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Time-of-flight studies of muon catalyzed fusion with a muonic tritium beam Fujiwara, Makoto C. 1999

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TIME-OF-FLIGHT STUDIES OF MUON CATALYZED FUSION WITH A MUONIC TRITIUM B E A M by Makoto C. Fujiwara M.A.Sc., The University of British Columbia, 1994 B.Eng., Yamanashi University, 1992 A.Eng., Kobe City College of Technology, 1990 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DE GRE E OF D O C T O R OF PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard The University of British Columbia April 1999 © Makoto C. Fujiwara, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, Canada V6T 1W5 Date: A b s t r a c t In this thesis, we establish a new approach in muon catalyzed fusion studies, the time-of-flight method with an atomic beam of muonic tritium, and report results for muonic tritium scattering and epithermal dfit resonant formation, providing the first quantitatve measurements on these reactions. Emission of muonic tritium from a solid hydrogen thin film into vacuum was observed via imaging of muon decay electrons, and the measurement of the position and the time of muon decay provided spectroscopic evidence for the Ramsauer-Townsend effect in the fit + p interaction. The RT minimum energy was determined to be 13.6 ±1.0 eV, in fair agreement with quantum three body calculations. Using this fit beam, we have confirmed theoretical fit + d scattering cross sections to an accuracy of 10% by measuring the attenuation of fit through a deuterium layer. The importance of p-wave scattering in the fit + d interaction, as suggested by the theory, was also confirmed by our data via comparisons with Monte Carlo calculations assuming different scattering angular distributions. The existence of a predicted resonance for dfit formation in fit + D2 collisions was directly confirmed for the first time. Our results correspond to a peak resonance rate of (8.7 ± 2.1) x 109 s _ 1 in Faifman's model, more than an order of magnitude larger than the room temperature rates, and indicate a resonance energy of 0.42 ± 0.04 eV for the F=l resonance peak in ortho deuterium. Assuming the theoretical [(<i^ t)dee] energy spectrum, these results imply sensitivity to the binding energy of the loosely bound state of the dfit molecule, with an accuracy approaching Jjhe magnitude of the relativistic a,nd._QED effects. Table of Contents Abstract ii List of Tables ix List of Figures xiii Acknowledgments xviii 1 Introduction 1 1.1 Muon Physics 1 1.2 Overview of the muon catalyzed fusion cycle 2 1.2.1 Muonic atom formation 4 1.2.2 Cascade 5 1.2.3 Muonic atom collisions 6 1.2.4 Processes of excited muonic atoms 10 1.2.5 Formation of muonic molecules 12 1.2.6 Nuclear fusion 14 1.2.7 Sticking and stripping 16 1.3 Measurement of resonant molecular formation 18 1.3.1 Cycling measurements in D/T mixture 18 1.3.2 Epithermal molecular formation 19 2 Theoretical aspects of pCF 25 2.1 Muonic molecule and the three body problem 25 in 2.1.1 The challenge 25 2.1.2 Three body coulomb problem 26 2.1.3 Adiabatic approaches 27 2.1.4 Variational approaches 34 2.1.5 Non-Coulombic corrections 39 2.2 Resonant molecular formation 40 2.2.1 The standard Vesman model 41 2.2.2 Subthreshold resonances 51 2.2.3 Condensed matter effects 52 3 Experimental Apparatus 54 3.1 Muon beam 54 3.2 Cryogenic target system 57 3.2.1 Gas handling and mixing 57 3.2.2 Gas deposition 59 3.2.3 Experimental vacuum space 60 3.2.4 Vacuum system 61 3.2.5 Tritium safety system 65 3.2.6 Operation 67 3.3 Target characterization 68 3.3.1 Characterization method 69 3.3.2 Film thickness and uniformity 75 3.3.3 Summary of target characterization 87 3.4 Detection System 88 3.4.1 Overview 88 3.4.2 Trigger 91 iv 3.4.3 Multi-wire proportional chambers 95 3.4.4 Silicon detector 97 3.4.5 Neutron detector 99 3.4.6 Germanium X-ray detector 101 3.4.7 Electron detection system 102 4 Experimental Runs 104 4.1 Run Series I 104 4.2 Run Series II 106 4.3 Histogramming and analysis tools 109 5 Monte Carlo simulation codes 111 5.1 SMC Monte Carlo code I l l 5.1.1 Muonic atom generation I l l 5.1.2 Reaction selection 112 5.1.3 Time and position evolution 113 5.1.4 Muonic processes 114 5.1.5 Muon decay and imaging 118 5.2 Other simulation codes 119 5.2.1 Full muonic processes 119 5.2.2 Specific processes 119 6 Analysis I — Absolute normalization 120 6.1 Effective target thickness 120 6.2 Silicon detector acceptance 127 6.3 Muon stopping fraction determination 130 6.4 Amplitude ratio method 130 v 6.4.1 Electron scintillator measurements 132 6.4.2 Electron telescope measurements 136 Aft 6.4.3 Corrections to SH in the amplitude ratio method 139 6.5 Absolute amplitude method 150 6.5.1 Delayed electron lifetime 153 6.5.2 Delayed electron cuts efficiency 156 6.6 Discussion of stopping fraction 159 6.6.1 The discrepancy 159 6.6.2 A solution 160 7 Analysis II — Emission of muonic tritium 165 7.1 MWPC imaging data 165 7.1.1 Cuts 165 7.1.2 Muonic tritium emission spectra 167 7.2 Measurement of the Ramsauer-Townsend effect 170 7.2.1 Monte Carlo parameter determination 170 7.2.2 The Ramsauer-Townsend minimum energy 171 8 Analysis III - Molecular formation 178 8.1 Si detector data 178 8.1.1 Si detector energy calibration 178 8.1.2 ADC dead time correction 181 8.1.3 Energy spectrum features 185 8.1.4 Time spectrum features 188 8.2 The dfit fusion measurements 190 8.2.1 Background subtraction 191 8.2.2 Yield measurements 192 v i 8.2.3 Systematic effects 207 8.2.4 dfid fusion 211 8.2.5 The yield results 217 8.2.6 Time-of-flight spectra 221 8.3 Monte Carlo analysis 224 8.3.1 Fusion yield analysis 224 8.3.2 Time-of-flight analysis 229 8.3.3 Monte Carlo uncertainties 238 8.3.4 Formation rate and resonance energy results 239 9 Discussion and Conclusion 242 9.1 Muonic tritium scattering 242 9.1.1 The Ramsauer-Townsend effect 242 9.1.2 fit + d scattering 245 9.2 Condensed matter and subthreshold effects 247 9.3 Resonant molecular formation 248 9.4 Concluding remarks 252 9.4.1 Improvements and Future directions 252 9.4.2 Summary 254 A Abbrev ia t ions and notat ion 256 A . l Abbreviations 256 A. 2 Error notation 259 B Resonant scattering of fit 260 B. l Effective formation model 262 B . l . l Validity 262 vii B.1.2 Epithermal Collisions 263 B.1.3 Comparison with the explicit back decay model 264 B.2 Back-decayed fit energy distribution 265 B.2.1 MMC recoil and thermalization 265 B.2.2 MMC ro-vibrational relaxation 267 B.2.3 Resonant excitation of DX molecule 268 B.2.4 Back-decayed fit energy 268 B.3 Implications for our measurements 272 C Muon decay electron time spectrum fit 274 Bibliography 277 vm List of Tables 1.1 Main properties of muonic atoms 7 1.2 Non-resonant muonic molecular formation rates by Faifman 13 1.3 Non-relativistic Coulomb molecular binding energies 14 2.1 Comparison of the binding energy for the loosely bound state of dpt. . . 40 3.1 Protium film thicknesses at various distance from the center of the film for system 3 81 3.2 Thickness of the protium film for system 1, deposited with standard dif-fuser position 82 3.3 Thickness calibration factors for a neon film at source positions -20 mm, 0 mm and 20 mm 83 4.1 Comparison of Run Series 104 4.2 Run summary for Series 1 105 4.3 Run summary for Series 2 (part 1) 107 4.4 Run summary for Series 2 (part 2) 108 6.1 The effective upstream layer thickness with different beam parameters. . 124 6.2 The effective thickness and the silicon detector acceptance of the down-stream layer 127 6.3 The effective thickness of the downstream layer with different upstream overlayer thickness, and the corresponding silicon detector acceptance. . . 127 ix 6.4 Variations in Si acceptance due to uncertainties in the geometry 128 6.5 Summary of effective average thickness and Si acceptance for all the target arrangements used in this thesis 129 6.6 The result of a fit to the electron time spectrum with two exponential functions with a constant background 134 6.7 The sensitivity of the stopping fraction to the shift of time zero 135 6.8 The fit results for the electron telescope spectrum 136 6.9 Measured and calculated probabilities for inclusive charged particle emis-sion reactions 141 6.10 Summary of high energy proton emission following muon capture 142 6.11 The electron detection efficiencies, calculated by the GEANT simulations. 149 6.12 Summary of correction factors 77, n needed in the derivation of stopping fraction via relative amplitude method 149 6.13 Comparison of the uncorrected and the corrected stopping fraction deter-mined from relative amplitude method 150 6.14 The lifetime of the first electrons after Si signal, fitted to a single expo-nential with a constant background 154 6.15 The lifetime of the first telescope after Si signal, fitted to a single expo-nential with and without a constant background term 155 6.16 The lifetime of 1st delayed electrons, and 1st delayed telescope, averaged over Sil and Si2, as well as Run A and Run B 156 6.17 The Del cut efficiency, e /^, the absolute electron detection efficiency, e eO e, and the muon stopping fraction, SF~BS, determined from the absolute yield method 157 6.18 Correction factors rj, K for the telescope measurements used in the deriva-tion of stopping fraction via relative amplitude method 162 x 6.19 Comparison of the uncorrected and the corrected stopping fraction deter-mined from relative amplitude method with the telescope detectors. . . . 162 6.20 Summary of stopping fractions determined via various methods 163 8.1 Comparison of different types of events and their signatures used in the correction for the ADC blocking of Si events 183 8.2 Summary of runs used for background subtraction 191 8.3 The target combinations for fusion yield measurements 192 8.4 Si yield per GMU for the US fusion measurements with different back-ground subtraction and energy cuts 196 8.5 Si yield per GMU for the MOD measurements with different background subtractions and energy cuts 197 8.6 Si yield per GMU for the TOF measurements with different background subtractions and energy cuts 199 8.7 Summary of the detector averaged Si yield for 3 T-l TOF measurement {ct = 0.1%) 204 8.8 The detector averaged Si yield for Series B (cj = 0.2%) 3 T-l TOF mea-surement 207 8.9 Corrections to the fit production yield due to nitrogen contamination in the target 211 8.10 dfid fusion proton yield in late time 214 8.11 The probability that the proton is produced after dt fusion is observed. . 