and past measurements of well known branching ratios. In addition, the simulation of a 252Cf source with this
67 104 Total Full p, Full 3H Energy Deposition Full p, Partial 3H Energy Deposition Counts / 1 keV Full 3H, Partial p Energy Deposition 103
102
10 0 100 200 300 400 500 600 700 800 900 1000 Deposited Energy [keV]
Figure 2.20. Deposited energy spectrum for 3He(n,p)3H demonstrating the wall effect. A full energy gaussian at 764 keV (teal) is produced when both products deposit their full energy. Additionally, two separate continua are produced if the proton escapes (green) or if the 3H escapes (yellow). detector configuration agreed with an efficiency measurement preformed prior to the experiment.
68 45
40
35
30
25
Neutron Detection Efficiency20 [%]
15
10 Total 5 Inner Ring Outer Ring 0 2 2 1 1 10− 2 10− 10− 2 10− 1 2 3 4 5 6 × × Neutron Energy [MeV]
Figure 2.21. BELEN-30 detection efficiency as a function of neutron energy. The total efficiency (black, triangles) is relatively flat up to &1 MeV. Inner (blue, circles) and outer rings (red, squares) also shown.
2.2.4 Triggering Scheme
During the experimental runs the data acquisition (DAQ) was activated by an accepted trigger. A schematic view of the electronic modules and triggers is shown in Figure 2.22. During the acquisition period, the system would not accept additional triggers, although those additional triggers are registered using scaler modules. There were two types of accepted trigger, implant and decay, as well as a correla- tion trigger to detect neutrons. The implant trigger was provided by a discriminator
69 Silicon IMplantation Beta Absorber (SIMBA)
1x 8ch Digitizers Struck SIS3302 3x 8ch Ch 0-7
3x 8ch 1x SAM3 Module 3x 60ch 3x GASSIPLEX ASICs, 3x 32ch ADC Multiplexer,ADC 3x Double Sided CAEN v785 Silicon Detectors (DSSDs) 6x 16ch Shaper / Disc. Gate 3x 16ch X: 60 Strips 6x 16ch Log Preamplifier Mesytec STM-16+ Ch 0-15 3x 32ch Mesytec MPR-16-LOG Y: 32 Strips Shaper 6x 16ch 3x 16ch Ch 16-32 Input Output Input Disc. 6x OR Disc. OR 1x 32ch ADC CAEN v785
Front Single Sided Gate 1x 16ch Silicon Detectors (SSSDs) 2x 16ch Shaper / Disc. Ch 0-15 2x 7 Strips 14ch Mesytec STM-16+ 2x 16ch Log Preamplifier 1x 16ch Ch 16-32 Mesytec MPR-16-LOG Shaper Back Single Sided 14ch 2x 16ch Input Output Input Disc. Silicon Detectors (SSSDs) 2x OR 2x 7 Strips Disc. OR 1x 32ch ADC CAEN v785
Gate 1x 16ch Shaper / Disc. 1x 16ch X Resistive Strip Single Ch 0-15 2ch Mesytec STM-16+ Sided Silicon Detector (Tracking) 1x 16ch Linear Preamplifier Mesytec MPR-16 Shaper Ch 16-32 2ch 1x 16ch Y Resistive Strip Single Input Output Input Disc. Sided Silicon Detector (Tracking) 1x OR Disc. OR Decay Trigger
Implant Identification Accepted Trigger Final Focal Plane Implant Trigger (S4) Scintillator Shaper / Discriminator Time to Analog Convertor Left PMT Input Disc.
Right PMT Input Start Disc. Output 32ch ADC Stop CAEN v785 Input Disc. Intermediate Focal Plane Start Gate Input Disc. (S2) Scintillator Output Stop Ch 0 4x Shaped Output AND Left PMT 4ch Ch 1 Right PMT Ch 2-5 24ch Shaper Ch 6-31 4x Focal Plane 4x 6ch Input Output Time Projection Chambers
BEta deLayEd Neutron (BELEN) Detector
Gate & Delay Trigger Delay: 200 µs Latch 1x 64ch TDC Start Start CAEN v767 ADC Gate Stop Reset Gate Generator Gate Out Ch 0-64 AND Start Delay 3x OR BELEN Trigger Gate
3x 16ch 2x 32ch ADC 3x 16ch Shaper / Disc. CAEN v785 Mesytec STM-16+ 3x 16ch Preamplifier Gate 2x 16ch Mesytec MPR-16-HV Disc. OR Ch 0-15 BELEN 3-He 30ch 3x 16ch Input Output Input Disc. Proportional Counters Ch 16-32 Shaper 1x 16ch 4x 8ch 4x 8ch Digitizers Struck SIS3302
Ch 0-7
Figure 2.22. The data acquisition (DAQ) system was triggered on an ac- cepted trigger, either a decay or implant trigger. For a detailed description see Section 2.2.4. Trigger types are highlighted by grey filled boxes. Modules highlighted in blue are read out when the DAQ is triggered.
70 signal from a coincidence between the two photo-multiplier tubes attached to the plastic scintillator located at the final focal plane. The discriminated value was cho- sen to maximize detection of the ions of interest. In addition to reading out the FRS parameters needed for particle identification, all of the channels of the SIMBA detector were also read out providing the ion position and energy deposition in the silicon layers. The decay trigger was provided by any of the linear-logarithmic preamplifiers signals coming from SIMBA. This included the tracking layers, SSSDs, and the horizontal strips in the DSSDs and excluded the vertical strips in the DSSDs. The level of the discriminator was set as low as possible to maximize the β-decay detection efficiency, while staying above the noise level. If the decay trigger was accepted an acquisition window was opened to allow for a BELEN trigger. The BELEN system was read by two DAQ systems, one analog and one digital. This campaign made use of the first coupling of the BELEN system with a digital DAQ. The analog system was used to confirm its proper operation and provide a backup system. As the systems are different their read out also differs. The BELEN analog DAQ system was initiated to record neutron events that oc- curred in coincidence with decay events. Following a decay trigger, a 200 µs window is opened which allowed time for emitted neutrons to be moderated in the polyethylene before being detected in one of the 3He proportional tubes. A BELEN trigger was produced by the discrimination of the neutron energy in any of the 30 3He counters. This trigger produced a gate to store the energy signal from the triggered propor- tional tube. Only the first neutron event energy was stored, all subsequent neutrons
71 were ignored by the analog ADC. The analog TDC recorded discriminated neutron event times over the entire window produced from the decay trigger. The BELEN digital DAQ was self-triggered on neutron events making use of a trapezoidal filter to save the neutron event time and energy. The digital DAQ has a small dead time relative to the analog system. Data from each event were stored in on board memory. The memory of the digital modules are read out and cleared when an accepted trigger was received. The clocks of the various modules were synchronized and the accepted trigger time is also recorded. This measurement of the accepted trigger time allows for reconstruction of when the neutrons are emitted relative to the decay events. This system permitted measurement of the detected neutrons at all times during the experiment. Its live time was not restricted to the 200 µs window that the analog system was. This allowed for a more complete neutron background study to be performed.
72 CHAPTER 3
ANALYSIS
After the data collection phase of the experiment, a series of multivariate analysis steps were necessary to extract the resulting half-lives and neutron branching ratios. First, a proper identification of event type, implant or decay, was necessary prior to performing a space-time correlation between these events. Once decay events have been associated with there corresponding implant event a sufficient background estimation is necessary to remove any additional time dependance of the decay curve. As the lifetimes are short, the resulting decay curve is fit with a multi-generational decay model that may include the parent nucleus and its children, grand-children and great grand-children. Finally, a correlation between detected neutrons and β-decays is performed to determine the probability of β-delayed neutron emission. Details of this analysis are discussed below.
3.1 Event Identification
Implant and decay events were triggered with relaxed conditions to ensure a high measurement efficiency. This was balanced against losses due to increased dead- time. Additional software gates were applied to clearly discriminate between the three types of events: implants, β-decays, and β-decay-like background events.
