A SEARCH FOR NOVEL DECAY MODES IN 11-BERYLLIUM By Jason Tyler Surbrook A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Physics—Doctor of Philosophy 2023 ABSTRACT About a decade ago, the measurement of the decay products of 11 Be yielded an unexpectedly large abundance of 10 Be. The presence of 10 Be daughters was interpreted as a signature of the novel 𝛽− -delayed proton emission. An alternative hypothesis soon followed, that free neutrons and loosely-bound neutrons in atomic nuclei may decay into unseen dark particles. This alternative hypothesis could explain the abundance of 10 Be daughters and the long-standing neutron lifetime puzzle. Such a hypothesis has impacts on the stability of neutron stars, elemental abundances from Big Bang nucleosynthesis, and physics beyond the Standard Model. In low-energy nuclear physics specifically, the new hypothesis brought a sharp focus to the behavior of nuclear systems near particle-emission thresholds and the mechanism of 𝛽− -delayed proton emission. More recently, a number of experiments have been conducted with beams of short-lived 11 Be (13.8 s) and long-lived 10 Be (1.5 MYr) to measure directly the 𝛽− -delayed proton and 𝛽− -delayed 𝛼 decays of 11 Be and related structure in 11 B. Similarly, many new theoretical calculations have made pre- and post-dictions on the energies and intensities of these decay channels in 11 Be. Most theoretical model calculations predict 𝛽− -delayed proton branching ratios smaller than reported by experiment. The branching ratio of the 𝛽− -delayed proton decay in 11 Be remains an open question. At the National Superconducting Cyclotron Laboratory (NSCL), on the campus of Michigan State University, we used the Gaseous Detector with Germanium Tagging (GADGET) system to measure the 𝛽− -delayed charged particle decay of 11 Be, in search of the 𝛽− -delayed proton decay. R-matrix calculations and Bayesian methods were employed to describe the competing (and far more frequent) 𝛽− -delayed 𝛼 channel. This dissertation covers the experimental and analysis methods used to produce the second reported measurement of the novel 𝛽− -delayed proton emission and early interpretation of this new result. Copyright by JASON TYLER SURBROOK 2023 ACKNOWLEDGMENTS Any significant undertaking is hardly the contribution of a single person and, believe me, finishing a Ph.D. is a significant undertaking. Significant, but incredibly meaningful. I have been given the opportunity to play scientist and push human knowledge, being the first to learn even a small part of this incredible world we find ourselves. In working on my Ph.D., I have had the privilege to meet so many talented and inquisitive people, while having incredible support from loved ones from before starting the program. Each of the following people has been part of my experience in pursuing this Ph.D. and for the better. This list is not and cannot be complete. I would first like to thank my advisor Chris Wrede. Chris has guided me professionally throughout most of my graduate school education and through the several projects before we found traction in 11 Be story in this document. He has afforded me a great deal of agency to take projects in new directions where solutions were not guaranteed. Thank you for your guidance and our many conversations. Next is my not-advisor, Ryan Ringle. While I was wary of afternoon coffee affecting my sleep, I hope you know just how important afternoon coffee was to me, especially toward the end of my Ph.D.. Coffee provides liquid willpower and facilitates conversation. Thank you for listening. Cheers to Scott Pratt and Pablo Guiliani, for teaching me Bayesian statistics when I needed it most. I credit the development of the Bayesian statistics code with the revitalization of the dissertation, and I could not have done it myself. I would like to thank my remaining committee members Laura Chomiuk and Joey Huston for your support and guidance. I need to acknowledge the incredible efforts of all the staff of the National Superconducting Cyclotron Laboratory. In particular, I want to highlight Tom Ginter, Chandana Sumithrarachchi, Mallory Smith, and Jill Berryman. A facility is hardly more than its people. Within the Department of Physics & Astronomy, I want to highlight Kim Crosslan, the patron saint of graduate students. As for the friends I met in graduate school, in no particular order, thank you for being yourselves: iv 50% Chris I., Alec, Isaac, Danny, Cathleen, Tamas, Moshe, Lijie, Ruchi, Tyler , Andrew, Jeremy, Tom-Erik, AAron, Fanurs, Adam, Jaclyn, John, Michael, TC, and Cami. Each of you have taught me something. I have grateful for the opportunity of having met you. Scott, Christian, Alex, Lexi, and Joe, I wish you the best of luck with the remainders of your degrees. I know you will do well. Thank you to the Lansing Makers Network. I have only recently learned of the sociological phrase ”third place,” and what a place you are. I commend the leadership, past and present, to have dreamed up and made the lovely, thoughtful, creative, caring place that is the Lansing Makers Network. Jimmy, Carl, Bryan, and Melody, you were my first introduction to the people of LMN and I couldn’t have asked for better people to get to know. As time went on, Evelyn, Tony, and Lou, you joined the ranks of friendly, familiar, inspired makers I hoped I might see in the space. LMN is a gem of the Lansing area. I wish you every opportunity to maintain such a wonderful community. I would like to thank my oldest friends for their camaraderie and support throughout so much of my life: Brenden, Lance, and Daniel. May we see each other more often and may you live long, richly-connected lives. I would like to thank the Cox and Currie families for inviting me in as their own while I was in Michigan. Aunt Christine, Uncle Steve, Jenny, Clark, Matt, Zach, Chloe, thank you for sharing your Roger’s City camping trips. Similarly, Uncle Bryan, Andrew, Alexa, Austin, Adam, and Ash, thank you for sharing with me Roscommon deer camp. I so cherish these trips and it breaks my heart to leave these traditions in Michigan. I want to thank my brother Joel and my sister Lauren for their friendship and support. May we also see each other more often. Thank you Bob for forgiving me after not going to Maryland. Thank you to Mary and Rick for their encouragement and support. Thank you Michael for always being excited about my work. I am the first in my family to earn a Ph.D.. I would like to thank my father for instilling an appreciation for science as a boy. I would like to acknowledge Adam Jensen, for personifying v curiosity and thoughtfulness. You played a large role in my realization that science and scientific thinking could be more than an esoteric triviality. To my mother, I cannot summarize your influence. You were endless in your support, and in finding avenues to make it all work. I credit you, uniquely, for my opportunity to attend undergrad straight from high school. I quickly involved myself there and more than a decade later, I am still doing Physics. Every day, I see opportunities because of your sacrifice. I don’t know how you did it all and so appreciate that you did. To my partner, Koryn. We have known each other since we were kids and we have grown so much in all these years. Your companionship and support have made all the difference as I pursued this demanding and rewarding degree. I look forward to the continued opportunities to learn, to experience, and to grow with you into the future. Thank you, always. vi TABLE OF CONTENTS CHAPTER 1 THE ATOMIC NUCLEUS: A LABORATORY FOR FUNDAMENTAL INTERACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Atomic Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Motivating a Measurement of the Novel 11 Be(𝛽− 𝑝) Decay . . . . . . . . . . . . . 5 1.3 Status of 11 Be(𝛽− 𝑝) Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 CHAPTER 2 EXPERIMENT: MEASURING THE NOVEL 11 Be(𝛽− p) RATE . . . . . 12 2.1 GADGET: The Gaseous Detector with Germanium Tagging . . . . . . . . . . . . . 12 2.2 Measurements at the National Superconducting Cyclotron Laboratory . . . . . . . 17 CHAPTER 3 ANALYSIS: DATA SORTING, INFERENCE WITH 𝜒2 -MINIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Data Sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 NSCL Experiment E18507 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 NSCL Experiment E19030 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 CHAPTER 4 ANALYSIS: R-MATRIX, GAS STOPPING, BAYESIAN INFERENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.1 Summary of what we do know . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 R-matrix Guidance to Describing the 𝛽− 𝛼 Background . . . . . . . . . . . . . . . 72 4.3 Low-Energy Ionization Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4 Bayesian Inference of 𝛽− 𝑝 with Markov Chain Monte Carlo . . . . . . . . . . . . . 97 CHAPTER 5 RESULTS AND DISCUSSION: EVIDENCE FOR NOVEL 𝛽− 𝑝 . . . . . 108 5.1 Results of Bayesian Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . 108 5.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 CHAPTER 6 CONCLUSIONS: INTERPRETING THIS WORK AND FUTURE EFFORTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 APPENDIX A SUPPLEMENTARY INFORMATION OF NSCL E18507 ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 APPENDIX B SUPPLEMENTARY INFORMATION OF NSCL E19030 ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 APPENDIX C SUPPLEMENTARY INFORMATION OF THE BAYESIAN ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 vii CHAPTER 1 THE ATOMIC NUCLEUS: A LABORATORY FOR FUNDAMENTAL INTERACTIONS The focus of this dissertation is measurement of the novel 𝛽− -delayed proton (𝛽− 𝑝) decay in 11- beryllium, 11 Be. Concurrent to the work presented in this dissertation, many related investigations by other researchers were undertaken and for a variety of scientific motivations. That is to say, the scientific interest in the 11 Be(𝛽− 𝑝) decay has been and continues to be multi-faceted. This chapter will introduce the atomic nucleus as a laboratory of physical phenomena and set the stage for the unique case of 11 Be. 1.1 The Atomic Nucleus The atom is the basic building block of the tangible, bulk matter that we interact with. Figure 1.1 shows a cartoon of the atom and its constituent structure. By volume, the atom is dominated by a diffuse cloud of electrons, 𝑒 − , orbiting a central, dense nucleus. By mass, the atom is dominated by the atomic nucleus, which is a collection of tightly bound, strongly interacting protons, p, and neutrons, n. The number of protons, carrying positive charge, is the atomic number, Z, and the element classification of a nucleus. The number of electrically-neutral neutrons is the isotone or neutron number, 𝑁, classification of a nucleus. A specific number of protons and neutrons uniquely defines a nuclide and all nuclei of a particular nuclide are identical. The expression of a particular nuclide is often abbreviated to 𝑍𝐴 𝑋 or 𝐴 𝑋, where the mass number 𝐴 = 𝑍 + 𝑁 denotes the total number of nucleons and 𝑋 is the chemical symbol. Thus 11-beryllium can be written as 11 Be. Beryllium, being the fourth element, has 𝑍 = 4 and 𝑁 = 𝐴 − 𝑍 = 11 − 4 = 7 neutrons. Naturally present beryllium is 9 Be and has 𝑍 = 4, 𝑁 = 5. More than 3000 unique isotopes have been observed, while some ∼7000 are predicted to exist 1 Figure 1.1: A cartoon diagram of a helium (He) atom and constituent subatomic particles with labels and approximate physical size. The atom is made of an electron cloud and an atomic nucleus. The atomic nucleus is made of protons and neutrons, which are composed of quarks. Protons are made of two up quarks and one down quark. Neutrons are made of one up quark and two down quarks. Figure Credit: Contemporary Physics Education Project, Lawrence Berkeley National Laboratory. 2 [1]. Each isotope has distinct properties such as mass, (charge) radius, magnetic moment, excitation spectra, and, for many, decay rates of various decay modes to more- stable isotopes. Many general features of nuclei can be predicted by placing a nucleus on the Chart of the Nuclides, which plots nuclei in a tile-mosaic fashion by their proton and neutron numbers. Figure 1.2 shows the Chart of the Nuclides, color coded for primary decay mode. For light nuclei of 𝐴 ≲ 40, nuclei are stable if 𝑍 = 𝑁 to within 1 unit or so. Above 𝐴 ∼ 40, the line of stability flattens and the stable nuclei tend to have greater number neutrons than protons. Nuclei that have greater 𝑍 than the stability line are called neutron deficient. For most of the chart, neutron deficient nuclides will undergo 𝛽+ or electron capture decay, converting a proton into a neutron by emission of a 𝛽+ particle (a high-energy positron) or capturing an electron, respectively, and emission of a neutrino. Similarly, most neutron-rich nuclei primarily decay by 𝛽− , converting a neutron into a proton by emission of a 𝛽− particle (a high-energy electron) and an anti-neutrino. Multi-step decays are possible. 𝛽-delayed particle emission is the emission of a nuclear particle (a proton, neutron, 𝛼/He-nucleus, or others) immediately following a 𝛽 decay. For 𝛽-delayed particle emissions, there are strong patterns in the type of 𝛽 decay and the emitted particle. The proton separation energy, or the energy required to remove a proton, is smaller when 𝑍 is large compared to 𝑁. Similarly, the neutron separation energy usually is large when 𝑍 is large compared to 𝑁. The converse of each of these statements is true (when 𝑍 is small relative to 𝑁). Putting all this together, 𝛽-delayed particle emissions are almost always 𝛽+ -delayed proton (𝛽+ 𝑝) decays, 𝛽− -delayed neutron (𝛽− 𝑛) decays, 𝛽+ -delayed (𝛽+ 𝛼) decays, or 𝛽− -delayed (𝛽− 𝛼) decays. In fact, prior to the observation of 11 Be(𝛽− 𝑝) reported in 2019, no 𝛽− 𝑝 decay had been measured in any nucleus [2]. Our first measurement was performed in May 2018 and, to our knowledge, was the very first data-taking of 𝛽− 𝑝 decays in the world. Those initial efforts, the follow-up measurement, and analysis lead us to report the second ever measurement of the 𝛽− 𝑝 decay here. 3 Figure 1.2: The Chart of the Nuclides. Nuclides are represented by colored squares, where the color depicts their primary decay mode. The colors represent: stable (black), 𝛽+ or electron capture (blue), 𝛽− (pink), 𝛼 (yellow), spontaneous fission (green), spontaneous proton (orange), spontaneous neutron (purple), and unknown (grey). Black-outlined, color-filled squares are radioactive, but very long lived, in excess of 1 billion years. Vertical and horizontal lines are so-called magic numbers. Figure Credit: National Nuclear Data Center (NNDC), Brookhaven National Laboratory. 4 1.2 Motivating a Measurement of the Novel 11 Be(𝛽− 𝑝) Decay In 2018, Fornal and Grinstein [3] noted that the decades-long, technique-specific discrepancy in neutron lifetime measurements [4, 5] could be reconciled alongside the then recent (2014) surprisingly large measurement of the the 11 Be−→10 Be transmutation [6] measured by Accelerator Mass Spectrometry (AMS). The reconciliation was that the neutron may have a ∼ 1% probability to decay into an ensemble of undetected particles with mass less than, but within 1.665 MeV, of the neutron mass. 1.2 The Neutron Lifetime Problem The lifetime of a particle is one of its most fundamental properties. Due to quark confinement, preventing the existence of a free quark, the proton and the neutron are among the simplest objects with which to study quark behavior. Protons and neutrons can exist as free particles or bound in atomic nuclei. The free proton is observationally stable against radioactive decay with a decay lifetime of > 7.7 · 1033 years [7]. The free neutron however, has slightly greater mass the proton and will decay to a final state containing a proton and leptons with a lifetime about 15 minutes. Furthermore, the exact value of the neutron lifetime has consequences for big bang nucleosynthesis [8] and quark mixing behavior in the Standard Model [9]. There are two general techniques in which the neutron lifetime is measured: the bottle technique and the beam technique [5]. In the bottle technique, the lifetime of neutrons is measured. Briefly, ultra-cold (𝑇 < 100neV) neutrons are confined in a material bottle or a magnetic bottle. Ideally, there are no mechanisms for the loss of the neutron except by the free decay of the neutron. A packet of neutrons is prepared and injected into this bottle and an amount of time Δ𝑡1 elapses, during which the neutrons can decay. The neutrons are then ejected and counted, yielding the number of remaining neutrons 𝑁1 . The 5 measurement is repeated for a different amount of holding time, Δ𝑡2 , yielding remaining neutrons 𝑁2 . If the bunches start with the same number of neutrons and the bottle is loss-less except that by neutron decay, then the decay lifetime can be calculated as: Δ𝑡2 − Δ𝑡1 𝜏= (1.1) ln(𝑁1 /𝑁2 ) Generally, however, to describe neutron losses to the bottle, the bottle dimensions are varied across different measurements in a measurement campaign. The free neutron lifetime is then extrapolated from the limit of an infinite bottle. In the beam method, the rate of proton generation is measured from a beam of cold neutrons entering a reflective electromagnetic trap. The neutron beam energy and length of the trap are known, so the time the neutrons spend in the electromagnetic trap is calculable. Within the Standard Model, neutrons decay and generate protons. These protons are reflected by the electromagnetic trap and counted. Un-reacted neutrons pass through the trap and are measured. The rate of proton creation is proportional to the neutron flux and inversely to the Standard Model lifetime: 𝑑𝑁 𝑁 𝑁¤ = =− (1.2) 𝑑𝑡 𝜏𝑝 If the only decay modes of the neutron are the usual Standard Model decays 𝑛 → 𝑝 + 𝑒 − + 𝜈¯ and 𝑛 → 𝑝 + 𝑒 − + 𝜈¯ + 𝛾, then the lifetimes measured by the disappearance of neutrons (bottle technique) and the appearance of protons (beam technique) should equal. However, as experiments using both techniques have improved and become more precise, there has emerged a discrepancy between the lifetimes extracted between the two measurements as large as 5𝜎 statistical significance. The most recent bottle measurement lifetime value is 6 𝜏bottle = 877.75 ± 0.28stat + 0.22/−0.16syst [10] and the most recent beam measurement is 𝜏beam = 887.7 ± 1.2stat ± 1.9syst [11]. If the assumption of the Standard Model decay is loosened, then the two measurements can be reconciled by a ∼ 1% branch into undetected particles. The bottle measurement would measure the total decay lifetime (𝜏total ∼ 877 s), the beam measurement would measure the Standard Model process (𝜏𝑝 ∼ 887 s), and the missing decay rate comes from the undetected or dark decay with lifetime of ∼ 8 · 104 s. 1/𝜏bottle = 1/𝜏beam + 1/𝜏dark (1.3) 1.2 The Large 11 Be−→10 Be Transmutation Rate In 2014, an Accelerator Mass Spectrometry measurement of the decay products of a sample of radioactive 11 Be (𝑡 1/2 = 13.8s) decay products was performed. They measured the presence of 10 Be consistent with a 11 Be−→10 Be decay probability of 8.3(9) · 10−6 [6], far in excess of the ∼ 3 · 10−8 decay probability predicted of the 11 Be(𝛽− 𝑝)10 Be decay beforehand by Baye and Tursunov [12]. The structure of 11 Be is unusual due to its low neutron separation energy of 𝑆𝑛 = 501.6(3) keV and valence s-wave neutron wavefunction. The confluence of these effects gives 11 Be an extended neutron halo that could enhance the 𝛽− 𝑝 rate if the neutron were able to decay into the continuum, in a ”quasi-free” way, and generate more 10 Be than previously expected. Similarly, if the valence neutron were to not decay by 𝛽− 𝑝 but instead by a dark mechanism, the observed 10 Be production rate would be enhanced as both the 𝛽− 𝑝 and dark decays produce 10 Be in their final states. 1.2 The Possibility of Dark Matter Decay in Nuclei In the original paper, Formal and Grinstein suggested the possibility of detecting 𝑛 → 𝜒dark in nuclei with low neutron separation energies, 𝑆𝑛 < 1.665 MeV. This was explored in greater detail 7 by Pfützner and Riisager [13] and 11 Be was identified as the ideal candidate. Its low neutron separation energy 𝑆𝑛 = 501.6(3) MeV leaves most of the allowed decay energy window and its relatively long 13.8 s half-life allows time for the hypothetical dark decay to have a branching ratio of up to 2.0 · 10−4 , assuming the dark decay rate inferred from the difference of the two measured lifetimes. Since both the 𝛽− 𝑝 and neutron dark decay create 10 Be in the final state, the distinguishing feature between the two channels is the emission (or not) of the visible 𝛽− particle, proton, and 10 Be recoil. The recommendation to pursue measurements to disentangle visible and dark decays in 11 Be was published in January 2018. We took our first 11 Be(𝛽− 𝑝) data, the first 11 Be(𝛽− 𝑝) dataset in May 2018. 1.3 Status of 11 Be(𝛽− 𝑝) Decay A great deal of work around the 11 Be(𝛽− 𝑝) decay has been performed since May 2018. The subsequent chapters will describe my own contributions, but here is a summary of the work done by others. Quickly after the neutron dark decay proposal, it was pointed out that this decay softened the nuclear equation of state and makes the existence of neutron stars above 0.7 𝑀⊙ (solar mass) stars difficult to explain [14, 15]. Neutron stars of 2𝑀⊙ are restored if the dark matter can interact with the visible neutron star [16]. A recent review of baryon number violation in neutron stars can be found in [17]. The Standard Model predicts relations between different neutron 𝛽-decay observables. One is the relationship between the neutron lifetime and the angular correlation coefficient, (so-called ”little”-) 𝑎, between the emitted 𝛽 and 𝜈¯𝑒 . A measurement of 𝑎 allows for a calculation of the 1−𝜆2 Standard Model term 𝜆 by the following relation: 𝑎 = 1+3·𝜆2 . The Standard Model neutron lifetime 8 also depends on the 𝛽-decay 𝐹𝑡 value, which is determined most precisely by the 0+ → 0+ 𝛽-decay in nuclei such as 10 C, 14 O, 22 Mg, and 26𝑚 Al [18]. The relation is 5172.3(31)s 𝜏= (1.4) 1 + 3 · 𝜆2 The world average of 𝜆 is 𝜆 = −1.2754(11) and yields a neutron lifetime consistent with the bottle method and inconsistent with the beam method [19, 20]. There is tension with a recent measurement of the 𝛽 − 𝜈¯𝑒 asymmetry yielding 𝜆 = −1.2677(28). This value implies the neutron lifetime to be consistent with the beam method and inconsistent with the bottle method [21]. In 2019, the first observation of 𝛽− 𝑝 was reported in 11 Be(𝛽− 𝑝) by Ayyad et al. [2]. They used a gaseous time projection chamber and measured a 178 keV proton from a stopped beam of 11 Be. Their 11 Be was produced by the Isotope Separation On Line (ISOL) technique at TRIUMF. From this proton measurement, they inferred a resonant state in 11 B at 196(20) keV above the proton separation threshold with width 12(5) keV and a 𝛽− 𝑝 branching ratio of 𝑏 𝑝 = 1.3(3) · 10−5 , consistent with the earlier AMS result of 8.3(9) · 10−6 intensity for 10 Be creation. In 2020, a new AMS measurement of the 11 Be→10 Be probability refuted the earlier claim of 11 Be→10 Be production at 8.3(9) · 10−6 and proposed a new limit for 10 Be production (by all decay mechanisms) of < 2.2 · 10−6 [22]. The authorship and facilities were similar between this new measurement and the 2014 paper [6]. This new limit is in conflict with the direct 𝛽− 𝑝 measurement of Ayyad et al. [2]. A number of nuclear structure theory efforts have been published to better clarify expectations around the Standard Model -allowed 𝛽− 𝑝 decay, especially in light of these conflicting claims of the measured 11 Be(𝛽− 𝑝) branching ratio. In 2020, Volya [23] performed shell model calculations using the psdu and fsu Hamiltonians 9 and reported no shell model states that could mediate the 𝛽− 𝑝 at 196(20) keV observed in [2]. Between the small 𝛽− decay energy window, competition from 𝛽− 𝛼 emission, and lack of a clear resonant state in 11 B, Volya concluded that 𝑏 𝑝 ∼ 5 · 10−10 , perhaps upwards of 𝑏 𝑝 ∼ 10−7 if a resonant state existed. Oko lowicz et al. [24] performed calculations of 11 B structure with shell model embedded in the continuum (SMEC) [25] and found a candidate resonant state that could mediate the 𝛽− 𝑝 at 𝐸 𝑝 ∼ 142 keV, slightly lower than that measured in Ayyad et al. [2]. In 2021, Elkamhawy et al. [26] published Effective Field Theory calculations that reproduce the Baye and Tursunov [12] branching ratio of 𝑏 𝑝 ∼ 3 · 10−8 to a factor of 2 for non-resonant 𝛽− 𝑝. By +4.0 assuming a resonance at 𝐸 𝑝 = 196(20) keV, a branch of 𝑏 𝑝 = 4.9+5.6 −2.9 (experiment)−0.8 (theory) · 10 −6 +5.3 with intrinsic width of Γ = 9.0+4.8 −3.3 (ex.)−2.2 (th.) keV was found, which is in excellent agreement with the measured branch 𝑏 𝑝 = 1.3(3) · 10−5 and width of 12(5) keV. In 2022, Oko lowicz et al. [24] calculated a 𝛽− 𝑝 branching ratio using SMEC as well as the interplay between the 𝛽 feeding and proton-penetrability. They concluded that the 𝛽− 𝑝 branching ratio and width measured in Ayyad et al. [2] are incompatible with the 3.3(1)% 𝛽− 𝛼 branch of [27]. They calculated the branch to be 𝑏 𝑝 ∼ 3 · 10−7 after model tuning to the 12(5) keV intrinsic width of [2]. Atkinson et al. performed a no-core shell model calculation and found resonant states that could mediate the 11 Be(𝛽− 𝑝), but calculate a small branching ratio of 𝑏 𝑝 = 1.3(5) · 10−6 [28]. Finally, Le Ahn et al. calculated scattering properties of 10 Be+𝑝 with the method of self-consistent Skyrme Hartree-Fock in the continuum and inferred a resonant state in 11 B with energy of 𝐸 𝑝 ∼ 182 keV and width Γ ∼ 6 keV [29]. Experimentally, another couple of publications are worth noting. Ayyad et al. [30] performed 10 Be(𝑝, 𝑝′) resonant scattering. They performed an R-matrix fit [31] to their measurement and reported a resonance at 𝐸 𝑝 = 171(20) keV with partial width Γ𝑝 = 4.5(11) keV and total width Γ𝑇 = 16(3) keV. Similarly, Lopez-Saavedra et al. [32] performed a transfer reaction to a 11 B 10 resonance with 10 Be(𝑑, 𝑛)10 Be+𝑝. They reported a resonant state at 211(40) keV with a small 𝛼 width component. This small 𝛼 component seems at odds with the small proton component of the proton resonance scattering measurement of Ayyad et al., unless there are multiple states at very similar energies that were preferentially probed by the choice of production mechanism. Generally, there seems to be building consensus between theory and experiment of a resonant state in 11 B capable of enhancing the 11 Be(𝛽− 𝑝) rate. There is a spread of possible resonance energies from 140 keV to 210 keV. There is some tension in the width of this resonance, with an associated proposal of not one new near-threshold resonance, but multiple to explain this tension [32]. Perhaps most importantly, there is still only the one direct measurement of the 11 Be(𝛽− 𝑝) branching ratio and it is incompatible with indirect measurements of the total 11 Be→ 10 Be decay rate. Theory predictions are not unified in their predictions of the 𝛽− 𝑝 branching ratio to even an order of magnitude. To clarify experimental inconsistencies and provide insight to the theory models that predict branching ratios, an independent measurement of the 11 Be(𝛽− 𝑝) branching ratio, by a search for the novel 𝛽− 𝑝 radiation, is sorely needed. 11 CHAPTER 2 EXPERIMENT: MEASURING THE NOVEL 11 Be(𝛽− p) RATE The measurement of 11 Be(𝛽− p) is hallmarked by three experimental challenges: the short lifetime of 11 Be, requiring online production at a rare-isotope facility; the low rate and rare incidence of such decays and the resulting large backgrounds from more common decay processes; the low energy of emitted protons restricting the use of several classes of detectors frequently used in nuclear 𝛽-decay studies. This chapter will describe the facilities and equipment used in measurements conducted to measure 11 Be(𝛽− p) and their solutions to each of these challenges. 2.1 GADGET: The Gaseous Detector with Germanium Tagging The Gaseous Detector with Germanium Tagging (GADGET) system [33] was developed to measure low-intensity, low-energy, 𝛽-delayed proton decays of radioactive nuclei, with the purpose of constraining thermonuclear reaction rates important to understanding explosive nucleosynthesis. It has conducted several measurements of 𝛽+ , specifically, -delayed proton (𝛽+ p) decays in nuclei of mass A=20-31 nuclei, measuring the emission of 𝛽-delayed protons as low as 200 keV [34, 35, 36] at the National Superconducting Cyclotron Laboratory (NSCL). The GADGET system is the synthesis of coupling the NSCL Segmented Germanium Array (SeGA), an array of high-resolution germanium crystal 𝛾-ray detectors, with the Proton Detector, a gaseous proportional counter. SeGA is installed about the body of the Proton Detector in its “barrel” configuration. As will be explained, the coupling of these two detector systems enables our search of rare 𝛽-delayed charged particle events. 12 2.1 Proton Detector: Principle of Operation The schematic of Figure 2.1 depicts the operational components of the Proton Detector. The detector is filled with a gas, which is selected for its desirable properties such as high electrical resistance and quenching of non-ionizing atomic and molecular excitations. The detector gas acts as both a beam stopper/catcher as well as a medium which is ionized in the presence of charged particle radiation. An electric field is applied within the detector to drift positive ions to the detector cathode and freed electrons toward the MICROMEGAS detection pads at a constant velocity and without amplification. Electrons reaching the MICROMEGAS will pass through the Micromesh and enter the charge amplification region, with an electric field capable of inducing a Townsend avalanche which produces a measurable amount of charge, inducing a signal on the MICROMEGAS anode. One important operational characteristic of the gas is its low density relative to other detectors such as plastic or glass scintillators and semiconductor materials such as silicon or germanium detectors. With their relativistic velocity and long tracks due to the low density of the gas, emitted 𝛽 radiations will typically deposit only a small fraction of their kinetic energy in their gas before reaching the detector boundaries. Low-energy protons, 𝛼-particles, and the complementary nuclear residues will stop near the site of the decay, losing all of their energy to interactions with the gas within several milli- or centimeters. This suppression of the 𝛽 spectrum allows measurement of 𝛽- delayed lights ions, such as protons and 𝛼 particles, to lower energies than possible in higher-density detector media. 2.1 Proton Detector: Design Figure 2.2 shows a cut-away mechanical drawing of the Proton Detector. Key operational compo- nents are labeled. The body of the Proton Detector is a cylinder of stainless steel with dimensions 13 of roughly 49 cm long by 16.5 cm in diameter. The detector active volume is roughly 40 cm in length and 10 cm in diameter. Radioactive beam enters through a 1.5 𝜇m thick, 50.2 mm diameter aluminized mylar Kapton window in the upstream end cap. This end cap is also equipped with a high voltage feedthrough, which is coupled to the detector cathode, and four small gas outlets, which allow for the removal of detector gas. Along the interior of the detector volume is the electric field cage, which electrically connects the cathode to the MICROMEGAS Micromesh by a resistive Kapton sheet with printed copper bands, rolled into a cylinder and held in place with insulating polyether ether ketone (PEEK) material. The alternating conductive-resistive sections act as a voltage divider and help ensure a uniform electric drift field in the detector bulk. The gating grid is a wire grid 1.5 cm before the MICROMEGAS and can be used to disrupt the electric drift field to protect the MICROMEGAS from large currents produced during beam deposition. The rear of the detector chamber has four gas inlets to introduce fresh gas, a valve for rough pumping of the detector, and voltage feedthroughs to control the MICROMEGAS anode and gating grid voltages. The downstream end is capped by the MICROMEGAS circuit board. The MICROMEGAS has two primary components: a gold plated copper anode and an electro-formed stainless steel micromesh supported by insulating pillars at 128 𝜇m above the anode plane. The set-up is cycled between beam-delivering and decay-measurement modes synchronously with the delivery of beam to the detector. The voltage on the gating grid is alternated to allow or prevent the passage of drifted electrons to the MICROMEGAS. In beam-delivery mode, a beam of radioisotopes is passed into the detector. Interactions between the impinging beam and the detector gas thermalize the beam. Nuclei of the beam may be neutralized and form neutral atoms, which then diffuse under Brownian motion until their radioactive decay. In the case of incomplete neutralization, positively-charged atoms will drift to the detector cathode. In beam-delivery mode, the gating grid voltage is set to 119 V, so ionization electrons created via beam stopping will drift 14 to the gating grid, but not pass to the MICROMEGAS. In decay-measurement mode, the beam is no longer deposited and the gating grid voltage is set to -225 V, to be electrically transparent with respect to the drift field and allow ionization electrons into the MICROMEGAS. These voltage settings are summarized in Table 2.1. Component z-axis position beam-delivery voltage decay-measurement voltage Cathode +40 cm -4.8 kV -4.8 kV Gating Grid +1.5 cm 119 V -225 V Micromesh 0 cm 0V 0V Anode -128 𝜇m 330 V 330 V Table 2.1: Voltage states for the Proton Detector components in beam-accepting and decay- measurement operation modes. The geometry of the MICROMEGAS anodes is shown in Figure 2.3. The anodes are split into three concentric circles of radius 1.4, 4.0, 5.0 cm. The ring between 1.4 and 4.0 cm is segmented fourfold, every 90◦ . The ring between 4.0 and 5.0 cm is segmented eightfold, every 45◦ . The pads are labeled alphabetically, inwards to outwards, from A to M. Pad A is the singular pad in the radius 1.4 cm segmentation. Pads B, C, D, and E are identical quadrant pads between the radius 1.4 cm and 4.0 cm segmentations. The remaining eight pads are identical octal pads between the 4.0 cm and 5.0 cm radial segmentations. The large areas of Pads A-E make these the primary measurement pads, while the thin radial sizes of the outer octal Pads F-M make them useful diagnostics to, in software, exclude counting of events corresponding to charged-particle tracks that are not confined in the sensitive volume of the Proton Detector. The measurement pads may be analysed independently for better energy resolution and greater suppression of the 𝛽-particle background or jointly as a single interior pad for higher efficiency. 