215 8.12 Summary of the correction due to protons from dfid fusion 217 8.13 Final results for fusion yields 219 8.14 Summary of the runs for the TOF time measurements 221 8.15 Summary of the attenuation analysis using the MOD yield data 229 xi 8.16 The results of x 2 fit for dfit formation rate 235 8.17 The results of \ 2 fit f ° r dfit formation resonance energy 235 8.18 Estimated effects of various systematic uncertainties on the T O F fusion yield 238 xn List of Figures 1.1 Greatly simplified cycle of muon catalyzed fusion in D/T mixture 3 1.2 Scattering cross sections of muonic tritium with a hydrogen isotope nucleus. 9 1.3 Summary of experimental results of the final sticking as a function of the target density, together with theoretical predictions 17 1.4 Resonant molecular formation rate at epithermal collision energies. . . . 20 1.5 Conceptual drawing of the time-of-flight measurement of resonant molec-ular formation rates 22 2.1 Co-ordinate system for muonic molecular ion ab\i used in the-Adiabatic Representation method 29 2.2 Three arrangement channels of the d\it system and their Jacobian co-ordinates used by Kamimura 36 2.3 Schematic energy diagram for resonant molecular formation 42 2.4 Molecular formation rates and fusion probability in the fit+D2 —> (d/j,t)dee collisions, calculated by Faifman 50 3.1 Schematic site drawing of TRIUMF facilities and beam lines 55 3.2 Layout of the M20B muon channel 56 3.3 Topology of important parts of the experimental vacuum and the gas han-dling systems 58 3.4 A view of the diffuser system 62 3.5 A perspective view of the central part of the target system 63 xiii 3.6 A pictorial view of vacuum system, roughly to scale 64 3.7 Schematic representation of the target safety enclosure 66 3.8 Schematic view of the target characterization setup 70 3.9 Counts versus vertical position of the silicon detector 72 3.10 Alpha particle energy spectra with different thicknesses of hydrogen film. 73 3.11 Test of the linearity of deposition 76 3.12 Comparision between sequential deposition of very thin films and the thick film deposition 78 3.13 Thickness profiles for different diffuser systems 79 3.14 Film thicknesses for system 3, plotted against the vertical distance from the diffuser center 80 3.15 Comparison of the measured thickness profile with Monte Carlo calcula-tions with different assumptions 85 3.16 Monte Carlo simulations comparing different distances between the dif-fuser surface and the cold foil surface 86 3.17 Schematic top view of the detector arrangement for Run Series 1 89 3.18 Schematic top view of the detector arrangement for Run Series 2 90 3.19 Schematic diagram for the trigger electronics, illustrating the key compo-nents 92 3.20 Schematic timing diagram for trigger electronics 94 3.21 Electronics diagram for the Multi-Wire Proportional Chamber imaging system 97 3.22 Electronics diagram for a silicon detector 98 3.23 Electronics diagram for a neutron detector 100 3.24 Electronics diagram for the germanium detector 101 3.25 Electronics diagram for the electron detection system 102 xiv 6.1 Dependence of the effective thickness and the effective deviation on the beam parameters and the radial cut-off values 121 6.2 The Y-distribution of decay electrons in the MWPC image, compared with Monte Carlo simulations assuming Gaussian beam distributions 122 6.3 The Y-distribution of decay electrons in the MWPC image, compared with Monte Carlo simulations assuming beams of Gaussian with flat top. . . . 123 6.4 Example of the simulated radial profile of the fit beam 125 6.5 An example of electron time spectrum and fit with two exponential func-tions with a constant background 133 6.6 Energy spectrum for telescope events and the dependence of stopping frac-tion on their energies 138 6.7 Electron decay energy spectra R(E) for the muon bound in lead and iron, as well as for free muon 144 6.8 Energy spectra for telescope events simulated with a GEANT simulation. 161 7.1 The time and extrapolated position in the ^-direction of the muon decay. 167 7.2 Time spectra of muon decay in the region z=[-10,10] mm with varying thickness of D 2 overlayer. The thicknesses given are approximate 169 7.3 Determination of MWPC resolution parameters 172 7.4 Background-subtracted time spectrum of emitted muonic tritium in vac-uum with a fit by Monte Carlo simulated histogram 173 7.5 Total x2 versus the energy scaling parameter K in the fit of fit emission data with Monte Carlo calculations 175 7.6 Total x2 versus K~1I2 near minimum 176 8.1 Silicon (Si2) energy spectra taken with 2 4 2 A m source 179 xv 8.2 Centroid values of ADC for 2 4 1 Am peak for different calibration runs, showing the stability of the system 180 8.3 Energy spectrum feature ofof Si events for standard TOF target with dif-ferent time cuts 186 8.4 Energy spectrum of Si events for pure H 2 target with different time cuts. 187 8.5 Time spectrum features of Si detector for standard TOF target with dif-ferent energy cuts 188 8.6 Time spectrum of Si detector for pure H 2 target with different energy cuts. 189 8.7 Si energy spectra with 3 T-l US D 2 layer in the early time range 193 8.8 Early time Si energy spectra with 14 T-l US D 2 194 8.9 Si energy spectra for the TOF yield measurement with 3 T-l DS D 2 in Series A (ct = 0.1%), with the time cut of 1 < t < 6 (is 200 8.10 Illustration of the background subtraction Method 2 201 8.11 Background subtraction Method 2-c 202 8.12 Si energy spectra for the TOF yield measurement with 3 T-l DS D 2 in Series B (ct = 0.2%), with the time cut of 1 < t < 6 (is 206 8.13 Energy cut efficiency determination using a dedicated Monte Carlo code. 209 8.14 Fusion yield per stopped muon for the US, MOD, and TOF measurements. 220 8.15 The time-of-flight spectrum for 3 T-l DS for Series A (ct = 0.1%) 222 8.16 The time-of-flight spectrum for 3 T-l DS for Series B (Q = 0.2%) 223 8.17 Monte Carlo simulations for US fusion yield compared to the experimental yield. 225 8.18 Experimental data and Monte Carlo results for the MOD yield measure-ments plotted on a log scale 228 xvi 8.19 The simulated correlations between the fusion time and the energy at which dfit molecular formation takes place for the TOF measurements using 3 T-l layer 230 8.20 The simulated time vs. energy for total events in a thick (20 T-Z) layer, where the correlation is weaker than the thin layer 231 8.21 Fit of the calculated MC spectrum to the experimental data for 3 T-l in Series A 233 8.22 Fit of the calculated MC spectrum to the experimental data for 3 T-l in Series B 234 8.23 Total x 2 versus dfit formation rate scaling factor S\, for Series A and Series B 236 8.24 Total x 2 versus the scaling factor of the resonant energy for dfit formation factor SE, for Series A and Series B 237 9.1 Allowed values for fit + p elastic scattering cross sections from the mea-surement 243 9.2 The time spectrum of fit decay in vacuum region compared with a Monte Carlo calculation assuming no RT minimum 244 9.3 Comparison of data and MC assuming a constant rate of molecular for-mation without resonance structures 252 xvn Acknowledgments I am most grateful to my supervisor and spokesperson of this experiment, Dr. Glen Mar-shall for his guidance and support he has given me throughout my years in Canada. Without his high standard for science and dedication to our experiment, this work would not have been possible. I would like to thank my advisor, Professor David Measday for his continuous support and advice on my studies at UBC and for valuable comments on this thesis. I'm always inspired by his deep knowledge of physics as well as his broad interests in other areas of scholarship. I appreciate their time for reading and correcting my thesis despite a stringent time constraint I imposed on them. I wish to thank my other local collaborators, Professors George Beer, Art Olin, and Ms. Tracy Pocerlli for their invaluable input and criticism of this work, and always lively (often heated) discussions at our weekly meetings. Contributions of Drs. Tom Huber, Francoise Mulhauser, and Paul Knowles, were ab-solutely essential to the experiment, particularly, in the design and the construction of our target system, and developing our Monte Carlo program, and I thank them immensely. I would also like to thank other member of Muonic Hydrogen Collaboration, Drs. John Bailey, Jack Beveridge, R. Jacot-Guillarmod, Peter Kammel, S. K. Kim, Ray. Kunselman, Jeff Martoff, G. R. Mason, Claude Petitjean, and Hannes Zmeskal, and Ms. Maureen Maier and Jessica Douglas for their invaluable contributions. I would like to thank my theoretical colleagues, Drs. Andrzej Adamczak, Mark Faif-man, Valeri Markushin, Volodja Melezhik, and Jan Wozniak, for taking time to answer xviii my many (often naive) questions. Thanks are due to Professors Jess Brewer, Elliot Burnell, Ken Crowe, Walter Hardy, Mike Hasinoff, and Anthony Merer, for their valuable comments and criticism of my thesis. Ray Kunselman also proofread my thesis. I am much indebted to Professors Eiko Torikai and Ken Nagamine for introducing me to the excitement of muon physics, and for their continuous encouragement and support through out my studies. My personal thanks go to Andre, Ermias, Hash, Jen, and Sarah for making my stay in Canada so special, and I thank my family and friends in Japan who have always supported me. Finally, I gratefully acknowledge the financial support from the Rotary Foundation, the University of British Columbia, Government of Canada, Green College, Northern Telecom, and Westcoast Energy Inc. xix Chapter 1 Introduction 1.1 Muon Physics Since its discovery in cosmic rays in 1937, the muon has played an important role in our understanding of nature. The muon has been extensively studied to determine its own properties and interactions and it has also been used as a probe to reveal the nature of other particles and systems. Indeed, it was the muon which gave us the first indication that nature replicates particles in a similar pattern, now known as generations. The existence of generations, and the muon itself for that matter, persists as a big mystery after over 50 years. With its 2.2 fis mean life, one of the longest of all elementary particles unstable under