73 First, implants were separated by requiring a valid MUSIC signal providing a Z measurement. Only ions with Z > 44 were considered for further analysis. In addition, valid signals in the scintillators and TPCs at the intermediate and final focal planes were required such that a A/Q measurement was provided. Finally, only ions which were implanted were of interest. An ion was considered an implant under the following conditions. It must first reach the SIMBA detector requiring a valid signal in both of the tracking layers. An ion was then considered implanted if and only if there is a large energy deposition above 50 MeV in a DSSD layer. The energy deposition in SIMBA provides a clear discrimination between decay and implant events, as shown in Figure 3.1. Implants were rejected if they punched through the DSSD layers. These punch through events were identified by a large energy deposition in one of the rear SSSD layers. The second type of event, decay events, must not resemble ions coming from the FRS. This requires that these events have MUSIC signals below the noise thresh- old. To provide a position correlation with the associated implant, a decay event with energy deposition in both a horizontal and vertical strip of a single DSSD was required. To maximize the resolution of the time-of-flight and atomic charge measurement the scintillator and MUSIC amplification gain was set such that signals from ions of interest could use a large fraction of the accepted range of the ADCs. Fission fragments with a smaller atomic charge will produce smaller signals with some signals being comparable to electronic noise. These small signals will not trigger the data acquisition through a implantation trigger since they are below the discrimination
74 ih sinfamns hs vnsaeo h hr type, third the of events. these are including events detector decay These the a fragments. in and fission particle low light charged very any set nearly was for threshold produced SIMBA was trigger the hand, other the On threshold. DSSD-0 Vert. Strip DSSD-1 Vert. Strip DSSD-2 Vert. Strip ihu eeto odto o hs vnste ol ecniee real considered be would they events these for condition rejection a Without eoiini vdn.Tecoe nryaoewiha vn sconsidered is line). event (green an indicated which above is energy implant energy chosen an high The with events evident. implant between is and DSSD-0 discrimination deposition deposition for clear energy layers low A DSSD with events (top). the decay of DSSD-2 strip (middle), each DSSD-1 into (bottom), energy Deposited 3.1. Figure 10 20 30 10 20 30 10 20 30 0 2 2.5 3 3.5 75 4 Depositied Energy(log 4.5 β dcylk background -decay-like 5 10 5.5 E) [arb.unit] 6 β-decay events. A real event would originate from one of the DSSD layers and propagate outwards. It is highly unlikely that a real event would deposit energy in both the front and rear SSSD layers. A light fission fragment will traverse all layers of SIMBA and deposit energy in every layer. An event depositing more than 300 keV in one of the front and one rear SSSD layers was considered a background event and was rejected for implant decay correlation.
3.2 Implant-Decay Correlation
The implant and decay events were recorded independently with no explicit cor- relation between them initially. The events must be correlated in both time and space. For every implant, decay events were searched for correlated events that are within the chosen space-time hyper-volume. The chosen physical correlation area was a square of five by five pixels centered on the implantation pixel of the implant. This area was then extended onto neighboring silicon layers to ensure that β-decay events corresponding to the ions that were implanted near the surfaces of the DSSD were not lost. This forms a correlation volume around the implant. The desired volume was a cuboid, but the DSSD layers had some small offsets between each layer in the vertical and horizontal directions. Figure 3.2 shows the probability of a decay event depositing its maximum energy as a function of radius from the implant. Any further increase in correlation volume did not significantly increase the β-decay detection efficiency. A time window relative to the implant of -1 to +4 seconds was chosen. This long window ensures that the nuclei of interest would undergo a number of half-lives.
76 45
40
35
30
25 Probability of Detection [%]
20
15
10
5
0 0 0.5 1 1.5 2 2.5 3 3.5 4 Distance from Implant Pixel [mm]
Figure 3.2. Probability of maximum energy deposition of a decay event as a function of radius from the implant event. A 1/r2 line (red) has been plotted against the data for comparison. Horizontal error bars are due to mapping a Cartesian grid into spherical coordinates.
The window is extended further in both positive and negative directions to allow that the background time-dependence was well understood. Any correlated events with a negative relative time could not be from the decay of the implant and must be background. At long times following the implant the probability of background is greater than the likelihood of a β-decay. This time window allows us to clearly see background correlated with the synchrotron spill structure of 1 s spills occurring ∼
77 every 3 s. ∼ To ensure that their is no ambiguity when correlating implants and β-decays, implants with overlapping correlation hyper-volumes were rejected. In other words, if a decay could be correlated with more than one implant, all implants were rejected. The time difference between the implant and decay event were recorded. A histogram was then constructed from the time differences of each correlated event. A single implant event was allowed to have multiple decay events to account for multiple generations of decays. This histogram can then be fitted with a GOED chain to determine the parent half-life.
3.2.1 β-Decay Neutron Emission Correlation
In addition to the implant-decay correlation, we must also correlate neutron emis- sions with β-decays. The BELEN detector system made use of a large polyethylene moderator to increase the detection efficiency of the emitted neutron. This modera- tor reduces the neutron energy by successive scattering throughout the material. As the neutron energy decreases the 3He capture cross-section increases. This modera- tion has some time dependance that can be described by an exponential decay with a time constant of 50 µs. Although the neutron emission is nearly instantaneous, the neutron detection may occur a significant amount of time after the decay event. The use of the digital acquisition system with BELEN allowed for an independent detec- tion of neutrons relative to the implant and decays events. These neutron detection must then be correlated with the proper decay event. The digital modules were read out with every accepted trigger and included a time stamp for the accepted trigger.
78 To ensure that the emitted neutrons were accounted for a time gate of 200 µs after the accepted trigger was used to identify detected neutrons. Any detected neutron was then associated with the respective decay event.
3.3 Background Estimation
Understanding the β-decay background in this type of experiment is difficult due to a time varying beam structure. One would like to know the rate at which background events occur within the correlation volume during the correlation time. One frequently used method to estimate this value is to measure the number of decay-like events within the correlation volume prior to implant arrival. A constant term is then added to the decay chain fitting for background events. This type of background estimation works well at facilities where beam is provided at a relatively constant rate over multiple correlation time windows. At the GSI facility, where this experiment was performed, the beam is delivered in spills with varying time structure within each spill. Spills are typically gaussian, with possible substantial deviations. The spill length and separation can be speci- fied. For this experiment, spills were provided with 1 s length and 3 s separation. ∼ ∼ For this reason, the method of background estimation above is not sufficient as it ignores any time dependence. Instead, an alternate method was used in which the exact time structure of the spill was preserved while varying the location on the detector. In this method, we created a virtual implant concurrently with the real implant and at a random location on the detector such that the correlation volume between the real and virtual implants did not overlap. During the implant β-decay
79 correlation, an additional correlation between the virtual implant and β-decays was also performed. Since the decay events correlated to the virtual implant were outside the real correlation volume it is very unlikely the event is a real decay. This background estimate can be further improved by randomly selecting the vir- tual implant according to the implant distribution. In this way, statistically the same regions of the detector were sampled. This is important since the background events were typically caused by to light fission fragments, which have a spatial distribution across the detector created by traversing the FRS. A KDE with an adaptive bandwidth, see Section 3.4, was then generated from the background event time distribution. The KDE has the advantage of being a continuous function and spreads the probability of an event over a time period longer than the bin width permitted in a histogram generation of the same data set. The wide bandwidth for regions with low count rate allow a correct treatment of the region between the spills. With this method, we have generated a background estimation that has precisely the same time structure and statistically the same positions as the real implants. This estimate is time-dependent and reproduces the expected spill time structure. Any time independent components were also incorporated into our estimate. To avoid biasing the background estimation due to virtual implant location, the position selection was repeated and a number of distributions were created. A density plot of the various distributions is shown in Figure 3.3. It shows the uncertainty in the background estimate due to choice of virtual implant location. The decay histogram fitting was performed with a Monte Carlo technique selecting a random background
80 location to incorporate this uncertainty.