15 2.1 SeGA: The Segmented Germanium Array The Segmented Germanium Array (SeGA) [37] is an array of high-resolution, coaxial high-purity germanium (HPGe), 𝛾-ray detectors. HPGe detectors work under high-voltage bias across the Ge crystal. As a 𝛾 ray interacts with the Ge, typically by a photoelectric effect complete absorption or a Compton-scatter which deposits only part of the incident 𝛾 energy, electrons are excited to the crystal’s conduction band producing electron-hole pairs. The bias voltage drifts these to cathode(s) and anode(s), respectively, where a current can be measured, ideally proportional to the number of liberated pairs and, in turn, the deposited 𝛾-ray energy. Ge is used for several desirable properties. First, Ge is a semiconductor, with electrical properties between that of common conductors and insulators. The band gap is the energy required to promote an electron in a crystal from the valence state to a conducting state. Ideal conductors have band gaps smaller than the Fermi energy of the valence electrons, so there always exist conduction electrons, and insulators have large band gaps on order the ionization energy, so there are never bound electrons in the conduction state. Semi-conductors, like Ge, have small band gaps above the Fermi energy, which allow application of biasing voltages to drift conduction electrons to readout electronics while also producing large numbers of charge carriers for an amount of deposited energy due to the low cost of promotion. The small band gap in (and large number of created charge carriers of) Ge gives HPGe detectors their excellent energy resolution as statistical fluctuations in the number of charge carriers shrink relative to the total as the total becomes large. One notable limitation of Ge’s small band gap is the electrical current produced by thermal excitation of valence electrons, so HPGe detectors must be operated at cryogenic temperatures involving refrigerators or liquid nitrogen (77K) cooling. Another desirable feature of HPGe is that it has a moderately high atomic number for a semi-conductor of 𝑍 = 32. The cross-section of MeV 𝛾 rays to interact with 16 a material scale roughly as 𝑍 2 of the material, so the intrinsic efficiency of Ge is higher than other semi-conductors including Si [38]. The segmentation of with SeGA gets its namesake is a 32-fold, 8-layer quadrant segmentation. This segmentation allows for position reconstruction of a 𝛾-ray event in the detector, which can be useful for Doppler reconstruction of 𝛾 rays produced in fast beam experiments. As our nuclei were stopped, only the central contact carrying the total current in the crystal was recorded in our measurements. 2.2 Measurements at the National Superconducting Cyclotron Laboratory 11 Be is a radionuclide with a half-live of just 13.76(7) s [39]. This short half-life requires that 11 Be must be produced in the same laboratory as experimental studies of it. Two measurements of the 𝛽 decay of 11 Be were conducted with GADGET at the now-retired National Superconducting Cyclotron Laboratory. The NSCL experiment numbers were E18507, which ran in May 2018, and E19030, which ran in July 2019. The NSCL operated the Coupled Cyclotron Facility and earlier cyclotron-enabled rare isotope production programs, and provided world-competitive rates of short-lived radioisotopes like 11 Be for over 50 years. 2.2 11 Be production by Projectile Fragmentation A beam of 11 Be was produced by projectile fragmentation at the Coupled Cyclotron Facility (CCF) at NSCL by accelerating a beam of stable 18 O to 120 MeV/u and impinging it on a 3196 mg/cm2 Be production target. The initial 18 O beam was created by Electron Cyclotron Resonance which ionizes gas with microwave-driven electrons trapped in a magnetic bottle. Ionized 18 O+ was extracted with an electrostatic nozzle and transported to the CCF. Cyclotrons provide ion acceleration by two primary elements: a magnetic field for confinement within the accelerator and radio frequency 17 high voltage to drive ions’ motion at their cyclotron frequency until reaching the desired energy. The CCF used the K500 to do primary acceleration before impinging beam upon a carbon foil to strip the beam of electrons down to bare nuclei. Acceleration of 18 O8+ is completed in the K1200. At ∼ 120 MeV/u, a variety of nuclear reactions on the Be target are possible, such as nuclear charge exchange, nucleon knockout, and collective fragmentation which can remove many nucleons, including the desired reaction(s) to produce 11 Be [40]. The A1900 fragment separator [41] used magnetic beam steering and position-selective slits to remove the unreacted 18 O8+ and reduce the cocktail of undesired reaction products. The radioisotope production up to this point is depicted in Figure 2.4 and further reading can be found in [41]. Beam was then delivered to the experimental setup in the S2 vault. 2.2 Silicon P.I.N. Detector The principle of particle identification by energy-loss and time-of-flight (ToF) measurements is as follows. Beam bunches are prepared to a particular magnetic rigidity 𝐵𝜌, giving particles in a beam bunch the same momentum to charge ratio, 𝑞𝑝 . As the beam travels to the experiment, the lighter mass species will have higher velocity and arrive sooner (having a shorter ToF). The instantaneous energy deposition in the silicon detector is well-modeled by the Bethe formula [38]: 𝑑𝐸 4𝜋𝑒 4 𝑧2 − = 𝑁𝐵 (2.1) 𝑑𝑥 𝑚0𝑣2 where 2𝑚 0 𝑣 2 𝑣2 𝑣2 𝐵 = 𝑍 [ln( ) − ln(1 − 2 ) − 2 ] (2.2) 𝐼 𝑐 𝑐 Here, 𝑣 and 𝑧 are the velocity and atomic number of the incident particle. 𝑁 and 𝑍 are the number density and atomic numbers of the detector material. 𝑚 0 is the electron mass and 𝑒 is the 18 electron charge. The MSX detector was selected for its thin 300𝜇m profile, so energy loss of the fast beam would be small relative to the total energy. In the approximation of small energy loss (no velocity change), that instantaneous energy deposition is proportional to the square ratio of the 𝑧2 beam-particle’s atomic charge to its velocity, 𝑣2 . Thus, beam species should have unique energy deposition and ToF profiles. 2.2 Electronics and Data Acquisition Figure 2.5 depicts the operational and readout electronics as instrumented in E19030. The two bias-voltage modules powered GADGET sub-systems and were controlled remotely by a dedicated network-connected Linux machine. SeGA electronics were installed and operated by the NSCL Gamma Group. The instrumentation of NSCL E18507 and E19030 were kept similar. Notable exceptions were the implementation of a beam time-of-flight timing circuit and implementation of the full Proton Detector MICROMEGAS pad plane in E19030. The ToF timing circuit was used to measure the time of flight of the beam specifically between the A1900 Focal Plane scintillator detector and our insertable Silicon PIN detector, less than 1 m upstream of the Proton Detector. The detector was inserted into the beam for short, dedicated measurements with attenuated beam between some full- intensity production runs to produce an independent measurement the delivered beam composition. Prior to E18507, several channels of the Proton Detector signal preamplifier were damaged and E18507 ran with only eight of the thirteen MICROMEGAS pads implemented. In E19030, all thirteen of the Proton Detector pads were implemented. 19 2.2 Implementation of GADGET electronics The drift field in the Proton Detector was imposed by a CAEN N1470B programable high voltage supply through the safe high voltage (SHV) feedthrough installed in the upstream end of the Proton Detector. To produce the 120 V/cm electric field in the drift region, the N1470B provided −4.8 kV. This field strength was calculated to provide a ∼ 8 𝜇s drift time from the detector cathode to the MICROMEGAS. To limit the risk of electrical discharges, the high voltage bias was slowly ramped (∼ 5 V/s) to the nominal voltage, where it remained for the duration of the experiment. The second high voltage supply module was the Mesytec MHV-4. The first channel produced the bias voltage for the charge-amplification field between the MICROMEGAS anodes and the Micromesh. The bias was set to 330 V to produce a field of ∼ 2.6 · 104 V/cm. The second channel produced the +40 V bias for the silicon PIN detector. The final two channels of the MHV-4 were used to operate the gating grid. The third channel was set to -225 V for the ”decay-measurement” operation mode and the fourth channel was set to +119 V for the ”beam-delivery” operation mode. The two gating-grid voltages were modulated by a CGC Instruments NIM-AMX500-3 model high- voltage switch module operating on the TTL standard for voltage logic set by the synchronizing circuit. The ”beam-delivery” and ”decay-measurement” cycle was set by a synchronizing circuit made of two LeCroy 222 NIM Dual Gate and Delay Generator modules [42]. Each module consists of two identical, independent sub-modules. The first module acts as a clock for the experiment cycle and the second module prepared an analog logic signal that was delivered to the facility to coordinate delivery of the pulsed 11 Be beam. The synchronizing circuit works as follows. In the first (”clock”) module, upon a receiving Start signal, the logic output of the first unit will turn positive for a screwdriver-programmable amount of time (from ∼ 100 nsec to ∼ 11 sec). At the end 20 of the logical-on period, a short (∼ 10 ns FWHM) NIM-level signal will originate from the DEL output. The DEL output of the first sub-module triggers the Start of the second and the the DEL of the second triggers the Start of the first. Each unit was programmed to 1 second intervals allowing for 1 second of ”beam-delivery” operation and 1 second of ”decay-measurement” operation. The length of the cycle was chosen to be short relative to the 13.8 s 11 Be halflife to minimize decay losses during the ”beam-delivery” operation, while long compared to the ∼ 10 − 100 ms short-lived contaminants. Longer cycles also reduce the relative detector dead time originating from switching between operation modes. The synchronizing circuit reset itself every 1+1=2 seconds. The second LeCroy 222 module was used to prepare the ”beam-on” signal to the facility. To ensure that the Proton Detector gating-grid voltage was set and beam would not be delivered before the safety of the GADGET electronics was ensured, the first unit of the second module was used to insert a ∼ 10 ms delay before the complementary unit indicated beam should be delivered. The length of this ”beam-on” logic pulse was short of 1 second (∼ 950 ms total) so that there would similarly be time between beam delivery and the transition of the gating grid to allow ionization to the MICROMEGAS again. A copy of the clock signal was sent to the data acquisition (Sec. 2.2.2.2) to correlate data events against the measurement cycle. A diagram of this logic is shown in Figure 2.6. 21 Figure 2.1: Schematic depicting the detection of a 𝛽+ p decay. 𝛽 particle and proton (and any other, non-depicted charged particles) create ionization in the detector media gas. Liberated electrons are drifted, amplified in number, and detected by the MICROMEGAS pad (unlabeled, gold and yellow). Electric field numbers are representative of nominal operating conditions. See text for current electronics settings. 22 Figure 2.2: An undimensioned mechanical design drawing of the Proton Detector. Key operational components are labeled. The bulk of the detector is 49 cm long by 16.5 cm in diameter, while the sensitive detection volume is roughly 40 cm by 10 cm in diameter. 23 Figure 2.3: Diagram of the detector pad plane. The MICROMEGAS anode is segmented into 13 pads labeled alphabetically A-M. Pads A-E are used for measurements while F-M are diagnostic. The pad geometry is segmented radially at 1.4, 4.0, and 5.0 cm. Figure 2.4: Schematic diagram of the Coupled Cyclotron Facility and the A1900 fragment separator. A variety of ion sources feed low-energy beams into the K500 and K1200 coupled cyclotrons for acceleration. Fast, ∼ 150 MeV/nucleon beams are impinged on the production target. The A1900 fragment separator uses magnetic rigidity separation to purify the beam before delivery to the experiment. Figure Credit: Ref. [41]. 24 Figure 2.5: Schematic of electronics, with some settings, and wiring for E19030. Dashed lines are meant to indicate devices, such as the Proton Detector, or various electronics or readout modules. Figure Credit: Lijie Sun 25 Figure 2.6: Schematic of electronics logical pulses controlling the Proton Detector and K1200 RF cycling for E19030. The numbers indicate the sub-module of the LeCroy222 the signal originates from. The first two sub-modules operate as a clock for the experiment. When NIM2/TTL2 are ”on”, the gating grid is set to its opaque setting. After a delay generated by DEL3, NIM4 signals ”on” for the CCF RF and beam delivery. Figure Credit: Lijie Sun 2.2 Data Acquisition Data acquisition of GADGET sub-systems was performed with NSCL’s Digital Data Acquisition System (DDAS). DDAS was developed to work with SeGA for the purposes of in-beam 𝛾 spec- 26 troscopy for detection of and identification of nuclear reaction residues [43] and later expanded to work with most familiar detectors [44]. It is meant to replace large portions of the catalog of traditionally-used analog Nuclear Instrument Modules (NIM) electronics used for signal shaping, timing, and other frequently used pre-analysis tools, such as coincidence matching between different detector (sub-) systems. In DDAS, analog electronic signals are fed into standard digitization elec- tronics and parameters such as peak voltages, voltage rise-times, and incident timing of radiation events are extracted from the digitized traces in software and saved to disk. Digitized traces may also be stored for further offline (re-)analysis. In stopped beam experiments, such as our 11 Be(𝛽− p) measurements, DDAS allows for essentially zero-dead-time measurements of rare decays. In both of our measurements, we utilized three 250-MHz, 16-channel PXI Digital Pulse Pro- cessors in a PXIE-CRATE-P16X14 from XIA, which will be referred to as Pixie0, Pixie1, and Pixie2. The first thirteen channels of Pixie0 recorded the Proton Detector’s MICROMEGAS volt- ages after passing through a Mesytec MPR-16-L preamplifier. Channel 14 of Pixie0 recorded the NIM output of the LeCroy222. Channel 15 of Pixie0 and Channels 0-14 of Pixie1 recorded the 16 SeGA central-contact, full-energy, signals. Pixie2 measured the signal of the silicon PIN detector in channel 0 and voltage from an Ortec566 Time To Charge NIM module [45] in channel 1. The output voltage of the Ortec566 maps to the timing between the A1900 Focal Plane scintilator and the upstream silicon detector. It returns a larger voltage for a greater delay. 2.2 NSCL Experiment E18507 Our first 11 Be(𝛽− p) measurement was NSCL Experiment in E18507 and ran in May 2018. To our knowledge, this measurement was the first of several across several groups world-wide to be conducted following the dark neutron hypothesis put forth by Fornal and Grinstein [3]. E18507 ran as an extension of E17023, the GADGET commissioning experiment [33], with 12 hours of 27 NSCL director-appointed discretionary beam time. Roughly six hours were used by the facility for beam-tuning and roughly six hours of beam was delivered. Figure 2.7 shows the facility-provided particle identification using the A1900 focal plane scintillator energy loss and timing with respect to the CCF RF. The beam was 85% pure 11 Be at a rate of 580000 particles per second. Contaminants were identified as 13 B, 12 B, and 9 Li. Of the contaminants, 13 B was the most intense. Figure 2.7: Particle identification plot provided by NSCL facility in E18507. The y-axis is an uncalibrated measurement of in-flight energy loss through a small detector. The x-axis is the particle time of flight through the A1900. 11 Be is the dominant species. The beam has low levels of contamination with 12 B, 13 B, and 9 Li. The repeated structure every ∼ 45 ns is caused by the arrival of the next RF beam bunch. Despite the 15% beam contamination, the measured charged-particle 𝛽 decay was essentially contamination free. The 1s-1s long on-off beam cycling removed most radioactive contaminants by way of letting them decay during beam delivery. The half-life of 13 B is 17.33(17)ms and the half-life of 12 B is 20.20(2)ms. The only contaminant with a half-live close to the 1 second on-off 28 cycle length is 178 ms 9 Li and it’s delivered on a level of just a couple percent. It may be present in the beginning of the 1 second measurement and will be entirely absent toward the end. 9 Li will decay by 𝛽− decay, but will appear in the detector virtually identically as those 𝛽− particles from 11 Be. (The 9 Li(𝛽− ) Q-value is 13606.45(20) keV and the 11 Be(𝛽− ) Q-value is 11509.46(24) keV.) There exists a 9 Li(𝛽− n)8 Be decay branch that will produce back-to-back 𝛼 particles from the breakup of the daughter 8 Be. The Q-value for 8 Be−→2𝛼 is 91.84(4) keV, which is dominated by the 11 Be(𝛽− ) background and not feasibly accessible with GADGET. As E18507 was a serendipitous opportunity after the commissioning of GADGET, the full MICROMEGAS was not able to be implemented. During GADGET commissioning, several preamplifier channels were damaged so Pads F, G, L, and M were not instrumented. Pads B and E were demoted to diagnostic veto pads and only Pads A, C, and D were used for measurements. 2.2 NSCL Experiment E19030 Following the analysis of NSCL E18507 described in Section 3.2, we proposed a subsequent measurement to gather greater statistics and improve the search sensitivity. Our proposal was awarded 96 hours of NSCL beam time, with roughly 76 hours of beam delivery to GADGET. The experimental setup for E19030 was nearly the same as E18507, but the full MICROMEGAS was instrumented as described in Section 2.1.2 and a MSX09 300𝜇m silicon PIN detector by Micron Semiconductor Ltd. was implemented upstream of GADGET. During dedicated runs with an attenuated beam, the silicon detector was inserted into the beam to perform energy-loss vs time-of-flight measurements between the silicon detector and the A1900 focal plane scintillator. These measurements were used for an independent particle identification of the incoming beam, but were of limited utility due to poor performance of our silicon detector. Figure 2.8 shows the beam particle identification measurements from the facility. In E19030, the delivered beam was 29 Figure 2.8: Particle identification plot provided by NSCL facility in E19030. The y-axis is an uncalibrated measurement of in-flight energy loss through a small detector. The x-axis is the particle time of flight through the A1900. 11 Be is the dominant species. The beam has low levels of contamination with 12 B, 13 B, and 9 Li. The repeated structure every∼ 45 ns is caused by the arrival of the next RF beam bunch. 89% pure 11 Be at a rate of 660,000 particles per second. Beam contaminants were similar to the previous measurement and were, in order of abundance, were 13 B, 12 B, and 9 Li. Similarly, the anticipated impact of the decays of these contaminants in the measured charged-particle spectrum was negligible. For the GADGET-side particle identification runs, to protect the silicon detector and related electronics, the beam was attenuated by a factor of 3000 to roughly 200 particles per second. 30 CHAPTER 3 ANALYSIS: DATA SORTING, INFERENCE WITH 𝜒2 -MINIMIZATION The analysis of data from NSCL experiments E18507 and E19030 share similarities from being conducted with the same experimental set up. Primitive data from DDAS was first extracted and combed for a number of correlations between the Proton Detector and SeGA detector systems to make histogramed charged-particle and 𝛾 spectra to which quantitative analysis was applied. A number of analysis techniques will be discussed as ultimately, the E18507 dataset was statistics limited and did not reach desired precision goals. It was found that applying the E18507 analysis to the larger E19030 dataset lead to a systematically limited result that also missed desired precision goals, and a new Bayesian Markov Chain Monte Carlo code (Chapter 4) was developed to handle model-fitting the domineering 11 Be(𝛽− 𝛼)7 Li background. 3.1 Data Sorting To save data with the NSCL DDAS described in Section 2.2.2.2, a common template was used to store values from the various detectors and detector sub-systems. That is, data was extracted from each channel and saved when any single channel triggered the threshold for a ”valid” event. Thus, most event files have a single entry from a single detector sub-system. This choice was made to reduce detector deadtime and provide greater flexibility to vary coincidence and threshold cuts in software, by scanning over events and selecting coincident events by their DDAS timestamps in an offline analysis. To organize the DDAS events into ”physical” events, a data file is searched chronologically for a charged-particle measurement on a MICROMEGAS pad. For every hit, the subsequent DDAS events are searched up to a period of 12 𝜇s later, which is in excess of the approximately 8 𝜇s 31 drift time for charges traveling the full length of the detector. The analysis software looks for other events on MICROMEGAS pads, veto (outer) and measurement (inner). Events that have triggered an outer veto pad are discarded as the charged-particle radiation was not confined to the sensitive volume of the Proton Detector. There are two data-sorting schemes to deal with events in the 12 𝜇s time window and stay within the Proton Detector sensitive volume: the composite pad scheme and the individual pad scheme. In the composite pad scheme, charge measurements from all five interior MICROMEGAS pads A-E are added together to create a single charged-particle spectrum. Contributions are gain matched, or aligned with a pre- energy calibration, to spectral features to standardize the outputs across the pads before summation and inclusion in the histogram to be analyzed. In the individual pad scheme, every other pad is treated as a veto pad and one histogram is created per pad. In this way, the individual pads each function as their own, smaller detector. The composite pad scheme has greater efficiency due to the larger functional-pad area, however has worse energy resolution due to differences in the individual pad electronics and a higher-energy 𝛽 background that reaches into the ∼ 200 keV energy region of interest from the large volume for 𝛽 particles to deposit energy. The individual pad scheme suppresses the 𝛽 background and has better resolution, but has worse detection efficiency, due to the more selective veto/exclusion cuts. 3.2 NSCL Experiment E18507 Figure 3.1 shows the unsorted charged-particle events on all instrumented pads. Individual features are difficult to identify, due to incomplete charge deposition per pad, except for a sharp feature around ADC channel 12000 in only the quadrant Pads B, C, D, and E. Application of the individual pad veto scheme yields Figure 3.2. The sharp features in the quadrant pads disappear and five distinct features remain. At the lowest energies, we have the 𝛽− spectrum, attenuated by the low-density detector gas. The other features are related to the ∼ 3% 11 Be(𝛽− 𝛼) branch [46, 47, 27]. 32 counts/40 channels padA padC padD 105 padB padE padH 104 padI padJ padK 103 102 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 ADC Channel Figure 3.1: The unsorted, raw spectra of all Pads in E18507. There are general features such as a large amount of counts below channel 1000, subtle peaks around channels 5000, 10000 and 15000, with fairly flat connections between peaks, and sharp peaks in some spectra around 12000. The behavior below 1000 channels comes from 𝛽 particles. The broad peaks at 5000, 10000, and 15000 are associated with the 11 Be(𝛽− 𝛼) decay. The sharp peaks near 12000 channels are thought to be electronic noise, perhaps from inefficiencies in the transitions of the gating grid. 3.2 Understanding the Li+𝛼 spectrum The nuclear structure information used in the first analysis came from the compilation and review of Ref. [39] and the 𝛽-decay feeding information from [47, 46]. The 11 Be(𝛽− 𝛼) proceeds through the 3/2+ state located at 9873(4) keV with a total intensity of 3.1(4)%. This 3/2+ state has an intrinsic width of 109(14) keV and decays via emission of an 𝛼 particle to 𝛼 and 7 Li, primarily to the 7 Li ground state. Some fraction of these decays populate the 478 keV excited state of 7 Li. Ref. [47, 46] measured this branch to populate the ground state 87.4(12)% and the 478 state 12.6(12)%. The 𝛼 separation energy of 11 B is 8664.310(10) keV, so 𝛼-decays to the 7 Li ground state have 33 counts/40 channels 105 padA 104 padC padD 103 102 10 1 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 ADC Channel Figure 3.2: The sorted, raw spectra of Pads A, C, D in E18507. Comparing to the unsorted histograms shown in Fig. 3.1, there is finer structure here that was revealed by the data sorting and event-vetoing algorithm. There remain a large amount of counts below channel 1000. The peaks around channels 5000, 10000 and 15000 are now clearly visible, in addition to a new peak around channel 3000. The behavior below 1000 channels comes from 𝛽 particles. The peaks peaks at 3000, 5000, 10000, and 15000 are associated with the 11 Be(𝛽− 𝛼) decay. The sharp peaks near 12000 channels are were present previously have been removed by the sorting and veto algorithm. 1209(4) keV center of mass kinetic energy and decays to the 7 Li excited state have 731(4) keV center of mass kinetic energy. To make sense of Fig. 3.2, one must consider additional detector effects coming from decays originating on the wall of the detector. Either by diffusion of 11 Be, by incomplete neutralization of the 11 Be beam, or both of these effects, a significant fraction of decays can originate from the detector cathode. The 𝛽− 𝛼 decay will produce a back-to-back 𝛼 and 7 Li pair, so decays originating on the interface of the detector gas and the detector body will lead to measurements of only one of the two nuclear-fragment charged-particle radiations. That is, for each of the two 7 Li final states, the following sets of charged-particle radiations will be measured: 34 𝛼 and 7 Li both, 𝛼 alone, or 7 Li alone. The energy sharing between the two fragments is given by kinematics determined by particle mass. Table 3.1 shows the breakdown of central energies for each detection possibility. Due to the broad nature (Γ = 109(14) keV) of the 9873 keV 𝛼-emitting state [39], the peaks from each of these radiations have substantial widths and considerable overlap in some cases. Description Center of Mass Energy 𝛼 contribution 7 Li contribution 11 Be(𝛽− 𝛼)7 Li(g.s.) 1209(4) 769(3) 440(2) 11 Be(𝛽− 𝛼)7 Li(𝛾) (478 keV) 731(4) 465(3) 266(2) Table 3.1: Central energies of decay events seen by the Proton Detector. Actual energies are a distribution determined by the 109(14) keV intrinsic width of the emitting 9874 3/2+ state in 11 B [39]. Energies are in keV. With these more complex spectra from wall-effects in mind, we can now make sense of the structures in Fig. 3.2. At ADC channels ¡2000, we have the suppressed 𝛽− spectrum. Around channel 3000, we have measurement of the cathode-originated 7 Li-recoil alone from the excited- state feeding of 7 Li (266 keV). The peak around channel 5000 is a composite peak made of two cathode-originating decays, the 𝛼-particle alone from the excited state feeding of 7 Li (465 keV) and the 7 Li-recoil alone of the ground state 7 Li feeding (440 keV). The peak around channel 10000 is another composite peak, consisting of the in-gas, full-energy measurement to the 7 Li excited state (731 keV) and the cathode-originating 𝛼-particle alone from the ground state feeding of 7 Li (769 keV). Finally, around channel 15000, we have the in-gas, full-energy decay to the 7 Li ground state (1209 keV). (Jumping ahead, see Fig. 4.9 for a theory calculation of all possible 𝛽− 𝛼 -component spectra before detector effects and note the considerable overlap between some different components.) This is demonstrated further in Fig. 3.3 and 3.4. Fig. 3.3 shows the charged-particle spectra for Pads A, C, and D in the individual pad veto scheme in coincidence with a 478 keV 𝛾 ray (470 35 to 480 keV). The in-gas, full-energy peak is strongly suppressed and only appears from accidental coincidences, arising from a 478 keV gamma from a decay outside of the Proton Detector or a Compton scatter event depositing 478 keV. Fig. 3.4 shows this same coincidence but subtracts a background of coincidence on 𝛾 rays from 482 to 492 keV. Now, the spectra are virtually pure 11 Be(𝛽− 𝛼)7 Li(𝛾). (The energies are roughly 266, 265, and 731 keV). Unfortunately, these spectra are severely statistically limited and have limited further utility. counts/12.0 keV 104 PadA 103 PadC PadD 2 10 10 1 0 200 400 600 800 1000 1200 1400 Energy (keV) Figure 3.3: Individual pad spectra for Pads A, C, and D in E18507 coincident with a 478 keV gamma from the deexcitation of 7 Li∗ fed from the 11 Be(𝛽− 𝛼) decay. Accidental coincidences with the 11 Be(𝛽− 𝛼)7 Li decay are present, but the low-energy portion of the spectra is enhanced relative to the high energy portion. 36 50 subtracted counts/12 keV PadA 40 PadC 30 20 PadD 10 0 − 10 − 20 0 200 400 600 800 1000 1200 1400 Energy (keV) Figure 3.4: Individual pad spectra for Pads A, C, and D in E18507 coincident with a 478 keV gamma from the deexcitation of 7 Li∗ fed from the 11 Be(𝛽− 𝛼) decay with random coincidence background subtraction. Only the 11 Be(𝛽− 𝛼)7 Li∗ emissions are visible here, with poor counting statistics. There is the 7 Li-alone at ∼ 265 keV, the 𝛼-alone at ∼ 465 keV, and the 7 Li+𝛼 full-energy decay at ∼ 731 keV. An important thing to note is that the properties of this state have been determined by a combination of 11 Be(𝛽) decays and 7 Li(𝛼,𝛼), 9 Be(3 He,𝑝), 12 C(𝑡,𝛼𝛾), and 14 N(𝑛,𝛼) reactions [39]. The energy and intrinsic width of the 𝛼 emitting state was determined by reaction measurements and are at odds with the recent 11 Be(𝛽− 𝛼) measurement in Ref. [27]. It was found in the the later E19030 analysis that the recent value of the intrinsic width Γ = 233(3)stat (3)syst , measured by 𝛽− 𝛼 decay and not nuclear reaction quantities, better agreed with what we measured in our 𝛽− 𝛼 measurement. (See Fig. 4.6.) The energy and width values in Ref. [27] were adopted in the E19030 analysis, but not the E18507 analysis. This change came as the new measurement was published in 2019, after the analysis for E18507 was largely completed. 37 3.2 Energy Calibration Application of the veto schemes and a linear energy calibration yields Figure 3.5. The energy calibration comes from fitting the peaks of the 7 Li ground state feeding decays at 440 (7 Li), 769 (𝛼), and 1209 keV (7 Li and 𝛼) to Gaussian profiles in each measurement pad. Due to the composite nature of the two lower-energy peaks, only the tops of the peaks (± ∼ 200 channels) were fit under the assertion that the peak maximum is determined by the ground state feeding peak energies, even if the full peak shape is harder to describe. 105 counts / 1.2 keV padA padC padD sum-A+C+D 104 padComp 103 102 10 1 0 200 400 600 800 1000 1200 1400 Energy (keV) Figure 3.5: The sorted and energy calibrated analysis spectra of E18507. Both individual pad and composite pad schemes are shown. The 𝛽 background is suppressed in the individual pad scheme, but at the cost of counting statistics. In the composite scheme, the recorded energy is the sum of energies across all measurement pads, within the correlation time window, corrected by the individual-pad-scheme calibrations. 38 There is a small shift of the composite pad spectrum to higher energy, as the larger effective pad yields greater 𝛽-particle energy deposition, but this effect is small (∼ 10 keV). 