weak decay, it also provides a rich variety of applications in diverse areas of science, including condensed matter physics and chemistry. From my somewhat biased point of view, currently there is renewed interest in basic muon physics, mainly on two fronts. High energy physicists are seriously considering the design and construction of a muon collider (the First Muon Collider), which if realized, has a potential to become a "Higgs Factory," as well as achieving a much higher collision energy than the existing electron colliders. At low energy, or the precision front as it is sometimes called, muonic processes forbidden (or highly suppressed) by the Standard Model, such as the decays /J, —> ej, and fj, —> eee, have drawn continuing and perhaps recently more intensified attention as a probe of the physics beyond the Standard Model. In particular, many Supersymmetic and/or Grand-Unified extensions of the Standard 1 Chapter 1. Introduction 2 Model, the current theoretical favorite, naturally require these processes to occur, which, if observed, would have a considerable impact on the way we see our Universe. Although its fundamental properties and interactions are of great interest, for the purpose of this thesis, which focuses on atomic and molecular aspects of muon physics, much of the behavior of the negative muon can be described by that of a heavy electron. Nevertheless, because of the muon's heavy mass, comparable to that of light nuclei, the muonic system exhibits many unique characteristics, including the catalysis of nuclear fusion among hydrogen isotopes. As we will see, the nuclear and particle physics as-pects of muonic processes become important experimentally in understanding various systematic effects and background processes encountered in our experiment. In addition, because of the use of solid targets, condensed matter physics comes into play. Indeed, this experiment allows me an opportunity to learn about a rich variety of physics at energy scales spanning over more than twelve orders of magnitude, ,from the hydrogen Debye temperature of 10-2 eV to the electroweak scale of 1011 eV. In the rest of this chapter, I will give a brief review of / J C F and describe the concept of our experiment. Chapter 2 focuses on the theoretical aspect, while the experiment apparatus is described in Chapter 3. Chapter 4 gives a summary of our experimental runs. After explaining our simulation codes in Chapter 5, the details of the analysis will be given in Chapters 6-8, which be followed by a discussion and conclusion. 1.2 Overview of the muon catalyzed fusion cycle Figure 1.1 illustrates a greatly simplified scheme of the muon catalyzed fusion (pCF) cycle in a D/T mixture1. The system has attracted the greatest interest because of 1 W h e n referring to a a mixture of hydrogen isotopes, a notation X / Y is used in this thesis, where X , Y are protium ( 1 H), deuterium or tritium. It is usually assumed that the mixture has an equilibrium molecular composition, i.e. , X2 '• XY : Y2 = cx : Icxcy : c\, where cx and cy denote the relative Chapter 1. Introduction 3 Figure 1.1: Greatly simplified cycle of muon catalyzed fusion in D / T mixture. its most favorable efficiency for fusion catalysis (see for a review [1-4]). Note that a homogeneous mixture, of mostly gas or liquid, has been used traditionally, as opposed to the inhomogeneous targets used in this thesis. A muon injected into the hydrogen target will slow down and form a small citomic system, muonic deuterium (fid) or muonic tritium (fit), by replacing the electron in the atom. If a fid is formed, the muon will be transferred to a triton forming a more tightly bound fit. The fit will then collide with a deuterium molecule and form the muonic molecule2 dfit. Molecular formation occurs predominantly via a resonant mechanism, in which the energy released from the formation of the dfit molecule is absorbed by the rotational and vibrational excitation of the molecular complex \(dfit)dee] where the compact object dfit acts as a pseudonucleus. Muonic molecule formation can also occur by releasing the energy via the Auger process (non-resonant formation), but this rate is much atomic concentrations. 2 T o be precise, this should be called a muonic molecular ion, but we shall simply denote it as a muonic molecule according to the convention in the field. Chapter 1. Introduction 4 smaller than resonant formation. Because the size of the muonic molecule is smaller than ordinary molecules by its mass ratio (m^/m e) in zeroth order, the internuclear distance in dfit is small enough that fusion takes place within 1 0 - 1 2 s. After fusion, the muon is released more than 99% of the time, but a small probability exists for a process known as sticking in which the muon becomes attached to the charged fusion product, in this case an a particle. If sticking occurs, the muon is lost from the cycle, and this indeed limits the ultimate number of fusions that one muon can catalyze. In the following sections, we will discuss each process involved in some detail. 1.2.1 Muonic atom formation When a muon stops in a hydrogen isotope target, it replaces the electron of an atom to form a muonic atom in an excited state. Understanding muonic atom formation and the cascade process to the ground state is important, since they affect muon transfer from excited states (the qis problem), as well as the creation of hot atoms via acceleration. The sequence is, however, one of the least studied processes in /^CF [2], partly due to the lack of direct experimental information. Recent theoretical developments on the formation processes are reviewed in Refs. [5-7]. In a simple picture 3, the muon is expected to be captured in an excited orbital state with the principal quantum number n ~ \fm^Jrne ~ 14, whose wave function has sim-ilar energy and spatial size to that of the ground state electron4. In reality, the target is molecular rather than atomic hydrogen, and muonic atom formation is predicted to occur via formation of an excited molecule [xy/j,~e~]*, and its decay, in semi-classical 3 Recal l that for the principle quantum number n, the orbital radius rn, the energy En and the orbital 2fc2 tJ 2 ryl 4 ry 2 velocity vn are: r „ = ^ze2 1 ^ n = ~W~ = ~"2n2h2 a n c ^ V n = nh> w n e r e ~ m M ^ s the reduced mass of the muon and the nucleus with the charge Z. 4 Note that the corresponding velocity of the electron differs from that of the muon by the factor Chapter 1. Introduction 5 calculations by Fesenko and Korenman [6], results in a broad distribution over quantum states n. More complete, but quasi-classical five-body dynamical calculations, including rotational and vibrational degrees of freedom of the molecule are being carried out by Cohen [7,8]. 1.2.2 Cascade Muonic atoms formed in highly excited states quickly de-excite to the ground state via several competing mechanisms [9,10]: 1. Radiative: (fix)i —>• (//a;)/+ 7 2. External Auger: (u-x)i + YZ -> (fxx)f + YZ+ + e~ 3. Stark Mixing: (fix)ni + Y —• (fix)nii + Y 4. Coulomb Collisions: (fj,x)i + y —»• (fix)f + y, nj < m 5. Elastic Scattering: (fix)n + YZ —+ (ux)n + YZ In addition, muon transfer can take place from an excited state before reaching the ground state, as will be discussed below. Each process has a different dependence on target density and collision energy, and competition between the processes dictates the energy distribution of muonic atoms in the ground state. Coulomb de-excitation (item 4 above) has received recent attention as a possible mechanism for accelerating muonic atoms to up to more than 100 eV. Calculations by Markushin predict as many as 50% of muonic atoms have energies above 1 eV at the instant of reaching the ground state at liquid hydrogen density [9]. Chapter 1. Introduction 6 1.2.3 Muonic atom collisions In the traditional description of the pCF cycle, it was assumed that the muonic atom was always thermalized at the target temperature [11], and the energy dependence of the reaction rates, if considered at all, was averaged out by integrating over the Maxwellian distribution. This type of analysis using the constant rates has proven very successful in particular for the pure liquid and gaseous deuterium system [12]. The importance of going beyond the constant rate approach was recognized by the suggestion of epithermal (i.e., non-thermalized) transient phenomena by Kammel [13] and by Cohen and Leon [14], but it was not until the complete set of theoretical cross sections [15-21] became available that an energy dependent analysis could be performed in fiCF [22]. It is absolutely essential for our measurement to take into account the energy depen-dence; or rather, our experiment is designed to take advantage of its sensitivity to the energy dependent cross sections. Theoretically, collisional processes of muonic atoms present challenging few-body problems. Because the muon mass is comparable to that of nuclei, the system is highly non-adiabatic, i.e., nuclear and muonic motions cannot be separated, as we shall see in detail in Section 2.1.2. The main collisional processes of muonic hydrogen isotope atoms include elastic scattering, muon transfer (also called charge exchange), and hyperfine transitions (spin flip). Table 1.1 lists the main properties of muonic atoms, while Fig. 1.2 compares the various cross sections of pd collisions with hydrogen isotope nuclei. Elastic scattering Elastic scattering processes dictate the deceleration of muonic atoms in the target, hence they determine the energy distribution at the time of the subsequent reaction (such as Chapter 1. Introduction 7 Table 1.1: Main properties of muonic atoms from Ref. [23]a. Nucleus Nucleus mass Ground state Hyperfine splitting Isotope splitting X Mx £ W (eV) ^EhJx* (eV) AExy (eV) P 1836.1515 -2528.517 0.1820 AEpd = 134.709 d 3670.481 -2663.226 0.0485 AEdt = 48.042 t 5496.918 -2711.268 0.2373 AEpt = 182.751 "The masses are given in units of the electron mass m e=0.5109991 M e V / c 2 . The muon mass is raM = 206.7686m e, the Rydberg energy, Ry = 13.605804 eV. These values are used in the calculations of muonic molecular enegy levels given in Table 2.1. muon transfer or molecular formation) in the /iCF cycle. There is a relatively limited amount of experimental information on muonic atom scattering cross sections [24]. In particular, there is hardly any data on energy dependent cross sections. The elastic scattering cross section can be written in terms of the sum of the contri-butions of the partial waves: / i where the physics is contained in the phase shift 8\. As one can see from Eq. 1.1, if for a given partial wave /, Si is approximately equal to nir (n = 1,2,3-••), then the partial cross section cr; goes to zero. For low energy collisions, where s wave (7 = 0) usually dominates, the total cross section can become very small, a process known as the Ramsauer-Townsend effect. If, on the other hand, Si ~ (n + |)7r, then the cross section becomes large and reaches a maximum value (Air/k2) called the unitarity limit. It is interesting to note that all the phase shifts at the zero energy limit are fixed at Si = J5/7T by Levinson's theorem [25,26], where Bi is the number of three-body bound states of given /, illustrating the intricate connection between bound states and scattering properties. As we shall see it is the Ramsauser-Townsend effect that makes it possible to produce a muonic atom beam for our experiments. Chapter 1. Introduction 8 M u o n transfer Due to the differences in the binding energies, the muon can be transferred to the heavier isotopes: (tix)ni + y -> x + (fiy)nii + — ( 1 . 