1
0.9 0.4
0.8 0.35
Probabilty Density 0.7 0.3 0.6 0.25 0.5 0.2 0.4 0.15 0.3 0.1 0.2
0.1 0.05
0 0 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Time Relative to Implant [s]
Figure 3.3. Variation of background time distribution due to location selec- tion for 127Ag. This represents the uncertainty in the background due to the choice of virtual implant location. The average probability density per time bin (blue) is overlayed.
81 3.4 Kernel Density Estimation
Kernel Density Estimation (KDE) is a part of a class of non-parametric density function estimators which includes the frequency histogram. The frequency his- togram is a non-continuous step-function defined by bin widths and heights. The heights are determined by the number of observations in a bin divided by the bin width. The KDE model was independently introduced by Rosenblatt [66] and Parzen [67] and is also referred to as the Parzen-Rosenblatt window method. A KDE is created from a set of n data points, xi, drawn from an unknown distribution f(x). The KDE is generated from a sum of kernels placed at each point xi. The estimated function fˆ(x) is written as n 1 X x xi fˆ(x) = K( − ) (3.1) nh i=1 h where K(x) is the chosen kernel and h is a smoothing parameter called the band- width. An example of a KDE is compared to a histogram in Figure 3.4. Both are generated from the same set of six observations. If the KDE is destined to be a probability density function (pdf), then the kernel must integrate to one. A number of kernels can be chosen including the typical normal, gaussian kernel. Other kernels include: uniform, triangular, cosine, quartic (bi-weight), tri-weight and Epanech- nikov [68]. The KDE inherits the properties of the kernel and depending on kernel choice may be a continuous and differentiable function. This inheritance property will also affect the smoothness. Choosing a smooth kernel will tend to smooth the entire KDE. Although the KDE is directly affected by the choice of kernel it is not
82 as critical to smoothing as the choice of bandwidth.
1 Histogram
Kernel Density Estimator 0.8 Individual Kernel
0.6 Probability Density Function
0.4
0.2
0 -3 -2 -1 0 1 2 3
Figure 3.4. Comparison of a frequency histogram (black, dotted line) and a Kernel Density Estimate (KDE), (blue solid line). The KDE is formed from the sum of individual normal kernels (red, dashed line) with adaptive bandwidths.
The term bandwidth is equivalent to the bin width when considering a histogram. A smaller bandwidth shows more detail, but can be subject to statistical noise, while
83 a large bandwidth will lead to over smoothing and a loss of detail. This trend is the same for the bandwidth of a KDE. Choosing an optimum bandwidth is the subject of a large body of research. For our purposes, we have started with the fixed bandwidth determined by Silverman’s Rule of Thumb [69], which specifies a bandwidth based on the assumption that the distribution is nearly normal. This gives the bandwidth parameter as 1/5 4 1/5 h = σn− (3.2) 3 where σ is an estimated from the provided data. This choice of bandwidth selector behaves poorly when the data is non-gaussian. For example, a bimodal distribution would not be modeled well with this global bandwidth. We can extend this definition and determine the bandwidth point-wise based on the local distribution of points by adaptive kernel estimation. We must redefine the estimated distribution fˆ(x) as an adaptive distribution fˆa(x).
n 1 X 1 x xi fˆa(x) = K( − ) (3.3) n i=1 hi hi
Where the bandwidth, hi, is now a variable quantity dependent on the data and point i and its neighbors. Abramson [45] proposed an adaptive bandwidth parameter given by the following expression : q hi = h/ f(xi) (3.4)
This relationship provides a adaptive bandwidth such that regions with high-density will have a small bandwidth and those with low density will require wide bandwidths to smooth out statistical fluctuations. This requires knowledge of the optimum band-
84 width, h, and the true density distribution, f(x). We start with the fixed bandwidth choice as our best guess giving a point-wise bandwidth of 1/5 s 4 σ 1/5 hi = ρ n− (3.5) 3 fˆ(xi) where the term ρ has been introduced to solve an issue in which the tails influence is overestimated may over smooth the kernel. This term can be ignored, set to unity, unless the standard deviation of the local structure, σlocal, is smaller than two orders of magnitude than the global structure, σ. This gives the following relationship for rho 2 1 σ/σlocal 10 ρ = ≤ (3.6) q σlocal 2 σ σ/σlocal > 10
This method removes almost any dependence on the original choice of bandwidth in the fixed kernel method as shown by Abramson [70]. With the adaptive kernel bandwidth multi-modal distribution can be correctly modeled. The discussion of other methods for optimum bandwidth determination are be- yond the scope of this text. For further discussion on bandwidth selection and of Kernel Density Estimators in general refer to Scott [71], Wand & Jones [72] and Härdle et al. [73]. We have shown that the KDE method makes some gains over frequency his- tograms such as smoothness. Both non-parametric models are positive definite. KDEs are more computational intensive, but unlike frequency histograms KDEs are bin-independent, continuous and defined everywhere. The method is insensitive to kernel choice. For our purposes we make use of a gaussian kernel and have provided
85 a simple method for estimating an optimum bandwidth.
3.5 Differential Decay Solution
To determine the half-life of a radioactive species, one must solve the system of differential equations that describe the decay chain
dNi(t) = λiNi(t) + λi 1Ni 1(t) (3.7) dt − − −
where λi is the decay constant for species i, and Ni(t) is the number of nuclei at time t. The first term corresponds to the destruction of species i with λi as the destruction decay constant. The second term corresponds to the production of species i due to the decay of species i 1. Traditionally, the Bateman equation[74] has been used as − the solution of this system of equations. The simplest case to consider is one in which the initial composition, Ni(t = 0), is purely the first species such that Ni=0(0) = 0. 6 Presented here is Bateman’s solution to that case:
i X λj t Ni(t) = N0(0) cje− j=1 (3.8) j Y λk c = j λ λ k=1 k − j
This decay chain and solution consider the situation in which there is no external production of activity and species i decays only to a single child, species i + 1, with no branching during decays. Although there is no external production of activity during the previously described experiment, there is a significant branching fraction.
86 This requires modification of Equation 3.7 to properly treat the branching ratios. We need to account for the fact that a child can be produced from multiple parents and adjust the production term accordingly. We add the branching ratio, ρi,j, describing the fraction of decays from species i j. This modification gives: →
m dNi X = λiNi + ρj,iλjNj (3.9) dt − j>i
To correctly solve this, we have adopted the methodology of Genealogically Ordered Exit-Only Decay (GOED) chains introduced by Yuan & Kernan [75]. A GOED is sorted with the parents being the first isotopes, having the lowest index, followed by its children, grand-children and so on. External production is not allowed thus the chain is exit-only. The GOED chain can be solved with a recursive algorithm as follows:
i X λit Ni(t) = Ci,je− j=1 (3.10) Pi 1 Ni(0) k−=1 Ci,k i = j Ci,j = − 1 Pi 1 − ρ λ C i = j λi λj k=j k,i k k,j − 6
This relationship properly treats the decay of a single species into multiple children and the production of a single child via multiple parents. It is required that each branching ratio is between zero and one
0 ρ 1 (3.11) ≤ i,j ≤
87 and that the branching ratios for a single species sum to one.
n X ρi,j 1 (3.12) j=i+1 ≡
The latter condition is often difficult to implement into a fitting routine as limits are often applied only to single parameters. To more easily enforce Equation 3.12 we used the following descriptions for the branching ratios:
0 j i ≤ ρ Pn i,j = 1 k=i+1,k=p ρi,k j = p (3.13) − 6 Pj 1 fi,j 1 k−=i+1,k=p ρi,k j = p − 6 6 where p is the index of the primary branch, chosen for convenience, and fi,j is a scaling factor where 0 f 1. This allows simple single parameter limits to ≤ i,j ≤ be set on fi,j, that enforce a limit on the sum of branching ratios, Equation 3.12.