3.2 Efficiency and Normalization to the 3% 𝛽− 𝛼 branch To deduce a branching ratio, a measurement of the total number of 11 Be decays is needed to normalize any measured non-zero intensity. For this, we used the intensity of the 3.30(10)% 11 Be(𝛽− 𝛼) background [27]. By integrating the charged-particle spectra and applying calculated efficiency corrections, we estimated the total number of 11 Be decays ”eligible” for measurement on each pad. The ratio of detected counts and this calculated number of atoms is the branching ratio. The first step in calculating efficiency corrections was to simulate the particle ranges in the detector gas using the code ”SRIM: The Stopping and Range of Ions in Matter” [48]. SRIM is a particle-based simulation code that calculates ion interactions in matter and can calculate a variety of useful descriptions, such as particle range, energy deposition profile, and fraction of energy lost to different ion stopping processes (ionization, lattice disruptions in solid targets, kinetic energy sharing with target). The simulated range distribution was modeled with a Gaussian profile. The average range as the central value and the calculated straggling was treated as the Gaussian width 𝜎. Although physical events will have a distribution of energies from the intrinsic width of the 𝛼-emitting 9874 keV state in 11 B, only the central energy was modeled. A Monte Carlo code was developed to estimate the efficiency of an event measured in a detector pad to survive the software veto cuts. This efficiency is normalized to each pad. To clarify, in calculating the pad efficiency, only tracks that project to the given pad are considered. In the limit of infinitesimal tracks, the efficiency of Pad A/C/D/ACD would be 100% as a track would originate and terminate in the same location, in only the same single pad. The SRIM-calculated ranges were input for several particle-energy combinations to be calculated and the distribution of 11 Be decays 39 Decay Type Location Pad A Efficiency (%) Pads C/D Efficiency (%) p+10 Be in-gas 77 86 cathode 88 93 𝛼+7 Li∗ in-gas 54 71 cathode 77 85 𝛼+7 Li in-gas 37 61 cathode 68 79 Table 3.2: Simulated charged-particle detection efficiency per pad for possible 11 Be(𝛽− 𝛼)7 Li decay radiations. Numbers are in percent. was assumed to be uniform in the detector, due to the long half-life allowing substantial diffusion before decaying. We simulated the efficiency of a 180 keV proton, a 18 keV 10 Be recoil, 𝛼-particles at 466 and 770 keV, and 7 Li recoils at 266 and 440 keV. Simulations were run for both in-gas decays, where the total event length is used (as the 7 Li+𝛼 decay is back to back) and the cathode-originating decays, where only one particle will be measured, leading to an shorter effective track. Table 3.2 lists the estimated detection efficiency of each type of decay. These values are weighted by the frequency of the in-gas vs cathode-originating fractions (∼ 60% vs ∼ 40%) and the 7 Li∗ vs 7 Li(g.s.) feedings (13% vs 87% [46]). The relative efficiency between 𝛽− p and 𝛽− 𝛼 was determined to be 1.49 for Pad A and 1.25 for Pads C and D. Integrating the spectrum, events depositing 200 keV of center of mass energy or greater are interpreted as 𝛽− 𝛼 events and used in normalization. Pad A recorded 112,285 events, Pad C recorded 223,966 events, and Pad D recorded 141,834 events. Assuming the 8 · 10−6 branch of Ref. [6], we would expect to observe 40 to 75 𝛽− p counts per pad. 3.2 Statistical Inference of 𝛽− p Statistical inference of the number of 11 Be(𝛽− 𝑝) counts was done on an individual pad basis. The inferred 𝛽− 𝑝 counts were normalized to the respective 11 Be(𝛽− 𝛼) spectra and efficiency calculations to calculate a branching ratio. The average branching ratio result(s) were averaged, weighted by 40 their standard 1 𝜎 error(s). Extraction of the observed 11 Be(𝛽− 𝑝) counts was performed by fitting an equation, which one parameter which represented the number of 11 Be(𝛽− 𝑝) counts, to the experimental histograms. The function parameters were optimized by a gradient-descent 𝜒2 minimization algorithm implemented by ROOT’s default histogram ”Fit()” function. Spectra were fit by a function consisting of the sum of an exponential function meant to capture the 𝛽-particle background, a quadratic polynomial meant to capture the 11 Be(𝛽− 𝛼) in the minimum around 𝐸 ∼ 200 keV, and a Gaussian peak meant to capture the possible 11 Be(𝛽− 𝑝) signal. In principle, the 𝛽− 𝑝 signal has contributions from each of the decay locations: a peak at ∼91% of the center of mass energy of the cathode-originated proton alone and second peak at higher energy from the in-gas proton and 10 Be recoil. Usually, gas ionization and ion energy loss processes are assumed to be linear, but SRIM estimated that only ∼ 50% of the 10 Be recoil energy made its way to ionization. (For all other particles, this fraction is ∼95+%.) Additionally, there is the cathode-originated 10 Be- alone radiation, but the estimated energy of such a decay is ∼ 20 keV and too low in energy to be detected by the Proton Detector. In this first and simplest analysis, these details were not yet fully investigated and the Gaussian peak was determined sufficient for a simple ”bump hunt” analysis. To search for the 11 Be(𝛽− 𝑝), we still need to further define our search function, namely the detector resolution for our Gaussian search function. Due to the broad nature of the 11 Be(𝛽− 𝛼) in our spectra, further complicated by the cathode-originating decays and feeding of the 7 Li∗ state, extracting the detector resolution was not straightforward. Instead, the detector resolution was parameterized as a function of energy using the following 𝛽+ 𝑝 peaks from previous GADGET measurements: 401, 555, 943 keV peaks from 25 Si [35]; 206, 267, 579, 866 keV peaks from 23 Al [34]; 259, 803, 1026 keV peaks from 31 Cl [36]. Data was fit per pad with a linear curve with 𝜒2 minimization. The adopted detector resolutions were calculated by the parameterization for each search energy, but were roughly 8.1%, 11.6%, and 6.4% for Pads A, C, and D, respectively. 41 With our search feature and background models defined, a series of fits to data were performed for center of mass energies from 160 keV to 250 keV on each of Pads A, C, and D. The fit window was from 60 keV below to 40 keV above the nominal search energy and the nominal energy was scanned over the search range. This asymmetry in the fit window is afforded by the exceptional quality of the fit of the high-energy portion of the 𝛽-particle background to an exponential curve in this data and in previous GADGET experiments [35, 33, 34, 36]. Fig. 3.6 shows one fit to Pad C, searching for a peak at 175 keV. Additional fits to data can be found in Appendix A. PadCHistCal Entries 2442745 1600 Mean 148.7 Std Dev 31.63 1400 χ2 / ndf 62.96 / 60 Prob 0.372 height 8.865 ± 5.848 1200 Counts/1.5 keV a 808.6 ± 140.8 b −8.013 ± 1.552 1000 c 0.02315 ± 0.00425 eH 18.93 ± 0.73 800 eM −0.1068 ± 0.0063 600 400 200 0 120 140 160 180 200 Energy (keV) Figure 3.6: Data from Pad C and a fit of the data to a modeling function. The model was the sum of an exponential function with parameters 𝑒𝐻 and 𝑒𝑀, a quadratic polynomial 𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 , and a Gaussian with fixed 𝐸 = 175 keV center and ∼ 11.6% width. The Gaussian peak is meant to extract the number of 𝛽− 𝑝 counts in the ℎ𝑒𝑖𝑔ℎ𝑡 parameter. This fit yields 9(6) 𝛽− 𝑝 counts. Fig. 3.7 shows the statistical error-weighted average of this search over all three pads. Systematic errors are not estimated. There are two regions of note. The first is a clear excess is seen at E∼170 keV. This excess is greater than zero at the 2 𝜎 level at its peak and exists over roughly the width of 42 a peak with the average detector resolution. Secondly, the final point at E=250 keV also reaches a 2 𝜎 significance above zero. The fit here incorporates the 7 Li∗ -alone peak to a large degree. Fitting a novel, small peak on top of a large peak may serve as an uncontrolled systematic error. This point deserves additional scrutiny before being taken too seriously. −3 ×10 2014AMS 0.06 zero 0.04 bp branching ratio 0.02 0 − 0.02 − 0.04 160 180 200 220 240 260 Energy (keV) Figure 3.7: Results of E18507, plotting the averaged branching ratio of Pads A, C, and D, against assumed resonance energy. Plotted also, in dashed lines, is zero and the then-only previous inference of the branching ratio by AMS [6]. 43 With these averaged measurements, upper limits were derived. Upper limits were defined by a Gaussian probability density function that is truncated at the non-physical negative values, and normalized so the physically-allowed zero and positive values integrate to unity. That is, the limit is determined by the following Eq. 3.1. Here, 𝐶 𝐼 is a number between 0 and 1 that is the desired degree of confidence, 𝑔(𝑥| 𝑥, ¯ 𝜎) is the Gaussian distribution with central value 𝑥¯ and standard 68% error bar 𝜎, and 𝑏 is the value of the upper limit. We plotted results for 90% and 95% confidence limits, shown in Figure 3.8. ∫ 𝑏 0 ¯ 𝜎)𝑑𝑥 𝑔(𝑥| 𝑥, 𝐶𝐼 = ∫∞ (3.1) 0 ¯ 𝜎)𝑑𝑥 𝑔(𝑥| 𝑥, 90% Upper Limit Region 10− 4 95% Upper Limit Region 8 Parts Per Million β-p branching ratio 10− 5 10− 6 160 180 200 220 240 Energy (keV) Figure 3.8: Upper limits derived from individual pad averaged results of E18507 in Fig. 3.7. The upper limit is defined in Eq. 3.1. The red-dashed line shows a branch of 8 · 10−6 , which was inferred from the indirect AMS technique [6]. Over most of the search space, there is no ability to exclude the 8 · 10−6 branching ratio. Exclusion occurs only for 185 ≲ 𝐸 ≲ 195 keV at ∼ 90% significance. 44 From this upper limits plot, there is only a small range of energies with clear distinction from zero and the 𝛽− 𝑝 rate of 8 · 10−6 inferred from the indirect AMS technique [6]. Only in the region of E=185-195 keV is there rejection of the finite value at a 90% but not 95% level. Given the short, roughly six hour, duration of beam delivery in E18507 and the statistical error bars similar in magnitude between the values of the available 𝛽− 𝑝 hypotheses, we concluded an order of magnitude greater statistics would likely allow discrimination between the then available 𝛽− 𝑝 predictions (𝑏 𝑝 ∼ 10−8 ) and inferences (𝑏 𝑝 = 8.3(9) · 10−6 ). This analysis was written up as part of the NSCL PAC proposal E19030 in PAC43 and was awarded the 95 hours requested beam time. 3.3 NSCL Experiment E19030 The setup of E19030 is described in 2.2.4. The primary difference between E18507 and E19030 was full implementation of all 13 MICROMEGAS pads in E19030. The initial analysis of E19030 was similar to the E18507 analysis described in Sec. 3.2.2, but expanded upon a great deal and will be described here. 3.3 GADGET-side Particle Identification In E19030, we also instrumented a dE-TOF circuit to conduct particle identification at the degrader and diagnostic cross, just upstream of GADGET. The structure of these data events was different than the usual GADGET events discussed before. In particle identification mode, only the MSX silicon detector and the ORTEC566 TAC signals were analyzed and values saved. In the DDAS event building, TAC signals were always stored as separate events. (The TAC signals were delayed by the signal travel time from the experimental vault to the DataU user area, where the ORTEC566 was installed.) The DDAS events were sorted into physical correlations in software by the signature of a silicon detector event followed by the TAC event in the subsequent DDAS event. The dE-TOF 45 matrix is shown in Fig. 3.9 and projections are shown in Fig. 3.10 (energy projection) and 3.12 (time projection). The red dashed lines show possible boundaries for the particle identification analysis. 46 4000 3500 102 3000 Energy Deposition (arb. units) 2500 2000 10 1500 1000 500 1 0 0 5000 10000 15000 20000 25000 Time Of Flight (arb. units) Figure 3.9: Silicon energy deposition and time of flight matrix color coded by intensity. The dashed lines are proposed thresholds that may be useful in particle identification. The primary observation is the dominance of one peak, assigned to the 11 Be component of the beam, at energy 𝐸 ∼ 1220 (arb. units) and time of flight 𝑡 ∼ 12000 (arb. units). The silicon detector histogram has several peak-like features with only limited separation, but there may be peaks at ∼500, ∼750, ∼1220, ∼2500 channels. The relative intensities of these regions is 0.3%, 2.7%, 96.8%, and 0.2% respectively. The number of peaks nicely matches the NSCL-provided PID shown in Fig. 2.8, but the relative abundances are discrepant. Perhaps the low-energy shoulder of the ∼1220 channel peak is another species. This is visualized in Fig. 3.11, which shows the silicon energy histogram and a fit with the sum of four Gaussian curves. While there is generally a nice match to data in shape, the extracted peak widths do not increase in a regular pattern, which one would expect as the energy increases. The realized silicon detector energy resolution was roughly 11% at FWHM. This was determined 47 by fitting the highest-intensity, (assumed) 11 Be energy peak with a Gaussian profile, shown in solid red in Fig 3.10. Offline tests showed energy resolution of down to ∼6%, which is still in excess of expectations for silicon detectors [38]. Additionally, LISE++ [49] calculations estimated energy depositions of 11(2) MeV, 19(3) MeV, 19(3) MeV, and 38(4) MeV for the 9 Li, 11 Be, 12 B, and 13 B, respectively. Taking the 19(3) MeV energy loss of 11 Be as the central value, perhaps the 11(2) MeV 9 Li is observed at roughly half the energy of the most intense-peak, but the 38(4) MeV 13 B at twice the energy of the most-intense peak is not. de Entries 521316 Mean 1200 Std Dev 125.6 103 χ2 / ndf 423 / 247 Prob 2.139e−11 counts 3356 ± 6.4 center 1220 ± 0.1 56.78 ± 0.08 counts / bin 102 sigma 10 1 0 500 1000 1500 2000 2500 3000 3500 4000 Energy Deposition (arb. units) Figure 3.10: Silicon energy deposition (blue) used in the particle identification electronics. The dashed red lines are proposed thresholds that may be useful in particle identification. The solid red line is a superimposed Gaussian profile, fit by 𝜒2 minimization to measure the peak center and resolution. Similarly, the TOF measurement was less insightful than expected. There are suggestive peaks at ∼ 6000, ∼12200, ∼ 15000, and ∼ 21000 channels with relative intensities of 0.3%, 96.1%,3.2%, 48 de Entries 521316 Mean 1198 Std Dev 109.1 103 counts / bin 102 10 1 400 600 800 1000 1200 1400 1600 Energy Deposition (arb. units) Figure 3.11: Silicon energy deposition (blue) used in the particle identification electronics with a smaller range than in previous Fig. 3.10. The solid red line is a superimposed sum of four Gaussian profiles, fit by 𝜒2 minimization. While the fit reproduces obvious features, all Gaussian parameters are all free parameters. It is not obvious what secondary features are from beam contaminants and what are from non-ideal behavior of the detector. and 0.5%, respectively. Again, the odd, now high-energy, shoulder of the most-intense peak may arise from a true measurement of a contaminant or may be an artifact of the particle identification circuit. The resolution of the most intense peak was measured to be 6.8% in channels, but this is somewhat harder to quantify due to a lack of timing calibration. Offline tests with pulser signals showed sub-ns timing resolution. The same LISE++ simulations as a paragraph before predict time separation of ∼40 ns between the first-arriving 12 B and the last-arriving 9 Li, with species-specific time spreads of ∼4 ns, which closely matches the facility provided PID, shown in Fig. 2.8. 49 104 TOF Entries 506712 Mean 1.231e+04 Std Dev 876.6 χ2 / ndf 231.5 / 147 103 Prob 1.07e−05 counts 5467 ± 10.4 center 1.219e+04 ± 0.6 352.1 ± 0.5 counts / bin sigma 102 10 1 0 5000 10000 15000 20000 25000 30000 Time of Flight (arb. units) Figure 3.12: Time of Flight (blue) used in the particle identification electronics. The dashed red lines are proposed thresholds that may be useful in particle identification. The solid red line is a superimposed Gaussian profile, fit by 𝜒2 minimization. In conclusion, while the supplementary GADGET-side particle identification was first imple- mented here in E19030, there are obvious limitations of the performance of the silicon detector and associated hardware during the online measurements. From arguments in Sec. 2.2.3 around the minimal impact of radioactive contaminants due to their short half-lives and non- charged-particle emitting behaviors, and very clear nature of the facility provided PID, this section is included for completeness only. Difficulties in this circuit are expected to have no bearing on the primary 𝛽− 𝑝 measurement in GADGET. 50 3.3 Summarizing the Remaining 𝜒2 -Minimization Analysis 3.3 Limitations of E18507 Analysis The analysis of E19030 using the E18507 method, accounting for the additional instrumented pads, was quickly found to be of limited utility. With the increased statistics, the simple background of the sum of an exponential curve and a quadratic polynomial no longer provided adequate quality fits to the data over many of the pads. The search range with reasonable 𝜒2 p-values was E ∼ 180 − 220 keV, which is rather limited in scope. Further, even for these reasonable p-values, there is the question of large (several 𝜎 significance) negative values for the 𝛽− 𝑝 peak height parameter. This is demonstrated in Fig. 3.13, which shows the fit of Pad C from 160 to 260 keV searching for a peak of 220 keV. (This ∼ 3𝜎 negative-value peak at E=220 keV is common across pads, but Pad C is shown here for its large p-value of ∼ 0.5.) Negative branching ratio is nonphysical, so this highly significant negative intensity and decent p-value is suspect. Additional fits across pads, every 15 keV, are included in Appendix B. 51 pad 2 - 220 keV pad2Cal 2200 Entries 4.017694e+07 2200 Mean 217.3 2000 Std 2000 Dev 33.18 Counts/1.0 keV χ 2 / ndf 1800 1800 93.29 / 94 Counts/1.0 keV Prob 0.5014 1600 1600 height − 43.49 ± 13.66 1400 a − 2.636e+04 ± 6.196e+02 1400 1200 b 132.7 ± 3.0 1200 c − 0.135 ± 0.007 1000 eH 10.99 ± 0.03 1000 160 180 200 220 240 260 eM − 0.01125 ± 0.00040 Energy (keV) 160 180 200 220 240 260 Energy (keV) Figure 3.13: Pad C data from E19030 and a fit of the data to a modeling function. The model was the sum of an exponential function with parameters 𝑒𝐻 and 𝑒𝑀, a quadratic polynomial 𝑎 + 𝑏 · 𝐸 +𝑐 · 𝐸 2 , and a Gaussian with fixed 𝐸 = 220 keV center and ∼ 11.6% width. The Gaussian peak is meant to extract the number of 𝛽− 𝑝 counts. This fit yields −44(13) 𝛽− 𝑝 counts. The quality of the fit, as judged by p-value (𝑃𝑟𝑜𝑏), is excellent, however, a negative amount of counts is nonphysical and this fit yields negative counts at nearly 3𝜎 significance. For completeness, Fig. 3.14 is the (statistical) error-weighted average branching ratio, deter- mined across pads A, B, C, D in E19030 using the analysis of E18507. In comparison to the results of E18507 shown in Fig. 3.7, there are two notable features. First, the E19030 results have statistical error bars of ∼ 2 · 10−6 as we sought to attain, but secondly, significant swaths of the search region exist in nonphysical negative branching ratios with incredible statistical significance. The situation with the E18507 analysis is as follows: we limit ourselves to the search range of E = 175 to 220 keV and have to contend with explaining the ∼ 4𝜎 nonphysical ”signal” at E = 220 keV or we can reduce our search range but not on the merits of the p-value as a quality of the fit. 52 Is this evidence that the background is unknowable? Hardly, but a more sophisticated, physically motivated model is needed and the idea of systematic errors needs to be forefront in the decision making. ×10−3 2014AMS zero 0.02 0.01 bp branching ratio 0 −0.01 −0.02 160 180 200 220 240 260 Energy (keV) Figure 3.14: Results of E19030 with the E18507 analysis, plotting the averaged branching ratio of Pads A, B, C, D, and E against assumed resonance energy. Plotted also, in dashed lines, is zero and the then-only previous inference of the branching ratio by AMS [6]. The minimal 𝜒2 value is improved by inclusion of additional polynomial terms. (One additional term pushes fits to reasonable p-values across most pads at most energies.) However, beginning with a cubic function to model the lowest energy portions of the 11 Be(𝛽− 𝛼) spectrum, we can no longer guarantee the same selectivity between the 𝛽− 𝛼 background and any possible 𝛽− 𝑝 signal. A search peak, whether a simple Gaussian or one more complicated, based out of greater inclusion of physics and detector features, as will be detailed soon, will be of mixed concavity. On the tails 53 of the peak, the second-derivative concavity will be positive. Near the maxima, the concavity will be negative. The choice of summed exponential and quadratic polynomial curves to describe the background provided some guarantee against background-selection of our search peak. While 𝜒2 -minimization fitting of a parameterized curve doesn’t explicitly select parameters based upon their influence on any order derivative of the curve (only the trivial zeroth), this is meant as a qualitative argument against increasing the polynomial number until reaching a satisfactory 𝜒2 p-value. A polynomial of sufficiently high order will perfectly describe our background and 𝛽− 𝑝 signal without differentiation of the two. Inclusion of a search peak will only improve such a fit, with the only evidence of accidental equivalence between the background and signal is an increase of the correlation matrix elements between the signal strength parameter and the sufficient-ly numerous polynomial parameters. Furthermore, such a correlation matrix is defined at a singular solution and may present somewhat different values depending on exactly the amount of signal that is captured by the background model and exact shape of the contours of 𝜒2 space around that solution, for a singular fit out of the many fits performed across pads and energies. 3.3 Fitting With Other Background Models One way to give greater detail to the background model while minimizing the risk of confusing signal for backgrounds (or vice versa), is to fit using not polynomials, but physically motivated functions that have more tightly-constrained shapes. In experiment and simulations, the high- energy portion of the 𝛽− background is already well modelled by an exponential curve. It is the 7 Li∗ peak (and low-energy tail of the ground state, 7 Li-alone and excited 7 Li∗ feeding 𝛼-alone peaks to a smaller degree) that need a better modelling function. The energy spectra of isolated resonances are well described by a Lorentzian (alternatively Breit-Wigner or Cauchy, with different notations of Γ) curve [50]. The general shape of this curve 54 is: 1 𝐿(𝐸) ∼ (3.2) ¯ 2 + Γ2 /4 (𝐸 − 𝐸) The simplest assumption of the detector response is a Gaussian response for a Delta function of a single energy. Previous measurements with GADGET [33, 34, 35, 36], have demonstrated that a Gaussian detector response is a fair approximation. The detector response of an input with non-zero width would be the mathematical convolution of the physical input and the Gaussian detector response as in Eq. 3.3. Here, 𝐷 (𝐸) is the observed distribution of energies measured by the detector, 𝑔 is the detector response at 𝐸 for a physical signal at energy 𝑥, and 𝑑 is the distribution of actual, physical energies. ∫ 𝐷 (𝐸) = 𝑔 ⊗ 𝑑 = 𝑔(𝐸, 𝑥)𝑑 (𝑥)𝑑𝑥 (3.3) The convolution of the Lorentzian and Gaussian distributions is the so-called Voigt profile. This function does not have a closed form except for particular values of the Gaussian width 𝜎 or Lorentzian width Γ. Evaluation of the Voigt profile was implemented by ROOT’s Voigt function, which is part of the TMath library. The next background model attempted was the sum of three parts: an exponential curve to model the 𝛽− particles, a Voigt profile to model the 7 Li∗ peak, and a constant term, from incomplete charge deposition, seen in previous measurements with narrow background peaks. A variety of fits were performed with this new background function with different fit ranges and assumptions for the correct Γ width of the 7 Li∗ peak. What will follow are fits over 145-250 keV in each of the quadrant pads. The width of the 7 Li∗ peak is left as a free parameter for two reasons. First, our background isn’t purely one peak. From the energies included in the fit range, we largely exclude the higher 55 energy peaks, but not entirely. Secondly, a R-matrix (Sec 4.2.1) analysis of 11 Be(𝛽− 𝛼) [27] was published during this analysis. The width of the 𝛼-emitting state in 11 B was significantly broader than the evaluated average that was determined from reaction and scattering measurements [39]. The 𝛽− 𝛼 R-matrix deduced width was reported to be Γ = 233(3) [3] keV, which is a factor of two larger than the evaluated 109(4) keV. Perhaps the 𝛽− -fed width is greater than the reaction and scattering width, as the earlier 𝛽− 𝛼 measurement by Alburger et al. [47] of ∼ 330 keV, but they claim no significance due to a poor energy calibration. The search signal in this model is a sum of two Voigt curves. The first curve is centered at the full center of mass energy, originating from decays in the volume and the second curve is at ∼ 10/11 of the full energy from the kinematics of the emitted proton and 10 Be, and comes from measuring just the cathode-originated protons. The Gaussian width 𝜎 of the peaks was assumed to be 7.5% of the full peak energy. The peaks are weighted by the measured split of in-gas and cathode decays, measured by the 𝛾-ray/𝛼 and 𝛾-ray/7 Li coincidences from the 11 Be(𝛽− 𝛼)7 Li∗ (𝛾) decay. The inclusion of a finite width to the 𝛽− 𝑝 signal is motivated by a measurement of the 11 Be(𝛽− 𝑝) at TRIUMF with a similar detector. The TRIUMF-measurement group measured the width to be Γ = 12(5) keV [2]. Figures 3.15, 3.16, and 3.17 show the unnormalized results of this peak search for the quadrant pads, assuming a narrow (Γ = 0 keV), Γ = 12keV, and Γ = 24 keV 𝛽− 𝑝 emitting resonance in 11 B, respectively. The range of Γ values shown here were informed by the TRIUMF measurement’s value of 12(5) keV. To guide reading the unnormalized counts, with the new E19030 dataset, each 100 counts is ∼ 1 · 10−6 branching ratio. At low widths, Pads B, C, and D each peak around E ∼ 175 keV, are largely consistent with zero, then at E ∼ 210 keV and greater report −2𝜎 negative values. Pad E has similar behavior, but is shifted to higher energies ∼ 10 keV. As the Γ width increases, the measurements lose this peak-like behavior and develop into a step function with very large branching ratios (∼ 20 to 100 · 10−6 ) for energies below E ∼ 200 keV and moderately large 56 negative branching ratios (∼ −20 · 10−6 ) at higher energies. It is hard to interpret these curves as a physical peak and the simplest description is that sufficiently wide search signals match some feature in the background better than the assumed background model. For an example, if we consider the behavior at E = 175 keV for Γ = 0, 12, and 24 keV. As the assumed proton intrinsic width Γ𝑝 increases, so does the 7 Li∗ width Γ𝐿𝑖 and total 7 Li∗ intensity while the flat background decreases. This effect is usually ≲ 1𝜎 in each of these parameters going from Γ𝑝 = 0 keV to Γ𝑝 = 24 keV, but this pattern is ubiquitous across all four pads. Further, the statistical error really is only relevant to this point in that it allows a comparison between statistical and systematic errors. The data is the same for each fit and choice of Γ𝑝 and not a re-measurement of the data itself. The variation of re-measurements of data is what the 1 𝜎 statistical error estimates. This correlation is just that, a correlation, but the shared behavior across pads is suggestive (beyond the absurd step-function curves) that our background model has additional systematic error. That is, if our background were modeled perfectly, the resulting intensity of the 𝛽− 𝑝 signal should not have wild dependencies on the assumed signal shape. 57 Gamma = 0.0 keV bp counts 3000 2000 1000 0 − 1000 − 2000 pad1 − 3000 pad2 − 4000 pad3 pad4 − 5000 170 180 190 200 210 220 Energy (keV) Figure 3.15: Inferred 𝛽− 𝑝-like counts from fitting the E19030 quadrant pads with the Exponen- tial+Flat+Voigt model. The 𝛽− 𝑝 peak shape is described in the text and the fits here assume a narrow, Γ𝑝 = 0 keV proton-emitting resonance. Generally, an excess is seen for 𝐸 ∼ 175 keV. At high energies, nonphysical negative counts are extracted. 58 Gamma = 12.0 keV bp counts 10000 8000 6000 4000 2000 0 − 2000 pad1 pad2 − 4000 pad3 pad4 − 6000 170 180 190 200 210 220 Energy (keV) Figure 3.16: Inferred 𝛽− 𝑝-like counts from fitting the E19030 individual pads with the Exponen- tial+Flat+Voigt model. The 𝛽− 𝑝 peak shape is described in the text and the fits here assume a Γ𝑝 = 12 keV proton-emitting resonance. Generally, below 𝐸 ∼ 200 keV, a somewhat flat amount of excess is observed. At higher energies, a somewhat flat ”excess” of nonphysical negative counts are extracted. 59 Gamma = 24.0 keV bp counts 15000 10000 5000 0 − 5000 pad1 pad2 pad3 − 10000 pad4 170 180 190 200 210 220 Energy (keV) Figure 3.17: Inferred 𝛽− 𝑝-like counts from fitting the E19030 individual pads with the Expo- nential+Flat+Voigt model. The 𝛽− 𝑝 peak shape is described in the text and the fits here assume a Γ𝑝 = 24 keV proton-emitting resonance. Generally, below 𝐸 ∼ 200 keV, a somewhat flat amount of excess counts is observed. Counting excesses of this size yield branching ratios of 𝑏 𝑝 ∼ 50 − 100 · 10−6 , in excess of any previous measurement by more than a factor of five. At higher energies, a somewhat flat ”excess” of nonphysical negative counts are extracted. Concluding this chapter, a few other background models were utilized. The flatness of the flat background component and the precise shape of the 7 Li∗ peak were tested with a variety of slightly different functional forms. One such model was a sum of two exponential functions (one steep for 𝛽− particles and one shallow) and two Voigt profiles (one for 7 Li∗ peak and one for the composite 7 Li-𝛼∗ peak). An example of one of these fits is in the Appendix Fig. B.37, where it is shown that a 𝜒2 optimization provides an excellent fit to data far in excess of our search region of 𝐸 𝑝 < 250 keV. This high-parameter model did away with the step-function in the results of the simpler Expoential+Flat+Voigt model, but became very hard to work with. Seemingly insignificant 60 changes in starting values for the 𝜒2 minimization or assumed peak energy could make extracted 𝛽− 𝑝 intensities move on order or even in excess of their 1𝜎 statistical error. (See Appendix Fig. B.39.) In Chapter 4, I will explain how moving the fitting procedure from ROOT’s 𝜒2 optimization by gradient descent into a from-scratch Markov Chain Monte Carlo method with Bayesian priors allowed for a sort of simplification of the analysis by doing away with inflexible Gaussian probability density functions and ”baking in” all of our statistical and systematic uncertainties into one self- consistent analysis procedure. 61 CHAPTER 4 ANALYSIS: R-MATRIX, GAS STOPPING, BAYESIAN INFERENCE Chapter 3 is a summary of how the data for E19030 came to exist and an exhibit of analysis difficulties of searching for a ∼ 10−5 𝛽− 𝑝 branch on the low-energy tails of the ∼ 0.03 𝛽− 𝛼 branch from the decay of 11 Be. This chapter will be a more direct account of the final analysis and motivations for the details of the analysis. The general data sorting from Chapter 3 will carry over, but other details will be revisited in this chapter with greater scrutiny. This additional scrutiny came from the implementation of the Bayesian analysis, which requires well defined uncertainty quantification of its inputs. This chapter is meant to show the complete state of the analysis for E19030. Results and discussion will continue in Chapter 5. The Markov Chain Monte Carlo used in this chapter was developed in close collaboration with Scott Pratt and Pablo Giuliani. 4.1 Summary of what we do know Recapping, here are the following elements needed to infer the 11 Be(𝛽− 𝑝) branching ratio: • Data sorting to produce analysis spectra • Determination of a background shape • Determination of the 𝛽− 𝑝 signal shape • Inference of the number of 𝛽− 𝑝 events in the analysis spectra • Normalization of the 𝛽− 𝑝 counts to generate a branching ratio The data sorting will not be revisited and the summary in Sec. 3.1 is complete. The determina- tion of the background shape will be revisited with information from the recent R-matrix analysis 62 of 11 Be(𝛽− 𝛼) decay [27] in Sec. 4.2. The shape of the assumed 𝛽− 𝑝 signal will be revisited through the lens of low-energy ion-stopping processes. This, in addition to the new recommended energy of the 𝛽− 𝛼 decay from [27], has carryover consequences for the energy calibration and the resonance energy values hereon should not be compared to the previous analysis of Chapter 3. Statistical inference took a large step in complexity; Sec. 4.4 will introduce both Bayes’ theorem, the Markov Chain Monte Carlo method to ”solve” Bayes’ theorem, and how these ideas were implemented to infer the 11 Be(𝛽− 𝑝) decay rate. 4.1 Some data-inferred details Before diving into new interpretation and analysis tools, there are a few important details that still need to be presented that can be pulled from the data alone: normalization of the 𝛽− 𝑝 events, spacial distribution of the 11 Be decays, and the energy resolution of the detector pads. 4.1 Normalization of the 𝛽− 𝑝 counts First, we will revisit the Proton Detector detection efficiency calculations which allow us to normal- ize to the well-studied 11 Be(𝛽− 𝛼) decay branch and calculate a branching ratio from a number of 𝛽− 𝑝 counts. Following the detailed work of Tamas Budner with his own GADGET analysis [51], the efficiency code described in Sec. 3.2.1.2 was updated to include diffusion of the ionization elec- trons as they drift through the detector gas toward the MICROMEGAS. Tyler Wheeler performed MAGBOLTZ calculations for each of us to estimate this effect. MAGBOLTZ performs numerical particle-tracking based simulations that solve the Boltzmann transport equation for electrons drift- ing in a medium, pushed by an external electric field. The simulations include atomic excitation such as bound-electron orbital promotion and vibrational excitation in polyatomic molecules like CH4 [52, 53]. 63 For our 11 Be measurements at 600 Torr, it was calculated that the electric-drift-field transverse diffusion coefficient of these electrons was 𝐷 𝑡 = 14060 cm2 /s. Decays were assumed to happen uniformly throughout the ∼ 8𝜇s drift length of the Proton Detector. The results of this new simulation for in-gas 7 Li(g.s.)+𝛼 decays are shown in Fig. 4.1. Simulations were run for cathode- originated 7 Li-alone and 𝛼-alone peaks, but are not included. The general behavior of the these cathode-originated efficiency curves were similar to that of in-gas decays shown here. For in-gas decays of 𝐸 ∼ 1200 keV, the detection efficiency for Pad A (∼ 31%) is similar to the quadrant pads (∼ 39%, shown as Pad B). This is roughly half of the estimated ∼ 72% for the composite ABCDE pad. The composite pad would be expected to have much greater detection efficiency due to the larger area compared to its edge, compared to the area/edge ratio of its constituent pads. Figure 4.1: Simulated detection efficiency of in-gas 7 Li(g.s.)+𝛼 decay for different pad schemes and respective geometries. The quadratic polynomial fits are meant to parameterize the output of the simulation for fast calculations of arbitrary energy. Figure 4.2 shows a comparison of the individual pads spectra, the sum of the individual pad spectra, and the composite ABCDE pad spectrum. Ignoring subtle differences in the energy cali- bration, the individual pad summed spectrum is nearly as intense as the composite pad spectrum. 64 This goes against expectations based on the efficiency calculation based off of SRIM and MAG- BOLTZ, which predicted nearly a factor of 2 between individual pad -summed efficiencies and the composite pad efficiency. Figure 4.3 shows the bin-by-bin ratio of the summed spectrum over the composite pad scheme spectrum. The efficiency is somewhat flat with respect to energy, with the individual pad scheme capturing ∼ 85% of the events that the composite pad scheme captured. This indicates that many more charged-particle events are confined to a single measurement pad than the simulation suggested. This suggests the SRIM calculated ranges or the MAGBOLTZ diffusion coefficient or both are too large, and simulated events cross over measurement pad boundaries in excess of real events. In-simulation veto-scheme thresholds, analogous to detector trigger thresholds and software veto-scheme thresholds in the experimental data, could explain this discrepancy. While the normalization of the 𝛽− 𝑝 decay intensity is performed against the 𝛽− 𝛼 decay intensity in the individual pad scheme, pad by pad, and not across different pad schemes as shown here, the takeaway is that the intensity of the sum of individual pad spectra is equivalent to the composite pad only when charged-particle tracks were shorter than calculated and activate only a single pad. For radiations that rarely activate other pads, the detection efficiency of every 𝛽− 𝛼 radiation in either scheme must then be quite high. Hereon, the detection efficiency ratio between 𝛽− 𝛼 radiations and 𝛽− 𝑝 decay both will be treated as unity with a ≲ 20% error. The number of 𝛽− 𝛼 decays measured in each pad are shown in Table 4.1. This is calculated by integrating the individual pad spectra above 𝐸 = 200 keV. This choice of the boundary includes minimal 𝛽 particles and captures all but the lowest-energy tails of the 𝛽− 𝛼 spectrum. The inference of 11 Be normalization comes from the 3.30(10)% value of the 𝛽− 𝛼 branching ratio reported in Refsgaard et al. [27]. We would expect just about 1000 counts of 𝛽− 𝑝 per pad, assuming a 10−5 branch, similar to that reported by Ayyad et al. [2]. This is a small signal when most of the 65 counts / 0.60 keV 106 Individual Pads Sum Of Ind. Pads 105 Composite Pad 104 103 102 0 200 400 600 800 1000 1200 1400 Energy (keV) Figure 4.2: Individual pad spectra (green), summed individual pad spectrum (blue), and composite pad spectrum (red). The sum of individual pad spectra is similar in counts to the composite pad spectrum over much of the spectrum which is suggestive of relatively flat detection efficiency curves as a function of energy. background in the search region is ∼ 1000 counts/bin with 1 keV binning and the search signal will be ∼ 20 keV wide. However, assuming Poisson counting statistics, the error in counts of a √ bin or a region of bins is 𝜎 ∼ 𝑁, where 𝑁 is the number of counts in the region. A 20 keV wide signal will cover roughly 20,000 background counts. We expect statistical limitations at about √ 𝜎stat ∼ 20, 000 ∼ 140 counts. Thus, a 1000 count signal is small, but can appear with ∼ 7𝜎stat in the limit of no systematic uncertainties. 