2 ) where AExy is the isotope splitting (Table 1.1) due to the reduced mass difference. The subscript n denotes the possibility of transfer from excited states (Section 1.2.4). In the D/T mixture target, the rate of muon transfer from the ground state of the deuterium atom to the triton is relatively slow (~ 3 x 108 s _ 1) and becomes one of the bottle-necks at low tritium concentrations. It should be noted that the released energy, divided between two projectiles, gives an acceleration to the muonic atom in the lab frame corresponding to the energy E(w) Lab = ™ x A E 2X V . (1.3) The energy dependent calculations predict that Xdt is significantly larger at higher energies (> 10 eV). This may be exploited, for example, by triple isotope mixture targets (H/D/T) [27], in which a faster reaction, muon transfer from a proton to a deuteron, creates an energetic fid, thus effectively speeding up the transfer to tritium. Hyperf ine transit ions Some muonic processes such as resonant molecular formation, fusion, and nuclear muon capture are known to depend on the hyperfine state of the muonic atom, though for different reasons. The most spectacular example is a drastic dependence of the dfid formation rate at low temperature on the fid hyperfine state [28]. Other cases where spin flip plays a crucial role include the Wolfenstein-Gershtein effect in pfid [29-31] and pfit fusion [32], and muon capture on a proton (Ref. [33] and references therein). Spin flip is Chapter 1. Introduction 9 CM O o co co co O o -18 10 ~ h 10 -19 10 -20 10 -21 ~i i i i i i i - i — i— I I I I - /xt + p - - /xt + d - Afct(0) + t f i i ( i ) + t 10 - l 10" 10* Lab energy (eV) 10" Figure 1.2: Scattering cross sections for fit with a hydrogen isotope nuclei from the Nuclear Atlas [16,17], showing the Ramsauer-Townsend minimum at around 10 eV for fit +p. fit(F) +t cross sections plotted include both elastic and spin exchange reactions, where (it(0) is the singlet state and fit(l) is the triplet state. usually considered in symmetric collisions: ^ ( i ? ) + x _> fix(F') + x + AEhJx\ (1.4) where the reaction is dominated by the exchange of the muon between two identical nuclei. There is also the possibility, though much smaller, of spin flip in asymmetric collisions: fJ-x(F) + y -> fix(F') + y + AE"'. (1.5) Chapter 1. Introduction 10 Obviously, unlike the symmetric case, this cannot be achieved by muon exchange, and a relativistic interaction is required to flip the spin so the cross sections are much lower. According to Cohen [21], who calculated these processes in the Improved Adiabatic ap-proach (Section 2.1.3), non-symmetric spin flip cross sections are several orders of mag-nitude smaller than symmetric spin flip cross sections. In addition, there is a prediction that spin flip in the pure deuterium system occurs via the back decay of the muonic molecular complex (resonant spin flip reaction) [34], but the experimental data become rather inconsistent with theoretical predictions, if the resonant spin flip is included [12,35]. 1.2.4 Processes of excited muonic atoms One of the primary questions in ^CF is the so called qls problem. In a D/T mixture, some fraction of the muons captured in the excited state of muonic deuterium can be transferred to the triton before reaching the ground state with the probability (1 — </is), where the probability for the muon to reach the ground Is state is denoted by qls. Since the transfer rate from excited states is very fast5, qu can affect the cycling rate (the number of fusions catalyzed by the same muon per second). Conversely, the extraction of /iCF parameters from the measurement of the cycling rates depends on the qis value (see Section 1.3.1). There has been a longstanding discrepancy between theoretical predictions, based on semi-classical calculations, and experimental data derived indirectly (see for example, 5 This is expected from a naive argument that the muonic atom with higher n is much more extended in size, hence has larger cross sections, but the bigger effect presumably is the long range behavior of the potential for the excited states. 2, Chapter 1. Introduction 11 Ref. [1,36]). Recent experimental developments [37,38] using state-of-the-art X ray de-tection technologies are producing more direct information about q\ds in the case of pd transfer in H/D mixtures, and it is hoped that they will give some insight into the more important D/T case. Processes involving excited states, n > 1, of the muonic atom have gained increas-ing theoretical attention. In addition to semi-classical calculations of excited transfer reactions [39,40], full quantal calculations are emerging [41] using the hyperspherical approach (Section 2.1.4). The transport cross sections of excited atoms including Stark transitions are being investigated mainly in semi-classical approaches [42,43]. Recently, a new process has been suggested by Froelich and Wallenius [44-46]. They predict a high rate of formation of a muonically excited metastable three-body state dpt* (associated with the adiabatic 3cr potential [47]) in the collision of excited pt(2s) with D 2 , which will then decay into /id(ls), effectively reversing the transfer reaction: pt(2s) + DX ->• [(dpt)*dee] -»• pd(ls) + TX, (1.7) where X = D or T. The excited molecule dpt* can decay into pt(ls) + d as well, but the decay width ratio is about 9 to 1 favoring the decay into pd(ls) + t, because the wave function of the latter channel is less orthogonal to, hence has a larger overlap with, the state dpt* [48]. The proposed side path model appears to give a better agreement with measurement of the cycling rates [49], but clearly more studies are desirable. Understand-ing of excited muonic atom processes is important also for QED and weak interaction experiments utilizing the possible meta-stable population of 2s muonic atoms [50]. We note that the side path model is unlikely to impact our measurements using pt emitted from layers, since the Ramsauer-Townsend effect is not expected for excited muonic atoms. Chapter 1. Introduction 12 1.2.5 Format ion of muonic molecules Non-resonant formation A straightforward process for muonic molecular formation is via the emission of an Auger electron: fix + YZ -»• [(xfiy)+ze] +e~. (1.8) The rate for this non-resonant process is typically of the order of 106 s _ 1 , hence one muon could catalyze not many more than one fusion, if this were the only process for formation. The most recent calculations of the non-resonant molecular formation rates were performed by Faifman for all the combinations of collisions [51]. Using the molecular bound state energy levels and wave functions derived from the improved two level ap-proximation in the Adiabatic Representation (Section 2.1.3), the rates were calculated taking into account monopole EO and dipole E l Auger transitions to all the bound states (earlier calculations considered only selected states). Note that the existence of weakly bound states, which is of paramount importance for resonant formation, also affects non-resonant formation. Resonant formation Observations were made in Dubna in the early 1960s for the temperature dependence of fusion yields in muon catalyzed dd fusion [52]; this was a strange phenomenon at the time since nuclear reactions of the MeV scale appeared to be affected by the target temperature which is of the meV scale. An Estonian graduate student, Vesman, proposed a resonant process for muonic molecular formation to explain the observations [53]. According to Chapter 1. Introduction 13 Reaction Collision energy (cms) [eV] 0.003 0.040 0.100 I. 78 0.22 2.57 5.54 0.15 II. 60 2.15 6.27 0.03 Table 1.2: Non-resonant muonic molecular formation rates in 106 s 1 by Faifman [51]. the Vesman mechanism, fix + YZViiKi -> [{xfiy)Sjvzee}Vf,Kr (1.9) where the kinetic energy of fix and the energy released upon formation of a loosely bound (xp,y)jv, with (J, v) the ro-vibrational quantum numbers, is absorbed in the rotational (K) and vibrational (y) excitation of the molecular complex [(xfiy)zee]*, a hydrogen like molecule with xfiy playing a role of one of the nuclei. Since the final states have discrete spectra corresponding to ro-vibrational levels (J//, KJ), the process takes place only when the collision energy matches the resonant condition. The temperature dependence is de-rived from the varying degree of overlap between the Maxwellian distribution (assuming fix is thermalized) of fix atoms with the resonance energy profile at different tempera-tures. For the Vesman mechanism to occur, the muonic molecule has to have a level with the binding energy smaller than the dissociation energy (~ 4.5 eV) of the host molecule. This is extremely loosely bound in comparison with the typical muonic energy scale of a few keV. Indeed such states have been shown to exist theoretically by Ponomarev and his co-workers for dfid and dfit molecules, with quantum numbers (J,v) = (1,1), where fip + p —> pfip fid + d —> dfid fit + t —> tfit fid + p —> pfid ftp + d —• pfid fit + d —dfit fid + t —> dfit fit + p —> pfit ftp + t —> put 1.81 1.80 0.03 0.10 2.71 2.64 5.69 5.63 0.17 0.16 0.56 6.07 2.26 2.21. 