This definition of ρi,j can be shown to obey Equation 3.11 through mathematical induction. First, we chose the case j = p and j = i + 1 6
i X ρ = f 1 ρ i,i+1 i,i+1 − i,k k=i+1 (3.14)
= fi,i+1 since we require 0 f 1, we must have 0 ρ 1. We now examine the case ≤ i,j ≤ ≤ i,i+1 ≤
88 of j = p and j = (i + 1) + 1 6
i+1 X ρ = f 1 ρ i,i+2 i,i+2 − i,k k=i+1 (3.15) = f (1 ρ ) i,i+2 − i,i+1
Again, we required that 0 f 1 and we have previously shown that 0 ρ ≤ i,j ≤ ≤ i,i+1 ≤ 1. Using these two statements, it can be shown that 0 ρ 1 therefore, ≤ i,(i+1)+1 ≤
0 ρ 1 j = p (3.16) ≤ i,j ≤ ∀ 6
For the case where j = p, we choose p = i + 1 and first examine the minimal case n = i + 2.
n X ρi,p = 1 ρi,k − k=i+1,k=1 6 i X (3.17) = 1 +2ρi,k − k=i+2 = 1 ρ − i,i+2
As shown above in Equation 3.16, we have 0 ρi,j=p 1 and therefore 0 ρi,p ≤ 6 ≤ ≤ ≤
89 1 for n = i + 2. For n = (i + 2) + 1 we find the following:
n=i+3 X ρi,p = 1 ρi,k − k=i+2 = 1 ρ ρ − i,i+2 − i,i+3 (3.18) = 1 ρ f (1 ρ ) − i,i+2 − i,i+3 − i,i+2 = 1 ρ f + f ρ − i,i+2 − i,i+3 i,i+3 i,i+2 = (1 ρ ) (1 f ) − i,i+2 − i,i+3
Since we require 0 fi,j>i 1 and have shown 0 ρi,j=p 1, it must be true that ≤ ≤ ≤ 6 ≤ 0 ρ 1 for n = (i + 1) + 1 therefore, ≤ i,j=p ≤
0 ρ 1 n (3.19) ≤ i,p ≤ ∀
Equation 3.16 and Equation 3.19 shows that the our definition of ρi,j, Equa- tion 3.13, satisfies the requirement that the branching ratio is between 0 and 1, Equation 3.11. This definition must also satisfy Equation 3.12. This can be shown by simply separating the sum into the terms from primary and secondary branches:
n n X X ρi,j = ρi,p + ρi,k j=i+1 k+1,k=p 6 n n X X (3.20) = 1 ρi,k + ρi,k − k+1,k=p k+1,k=p 6 6 = 1
We can now use this definition of the Pn values to enforce that the sum of decay
90 branching ratios of one species is equal to one. This allows us to extend the decay chain from two daughters per decay to an arbitrary number. This definition also permits setting limits on known branching ratios. Using this in conjunction with the recursive solution to the GOED chains provides a parametric model, which can be used to extract half-life values.
3.6 Half Life Analysis
To determine the half-lives of the radioactive species implanted into the SIMBA detector a fit to a histogram of decay times was performed. The time of each decay was extracted as the time difference of an event clock between the implant and decay event. These decays time were then used to generate a histogram in which multiple decay times per implant were permitted. This allows for a fit to be performed with components from each subsequent generation. The fitting was performed with a Monte Carlo method to properly account for systematic uncertainties in both the daughter decay properties and the background determination. The adopted GOED decay model was discussed above in Section 3.5. The model permitted the parent, (Z, N), to decay to a daughter, (Z + 1, N 1), − and step-daughter, (Z + 1, N 2), the species created from a β-decay and neutron − emission. Each daughter was permitted to repeat this and decay into its daughter and step-daughter. Each subsequent decay half-life was chosen from a gaussian distribution centered at the adopted literature value with a width corresponding to the 1σ uncertainty. Another gaussian distribution was used for the neutron branching ratio of the offspring species.
91 An additional component for the background was included. In most cases, there were sufficient statics to determine the background component without the need for a scaling factor. The background determination random generator seed was varied during the Monte Carlo analysis. This resulted in choosing virtual implants from different regions if the detector producing slight variations in the background time dependence. In this way, the systematic uncertainty due to virtual implant choice is determined. The best decay curve fits with a randomly chosen background seed are shown for the palladium cases in Figure 3.5 and Figure 3.6 and for the silver case in Figure 3.7 and Figure 3.8.
The β-detection efficiency, β, can be determined by the ratio of the number of detected parent β-decays, Nβ, to the number of implant events, Nimplants. The determination of Nβ is performed in the fit using the following relation:
N N t N = 0 = 0 1/2 (3.21) β λ∆t ln(2) ∆t
where N0 is the initial activity found from the fit minimization and ∆t is the bin width of the fitted histogram. We then find the efficiency as
Nβ 1 N0t1/2 β = = (3.22) Nimplants Nimplants ln(2) ∆t
The β-detection efficiency for each isotopic case is shown in Figure 3.9. The average efficiency of SIMBA is roughly 60 %. The figure shows a drop in efficiency for 123Pd and 124Pd. The chosen FRS settings physically centered more neutron-
92 123Pd 25 123Pd 10 20 15 1 10
Counts /0.1 50 ms Counts / 505 ms
100 Time124 (sec)Pd 80 Time124 (sec)Pd
60 10 40 1 20 Counts / 50 ms Counts / 50 ms 0.1 Time125 (sec) Time125 (sec) Pd 50 Pd 10 40 30 1 20 Counts / 25 ms Counts /10 25 ms 0.1 -1 0 1 2 3 4 -1 0 1 2 3 4 Decay Time [s] Decay Time [s]
123 125 Figure 3.5. Decay curve histograms (grey) are fit for − Pd cases shown in logarithmic (left) and linear (right) scales. The fit function (black) is com- prised of a genealogically ordered exit-only decay chain (green) accounting for components (red) from the parent generation (red, solid line), daugh- ters (dashed line), grand-daughters (dotted line) and great grand-daughters (dashed-dotted line) as well as a background component (blue).
93 126 126 Pd 25 Pd 10 20
1 15 10
Counts / 250.1 ms Counts / 255 ms 10 Time127 (sec)Pd 6 Time127 (sec)Pd
1 4
0.1 2
Counts /0.01 12.5 ms Counts / 12.5 ms Time128 (sec)Pd Time128 (sec)Pd 3 1
2 0.1
1 Counts / 25 ms 0.01 Counts / 25 ms
-1 0 1 2 3 4 -1 0 1 2 3 4 Decay Time [s] Decay Time [s]
126 128 Figure 3.6. Decay curve fits for − Pd shown in logarithmic (left) and linear (right) scales. Fit function (black) is comprised of decay components (red) from the parent generation (red, solid line), daughters (dashed line), grand-daughters (dotted line) and great grand-daughters (dashed-dotted line) as well as a background component (blue). The background subtracted fit (green) is plotted for comparison.
94 124Ag 50 124Ag 40 10 30 1 20
Counts / 50 ms 10 0.1 Time125 (sec) 1500 Time125 (sec) 1000 Ag Ag
1000 100
500 10 Counts / 50 ms
Time126 (sec)Ag Time126 (sec)Ag 1000 1500
100 1000
500 Counts / 50 ms 10
-1 0 1 2 3 4 -1 0 1 2 3 4 Decay Time [s] Decay Time [s]
124 126 Figure 3.7. Decay curve fits for − Ag shown in logarithmic (left) and linear (right) scales. Fit function (black) is comprised of decay components (red) from the parent generation (red, solid line), daughters (dashed line), grand-daughters (dotted line) and great grand-daughters (dashed-dotted line) as well as a background component (blue). The background subtracted fit (green) is plotted for comparison.