4.1 Cathode-volume fraction, decay location determination The relative intensity of the 10 Be+proton and 𝑝-alone peaks is determined by the physical distribu- tion of the 11 Be decays. This distribution was determined by the relative intensity of 7 Li+𝛼 against 66 ratio of sum-of-singles & ABCDE arb. units / 0.60 keV 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 200 400 600 800 1000 1200 1400 energy (keV) Figure 4.3: Bin-by-bin ratio of the summed individual pad spectra and the composite pad spectrum shown in Fig. 4.2. Overall, the sum of individual pads tend to have ∼ 85% of the total counts of the composite ABCDE pad. 7 Li-alone and 𝛼-alone peaks. Identification was done by considering charged-particle energy de- position and timing between detection of charge in the Proton Detector and a 478 keV 𝛾 ray in the SeGA array. Figure 4.4 shows a two dimensional histogram that was used to determine the location of 11 Be decays along the 𝑧-axis in Pad B. At all times, there is an intense band above ∼ 400 keV, another below ∼ 800 keV, and another at ∼ 1200 keV. These are from accidental coincidences. This is shown by the similar intensity of these bands after the sharp spike of events at times ∼ 7.2𝜇s. The detector drift field was set so that the drift time of the full length of the detector would be 7 − 8𝜇s, so the peak at ∼ 7.2𝜇s is interpreted as decays originating from the upstream detector cathode. Similarly, there are two spots in this time spike, one at 𝐸 ∼ 260 keV and one 𝐸 ∼ 440 keV. The peak at 𝐸 ∼ 260 keV is the 7 Li-alone detection and the 𝐸 ∼ 440 keV peak is the 𝛼-alone 67 Pad Label Integrated Counts Inferred 11 Be Normalization Inferred 𝛽− 𝑝 for 𝑏 𝑝 = 10−5 B 3.825 · 106 1.16(4) · 108 1.16(4) · 103 C 3.080 · 106 9.3(3) · 107 9.3(3) · 102 D 3.244 · 106 9.8(3) · 10 7 9.8(3) · 102 E 3.419 · 106 1.04(3) · 108 1.04(3) · 103 Table 4.1: Per Pad, integrated number of observed 𝛽− 𝛼 counts, inferred 11 Be normalization, and inferred 11 Be(𝛽− 𝑝) counts assuming a branch of 𝑏 𝑝 = 10−5 . detection. At lower time, there is a blob of intensity that comes from the in-gas decay and detection of both the 7 Li and 𝛼 together. With this understanding of the feature at ∼ 7.2𝜇s as the cathode, the events at greater times are certainly accidental coincidences. One can perform a background subtraction to remove these at earlier times. There are several possible methods to perform this, but the simplest is to project the coincident-events from this histogram to a one dimension histogram of energy. The formula for this background subtraction is below, if 𝐸 and 𝑡 are discrete bin values. Note that Δ𝑡 is then a number of bins. The background subtracted histogram calculated bin-by-bin from the original spectrum and the time-integrated projection of the late-time spectrum. ℎacc. coinc. (𝐸) ℎsub (𝐸, 𝑡) = ℎ 𝛾−coinc. (𝐸, 𝑡) − (4.1) Δ𝑡 This subtraction yields Fig. 4.5. The features that remain are the diffuse in-gas decays that measure the full charged-particle energy from the 11 Be(𝛽− 𝛼)7 Li∗ decay that arrive up to ∼ 7.2𝜇s after the 𝛾 ray, the measurement of 7 Li-alone and 𝛼-alone originating from the cathode arriving ∼ 7.2𝜇s after the 𝛾 ray, and the intense 𝛽-particle background at low energy. Due to the long tracks of the 𝛽 particles, the ionization from these radiations appears more diffusely in time than the heavy ions with short ranges and more localized tracks. The fraction of cathode and volume decays was taken by integrating the heavy ion 𝐸 > 200 68 timeBetween478andMMpad1 2000 MM energy (keV), binWid = 30 keV 1800 103 1600 1400 1200 102 1000 800 600 10 400 200 0 1 0 2000 4000 6000 8000 10000 12000 14000 time after 478 keV arrival, binWid = 100 ns Figure 4.4: Coincidence histogram of the Pad B energy and pad signal arrival time when coincident with a 478 keV 𝛾 ray. Events after ∼ 7.2𝜇s are accidental coincidences. The spike at ∼ 7.2𝜇s comes from detection of only one of the light ions from cathode-originated 11 Be(𝛽− 𝛼)7 Li∗ decay. The diffuse cloud of events at time < 7.2𝜇s and energy ∼ 700 keV are from full-energy 7 Li+𝛼 events originating in the detector gas. The low-energy events at all times are 𝛽− particles. keV coincident events in the background-subtracted spectra. Events recorded for 0 < 𝑡 < 6.8𝜇s were deemed in-gas volume events. Events recorded for 6.8 < 𝑡 < 7.6𝜇s were deemed cathode- originated. Pads B, C, D, and E had cathode-originated fractions of 63.4%, 71.6%, 73.1%, and 67.4% respectively. The choice of the energy threshold creates a ∼ 1.5% error in these numbers. The counting-statistics error is small compared to this threshold systematic error. 4.1 Detector Resolution Measurements In the previous analysis, the anticipated detector resolution was determined by fitting historical resolution data in other GADGET experiments that used different gas pressures and operational 69 timeBetween478andMMpad1 2000 MM energy (keV), binWid = 30 keV 1800 103 1600 1400 1200 102 1000 800 600 10 400 200 0 1 0 2000 4000 6000 8000 10000 12000 14000 time after 478 keV arrival, binWid = 100 ns Figure 4.5: Coincidence histogram of the Pad B energy and pad signal arrival time when coincident with a 478 keV 𝛾 ray after subtraction of accidental coincidences. Events after ∼ 7.2𝜇s are statistically consistent with zero. The spike at ∼ 7.2𝜇s comes from detection of only one of the light ions from cathode-originated 11 Be(𝛽− 𝛼)7 Li∗ decay. The diffuse cloud of events at time < 7.2𝜇s and energy ∼ 700 keV are from full-energy 7 Li+𝛼 events originating in the detector gas. voltages. Ideally, these settings were scaled proportionally to give historically similar behavior (see Figures A.1,A.2,A.3), but using data from within the E19030 experiment would still be best. Additionally, the Bayesian analysis that was developed does not strictly require Gaussian distributed errors and more general forms can be chosen. Figure 4.6 shows a fit of the in-gas, full energy 7 Li+𝛼 peak. In Pad B, the measured resolution is 5.0(8)% FWHM and intrinsic width is 225(2) keV. For Pads C, D, and E, the detector resolution was measured to be 5.4(4)%, 6.4(4)%, and 5.2(4)% FWHM respectively. The intrinsic widths were 226(2), 230(2), and 222(2) keV respectively. Figure 4.7 shows a fit to the low energy spectrum of Pad B, to extract detector resolution from 70 pad1Cal pad1Cal 3000 counts / 1.0 keV Entries 4.41809e+07 Mean 1203 Std Dev 111 2500 χ2 / ndf 522.4 / 494 Prob 0.1821 h 1.036e+06 ± 4.213e+03 2000 cent 1182 ± 1.3 sigm 24.95 ± 3.76 gama 224.7 ± 2.9 1500 exp1 24.27 ± 0.05 exp2 −0.01873 ± 0.00005 1000 500 1000 1100 1200 1300 1400 1500 Energy (keV) Figure 4.6: Pad B full-energy 11 Be(𝛽− 𝛼)7 Li decay peak with energy 1182 keV fit by a Voigt curve and an exponential curve, meant to model the high-energy tail of the 𝛽− 𝛼 spectrum to lower energy. The intrinsic width, 𝑔𝑎𝑚𝑎, is 225(4) keV and the detector resolution, 𝑠𝑖𝑔𝑚, is 5.0(8)% FWHM. The other fit parameters are: the integrated peak counts, ℎ; the peak center, 𝑐𝑒𝑛𝑡; an exponential height, 𝑒𝑥 𝑝1, and decay rate, 𝑒𝑥 𝑝2 that take the shape of exp(𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸). the other isolated peak, the 7 Li∗ peak. The measured detector resolution here is zero, but is highly correlated with the assumed intrinsic width of the 7 Li∗ peak, which was fixed to 82 keV as a kinematic rescaling of the width measurement of the full decay (the parameter 7𝑠𝑖𝑔𝑚 in Fig. 4.6). Freeing this parameter allowed fits returning up to ∼ 30% FWHM resolution and hard-to-believe small intrinsic widths. The other pads had low-energy resolutions of 4.6(4)%, zero, and 8.6(14)% FWHM, respectively. With obvious caveats of the correlation between the detector width and intrinsic width, this is suggestive that the detector resolution at low energy is somewhere in the range of 4.5-10% FWHM, which is reasonable for such a detector [38] and consistent with previous GADGET measurements. 71 7000 pad1Cal counts/1.00 keV Entries 4.41809e+07 Mean 243.8 6000 Std Dev 59.87 χ2 / ndf 211.1 / 200 Prob 0.282 5000 exp1 19.11 ± 0.18 exp2 − 0.1013 ± 0.0015 flat − 74.56 ± 27.44 4000 7hite 1.992e+05 ± 3.358e+03 7cent 238.6 ± 0.4 7sigm 0.05387 ± 0.00001 3000 72hite 2.751e+06 ± 1.525e+05 72sigm 41.88 ± 3.79 2000 exp21 9.574 ± 0.111 exp22 − 0.02369 ± 0.00081 1000 0 120 140 160 180 200 220 240 260 280 300 320 Energy (keV) 200 100 0 −100 100 150 200 250 300 Figure 4.7: (Top) The low-energy spectrum of Pad B. The fit function is the sum of two exponential functions, a constant, a Voigt peak (7 Li∗ ), and a semi-restricted Voigt peak (7 Li). The intrinsic width is fixed to 82 keV scaled from the width measured in Fig. 4.6. The fit resolution (from the parameter 7𝑠𝑖𝑔𝑚) for Pad B is zero. Other pads show up to 5% FWHM. The other fit parameters are: an exponential height, 𝑒𝑥 𝑝1, and decay rate, 𝑒𝑥 𝑝2 that take the shape of exp(𝑒𝑥 𝑝1+𝑒𝑥 𝑝2· 𝐸); a second exponential height, 𝑒𝑥 𝑝21, and decay rate, 𝑒𝑥 𝑝22 that take the shape of exp(𝑒𝑥 𝑝21 + 𝑒𝑥 𝑝22 · 𝐸); a constant background term, 𝑓 𝑙𝑎𝑡; the integrated peak counts in the 7 Li∗ peak, 7ℎ𝑖𝑡𝑒; the 7 Li∗ peak center, 7𝑐𝑒𝑛𝑡; the integrated peak counts in the 7 Li&𝛼∗ peak, 72ℎ𝑖𝑡𝑒, and the 7 Li&𝛼∗ peak resolution, 72𝑠𝑖𝑔𝑚. (Bottom) Fit residuals. 4.2 R-matrix Guidance to Describing the 𝛽− 𝛼 Background The R-matrix theory is a non-relativistic quantum mechanical scattering theory that was developed largely independently for atomic and nuclear physics [54]. The present utility of the R-matrix theory comes from the recent R-matrix characterization of the 11 Be(𝛽− 𝛼) decay spectrum by Refsgaard et al. [27]. R-matrix theory calculations using the Refsgaard-suggested values were used to better understand the 11 Be(𝛽− 𝛼) background in GADGET. 72 4.2 Introduction to R-matrix theory The R-matrix is a phenomenological multi-channel scattering theory, developed by numerous au- thors, although the fundamental inception is credited to Kapur and Peierls [55, 54]. The framework has been used widely to relate nuclear scattering observables to nuclear structure. Modern formu- lations, such as the Brune formalism [56] have theory inputs with clear physical analogs such as nuclear state energy, spin, parity, and mixing parameters with other states. Furthermore, the Brune convention is used in the R-matrix code Azure2 [31] which was used to perform the R-matrix calculations of the 11 Be(𝛽− 𝛼) background. In the R-matrix theory, the nuclear wavefunction is described with in two regions. The wave- function of the internal region has well defined angular momentum 𝐽 and 𝑀 and can be expanded into a complete set of basis eigenstates. The wavefunction of the external region is assumed to be subject to just Coulomb interactions and well defined by allowed coupled-particle channels. The external wavefunction, then, is described by inbound and outbound Coulomb functions. These Coulomb functions are fully analytic and have well defined energy, charges, masses, and angular momenta. The utilization of the theory falls out of matching the external Coulomb functions and their derivatives to solution to the interior wavefunction at the boundary surface. The R-matrix is defined as: ∑︁ 𝛾𝜆𝑐 𝛾𝜆𝑐′ 𝑅𝑐𝑐′ = (4.2) 𝐸𝜆 − 𝐸 where 𝛾𝜆𝑐 is the integral of the internal eigenstate wavefunction of the 𝜆th level and the external wavefunction corresponding to the 𝑐-channel. Thus, 𝛾𝜆𝑐 is a measure of the contribution of the 𝜆th level to the observation of the 𝑐-channel. 𝐸𝜆 is the energy of the 𝜆𝑡ℎ level and 𝐸 is the scattering energy. With this R-matrix defined, calculations can be performed with the collision matrix, which is calculated from the following inputs: the R-matrix, channel energies, wavefunction-matching 73 boundary radius, and the incoming and outbound Coulomb wavefunctions. Alternatively, to simplify computation of the collision matrix in the case of few levels, one may use the A-matrix formalism. The A-matrix is defined by its inverse: Γ𝜆𝜆′ ( 𝐴−1 )𝜆𝜆′ = (𝐸𝜆 − 𝐸)𝛿𝜆𝜆′ + Δ𝜆𝜆′ − (4.3) 2 with the following definitions: ∑︁ Δ𝜆𝜆′ = − 𝛾𝜆𝑐 𝛾𝜆′ 𝑐 (𝑆 𝑐 − 𝐵𝑐 ) (4.4) 𝑐 2 Γ𝜆𝑐 = 2𝑃𝑐 𝛾𝜆𝑐 (4.5) ∑︁ Γ𝜆𝜆′ = 2𝑃𝑐 𝛾𝜆𝑐 𝛾𝜆′ 𝑐 (4.6) 𝑐 Where 𝑆 𝑐 is the shift factor, 𝑃𝑐 is the penetration factor and 𝐵𝑐 is the boundary condition parameter. These values are related to the wavefunction and the logarithmic derivative matching at the boundary radius [57]. Barker and Warburton [58] write the 𝛽-fed spectral function in terms of this mathematically equivalent A-matrix: ∑︁ 𝑁𝑐 (𝐸) = 𝑓 𝛽 𝑃𝑐 | 𝐵𝜆 𝛾 𝜇𝑐 𝐴𝜆𝜇 | 2 (4.7) 𝜆𝜇 Here, 𝑓 𝛽 is the phase-space factor for the 𝛽 decay, 𝐵𝜆 is the 𝛽 decay feeding strength, and 𝐴𝜆𝜇 is an element of the A-matrix. The form of Eq. 4.7 is simpler than in Barker and Warburton, but this is due to the consideration of Gamow-Teller decays only in the Refsgaard analysis of 11 Be(𝛽− 𝛼). Gamow-Teller decays have Δ𝑆 = 1 from spin alignment of the emitted electron and anti-neutrino. Fermi decays are the opposite, having Δ𝑆 = 0 and anti-alignment of electron and anti-neutrino spin. 74 The ground state of 11 Be is 1/2+ and the 𝛽− 𝛼 emitting state in 11 B has been identified as 3/2+. Allowed 𝛽 decays impart no angular momentum (that is, Δ𝐿 = 0). Angular momentum algebra requires a 1/2+ −→ 3/2+ decay to be Gamow-Teller, Δ𝑆 = 1. The Azure2 code performs these calculations, taking as input structure information and 𝛽 decay feeding information. One output of the Azure2 calculations is the spectral shape from each channel, that is the shape as a function of energy of the 11 Be(𝛽− 𝛼)7 Li and 11 Be(𝛽− 𝛼)7 Li∗ decays. 4.2 R-matrix Calculations of 11 Be(𝛽− 𝛼) The suggested R-matrix parameters from [27] are summarized in Table 4.2. They report statistical and systematic errors both, and we consider parameters in these values as the root of the sum of squared errors. The radial boundary term 𝑟 0 , which sets the R-matrix boundary radius at 𝑅 = 𝑟 0 ( 𝐴11/3 + 𝐴21/3 ), is adopted to be 1.6 fm. 𝐴1 and 𝐴2 are the mass numbers of the outgoing channel particles, or 7 and 4 here. 𝜆=1 𝜆=2 𝐸𝜆 (keV) 9846(1)[10] 11490(80)[50] Γ𝜆1 (keV) 233(3)[3] 430(150)[50] Γ𝜆2 (keV) 20.4(3)[3] 50(60)[50] M(GT) 0.717(12)[7] 1.05(17)[5] B(GT) 0.318(11)[6] 0.7(2)[1] ln(ft) 4.08(3)[2] 3.8(3)[1] Table 4.2: R-matrix parameters adopted from [27]. Values in parentheses are their statistical errors and values in square brackets are their systematic errors. Figure 4.8 shows the output of the R-matrix calculation with the central values of Tab. 4.2. The energy scale is relative to the 8.664 MeV 𝛼 separation energy in 11 B. The two output channels and sum are color coded. It is the primary peak of each channel that is seen in the GADGET data. The upturn in the R-matrix output above 2 MeV of excitation energy is not seen in the individual pad data, but is somewhat in the composite pad scheme, possibly due to detection efficiency effects at 75 these larger energies. Rather, there is an absence of events above ∼ 2750 keV. The ”turn-off” energy in the R-matrix is ∼ 2800 keV. The Q-value for the 𝛽− 𝛼 is 2.834 MeV and drives this ”turn-off” of the spectral strength. The small difference in the ”turn-off” energies is a nice test of our energy calibration extrapolation from ∼ 1.2 MeV. (The energy calibration will be revisited in light of the exploration of very-low energy ionization processes, Sec. 4.3.2.) spectral strength (arb. units) 102 A+7Li* (478 keV) 10 A+7Li (g.s.) All Channels 1 10− 1 10− 2 10− 3 10− 4 0 500 1000 1500 2000 2500 3000 3500 Excitation Energy (keV) Figure 4.8: The R-matrix calcuated spectral strength of the 11 Be(𝛽− 𝛼) with the central values of Table 4.2 [27]. The curves are the 7 Li∗ -fed channel (red), 7 Li grounds state -fed channel (blue), and the sum of the two channels (black). The two-state structure is evident in the two-peak behavior of the spectral output, despite the strong 𝛽-feeding preference of the lower-energy state. Figure 4.9 shows the R-matrix output representing the physical spectra and also the inferred individual 𝛼 and 7 Li spectra. The 𝛼 and 7 Li individual spectra are not output from Azure2, but are calculated by rescaling of the energy of the output. The rescaling comes from the two-body kinematics of the 11 B(𝛼)7 Li breakup. 76 spectral strength (arb. units) 102 A+7Li* (478 keV) A+7Li (g.s.) A-alone (Li*, 478 keV) Li-alone (Li*, 478 keV) A-alone (Li, g.s.) 10 Li-alone (Li, g.s.) 1 10− 1 10− 2 0 500 1000 1500 2000 2500 3000 3500 Excitation Energy (keV) Figure 4.9: The R-matrix calculate strength of 11 Be(𝛽− 𝛼) with the central values of Table 4.2 [27]. The thick curves are the 7 Li∗ -fed channel (red), 7 Li ground state -fed channel (blue). The thin curves show the kinematically scaled distributions for the 7 Li-alone (lower energy of a color) and the 𝛼-alone (higher energy of a color). Note the strong overlap of the 7 Li(g.s.)-alone and 𝛼-alone from the 7 Li∗ feeding, and the 𝛼-alone from the 7 Li(g.s.) feeding and the full-energy 𝛼+7 Li∗ . The initial thought was to use the R-matrix calculations directly in the fit to the GADGET spectra. However, it was quickly determined that such a scheme would be computationally challenging. In this scheme, we would want to smear the output by an assumed detector response, but early tests showed this taking prohibitively long if the smearing would be parameterized and included in the fit and not merely calculated once at the beginning of a fitting code. Furthermore, which R-matrix curve should be used? Refsgaard et al. [27] gives individual R-matrix parameters and associated errors. It is not these parameters that are comparable to GADGET data, but instead the curves they generate, which add additional computational overhead. Due to these challenges in using actual R-matrix calculations in the fit to data, we investigated if the R-matrix output could be 77 approximated locally, in the region of interest, with a quickly-evaluated curve that could take its place in the analysis. The lowest energy peak, the 7 Li∗ peak with 𝐸 ∼ 256 keV of energy is the primary 𝛽− 𝛼 background in the region of interest. The 7 Li∗ peak calculated by Azure2 was fit by a Lorentzian. The fit is shown in Fig. 4.10. This is quite similar to and actually a test of the previous fit of GADGET data with a Voigt curve, assuming a Lorentzian-described physical spectrum smeared by a Gaussian detector response. Qualitatively, the two curves are similar, but this discrepancy is unacceptable for a search of our precision. Local fits with a Lorentzian were better, such as just the region lower in energy than the peak, but even this isn’t sufficient. To quantify these claims, we first need to quantify the quality of the fit. For a rough estimate of the precision needed, we first need to estimate the branching ratio into the 7 Li∗ -alone peak measured in GADGET: 0.5 for the cathode efficiency of 7 Li-alone and not 𝛼-alone, ∼ 0.6 for the fraction of decays originating on the cathode, ∼ 0.1 for the fraction of 𝛽− 𝛼 decays that populate the 7 Li 478 keV excited state, and ∼ 0.03 for the 𝛽− 𝛼 branching ratio. All together, the 7 Li∗ -alone decay from the detector cathode occurs with intensity of ∼ 9 · 10−4 , which is large compared to the ∼ 1 · 10−5 nominal branch of the signal. The integral of a peak is roughly proportional to the product of its height and its width. The R-matrix spectrum is peaked at ∼ 20 strength units (SU) and has width ∼ 80 keV. The area then is ∼ 1600 SU·keV. If we take the branching ratio intensity of this peak as the ∼ 9 · 10−4 from 1600SU·keV before, the spectral intensity per unit of 𝛽 decay intensity is ∼ 9·10−4 ≃ 1.8 · 106 SU·keV. If our approximation is to be useful, we want the mis-match to introduce not more than ∼ 1 · 10−6 branching ratio error. This corresponds to a spectral intensity error of ∼ 1.8 SU·keV. If we assume a 𝛽− 𝑝 signal peak with width of 20 keV and a constant residual error between the approximation and the R-matrix output over the signal-peak width, the magnitude of the residue needs to be less 1.8SU·keV than ∼ 20 keV ≃ 0.09 SU. The Lorentzian fit of Fig. 4.10 is different from the R-matrix curve by 78 20 spectral strength (arb. units) 18 16 14 12 R-matrix output 10 Lorentzian fit 8 6 4 2 100 200 300 400 500 600 700 Energy (keV) Figure 4.10: R-matrix calculation of the isolated spectrum of 7 Li∗ -alone (black) and comparison of a Lorentzian curve (red). Despite the reasonable fit to the experimental data using a Voigt curve (see Sect. 3.3.2.2), the R-matrix calculation output spectrum has notable differences than the Lorentzian that is implicit in the Voigt curve. up to ∼ 2 SU in some places, which could show up at a ∼ 2 · 10−5 systematic error, twice that of the 𝛽− 𝑝 branching ratio reported in Ayyad et al. [2]! To improve the estimation of the R-matrix output, the output was fit with the product of a Voigt profile and a polynomial function. The idea here is that the Voigt curve is peak-like, but symmetric. Multiplication by a polynomial can make the approximate function asymmetric and behave closer to the R-matrix output. If the general shape of the Voigt is close to the R-matrix output, then the polynomial would make small corrections to the Voigt to bring it in line with the R-matrix curve. However, the Lorentzian is tricky as starting with a polynomial with degree two, the polynomial can overwhelm the ∼ 1/𝑥 2 behavior of the curve. The Gaussian is safe and and will overcome any polynomial of any order. (As an aside, this peak-like requirement can be enforced in a Bayesian 79 1 fitting analysis so even the polynomial order constraint from the ∼ 𝑥2 Lorentzian form is not a true a constraint.) With this mapping between the residual in the R-matrix fit and the systematic-error induced in the inferred branching ratio, the order of the polynomial in the approximation function was increased until the residual error was less than 0.09 SU. This first occurred for a linear ”correcting” polynomial. This fit and its residuals are shown in Fig. 4.11. The linear corrective fit has residual errors of less than 0.06 SU, so systematic shifts in the 𝛽− 𝑝 branching ratio from this approximation should be held under ∼ 1 · 10−6 . 20 spectral strength (arb. units) 18 16 14 12 R-matrix output 10 Voigt(X)Poly fit 8 6 R-matrix output 4 2 Voigt(X)Poly fit 140 160 180 200 220 240 260 Energy (keV) 0.06 0 − 0.06 140 160 180 200 220 240 260 Figure 4.11: (Top) in black, the R-matrix output for the excited state spectra, energy-scaled to give the shape of just the 7 Li∗ -alone peak, but this is largely obscured by the fit in red. The fit is the product of a Voigt curve and a polynomial. (Bottom) fit residuals, in the same scale as the top. 80 4.3 Low-Energy Ionization Processes Gaseous proportional counters, like the Proton Detector in GADGET, ideally have a linear detector response. That is, the energy of radiation lost as an ion traverses the detector gas, ideally, is proportional to the charge measured at the end of the detector. Non-linearity can be caused by a multitude of effects, but can generally be described as detector physics or signal processing effects. Physics-driven non-linearity in the detector’s response is generically described as the pulse height defect. The first source is related to the detector dead layer, the surface layer of the detector that engages in ion energy-loss processes, but is not part of the sensitive portion of the detector. Another source of the pulse height defect is due to charge recombination of the radiation-created ion pairs. Decays in the Proton Detector are internal to the gas, so there are no windows or dead layers. As for charge recombination, the low density of the detector gas leads to long particle tracks and low charge-pair density, making recombination less likely than in a solid state detector. One contribution to the pulse height defect that is equally present in the gaseous detectors is related to a ion-species dependence on the stopping, with partitioning of energy-loss processes depending on the ion’s mass and charge [38]. In addition to producing ionization, impinging ions can perform non-ionizing electronic excitation, impart kinetic energy to target atoms with elastic scattering, and (primarily reserved for electron radiations) produce ”breaking” Bremsstrahlung photons, which may or may not be reabsorbed by the detector in a different location. This is notable for energies of ≲ 200 keV/u, where ion velocity has a dramatic effect on the stopping and ionization [59]. This threshold is several times the Bohr velocity ∼ 25 keV/u, where the ion- stopping physics is very different than at higher energies. In the high energy regime, the ion and target interaction is eikonal. In a classical picture, the target atom is functionally ”frozen” in space for the duration of the interaction. As the ion velocity decreases, the electron cloud of the target 81 atom is capable of responding to the presence of the bombarding ion, providing more effective charge screening of the target atom’s nuclear charge. Generally, at lower velocities, ions more easily charge-exchange electrons with their environment changing their charge state ([60] proton on argon gas, [61] helium on argon gas, [62] lithium on argon gas), stronger charge screening changes the weights of interactions between the targets’ nuclei and electrons, and interaction times are longer. These effects generate a pulse height defect as the detector only measures liberated electrons, which is only one of the many channels by which the ion can lose energy. Understanding the pulse height defect is important for our case. It influences the energy calibration as our data has no internal reference of 𝛽− 𝑝 to calibrate against, only 𝛽− 𝛼, of which the 7 Li radiations have ∼ 35 and ∼ 65 keV/u, which is quite near this Bohr velocity threshold. It influences the shape of our assumed 𝛽− 𝑝 signal in that the spacing of the 𝑝-alone cathode peak and the 𝑝+10 Be in-gas peak depends on both the proton ionization and the 10 Be ionization. The 10 Be energy is ∼ 2 keV/u, far below the Bohr velocity. The proton and 10 Be recoil will have about an order of magnitude different energy, (because of) an order of magnitude different mass, and a factor of four different nuclear charge. In this section, I will show evidence that each of these are important to understand. 4.3 Estimating Ionization in P-10 Gas There exists a dearth of experimental data on low-energy, energy-loss partitioning of ions traversing of P-10 gas. A great number of measurements were conducted in the 1970s and before to measure the ionization yields of different ions in a variety of gasses, mostly air, air constituents, noble gases, and so-called tissue-equivalent gases meant to model the stoichiometric makeup of human tissue. Much of this is summarized in the ICRU Report 31 [63]. Despite being published more than 40 years ago, it remains an influential compilation of this work [38], but it lacks information on gas 82 mixtures used in low-energy nuclear physics like P-10 and CO2 /He mixtures, as well as impinging ions heavier than 𝛼 particles. This is not to say that P-10 gas is not characterized. In fact, it is a well studied gas and the pulse height defect is well-characterized for a wide range of masses and charges at energies of hundreds of keV and up with the LSS theory by Linehard et al. [64], demonstrated nicely with a recent measurement of 252 Cf spontaneous fission in P-10 gas [65]. Fragments are peaked at ∼ 110u and ∼ 145u and have ∼ 185 MeV to share. Assuming a two-body decay, the lowest energy fragments would have ∼ 500 keV/u or about five times the Bohr velocity. The data that are missing are these Bohr velocity ionization yields such as those measured for a variety of low-energy ion beams impinged on argon gas [66, 67, 68, 69, 70, 71, 72, 73] or on methane gas [74, 75, 76, 77, 78]. In light of a lack of experimental information, two approaches were taken to approximate the pulse height defect of the 𝛽− 𝛼 spectrum and set the shape of the assumed 𝛽− 𝑝 signal. The first approach was calculation using the now familiar ion-stopping simulation software SRIM. The second was a compilation, parameterization, and uncertainty quantification of the literature of the ionization yields (measured as W values, see Sec. 4.3.1.2) of beams impinged on pure argon and methane gas that was used with a gas mixing calculation to estimate unmeasured gas properties of P-10. 4.3 Estimating Ionization Yields with SRIM The SRIM code was first introduced in Sec. 3.2.1.2. There, we used it to calculate particle ranges in P-10 gas. We use this code again to study the partitioning of energy loss processes. SRIM categorizes the energy loss into six bins. There is energy lost to: ionization, vacancies, and phonons, for each the primary ion and for secondary ions. A secondary ion is an ion that receives kinetic energy (above a threshold, for computation speed) from the primary ion. Energy 83 ion recoils sum ionization 99.63 0.05 99.68 vacancies 0.02 0.03 0.05 phonons 0.01 0.25 0.26 Table 4.3: SRIM calculated energy loss partioning for a 170 keV proton. The vast majority (99.7%) of the energy loss goes into electronic stopping and ionization production. Numbers are in percent. ion recoils sum ionization 38.87 10.53 49.40 vacancies 0.97 7.38 8.35 phonons 0.36 41.88 42.24 Table 4.4: SRIM calculated energy loss partioning for a 20 keV 10 Be. The energy loss is spread out between primary interactions and recoil interactions. Only about 50% of the energy goes into electronic stopping and ionization production. Numbers are in percent. lost to ionization is energy that is spent creating ion pairs in the target material. Energy lost to vacancies is energy that is spent when a target atom receives a kick that removes it from the atomic lattice it occupies. (SRIM was developed to estimate exactly this, the damage, in solid targets and electronics.) Finally, the energy into phonons is a catch-all bucket for energy conservation in the code. Energy in this bucket comes from target atoms that receive energy, but not sufficient to leave their lattice location. SRIM uses a interatomic potential developed by its authors [79] to calculate the energy sharing of ions with target atoms (nuclear stopping) and electrons (electronic stopping) [59]. At typical nuclear physics experimental energy and higher, the electronic stopping dominates the total stopping. At very low energies, the nuclear stopping and electronic stopping are similar. SRIM stopping powers are quoted to have an error of 4.3% from experimental stopping values [48]. The breakdown of energy loss for various ions follows: 170 keV proton in Tab. 4.3, 20 keV 10 Be in Tab. 4.4, 266 keV 7 Li in Tab. 4.5, 440 keV 7 Li in Tab. 4.6, and 770 keV 𝛼 in Tab. 4.7. The exact meaning of vacancies and phonons values are hard to interpret in the P-10 gas, but taking the sum of the ion and recoil ionization values, we have that most particles will deposit the vast majority 84 ion recoils sum ionization 91.89 1.52 93.41 vacancies 0.14 0.96 1.10 phonons 0.04 5.45 5.49 Table 4.5: SRIM calculated energy loss partioning for a 266 keV 7 Li. The bulk of the energy loss is goes into primary ionization, but some ionization from recoiling ions is recovered. About 93% of the energy ends up going to ionization. Compare to Tab. 4.6 to see the effect of ion energy on energy loss partitioning. Numbers are in percent. ion recoils sum ionization 94.76 1.00 95.76 vacancies 0.09 0.62 0.71 phonons 0.03 3.51 3.54 Table 4.6: SRIM calculated energy loss partioning for a 440 keV 7 Li. The bulk of the energy loss is goes into primary ionization, but some ionization from recoiling ions is recovered. About 96% of the energy ends up going to ionization. Compare to Tab. 4.5 to see the effect of ion energy on energy loss partitioning. Numbers are in percent. ion recoils sum ionization 98.87 0.19 99.06 vacancies 0.03 0.12 0.15 phonons 0.01 0.77 0.78 Table 4.7: SRIM calculated energy loss partioning for a 770 keV 𝛼. The vast majority (99.1%) of the energy loss goes into electronic stopping and ionization production. Numbers are in percent. 85 of their energy in the form of ionization: 170 keV protons 99.7%, 20 keV 10 Be 49%, 266 keV 7 Li 93%, 440 7 Li 96%, and 770 keV 𝛼 99%. The efficiency of an in-gas 7 Li+𝛼 decay of energy 𝐸 should then be ∼ 4/11 · 96% + 7/11 · 99% ∼ 98%. The large pulse height defect calculated for 10 Be was somewhat surprising and SRIM provides no error quantification. In private communication, Ziegler recommended referencing experimental results from similar experiments, if they existed, as the 10 Be energy is low, suggesting errors in stopping could be as large as 30%. 4.3 Estimating Ionization Yields of Gas Mixtures The more detailed approach and that which was adopted in the final analysis was to consider data on the ionization yields of the P-10 components and average them to create an estimate for the P-10. This is not merely calculating an average ionization, molecule for molecule, according to the gas mixture, as the gas has properties from both its constituents and the interactions of its constituents. The role of the methane in P-10 is that of a quench gas. The quench gas has a lower ionization potential than the primary gas (12.61 eV [80] and 15.76 eV [81], respectively) and helps with localization of charge production in the vicinity of radiations by absorbing UV radiations and by the Penning Ionization mechanism, where excited states of argon gas can de-excite by collisions with the methane, producing an ion pair off the methane [38]. Ionization yields will now be discussed by their inverse, normalized to ion pair. That is, the following discussion will involved the W value, which has units of energy per pair and has some analog to the work function of the photoelectric effect. It is the average amount of energy required to liberate a single ion pair and is often written as 𝑊 (𝐸), emphasising the role of the energy of the impinging ion. This is incomplete as the W value is sensitive to ion species as well. The form of the W value averaging equation was first described in the 1950s by the Basel group [82, 83] and extended in 1987 by Tawara et al.