6.47 6.38 0.03 0.03 Chapter 1. Introduction 14 Table 1.3: Non-relativistic Coulomb molecular binding energies -Cjrv (eV). J, V ppp ppd pp,t dfid dfit tfit 0,0 253.15 221.55 213.84 325.07 319.14 362.91 0,1 35.84 34.83 83.77 1,0 107.27 97.50 99.13 226.68 232.47 289.14 1,1 1.97 0.66 45.21 2,0 86.45 102.65 172.65 3,0 48.70 J and v denote rotational and vibrational states, respectively. Muonic molecular ions are three-body systems interacting with both electromagnetic and strong forces, but to a good approximation (^ $10~4 eV), direct effects of the strong force can be neglected in calculating energy levels of loosely bound states, although other nuclear properties such as the charge form factors and polarizabilities play non-negligible roles. Now modern calculations have achieved amazing accuracies, as discussed in Section 2.1. Table 1.3 shows the nonrelativistic Coulomb bound state energy levels of muonic molecules. The molecular complex [(xfiy)dee]* is metastable, and can decay into several chan-nels. An Auger transition of the xpy molecule is an important step, which stabilizes the molecule, leading to fusion. This competes with the back decay process, which returns the excited molecular complex to fix and YZ. The details of the formation and back decay processes are discussed further in Section 2.2 and in Appendix B. 1.2.6 Nuclear fusion The rate of nuclear fusion in the dfit molecule is of the order of 1012 s _ 1 , hence it is too fast to measure experimentally. The reaction is strongly dominated by the resonance 5 H e J , 7 r = 2 + , which lies about 55 keV above fit + d threshold. Because of the centrifugal Chapter 1. Introduction 15 barrier of the J = 1 states of the molecule, fusion is expected to take place mainly from J = 0 states, i.e., (J,v) = (0,1) and (0,0), into which the initial state (1,1) is quickly converted via fast El Auger transitions. For dpd, on the other hand, the transition from J = 1 —* 0 is strongly suppressed (since it requires spin flip, analogous to the ortho-para transitions in the homonuclear hydrogen molecule), and fusion takes place mainly from the J = 1 states, if the molecule is resonantly formed in the (1,1) state. This in fact offers a unique opportunity to study the p wave fusion reaction at very low energy, which is difficult to do in a beam experiment. An interesting feature of this reaction is that the fusion width ratio of the (n+3He) channel to the (p + t) channel is about 1.4, apparently violating charge symmetry. In a simple approach, the fusion rate is calculated by treating three-body Coulomb physics (molecular wave functions) and nuclear physics separately [54], a method known as factorization. For dpt, we have where A is related to the zero energy limit of the astrophysical S factor, with Mdt being the reduced mass of d and t. S is obtained from the fusion cross section with Coulomb repulsion factored out, hence it contains all the nuclear physics. Thus in this approach, the overlap of the d — t wave functions in the united-atom limit is multiplied by a single number describing the strong interaction. More accurate approaches treat the nuclear force dynamically by directly incor-porating it as a complex potential in the three-body Hamiltonian (optical potential method) [55,56], or as complex boundary conditions at the nuclear surface (i?-matrix Chapter 1. Introduction 16 method) [57-59]. These elaborate approaches agree rather well with each other, as well as with the simple factorization approach. In fix + y collisions, fusion "in flight", i.e., without forming a bound x\iy molecule, recently, Faddeev equations (see [26] and references therein). The reactions are enhanced, compared to bare nuclear collisions, especially if virtual states exist, due to the increased overlap of the nuclear wave functions. 1.2.7 Sticking and stripping Sticking is a process in which the muon becomes attached to a positively charged fusion product such as an a particle. The probability of the process, denoted by u>s, imposes a more stringent limit on the efficiency of fiCF than the short lifetime of the muon itself, hence it has attracted much attention, but discrepancy between experiments and theory persists, motivating further investigation. Sticking takes place in two distinct steps: where fia* refers to the muonic helium ion in a bound state n, I recoiling with kinetic energy 3.46 MeV. The initial sticking is the intrinsic branching ratio of the sticking re-action channel immediately after the fusion, while stripping can occur during the slowing down of the fia, that is the muon is detached from the a with probability R, in collisions is also possible, and its rates have been calculated by various methods including, most 1-u;1 ,o Chapter 1. Introduction 17 1.2 g 1.0 3 0.8 75 0.4 0.2 0.0 D C ~\ i i 1 1 r initial sticking final sticking - 1 I — • PSI 87,88,90,93 * LAMPF 87,93 -0 KEK 87 • PSI Direct 93 -0.0 0.2 0.4 0.6 0.8 1.0 Density 0 1.2 1.4 1.6 Figure 1.3: Summary of experimental results of the final sticking LOS as a function of the target density <f>, plotted with theoretical predictions, where <f) = 1 is the liquid hydrogen density (4.25 x 1022 atoms/cm3). with the neighbouring atoms. Hence the final sticking UJS is: us=u°s(l-R). ; i . i2) The experimental results on sticking have been in disagreement with theory and, sometimes, with each other. The challenge lies in its low branching ratio and in the handling of tritium. Figure 1.3 summarizes the previous measurements of final sticking plotted against target density <j>. Apart from a longstanding discrepancy on the density dependence between two major experimental groups at PSI and LAMPF, in which the analysis of the latter now appears to be unreliable at low densities, most experimental values are systematically and substantially smaller than the theoretical prediction, even though different methods have been used. For a recent review, see Ref. [60]. Chapter 1. Introduction 18 1.3 Measurement of resonant molecular formation In this section, we shall describe the concept of our experiment using an atomic pt beam and discuss its expected advantages. But first, let us make a few remarks on the conventional experiments in D/T mixtures. 1.3.1 Cycling measurements in D / T mixture In conventional experiments [61-69], the molecular formation rates are determined from the measurements of the cycling rate A c, which is l / r c , where r c is the mean time between the 14 MeV neutrons from fusions catalyzed by the same muon. In this method, the cycling rates, measured at various target conditions (such as the tritium concentration c t, temperature T, and the density (f), are fitted to an expression describing the kinetics model. Molecular formation rates Xd t , together with quantities such as the d —> t transfer rate A^ and g l s , are the fitting parameters. A simplified expression for Ac reads [67]: 1 w cdqu | 3/4 | 1/4 + 3/4X! ^ A c ~ Ct\dt CtXW + Cd\XM CdX°dfltX2 Xi = , . u , (1-13) C t A i o + CdXdlit X2 = Atfit ct<f>Xtij,t + A/ M i where x i is the fraction of fit, F = 1 —> 0 transition with the spin flip rate A i 0 , and X 2 is a correction factor taking into account the tfit fusion time, with Xtllt and X{ t being tfit formation and fusion rates, respectively (note tfit formation is non-resonant, hence assumed to be independent of F). An improved formula would include a further correction term due to the muon recycling in dfid and tfit branches without producing the 14 MeV neutrons, which depends on parameters like dfid formation and spin flip Chapter 1. Introduction 19 rates. The molecular formation rate depends not only on the fit hyperfine state F, but also on the target molecular species. Thus, A^ t consists of different components CdX^t = CDXH-* + ^Xd,t-t + Cd*!,* f°r F=0,1, (1.14) which have to be disentangled in the fit. Often, assumptions have to be made in actual fits with guidance from theory. For example, at low temperatures all the terms in \ d t except A°M i_ d are assumed to be zero [22,67]. We shall come back to the implications of these assumptions later in this thesis. As Cohen points out [4], further complication in the conventional method comes from the lack of precise characterization of the parameter qls (the probability that the muon reaches the ground state of dfi before transferring to a triton). Its behavior suggested by the cycling fit is in large disagreement with theoretical expectations of its ct and <f> dependence. Independent measurements of q\ds for pd transfer are emerging [37], but no direct information is yet available for the dt case. Furthermore, the recent suggestion by Froelich and Wallenius of a resonant side-path in D/T cycling [44,49] (see Section 1.2.4) could have significant impact in the extraction of the dfit formation rate from the cycling rate. Finally, it should be noted that in these analyses using single rates, the energy de-pendences of the processes are averaged out. 1.3.2 Epithermal molecular formation Muonic molecule formation with epithermal (non-thermalized) fit atoms has attracted much recent interest because of the very large rate predicted by theory [70]. With the maximum rate approaching IO1 0 s _ 1 , this is nearly two orders of magnitude larger than Chapter 1. Introduction 20 12 10 s—N | CO 8 05* o 6 ^ — ' CO 0) 4 c d PH 2 0 /ut+D2->(d/ut)dee / x t + d s c a t t e r i n g 0.0 0.5 1.0 1.5. yu,t Lab energy (eV) 2.0 Figure 1.4: Resonant molecular formation rate in pt + Di collisions calculated for a 3 K target [71,72]. The rates are normalized to LHD and averaged over F. More detailed structure will be given in Fig. 2.4. Also shown is the pt elastic scattering rate on the d nucleus [17]. formation rates with pt in thermal equilibrium at room temperature (~ 108 s _ 1 ). Fig-ure 1.4 shows the recently calculated epithermal resonant molecular formation rate in pt + D2 collisions [71,72], normalized to liquid hydrogen density (LHD). Also plotted is the elastic scattering rate of pt on the d nucleus [17]. Epithermal molecular formation, in fact, was first suggested by Cohen and Leon [14], and independently by Kammel [13], to explain the fusion time spectra observed in the PSI experiment with low density gas [64], in which an unexpected prompt peak was ini-tially misinterpreted as due to the hyperfme effect6. Later