95 127Ag 400 127Ag 100 300
10 200 100 Counts / 25 ms 1 100 Time128 (sec) Time128 (sec) Ag 60 Ag
10 40
1 20 Counts / 25 ms 0.1 129 129 10 Time (sec)Ag 8 Time (sec)Ag
1 6 4 0.1 2 Counts / 12.5 ms 0.01 -1 0 1 2 3 4 -1 0 1 2 3 4 Decay Time [s] Decay Time [s]
127 129 Figure 3.8. Decay curve fits for − Ag shown in logarithmic (left) and linear (right) scales. Fit function (black) is comprised of decay components (red) from the parent generation (red, solid line), daughters (dashed line), grand-daughters (dotted line) and great grand-daughters (dashed-dotted line) as well as a background component (blue). The background subtracted fit (green) is plotted for comparison.
96 rich species onto the DSSD. These two cases have implant positions on the edge of the DSSD detectors β particles may be emitted such that do not interact with the silicon layers. Thus the efficiency should decrease drastically at as the mass number decreases.
100
90
80 Efficiency [%]
70
60
50
40
30
20
10
0 123 124 125 126 127 128 129 Mass Number (A)
Figure 3.9. β-detection efficiency as a function of mass number for palladium (red) and silver (blue) isotopes.
The final determination of the half-life was drawn from the mean value of the
97 Monte Carlo results. The upper and lower statistical uncertainties varied for each Monte Carlo run and final values were chosen in the same manner. The systematic uncertainty was then determined by the width of the resulting half-life distribu- tion. The resulting values are tabulated and compared against literature values in Table 3.1. In addition, a comparison between the half-lives from this work and theo- retical values from Fang et al. [76], Niu et al. [77], Borzov [78], and Möller et al. [48] is shown in Table 3.2.
98 TABLE 3.1
COMPARISON OF MEASURED HALF-LIVES TO PREVIOUS EXPERIMENTAL VALUES
Isotope This Work** Montes et al. [50] Kratz et al.[79–81] Wohr et al. [82] Fedoseyev et al. [83] Hill et al. [84] Reeder et al. [85]
123 +45 +38 Pd 170 38(18) 174 34 − − 124 +17 *** +38 Pd 144 15(18) 38 19 − − 125 +8 Pd 61 7 (1) − 126 +11 Pd 56 9 (2) − 99 127 +21 Pd 73 19(12) − 128Pd < 262
124 +27 **** +15 Ag 230 24 (6) 187 15 172 (5) 170(3) 540(80) − − 125 +9 Ag 163 6 (6) 166 (7) − 126 +3 Ag 114 3 (1) 107(12) − 127 +4 Ag 102 4 (3) 109(25) 3690(50) − 128 +10 Ag 73 9 (3) 58(5) − 129 +34 +5 Ag 95 27 (9)† 46 9 − − * All half-lives reported in milliseconds. ** +stat Uncertainties listed: −stat(sys). *** Possible mixture of half-lives from the ground state and isomer identified by Kameda et al. [86]. **** Possible mixture of half-lives from ground state and isomer identified by Kautzsch et al. [87]. † Possibile mixture of half-lives from the ground state and of isomer identified by Kratz et al.[79–81]. 300
250
200
150
100
Palladium Half Lives [ms] 50
123 124 125 126 127 128 129 250 AtomicThis Work mass Number (A) Literature Value QRPA, Moller 2003 200 QRPA, Fang 2013 QRPA, Borzov 2003 150 QRPA, Niu 2013
100
Silver Half Lives [ms] 50
123 124 125 126 127 128 129 Atomic Mass Number (A)
Figure 3.10. Comparison of half-lives measured in this work (blue) versus previous literature values (black), Table 3.1, as well as theoretical values, Table 3.2.
We report the first half-life measurements for 125Pd, 126Pd and 127Pd. Other reported values agree with recent literature measurements, except for two deviating cases: 124Pd and 129Ag. We also find that the theoretical calculations by Möller
100 et al. [48] overestimate the palladium half-lives. For the silver half-lives, the the- oretical calculation seems to initially under estimate the half-lives, for the case of 124Ag and 125Ag. We then observe a trend similar to that seen with the experimental values, except for 129Ag. This discrepancy may be due to a possible isomeric state, which is explained below. The new values for palladium isotopes will have the effect of increasing the speed of the r-process as the flow can surpass the waiting-point more quickly. This will tend to increase the abundances of heavier nuclei and decrease that of lighter nuclei. This effect may help resolve the large under-abundances observed just prior to the A = 130 mass peak.
101 TABLE 3.2
COMPARISON OF MEASURED HALF-LIVES TO THEORETICAL CALCULATIONS
Isotope This Work** Fang et al. [76] Niu et al. [77] Borzov [78] Möller et al. [48]
123 +45 Pd 170 38(18) 187.3 143 396.64 − 124 +17 *** Pd 144 15(18) 138.5 114.6 105 288.69 − 125 +8 Pd 61 7 (1) 104.5 76 265.44 − 126 +11 Pd 56 9 (2) 83.62 70.9 76 221.69 − 127 +21 Pd 73 19(12) 62.38 51 210.12 − 128Pd < 262 33.08 48.55 43 74.23
124 +27 **** Ag 230 24 (6) 171 111.36 − 125 +9 Ag 163 6 (6) 124.6 157 110.79 − 126 +3 Ag 114 3 (1) 98 109.99 − 127 +4 Ag 102 4 (3) 61.25 90 79.97 − 128 +10 Ag 73 9 (3) 60 79.64 − 129 +34 Ag 95 27 (9)† 29.46 56 31.66 − * All half-lives reported in milliseconds. ** +stat Uncertainties listed: −stat(sys). *** Possible mixture of half-lives from the ground state and isomer identified by Kameda et al. [86]. **** Possible mixture of half-lives from ground state and isomer identified by Kautzsch et al. [87]. † Possibile mixture of half-lives from the ground state and of isomer identified by Kratz et al.[79–81].
102 3.6.1 124Pd Half-Life
The discrepancy observed for 124Pd between our measurement and the work done by Montes et al. [50] may be explained by the existence of a long lived isomer. Our production method at higher energies may be able to access this isomeric level. This may result in the measurement of decays from a mixture of two states, the ground- state and the long-lived isomer. A possible isomeric state was identified by Kameda et al. [86]. They were only sensitive to half-lives up to 20 µs and were only able to provide this as a lower limit. A possible explanation for this isomer can be found in nearby even-even isotopes. A trend is observed in which the 7- level excitation energy drops with increasing neutron number while that of both the 6+ and 5- increase, see Figure 3.11. When the 7- level drops below the 6+ and 5- level the possibility of a M1 or E2 transition is lost. The next most likely transition is an E3 transition from 7- to 4+. Examining the specific cases in the tin isotopes we find 126Sn has a 7- isomer with half-life, 6.6 µs, that decays via an E2 transition to the available 5- level [88]. Moving to more neutron-rich nucleus, 128Sn, we find the 7- level has dropped below the 5- level [88, 89]. The next accessible transition is an E3 transition to the 4+ level. This causes the 7- isomer half-life to increase significantly to 6.5 s. We see a similar trend in the palladium isotopes although data is unavailable for the most neutron-rich nuclei. Assuming that the measured half-life is purely the isomeric state we can estimate the excitation energy, Ex, of the 7- state. The
103 3000
2500
2000 Excitation Energy [keV]
1500
1000
Pd 7- Pd 6+ Pd 5- Pd 4+ 500 This Work Cd 7- Cd 6+ Cd 5- Cd 4+ Montes et al. Sn 7- Sn 6+ Sn 5- Sn 4+
0 64 66 68 70 72 74 76 78 80 82 84 Neutron Number (N)
Figure 3.11. Excitation energies for levels with spin 7- (solid line), 6+ (dashed line), 5- (dotted line) and 4+ (dash-dotted line) as a function of neutron number are shown for even-N isotopes of palladium (blue), cadmium (red) and silver (yellow). Calculated Weisskopf estimates of the required en- ergy for a 7- isomer in 124Pd with half-life from this work (circle) and Montes et al. [50] (square). This proposed isomer will not become significant unless the 7- level lies below both the 6+- and 5- level as in 128Sn (N = 78).