[84]. Tawara et al. parameterized the W value 86 function curve for 𝛼 particles of ∼ 1 − 5 MeV in various mixtures of argon and methane detector gas. The W value of a mixture of a primary component 𝑥 with a quench gas 𝑦 can be calculated by the following equation: 𝑊𝑥𝑦 (𝐸) −1 = (𝑊𝑥 (𝐸) −1 − 𝑊𝑦 (𝐸) −1 ) · 𝑍 (𝐸) + 𝑊𝑦 (𝐸) −1 + 𝑍 (𝐸) · 𝐵𝑥𝑦 (4.8) where 𝑍 (𝐸) is the fractional stopping between gas components with stopping powers 𝑆𝑖 and partial pressures 𝑃𝑖 𝑆 𝑦 𝑃 𝑦 −1 𝑍 (𝐸) = (1 + ) (4.9) 𝑆 𝑥 𝑃𝑥 and the mixture-dependent parameter 𝐵𝑥𝑦 = 0.09 for P-10 gas. The 𝐵𝑥𝑦 term is new with Tawara et al. [84]. Due to the absence of data on the W value of P-10 for low-energy ions, the W value was estimated by the Tawara formula. The pulse height defect is related to the W value in that as the W value increases, fewer ion pairs are created for the same impinging energy. 87 Parameterization of Argon Gas W value First, argon was studied. The nominal W value for argon gas is 26.3 eV/pair in [38] and 27(1) eV/pair in [63]. The following W values for ions impinging on argon were compiled and are shown in Fig. 4.12: • Phipps et al. [70]: H, He, C, N, O, Ar on Ar • Chemtob et al. [72]: H, He, Ar on Ar • Chappell and Sparrow [71]: He on Ar • Leimgruber et al. [73]: N, O on Ar • Nguyen et al. [69]: H, He, Ar on Ar • Larson [68]: H on Ar • Lowry and Miller [66]: H on Ar 88 W(E) vs incident energy in argon gas 90 Ph64 Ch78 80 Sp67 Le65 70 La58 W(E ) (eV/pair) Lo58 60 50 40 30 101 102 103 E (keV) Figure 4.12: The W value against incident ion energy into argon gas. The data are color coded according to their publication. The W value is multi-valued for a particular value of energy and this is due to the impinging ion species, shown later. There is overlap in energies between most sets of measurements, so consistency of W values can be checked. Error bars aren’t shown. The spread in data is due to different impinging species, shown better in Fig. 4.13. In general, it looks as if all values may converge to the ∼ 27 eV/pair from the compilations at sufficiently high energy, but the W value of heavy ions is a function of energy. The heavier the ion, the stronger the energy dependence. Figure 4.14 shows much of the complexity in √︁ √︁ the W value is actually due to the ion’s mass. Velocity is calculated as 𝑣 = 𝐸/𝐴( (keV/u)). This velocity is the incident velocity, so these W values are integrated over the whole energy loss. The fit shown is a power law, with the high-energy behavior reducing to the nominal W value of 26.3 eV/pair. 𝑊 (𝑣) = 26.3 + 𝛼 · 𝑣 𝛽 (4.10) 89 W(E) vs incident energy in argon gas 90 H+ He+ 80 C+ N+ 70 O+ W(E ) (eV/pair) Ar+ 60 50 40 30 101 102 103 E (keV) Figure 4.13: The W value against incident ion energy into argon gas. The data are color coded according to the impinging ion species. The W value is not a function of energy, or at least is a very weak function of energy, for impinging H+ and He+ . There W value a strong dependence on energy for all other chemical species shown. 90 ion W value (eV/pair) ∼ 1𝜎 error (eV/pair) 10 Be, 20 keV 56.6 5.8 7 Li, 256 keV 27.9 0.7 7 Li, 430 keV 27.2 0.6 𝛼, 752 keV 26.6 0.4 Table 4.8: Estimated W values and associated 1𝜎 errors for relevant charged-particle radiations, in argon gas. See text and Fig. 4.14 for determination of these estimates. While the velocity isn’t a perfect predictor of W value and clear patterns exist in the fit residuals, it is quite suggestive. This matches some of the qualitative arguments about ion velocity determining stopping physics in the target from [59] and lends some credence to the ∼ 0.5 ionization of the 10 Be calculated by SRIM. In these units, a 20 keV 10 Be has velocity of ∼ 1.4, a 250 keV 7 Li has velocity of ∼ 6, a 430 keV 𝛼 has velocity of ∼ 10. Thus, of the particles in our data, the largest pulse height defect occurs for the 10 Be recoil. Additional terms for the 𝑍 of the impinging ion can improve the fit some, but a simple function seems sufficient for our purposes. The fit yields parameters of 𝛼 = 61.6 and 𝛽 = −2.05. Parameter errors are not shown as the fit was performed without defined error bars to data. Error in the parameterization was estimated by taking the standard deviation of the fit residuals over the interval (-25%,+30%) (∼ ±0.26 in natural log scale). With these definitions, we have the following description of the W value of our spectra in pure argon gas. 91 Figure 4.14: √ (Above) The W value against incident ion velocity into argon gas. Velocity is in units of keV/u. The data are color coded according to the impinging ion species and markers represent data source. By plotting in velocity and not energy, much of the spread in W values has disappeared. The smooth line is a power-law equation Eq. 4.10. (Below) Residuals between data and the fit. 92 Parameterization of Methane Gas W value Similarly, the available W value data for methane gas was collected. The nominal W value for methane is 29.1 eV/pair in [38] and 29(1) eV/pair in [63]. The following W values for ions impinging on methane were compiled and shown in Fig 4.15: • Nguyen et al. [77]: H, He, C, O, N on methane • MacDonald and Sidenius [78]: 𝑍 ⩽ 22 on methane (with select elements shown) • Jesse [74]: He on methane • Varma and Baum [75] He on methane • Tawara et al. [76] He on methane 93 ion W value (eV/pair) ∼ 1𝜎 error (eV/pair) 10 Be, 20 keV 68.3 9.8 7 Li, 256 keV 35.8 3.8 7 Li, 430 keV 34.0 3.8 𝛼, 752 keV 31.6 2.9 Table 4.9: Estimated W values and associated 1𝜎 errors for relevant charged-particle radiations, in methane gas. See text and Fig. 4.15 for determination of these estimates. Here, we see similar behavior to the argon data, where the W value as an asymptotically decreasing function of velocity. A similar power law function was fit to data, but with the high energy W value of methane in place of that for argon. 𝑊 (𝑣) = 29.1 + 𝛼 · 𝑣 𝛽 (4.11) In the methane data, 𝛼 = 59.7 and 𝛽 = −1.22. With the same error assesment as in the argon data, we have the following W values for methane: Hidden in plain sight is a strong dependence on the chemical group of the inpinging ion. Group I, alkali metals have larger W values than other groups. This pattern was noted in the original paper, but no mechanism was prescribed to this behavior [78]. Given the richness of this data (all elements for 𝑍 ⩽ 22 and very low energy/mass ratios), some time was spent modeling this behavior in light of chemical electronegativity and only weak patterns were found beyond the Group I elements, low electronegativity, and large W values. (Fluorine, with near maximal electronegativity falls under the fit, but not by much and other elements with modest electronegativity have still lower W values.) This effect likely is the culmination of many related effects, each with ties to atomic shell structure. 94 Figure 4.15: √ (Above) The W value against incident ion velocity into methane gas. Velocity is in units of keV/u. The data are color coded according to the impinging ion species and marker shapes represent data source. The smooth line is a power-law equation Eq. 4.11. (Below) Residuals between data and the fit. 95 radiation W value (eV/pair) 1𝜎 error (eV/pair) W value fraction 1𝜎 error 10 Be, 20 keV 39.4 3.0 0.56 0.08 7 Li, 256 keV 23.2 0.7 0.94 0.03 𝛼+7 Li, 1182 keV 22.4 0.6 0.98 0.02 nominal 21.9 0.5 1.00 0.02 Table 4.10: Estimated W values and associated 1𝜎 errors for relevant charged-particle radiations, in P-10 gas. See text for determination of these estimates. Estimation of P-10 Gas W value With each element of the Tawara formula [84] defined, P-10 W values were estimated. To perform error propagation, a Monte Carlo calculation was performed for each ion of interest in the previous tables. Values were sampled according to a Gaussian probability density function centered on the power-law fit function values and the width defined by the standard deviation of the fit residuals, the same as the tables above. Samples were drawn 100,000 times and applied in the Tawara formula. Results were saved. The mean of the results is the central value of the final (Gaussian-enough) probability density function and the standard deviation is the 1𝜎 error. Most of the W values calculated in the Tawara formula are a bit lower than the nominal value of P-10 (26.5 eV/pair). As done in Tawara et al. [84], these calculated W values are normalized to the ”nominal” high-energy value calculated in the Tawara formula using the high-energy values of the constituent gases. Thus, the full energy 7 Li+𝛼 peak will create 98(2)% of the charge per unit energy if it was higher in energy. The other isolated peak, the 256 keV 7 Li ∗ -alone peak, creates 94(3)% of the ionization a higher-energy particle would. The 10 Be peak does, in fact, produce only about half the ionization than one would expect assuming linearity in the detector gas, as was calculated by SRIM. 96 4.3 Effects on Analysis With a more thorough accounting of the behavior of the P-10 gas pulse height defect, several items will be reevaluated. First, the energy calibration of the E18507 analysis and E18507-style analysis needs to change. In light of these gas effects and the publication of Refsgaard et al. [27], which provides a ∼ 30 keV lower-energy state for the 𝛽− 𝛼-emitting state in 11 B, the energy calibration was redone. The energy scale was set to the physical center of mass energy, so the 1182 keV 7 Li+𝛼 in-gas, full energy peak appears with 1182 · 0.98(2) = 1158(24) keV and the 256 keV 7 Li∗ -alone cathode-originated peak appears with 256 · 0.94(3) = 241(8) keV. Secondly, the large pulse height defect of the 10 Be changes the shape of the assumed 𝛽− 𝑝 search peak. Assuming a 200 keV 𝛽− 𝑝 resonance, the proton will carry 10/11 · 200keV = 182 keV and the 10 Be will carry the remaining 1/11 · 200keV = 18 keV. However, while the proton will efficiently create charge, the 10 Be will create some 18 keV · 0.56(8) = 10(2) ”keV” of charge. This pushes the two constituent peaks closer together toward a single peak appearing at ∼ 96% the center of mass energy. 4.4 Bayesian Inference of 𝛽− 𝑝 with Markov Chain Monte Carlo With many of the fit parameters being constrained in intuitive, but highly relational ways like a prohibition on the very flexible R-matrix approximation function creating a local maximum that could fit and hide a 𝛽− 𝑝 signal, a more flexible fitting scheme was needed. Additionally, some parameters, such as the 10 Be ionization efficiency have well-described, Gaussian errors. Others, like the detector resolution at ∼200 keV are less clear, but experience and some data from the spectra inform us that the resolution is not worse than 10% FWHM. In the case of the detector resolution, a Gaussian probability density function requires a choice of a central value, but it is unclear which value to pick and it really needs a fairly sharp cut-off past 10%. Gaussian probability 97 density functions have non-zero value at all values. All these points can be solved by the marriage of Bayes’ Theorem and a Markov Chain Monte Carlo (MCMC) sampling algorithm. 4.4 Introduction to Bayes’ Theorem Bayes’ Theorem is 𝑝(d—H) · 𝑝(H) 𝑝(H—d) = (4.12) 𝑝(d) It says that the posterior, or updated, probability density function of a hypothesis 𝐻, in light of new data 𝑑, is the confluence of the quality of agreement of your data to the assumed hypothesis 𝑝(d—H), your prior belief in the hypothesis 𝑝(H), and your belief in the ability to obtain data 𝑑 across all possible hypotheses 𝑝(d). While belief in a hypothesis strains some desire for absolute objectivity in analysis, this is merely semantics. Taking 𝑝(𝑑) = 𝑝(𝐻) = 1 yields the typical statistical inference relation 𝑝(H—d) = 𝑝(d—H) (4.13) from which classical parameter estimation are inferred. It was this inference relation that was implicit in the E18507 style analysis that sometimes yielded large negative values for the 𝛽− 𝑝, clearly driven by a systematic error, likely by poor choice of a background model, as negative event counts are not physical. Presented another way, the form of Eq. 4.13 is a statement of statistical uncertainty, within a particular model, alone. Varying the model assumptions and repeating the analysis to characterize the results’ influence by model assumptions is a common way to estimate systematics of the analysis. This distinction is unnecessary and, in fact, can complicate such a systematic search if multiple parameters are correlated and not varied according to to this correlation, which itself is a complicating factor. Rather, the classical form of the inference equation 98 is an artifact of a historical lack of computational power required to numerically ”solve” Bayes’ Theorem and extract the posterior probability distributions. The model limits that are tested to infer the influence of systematic errors are a choice. Model selection, including model complexity (such as the degree of a polynomial or the choice between a Gaussian, Lorentzian, or Voigt), is a choice. The classical inference equation 4.13 does not exist separate from systematic-influencing choices based on model beliefs, but is embedded within them with the details hidden away from a reader and maybe even an incurious practitioner. While the discussion has grown into the unification of statistical and systematic errors in the Bayesian scheme, here is an example that shows the application of a prior distribution 𝑝(𝐻) is a standard tool in the analyst’s tool-belt and not something entirely new. An Example: When reducing multiple independent measurements with central values 𝑥𝑖 and Gaussian distributed errors 𝜎𝑖 , one performs the calculation Í 𝑥𝑖 𝜎𝑖2 𝑥¯ = Í 1 (4.14) 𝜎𝑖2 to find the error-weighted average. If we considered the probability density functions of each of these measurements, they would be of the form −(𝑥 − 𝑥𝑖 ) 2   𝑝𝑖 (𝑥) ∼ exp . (4.15) 2𝜎𝑖2 The probability of a set of independent events is simply the product of the probability of each, so the composite probability is 1 ∑︁ (𝑥 − 𝑥𝑖 ) 2 Ö   𝑃(𝑥) = 𝑝𝑖 (𝑥) ∼ exp − . (4.16) 2 𝜎𝑖2 99 Expanding the sum and collecting terms in powers of 𝑥 yields  1 ∑︁ 1 ∑︁ 𝑥𝑖 ∑︁ 𝑥 2  2 𝑖 𝑃(𝑥) ∼ exp − · (𝑥 − 2𝑥 + ) . (4.17) 2 𝜎𝑖2 𝜎𝑖2 𝜎𝑖2 Completion of the square yields (and dropping constant terms that are wrapped up in the propor- tionality symbol)  ∑︁ 1  1 ∑︁ 1 ∑︁ 𝑥𝑖 −1  2 𝑃(𝑥) ∼ exp − · · (𝑥 − · ) (4.18) 2 𝜎𝑖2 𝜎𝑖2 𝜎𝑖2 The joint probability is maximized for Í 𝑥𝑖 𝜎𝑖2 𝑥¯ = Í 1 , (4.19) 𝜎𝑖2 which is the usual result for error-weighted averaging. To make the example clear, we invoke multiplicative associativity applied to Eq. 4.16. That is, the joint probability can be written as the product of new measurements and old (prior) measurements Ö Ö Ö 𝑃(𝑥) = 𝑝𝑖 (𝑥) = 𝑝𝑖 (𝑥) 𝑝 𝑗 (𝑥) (4.20) 𝑖,new 𝑗,old or new measurements against theory Ö Ö Ö 𝑃(𝑥) = 𝑝𝑖 (𝑥) = 𝑝𝑖 (𝑥) 𝑝 𝑗 (𝑥) (4.21) 𝑖,experiment 𝑗,theory or equivalently and more generally, 𝑃(𝑥) = 𝑝 data · 𝑝 prior (4.22) 100 which is identically Bayes’ Theorem, Eq. 4.12, without the constant denominator. 4.4 Introduction to Markov Chain Monte Carlo The Markov Chain Monte Carlo is a class of algorithms that is useful in exploring a probabilistic parameter space. MCMC algorithms are numerical in nature and the results are numerical strings or chains of sampled parameters. That is, instead of solving an integral or solving a 𝜒2 -minimization analytically and calculating the solution and errors in the solution by the shape of the probability surface around the solution, the algorithm is run to generate samples of interest filling the chain, and upon completion, the values of the chain can be interpreted as a sampled set of solutions to the problem of interest, with more frequently sampled values being preferred. Less abstractly, the Metropolis-Hastings algorithm is a MCMC algorithm that performs a ”random walk” through a probability-like space, 𝑓 (® 𝑥 ), which is proportional to the probability space, 𝑝(® 𝑥 ), and the generated chains will be sampled according to the probability space. The basic steps of the algorithm follow: • Assuming 𝑥®𝑖 , compute a test value 𝑡® according to some generator 𝑔( 𝑡®|® 𝑥𝑖 ) 𝑓 ( 𝑡®) 𝑝( 𝑡®) • Calculate the probability fraction 𝛼 = 𝑓 ( 𝑥®𝑖 ) = 𝑝( 𝑥®𝑖 ) and a random number 𝑟 from a uniform distribution • Compare 𝛼 and 𝑟. Accept the trial value if 𝛼 > 𝑟 and set 𝑥®𝑖+1 = 𝑡®. Reject the trial value 𝑥®𝑖 if 𝛼 < 𝑟 and set 𝑥®𝑖+1 = 𝑥®𝑖 • Repeat with 𝑥®𝑖+1 The stored values of 𝑥® make the Markov chain and these values will have the same properties as a sample set sampled directly from 𝑝(® 𝑥 ). 101 While simple at a high level, there are some assumptions and tricks related to the generator 𝑔( 𝑡®| 𝑥®𝑖 ). As written, the generator must be symmetric 𝑔(® 𝑥 | 𝑦®) = 𝑔( 𝑦® |® 𝑥 ). In principle, asymmetric generators can be used, but this biases your trial samples and must be accounted for to correctly sample the probability distribution 𝑝(𝑥). In a large number of sampled dimensions, the odds of performing a ”bad” step in one or more dimension increases. To solve for this, a common generator to use is the multi-variate Gaussian, which is a multi-dimensional Gaussian defined by a covariance matrix. An alternative is Gibbs sampling, which is a subset of Metropolis-Hasting schemes and is implemented similarly, but only one component of the parameter space is varied in each step. The individual components are updated cyclically. The Gibbs-sampled component may be an individual parameter of the probability space or a linear combination of parameters. This sampling reduces the dimensionality of each step, but increases the number of steps required to achieve the same coverage in the entire probability space. These MCMC solutions are robust to high-dimensional problems that may have local minima, as solutions will enter and exit minima in the probability distributions during the parameter walk. Furthermore, they allow for non-Gaussian-distributed probability surfaces which can more closely model the inference problem at hand. 4.4 Implementation of Bayesian Inference with MCMC A Bayesian Markov Chain Monte Carlo code was written to perform the fit of the E19030 spectra for Pads B, C, D, and E. Trial steps across the model parameter space were generated with ROOT’s RooMultiVar object that provides the mathematics of a multi-variate Gaussian. It is easily trained by passing data to other ROOT objects. Historical steps were stored in a TPrincipal object that was used to generate the data covariance matrix (TMatrixD) which the multi-variate Gaussian object (RooMultiVar) 102 was trained to. New trial parameter steps were (pseudo-)randomly generated and passed to the step-evaluation portion of the code. The probability-proportional function ( 𝑓 (® 𝑥 ) in Sec. 4.4.2) for a particular set of model param- eters was done in two steps: fitting to data and evaluation against the Bayesian priors. 4.4 Fitting To Data The data was fit from 120 keV to 245 keV in the 100% ionization efficiency center of mass energy scale and 𝛽− 𝑝 peaks were tested for center mass energy from 145 keV to 215 keV. The low energy range limit comes from the shape of the 𝛽 background becoming less exponential in shape and therefore too complex to model. The high energy range limit is roughly the peak of the 256 keV 7 Li∗ after accounting for incomplete ionization processes of the slow 7 Li. The fit background function was the sum of an exponential curve, a flat offset, and the product of a Voigt and quadratic polynomial, written below 𝑓back (𝐸) = exp(𝑎 + 𝑏 · 𝐸) + 𝑐 + Voigt(𝐸, 𝐸¯ 7 , 𝜎7 , 𝛾7 ) · ( 𝐴0 + 𝐴1 · 𝐸 + 𝐴2 · 𝐸 2 ) (4.23) and the peak function was  1 𝑓peak (𝐸) = 𝑁 · 𝑐 frac Voigt(𝐸, 𝐸¯ = 𝑚 10 𝑞 𝑝 𝐸 𝑝 , 𝜎, 𝛾 = 𝑚 10 𝑞 𝑝 Γ) 2  + (1 − 𝑐 frac )Voigt 𝐸, 𝐸¯ = (𝑚 10 𝑞 𝑝 + 𝑚 𝑝 𝑞 10 )𝐸 𝑝 , 𝜎, 𝛾 = (𝑚 10 𝑞 𝑝 + 𝑚 𝑝 𝑞 10 Γ)  (4.24) where 𝑐 frac is the fraction of decays originating on the Proton Detector cathode, 𝑚 10 is the mass fraction of the 10 Be to the 10 Be+proton system, 𝑚 𝑝 is the mass fraction of the 10 Be to the 10 Be+proton system, 𝑞 𝑝 was the proton ionization efficiency, 𝑞 10 was the 10 Be ionization efficiency, 𝐸 𝑝 was the 103 center of mass energy of the 𝛽− 𝑝 resonance, 𝜎 was the Gaussian width detector resolution, and Γ was the intrinsic width of the 𝛽− 𝑝 resonance. The total (background and peak-search both) model has 14 parameters. The fit was judged by the 𝜒2 sum of the squared difference between data and the model function, divided by the square of the expected data error, which was the root of the data in the bin, from Poisson counting statistics. The probability of a particular data histogram given the model is then 𝑝(d|H) = exp(−𝜒2 /2). 4.4 Evaluation of model against priors The next part was to evaluate a particular model against the priors. Strictly speaking, many of these priors are un-normalizable, so these priors are not true probabilities that integrate to 1. This is alright as the MCMC code requires only a function that is proportional to probability, so un- normalized priors are fine if not true probabilities. Here is a list of individual priors 𝑝𝑖 that are Î multiplied to generate the entire model prior 𝑝 𝑀 = 𝑝𝑖 : • 𝑝 0 = 1 if the exponential slope meant for the 𝛽 background is negative, 𝑏 < 0 • 𝑝 1 = 1 if the constant flat background is positive AND less than the minimum data bin, 0 ≤ 𝑐 ≲ 1200 2 • 𝑝 2 = exp − (𝑐2·0.006 frac −𝑐¯frac ) 2 , so the cathode fraction is within ∼1.5% of the value measured using the 𝛽− 𝛼 − 𝛾(478keV) coincidence • 𝑝 3 = 1 if proton counts, 𝑁, are positive • 𝑝 4 = 1 if the detector resolution is between 4.5% and 10% of the center of mass energy 𝐸 𝑝 • 𝑝 5 = 1 if the 𝛽− 𝑝 intrinsic width Γ is less than 30 keV 104 (𝑞 10 −𝑞¯ 10 ) 2  • 𝑝 6 = exp − 2·0.082 , so the 10 Be ionization efficiency matches that calculated in Sec. 4.3.1.2 • 𝑝 7 = 1 if the nominal Voigt center is between 210 and 260 keV, 210 < 𝐸¯ 7 < 260 • 𝑝 8 = 1 if the Voigt·Polynomial is positive, that is 𝐴0 + 𝐴1 · 𝐸 + 𝐴2 · 𝐸 2 over the fit window • 𝑝 9 = 1 if the nominal Voigt widths are less than 300 keV, that is 𝜎7 < 300 and 𝛾7 < 300 • 𝑝 10 = 1 if the Voigt· Polynomial curve is increasing on the interval 120-150 keV. This guarantees the peak-like behavior far from the peak otherwise, 𝑝𝑖 = 0. That is, most of our priors are ”ignorant” to model parameters, granting the model great flexibility, and instead only excludes unphysical behavior of the background model. This point is critical. We did not and do not have access to a 11 Be(𝛽− 𝛼)7 Li ”calibration” that guarantees no 11 Be(𝛽− 𝑝)10 Be. We need a background model that can accommodate the convolution of real 11 Be(𝛽− 𝛼)7 Li and the detector response to a precision exceeding that of any previous GADGET experiment, due to asymmetry of the nominal ∼ 1·10−5 𝛽− 𝑝 branch against the ∼ 3·10−2 𝛽− 𝛼 background. In the scenario of an incorrectly simple model, the Bayesian posterior distribution will still incur a systematic shift from the signal compensating for a deficiency in the background model. However, in the scenario of a too-complicated model, where one has model redundancies between signal and nuisance parameters, all confusions between signal and background will be represented with their quality-of-fit probabilistic weights. Complicated correlations are explored across the entirety of the parameter space and not merely at the ”most probable” point, and are easily visualized in 2D plots of signal vs nuisance-parameter. 105 4.4 The MCMC Algorithm This next section outlines the explicit coding details of the MCMC code. Initial parameter values were determined by a 𝜒2 fit of the data with the background function alone, Eq. 4.23. Initial step sizes were selected by the 𝜒2 fit error bars. Each of the Pads B, C, D, and E were treated independently, so the algorithm was run for each pad and assumed resonance energy 𝐸 𝑝 on the interval 145 to 215 keV with 1 keV spacing for a total of 280 implementations. There were five burn in periods of the MCMC. Each burn in consisted of 10,000 saved samples spaced every 20 steps along the chain. The burn-in chains were thus 200,000 samples long. As many of the model parameters are correlated and high-dimensionality steps are less likely to succeed than single-dimension steps, the first burn in was sampled according to the Gibbs scheme with individual parameters. That is, the first parameter was updated by 𝑡0 = 𝑥 0 + 𝑟 (0, 𝜎0 ), where 𝑟 is a random number generated from a Gaussian distribution with mean zero and width 𝜎0 . The step was evaluated and either accepted or rejected, then the second parameter was varied by 𝑡1 = 𝑥 1 + 𝑟 (0, 𝜎1 ) and evaluated, and so on. Once this primary chain completed, four additional burn in chains were sampled. These chains proposed steps that updated all model parameters using steps sampled from a multi-variate Gaussian defined by the covariance of the previous burn in chain, 𝑑𝑥. That is, the trial steps were calculated ® 𝑖 . The burn in periods allowed the MCMC code to reach the statistical distribution by 𝑡® = 𝑥®𝑖 + 𝑑𝑥 if the initial parameter values or step sizes were not in equilibrium with the posterior distribution. This was nearly guaranteed. The ”data-taking” chain followed. Here, the step sizes are the values drawn from the multi- variate Gaussian distribution scaled by 0.25. From this chain 1 million samples were drawn, spaced every 20 steps, for a total of 20 million steps. The covariance matrix from which the trial steps 106 are generated was reevaluated every 500 saved samples (10,000 steps) in the chain. If a batch of 500 samples had poor update success (> 10% of saved samples were identical to the previous saved sample), then that block of steps was repeated with 33% smaller step sizes. Resampling allowed more efficient motion throughout the parameter space. The check against high failure rates of the proposed trial steps safeguarded against the MCMC ”getting stuck”, leading to a singular covariance matrix and breaking at the next new trial step generation. 107 CHAPTER 5 RESULTS AND DISCUSSION: EVIDENCE FOR NOVEL 𝛽− 𝑝 The Bayesian Markov Chain Monte Carlo procedure described in Sec. 4.4 was applied to GADGET Pads B, C, D, and E. A number of metrics to judge the quality of the performance of the MCMC will be presented and then the inferred 𝛽− 𝑝 branching ratio. This branching ratio will be compared to existing theory work and the other experimental work. 5.1 Results of Bayesian Markov Chain Monte Carlo The first results we will show are related to the behavior of the MCMC. The MCMC generates a great deal of data. To store 1,000,000 double precision floating point numbers requires 8M bytes. 1 million samples over 14 parameters requires 112 MB of storage per fit. We can study 1D parameter distributions, 2D parameter correlation distributions, conditional distributions (e.g. ”What is the branching ratio and detector resolution correlation, assuming a 𝛽− 𝑝 intrinsic width Γ = 5 keV?”), as well as the actual path of the MCMC through the parameter space, which can be used to ensure the chain is exploring the space efficiently. These distributions and tests exist for all fits, across all pads and energies. 5.1 Convergence of the Markov Chain A first test of the MCMC is to look at the parameter traces. (These are sometimes called caterpillar plots for aesthetic reasons.) If the MCMC is sampling the parameter space efficiently, then samples are fairly uncorrelated, as random samples ought to be if they are truly random. Figures 5.1, 5.2, 5.3, 5.4 shows the trace from the MCMC fit of Pad B at 𝐸 = 160 keV. The sampling here is showing every 100th saved point. Most traces are fairly random in appearance, but to demonstrate some behavior of correlation, consider the 7 Li∗ center parameter in the middle of Fig. 5.3. The sampling 108 appears fairly random around 𝐸 7 ∼ 230 keV most of the time. However, there is another solution for 𝐸 7 ∼ 250 keV. By considering the 𝜒2 value at different times (Fig. 5.5), we can see these transitions from 𝐸 7 ∼ 230 keV to 𝐸 7 ∼ 250 keV are not met with any sort of decrease in the quality of the fit. However, there is still correlation in the sample as once the value of the 𝐸 7 parameter jumps, it fluctuates about this new value for some thousands of samples. While this correlation is not ideal, the MCMC steps into and out of this region several times, and does not become ”stuck” in a bad region nor does it become ”un-stuck” and continually move toward a new, better, region. That is, this motion is probably fairly statistical. 109 19 18.5 18 17.5 exp1 17 16.5 16 15.5 15 ×103 0 200 400 600 800 1000 −0.065 −0.07 −0.075 −0.08 exp2 −0.085 −0.09 −0.095 −0.1 ×103 0 200 400 600 800 1000 1000 800 flat 600 400 200 ×103 0 200 400 600 800 1000 Figure 5.1: MCMC parameter traces for Pad B and 𝐸 𝑝 = 160 keV. The parameters are the exponential scaling (𝑒𝑥 𝑝1), exponential slope (𝑒𝑥 𝑝2), and constant background ( 𝑓 𝑙𝑎𝑡). Parameter values are plotted in the y-axis and the sample number along the MCMC are plotted along the x-axis. (1/5) 110 12000 10000 pCounts 8000 6000 4000 2000 ×103 0 200 400 600 800 1000 7 6.5 6 5.5 pSigma 5 4.5 4 3.5 3 ×103 0 200 400 600 800 1000 30 25 pGamma 20 15 10 5 ×103 0 200 400 600 800 1000 Figure 5.2: MCMC parameter traces for Pad B and 𝐸 𝑝 = 160 keV. The parameters are the 𝛽− 𝑝 counts (𝑝𝐶𝑜𝑢𝑛𝑡𝑠), detector resolution applied to the 𝛽− 𝑝 signal (𝑝𝑆𝑖𝑔𝑚𝑎), and assumed intrinsic width Γ𝑝 (𝑝𝐺𝑎𝑚𝑚𝑎). Parameter values are plotted in the y-axis and the sample number along the MCMC are plotted along the x-axis. (2/5) 111 0.8 0.7 10Be-IonEff 0.6 0.5 0.4 0.3 ×103 0 200 400 600 800 1000 260 250 7LiCent 240 230 220 210 ×103 0 200 400 600 800 1000 45 40 35 7LiSigma 30 25 20 15 10 5 ×103 0 200 400 600 800 1000 Figure 5.3: MCMC parameter traces for Pad B and 𝐸 𝑝 = 160 keV. The parameters are the 10 Be ionization efficiency (10𝐵𝑒 − 𝐼𝑜𝑛𝐸 𝑓 𝑓 ), the Voigt center of the 7 Li∗ peak (7𝐿𝑖𝐶𝑒𝑛𝑡), and the Gaussian width of the 7 Li∗ peak 𝜎7 (7𝐿𝑖𝑆𝑖𝑔𝑚𝑎). Parameter values are plotted in the y-axis and the sample number along the MCMC are plotted along the x-axis. (3/5) 112 120 100 7LiGamma 80 60 40 20 ×103 0 200 400 600 800 1000 ×103 9000 8000 7000 6000 pol0 5000 4000 3000 2000 1000 ×103 0 200 400 600 800 1000 10000 0 −10000 −20000 pol1 −30000 −40000 −50000 −60000 −70000 ×103 0 200 400 600 800 1000 Figure 5.4: MCMC parameter traces for Pad B and 𝐸 𝑝 = 160 keV. The parameters are the Lorentzian width of the 7 Li∗ peak 𝛾7 (7𝐿𝑖𝐺𝑎𝑚𝑚𝑎), the zeroth term in the polynomial applied to the 7 Li∗ peak Voigt (𝑝𝑜𝑙0), and the linear term in the polynomial applied to the 7 Li∗ peak Voigt (𝑝𝑜𝑙1). Parameter values are plotted in the y-axis and the sample number along the MCMC are plotted along the x-axis. (4/5) 113 140 120 100 80 pol2 60 40 20 0 −20 ×103 0 200 400 600 800 1000 150 145 140 chisq 135 130 125 120 ×103 0 200 400 600 800 1000 Figure 5.5: MCMC parameter traces for Pad B and 𝐸 𝑝 = 160 keV. The parameters are the quadratic term in the polynomial applied to the 7 Li∗ peak Voigt (pol2) and (not a parameter) the Bayesian- modified 𝜒2 of the fit (chisq). Parameter values are plotted in the y-axis and the sample number along the MCMC are plotted along the x-axis. (5/5) 114 One example of unambiguously poor behavior is walking behavior, with high correlation and a clear direction. Such behavior can be seen in the burn-in chains, moving from non-ideal fit parameters toward the statistical distribution of fit parameters. As these walking samples are never revisited, they arguably are not truly part of the true probability distribution, but are instead an artifact of the MCMC implementation. Similarly, parameters that rarely update are unfavorable. A stepping scheme that rarely makes a successful step should still have the desired statistical properties of a stepping scheme that has more successful stepping, but will take a very long time to converge to a desirable distribution. In shorter chains, the final sample distributions will have ”spikes” in the distributions because of the rarity of steps and, when steps are rare, the same step is ”sampled” and saved repeatedly. This is to argue that while obvious correlations exist in this chain, every unusual space is sampled several distinct times and these traces still exhibit nice statistical properties in both regions. One quantitative metric of this is the success rate of the sampling. If the success rate is too high, then the chain likely is taking very small steps relative to the features in the N-dimensional probability space. These small steps mean the whole space is not explored as quickly as if the trial steps were larger. Similarly, if the trial-step success rate is too low, then the trial steps are likely too large, or with a correlation different than the shape of the probability space. Gelman et al. showed that for high dimensional systems, such as our 14 parameter model that the convergence of the sampling is optimized for a trial step success rate of 23.4% [85]. These traces are for just the fit of Pad B with the assumed energy 𝐸 𝑝 = 160 keV, but all traces that were visually inspected appear similar in their sampling and across Pads and across assumed energies. All MCMC iterations had success rates of 22 − 26%, which is close to the ideal success rate. One benefit of the MCMC that was discussed in 4.4.3 was that complicated parameter corre- lations can be investigated over the entirety of the probability space and not merely in the vicinity 115 of the 𝜒2 -minimum location. Figures 5.6, 5.7, 5.8, 5.9, 5.10, 5.11, 5.12, 5.13, 5.14 show the full correlation output of the MCMC run on Pad B at 𝐸 𝑝 = 160 keV. The nine pages are sections of the same type of graphic. The parameters are plotted row-wise and column-wise to make a 14x14 ma- trix of the two-parameter correlations and the two-dimensional, color-coded histograms are filled with the specified parameter values along the Markov Chain. More intense regions are regions where the MCMC spent more ”time”/samples and are to be interpreted as proportionally more probable. (The histograms are not normalized to integrate to unity, so the intensity is proportional to probability.) Uncorrelated parameters will have ellipse-like shapes in their sample distributions and highly correlated parameters, like the exponential background scaling and decay parameters, will appear as tight lines or curves. Parameters within a component of the background tend to be highly correlated. This can be seen in the exp1 and exp2 exponential parameters modelling the 𝛽 background. Similarly, the 7 Li∗ peak’s 𝜎 and 𝛾 are well correlated along an elliptical contour, even if 𝛾 ∼ 70 keV and 𝜎 ∼ 10 keV are the favored values. Finally, the polynomial terms of the same peak pol0, pol1, pol2 are all highly correlated and in complicated ways between all three parameters. What is not strongly correlated with background parameters is the inferred proton counts. Proton counts are shown in the fourth row of Fig. 5.6, 5.7, and 5.8. A lack of correlation with background parameters is suggestive that our model description of the background is modeling the background and our model description of the signal is modeling the signal, with minimal ”mixing” of the two. (Again, a great example of ”mixed” parameters is the interplay of the two Voigt widths.) The proton counts parameter, however, is not entirely uncorrelated with parameters associated with the shape of the 𝛽− 𝑝 peak. The strongest example of this is the correlation between the counts parameter and the intrinsic width parameter, shown in Fig. 5.15. This correlation shows that at 𝐸 𝑝 = 160 keV, as measured by Pad B, for all values of Γ𝑝 , there is an inference of non-zero proton 116 counts. Secondly, if we assume a large value of Γ, we must infer a large number of 𝛽− 𝑝 counts. Conversely, an assumed large number of 𝛽− 𝑝 counts implies a large intrinsic width Γ𝑝 . Finally, one may refer to Appendix C, which shows a collection of MCMC samples in their function, not parameter, forms compared to the experimental data. The figures within show examples of the complex interplay show in the histograms of Figures 5.6, 5.7, 5.8, 5.9, 5.10, 5.11, 5.12, 5.13, 5.14, and the flexibility of different components of the background model. 