Jeitler et al. [22, 74] carried 6 A qualitative remark of the possibility of epithermal formation was given earlier by Rafelski. [73]. Chapter 1. Introduction 21 out Monte Carlo studies in the homogeneous mixture of D / T 7 . They obtained quali-tative agreement between the experimental data and the simulations assuming strong epithermal resonances, but neither the strength nor the position the resonances could be determined by their analysis, because (a) the neutron time spectra are essentially insensi-tive to the position of epithermal resonances, and (b) the magnitude of the prompt peak is dependent both on the formation rate strength and the poorly known initial population of epithermal pt, hence with the lack of knowledge of the latter, the former cannot be extracted reliably. In order to access these resonances with thermal energies in a conventional target, one would require a temperature of several thousand degrees. The Dubna group has recently developed a target which can operate at up to 800 K, but this is still substantially smaller than the required temperature for the main resonances. Efforts are being made by the PSI/Vienna group to develop an ambitious target system for 2000 K, but the technical difficulties in handling tritium at such a high temperature have so far prevented the realization of such a system. In this thesis, we will develop an alternative approach [76-79] for investigating the epithermal molecular formation, as illustrated in Fig. 1.5. Taking advantage of the Ramsauser-Townsend effect, a neutral atomic beam of muonic tritium is obtained [80]. By placing a second interaction layer, separated by a drift distance in vacuum, reaction measurements are possible in an event-by-event fashion. This is in sharp contrast to conventional methods where a muon is stopped in a bulk gas or liquid target in which complex, and inevitably interconnected chains of reactions take place, as seen in Sec-tion 1.3.1. The use of the muonic atom beam would thus help us to isolate the process of interest from the rest of the "mess." Furthermore, the time of flight of pt across the drift distance provides information on the pt kinetic energy, hence the energy dependence of 7Markushin et al. also performed a Monte Carlo analysis of the triple mixture H / D / T [27,75] Chapter 1. Introduction the formation rates can be tested. 22 Figure 1.5: Conceptual drawing of the time-of-flight measurement of resonant molecular formation rates using the fit atomic beam. The Basic processes involved in our experiments are as follows [81,82]: P roduc t ion : Muonic atoms are produced by stopping low-momentum muons ( ~ 27 MeV/c) in a thin layer (~ 0.4 mm) of hydrogen with a small tritium concentration. The knowledge of the fraction of muons which stops in the hydrogen layer is of great importance for our measurements, and will be discussed in Section 6.3. Accelera t ion: The muon transfer reaction pp+t —> p+pt, taking place in ~ 100 ns [83], is used to create a fast pt with a kinetic energy of ~ 45 eV, which results from the reduced mass difference in the binding energy. Chapter 1. Introduction 23 E x t r a c t i o n : fit loses its energy v ia scattering mainly w i th protons, unt i l i t reaches around 10 eV, where due to the Ramsauer-Townsend (RT) effect, the fit + p elastic cross section drops by a few orders of magnitude, and the rest of the target becomes nearly transparent. Since there is no RT effect in the fip + p scattering 8 , fit is selectively extracted f rom the layer w i th the characteristic energy of ~ 10 eV. The understanding of the fit emission mechanism is essential to our measurements, and w i l l be discussed in Chapter 7. M o d e r a t i o n : The fit beam energy can be degraded, when necessary, by placing a mod-eration layer on top of the production layer. For the epithermal formation rate measurement, where the largest peak lies at around 0.5 eV, the beam energy of ~ 10 eV is indeed too high, hence a D2 overlayer was used as a moderator (not shown in Fig. 1.5). A th in layer (~ 5 fim) of D 2 is sufficient to slow down the fit, due to the large elastic cross section(see Fig. 1.2). Fusion events also occur at the upstream (US) moderation layer, which become background to the fusion at the downstream (DS) reaction layer, but because of their mostly prompt t ime structure, the US events can be separated f rom the DS signal. R e a c t i o n : The fit, after travell ing across the drif t distance, collides w i th a target D2 molecule. Resonant dfit formation can take place in two different manners: (a) the direct process, where dfit is formed directly before fit is scattered, and (b) the indirect process, in which fit first loses energy in collision w i th another D2 molecule, before molecular formation (see Fig. 1.4 for a comparison of the rates). The process (a) is most interesting because i t preserves the correlation between the fit t ime of flight (which is roughly equal to the t ime between the muon stop and the detection of a fusion product, due to short emission and fusion times) and the energy at which 8 T h i s is due in part to the existence of a virtual (nearly bound) state near ftp -f p threshold. Chapter 1. Introduction 24 molecular formation takes place. On the other hand, in the process (b), while the correlation is obscured (e.g., a fast fit with a short time-of-flight can slow down significantly in the reaction layer and form dfit at much lower energy), it can still give us useful information, such as the magnitude of the resonance. The use of a thin layer, together with a moderation of the pt energy mentioned above, reduces the fraction of the indirect process, thereby enhancing our sensitivity. There are, of course, difficulties and disadvantages involved in our approach. In addition to the obvious technical challenges in dealing with cryogenic and ultra-high vacuum targets with the potential hazard of tritium, there are difficulties and limitations which we did not foresee in advance (which are reflected in the volume of this thesis!). They will be discussed in due course. Before describing the details of our measurements and the analysis, we shall discuss some theoretical aspects of muon catalyzed fusion in the following chapter, with partic-ular emphasis on the muonic three body problem and molecular formation, in order to highlight the importance of the physics involved in our experiment. Chapter 2 Theoretical aspects of //CF 2.1 Muonic molecule and the three body problem 2.1.1 The challenge Accurate descriptions of muonic molecule properties are essential to an understanding of fiCF processes. The energy levels of the loosely bound state (J,v) = (1,1) affect the temperature dependence of molecular formation rates; since 1 meV ~ 10 K, it is desirable to achieve an accuracy of better than 1 meV for the energy level, which should be compared to the three body break up energy of muonic molecular ions of ~ 3 keV. In addition to the energy levels, the molecular formation matrix elements are sensitive to details of the dfit wave function, particularly in its asymptotic region, the formation rates being proportional to the square of a wave function parameter C (see Section 2.2). Furthermore, the accuracy of the three-body wave function is important for fia sticking as well. The convergence in variational calculations of the sticking probability us is much slower than that of the energy biding energy e; the general trend is that when e is converged to nc digits, us is accurate only to n s/3 significant figures [3,84]. Thus, understanding two of the most important processes in fiCF, i.e., molecular formation and sticking, demands the solution of the three-body problem to an accuracy of better than 10 - 7 , a challenging task to the theorists. Indeed, in 1993, Szalewicz, one of the experts in the field, asserted these calculations to be "some of the most demanding few-body 25 Chapter 2. Theoretical aspects of pCF 26 calculations, taxing the most powerful supercomputers" [85]. Because the three body problem is central to processes involved in this thesis, we shall review its theoretical framework in some detail in the sections that follow, focusing on understanding the underlying physical concepts. 2 . 1 . 2 Three body coulomb problem It is well known that the three body problem does not have a general analytical solution, and it has been the subject of considerable academic efforts since the 17th century1. The history as well as recent progress on the oldest three body problem, the Moon-Earth-Sun system have been reviewed by Gutzwiller [87]. The Coulomb three body problem, in particular, has been a difficult one to solve accurately, due in part to the long range nature of the interaction, in comparison to few nucleon systems in which the interaction is short-ranged. It is an active area of research, as indicated by a number of recent developments and refinements of theoretical techniques with the help of ever-increasing computing resources2. Efforts are being made to extend the calculations to full four-body muonic problems [92,93]. In general, the three body Hamiltonian can be written as 7V=3 N=3 H = T + V = J2 — A , + £ ViApi - Pj\), (2.1) where pi is the particle position vector [3]. The center of mass motion can be separated out, leaving a six dimensional problem in r i , v-z space. The exact form of the Hamiltonian depends on the choice of the co-ordinates r*i, r^. In terms of kinematics, there is no unique choice of the co-ordinates, hence choosing suitable ones, which will give accurate results, is one of the challenges that theorists face. 1It is said that there were over 800 papers published on the subject between 1750 and 1920s [86]. 