Weisskopf estimate for an E3 transition is
14 2 7 λW (E3)~ = 2.3 10− A Eγ (3.23) ×
104 π π E (MeV)J T E (MeV)J T x - 1/2 x 1/2 2.219 7 6.6(14) us ? 2.194 ?
- 2.162 5 10.8(7) ns
? - 2.130 ? 2.121 (5 ) 8.6(8) ns + (+) 2.104 (2) 2.111 2 - 2.091 (7 ) 6.5(5) s
+ 2.050 4
+ 2.000 (4 )
+ + 1.169 2 1.141 2
+ + 0.00 0 0.23(14) My 0.00 0 59.07(14) m 126 128 Sn Sn
Figure 3.12. Level schemes of 126Sn and 128Sn. The 7- state drops below the 5- state in 128Sn increasing the isomeric half-life.
Replacing the decay constant with ln(2)/t1/2 and solving for Eγ we find:
!1/7 ln(2)~ E = (3.24) γ 2.3 10 4A2t × − 1/2
This gamma energy plus the recently measured excitation energy of the 4+ state,
Ex(4+) = 1300(22) keV [90], allows us to estimate the required 7- excitation energy. Assuming the half-life from this work is completely due to the isomeric state we find E (7 ) = 1332 keV. Assuming Montes et al. [50] measured only the isomeric x −
105 state we find E (7 ) = 1326 keV. Both of these values are reasonably close to the x − measured line at Eγ = 62.2(16) keV from Kameda et al. [86].
3.6.2 129Ag Half-Life
For the 129Ag case, Kratz et al.[79–81] indicated that a long-lived isomer may exist. They suggest a πp1/2 isomer and using a QRPA model, which accounted for first forbidden decays, calculate a half-life value of 125 ms. After performing a reexamination of their experimental measurement, they proposed a rough value for the isomeric half-life of 160 ms. In this work, a fit considering the isomer was performed. The ground-state half-
+5 life was fixed to the reported value of 46 9 ms from Kratz et al.. The initial pop- − ulation was taken as a mixture of both the ground-state and isomeric-state. The
+90 isomeric half-life was extracted as 130 50(20) ms. This value is in agreement with − the 160 ms value suggested by Kratz et al..
3.7 Detection Limits
In measurements with low counting statistics it is important to define the de- tection limits. The most widely used criteria to specify these limits was described by Currie [91]. First, one must decide if the measurement of a sample indicated a detection, this level is known as the critical level. At the critical level the number of false-positives should be kept small. A typical measurement of this type will provide the number of measured events, Nm, and the number of background events, Nb. The number of signal events would then be, N = N N . s m − b
106 For multiple measurements of zero-activity, N¯m = N¯b, and Ns will follow a gaus- sian distribution centered at 0 with a width of
q 2 σs=0 = 2σb (3.25)
We would like to define a critical level, Lc, such that 95% of the measurements are below Lc indicating a zero activity measurement. Since the distribution is gaussian we find the critical limit
Lc = 1.645σs=0 = 2.326σB (3.26) will provide a false-positive probability of less than 5%. The second limit of interest ensures that the probability of false-negative mea- surements is minimized. This limit suggests a minimum detectable activity that could be achieved for a measurement. This limit of detection, Ld, would be chosen such that a distribution of Ns would be centered at Ld with a width σs>0 providing the probability of a measurement below Lc is less than 5%. This requires that
Ld = Lc + 1.645σs>0 (3.27)
If we first take the approximation σs > 0 = σs = 0 = √2σb we find
Ld = Lc + 1.645√2σb
= Lc + 2.326σb (3.28)
= 4.653σb
107 This first approximation can be improved by using the correct form of σs>0, defined by the width of the distribution of the measured signal and the background signal as
q σs>0 = Nm + Nb (3.29) q = Ns + 2Nb
Expanding for N N and using the first approximation for L from Equation 3.28 s b d we have
q Ns σs>0 ∼= 2Nb 1 + 4Nb q 4.653σb = 2Nb 1 + (3.30) 4Nb
= √2σb + 1.645
Plugging this and Equation 3.26 into Equation 3.27 we find the Currie Equation:
Ld = 4.653σb + 2.706 (3.31)
This limit can be used as the minimum number of counts needed to ensure a signal is not a false-negative when the critical limit of activity is Lc.
3.8 Determination of Neutron Branching Probabilities
The probability of neutron branching, Pn, is determined by a simple ratio be- tween the number of events detected with only a β-decay, Nβ, and those with a
108 correlated β-decay and neutron emission, Nβ,n. Correcting for the number of back- ground correlations, Bβ,n, and the neutron detection efficiency, n, gives the following relation:
(Nβ,n Bβ,n) f Pn = − (3.32) Nβ n
Where f is a small correction due to a counting bias, explained below. Making use of the Monte Carlo simulations performed for the half-life analysis (Section 3.6) we have already determined a distribution of the number of detected β-decays from each performed fit. The β-detection efficiency, β, drops out as both Nβ and Nβ,n are corrected by this value. This leaves only the determination of values related to the neutron detection. After performing the correlation of detected neutrons and β-decays (Section 3.2.1) a simple accounting of the number of events with a β-neutron correlation was per- formed. The β-neutron correlated events were counted within a time frame of ten half-lives after the implant event. This count was repeated for each Monte Carlo run, as the half-life varied, providing a distribution of Nβ,n. In addition, we find the number of background events, Bβ,n, with correlated neutron emission within the same 10 t1/2 time window. Knowing the width of the background distribution, σB, we then apply the Currie equation, Equation 3.31, to determine that the distribu- tion of N B must be above the critical limit, L , to allow for a quantitative β,n − β,n d analysis. This leaves only the neutron detection efficiency unknown. An efficiency mea- surement was performed with a 252Cf neutron emission source. 252Cf decays by spontaneous fission emitting on average 3.735 neutrons per fission [92]. The source
109 provided a known activity, but unfortunately emitted neutrons at a large range of energies up to 15 MeV [93]. This exceeds the linear region of the detector efficiency.
By using previously measured Pn values and Geant4 simulations [65] an efficiency could be determined. A similar detection system was used to previously perform measurements of well known branching ratios. A Geant4 simulation of these measurements was performed showing good agreement to experimental results. Another Geant4 simulation com- paring an efficiency run with a 252Cf source and its large neutron energy distribution in the same system was performed and also showed good agreement. Using the new BELEN-30 configuration, another set of simulations were performed. The first of a 252Cf source matched our experimental measurement of the neutron detection
252 efficiency, n( Cf) = 40.1(4)%. Knowing that we find good agreement with the simulations we can adopt the simulation’s value of neutron detection efficiency in the linear region, n = 40(2)%. Finally, a small correction was necessary due to a bias toward detecting events with smaller Q-values. This bias is an artifact of the inability to set a non-zero threshold in the DSSD detectors of SIMBA. A β-decay to the a level of the daughter below the neutron separation energy will emit an electron with larger energy than a decay to a level above the neutron separation energy. Energy deposition into the strips of the DSSD is inversely proportional to the total electron energy, see Equation 2.4. Thus events emitting neutrons are more likely to deposit an energy above the DSSD threshold than an event not emitting a neutron. This bias was corrected by performing simulations that were not restricted to a
110 non-zero bias. Using the Geant4 simulation tool kit [65], the β-decays were simulated across the DSSD surface reproducing the implant position profile. The maximum energy deposited within the correlation volume was then determined and plotted in a histogram, see Figure 3.13. The simulations were repeated for multiple Q- values covering a range from 2-13 MeV. The ratio of the number of events above the threshold between the β-decay at the Qβ and Qβ,n then provides a correction value, f, as a function of threshold energy. In addition, a decay to an excited state above the neutron separation energy is unlikely to occur exactly at the separation energy.