117 1D-exp1 ×106 ×103 45000 140 14000 12000 40000 0.1 20 120 20 12000 20 35000 20 10000 100 10000 30000 8000 exp1 exp1 exp1 exp1 80 8000 25000 20000 6000 60 6000 0.05 40 4000 15000 4000 10000 15 20 15 2000 15 5000 15 2000 0 0 0 0 0 ×10−3 ×103 ×106 exp1 (unitless) −114.9 −49.1 0 1.2 0 0.02 2.8 7.5 exp2 flat pCounts pSigma 1D-exp2 ×10−3 ×103 ×106 ×10−3 ×10−3 40000 ×10−3 140 0.1 12000 −60 −60 −60 −60 35000 10000 120 30000 10000 8000 100 25000 8000 exp2 exp2 exp2 exp2 −80 −80 −80 −80 80 20000 6000 6000 60 0.05 15000 4000 4000 40 10000 −100 −100 −100 −100 2000 2000 20 5000 0 0 0 0 0 ×103 ×106 12.7 21.8 exp2 (1/keV) 0 1.2 0 0.02 2.8 7.5 exp1 flat pCounts pSigma 1D-flat ×103 ×103 ×106 ×103 ×103 12000 3500 14000 1 1 12000 0.03 1 1 3000 12000 10000 10000 10000 2500 8000 8000 8000 0.02 2000 flat flat flat flat 6000 6000 6000 1500 0.5 0.5 4000 0.5 4000 0.5 1000 4000 2000 2000 0.01 2000 500 0 0 0 0 −3 ×10 0 ×10 6 12.7 21.8 −114.9 −49.1 flat (counts) 0 0.02 2.8 7.5 exp1 exp2 pCounts pSigma 1D-pCounts ×106 ×106 40000 ×106 ×106 ×106 10000 45000 12000 0.1 0.015 40000 0.015 35000 0.015 10000 0.015 35000 8000 30000 30000 pCounts pCounts pCounts pCounts 25000 8000 6000 0.01 25000 0.01 20000 0.01 6000 0.01 20000 15000 0.05 4000 15000 4000 10000 0.005 10000 0.005 5000 0.005 2000 0.005 2000 5000 0 0 0 0 ×10−3 ×103 0 12.7 21.8 −114.9 −49.1 0 1.2 Proton Counts (counts) 2.8 7.5 exp1 exp2 flat pSigma 1D-pSigma ×106 3500 10000 0.03 12000 10000 3000 8000 10000 8000 2500 6 6 6 6 0.02 pSigma pSigma pSigma pSigma 8000 6000 2000 6000 6000 1500 4000 4000 4000 1000 2000 0.01 4 2000 4 2000 4 500 4 0 0 0 0 ×10−3 ×103 ×106 0 12.7 21.8 −114.9 −49.1 0 1.2 0 0.02 Cathode Fraction (unitless) exp1 exp2 flat pCounts Figure 5.6: (1/9) 1-D distributions and 2-D correlations for Pad B and 𝐸 𝑝 = 160 keV. 118 18000 22000 35000 25000 50000 20000 16000 20 14000 20 30000 20 20 18000 20 20000 40000 16000 12000 25000 14000 exp1 exp1 exp1 exp1 exp1 10000 30000 15000 20000 12000 8000 10000 15000 20000 10000 6000 8000 10000 6000 4000 5000 15 2000 15 5000 15 10000 15 4000 2000 15 0 0 0 0 0 ×103 ×103 0 33 0.2 1 0.19 0.29 51.6 0 0.2 Gamma 10Be-IonEff 7LiCent 7LiSigma 7LiGamma ×10−3 ×10−3 ×10−3 50000 ×10−3 ×10−3 24000 16000 20000 22000 18000 −60 −60 −60 −60 −60 14000 30000 20000 40000 16000 12000 25000 18000 14000 16000 10000 20000 30000 12000 14000 exp2 exp2 exp2 exp2 exp2 −80 8000 −80 15000 −80 20000 −80 10000 8000 −80 12000 10000 6000 8000 10000 6000 4000 6000 10000 4000 −100 −100 −100 −100 −100 5000 4000 2000 2000 2000 0 0 0 0 0 ×103 ×103 0 33 0.2 1 0.19 0.29 51.6 0 0.2 Gamma 10Be-IonEff 7LiCent 7LiSigma 7LiGamma ×103 ×103 ×103 ×103 ×103 7000 5000 12000 6000 10000 1 1 1 10000 1 5000 1 6000 4000 8000 5000 8000 4000 3000 6000 4000 flat 2000 flat flat 6000 flat 3000 flat 3000 0.5 0.5 4000 0.5 4000 0.5 2000 0.5 2000 1000 2000 2000 1000 1000 0 0 0 0 0 ×10 3 ×103 0 33 0.2 1 0.19 0.29 51.6 0 0.2 Gamma 10Be-IonEff 7LiCent 7LiSigma 7LiGamma ×106 ×106 ×106 ×106 20000 ×106 10000 18000 22000 40000 0.015 0.015 30000 0.015 0.015 16000 0.015 20000 8000 35000 18000 25000 30000 14000 16000 pCounts pCounts pCounts pCounts pCounts 6000 20000 12000 14000 25000 0.01 0.01 0.01 0.01 10000 0.01 12000 15000 20000 10000 4000 8000 15000 8000 10000 6000 10000 6000 0.005 2000 0.005 5000 0.005 0.005 4000 0.005 4000 5000 2000 2000 0 0 0 0 0 ×103 ×103 0 33 0.2 1 0.19 0.29 51.6 0 0.2 Gamma 10Be-IonEff 7LiCent 7LiSigma 7LiGamma 10000 7000 4500 10000 5000 4000 6000 8000 3500 4000 5000 8000 6 3000 6 6 6 6 pSigma pSigma pSigma pSigma pSigma 6000 4000 2500 6000 3000 2000 4000 3000 4000 2000 1500 2000 1000 2000 4 500 4 4 2000 4 1000 4 1000 0 0 0 0 0 ×103 ×103 0 33 0.2 1 0.19 0.29 51.6 0 0.2 Gamma 10Be-IonEff 7LiCent 7LiSigma 7LiGamma Figure 5.7: (2/9) 1-D distributions and 2-D correlations for Pad B and 𝐸 𝑝 = 160 keV. 119 35000 35000 70000 30000 30000 30000 60000 20 20 25000 20 25000 20 25000 50000 20000 20000 exp1 exp1 exp1 exp1 20000 40000 15000 15000 15000 30000 10000 10000 10000 20000 15 5000 15 5000 15 5000 15 10000 0 0 0 0 ×10 9 ×10 6 ×10 3 ×103 0 0.01 −0.1 0 0.2 0.11 0.18 pol0 pol1 pol2 chisq ×10−3 ×10−3 30000 ×10−3 ×10−3 30000 60000 25000 −60 25000 −60 25000 −60 −60 50000 20000 20000 20000 40000 exp2 exp2 exp2 exp2 15000 −80 15000 −80 15000 −80 −80 30000 10000 10000 10000 20000 −100 −100 −100 −100 5000 5000 5000 10000 0 0 0 0 ×109 ×106 ×103 ×103 0 0.01 −0.1 0 0.2 0.11 0.18 pol0 pol1 pol2 chisq ×103 ×103 ×103 ×103 22000 18000 18000 14000 1 20000 18000 1 16000 1 12000 1 16000 14000 14000 16000 10000 12000 12000 14000 12000 10000 8000 10000 flat 10000 flat 8000 flat 6000 flat 8000 0.5 8000 0.5 6000 0.5 0.5 6000 6000 4000 4000 4000 4000 2000 2000 2000 2000 0 0 0 0 ×10 9 ×10 6 ×10 3 ×103 0 0.01 −0.1 0 0.2 0.11 0.18 pol0 pol1 pol2 chisq ×106 30000 ×106 ×106 ×106 60000 25000 0.015 25000 0.015 25000 0.015 0.015 50000 20000 20000 20000 pCounts pCounts pCounts pCounts 40000 0.01 15000 0.01 15000 0.01 15000 0.01 30000 10000 10000 10000 20000 0.005 5000 0.005 5000 0.005 5000 0.005 10000 0 0 0 0 ×109 ×106 ×103 ×103 0 0.01 −0.1 0 0.2 0.11 0.18 pol0 pol1 pol2 chisq 9000 9000 8000 8000 16000 8000 7000 7000 14000 7000 6000 12000 6 6000 6 6000 6 6 pSigma pSigma pSigma pSigma 5000 10000 5000 5000 4000 8000 4000 4000 3000 3000 6000 3000 2000 2000 2000 4000 4 1000 4 1000 4 1000 4 2000 0 0 0 0 ×10 9 ×10 6 ×10 3 ×103 0 0.01 −0.1 0 0.2 0.11 0.18 pol0 pol1 pol2 chisq Figure 5.8: (3/9) 1-D distributions and 2-D correlations for Pad B and 𝐸 𝑝 = 160 keV. 120 16000 5000 10000 18000 4500 30 16000 30 14000 30 30 30 4000 4000 8000 14000 12000 3500 Gamma Gamma Gamma Gamma Gamma 12000 10000 3000 6000 20 10000 20 8000 20 3000 20 20 2500 8000 4000 2000 6000 2000 6000 1500 4000 10 4000 10 10 1000 10 2000 10 1000 2000 2000 500 0 0 0 0 0 −3 ×10 ×10 3 ×10 6 12.7 21.8 −114.9 −49.1 0 1.2 0 0.02 2.8 7.5 exp1 exp2 flat pCounts pSigma 10000 1 1 1 10000 1 30000 1 35000 30000 8000 30000 25000 10Be-IonEff 10Be-IonEff 10Be-IonEff 10Be-IonEff 10Be-IonEff 25000 8000 25000 20000 20000 6000 6000 20000 15000 15000 15000 4000 4000 0.5 10000 0.5 10000 0.5 0.5 10000 0.5 2000 2000 5000 5000 5000 0 0 0 0 0 −3 ×10 ×10 3 ×10 6 12.7 21.8 −114.9 −49.1 0 1.2 0 0.02 2.8 7.5 exp1 exp2 flat pCounts pSigma ×103 ×103 50000 ×103 ×103 ×103 12000 50000 40000 10000 40000 10000 35000 40000 30000 8000 7LiCent 7LiCent 7LiCent 7LiCent 7LiCent 8000 0.25 30000 0.25 30000 0.25 0.25 25000 0.25 6000 6000 20000 20000 20000 15000 4000 4000 10000 10000 10000 2000 2000 5000 0.2 0 0.2 0 0.2 0 0.2 0 0.2 0 ×10−3 ×103 ×106 12.7 21.8 −114.9 −49.1 0 1.2 0 0.02 2.8 7.5 exp1 exp2 flat pCounts pSigma 20000 20000 22000 6000 18000 18000 5000 20000 16000 5000 16000 40 18000 16000 40 14000 40 40 14000 40 4000 7LiSigma 7LiSigma 7LiSigma 7LiSigma 7LiSigma 14000 4000 12000 12000 12000 10000 3000 10000 3000 10000 8000 8000 8000 2000 20 6000 20 6000 20 2000 20 6000 20 4000 4000 1000 4000 1000 2000 2000 2000 0 0 0 0 0 0 0 ×10−3 0 ×103 0 ×106 0 12.7 21.8 −114.9 −49.1 0 1.2 0 0.02 2.8 7.5 exp1 exp2 flat pCounts pSigma 0.15 ×10 0.15 ×10 0.15 ×10 0.15 ×10 0.15 ×10 3 3 3 3 3 24000 7000 7000 25000 22000 22000 20000 6000 20000 6000 18000 18000 20000 7LiGamma 7LiGamma 7LiGamma 7LiGamma 7LiGamma 5000 16000 5000 16000 0.1 15000 0.1 14000 0.1 4000 0.1 14000 0.1 4000 12000 12000 10000 3000 10000 3000 10000 8000 8000 2000 2000 0.05 5000 0.05 6000 4000 0.05 0.05 6000 4000 0.05 1000 1000 2000 2000 0 0 0 0 0 ×10−3 ×103 ×106 12.7 21.8 −114.9 −49.1 0 1.2 0 0.02 2.8 7.5 exp1 exp2 flat pCounts pSigma Figure 5.9: (4/9) 1-D distributions and 2-D correlations for Pad B and 𝐸 𝑝 = 160 keV. 121 1D-Gamma ×106 14000 16000 8000 10000 30 30 30 30 0.04 12000 14000 7000 8000 12000 6000 10000 Gamma Gamma Gamma Gamma 10000 5000 6000 20 8000 20 8000 20 20 4000 6000 4000 0.02 4000 6000 3000 4000 2000 10 2000 10 10 10 2000 2000 1000 0 0 0 0 0 ×10 3 ×103 Proton Energy (keV) 0.2 1 0.19 0.29 51.6 0 0.2 10Be-IonEff 7LiCent 7LiSigma 7LiGamma 1D-10Be-IonEff ×106 35000 1 14000 1 1 16000 1 20000 30000 18000 12000 14000 16000 10Be-IonEff 10Be-IonEff 10Be-IonEff 10Be-IonEff 25000 12000 10000 14000 20000 10000 12000 8000 0.05 15000 8000 10000 6000 8000 6000 0.5 4000 0.5 10000 0.5 4000 0.5 6000 4000 2000 5000 2000 2000 0 0 0 0 0 ×103 ×103 0 33 Proton Resolution Sigma (keV) 0.19 0.29 51.6 0 0.2 Gamma 7LiCent 7LiSigma 7LiGamma 1D-7LiCent ×103 ×103 35000 ×106 ×103 ×103 16000 22000 35000 14000 30000 0.1 20000 18000 30000 12000 25000 16000 25000 7LiCent 7LiCent 7LiCent 7LiCent 14000 0.25 10000 0.25 20000 0.25 12000 0.25 20000 8000 15000 10000 6000 0.05 8000 15000 10000 6000 10000 4000 4000 2000 5000 5000 2000 0.2 0 0.2 0 0.2 0 0.2 0 0 ×103 0 33 0.2 1 Proton Emitting Resonance Intrinsic Width (keV) 51.6 0 0.2 Gamma 10Be-IonEff 7LiSigma 7LiGamma 1D-7LiSigma ×106 8000 22000 22000 16000 20000 20000 7000 14000 18000 18000 40 6000 40 12000 40 16000 0.04 40 16000 7LiSigma 7LiSigma 7LiSigma 7LiSigma 5000 14000 14000 10000 12000 12000 4000 8000 10000 10000 3000 6000 8000 8000 20 2000 20 4000 20 6000 0.02 20 6000 4000 4000 1000 2000 2000 2000 0 0 0 0 0 0 0 ×10 3 0 0 ×103 0 33 0.2 1 0.19 0.29 Proton Ionizaton Efficiency (unitless) 0 0.2 Gamma 10Be-IonEff 7LiCent 7LiGamma 1D-7LiGamma 0.15 ×10 0.15 ×10 0.15 ×10 0.15 ×10 ×106 3 3 3 3 10000 20000 22000 18000 35000 20000 0.06 8000 30000 18000 16000 7LiGamma 7LiGamma 7LiGamma 7LiGamma 14000 25000 16000 0.1 6000 0.1 12000 0.1 0.1 14000 0.04 20000 12000 10000 10000 4000 8000 15000 8000 6000 10000 0.05 2000 0.05 4000 0.05 0.05 6000 4000 0.02 5000 2000 2000 0 0 0 0 ×103 0 0 33 0.2 1 0.19 0.29 51.6 10Be Ionization Efficiency (unitless) Gamma 10Be-IonEff 7LiCent 7LiSigma Figure 5.10: (5/9) 1-D distributions and 2-D correlations for Pad B and 𝐸 𝑝 = 160 keV. 122 14000 12000 25000 30 12000 30 12000 30 30 10000 20000 10000 10000 8000 Gamma Gamma Gamma Gamma 8000 15000 20 8000 20 20 6000 20 6000 6000 10000 4000 4000 4000 10 2000 10 2000 10 2000 10 5000 0 0 0 0 ×10 9 ×10 6 ×10 3 ×103 0 0.01 −0.1 0 0.2 0.11 0.18 pol0 pol1 pol2 chisq 25000 1 25000 1 25000 1 1 50000 20000 20000 10Be-IonEff 10Be-IonEff 10Be-IonEff 10Be-IonEff 20000 40000 15000 15000 15000 30000 10000 10000 10000 20000 0.5 0.5 0.5 0.5 5000 5000 5000 10000 0 0 0 0 ×109 ×106 ×103 ×103 0 0.01 −0.1 0 0.2 0.11 0.18 pol0 pol1 pol2 chisq ×103 ×103 ×103 ×103 35000 35000 35000 60000 30000 30000 30000 50000 25000 25000 25000 40000 7LiCent 7LiCent 7LiCent 7LiCent 0.25 20000 0.25 20000 0.25 20000 0.25 30000 15000 15000 15000 20000 10000 10000 10000 5000 5000 5000 10000 0.2 0 0.2 0 0.2 0 0.2 0 ×109 ×106 ×103 ×103 0 0.01 −0.1 0 0.2 0.11 0.18 pol0 pol1 pol2 chisq 14000 14000 30000 12000 12000 12000 25000 40 10000 40 10000 40 10000 40 7LiSigma 7LiSigma 7LiSigma 7LiSigma 20000 8000 8000 8000 15000 6000 6000 6000 20 4000 20 4000 20 4000 20 10000 2000 2000 2000 5000 0 0 0 0 0 ×109 0 ×106 0 ×103 0 ×103 0 0.01 −0.1 0 0.2 0.11 0.18 pol0 pol1 pol2 chisq 0.15 ×10 0.15 ×10 0.15 ×10 0.15 ×10 3 3 3 3 20000 20000 18000 35000 18000 18000 16000 30000 16000 16000 14000 7LiGamma 7LiGamma 7LiGamma 7LiGamma 14000 14000 25000 0.1 12000 0.1 12000 0.1 12000 0.1 10000 20000 10000 10000 8000 8000 15000 8000 6000 6000 6000 10000 0.05 4000 0.05 4000 0.05 4000 0.05 2000 5000 2000 2000 0 0 0 0 ×10 9 ×10 6 ×10 3 ×103 0 0.01 −0.1 0 0.2 0.11 0.18 pol0 pol1 pol2 chisq Figure 5.11: (6/9) 1-D distributions and 2-D correlations for Pad B and 𝐸 𝑝 = 160 keV. 123 ×106 ×106 ×106 ×106 30000 ×106 35000 30000 9000 22000 30000 20000 25000 8000 25000 10 25000 10 10 18000 10 10 7000 16000 20000 20000 6000 14000 20000 pol0 pol0 pol0 pol0 pol0 12000 5000 15000 15000 15000 10000 4000 5 10000 5 10000 5 8000 6000 5 10000 5 3000 2000 5000 4000 5000 5000 2000 1000 0 0 0 0 0 ×10−3 ×103 ×106 12.7 21.8 −114.9 −49.1 0 1.2 0 0.02 2.8 7.5 exp1 exp2 flat pCounts pSigma ×106 35000 ×106 30000 ×106 ×106 ×106 9000 0 30000 0 0 18000 0 25000 0 8000 25000 16000 7000 25000 14000 20000 20000 6000 12000 20000 5000 pol1 pol1 pol1 pol1 pol1 15000 10000 15000 −0.05 −0.05 −0.05 −0.05 −0.05 15000 8000 4000 10000 10000 3000 10000 6000 4000 2000 5000 5000 5000 2000 1000 0 0 0 0 0 ×10−3 ×103 ×106 12.7 21.8 −114.9 −49.1 0 1.2 0 0.02 2.8 7.5 exp1 exp2 flat pCounts pSigma ×103 ×103 ×103 ×103 ×103 30000 25000 8000 25000 14000 7000 25000 12000 20000 20000 6000 10000 20000 5000 15000 15000 pol2 pol2 pol2 pol2 pol2 0.1 15000 0.1 0.1 8000 0.1 0.1 4000 6000 10000 10000 3000 10000 4000 2000 5000 5000 5000 2000 1000 0 0 0 0 0 0 0 0 0 0 ×10−3 ×103 ×106 12.7 21.8 −114.9 −49.1 0 1.2 0 0.02 2.8 7.5 exp1 exp2 flat pCounts pSigma 0.18 ×10 0.18 ×10 0.18 ×10 0.18 ×10 0.18 ×10 3 3 3 3 3 70000 60000 60000 18000 16000 60000 16000 50000 50000 14000 14000 0.16 50000 0.16 40000 0.16 12000 0.16 40000 0.16 12000 10000 chisq chisq chisq chisq chisq 40000 10000 30000 30000 8000 30000 8000 0.14 0.14 20000 0.14 6000 0.14 20000 0.14 6000 20000 4000 4000 10000 10000 10000 2000 2000 0.12 0 0.12 0 0.12 0 0.12 0 0.12 0 ×10−3 ×103 ×106 12.7 21.8 −114.9 −49.1 0 1.2 0 0.02 2.8 7.5 exp1 exp2 flat pCounts pSigma Figure 5.12: (7/9) 1-D distributions and 2-D correlations for Pad B and 𝐸 𝑝 = 160 keV. 124 ×106 ×106 ×106 ×106 14000 ×106 14000 35000 20000 25000 12000 12000 18000 30000 10 10000 10 20000 10 10 10000 10 16000 25000 14000 8000 8000 12000 15000 20000 pol0 pol0 pol0 pol0 pol0 10000 6000 15000 6000 10000 8000 5 4000 5 5 10000 5 4000 5 6000 5000 4000 2000 5000 2000 2000 0 0 0 0 0 ×10 3 ×103 0 33 0.2 1 0.19 0.29 51.6 0 0.2 Gamma 10Be-IonEff 7LiCent 7LiSigma 7LiGamma ×106 ×106 ×106 ×106 14000 ×106 20000 0 12000 0 25000 0 35000 0 12000 0 18000 30000 16000 10000 20000 10000 25000 14000 8000 15000 8000 12000 pol1 pol1 pol1 pol1 pol1 20000 10000 −0.05 −0.05 −0.05 −0.05 −0.05 6000 6000 15000 8000 10000 4000 10000 4000 6000 5000 4000 2000 5000 2000 2000 0 0 0 0 0 ×103 ×103 0 33 0.2 1 0.19 0.29 51.6 0 0.2 Gamma 10Be-IonEff 7LiCent 7LiSigma 7LiGamma ×103 ×103 25000 ×103 ×103 ×103 12000 35000 18000 12000 30000 16000 10000 20000 10000 14000 25000 8000 12000 15000 8000 20000 pol2 pol2 pol2 pol2 pol2 0.1 6000 0.1 0.1 0.1 6000 0.1 10000 10000 15000 8000 4000 4000 6000 10000 5000 4000 2000 5000 2000 2000 0 0 0 0 0 0 0 0 0 0 ×103 ×103 0 33 0.2 1 0.19 0.29 51.6 0 0.2 Gamma 10Be-IonEff 7LiCent 7LiSigma 7LiGamma 0.18 ×10 0.18 ×10 0.18 ×10 0.18 ×10 0.18 ×10 3 3 3 3 3 25000 30000 60000 35000 50000 25000 30000 20000 50000 40000 0.16 0.16 0.16 40000 0.16 20000 0.16 25000 15000 chisq chisq chisq chisq chisq 30000 20000 30000 15000 10000 15000 0.14 0.14 20000 0.14 20000 0.14 10000 0.14 10000 5000 10000 10000 5000 5000 0.12 0 0.12 0 0.12 0 0.12 0 0.12 0 ×103 ×103 0 33 0.2 1 0.19 0.29 51.6 0 0.2 Gamma 10Be-IonEff 7LiCent 7LiSigma 7LiGamma Figure 5.13: (8/9) 1-D distributions and 2-D correlations for Pad B and 𝐸 𝑝 = 160 keV. 125 1D-pol0 ×106 ×106 ×103 ×106 ×106 100 45000 70000 40000 10 80 10 60000 10 35000 50000 30000 0.05 60 pol0 pol0 pol0 40000 25000 20000 40 30000 5 5 20000 5 15000 20 10000 10000 5000 0 0 0 0 ×106 ×103 ×103 7Li Energy (keV) −0.1 0 0.2 0.11 0.18 pol1 pol2 chisq 1D-pol1 ×106 ×103 ×106 ×106 ×106 45000 0 100 0 90000 0 40000 80000 80 35000 70000 30000 60000 60 0.05 pol1 pol1 pol1 50000 25000 −0.05 −0.05 −0.05 40000 20000 40 30000 15000 20000 10000 20 10000 5000 0 0 0 ×109 0 ×103 ×103 0 0.01 7Li Resolution Sigma (keV) 0.2 0.11 0.18 pol0 pol2 chisq 1D-pol2 ×103 ×103 ×106 ×103 70000 90000 40000 80000 35000 60000 70000 0.06 30000 50000 60000 25000 pol2 pol2 pol2 0.1 40000 0.1 50000 0.04 0.1 20000 30000 40000 15000 30000 20000 10000 20000 10000 10000 0.02 5000 0 0 0 0 0 0 ×109 ×106 0 ×103 0 0.01 −0.1 0 7Li Intrinsic Width (keV) 0.11 0.18 pol0 pol1 chisq 1D-chisq 0.18 ×10 0.18 ×10 0.18 ×10 ×106 3 3 3 45000 45000 40000 40000 0.15 40000 35000 35000 35000 0.16 30000 0.16 30000 0.16 30000 25000 0.1 chisq chisq chisq 25000 25000 20000 20000 20000 0.14 15000 0.14 15000 0.14 15000 10000 10000 10000 0.05 5000 5000 5000 0.12 0 0.12 0 0.12 0 ×109 ×106 ×103 0 0 0.01 −0.1 0 0.2 Pol0 (counts) pol0 pol1 pol2 Figure 5.14: (9/9) 1-D distributions and 2-D correlations for Pad B and 𝐸 𝑝 = 160 keV. 126 Proton Counts (counts) 16000 2500 14000 2000 12000 10000 1500 8000 1000 6000 4000 500 2000 0 5 10 15 20 25 30 Proton Emitting Resonance Intrinsic Width (keV) Figure 5.15: 2D parameter sample distribution between the intrinsic width Γ of the proton-emitting resonance and proton counts for Pad B at 𝐸 𝑝 = 160 keV. For large values of Γ, a large number of 𝛽− 𝑝 counts is inferred. Conversely, a large number of 𝛽− 𝑝 counts implies a large intrinsic width Γ. 5.1 𝛽− 𝑝 intensity maps The previous section was a deep dive into just one fit to the data to test the convergence of the MCMC code. Ultimately, the primary observable of interest is the intensity of the 11 Be(𝛽− 𝑝) decay and its energy. The MCMC algorithm gives us a Markov Chain of parameter samples assuming a proton resonance energy 𝐸 𝑝 . By plotting the proton counts parameter for each 𝐸 𝑝 in a 2D histogram, we arrive at an intensity map showing the inferred 𝛽− 𝑝 branching ratio as a function of assumed resonance energy. Figure 5.16 shows this map for Pads B and C. Figure 5.17 shows this map for Pads D and E. These intensity maps are probability density profiles. Since each pad is operated independently 127 −3 × 10 5 10 0.1 Inferred Branching Ratio 104 0.08 3 10 0.06 102 0.04 10 0.02 0 150 160 170 180 190 200 210 1 Energy of Resonance (keV) −3 × 10 5 10 0.14 Inferred Branching Ratio 0.12 104 0.1 3 10 0.08 0.06 102 0.04 10 0.02 0 150 160 170 180 190 200 210 1 Energy of Resonance (keV) Figure 5.16: Intensity map from the MCMC sampling for Pad B (top) and Pad C (bottom). Inferred proton counts are normalized by the 𝛽− 𝛼 branch to calculate a branching ratio and this branching ratio is plotted against resonance energy. The black curve shows the most probable value; red shows the 68% confidence interval boundaries; green shows the 95% confidence interval boundaries; purple shows the 99.7% confidence interval boundaries. Pad B is suggestive of a finite value but consistent with a zero value for 𝐸 𝑝 < 170 keV. Above 𝐸 𝑝 = 170 keV, there is no indication of an excess caused by 𝛽− 𝑝. Pad C is suggestive of a finite value but consistent with a zero value for 150 < 𝐸 𝑝 < 170 keV and 𝐸 𝑝 ≳ 200 keV. Elsewhere, there is no indication of an excess caused by 𝛽− 𝑝. of the others, each map from each pad is an independent measurement of the inferred branching ratio 𝑏 𝑝 . To combine, a bin-by-bin multiplication of the intensity map for each of the four pads was performed. To keep the interpretation of probability for an assumption of 𝐸 𝑝 , the combined probability distribution was normalized to unity for each 𝐸 𝑝 . The result of this multiplication is shown in Fig. 5.18. 128 −3 0.14 × 10 10 5 Inferred Branching Ratio 0.12 104 0.1 3 0.08 10 0.06 102 0.04 10 0.02 0 150 160 170 180 190 200 210 1 Energy of Resonance (keV) −3 × 10 10 5 0.1 Inferred Branching Ratio 104 0.08 3 10 0.06 102 0.04 10 0.02 0 150 160 170 180 190 200 210 1 Energy of Resonance (keV) Figure 5.17: Intensity map from the MCMC sampling for Pad D (top) and Pad E (bottom). Inferred proton counts are normalized by the 𝛽− 𝛼 branch to calculate a branching ratio and this branching ratio is plotted against resonance energy. The black curve shows the most probable value; red shows the 68% confidence interval boundaries; green shows the 95% confidence interval boundaries; purple shows the 99.7% confidence interval boundaries. Pad D is suggestive of a finite value but consistent with a zero value for 150 < 𝐸 𝑝 < 165 keV and 185 < 𝐸 𝑝 < 210. Pad E is suggestive of a finite value but consistent with a zero value for 𝐸 𝑝 < 150 keV and 165 < 𝐸 𝑝 < 185 keV. Above 𝐸 𝑝 = 180 keV, there is no indication of an excess caused by 𝛽− 𝑝. The energy uncertainty in the region of interest comes primarily from the uncertainty in the pulse height defect of the 7 Li∗ peak which was determined to be ∼ 3% in Sec. 4.3, or ∼ 8 keV. Additional uncertainty comes from the extrapolation of the energy calibration between the 7 Li∗ peak and full-energy 7 Li+𝛼 peak. The relative uncertainty from the extrapolation comes from the uncertainty of the pulse height defect for the full-energy peak, or ∼ 2%, measured from the 7 Li∗ peak at 𝐸 𝑝 = 241(8) keV. The energy uncertainty from the energy extrapolation to a peak at 𝐸 𝑝 = 150 keV would be be less than ∼ 2 keV, which is small relative to the ∼ 8 keV uncertainty 129 ×10−3 10−1 0.05 Inferred Branching Ratio 0.04 10−2 0.03 0.02 10−3 0.01 0 150 160 170 180 190 200 210 Energy of Resonance (keV) Figure 5.18: Combined intensity map of all four pads. Probability density below ∼ 10−4 /bin is truncated. The black curve shows the most probable value; red shows the 68% confidence interval boundaries; green shows the 95% confidence interval boundaries; purple shows the 99.7% confidence interval boundaries. There is evidence of a 𝛽− 𝑝 -like excess at 𝐸 𝑝 ∼ 160 keV. from the location of the 7 Li∗ peak. 5.2 Discussion With the composite heatmap of Fig. 5.18, there is a clear excess at 𝐸 𝑝 = 160(8) kev with a maximum in the branching ratio intensity of 𝑏 𝑝 = 8+5 −6 −4 · 10 . The central value comes from the most intense value and the asymmetric errors come from integrating the most likely values out to 68% of the probability space for the assumed 𝐸 𝑝 . This is done to preserve the most probable regions in the confidence interval integral. For a further argument for this construction of a confidence interval, see Appendix Chapter C. 130 The choice of intrinsic width Γ is highly correlated with the inferred 𝛽− 𝑝 branching ratio. For all fits, the smaller the Γ, the smaller the inferred branching ratio. The branching ratios shown in Fig. 5.18 are integrated over all values of Γ, some of which are sizable. That is, much of the uncertainty in this result is driven by our choice of a flat prior in the width Γ for Γ < 30 keV. It is also worth noting that Pad E yields slightly different results than the other pads. In Pads B, C, and D, there are small excesses peaking at 𝑏 𝑝 ∼ 10−5 from 150 ≲ 𝐸 𝑝 ≲ 170 keV and Pads C and D have similar or smaller excesses for 𝐸 𝑝 > 190 keV. Pad E, however, has a 𝑏 𝑝 ∼ 10−5 excess at 165 ≲ 𝐸 𝑝 ≲ 185 keV. All pads were treated with the same analysis procedure, from data sorting, to the energy calibration, to statistical inference, so it is hard to ascertain the source of this, if it is significant beyond statistical fluctuations. The effect of a non-statistical energy shift in Pad E would be to lower the composite branching ratio at 𝐸 𝑝 = 160(8) keV. 5.2 Comparison To Other Experiments Several other experiments have been conducted to understand the 11 Be(𝛽− 𝑝) branching ratio. Most directly comparable to our results are those of Ayyad et al. (2019) [2], where they reported 𝐸 𝑝 = 196(20) keV, Γ = 12(5) keV and 𝑏 𝑝 = 1.3(3) · 10−5 in a gaseous time projection chamber. Our energy is ∼ 2𝜎 lower than that reported in Ayyad et al. (2019) and our 𝛽− 𝑝 branching ratios are consistent. Next, we consider the indirect branching ratio determined by AMS of the 11 Be decay products. The newest AMS measurement can only measure a total 10 Be production rate and their suggested value of the branching ratio was < 2.2 · 10−6 , which is lower than our measured value. Additionally, two reaction measurements have been published that provide 11 B resonance structure information for comparison. A measurement of the 10 Be(𝑝, 𝑝′)10 Be scattering reaction by Ayyad et al. (2022) [30] reported observation of a resonance with energy 𝐸 𝑝 = 171(20) keV, proton partial width Γ𝑝 = 4.5(11) keV, total width Γ𝑇 = 16(3) keV. Our energy agrees with this 131 measurement and is more than a factor of two more precise. A measurement of 10 Be(𝑑, 𝑛)10 Be+𝑝 reported observation of a resonance with energy 𝐸 𝑝 = 211(40) keV. We are consistent with this second measurement, but that is due to the large 40 keV uncertainty in the resonance energy. The comparable experimental values are in Tab. 5.1. Our resonance energy 𝐸 𝑝 = 160(8) keV is consistent with all literature resonance energies. Our measurement of the branching ratio 𝑏 𝑝 = 8+5 −6 is consistent with the Ayyad et al. (2019) 𝛽− 𝑝 measurement and are somewhat at −4 · 10 odds with the most recent AMS inferred limit by Riisager et al.. We are not able to make strong claims about the intrinsic width of the state, however, if the width is determined precisely in the future, then the probability density plots from the analysis of our experimental data can be used to refine the experimental branching ratio. Reference 𝐸 𝑝 (keV) Γ𝑝 (keV) BR(𝛽− 𝑝) production mechanism Riisager et al. (2014) [6] - - 8.3(9) · 10−6 10 Be AMS Ayyad et al. (2019) [2] 196(20) 12(5) 1.3(3) · 10−5 𝛽− 𝑝 Riisager et al. (2020) [22] - - < 2.2 · 10−6 10 Be AMS Ayyad et al. (2022) [30] 171(20) 4.5(11) - 10 Be(𝑝, 𝑝 ′) Lopez-Saavedra et al. [32] 211(40) - - 10 Be(𝑑, 𝑛)10 Be+𝑝 Surbrook et al. (this work) 160(8) - 8+5 −4 · 10 −6 𝛽− 𝑝 Table 5.1: Summary of experimental results for the branching ratio of the 11 Be(𝛽− 𝑝) decay and candidate resonant state properties. Resonance energies, 𝐸 𝑝 , are in reference to the proton emission threshold. 5.2 Comparison To Theory A number of theory models have been applied to this question of a ∼ 10−5 𝛽− 𝑝 branching ratio. Different models can calculate different physical observables and patterns between observables. The following papers report calculations of the 𝛽− 𝑝 branching ratio. Baye and Tursunov calculated 𝛽− 𝑝 decay through a structure-less continuum and calculated 𝑏 𝑝 ∼ 3 · 10−8 [12]. Volya performed shell model calculations of the 11 B nucleus with the psdu and fsu interactions, to search for undiscovered 132 near-threshold resonances that could participate in allowed 𝛽− 𝛼 and 𝛽− 𝑝 decays. He calculated 𝑏 𝑝 ∼ 5 · 10−10 , unless there exists a resonant state which was not predicted by the shell model calculations, then 𝑏 𝑝 < 1 · 10−7 [23]. Oko lowicz et al. performed shell model calculations with a continuum coupling and found a near-threshold resonance that could participate in an allowed 𝛽− 𝑝 decay with a small intensity 𝑏 𝑝 ∼ 3 · 10−7 [86]. Elkamhawy et al. performed Halo effective field theory calculations with an assumed resonance of the energy reported in Ayyad et al. (2019) [2] and +4.0 found an experimentally consistent branch of 𝑏 𝑝 = 4.9+5.6−2.9 (ex.)−0.8 (th.) · 10 −6 [26, 87]. Atkinson et al. performed ab initio calculations of a participating resonance in 11 B and find 𝑏 𝑝 = 1.3(5) · 10−6 [28]. These model predictions are summarized in Table 5.2. Reference 𝐸 𝑝 (keV) Γ𝑝 (keV) BR(𝛽− 𝑝) Baye and Tursunov [12] ∼ 150 - ∼ 3 · 10−8 Volya [23] - - ∼ 5 · 10−10 Oko lowicz et al. (2020) [24] ∼ 142 - - Oko lowicz et al. (2022) [86] ∼ 160 - ∼ 3 · 10−7 +5.3 +4.0 Elkamhawy et al. (2021) [26] (196)∗ 9.0+4.8 −3.3 (ex.)−2.2 (th.) 4.9+5.6 −2.9 (ex.)−0.8 (th.) · 10 −6 Atkinson et al. [28] - - 1.3(5) · 10−6 Le Ahn et al. [29] ∼ 182 ∼6 - Table 5.2: Summary of theory predictions for the branching ratio of the 11 Be(𝛽− 𝑝) decay and candidate intermediary resonant state properties. Resonance energies, 𝐸 𝑝 , are in reference to the proton emission threshold. ∗ The energy of Elkamhawy et al. (2021) was set to the experimental value of [2]. Generally, theory predicts small widths Γ < 10 keV and small branching ratios 𝑏 𝑝 < 10−5 . Many of these branching ratios are smaller than the precision of our measurement and are effectively zero-valued, which our result excludes at 94% confidence at 𝐸 𝑝 = 160 keV. Elkamhawy et al. (2022) [87] is a follow-up paper to the 2021 paper [26] that goes into detail regarding these effective field theory calculations, which predict the relationship between the resonance energy 𝐸 𝑝 , width Γ𝑝 , and the branching ratio 𝑏 𝑝 . Considering our result’s relationship between the width Γ and branching 133 ratio 𝑏 𝑝 , we may consider testing the consistency between our result and their calculations by more than just the branching ratio 𝑏 𝑝 . In terms of energy predictions, the 𝛽− 𝑝-like excess in our data is found at 𝐸 𝑝 ∼ 160(8) keV. The Baye and Tursunov value is not a resonance energy and comes from broad continuum behavior, so we’ll disregard. Our resonance energy is similar to Oko lowicz et al., especially the 2022 value of 𝐸 𝑝 ∼ 160 keV, which includes 𝛽 decay feeding weights. Our value is somewhat low compared to the value of Le Ahn et al. (𝐸 𝑝 ∼ 182 keV), but there is no uncertainty reported from that calculation. 134 CHAPTER 6 CONCLUSIONS: INTERPRETING THIS WORK AND FUTURE EFFORTS 6.1 Conclusions Concluding, the determination of the 11 Be(𝛽− 𝑝) branching ratio is strongly motivated. The novel 𝛽− 𝑝 decay is a probe of the three-fold near-threshold structure in 11 B. (The thresholds for 𝑛, 𝑝, and 3H are all similar in 11 B.) This near-threshold structure likely has complex behavior, explaining the difficulty in theoretical calculations to report a consistent 𝛽− 𝑝 branching ratio and the limited number of experimental results to report consistent partial widths of the participating state in 11 B. Prior to our measurement, only one direct observation of the 𝛽− 𝑝 from 11 Be(𝛽− 𝑝) had been reported and it suggested a finite value of the branching ratio. This is inconsistent with the most recent indirect measurement result inferred by counting the 10 Be residues which sets an upper limit on the production of 10 Be from 11 Be. The 11 Be(𝛽− 𝑝) decay yields a 10 Be residue, so 11 Be(𝛽− 𝑝) decay can explain the presence of 10 Be in the decay products of a sample of 11 Be. An excess of 10 Be production relative to 11 Be(𝛽− 𝑝) decay is a signature of the hypothetical dark decay of loosely-bound neutrons. The converse, where 11 Be(𝛽− 𝑝) is observed in excess of 10 Be production, defies physical explanation. As new measurements are performed, should the 10 Be production rate be deemed in excess of the 11 Be(𝛽− 𝑝) decay rate, one can again consider the hypothetical dark decay of the halo neutron in 11 Be. The 𝛽− -delayed charged particle spectrum of 11 Be was measured using the GADGET system at the National Superconducting Cyclotron Laboratory. We carefully characterized portions of the significant 11 Be(𝛽− 𝛼) background and our expected 𝛽− 𝑝 signal, guided by R-matrix calculations and 𝛾-ray coincidences with 𝛽− 𝛼 decays. We performed a search for a 𝛽− 𝑝 -like excess in the 𝛽− -delayed charged particle spectrum using a Bayesian statistical model, sampled by a Markov Chain 135 Monte Carlo algorithm. Here, we report the second direct measurement of 11 Be(𝛽− 𝑝). We found a ∼ 2𝜎 𝛽− 𝑝 excess at resonant energy 𝐸 𝑝 = 160(8) keV and branching ratio 𝑏 𝑝 = 8+5 −6 −4 · 10 , which is consistent with the experimental literature. The energy of the 𝛽− 𝑝 excess is compatible with resonances predicted by many structure models and, similar to the other 𝛽− 𝑝 measurement, the measured branching ratio is larger than most models predict. 6.2 Outlook Looking forward, several types of measurements could be performed to further investigate the interesting 11 Be(𝛽− 𝑝) decay: Old 7 Li(𝛼,𝛼′) scattering measurements could be repeated with very fine precision around 150- 200 keV above the 11 B proton threshold. A recent Ph.D. project proposal at Michigan State University discussed performing such a measurement at Notre Dame in the near future. If the structure of 11 B around 𝐸 𝑥 = 11.4 MeV is multi-faceted, state selection may be probe- sensitive and the (𝛼, 𝛼′) scattering may yield not just additional, but unique information. The 𝛽− 𝑝 from 11 Be could be remeasured. Our present measurement was systematically limited. The MICROMEGAS pad plane of GADGET was recently upgraded from the 13 pixel design discussed here to a new pad plane geometry with 1024 pixels. GADGET II, as it is called, can operate as a time projection chamber, which has demonstrated particle identification. Utilization of GADGET II to perform another measurement of 11 Be may reduce the large 𝛽− and 𝛽− 𝛼 backgrounds we contended with and limit the systematic uncertainties related to their modeling. The indirect measurements of the 11 Be(𝛽− 𝑝) branching ratio with AMS could be improved. The indirect AMS 11 Be(𝛽− 𝑝) branching ratio limit comes from the collection sample yielding the smallest branching ratio; other samples measured more than two orders of magnitude larger abundances of 10 Be, which the authors contend was due to molecular contamination of the 11 Be+ 136 sample beam with 10 BeH+ , which has a relative mass difference of ∼ 3 · 10−5 from 11 Be+ . Standard magnetic cleaning of cocktail beams cannot separate these two ions and the techniques used in the past to produce intense beams of 11 Be will also produce 10 Be. An improvement to the AMS measurement could be to use some technique to purify sample beam against 10 BeH+ . Two such techniques could be mass filtering using a Multi-Reflection Time-Of-Flight mass spectrometer or charge breeding the 11 Be beam to a high charge state, which would break any possible molecules. Such schemes suffer beam intensity losses, but new rare isotope production facilities may have the 11 Be production capability to suffer these losses and still deliver enough 11 Be to measure the minuscule branching ratios predicted by many theoretical models. Changing focus, one could perform direct measurements of the W-value of modern detector gas mixtures at ∼ keV/u beam energies, similar to the work done in the 1960s, 1970s, and 1980s on chemically pure and tissue equivalent gas. It would seem that there is a gap in the scientific literature here and the model to parameterize detector gas behavior from its constituents is not well tested at very low energies or across elemental beam species. At these low beam energies, the beam velocity approaches that of the classical bound-electron velocity, so it is hard to imagine the absence of chemical effects in the beam stopping and ionization production. Finally, another bound-neutron dark decay candidate was 15 C. One nice feature of 15 C is that the 15 C(𝛽− 𝑝) decay is energetically forbidden, so the generation of 14 C from 15 C is a clear signal of a bound-neutron dark decay. However, 14 C is naturally abundant and presents an obvious hazard of a false detection. Additionally, the relative mass difference between 14 CH+ and 15 C+ is also ∼ 3 · 10−5 , so the difficulties from the 11 Be AMS measurements of removing a daughter-containing molecule from the beam holds true in 15 C. The AMS sample beam preparation techniques proposed for a re-measurement of 11 Be−→10 Be could also be utilized with 15 C to reduce 14 C background in AMS sample collection. 