2See recent calculations of muonic and other three-body systems, [36,88-91]. Chapter 2. Theoretical aspects of /iCF 27 2.1.3 Adiabatic approaches Born-Oppenheimer approximation In the conventional method for a molecular system, known as the Born-Oppenheimer approximation, nuclear motions are decoupled from electronic (or muonic) motions, i.e., the total wave functions are assumed to have the form V(r,R) = x(R)i>(r,R), (2.2) where ib(r, R), which is obtained from H°ip(r, R) = E°(R)i/>(r, R), (2.3) are the eigenfunctions for the Hamiltonian for fixed nuclei, ri°, with E° its eigenvalues. Here, the internuclear distance R is just a parameter. The nuclear wave function, x{R), is solved using E°(R) as the effective potential for the nuclear motion; X(R) = 0. (2.4) Born's Adiabatic method The Born method [94] extends the Born-Oppenheimer approximation by including cor-rections to the effective potential E°(R) in Eq. 2.4. The full Hamiltonian, separated in two parts, includes the term Ti' describing the relative motion of the two nuclei cis well as the coupling between the electronic (muonic) and the nuclear motions, in addition to the term 7i° for the fixed nuclear problem; H = H° + W. (2.5) Note that rt° includes nuclear repulsion in addition to the electronic interaction with the nuclei. The total wave function is expanded in the complete set of ipn(r, R): V(r,R) = Y,Xn(R)Mr,R), (2-6) n - ^ 2 R + E°(R)-E Chapter 2. Theoretical aspects of pCF 28 where ^>n(r, R) are the eigenstates of the fixed nuclei problem Eq. 2.3. The solution to the problem {'H? + 'H')V(r,R) = EV(r,R) (2.7) is obtained by substituting (2.6) into (2.5), multiplying the result by ip*(r,R), and inte-grating over r, which yields Xn(R) = -J2n'mn(R)Xm(R) (2.8) where « L = J Mr,R)H'Mr.R)dr. (2.9) Note that in (2.8), the diagonal term 7inn is moved to the left-hand side, hence the summation is for off-diagonal terms (ra ^ n). In the Born adiabatic approximation, the right hand side of (2.8), i.e. the coupling of different electronic states m, is neglected, leaving only the first term in the expansion in Eq. (2.6) $ = Xnfin- Thus, the equation for nuclear motion reads Xn = 0, (2.10) with the effective potential for a given electronic state n given by K(R) = E°n(R) + H'nn(R), (2-11) which includes the diagonal correction to the zero order potential, taking partly into account the coupling between the nuclear and electronic (muonic) motion. The Born-Oppenheimer approximation, on the other hand, neglects this coupling, and assumes U'n(R) = E°(R), as was seen in Eq. 2.4. The validity of the Born-Oppenheimer approximation is dependent on the smallness of the expansion parameter K ~ (m/M)1^4 where M is the reduced mass of the two nuclei, - - A R + E°n(R) + Kn-E - - A R + U^(R)-E Chapter 2. Theoretical aspects of fiCF 29 a R b Figure 2.1: Co-ordinate system for muonic molecular ion ab/j, used in the Adiabatic Representation method. and m the mass of electron or muon. The parameter K reflects the ratio of the amplitudes of the nuclear motion in comparison to the equilibrium internuclear distance, hence the assumption that K <C 1 allows one to neglect the term Ti' in (2.8), which contains derivatives of <j)(r,R) with respect to R [25]. Dependence of the Born approximation on (m/M) is less clear and somewhat qualitative, and the accuracy depends on specific characteristics of the problem including the strength of off-diagonal coupling in Eq. 2.8. In the case of the muonic molecule, m/M is not small. For its loosely bound states, which are our main interest, the vibrational amplitudes are not small compared to the internuclear distance. Hence both the Born-Oppenheimer and the Born Adiabatic ap-proximation fail to describe even qualitative characteristics of the muonic molecule, such as the number of the bound states [3]. As for loosely bound states, the former method gives a state bound much too deeply, while with the latter method the state is not bound at all [4]. Adiabatic Representation method The Dubna group, led by Ponomarev, undertook extensive efforts in the 1970s to solve Chapter 2. Theoretical aspects of fxCF 30 the muonic three body problem with high accuracy. Their method, known as the Adia-batic Representation method, is based on the expansion of the three-body wave function (Eq. 2.12) in the basis of solutions to the two-centre problem (Eq. 2.13) describing the muonic motion in the field of two fixed nuclei: ¥ ( r , i 2 ) = ^ Mr;R)^xAR), (2-12) where NT indicates summation over the discrete states and integration over continuum states, and r and R are given in the co-ordinates shown in Fig. 2.13. The basis <f>j is given by solving (in the units e = h = 1) 1 A 1 1 ' - A r R), (2.13) 2m a r a rb_ where the reduced mass is defined with respect to one of the nuclei Ma (Ma > Mi,): 1 1 1 , — = + TT> 2.14 ma Ma such that the energy e is measured from the ground state of an isolated atom u-d. The right hand side of Eq. 2.13 is replaced by \k2<f>(R; k, R) for the continuum spectrum. The equation for the nuclear wave functions Xj{R) i s given by an infinite set of coupled integro-difFerential4 equations: d2 dR2 + A - UU{R) Xl(R) = Ul3(R)X3{R). (2.15) where Uij are the effective potentials due to the kinetic energy of the nuclear motion and its coupling to the muon motion, and A are the eigenvalues, from which the binding energies are derived. Note that simplified notations given in Ref. [95] are shown here; for 3 Taking out the factor l/R from the definition of X(R) is a standard procedure which simplifies the form of the operator. integrat ion comes from the operator Uij. Chapter 2. Theoretical aspects of pCF 31 the complete formulae, see a detailed review [96]. Equation 2.15 is similar to the starting point of the Born approximation derivation (Eq. 2.8), but contrary to the approximation made in Eq. 2.10, the non-adiabacity is fully taken into account by the off-diagonal coupling terms Uij. Ponomarev and his colleagues overcame the computational challenges (and limited computer resources at the time) and solved Eq. 2.15 via systematic expansion in the finite set of the basis functions where the continuum spectra are discretized via: which gives a truncated system of coupled integro-differential equations with finite num-ber N. In contrast to the Born Adiabatic method, where accuracy was not well deter-mined, the convergence of the expansion can be tested, and the solution was obtained with a controlled accuracy [96]. With a few hundred basis functions, the solution con-verged, giving an accuracy of 10 - 1 eV [97]. Thus, they showed for the first time the existence of the loosely bound state for the dpd and dpt molecules. In fact, for the loosely bound states, contributions from off-diagonal terms from both discrete and con-tinuum states are significant [97], and this is why earlier adiabatic calculations failed to see such states. Scattering calculations with the Adiabatic Representation The Adiabatic Representation method can be applied to three body scattering problems. For scattering cross sections, the need for accuracy, from the point of view of pCF physics, is much less strict; the few percent level in cross sections is often sufficient (compared to the 10~7 level for the energy levels). (2.16) Chapter 2. Theoretical aspects of pCF 32 Effective two level approximation Bubak and Faifman performed the first compre-hensive calculations for all combinations (except for the asymmetric spin flip) of muonic atom scattering in a broad energy range [15]. They used an improved form of the two level approximation of the Adiabatic Representation method in which the first two states are used in the expansion of the three-body wave function in Eq. 2.12 (i.e., N=2 in Eq. 2.16), hence Eq. 2.15 reduced to the system of two equations coupled with off-diagonal effective potentials U12 and Ui\- In order to overcome the shortcomings of the Adiabatic Represen-tation method with a small (see Section 2.1.3), a correction was made to the reduced masses, giving an appropriate asymptotic behavior of the solutions [98]. A considerable reduction of computational complexity of using only two levels allowed crude, but very useful, estimates of various cross sections. They quoted accuracies of typically 10-20% for most cases. It should be noted that the effective two level approximation, when applied to the muonic three body bound state problem, correctly predicts the existence of loosely bound states, which conventional two level approaches have failed to find. Multi-level calculations An extensive effort was put together by Ponomarev and his colleagues to perform the task of truly multi-level calculations in the Adiabatic Represen-tation [99]. The results for scattering on a bare nucleus have been published in an atlas of cross sections, referred in this thesis as the "Nuclear Atlas" [16-18]. The accuracy for the purely three-body problems (i.e., atomic and molecular effects ignored) is claimed to be ~ 3% [24]. For a recent review of multi-level calculations and a comparison with other methods, see [100,101]. Improved Adiabatic methods The difficulties associated with the Adiabatic Representation method have been discussed by many authors (for example, see Ref. [102]). Ponomarev summarizes its disadvantages Chapter 2. Theoretical aspects of fiCF 33 in treating boundary conditions [103,104]: 1. Incorrect dissociation limit of the system (a,b,(i), when it decays into (a,fi) + b and a + (b, n). 2. Incorrect value of momentum in the reaction channels. 3. Infinite and long-range behavior of the non-diagonal potential Uij(R) as R —> oo. Cohen and his co-workers at Los Alamos used what they call the "Improved Adia-batic" approach [105] to address some of the above limitations. In the standard Adiabatic Representation method, the Hamiltonian was separated into two parts (Eq. 2.5); the zero order term describing the two centre Coulomb problem (H°TC), and the perturbative term for the relative motions of the nuclei as well as nuclear-muonic motion coupling {Ti'AR). In the asymmetric nuclear system (e.g., Ma > Mb), since infinite mass for both the nuclei is assumed in Ti°TC, the zeroth order solution cannot distinguish the channel afj, from bfi, into the former of which the ground muonic state (lsa) should dissociate in the limit R —> oo (recall the afi is more deeply bound than the bfi due to the reduced mass difference). The Improved Adiabatic approach instead partitions the Hamiltonian into where the mass polarization (i.e. nuclear-muonic motion coupling) term Timp and the internuclear orientation term Tiang, which were part of the perturbation Ti'AR in the Adiabatic Representation, are now included in the zero order Hamiltonian. The term Tiang oc L2R acting in the nuclear centre of mass frame (as opposed to the geometric centre of nuclei frame) breaks the symmetry, and causes the zero order muon wave function to Chapter 2. Theoretical aspects of pCF 34 move to the heavier nuclei as R —* 0 [105]. Therefore, the Improved Adiabatic method is expected to give more accurate results for the asymmetric nuclear case, compared to the standard Adiabatic Representation approximation, for the same number of basis functions N, hence achieving faster convergence. Of course, in the limit N —> oo, both approaches should give the same results. The two level approximation of the Improved Adiabatic method has been applied by the Los Alamos group to the bound state calcu-lations of H D + [105] and dpd [57], as well as in the scattering calculations [19—21], The latter cover all the possible combinations of muonic hydrogen isotope scattering, including the only calculation to date of the hyperfme transitions in asymmetric collisions. Despite the criticisms of the Adiabatic Representation, it should be noted that the effective two level approximation used by Bubak and Faifman (p. 31) also deals with the dissociation problem within the framework of the Adiabatic Representation, and appears to give a reasonable accuracy for only N = 2. 