An additional 500 keV was added to the Qβ,n to account for the additional excitation energy of this level above the separation energy. Plots of the resulting correction factors for each isotopic case are shown in Fig- ure 3.14 and Figure 3.15. The correction factor has some uncertainty due to the unknown Q-value. We have adopted the uncertainties from the Atomic Mass Evalu- ation 2012 [16] extrapolations. Finally we can determine the correction for our DSSD thresholds set near 150 keV. After establishing the efficiency and the bias correction factor, we can apply
Equation 3.32 and find a distribution of Pn values. This distribution indicates the most likely Pn value, which was used as the adopted value. The statistical uncer- tainty was calculated in quadrature from √N uncertainties for both the number of detected correlated β-neutrons, Nβ,n, and the background, Bβ,n. The uncertainty from the efficiency and the bias correction was also considered. The width of the Pn distribution is a consequence of the Monte Carlo simulations of variations of known daughter parameters and thus are related to the systematic uncertainty. Including
111 0.4 Q = 3 MeV 0.35 Q = 5 MeV
0.3 Counts / 1 keV [%] 0.25
0.2
0.15
0.1
0.05
0 0 50 100 150 200 250 300 350 400 450 500 Maximum Energy Deposited in a Single Pixel within Correlation Volume [keV]
Figure 3.13. Histogram of Deposited Energy within Correlation Volume for various Q-values. In our region of interest the β-decay values are larger than the neutron separation energies giving a positive β-delayed neutron emission Q-value. The change in Q-value between a decay to ground-state or an excited change above the separation energy shifts the electron energy spectrum. More electrons are detected at low energy, less than 150 keV, at a lower Q-value, decay to higher lying energy state (blue) than at a higher Q-value, decaying to ground-state (red).
the width of the distribution in the uncertainty accounts for systematic uncertainties generated from the half-life determination. The parameters relevant to determina- tion of the Pn values, Nimpalnts, Nβ, Nbeta,n, and Bbeta,n are tabulated in Table 3.3.
The resulting Pn values are presented in Table 3.4.
112 1.05 1.05 1.05
1 1 1
0.95 0.95 0.95 Pd Correction Factor Pd Correction Factor Pd Correction Factor
123 0.9 124 0.9 125 0.9 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Energy Threshold [keV] Energy Threshold [keV] Energy Threshold [keV]
1.05 1.05 1.05
1 1 1
0.95 0.95 0.95 Pd Correction Factor Pd Correction Factor Pd Correction Factor
126 0.9 127 0.9 128 0.9 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Energy Threshold [keV] Energy Threshold [keV] Energy Threshold [keV]
Figure 3.14. Palladium neutron emission probability correction factors as a function of threshold energy. The DSSD detectors are biased toward detec- tion of β correlated with neutron emission due to their lower energy. Geant4 [65] simulations were performed to determine the necessary correction fac- tors. Each panels shows a correction for a specific isotope of palladium. The gray band shows the variation in correction factor due to uncertainty in Q-value.
123 126 125 128 We report the first measurements of the Pn values for − Pd and − Ag.
Due to low statistics we were only able to assign upper limits to the Pn values for
127 124 129 Pd, Ag and Ag. The only previously measured Pn value in this region was 127Ag from Montes et al. [50]. Our upper limit for this isotope agrees with their reported value.
113 1.05 1.05 1.05
1 1 1
0.95 0.95 0.95 Ag Correction Factor Ag Correction Factor Ag Correction Factor
124 0.9 125 0.9 126 0.9 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Energy Threshold [keV] Energy Threshold [keV] Energy Threshold [keV]
1.05 1.05 1.05
1 1 1
0.95 0.95 0.95 Ag Correction Factor Ag Correction Factor Ag Correction Factor 129 127 0.9 128 0.9 0.9 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Energy Threshold [keV] Energy Threshold [keV] Energy Threshold [keV]
Figure 3.15. Silver neutron emission probability correction factors as a func- tion of threshold energy. The DSSD detectors are biased toward detection of β correlated with neutron emission due to their lower energy. Geant4 [65] simulations were performed to determine the necessary correction factors. Each panels shows a correction for a specific isotope of silver. The gray band shows the variation in correction factor due to uncertainty in Q-value.
We find that the theoretical estimates of Möller et al. [48] underestimate the neutron branching ratios for all cases in both palladium and silver. The use of our new values over those of Möller et al. [48] will tend to smooth the odd-even effect observed in the final abundances determined by r-process models.
114 TABLE 3.3
NUMBER OF IMPLANTS, Nimplants
DETECTED β-DECAY, Nβ
NUMBER OF CORRELATED β-NEUTRON EVENTS, Nβ,n
NUMBER OF BACKGROUND CORRELATED β-NEUTRON, Bβ,n
Isotope Nimplants Nβ Nβ,n Bβ,n 123Pd 228 78 7 3
124Pd 777 255 27 8
125Pd 420 190 14 4
126Pd 115 73 8 1
127Pd 33 23 1 0
124Ag 388 233 10 4
125Ag 12344 6728 484 143
126Ag 10267 6405 504 124
127Ag 3563 2238 177 43
128Ag 361 213 20 3
129Ag 24 1 1
115 TABLE 3.4
RESULTING NEUTRON BRANCHING RATIO, Pn
Pn VALUES FROM LITERATURE
Experimental* Theoretical* Isotope This work Montes et al. [50] Fang et al. [76] Borzov [78] Möller et al. [48]
123Pd 10 (6) 1.5 0.034 0.35
124Pd 17 (5) 2.8 0.075 0.77
125Pd 12 (4) 2.5 0.14 2.91
126Pd 22 (9) 5.7 0.75 3.98
127Pd < 19 4.6 4.01 8.15
124Ag < 9 1.3(9) 0.895 1.26
125Ag 11.8(10) 4.8 1.8 4.10
126Ag 13.7(11) 3.12 3.99
127Ag 14.6(15) 5.5 4.3 6.91
128Ag 20 (5) 9.07 7.14
129Ag < 20 6.8 7.1 9.84 * All branching ratios reported as percentages.
116 35 This Work 30 Literature Value QRPA, Moller 2003 25 QRPA, Fang 2013 QRPA, Borzov 2003 Values [%] 20 n P 15 10 5 Palladium
123 124 125 126 127 128 129 25 Atomic mass Number (A)
20
15 Values [%] n P 10
Silver 5
123 124 125 126 127 128 129 Atomic Mass Number (A)
Figure 3.16. Comparison of half-lives measured in this work (blue) versus previous literature values (black) as well as theoretical calculations, Table 3.4
117 CHAPTER 4
R-PROCESS IMPLICATIONS
4.1 High Entropy Wind Model
To explore the impact of these half-life and branching ratio measurements an astrophysical model of the high entropy wind conditions thought to be found near the surface of the proto-neutron star in a core collapse supernova, as described in Section 1.4.2.2, was used. As a first look at the impact we use a simplified para- metric model described by [36, 94, 95]. This model follows a number of adiabatic expansions described by an entropy per baryon, S, an initial electron fraction, Ye, and an expansion velocity v. A full reaction network is coupled to this model which includes β-decay, neutron capture and photo-dissociation reactions. The network neglects the neutrino captures and fission. The astrophysical parameters are tuned to reproduce an r-process. Each expansion used fixed realistic values of Ye = 0.45
1 and v = 7500 km s− . A set of equidistant entropy values were used with values ranging from S/k = 10 260 in steps of 10. Although these values are larger than − may be expected in a supernova they have been shown to reproduce a reasonable r-process [36, 95]. β-decay rate and Pn values were chosen from experimental values where available and QRPA calculations otherwise [48]. Neutron capture reaction
118 rates were chosen from the NON-SMOKER statistical model [96] using the FRDM mass model [29]. The astrophysical parameters of the model were chosen to reproduce the r-process peak found at A = 130. A comparison of the final abundances from to model compared to solar abundances is shown in Figure 4.1. This simple model reproduces the expected peaks at A = 130 and 195, but under produces the rare-earth element peak.