137 BIBLIOGRAPHY [1] Jochen Erler, Noah Birge, Markus Kortelainen, Witold Nazarewicz, Erik Olsen, Alexander M Perhac, and Mario Stoitsov. The limits of the nuclear landscape. Nature, 486(7404):509–512, 2012. [2] Y Ayyad, B Olaizola, W Mittig, G Potel, V Zelevinsky, M Horoi, S Beceiro-Novo, M Alcorta, C Andreoiu, T Ahn, et al. Direct observation of proton emission in be 11. Physical Review Letters, 123(8):082501, 2019. [3] Bartosz Fornal and Benjamı́n Grinstein. Dark matter interpretation of the neutron decay anomaly. Physical review letters, 120(19):191801, 2018. [4] Pieter Mumm. Resolving the neutron lifetime puzzle. Science, 360(6389):605–606, 2018. [5] Fred E. Wietfeldt and Geoffrey L. Greene. Colloquium: The neutron lifetime. Rev. Mod. Phys., 83:1173–1192, Nov 2011. doi: 10.1103/RevModPhys.83.1173. URL https://link.aps. org/doi/10.1103/RevModPhys.83.1173. [6] K Riisager, O Forstner, MJG Borge, JA Briz, M Carmona-Gallardo, LM Fraile, HOU Fynbo, T Giles, A Gottberg, A Heinz, et al. Be11 (𝛽p), a quasi-free neutron decay? Physics Letters B, 732:305–308, 2014. [7] K Abe, Y Haga, Y Hayato, M Ikeda, K Iyogi, J Kameda, Y Kishimoto, M Miura, S Moriyama, M Nakahata, et al. Search for proton decay via p→ e+ 𝜋 0 and p→ 𝜇+ 𝜋 0 in 0.31 megaton· years exposure of the super-kamiokande water cherenkov detector. Physical Review D, 95(1): 012004, 2017. [8] Robert E. Lopez and Michael S. Turner. Precision prediction for the big-bang abundance of primordial 4 He. Phys. Rev. D, 59:103502, Mar 1999. doi: 10.1103/PhysRevD.59.103502. URL https://link.aps.org/doi/10.1103/PhysRevD.59.103502. [9] Particle Data Group, RL Workman, VD Burkert, V Crede, E Klempt, U Thoma, L Tiator, K Agashe, G Aielli, BC Allanach, et al. Review of particle physics. Progress of theoretical and experimental physics, 2022(8):083C01, 2022. [10] FM Gonzalez, EM Fries, C Cude-Woods, T Bailey, M Blatnik, LJ Broussard, NB Callahan, JH Choi, SM Clayton, SA Currie, et al. Improved neutron lifetime measurement with ucn 𝜏. Physical review letters, 127(16):162501, 2021. [11] A. T. Yue, M. S. Dewey, D. M. Gilliam, G. L. Greene, A. B. Laptev, J. S. Nico, W. M. Snow, and F. E. Wietfeldt. Improved determination of the neutron lifetime. Phys. Rev. Lett., 111: 222501, Nov 2013. doi: 10.1103/PhysRevLett.111.222501. URL https://link.aps.org/doi/10. 1103/PhysRevLett.111.222501. [12] D. Baye and E.M. Tursunov. 𝛽 delayed emission of a proton by a one-neutron halo nu- cleus. Physics Letters B, 696(5):464–467, 2011. ISSN 0370-2693. doi: https://doi. org/10.1016/j.physletb.2010.12.069. URL https://www.sciencedirect.com/science/article/pii/ S0370269311000281. 138 [13] M Pfützner and K Riisager. Examining the possibility to observe neutron dark decay in nuclei. Physical Review C, 97(4):042501, 2018. [14] Gordon Baym, D. H. Beck, Peter Geltenbort, and Jessie Shelton. Testing dark decays of baryons in neutron stars. Phys. Rev. Lett., 121:061801, Aug 2018. doi: 10.1103/PhysRevLett. 121.061801. URL https://link.aps.org/doi/10.1103/PhysRevLett.121.061801. [15] David McKeen, Ann E. Nelson, Sanjay Reddy, and Dake Zhou. Neutron stars exclude light dark baryons. Phys. Rev. Lett., 121:061802, Aug 2018. doi: 10.1103/PhysRevLett.121. 061802. URL https://link.aps.org/doi/10.1103/PhysRevLett.121.061802. [16] Benjamı́n Grinstein, Chris Kouvaris, and Niklas Grønlund Nielsen. Neutron star stability in light of the neutron decay anomaly. Phys. Rev. Lett., 123:091601, Aug 2019. doi: 10.1103/ PhysRevLett.123.091601. URL https://link.aps.org/doi/10.1103/PhysRevLett.123.091601. [17] Jeffrey M Berryman, Susan Gardner, and Mohammadreza Zakeri. Neutron stars with baryon number violation, probing dark sectors. Symmetry, 14(3):518, 2022. [18] JC Hardy and IS Towner. Superallowed 0+→ 0+ nuclear 𝛽 decays: 2020 critical survey, with implications for v ud and ckm unitarity. Physical Review C, 102(4):045501, 2020. [19] D Dubbers, H Saul, B Märkisch, T Soldner, and H Abele. Exotic decay channels are not the cause of the neutron lifetime anomaly. Physics Letters B, 791:6–10, 2019. [20] Dirk Dubbers and Bastian Märkisch. Precise measurements of the decay of free neutrons. Annual Review of Nuclear and Particle Science, 71:139–163, 2021. [21] M. Beck, F. Ayala Guardia, M. Borg, J. Kahlenberg, R. Muñoz Horta, C. Schmidt, A. Wunderle, W. Heil, R. Maisonobe, M. Simson, T. Soldner, R. Virot, O. Zimmer, M. Klopf, G. Konrad, S. Baeßler, F. Glück, and U. Schmidt. Improved determination of the 𝛽−𝜈 𝑒 angular correlation coefficient 𝑎 in free neutron decay with the 𝑎SPECT spectrom- eter. Phys. Rev. C, 101:055506, May 2020. doi: 10.1103/PhysRevC.101.055506. URL https://link.aps.org/doi/10.1103/PhysRevC.101.055506. [22] K Riisager, MJG Borge, José Antonio Briz, M Carmona-Gallardo, O Forstner, LM Fraile, HOU Fynbo, A Garzon Camacho, JG Johansen, B Jonson, et al. Search for beta-delayed proton emission from 11 be. The European Physical Journal A, 56(3):1–8, 2020. [23] A. Volya. Assessment of the beta-delayed proton decay rate of 11be. Europhysics Letters, 130 (1):12001, may 2020. doi: 10.1209/0295-5075/130/12001. URL https://dx.doi.org/10.1209/ 0295-5075/130/12001. [24] J. Oko lowicz, M. P loszajczak, and W. Nazarewicz. Convenient location of a near-threshold proton-emitting resonance in 11 B. Phys. Rev. Lett., 124:042502, Jan 2020. doi: 10.1103/ PhysRevLett.124.042502. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.042502. [25] J. Oko lowicz, M. P loszajczak, and I. Rotter. Dynamics of quantum systems embedded in a continuum. Physics Reports, 374(4):271–383, 2003. ISSN 0370-1573. doi: https://doi. org/10.1016/S0370-1573(02)00366-6. URL https://www.sciencedirect.com/science/article/ pii/S0370157302003666. 139 [26] Wael Elkamhawy, Zichao Yang, Hans-Werner Hammer, and Lucas Platter. 𝛽-delayed proton emission from 11be in effective field theory. Physics Letters B, 821:136610, 2021. ISSN 0370- 2693. doi: https://doi.org/10.1016/j.physletb.2021.136610. URL https://www.sciencedirect. com/science/article/pii/S0370269321005505. [27] J. Refsgaard, J. Büscher, A. Arokiaraj, H. O. U. Fynbo, R. Raabe, and K. Riisager. Clarification of large-strength transitions in the 𝛽 decay of 11 Be. Phys. Rev. C, 99:044316, Apr 2019. doi: 10.1103/PhysRevC.99.044316. URL https://link.aps.org/doi/10.1103/PhysRevC.99.044316. [28] M. C. Atkinson, P. Navrátil, G. Hupin, K. Kravvaris, and S. Quaglioni. Ab initio calculation of the 𝛽 decay from 11 Be to a 10 Be + 𝑝 resonance. Phys. Rev. C, 105:054316, May 2022. doi: 10.1103/PhysRevC.105.054316. URL https://link.aps.org/doi/10.1103/PhysRevC.105. 054316. [29] Nguyen Le Anh, Bui Minh Loc, Naftali Auerbach, and Vladimir Zelevinsky. Single-particle properties of the near-threshold proton-emitting resonance in 11 B. Phys. Rev. C, 106:L051302, Nov 2022. doi: 10.1103/PhysRevC.106.L051302. URL https://link.aps.org/doi/10.1103/ PhysRevC.106.L051302. [30] Y. Ayyad, W. Mittig, T. Tang, B. Olaizola, G. Potel, N. Rijal, N. Watwood, H. Alvarez-Pol, D. Bazin, M. Caamaño, J. Chen, M. Cortesi, B. Fernández-Domı́nguez, S. Giraud, P. Gueye, S. Heinitz, R. Jain, B. P. Kay, E. A. Maugeri, B. Monteagudo, F. Ndayisabye, S. N. Paneru, J. Pereira, E. Rubino, C. Santamaria, D. Schumann, J. Surbrook, L. Wagner, J. C. Zamora, and V. Zelevinsky. Evidence of a near-threshold resonance in 11 B relevant to the 𝛽-delayed proton emission of 11 Be. Phys. Rev. Lett., 129:012501, Jun 2022. doi: 10.1103/PhysRevLett. 129.012501. URL https://link.aps.org/doi/10.1103/PhysRevLett.129.012501. [31] RE Azuma, E Uberseder, EC Simpson, CR Brune, H Costantini, RJ De Boer, J Görres, M Heil, PJ LeBlanc, C Ugalde, et al. Azure: An r-matrix code for nuclear astrophysics. Physical Review C, 81(4):045805, 2010. [32] E. Lopez-Saavedra, S. Almaraz-Calderon, B. W. Asher, L. T. Baby, N. Gerken, K. Hanselman, K. W. Kemper, A. N. Kuchera, A. B. Morelock, J. F. Perello, E. S. Temanson, A. Volya, and I. Wiedenhöver. Observation of a near-threshold proton resonance in 11 B. Phys. Rev. Lett., 129:012502, Jun 2022. doi: 10.1103/PhysRevLett.129.012502. URL https://link.aps.org/doi/ 10.1103/PhysRevLett.129.012502. [33] M. Friedman, D. Pérez-Loureiro, T. Budner, E. Pollacco, C. Wrede, M. Cortesi, C. Fry, B. Glassman, M. Harris, J. Heideman, M. Janasik, B.T. Roeder, M. Roosa, A. Saas- tamoinen, J. Stomps, J. Surbrook, P. Tiwari, and J. Yurkon. Gadget: a gaseous de- tector with germanium tagging. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 940:93– 102, 2019. ISSN 0168-9002. doi: https://doi.org/10.1016/j.nima.2019.05.100. URL https://www.sciencedirect.com/science/article/pii/S0168900219308162. [34] M. Friedman, T. Budner, D. Pérez-Loureiro, E. Pollacco, C. Wrede, J. José, B. A. Brown, M. Cortesi, C. Fry, B. Glassman, J. Heideman, M. Janasik, M. Roosa, J. Stomps, J. Surbrook, 140 and P. Tiwari. Low-energy 23 Al 𝛽-delayed proton decay and 22 Na destruction in novae. Phys. Rev. C, 101:052802, May 2020. doi: 10.1103/PhysRevC.101.052802. URL https: //link.aps.org/doi/10.1103/PhysRevC.101.052802. [35] L. J. Sun, M. Friedman, T. Budner, D. Pérez-Loureiro, E. Pollacco, C. Wrede, B. A. Brown, M. Cortesi, C. Fry, B. E. Glassman, J. Heideman, M. Janasik, A. Kruskie, A. Magilligan, M. Roosa, J. Stomps, J. Surbrook, and P. Tiwari. 25 Si 𝛽+ -decay spectroscopy. Phys. Rev. C, 103:014322, Jan 2021. doi: 10.1103/PhysRevC.103.014322. URL https://link.aps.org/doi/ 10.1103/PhysRevC.103.014322. [36] T. Budner, M. Friedman, C. Wrede, B. A. Brown, J. José, D. Pérez-Loureiro, L. J. Sun, J. Surbrook, Y. Ayyad, D. W. Bardayan, K. Chae, A. A. Chen, K. A. Chipps, M. Cortesi, B. Glassman, M. R. Hall, M. Janasik, J. Liang, P. O’Malley, E. Pollacco, A. Psaltis, J. Stomps, and T. Wheeler. Constraining the 30 P( 𝑝, 𝛾) 31 S reaction rate in one novae via the weak, low-energy, 𝛽-delayed proton decay of 31 Cl. Phys. Rev. Lett., 128:182701, May 2022. doi: 10.1103/PhysRevLett.128.182701. URL https://link.aps.org/doi/10.1103/PhysRevLett.128. 182701. [37] W.F. Mueller, J.A. Church, T. Glasmacher, D. Gutknecht, G. Hackman, P.G. Hansen, Z. Hu, K.L. Miller, and P. Quirin. Thirty-two-fold segmented germanium detectors to identify 𝛾- rays from intermediate-energy exotic beams. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 466 (3):492–498, 2001. ISSN 0168-9002. doi: https://doi.org/10.1016/S0168-9002(01)00257-1. URL https://www.sciencedirect.com/science/article/pii/S0168900201002571. [38] Glenn F Knoll. Radiation detection and measurement. John Wiley & Sons, 2010. [39] J.H. Kelley, E. Kwan, J.E. Purcell, C.G. Sheu, and H.R. Weller. Energy levels of light nuclei a=11. Nuclear Physics A, 880:88–195, 2012. ISSN 0375-9474. doi: https://doi. org/10.1016/j.nuclphysa.2012.01.010. URL https://www.sciencedirect.com/science/article/ pii/S0375947412000413. [40] Kenneth S Krane. Introductory nuclear physics. John Wiley & Sons, 1991. [41] A. Stolz, T. Baumann, T.N. Ginter, D.J. Morrissey, M. Portillo, B.M. Sherrill, M. Steiner, and J.W. Stetson. Production of rare isotope beams with the nscl fragment separator. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 241(1):858–861, 2005. ISSN 0168-583X. doi: https://doi.org/10.1016/j.nimb. 2005.07.168. URL https://www.sciencedirect.com/science/article/pii/S0168583X0501339X. The Application of Accelerators in Research and Industry. [42] LeCroy Corporation. 222 nim dual gate and delay generator, 0000. URL https://teledynelecroy. com/lrs/dsheets/222.htm. [43] K. Starosta, C. Vaman, D. Miller, P. Voss, D. Bazin, T. Glasmacher, H. Crawford, P. Mantica, H. Tan, W. Hennig, M. Walby, A. Fallu-Labruyere, J. Harris, D. Breus, P. Grudberg, and W.K. Warburton. Digital data acquisition system for experiments with segmented detectors at national superconducting cyclotron laboratory. Nuclear Instruments and Methods in Physics 141 Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 610 (3):700–709, 2009. ISSN 0168-9002. doi: https://doi.org/10.1016/j.nima.2009.09.016. URL https://www.sciencedirect.com/science/article/pii/S0168900209017392. [44] C.J. Prokop, S.N. Liddick, B.L. Abromeit, A.T. Chemey, N.R. Larson, S. Suchyta, and J.R. Tompkins. Digital data acquisition system implementation at the national super- conducting cyclotron laboratory. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 741:163– 168, 2014. ISSN 0168-9002. doi: https://doi.org/10.1016/j.nima.2013.12.044. URL https://www.sciencedirect.com/science/article/pii/S0168900213017488. [45] Inc Advanced Measurement Technology. Model 566 time-to-amplitude converter (tac) op- erating and service manual, 0000. URL https://www.ortec-online.com/-/media/ametekortec/ manuals/5/566-mnl.pdf. [46] D. J. Millener, D. E. Alburger, E. K. Warburton, and D. H. Wilkinson. Decay scheme of 11 Be. Phys. Rev. C, 26:1167–1185, Sep 1982. doi: 10.1103/PhysRevC.26.1167. URL https://link.aps.org/doi/10.1103/PhysRevC.26.1167. [47] D. E. Alburger, D. J. Millener, and D. H. Wilkinson. Delayed particles from the beta decay of 11 Be. Phys. Rev. C, 23:473–479, Jan 1981. doi: 10.1103/PhysRevC.23.473. URL https://link.aps.org/doi/10.1103/PhysRevC.23.473. [48] James F Ziegler, Matthias D Ziegler, and Jochen P Biersack. Srim–the stopping and range of ions in matter (2010). Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 268(11-12):1818–1823, 2010. [49] OB Tarasov and D Bazin. Lise++: Radioactive beam production with in-flight separators. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 266(19-20):4657–4664, 2008. [50] Christian Iliadis. Nuclear physics of stars. John Wiley & Sons, 2015. [51] Tamas Aleksei Budner. 31cl Beta-Delayed Proton Decay and Classical Nova Nucleosynthesis. PhD thesis, Michigan State University, 2022. [52] S.F. Biagi. A multiterm boltzmann analysis of drift velocity, diffusion, gain and magnetic-field effects in argon-methane-water-vapour mixtures. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 283 (3):716–722, 1989. ISSN 0168-9002. doi: https://doi.org/10.1016/0168-9002(89)91446-0. URL https://www.sciencedirect.com/science/article/pii/0168900289914460. [53] S.F. Biagi. Monte carlo simulation of electron drift and diffusion in counting gases under the influence of electric and magnetic fields. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 421 (1):234–240, 1999. ISSN 0168-9002. doi: https://doi.org/10.1016/S0168-9002(98)01233-9. URL https://www.sciencedirect.com/science/article/pii/S0168900298012339. 142 [54] Pierre Descouvemont and D Baye. The r-matrix theory. Reports on progress in physics, 73 (3):036301, 2010. [55] PL Kapur and Ro Peierls. The dispersion formula for nuclear reactions. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 166(925):277–295, 1938. [56] Carl R Brune. Alternative parametrization of r-matrix theory. Physical Review C, 66(4): 044611, 2002. [57] AM Lane and RG Thomas. R-matrix theory of nuclear reactions. Reviews of Modern Physics, 30(2):257, 1958. [58] FC Barker and EK Warburton. The beta-decay of 8he. Nuclear Physics A, 487(2):269–278, 1988. [59] James F Ziegler, Jochen P Biersack, and Matthias D Ziegler. Srim, 2003. [60] M Eugene Rudd, RD DuBois, LH Toburen, CA Ratcliffe, and TV Goffe. Cross sections for ionization of gases by 5-4000-kev protons and for electron capture by 5-150-kev protons. Physical Review A, 28(6):3244, 1983. [61] Claudia Carmen Montanari and Jorge Esteban Miraglia. Multiple ionization of argon by helium ions. Journal of Physics B: Atomic, Molecular and Optical Physics, 49(17):175203, 2016. [62] ALC Losqui, F Zappa, GM Sigaud, W Wolff, MM Sant’Anna, ACF Santos, H Luna, and WS Melo. Absolute cross sections for electron loss, electron capture, and multiple ionization in collisions of li2+ with argon. Journal of Physics B: Atomic, Molecular and Optical Physics, 47(4):045202, 2014. [63] A ICRU. Average energy required to produce an ion pair. ICRU Report 31, 1979. [64] Jens Lindhard, Morten Scharff, and Hans E Schiøtt. Range concepts and heavy ion ranges, volume 33. Munksgaard Copenhagen, 1963. [65] M. Peck, A. Göök, J. Enders, F.-J. Hambsch, and S. Oberstedt. Pulse-height defect of ar+cf4 mixtures as a counting gas for fission-fragment detectors. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equip- ment, 919:105–112, 2019. ISSN 0168-9002. doi: https://doi.org/10.1016/j.nima.2018.12.060. URL https://www.sciencedirect.com/science/article/pii/S0168900218318576. [66] Ralph A Lowry and Glenn H Miller. Ionization yield of protons in nitrogen and argon. Physical Review, 109(3):826, 1958. [67] JA Dennis. Ionization by low energy heavy ions in nitrogen, carbon dioxide, argon and hydrocarbon gases. radiation effects, 8(1-2):87–98, 1971. [68] HV Larson. Energy loss per ion pair for protons in various gases. Physical Review, 112(6): 1927, 1958. 143 [69] Van Dat Nguyen and Maurice Chemtob. Calculated and measured w values in n 2, ar, ch 4 and co 2 gases for ions h+, he+, c+, n+, o+ and ar+ in the energy range 25 kev to 375 kev. Technical report, CEA Centre d’Etudes Nucleaires de Fontenay-aux-Roses, 1979. [70] JA Phipps, JW Boring, and RA Lowry. Total ionization in argon by heavy ions of energies 8 to 100 kev. Physical Review, 135(1A):A36, 1964. [71] SE Chappell and JH Sparrow. The average energy required to produce an ion pair in argon, nitrogen, and air for 1-to 5-mev alpha particles. Radiation research, 32(3):383–403, 1967. [72] M Chemtob, N Parmentier, and VD Nguyen. Some experimental results on w-values for heavy particles. Physics in Medicine & Biology, 23(6):1197, 1978. [73] R LEIMGRUBER, P HUBER, and E BAUMGARTNER. Messung der arbeit pro ionenpaar in verschiedenen gasen für stickstoff- und sauerstoffionen im energiegebiet von 0,14 bis 0,7 mev. HELVETICA PHYSICA ACTA, 38(5):499–513, 1965. [74] William P Jesse. Alpha-particle ionization in argon-methane mixtures and the energy depen- dence of the ion pair-formation energy. Physical Review, 174(1):173, 1968. [75] MN Varma and JW Baum. Energy dependence of w for alpha particles in n2, co2, ch4, ar, h2 and rossi-type tissue-equivalent gases. Physics in Medicine & Biology, 23(6):1162, 1978. [76] H Tawara, N Ishida, J Kikuchi, and T Doke. Measurements of the w values in argon, nitrogen, and methane for 0.93 to 5.3 mev alpha particles. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 29(3):447–455, 1987. [77] VD Nguyen, M Chemtob, J Chary, F Posny, and N Parmentier. Recent experimental results on w-values for heavy particles. Physics in Medicine & Biology, 25(3):509, 1980. [78] JR Macdonald and G Sidenius. The total ionization in methane of ions with 1¡= z¡= 22 at energies from 10 to 120 kev. Physics Letters A, 28(8):543–544, 1969. [79] JP Biersack and JF Ziegler. Refined universal potentials in atomic collisions. Nuclear Instruments and Methods in Physics Research, 194(1-3):93–100, 1982. [80] SG Lias. Ionization energy evaluation in nist chemistry webbook, nist standard reference database, number 69. Linstrom PJ, Mallard WG, Eds, page 20899, 2016. [81] A. Kramida, Yu. Ralchenko, J. Reader, and and NIST ASD Team. NIST Atomic Spectra Database (ver. 5.10), [Online]. Available: https://physics.nist.gov/asd [2022, December 12]. National Institute of Standards and Technology, Gaithersburg, MD., 2022. [82] W Haeberli, P Huber, and E Baldinger. Arbeit pro ionenpaar in gasgemischen fur alpha- teilchen (work per ion pair in gas mixtures for alpha-particle). HELVETICA PHYSICA ACTA, 23(5):481–484, 1950. [83] Willy Haeberli, P Huber, and E Baldinger. Arbeit pro ionenpaar von gasen und gasmis- chungen fur alpha-teilchen (work per ion pair of gases and gas mixtures for alpha particles). HELVETICA PHYSICA ACTA, 26(2):145–180, 1953. 144 [84] H Tawara, J Kikuchi, and T Doke. Energy dependences of w values for mev alpha particles in ar-xe, ar-n2, and ar-ch4 mixtures. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 29(3):456–460, 1987. [85] A. Gelman, W. R. Gilks, and G. O. Roberts. Weak convergence and optimal scaling of random walk Metropolis algorithms. The Annals of Applied Probability, 7(1):110 – 120, 1997. doi: 10.1214/aoap/1034625254. URL https://doi.org/10.1214/aoap/1034625254. [86] J Oko lowicz, M P loszajczak, and W Nazarewicz. 𝛽-p and 𝛽 − 𝛼 decay of the 11be neutron halo ground state. Journal of Physics G: Nuclear and Particle Physics, 49(10):10LT01, aug 2022. doi: 10.1088/1361-6471/ac8948. URL https://dx.doi.org/10.1088/1361-6471/ac8948. [87] Wael Elkamhawy, Zichao Yang, Hans-Werner Hammer, and Lucas Platter. Weak decay of halo nuclei. arXiv, December 2022. doi: 10.48550/arXiv.2212.05087. URL https: //arxiv.org/abs/2212.05087. 145 APPENDIX A SUPPLEMENTARY INFORMATION OF NSCL E18507 ANALYSIS Historical GADGET Individual-Pad Resolution This appendix section shows the historical GADGET narrow-peak detector resolution used to estimate the detector response in E18507. pad A resolution 12 25 Si 10 23 Al 31 8 Cl ∆E/E (%) 6 4 2 0 0 200 400 600 800 1000 1200 energy (keV) Figure A.1: Full width at half maximum energy resolution of GADGET Pad A in previous experiments [35, 34, 36]. Error bars not visible are smaller than the marker. Figure credit: Moshe Friedman. 146 pad C resolution 20 25 18 Si 23 16 Al 14 31 Cl ∆E/E (%) 12 10 8 6 4 2 0 0 200 400 600 800 1000 1200 energy (keV) Figure A.2: Full width at half maximum energy resolution of GADGET Pad C in previous experiments [35, 34, 36]. Error bars not visible are smaller than the marker. Figure credit: Moshe Friedman. 147 pad D resolution 12 25 Si 10 23 Al 31 8 Cl ∆E/E (%) 6 4 2 0 0 200 400 600 800 1000 1200 energy (keV) Figure A.3: Full width at half maximum energy resolution of GADGET Pad D in previous experiments [35, 34, 36]. Error bars not visible are smaller than the marker. Figure credit: Moshe Friedman. 148 Individual Pad Fits in E18507 This appendix section shows a sample of spectra and their statistical model fitting used to extract the number of 𝛽− 𝑝 -like events in different detector pads and assumed resonance energies. It is included to show the quality of the E18507 analysis applied to the E18507 dataset. The following section shows the (poor) quality of this analysis applied to the E19030 dataset. 2400 PadAHistCal Entries 1184993 2200 Mean 122.3 Std Dev 27.21 2000 χ2 / ndf 69.04 / 60 1800 Prob 0.1985 height 10.64 ± 5.76 Counts/1.5 keV 1600 a 530 ± 115.9 b −4.857 ± 1.387 1400 c 0.01378 ± 0.00412 eH 19.68 ± 0.33 1200 eM −0.1203 ± 0.0032 1000 800 600 400 200 0 100 120 140 160 180 200 Energy (keV) Figure A.4: Data of Pad A from E18507 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 160 keV. 149 PadCHistCal Entries 2442745 Mean 119.2 3500 Std Dev 24.39 χ2 / ndf 63.02 / 60 3000 Prob 0.3702 height 9.851 ± 6.609 Counts/1.5 keV a 150.2 ± 204.0 2500 b − 0.7792 ± 2.3555 c 0.003368 ± 0.006794 2000 eH 17.44 ± 0.22 eM − 0.09312 ± 0.00223 1500 1000 500 0 100 120 140 160 180 200 Energy (keV) Figure A.5: Data of Pad C from E18507 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 160 keV. 150 PadDHistCal Entries 2182386 3000 Mean 116.2 Std Dev 21.64 χ2 / ndf 51.84 / 60 2500 Prob 0.7642 height − 2.041 ± 5.109 Counts/1.5 keV a − 189.2 ± 103.9 2000 b 2.804 ± 1.184 c − 0.007594 ± 0.003393 eH 17.43 ± 0.17 1500 eM − 0.09391 ± 0.00167 1000 500 0 100 120 140 160 180 200 Energy (keV) Figure A.6: Data of Pad D from E18507 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 160 keV. 151 PadAHistCal 300 Entries 1184993 Mean 178.9 Std Dev 31.17 250 χ2 / ndf 71.79 / 60 Prob 0.1417 height 0.3716 ± 5.0927 Counts/1.5 keV 200 a 649.3 ± 74.5 b −6.005 ± 0.822 c 0.01657 ± 0.00222 eH 36.34 ± 0.37 150 eM −0.25 ± 0.16 100 50 0 140 160 180 200 220 Energy (keV) Figure A.7: Data of Pad A from E18507 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 190 keV. 152 PadCHistCal Entries 2442745 500 Mean 177.8 Std Dev 32.34 χ2 / ndf 56.03 / 60 Prob 0.6217 400 height −5.764 ± 4.456 Counts/1.5 keV a 971.6 ± 10.9 b −9.516 ± 0.101 c 0.0267 ± 0.0003 300 eH 32.39 ± 1.03 eM −0.2097 ± 0.0077 200 100 0 140 160 180 200 220 Energy (keV) Figure A.8: Data of Pad C from E18507 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 190 keV. 153 PadDHistCal 400 Entries 2182386 Mean 176.1 Std Dev 33.06 350 χ2 / ndf 69.59 / 60 Prob 0.1861 300 height −3.776 ± 3.950 Counts/1.5 keV a 698 ± 8.6 250 b −7.007 ± 0.079 c 0.01953 ± 0.00025 eH 27.17 ± 0.99 200 eM −0.1716 ± 0.0074 150 100 50 0 140 160 180 200 220 Energy (keV) Figure A.9: Data of Pad D from E18507 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 190 keV. 154 220 200 180 160 Counts/1.5 keV 140 120 100 PadAHistCal Entries 1184993 80 Mean 215.3 Std Dev 29.8 χ 2 / ndf 60.49 / 60 60 Prob 0.458 height − 7.473 ± 5.999 a − 2.545e+04 ± 1.973e+02 40 b 102.9 ± 12.9 c − 0.1239 ± 0.0272 eH 10.26 ± 0.05 20 eM − 0.005267 ± 0.000893 0 160 180 200 220 240 260 Energy (keV) Figure A.10: Data of Pad A from E18507 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 220 keV. 155 300 250 Counts/1.5 keV 200 150 PadCHistCal Entries 2442745 Mean 217.6 Std Dev 29.71 100 χ 2 / ndf 48.07 / 60 Prob 0.8662 height − 0.7824 ± 5.7905 a − 2.556e+04 ± 2.309e+02 b 101.7 ± 0.8 50 c − 0.1163 ± 0.0020 eH 10.29 ± 0.01 eM − 0.005383 ± 0.000081 0 160 180 200 220 240 260 Energy (keV) Figure A.11: Data of Pad C from E18507 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 220 keV. 156 180 160 140 Counts/1.5 keV 120 100 80 PadDHistCal Entries 2182386 Mean 218.2 Std Dev 29.49 60 χ 2 / ndf 82.05 / 60 Prob 0.0309 height 2.233 ± 4.684 40 a − 2.585e+04 ± 6.113e+01 b 95.92 ± 0.43 c − 0.1065 ± 0.0011 20 eH 10.26 ± 0.00 eM − 0.004764 ± 0.000021 0 160 180 200 220 240 260 Energy (keV) Figure A.12: Data of Pad D from E18507 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 220 keV. 157 200 180 160 Counts/1.5 keV 140 120 100 PadAHistCal Entries 1184993 80 Mean 244.6 Std Dev 28.63 χ 2 / ndf 64.65 / 60 60 Prob 0.3176 height 6.415 ± 6.311 a − 3.393e+04 ± 2.495e+02 40 b 125.9 ± 13.6 c − 0.141 ± 0.025 eH 10.53 ± 0.05 20 eM − 0.004764 ± 0.000733 0 200 220 240 260 280 Energy (keV) Figure A.13: Data of Pad A from E18507 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 250 keV. 158 300 250 Counts/1.5 keV 200 150 PadCHistCal Entries 2442745 Mean 246.9 100 Std Dev 27.82 χ 2 / ndf 41.19 / 60 Prob 0.9697 height 13.98 ± 10.29 a − 3.347e+04 ± 1.091e+03 b 136.7 ± 20.9 50 c − 0.1637 ± 0.0390 eH 10.55 ± 0.10 eM − 0.005441 ± 0.001079 0 200 220 240 260 280 Energy (keV) Figure A.14: Data of Pad C from E18507 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 250 keV. 159 160 140 120 Counts/1.5 keV 100 80 PadDHistCal Entries 2182386 Mean 246.9 60 Std Dev 27.57 χ 2 / ndf 75.94 / 60 Prob 0.0803 40 height 2.319 ± 5.391 a − 3.645e+04 ± 1.352e+02 b 134.7 ± 0.4 c − 0.152 ± 0.001 20 eH 10.59 ± 0.00 eM − 0.004637 ± 0.000032 0 200 220 240 260 280 Energy (keV) Figure A.15: Data of Pad C from E18507 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 250 keV. 160 APPENDIX B SUPPLEMENTARY INFORMATION OF NSCL E19030 ANALYSIS Limitations of E18507 Analysis This appendix section shows a sample of spectra and their statistical model fitting used to extract the number of 𝛽− 𝑝 -like events in different detector pads and assumed resonance energies from the E19030 dataset. The fitting model here was the same as E18507, and these samples are shown to show a range of behaviors, some undesirable, of the E18507 analysis applied to the E19030 dataset. In general, the fits to data are poor, judged by the 𝜒2 per degree of freedom and associated p-value (𝜒2 /ndf and 𝑃𝑟𝑜𝑏 in the plot information boxes, respectively) and/or have unphysical negative values of the 𝛽− 𝑝 counts. The range of detector pads and energies that satisfy these two requirements is quite small, suggesting that this choice of fitting model is poor and should be reconsidered. pad 1 - 160 keV 3 ×10 160 pad1Cal 3 × 10 Entries 160 4.41809e+07 140 Mean 108.9 140 120 Std Dev 16.4 Counts/1.0 keV 120 χ 2 / ndf 299.1 / 94 100 100 Counts/1.0 keV Prob 3.215e − 23 80 80 height 144.6 ± 20.5 60 a − 1.624e+04 ± 1.780e+02 60 b40 178.7 ± 1.4 40 c20 − 0.4609 ± 0.0027 19.96 ± 0.02 20 eH 0 100 120 140 160 180 200 eM − 0.08256 ± 0.00018 Energy (keV) 0 100 120 140 160 180 200 Energy (keV) Figure B.1: Data of Pad B from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 160 keV. 161 pad 2 - 160 keV 3 ×10 pad2Cal 140 × 10 3 Entries 4.017694e+07 140 120 Mean 108.7 120 Std Dev 16.35 Counts/1.0 keV 100 χ / ndf 1002 288.4 / 94 Counts/1.0 keV Prob 1.308e − 21 80 80 height 106 ± 17.6 60 60 a − 1.202e+04 ± 1.854e+02 40 b 133.7 ± 1.9 40 c20 − 0.3439 ± 0.0054 20 eH 20.1 ± 0.0 0 100 120 140 160 180 200 eM − 0.08507 ± 0.00016 Energy (keV) 0 100 120 140 160 180 200 Energy (keV) Figure B.2: Data of Pad C from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 160 keV. 162 pad 3 - 160 keV 3 ×10 160 pad3Cal 3 × 10 160 Entries 4.266081e+07 140 Mean 140 108.7 120 Std Dev 16.21 Counts/1.0 keV 120 χ 2 / ndf 308.5 / 94 100 100 Counts/1.0 keV Prob 1.181e − 24 80 80 height 160.3 ± 17.5 60 a − 1.62e+04 ± 4.77e+01 60 b40 176.6 ± 0.3 40 c20 − 0.4532 ± 0.0020 20 eH 19.98 ± 0.01 0 100 120 140 160 180 200 eM − 0.08295 ± 0.00013 Energy (keV) 0 100 120 140 160 180 200 Energy (keV) Figure B.3: Data of Pad D from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 160 keV. 163 pad 4 - 160 keV 3 ×10 pad4Cal 3 × 10 140 Entries 4.146628e+07 140 Mean 108.9 120 120 Std Dev 16.52 Counts/1.0 keV 100 χ2 100 / ndf 295.9 / 94 Counts/1.0 keV Prob 9.942e − 23 80 80 height 140.6 ± 18.4 60 60 a − 1.458e+04 ± 2.726e+02 40 b 160.2 ± 3.0 40 c20 − 0.4111 ± 0.0083 20 eH 0 19.93 ± 0.02 100 120 140 160 180 200 eM − 0.08308 ± 0.00016 Energy (keV) 0 100 120 140 160 180 200 Energy (keV) Figure B.4: Data of Pad E from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 160 keV. 164 pad 1 - 175 keV 50000 pad1Cal 50000 Entries 4.41809e+07 Mean 129 40000 40000 Std Dev 24.28 Counts/1.0 keV χ2 / ndf 115.4 / 94 30000 30000 Counts/1.0 keV Prob 0.0667 height 38.77 ± 15.62 20000 20000 a 2823 ± 231.4 b −24.22 ± 2.47 10000 c 0.07935 ± 0.00673 10000 eH 21 ± 0.0 0 120 140 160 180 200 220 eM − (keV) ± 0.00039 0.09226 Energy 0 120 140 160 180 200 220 Energy (keV) Figure B.5: Data of Pad B from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 175 keV. 165 pad 2 - 175 keV 45000 pad2Cal 45000 40000 Entries 4.017694e+07 40000 Mean 129.1 35000 35000 Std Dev 24.59 Counts/1.0 keV 30000 300002 χ / ndf 102.5 / 94 25000 Counts/1.0 keV 25000 Prob 0.2576 20000 height 25 ± 12.6 20000 15000 a 3628 ± 34.7 15000 b −33.73 ± 0.30 10000 10000 c 0.104 ± 0.001 5000 eH 21.14 ± 0.02 5000 0 120 140 160 180 200 220 eM − (keV) ± 0.00017 0.09474 Energy 0 120 140 160 180 200 220 Energy (keV) Figure B.6: Data of Pad C from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 175 keV. 166 pad 3 - 175 keV 50000 pad3Cal 50000 Entries 4.266081e+07 Mean 128.7 40000 40000 Std Dev 24.07 Counts/1.0 keV χ2 / ndf 107.8 / 94 30000 30000 Counts/1.0 keV Prob 0.1569 height 65.82 ± 15.66 20000 20000 a 4091 ± 172.3 b −38.92 ± 1.71 10000 c 0.1193 ± 0.0045 10000 eH 21.14 ± 0.05 0 120 140 160 180 200 220 eM −0.09371 Energy (keV) ± 0.00046 0 120 140 160 180 200 220 Energy (keV) Figure B.7: Data of Pad D from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 175 keV. 167 pad 4 - 175 keV pad4Cal 45000 Entries 4.146628e+07 45000 40000 Mean 129.2 40000 35000 Std Dev 24.58 Counts/1.0 keV 35000 30000 χ2 / ndf 102.9 / 94 30000 Counts/1.0 keV Prob 0.2485 25000 25000 height −4.954 ± 15.017 20000 20000 a 1305 ± 148.7 15000 15000 b −7.998 ± 1.573 10000 10000 c 0.03499 ± 0.00435 5000 eH 20.94 ± 0.04 5000 0 120 140 160 180 200 220 eM − (keV) ± 0.00037 0.09244 Energy 0 120 140 160 180 200 220 Energy (keV) Figure B.8: Data of Pad E from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 175 keV. 168 pad 1 - 190 keV 14000 pad1Cal 14000 Entries 4.41809e+07 12000 12000 Mean 158.9 Std Dev 34.06 10000 Counts/1.0 keV 10000 χ 2 / ndf 103 / 94 8000 Counts/1.0 keV 8000Prob 0.2467 height 2.17 ± 13.96 6000 6000 a 8411 ± 295.5 4000 b − 80.63 ± 2.96 4000 2000 c 0.2218 ± 0.0075 2000 eH 22.44 ± 0.19 140 160 180 200 220 eM − 0.1038 ± 0.0014 Energy (keV) 140 160 180 200 220 Energy (keV) Figure B.9: Data of Pad B from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 190 keV. 169 pad 2 - 190 keV 12000 pad2Cal 12000 Entries 4.017694e+07 10000 Mean 159.7 10000 Std Dev 34.41 Counts/1.0 keV 8000 8000 χ 2 / ndf 102.4 / 94 Counts/1.0 keV Prob 0.2598 6000 6000 height 6.068 ± 11.541 a 6413 ± 30.9 4000 4000 b − 61.6 ± 0.3 2000c 0.1737 ± 0.0008 2000 eH 21.96 ± 0.05 140 160 180 200 220 eM − 0.1014 ± 0.0004 Energy (keV) 140 160 180 200 220 Energy (keV) Figure B.10: Data of Pad C from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 190 keV. 170 pad 3 - 190 keV pad3Cal Entries 4.266081e+07 12000 12000 Mean 158.5 10000 Std Dev 33.9 Counts/1.0 keV 10000 χ 2 / ndf 98.95 / 94 8000 Counts/1.0 keV 8000 Prob 0.3433 height 9.412 ± 11.211 6000 6000 a 7572 ± 31.4 4000 b − 71.99 ± 0.27 4000 2000 c 0.1977 ± 0.0008 2000 eH 22.53 ± 0.05 140 160 180 200 220 eM − 0.1048 ± 0.0004 Energy (keV) 140 160 180 200 220 Energy (keV) Figure B.11: Data of Pad D from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 190 keV. 171 pad 4 - 190 keV pad4Cal 12000 Entries 4.146628e+07 12000 Mean 159.4 10000 10000 Std Dev 34.19 Counts/1.0 keV χ 2 / ndf 89.8 / 94 8000 8000 Counts/1.0 keV Prob 0.6035 6000height 38.16 ± 12.67 6000 a 7243 ± 31.8 4000 4000 b − 69.01 ± 0.27 2000c 0.1912 ± 0.0009 2000 eH 22.34 ± 0.05 140 160 180 200 220 eM − 0.1038 ± 0.0004 Energy (keV) 140 160 180 200 220 Energy (keV) Figure B.12: Data of Pad E from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 190 keV. 172 pad 1 - 220 keV 2600 pad1Cal 2600 Entries 4.41809e+07 2400 Mean 2400 216.8 2200 Std Dev 33.35 Counts/1.0 keV 2200 2000 χ 2 / ndf 114.5 / 94 2000 Counts/1.0 keV Prob 0.07364 1800 1800 height − 45.74 ± 15.53 1600 1600 a − 1.87e+04 ± 6.38e+03 1400 b 101.1 ± 35.5 1400 1200 