125 127 To understand the impact of the new experimental measurements of − Pd half-lives a series of calculations were performed, shown in Figure 4.2. First, the nominal calculation using all previous theory values was performed. The half-lives corresponding to the isotopes with new measurements were then varied up and down by a factor of five. This factor was adopted as a reasonable uncertainty in the theoretical half-life. This provided a band in which the final abundances may lay with the uncertainty from the nuclear theory. The calculations were then repeated with the new half-life values and their upper and lower limits. We observe an impact in the A = 126 to 129 region. The uncertainty in the final abundances decrease from 40% to 10%. We also observe that the nominal value shifted by 10% reducing the under and over production at A = 126 and 128 respectively. To understand the impact of the 128Pd upper limit another set of calculations similar to the previously described set were performed, shown in Figure 4.3. Again the theory values were increased and decreased by a factor of five. With only a measured upper limit available we performed a single calculation at that limit. This only slightly reduced the uncertainty in the half-life and the impact in the r-process
119 102 Solor r-Process Abundances
Abundance 10 High Entropy Wind Model
1
1 10−
2 10−
3 10− 40 60 80 100 120 140 160 180 200 220 Atomic Mass Number (A)
Figure 4.1. A simple parametrized model of the high entropy wind conditions thought to be found during core collapse supernovae (black line) is compared to the solar r-process abundances (grey points) [6, 22].
abundances was also small.
Finally, the impact of newly measured neutron branching ratios, Pn values, on the r-process was evaluated. A set of calculations, Figure 4.4, where the theoretical
123 126 125 128 Pn values of isotopes with first time Pn measurements, − Pd and − Ag, were increased to 100% and decreased to 0% were performed to provide an initial band of possible r-process abundances. Using the new measurement values and their uncertainties a new band was determined. We observed an impact in the mass region of the new Pn values as well as for lighter abundances from A = 111 to 128. We
120 found a significant constraint on the possible final abundance pattern with the new experimental values reducing the largest abundance uncertainty from a factor of 6 to 0.4. The nominal abundance value was found to increase for A = 122 125 and − decrease for A = 125 129. −
121 rybn)cmae otoecluae ihtertclvalues. theoretical with (dark uncertainty calculated the those in to line) reduction compared red a factor show band) (dashed limits grey upper a line) corresponding blue by the (dashed from line) lower and and measurements line) blue black experimental (solid (dashed new work with decreased this determined and Abundances line) five. red of (solid increased were stpswt rttm aflf esrmnsfo hswork, this from for measurements half-lives cal- (bottom half-life theoretical was shown time The which are first line) values. ratios with black theoretical addition, (solid isotopes nominal calculation In the baseline using the culated band). to grey relative (light pane) shown is lives on wind entropy effect high solar The a to from compared determined (lines) pane) model (upper Abundances 4.2. Figure
Ratio 10 Abundance 0.8 1.2 1.4 − 1 1 1 105 r poesaudne rmucranisi hoeia half- theoretical in uncertainties from abundances -process 110 115 r ge points). (grey ] 22 [6, abundances -process 122 120 tmcMs ubr(A) Number Mass Atomic 125 hsWork This Work This Work This 2003 QRPA 2003 QRPA 2003 QRPA Abundances r-Process Solar 130 t t t 1 1 1 5 1 t / / / 1 / 2 2 2 / × 5 2 pe Limit Upper Limit Lower Values × t Values 1 / t 2 1 / 2 125 135 − 127 Pd, ad oprdt hs acltdwt hoeia values. theoretical grey with (dark calculated uncertainty five. those the work in to of reduction this compared factor slight from band) a a only limit by show upper line) line) black experimental blue (dashed new (solid with decreased calculated for determined pane) and was half-life Abundances (bottom line) theoretical which red shown The line) (solid are values. black increased theoretical ratios (solid nominal addition, calculation the In baseline using the band). to grey relative (light shown is wind entropy on high effect solar The a to from compared determined (lines) pane) model (upper Abundances 4.3. Figure
Ratio 10 Abundance 0.6 0.8 1.2 1.4 − 1 1 1 105 r poesaudne rmucranisi hoeia half-lives theoretical in uncertainties from abundances -process 110 115 r ge points). (grey ] 22 [6, abundances -process 123 120 tmcMs ubr(A) Number Mass Atomic 125 hsWork This Work This 2003 QRPA 2003 QRPA 2003 QRPA Abundances r-Process Solar 130 t t 1 1 5 1 t / / 1 / 2 2 / × 5 2 oe Limit Lower Values × t Values 1 / t 2 1 / 2 128 135 dwas Pd iue44 bnacs(pe ae eemndfo ihetoywind entropy on effect high solar The a to from compared determined (lines) pane) model (upper Abundances 4.4. Figure rnhn ratios, branching ihtertclvle n hf rmtetertclnmnlvalue. nominal theoretical the from calculated shift those to a compared and corresponding values band) substantial theoretical grey the a with (dark show uncertainty and limits line) the blue line) in (dashed lower reduction black and line) (dashed red experimental (dashed work new upper this with determined from Abundances measurements respectively. 0% and 100% first 125 theoretical with The isotopes values. for line) theoretical values black nominal (solid the calculation using baseline calculated the was which to relative pane) (bottom shown are − 10
128 Ratio Abundance 0.8 1.2 1.4 − 1 g eeicesd(oi e ie n erae sldbu ie to line) blue (solid decreased and line) red (solid increased were Ag, 1 1 105 r poesaudne rmucranisi hoeia neutron theoretical in uncertainties from abundances -process P 110 n aus sson(ih rybn) nadto,ratios addition, In band). grey (light shown is values, 115 P n esrmnsfo hswork, this from measurements r ge points). (grey ] 22 [6, abundances -process 124 120 tmcMs ubr(A) Number Mass Atomic 125 hsWork This Work This Work This 2003 QRPA 2003 QRPA 2003 QRPA Solar r -Process 130 P P P P P P n n n n n n pe Limit Upper Limit Lower Values 100% = 0% = Values 123 − 126 135 dand Pd P n 4.2 Conclusion
125 127 We have reported on the first measurements of the half-lives of − Pd as
123,124 124 129 well as previously measured values of Pd and − Ag, some with significant improvements in their uncertainties. The only deviations observed may be explained by the possible existence of long-lived isomers. In addition, we have reported first
123 127 time measurement or upper limits for neutron branching ratios of − Pd and
125 129 124 − Ag as well as a repeated measurement of Ag. These bulk properties of the nuclei can provide the first constraints on shape, masses and β-decay strength. Through systematic trends of the half-lives and branch- ing ratios a constraint on both mass models, as well as QRPA models, can be ob- tained. These constraints lead to better nuclear modeling of even more neutron-rich nuclei, which in turn improves r-process modeling. Comparison between the experimental values and QRPA calculations has shown some deviations for palladium isotopes and a surprising relative agreement for the silver isotopes. These measured values are also used as direct inputs into r-process models help- ing to constrain the impact from uncertainties in nuclear properties. Half-lives affect the waiting-point nuclei and thus how much material is able to reach heavy masses. Branching ratios play a role after freeze-out smoothing the final abundance pattern of astrophysical calculations that is compared to the solar r-process pattern. The- oretical models also play an important role in the r-process, as the path runs far from stability through very neutron-rich nuclei that have been difficult to access ex- perimentally. We have shown the impact on a high entropy wind model using our
125 newly measured half-lives and neutron branching ratios. We observe a reduction in the uncertainty in the final abundance pattern from A = 124 to 130 as well as a shift in the abundance pattern. Continued investigation of nuclear parameters important to the r-process will help reduce the uncertainty due to nuclear data. The remaining sensitivity of mod- els will be to uncertainties in astrophysical parameters. This will allow for better understanding of the astrophysical scenarios responsible for the r-process.
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