c − 0.09666 ± 0.05387 1200 1000 eH 11.23 ± 0.10 160 180 200 220 240 260 1000 eM − 0.01533 ± 0.00233 Energy (keV) 160 180 200 220 240 260 Energy (keV) Figure B.13: Data of Pad B from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 220 keV. 173 pad 2 - 220 keV pad2Cal 2200 Entries 4.017694e+07 2200 Mean 217.3 2000 Std 2000 Dev 33.18 Counts/1.0 keV χ 2 / ndf 1800 1800 93.29 / 94 Counts/1.0 keV Prob 0.5014 1600 1600 height − 43.49 ± 13.66 1400 a − 2.636e+04 ± 6.196e+02 1400 1200 b 132.7 ± 3.0 1200 c − 0.135 ± 0.007 1000 eH 10.99 ± 0.03 1000 160 180 200 220 240 260 eM − 0.01125 ± 0.00040 Energy (keV) 160 180 200 220 240 260 Energy (keV) Figure B.14: Data of Pad C from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 220 keV. 174 pad 3 - 220 keV 2400 pad3Cal Entries 2400 4.266081e+07 2200 Mean 216.8 2200 2000 Std Dev 33.41 Counts/1.0 keV 2000 χ 2 / ndf 108 / 94 1800 1800 Counts/1.0 keV Prob 0.1538 height 1600 − 37.37 ± 13.95 1600 a − 3.799e+04 ± 5.574e+02 1400 1400 b 161.1 ± 2.6 1200 c − 0.1332 ± 0.0065 1200 1000 eH 11.07 ± 0.01 1000 160 180 200 220 240 260 eM −Energy 0.008287 (keV) ± 0.000188 160 180 200 220 240 260 Energy (keV) Figure B.15: Data of Pad D from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 220 keV. 175 pad 4 - 220 keV 2400 pad4Cal 2400 Entries 4.146628e+07 2200 2200 Mean 216.8 2000 Std Dev 33.28 Counts/1.0 keV 2000 χ 2 / ndf 114 / 94 1800 1800 Counts/1.0 keV Prob 0.07886 − 43.76 ± 15.21 1600 height 1600 a − 3.179e+04 ± 3.385e+03 1400 1400 b 152.9 ± 21.1 1200 c − 0.1508 ± 0.0377 1200 1000 eH 11.07 ± 0.09 160 180 200 220 240 260 1000 eM − 0.01019 ± 0.00059 Energy (keV) 160 180 200 220 240 260 Energy (keV) Figure B.16: Data of Pad E from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 220 keV. 176 pad 1 - 235 keV pad1Cal 2600 Entries 4.41809e+07 2600 2400 Mean 234.6 2400 Std Dev 31.3 2200 Counts/1.0 keV 2200 χ 2 / ndf 162.3 / 94 2000 2000 Counts/1.0 keV Prob 1.558e − 05 1800 1800 height − 75.68 ± 18.23 1600 − 1.689e+05 ± 6.645e+03 1600 a 1400 b 868.3 ± 39.0 1400 1200 c − 1.198 ± 0.062 1200 1000 eH 12.41 ± 0.04 180 200 220 240 260 280 1000 eM −Energy 0.008558 (keV) ± 0.000130 180 200 220 240 260 280 Energy (keV) Figure B.17: Data of Pad B from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 235 keV. 177 pad 2 - 235 keV 2400 pad2Cal 2400 Entries 4.017694e+07 2200 Mean 2200 234.6 2000 Std Dev 31.15 Counts/1.0 keV 2000 χ 2 / ndf 155.3 / 94 1800 1800 Counts/1.0 keV Prob 7.168e − 05 1600 1600 height − 82.38 ± 14.39 1400 a − 2.177e+05 ± 4.387e+03 1400 1200 b 1049 ± 17.3 1200 c − 1.396 ± 0.021 1000 eH 12.56 ± 0.02 1000 180 200 220 240 260 280 eM −Energy 0.007427 (keV) ± 0.000075 180 200 220 240 260 280 Energy (keV) Figure B.18: Data of Pad C from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 235 keV. 178 pad 3 - 235 keV 2600 pad3Cal 2600 2400 Entries 4.266081e+07 2400 Mean 234.5 2200 Std 2200 Dev 31.4 Counts/1.0 keV 2000 χ2 2000 / ndf 148.3 / 94 Counts/1.0 keV Prob 0.0003005 1800 1800 height − 186.4 ± 14.0 1600 1600 a − 2.166e+05 ± 2.707e+02 1400 1400 b 1121 ± 1.2 1200 c − 1.576 ± 0.003 1200 1000 eH 12.64 ± 0.00 1000 eM 180 200 220 −Energy 240 0.008479 260 280 ± 0.000023 (keV) 180 200 220 240 260 280 Energy (keV) Figure B.19: Data of Pad D from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 235 keV. 179 pad 4 - 235 keV pad4Cal 2400 Entries 4.146628e+07 2400 2200 Mean 234.4 2200 Std Dev 31.31 Counts/1.0 keV 2000 2000 2 χ / ndf 184.3 / 94 1800 Counts/1.0 keV Prob 1800 7.893e − 08 height − 61.89 ± 18.43 1600 1600 a − 2.387e+05 ± 2.484e+03 1400 1400 b 1054 ± 34.0 1200 c − 1.31 ± 0.07 1200 1000 eH 12.59 ± 0.02 180 200 220 240 260 280 1000 eM − 0.006413 ± 0.000224 Energy (keV) 180 200 220 240 260 280 Energy (keV) Figure B.20: Data of Pad E from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 235 keV. 180 pad 1 - 250 keV pad1Cal 2600 Entries 4.41809e+07 2600 2400 Mean 249 2400 Std Dev 30.08 2200 Counts/1.0 keV 2200 χ 2 / ndf 136.1 / 94 2000 2000 Counts/1.0 keV Prob 0.002996 1800 1800 height 38.09 ± 19.04 1600 − 4.361e+05 ± 6.103e+03 1600 a 1400 b 1925 ± 35.2 1400 1200 c − 2.42 ± 0.06 1200 1000 eH 13.19 ± 0.02 200 220 240 260 280 1000 eM −Energy 0.006399 (keV) ± 0.000116 200 220 240 260 280 Energy (keV) Figure B.21: Data of Pad B from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 250 keV. 181 pad 2 - 250 keV 2400 pad2Cal 2400 Entries 4.017694e+07 2200 Mean 2200 248.9 2000 Std Dev 30.07 Counts/1.0 keV 2000 χ 2 / ndf 126.1 / 94 1800 1800 Counts/1.0 keV Prob 0.01509 1600 1600 height 56.95 ± 14.51 1400 a − 4.465e+05 ± 2.003e+02 1400 1200 b 1842 ± 0.9 1200 c − 2.209 ± 0.002 1000 eH 13.17 ± 0.00 1000 200 220 240 260 280 eM −Energy 0.005697 (keV) ± 0.000006 200 220 240 260 280 Energy (keV) Figure B.22: Data of Pad C from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 250 keV. 182 pad 3 - 250 keV 2600 pad3Cal 2600 2400 Entries 4.266081e+07 2400 Mean 249 2200 Std 2200 Dev 30.18 Counts/1.0 keV 2000 χ2 2000 / ndf 173.1 / 94 Counts/1.0 keV Prob 1.282e − 06 1800 1800 height 74.14 ± 23.18 1600 1600 a − 4.497e+05 ± 1.672e+03 1400 1400 b 1869 ± 37.1 1200 c − 2.248 ± 0.072 1200 1000 eH 13.18 ± 0.01 1000 eM 200 220 240 −Energy 0.005792 260 280 ± 0.000153 (keV) 200 220 240 260 280 Energy (keV) Figure B.23: Data of Pad D from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 250 keV. 183 pad 4 - 250 keV pad4Cal 2400 Entries 4.146628e+07 2400 Mean 248.8 2200 2200 Std Dev 30.24 Counts/1.0 keV 2000 χ2 2000 / ndf 120.6 / 94 Counts/1.0 keV 1800 Prob 1800 0.03382 height 55.56 ± 15.67 1600 1600 a − 4.499e+05 ± 2.105e+02 1400 1400 b 1921 ± 0.9 1200 c − 2.357 ± 0.002 1200 1000 eH 13.2 ± 0.0 200 220 240 260 280 1000 eM − 0.006053 ± 0.000007 Energy (keV) 200 220 240 260 280 Energy (keV) Figure B.24: Data of Pad E from E19030 (blue) and model fit (red). The fit function is the sum of an exponential (exp[𝑒𝐻 + 𝑒𝑀 · 𝐸]), quadratic polynomial (𝑎 + 𝑏 · 𝐸 + 𝑐 · 𝐸 2 ), and Gaussian peak with integrated counts ℎ𝑒𝑖𝑔ℎ𝑡. The peak width is described in text and peak center is fixed to 250 keV. 184 Model 1: Exponential + Flat + Voigt This appendix section shows a sample of spectra and their statistical model fitting used to extract the number of 𝛽− 𝑝 -like events in different detector pads and assumed resonance energies from the E19030 dataset. The fitting model here was the sum of an exponential curve, a constant term, a Voigt profile, and a signal-like peak with height as a parameter. In general, the fits to data are reasonable, judged by the 𝜒2 per degree of freedom and associated p-value (𝜒2 /ndf and 𝑃𝑟𝑜𝑏 in the plot information boxes, respectively). However, the results have difficult to explain behaviors. See the discussion in the main text at Section 3.3.2.2. These figures are included here to supplement that text without affecting readability. 185 pad1Cal - pCounts: 1737(1290), LiGamma: 76(4) pad1Cal counts/1 keV Entries 4.41809e+07 6000 χ2/ndf: 129.7/118 Mean Std Dev 212.4 39.53 χ 2 / ndf 129.7 / 118 prob : 0.218 Prob exp1 0.218 20.26 ± 1.40 exp2 − 0.08707 ± 0.00950 4000 flat 717.2 ± 44.5 vHeight 2.092e+05 ± 1.722e+04 vCent 267.1 ± 1.2 vLG 75.76 ± 4.06 pCounts 1737 ± 1290.4 2000 0 150 200 250 200 counts/1 keV 100 0 −100 −200 150 200 250 Figure B.25: Data of Pad B from E19030, fit to data (red), and fit components (dashed black). The fit function is the sum of an exponential(exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸]), constant ( 𝑓 𝑙𝑎𝑡), Voigt to model the 7 Li∗ peak (with counts 𝑣𝐻𝑒𝑖𝑔ℎ𝑡, center 𝑣𝐶𝑒𝑛𝑡, and Γ width 𝑣𝐿𝐺), and a Voigt search peak with integrated counts 𝑝𝐶𝑜𝑢𝑛𝑡𝑠. The detector resolution is given in text and proton resonance width is fixed Γ = 0.0 keV. The search peak center is fixed to 175 keV. 186 pad1Cal - pCounts: 4516(1396), LiGamma: 79(4) pad1Cal counts/1 keV Entries 4.41809e+07 6000 χ2/ndf: 125.5/118 Mean Std Dev 212.4 39.53 χ 2 / ndf 125.5 / 118 prob : 0.302 Prob exp1 0.3016 20.77 ± 0.64 exp2 − 0.09054 ± 0.00434 4000 flat 665.8 ± 44.0 vHeight 2.259e+05 ± 1.859e+04 vCent 267.8 ± 1.3 vLG 79.35 ± 4.25 pCounts 4516 ± 1396.2 2000 0 150 200 250 200 counts/1 keV 100 0 −100 −200 150 200 250 Figure B.26: Data of Pad B from E19030, fit to data (red), and fit components (dashed black). The fit function is the sum of an exponential(exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸]), constant ( 𝑓 𝑙𝑎𝑡), Voigt to model the 7 Li∗ peak (with counts 𝑣𝐻𝑒𝑖𝑔ℎ𝑡, center 𝑣𝐶𝑒𝑛𝑡, and Γ width 𝑣𝐿𝐺), and a Voigt search peak with integrated counts 𝑝𝐶𝑜𝑢𝑛𝑡𝑠. The detector resolution is given in text and proton resonance width is fixed Γ = 12.0 keV. The search peak center is fixed to 175 keV. 187 pad1Cal - pCounts: 8038(2389), LiGamma: 83(5) pad1Cal counts/1 keV Entries 4.41809e+07 6000 χ2/ndf: 124.1/118 Mean Std Dev 212.4 39.53 χ 2 / ndf 124.1 / 118 prob : 0.333 Prob exp1 0.3326 20.84 ± 0.62 exp2 − 0.09091 ± 0.00421 4000 flat 602 ± 61.3 vHeight 2.442e+05 ± 2.437e+04 vCent 268.4 ± 1.5 vLG 82.84 ± 5.17 pCounts 8038 ± 2388.9 2000 0 150 200 250 200 counts/1 keV 100 0 −100 −200 150 200 250 Figure B.27: Data of Pad B from E19030, fit to data (red), and fit components (dashed black). The fit function is the sum of an exponential(exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸]), constant ( 𝑓 𝑙𝑎𝑡), Voigt to model the 7 Li∗ peak (with counts 𝑣𝐻𝑒𝑖𝑔ℎ𝑡, center 𝑣𝐶𝑒𝑛𝑡, and Γ width 𝑣𝐿𝐺), and a Voigt search peak with integrated counts 𝑝𝐶𝑜𝑢𝑛𝑡𝑠. The detector resolution is given in text and proton resonance width is fixed Γ = 24.0 keV. The search peak center is fixed to 175 keV. 188 pad2Cal - pCounts: 1817(863), LiGamma: 72(3) pad2Cal counts/1 keV Entries 4.017694e+07 6000 χ2/ndf: 109.4/118 Mean Std Dev 213 39.32 χ 2 / ndf 109.4 / 118 prob : 0.702 Prob exp1 0.7016 21.82 ± 0.88 exp2 − 0.0989 ± 0.0060 4000 flat 686.4 ± 26.7 vHeight 1.725e+05 ± 1.116e+04 vCent 265.9 ± 1.1 vLG 71.71 ± 3.23 pCounts 1817 ± 862.8 2000 0 150 200 250 200 counts/1 keV 100 0 −100 −200 150 200 250 Figure B.28: Data of Pad C from E19030, fit to data (red), and fit components (dashed black). The fit function is the sum of an exponential(exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸]), constant ( 𝑓 𝑙𝑎𝑡), Voigt to model the 7 Li∗ peak (with counts 𝑣𝐻𝑒𝑖𝑔ℎ𝑡, center 𝑣𝐶𝑒𝑛𝑡, and Γ width 𝑣𝐿𝐺), and a Voigt search peak with integrated counts 𝑝𝐶𝑜𝑢𝑛𝑡𝑠. The detector resolution is given in text and proton resonance width is fixed Γ = 0.0 keV. The search peak center is fixed to 175 keV. 189 pad2Cal - pCounts: 3135(1523), LiGamma: 74(4) pad2Cal counts/1 keV Entries 4.017694e+07 6000 χ2/ndf: 109.9/118 Mean Std Dev 213 39.32 χ 2 / ndf 109.9 / 118 prob : 0.690 Prob exp1 0.6901 21.79 ± 0.85 exp2 − 0.09868 ± 0.00581 4000 flat 659.2 ± 35.1 vHeight 1.802e+05 ± 1.376e+04 vCent 266.2 ± 1.2 vLG 73.61 ± 3.80 pCounts 3135 ± 1522.9 2000 0 150 200 250 200 counts/1 keV 100 0 −100 −200 150 200 250 Figure B.29: Data of Pad C from E19030, fit to data (red), and fit components (dashed black). The fit function is the sum of an exponential(exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸]), constant ( 𝑓 𝑙𝑎𝑡), Voigt to model the 7 Li∗ peak (with counts 𝑣𝐻𝑒𝑖𝑔ℎ𝑡, center 𝑣𝐶𝑒𝑛𝑡, and Γ width 𝑣𝐿𝐺), and a Voigt search peak with integrated counts 𝑝𝐶𝑜𝑢𝑛𝑡𝑠. The detector resolution is given in text and proton resonance width is fixed Γ = 12.0 keV. The search peak center is fixed to 175 keV. 190 pad2Cal - pCounts: 4298(2622), LiGamma: 75(4) pad2Cal counts/1 keV Entries 4.017694e+07 6000 χ2/ndf: 110.6/118 Mean Std Dev 213 39.32 χ 2 / ndf 110.6 / 118 prob : 0.672 Prob exp1 0.6719 21.52 ± 0.92 exp2 − 0.09682 ± 0.00628 4000 flat 639 ± 47.1 vHeight 1.849e+05 ± 1.670e+04 vCent 266.4 ± 1.2 vLG 74.62 ± 4.38 pCounts 4298 ± 2621.7 2000 0 150 200 250 200 counts/1 keV 100 0 −100 −200 150 200 250 Figure B.30: Data of Pad C from E19030, fit to data (red), and fit components (dashed black). The fit function is the sum of an exponential(exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸]), constant ( 𝑓 𝑙𝑎𝑡), Voigt to model the 7 Li∗ peak (with counts 𝑣𝐻𝑒𝑖𝑔ℎ𝑡, center 𝑣𝐶𝑒𝑛𝑡, and Γ width 𝑣𝐿𝐺), and a Voigt search peak with integrated counts 𝑝𝐶𝑜𝑢𝑛𝑡𝑠. The detector resolution is given in text and proton resonance width is fixed Γ = 24.0 keV. The search peak center is fixed to 175 keV. 191 pad3Cal - pCounts: 1806(1097), LiGamma: 64(3) pad3Cal counts/1 keV Entries 4.266081e+07 6000 χ2/ndf: 119.8/118 Mean Std Dev 212.2 39.62 χ 2 / ndf 119.8 / 118 prob : 0.435 Prob exp1 0.4352 21.76 ± 1.06 exp2 − 0.09783 ± 0.00719 4000 flat 786.2 ± 22.2 vHeight 1.531e+05 ± 8.602e+03 vCent 265 ± 0.9 vLG 63.79 ± 2.61 pCounts 1806 ± 1096.8 2000 0 150 200 250 200 counts/1 keV 100 0 −100 −200 150 200 250 Figure B.31: Data of Pad D from E19030, fit to data (red), and fit components (dashed black). The fit function is the sum of an exponential(exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸]), constant ( 𝑓 𝑙𝑎𝑡), Voigt to model the 7 Li∗ peak (with counts 𝑣𝐻𝑒𝑖𝑔ℎ𝑡, center 𝑣𝐶𝑒𝑛𝑡, and Γ width 𝑣𝐿𝐺), and a Voigt search peak with integrated counts 𝑝𝐶𝑜𝑢𝑛𝑡𝑠. The detector resolution is given in text and proton resonance width is fixed Γ = 0.0 keV. The search peak center is fixed to 175 keV. 192 pad3Cal - pCounts: 2975(1665), LiGamma: 65(3) pad3Cal counts/1 keV Entries 4.266081e+07 6000 χ2/ndf: 120.3/118 Mean Std Dev 212.2 39.62 χ 2 / ndf 120.3 / 118 prob : 0.423 Prob exp1 0.4229 21.7 ± 0.9 exp2 − 0.09743 ± 0.00600 4000 flat 764.7 ± 28.4 vHeight 1.586e+05 ± 1.036e+04 vCent 265.2 ± 0.9 vLG 65.2 ± 3.0 pCounts 2975 ± 1664.7 2000 0 150 200 250 200 counts/1 keV 100 0 −100 −200 150 200 250 Figure B.32: Data of Pad D from E19030, fit to data (red), and fit components (dashed black). The fit function is the sum of an exponential(exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸]), constant ( 𝑓 𝑙𝑎𝑡), Voigt to model the 7 Li∗ peak (with counts 𝑣𝐻𝑒𝑖𝑔ℎ𝑡, center 𝑣𝐶𝑒𝑛𝑡, and Γ width 𝑣𝐿𝐺), and a Voigt search peak with integrated counts 𝑝𝐶𝑜𝑢𝑛𝑡𝑠. The detector resolution is given in text and proton resonance width is fixed Γ = 12.0 keV. The search peak center is fixed to 175 keV. 193 pad3Cal - pCounts: 3922(2816), LiGamma: 66(3) pad3Cal counts/1 keV Entries 4.266081e+07 6000 χ2/ndf: 120.9/118 Mean Std Dev 212.2 39.62 χ 2 / ndf 120.9 / 118 prob : 0.409 Prob exp1 0.4085 21.44 ± 0.97 exp2 − 0.09562 ± 0.00663 4000 flat 748.6 ± 38.0 vHeight 1.621e+05 ± 1.235e+04 vCent 265.3 ± 1.0 vLG 65.97 ± 3.41 pCounts 3922 ± 2815.8 2000 0 150 200 250 200 counts/1 keV 100 0 −100 −200 150 200 250 Figure B.33: Data of Pad D from E19030, fit to data (red), and fit components (dashed black). The fit function is the sum of an exponential(exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸]), constant ( 𝑓 𝑙𝑎𝑡), Voigt to model the 7 Li∗ peak (with counts 𝑣𝐻𝑒𝑖𝑔ℎ𝑡, center 𝑣𝐶𝑒𝑛𝑡, and Γ width 𝑣𝐿𝐺), and a Voigt search peak with integrated counts 𝑝𝐶𝑜𝑢𝑛𝑡𝑠. The detector resolution is given in text and proton resonance width is fixed Γ = 24.0 keV. The search peak center is fixed to 175 keV. 194 pad4Cal - pCounts: -275(1116), LiGamma: 67(3) pad4Cal counts/1 keV Entries 4.146628e+07 6000 χ2/ndf: 111.6/118 Mean Std Dev 212.5 39.37 χ 2 / ndf 111.6 / 118 prob : 0.648 Prob exp1 0.6475 19.79 ± 1.13 exp2 − 0.0848 ± 0.0077 4000 flat 771.4 ± 29.5 vHeight 1.61e+05 ± 1.05e+04 vCent 265.1 ± 0.9 vLG 67.1 ± 3.0 pCounts − 275 ± 1115.5 2000 0 150 200 250 200 counts/1 keV 100 0 −100 −200 150 200 250 Figure B.34: Data of Pad E from E19030, fit to data (red), and fit components (dashed black). The fit function is the sum of an exponential(exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸]), constant ( 𝑓 𝑙𝑎𝑡), Voigt to model the 7 Li∗ peak (with counts 𝑣𝐻𝑒𝑖𝑔ℎ𝑡, center 𝑣𝐶𝑒𝑛𝑡, and Γ width 𝑣𝐿𝐺), and a Voigt search peak with integrated counts 𝑝𝐶𝑜𝑢𝑛𝑡𝑠. The detector resolution is given in text and proton resonance width is fixed Γ = 0.0 keV. The search peak center is fixed to 175 keV. 195 pad4Cal - pCounts: 1090(1791), LiGamma: 68(3) pad4Cal counts/1 keV Entries 4.146628e+07 6000 χ2/ndf: 111.2/118 Mean Std Dev 212.5 39.37 χ 2 / ndf 111.2 / 118 prob : 0.658 Prob exp1 0.6579 20.48 ± 1.01 exp2 − 0.08952 ± 0.00685 4000 flat 760.8 ± 30.4 vHeight 1.645e+05 ± 1.123e+04 vCent 265.3 ± 1.0 vLG 68.1 ± 3.2 pCounts 1090 ± 1791.0 2000 0 150 200 250 200 counts/1 keV 100 0 −100 −200 150 200 250 Figure B.35: Data of Pad E from E19030, fit to data (red), and fit components (dashed black). The fit function is the sum of an exponential(exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸]), constant ( 𝑓 𝑙𝑎𝑡), Voigt to model the 7 Li∗ peak (with counts 𝑣𝐻𝑒𝑖𝑔ℎ𝑡, center 𝑣𝐶𝑒𝑛𝑡, and Γ width 𝑣𝐿𝐺), and a Voigt search peak with integrated counts 𝑝𝐶𝑜𝑢𝑛𝑡𝑠. The detector resolution is given in text and proton resonance width is fixed Γ = 12.0 keV. The search peak center is fixed to 175 keV. 196 pad4Cal - pCounts: 2884(2719), LiGamma: 70(4) pad4Cal counts/1 keV Entries 4.146628e+07 6000 χ2/ndf: 110.1/118 Mean Std Dev 212.5 39.37 χ 2 / ndf 110.1 / 118 prob : 0.684 Prob exp1 0.6844 20.77 ± 0.95 exp2 − 0.09146 ± 0.00648 4000 flat 736.5 ± 41.7 vHeight 1.709e+05 ± 1.390e+04 vCent 265.5 ± 1.1 vLG 69.67 ± 3.77 pCounts 2884 ± 2718.9 2000 0 150 200 250 200 counts/1 keV 100 0 −100 −200 150 200 250 Figure B.36: Data of Pad E from E19030, fit to data (red), and fit components (dashed black). The fit function is the sum of an exponential(exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸]), constant ( 𝑓 𝑙𝑎𝑡), Voigt to model the 7 Li∗ peak (with counts 𝑣𝐻𝑒𝑖𝑔ℎ𝑡, center 𝑣𝐶𝑒𝑛𝑡, and Γ width 𝑣𝐿𝐺), and a Voigt search peak with integrated counts 𝑝𝐶𝑜𝑢𝑛𝑡𝑠. The detector resolution is given in text and proton resonance width is fixed Γ = 24.0 keV. The search peak center is fixed to 175 keV. 197 Model 2: 2Expoential + Flat + 2 Voigt This appendix section shows a sample of spectra and their statistical model fitting used to extract the number of 𝛽− 𝑝 -like events in different detector pads and assumed resonance energies from the E19030 dataset. The fitting model here was the sum of an two exponential curves, a constant term, two Voigt profile, and a signal-like peak with height as a parameter. In general, the fits to data are reasonable, judged by the 𝜒2 per degree of freedom and associated p-value (𝜒2 /ndf and 𝑃𝑟𝑜𝑏 in the plot information boxes, respectively). However, the results have difficult to explain behaviors. See the discussion in the main text at Section 3.3.2.2 and Fig. B.39. These figures are included here to supplement that text without affecting readability. 7000 pad1Cal counts/1.00 keV Entries 4.41809e+07 Mean 243.8 6000 Std Dev 59.87 χ2 / ndf 211.1 / 200 Prob 0.282 5000 exp1 19.11 ± 0.18 exp2 − 0.1013 ± 0.0015 flat − 74.56 ± 27.44 4000 7hite 1.992e+05 ± 3.358e+03 7cent 238.6 ± 0.4 7sigm 0.05387 ± 0.00001 3000 72hite 2.751e+06 ± 1.525e+05 72sigm 41.88 ± 3.79 2000 exp21 9.574 ± 0.111 exp22 − 0.02369 ± 0.00081 1000 0 120 140 160 180 200 220 240 260 280 300 320 Energy (keV) 200 100 0 −100 100 150 200 250 300 Figure B.37: (Top) Here shows the 2Exponential + Flat + 2Voigt fit to Pad B from E19030 with no 𝛽− 𝑝 peak. The red line is the full fit and black dashed lines are individual components that are summed to create the red. The background is the sum of two exponentials (exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸] + exp[𝑒𝑥 𝑝21 + 𝑒𝑥 𝑝22 · 𝐸]), a constant term ( 𝑓 𝑙𝑎𝑡), a Voigt to model the 7 Li∗ peak (with integrated counts 7ℎ𝑖𝑡𝑒, center 7𝑐𝑒𝑛𝑡, detector resolution 7𝑠𝑖𝑔𝑚, and fixed intrinsic width from text), and a Voigt to model the peak at 𝐸 ∼ 440 keV (with fixed center at 440 keV and intrinsic width from text, integrated height 72ℎ𝑖𝑡𝑒, and detector resolution-like term 72𝑠𝑖𝑔𝑚). (Bottom) Fit residuals. The fit quality is reasonable. For this choice of fit range, the 𝜒2 per degree of freedom is 229/200 and the p-value is 0.177. Compare to Fig. B.38. The removal of the 𝛽− 𝑝 peak hardly influences the quality of the fit. 198 pad1Cal,lo=130,hi=350,pE=200,sig=5.0,gam=0.0 pad1Cal 7000 counts/1.00 keV Entries 4.41809e+07 Mean 242.6 Std Dev 71.89 6000 χ2 / ndf 227.5 / 209 Prob 0.1814 exp1 22.45 ± 0.19 5000 exp2 −0.1042 ± 0.0015 flat 128.7 ± 30.8 7hite 2.04e+05 ± 3.47e+03 4000 7cent 266.3 ± 0.4 7sigm 0.1024 ± 0.0054 3000 pCounts 656.3 ± 480.7 72hite 1.603e+06 ± 4.774e+04 72sigm 42.01 ± 2.36 2000 exp21 11.13 ± 0.15 exp22 −0.02992 ± 0.00101 1000 0 150 200 250 300 350 Maybe-Real (TM) Energy (keV) Graph 200 100 0 −100 150 200 250 300 350 Figure B.38: (Top) Here shows the 2Exponential + Flat + 2Voigt fit to Pad B from E19030 with the height of a 𝛽− 𝑝 peak as a parameter (𝑝𝐶𝑜𝑢𝑛𝑡𝑠). The red line is the full fit and black dashed lines are individual components that are summed to create the red. The signal’s assumed resonance energy is 𝐸 = 200 keV. The background is the sum of two exponentials (exp[𝑒𝑥 𝑝1 + 𝑒𝑥 𝑝2 · 𝐸] + exp[𝑒𝑥 𝑝21 + 𝑒𝑥 𝑝22 · 𝐸]), a constant term ( 𝑓 𝑙𝑎𝑡), a Voigt to model the 7 Li∗ peak (with integrated counts 7ℎ𝑖𝑡𝑒, center 7𝑐𝑒𝑛𝑡, detector resolution 7𝑠𝑖𝑔𝑚, and fixed intrinsic width from text), and a Voigt to model the peak at 𝐸 ∼ 440 keV (with fixed center at 440 keV and intrinsic width from text, integrated height 72ℎ𝑖𝑡𝑒, and detector resolution-like term 72𝑠𝑖𝑔𝑚). (Bottom) Fit residuals. The fit quality is reasonable. For this choice of fit range, the 𝜒2 per degree of freedom is 228/200 and the p-value is 0.181. Compare to Fig. B.37. The addition of the 𝛽− 𝑝 peak hardly influences the quality of the fit. 199 bp counts G=0 keV 15000 G=12 keV G=24 keV 10000 5000 0 − 5000 − 10000 160 170 180 190 200 210 220 230 240 250 Energy (keV) Figure B.39: Here shows inferred 𝛽− 𝑝 counts from scanning the assumed energy and proton emitting state’s width Γ for the 2Exponential + Flat + 2Voigt fit model for Pad B from E19030. Error bars are statistical. I speculate that the 𝜒2 surface has both bulk and small-scale features, especially for larger values of Γ. Note the non-statistical spread in data points that are 1 keV apart and sometimes wildly fluctuating error bars. (The detector resolution is 5-10% or 10-20 keV, so changes in results from changes in fit centers of 1 keV difference are artifacts of the fitting.) With the very similar quality of fits of Fig. B.37 and B.38 in mind, also note the behavior around 𝐸 ∼ 180 keV. Values of ∼ 1000 counts are extracted assuming Γ = 0 keV. Values of ∼ 1000 or ∼ 5000 for Γ = 12 keV. Is one measurement correct and the other a systematic shift from fitting? Is the difference 𝛿 ∼ 4000 an estimate of the systematic error? Nevertheless, the minimization terminating point is a reasonable value. The 𝜒2 minimization chooses a single value and it seems that not a single value, but a variety of values are all reasonable. 200 APPENDIX C SUPPLEMENTARY INFORMATION OF THE BAYESIAN ANALYSIS Visualizing Bayesian MCMC Fits to Data Some results of the Bayesian MCMC applied in the E19030 experiment are shown using single- parameter histograms and parameter-correlation, two-dimensional histograms in Chapter 5. This Appendix section will show a sample collection of fits to the E19030 histogramed data. The parameters for the functions shown in the following graphics are drawn from the MCMC chain. Each graphic shows the experimental data and the fit function (and its components). The graphics shown here are data from Pad B and the posterior distribution samples assuming a 𝛽− 𝑝 resonance energy of 𝐸 𝑝 = 160 keV. 201 Figure C.1: The 0th sample of the Bayesian Markov Chain Monte Carlo algorithm, after a burn- in period described in text. The jagged dark blue is the experimental data; the black line is the exponential function component to model the 𝛽− background; the green line is the Quadratic⊗Voigt function to model the 𝛽− 𝛼 background; the red constant line is the constant component to model the incomplete energy deposition of the 𝛽− 𝛼 background; the smooth solid blue shows a 20 times magnified 𝛽− 𝑝 signal; the dashed blue lines show the 20 times magnified contributions of the 𝛽− 𝑝 signal; and, the smooth red line is the sum of all components, which was compared against the experimental data. 202 Figure C.2: The 100 000th sample of the Bayesian Markov Chain Monte Carlo algorithm, after a burn-in period described in text. The jagged dark blue is the experimental data; the black line is the exponential function component to model the 𝛽− background; the green line is the Quadratic⊗Voigt function to model the 𝛽− 𝛼 background; the red constant line is the constant component to model the incomplete energy deposition of the 𝛽− 𝛼 background; the smooth solid blue shows a 20 times magnified 𝛽− 𝑝 signal; the dashed blue lines show the 20 times magnified contributions of the 𝛽− 𝑝 signal; and, the smooth red line is the sum of all components, which was compared against the experimental data. 203 Figure C.3: The 200 000th sample of the Bayesian Markov Chain Monte Carlo algorithm, after a burn-in period described in text. The jagged dark blue is the experimental data; the black line is the exponential function component to model the 𝛽− background; the green line is the Quadratic⊗Voigt function to model the 𝛽− 𝛼 background; the red constant line is the constant component to model the incomplete energy deposition of the 𝛽− 𝛼 background; the smooth solid blue shows a 20 times magnified 𝛽− 𝑝 signal; the dashed blue lines show the 20 times magnified contributions of the 𝛽− 𝑝 signal; and, the smooth red line is the sum of all components, which was compared against the experimental data. 204 Figure C.4: The 300 000th sample of the Bayesian Markov Chain Monte Carlo algorithm, after a burn-in period described in text. The jagged dark blue is the experimental data; the black line is the exponential function component to model the 𝛽− background; the green line is the Quadratic⊗Voigt function to model the 𝛽− 𝛼 background; the red constant line is the constant component to model the incomplete energy deposition of the 𝛽− 𝛼 background; the smooth solid blue shows a 20 times magnified 𝛽− 𝑝 signal; the dashed blue lines show the 20 times magnified contributions of the 𝛽− 𝑝 signal; and, the smooth red line is the sum of all components, which was compared against the experimental data. 205 Figure C.5: The 400 000th sample of the Bayesian Markov Chain Monte Carlo algorithm, after a burn-in period described in text. The jagged dark blue is the experimental data; the black line is the exponential function component to model the 𝛽− background; the green line is the Quadratic⊗Voigt function to model the 𝛽− 𝛼 background; the red constant line is the constant component to model the incomplete energy deposition of the 𝛽− 𝛼 background; the smooth solid blue shows a 20 times magnified 𝛽− 𝑝 signal; the dashed blue lines show the 20 times magnified contributions of the 𝛽− 𝑝 signal; and, the smooth red line is the sum of all components, which was compared against the experimental data. 206 Figure C.6: The 500 000th sample of the Bayesian Markov Chain Monte Carlo algorithm, after a burn-in period described in text. The jagged dark blue is the experimental data; the black line is the exponential function component to model the 𝛽− background; the green line is the Quadratic⊗Voigt function to model the 𝛽− 𝛼 background; the red constant line is the constant component to model the incomplete energy deposition of the 𝛽− 𝛼 background; the smooth solid blue shows a 20 times magnified 𝛽− 𝑝 signal; the dashed blue lines show the 20 times magnified contributions of the 𝛽− 𝑝 signal; and, the smooth red line is the sum of all components, which was compared against the experimental data. 207 Figure C.7: The 600 000th sample of the Bayesian Markov Chain Monte Carlo algorithm, after a burn-in period described in text. The jagged dark blue is the experimental data; the black line is the exponential function component to model the 𝛽− background; the green line is the Quadratic⊗Voigt function to model the 𝛽− 𝛼 background; the red constant line is the constant component to model the incomplete energy deposition of the 𝛽− 𝛼 background; the smooth solid blue shows a 20 times magnified 𝛽− 𝑝 signal; the dashed blue lines show the 20 times magnified contributions of the 𝛽− 𝑝 signal; and, the smooth red line is the sum of all components, which was compared against the experimental data. 208 Figure C.8: The 700 000th sample of the Bayesian Markov Chain Monte Carlo algorithm, after a burn-in period described in text. The jagged dark blue is the experimental data; the black line is the exponential function component to model the 𝛽− background; the green line is the Quadratic⊗Voigt function to model the 𝛽− 𝛼 background; the red constant line is the constant component to model the incomplete energy deposition of the 𝛽− 𝛼 background; the smooth solid blue shows a 20 times magnified 𝛽− 𝑝 signal; the dashed blue lines show the 20 times magnified contributions of the 𝛽− 𝑝 signal; and, the smooth red line is the sum of all components, which was compared against the experimental data. 209 Figure C.9: The 800 000th sample of the Bayesian Markov Chain Monte Carlo algorithm, after a burn-in period described in text. The jagged dark blue is the experimental data; the black line is the exponential function component to model the 𝛽− background; the green line is the Quadratic⊗Voigt function to model the 𝛽− 𝛼 background; the red constant line is the constant component to model the incomplete energy deposition of the 𝛽− 𝛼 background; the smooth solid blue shows a 20 times magnified 𝛽− 𝑝 signal; the dashed blue lines show the 20 times magnified contributions of the 𝛽− 𝑝 signal; and, the smooth red line is the sum of all components, which was compared against the experimental data. 210 Figure C.10: The 900 000th sample of the Bayesian Markov Chain Monte Carlo algorithm, after a burn-in period described in text. The jagged dark blue is the experimental data; the black line is the exponential function component to model the 𝛽− background; the green line is the Quadratic⊗Voigt function to model the 𝛽− 𝛼 background; the red constant line is the constant component to model the incomplete energy deposition of the 𝛽− 𝛼 background; the smooth solid blue shows a 20 times magnified 𝛽− 𝑝 signal; the dashed blue lines show the 20 times magnified contributions of the 𝛽− 𝑝 signal; and, the smooth red line is the sum of all components, which was compared against the experimental data. 211 Confidence Interval Calculations Converting a non-Gaussian probability density function to a confidence interval is not as straight- forward as the Gaussian interval. In the Gaussian probability density function, the distribution is such that 1) the most likely values are toward the median of the distribution and 2) the probability of increasingly/decreasingly extreme values diminishes symmetrically away from the median. How- ever, in general, these properties are not guaranteed in probability density functions and are both untrue in the case of an exponential probability density function that might be used in describing an upper limit on a value. Figure C.11, starts at the median of the distribution and moves symmetrically outward. Math- ematically, this symmetry condition can be written as: 𝑑𝑝(𝑥 left ) 𝑑𝑝(𝑥 right ) · Δ𝑥left = · Δ𝑥 right (C.1) 𝑑𝑥 𝑑𝑥 This symmetry only serves nicely in symmetric probability density functions. In asymmetric ones, one can find unfortunate circumstances such as physical boundaries never being included in a limit, no matter how probable, as the symmetric scheme requires probability be outside of the interval in equal amounts. This is undesirable in the case of an upper-limit against a physical boundary, where the exponential curve may nicely describe the probability density function, but the median-scheme never allows for an inference of a value of zero. Now consider Fig. C.12. The scheme used here starts at the most probable value and integrates outwards in the probabilistically preferred direction. In the case of an exponential probability density function, this is necessarily one-sided. As the confidence interval increases, zero is always included, as is desired for an upper limit probability distribution. The final example will be a suggestive, but not conclusive finite measurement, similar to some of the probability density functions that were extracted in the Bayesian analysis of Sec. 4.4 and Chapter 5. This distribution was generated by the sum of an arctangent function and a Gaussian curve. It is meant to have a clear peak representing a clearly favored value, but still substantial probability close to zero. The symmetric median scheme will return confidence intervals centered on the low-value shoulder of the peak. Again, at even high confidence, it excludes zero, which has substantial probability density. The mode scheme always includes the peak. Furthermore, for low confidence values, it reports a finite value with an upper and lower limit. At some degree of confidence, however, the interval reverts to being an upper limit. This ability to change from a finite measurement to a limit is valuable with these suggestive, in-between distributions that are not clearly limits nor finite valued; at low confidence, the interval is a finite value; at high confidence, the interval is a limit. 212 5000 99% CI 95% CI 90% CI probability density of x (1/ arb. units) 4000 50% CI 33% CI 3000 2000 1000 0 0 10 20 30 40 50 60 70 80 90 100 parameter x (arb. units) Figure C.11: Confidence intervals of an upper-limit-like exponential probability density function, taken symmetrically about the median. At no confidence level does the symmetric confidence interval include zero, which would likely be desirable. 213 5000 99% CI 95% CI 90% CI probability density of x (1/ arb. units) 4000 50% CI 33% CI 3000 2000 1000 0 0 10 20 30 40 50 60 70 80 90 100 parameter x (arb. units) Figure C.12: Confidence intervals of an upper-limit-like exponential probability density function, taken asymmetrically about the mode and incremented according to the most probable direction. The asymmetric calculation nicely provides natural interpretations of a statistical limit of a param- eter. 214 2500 99% CI 95% CI 90% CI probability density of x (1/ arb. units) 2000 50% CI 33% CI 1500 1000 500 0 0 10 20 30 40 50 60 70 80 90 100 parameter x (arb. units) Figure C.13: Confidence intervals of a suggestive, but not conclusively non-zero probability density function, taken symmetrically about the median. At low values for the confidence interval, the peak in the probability density function is missed. Similarly to the median-scheme with the exponential distribution in Fig. C.11, zero is never included, even at high confidence. 215 2500 99% CI 95% CI 90% CI probability density of x (1/ arb. units) 2000 50% CI 33% CI 1500 1000 500 0 0 10 20 30 40 50 60 70 80 90 100 parameter x (arb. units) Figure C.14: Confidence intervals of a suggestive, but not conclusively non-zero probability density function, taken symmetrically about the mode. At low values for the confidence interval, the peak in the probability density function is inferred. Similarly to the mode-scheme with the exponential distribution in Fig. C.12, at high confidence a limit-like confidence interval is retrieved. 216