2.1 A Variational approaches More accurate results for the bound state energy levels of muonic molecules have been ob-tained with variational calculations. Szalewicz reviewed recent progress on the variational approaches [85]. In these approaches, the exact wave function ^ for the Hamiltonian TL is approximated by an expansion with a finite set of suitable basis functions The approximate energy levels E N , and approximate eigenfunctions C{tpi are found by diagonalizing an N x N matrix. The variational principle states N (2.18) i=l >E. ' 0 , (2.19) Chapter 2. Theoretical aspects of fiCF 35 where E0 is the ground state energy, hence, all the approximate energies En are upper bounds to the exact energies. The basis functions, also called trial functions, contain some non-linear parameters, which are to be optimized by repeating the diagonalization and finding En with a systematic variation of the parameters. The choice of the basis functions is a major factor which determines the accuracy and convergence of the calculations. Physical intuition and computational experience in choosing the functions can be rewarded with faster convergence and better accuracy. Hylleraas type basis expansion A popular choice are Hylleraas type functions [106-109], which for J = 0 states read <fin = (r^t)kn(rlid)ln(rdt)mn exp(-a n r^ - f3nr^d - lnrdt), (2.20) and are expressed in the interparticle co-ordinates, where rxy denotes the distance be-tween the particle x and y. The exponential parameters an,i3n,^n are often taken to be the same for all n to reduce the number of parameters to be optimized. For loosely bound states (1,1) of dfid and dfit, similar functions with k{ — U = m;, known as Slater gemials were found to be more useful in representing the large physical size of the states [84,110]. The disadvantage with Slater gemials is that one has to optimize a large number of ex-ponential parameters, which is very time consuming. Another disadvantage is its "linear dependencies" problem. This is due to the fact that the basis in this set is nearly linearly dependent, i.e. rather non-orthogonal. Since the functions differ only by their expo-nents, an optimized basis set sometimes has several functions with close values of the exponents [85]. Thus, the use of extended arithmetic precision (~ 30 decimal digits) is necessary. Chapter 2. Theoretical aspects of ftCF 36 Coupled rearrangement channel method A very appealing method in calculating three body problems, known as the non-adiabatic coupled rearrangement channel method [111], was developed by Kamimura. The method has been applied to many muonic and other three-body problems [44-48,112-115]. In his approach, the total three body-wave function is expanded in basis functions spanned over three rearrangement channels in the Jacobian co-ordinate system, which is illustrated in Fig. 2.2 for the case of the dfit system. The method treats the muon on an equal footing with the other two nuclei. All three channels are explicitly employed with each having its own importance; since the (J,i;)=(l,l) state lies only ~0.66 eV below the [pt]is-d (c = 1 in Fig 2.2) break up threshold, and ~48 eV below the [fid]is-t (c = 2) threshold5, the large components of the three body wave functions are expected to be in the c = 1 and c = 2 configurations. The [dt]-fi component (c = 3) is important for calculations of fusion and sticking where the details of the wave functions at small r 3 are needed. Gaussian gemials are used as basis set functions to describe radial dependences of the 5These should be compared to, for example, the [fii\2s,p~d threshold which is ~ 2 keV higher. Chapter 2. Theoretical aspects of pCF 37 wave function. For each channel c, the basis has the form = <j>i(rc)xi(Rc) = rc exp Rccexp Rc Ric i = l,n 1 = 1,N (2.21) where r,-c and Ric are the nonlinear parameters to be optimized, and lc, Lc stand for the relative angular momentum associated with co-ordinates rc,Rc respectively. The use of a Gaussian basis allowed straightforward analytical integrations in calculations of kinetic and potential energy matrix elements. Some detailed examples are given in Ref. [46]. A "faster damping" of the Gaussian tail, compared to the Hylleraas type functions with exponential tails, is not a problem even when representing a diffuse (1,1) state, since the parameters r; c and Ric can be made much larger than the muonic molecular size. In addition to the philosophical appeal of treating the muon on the same footing as the nuclei, as mentioned above, this method possesses some practical advantages, which include the following. First, because the basis spans three channels, linear dependencies among the basis functions is smaller, i.e. non-orthogonality is not too severe, compared to the single-channel basis function. This permitted calculations in double precision (64 bits, 14~16 decimal digits), which resulted in a very short computation time. Second, due to the explicit use of Jacobian co-ordinates with the rearrangement channels, the method is suitable for scattering calculations, where correct boundary conditions are satisfied (cf. recall the problems with the Adiabatic Representation). This feature is also useful in describing the dpt wave function in the calculation of molecular formation rates (see Section 2.2). The relative contributions of each channel can be tested with this method by removing one channel from the calculations, which proves the relative importance of the channels, in the order (c=l) > (c=2) > (c=3), as expected above, but also shows that to achieve the 10 - 3 eV level of accuracy in en, it is necessary to include c = 3. With the basis set Chapter 2. Theoretical aspects of pCF 38 of Nb ~ 2600, the final accuracy of about 2 x 10 5 eV was achieved, which includes the estimated uncertainty in extrapolation to Nb —> o o [111]. Among other approaches for the three-body Coulomb problem, two of them, which have received recent attention in pCF theory, are briefly discussed below. Hyperspherical approach Hyperspherical co-ordinates in the three body system are chosen in terms of one "hyperradius," which measures the size of the system, and the rest of the five variables as angles. In the simplest case, two independent vectors r\,T2 (see Section 2.1.2) are replaced by p=[rl + rl]*, a = tan- 1—, (2.22) where p is the hyperradius and the a hyperangle. Thus we have (ri, r 2) —> (p, fi), where = (a, 0i,4>i, 92, fa) denotes five angles with $i, fa being the spherical angles of the two vectors. A thorough review by Lin exists for this approach [116]. He argues that the hyperspherical approach provides a unified view of the three body problem regardless of the masses of the particles involved6. When applied with the adiabatic approximations, this approach gives faster convergence than the Adiabatic Representation [39,117,118]. Abramov et al. discuss the physical relationship between the two approaches [119]. The hyperspherical approaches have been recently applied to excited muon transfer [41], sticking in dpt [120], properties of dp3He molecules [90], and the calculation of dpt formation rates [121]. Faddeev equations Faddeev equations, which provide a rigorous approach to three-body problems, have been popularly used in nuclear few-body calculations [86], yet for 6 W e note that an alternative term Hyperradius approach has been suggested by Alexander Matveenko as a more general name. Chapter 2. Theoretical aspects of pCF 39 purely Coulombic systems, the method had been considered impractical until recently especially for scattering problems, due to the long-range nature of the interaction [122]. Kvitsinsky, Hu and their co-workers have solved some of the difficulties, improving upon a previously proposed modification to the Faddeev equations [123], and have calculated the bound state properties, e.g., ppp [124] and dpt [125], as well as scattering and fusion in flight cross sections [26,122]. The calculations for the non-relativistic point-like three body problem have reached impressive accuracies. Some selections from the results are summarized in Table 2.1. At this level of accuracy, the results are sensitive to the variations in the physical constants used in the calculation, such as the masses of the particles, particularly that of the triton [126]. Even a relatively small change in the masses can influence the binding energies as much as 0.5 meV. Table 2.1 is compiled using the constants given in Table 1.1 on page 7, in which I have made corrections when a different mass set was used in the calculations. 2.1.5 Non-Coulombic corrections At the level of accuracy required (< 1 meV), the effects beyond the Coulombic, nonrel-ativistic point-like description of the muonic molecules are important. Because of the heavy mass of the muon, muonic hydrogen atoms and molecules are sensitive to effects such as QED vacuum polarization and the nuclear charge form factor as well as hyperfine effects. However, it is the difference between the corrections to the atomic and molecular energy levels which affects the binding energy ejv, AeJv = AEJ°l - AE?som. (2.23) Chapter 2. Theoretical aspects of pCF 40 Table 2.1: Comparison of the binding energy for the (J,v) = (1,1) state of dpt, —e"[ (in meV). Authors Year Method N fcll Vinitskii et al. [97,127] 1978-80 Adiabatic 264 640 Gocheva et al. [128] 1985 Adiabatic 884 656 Bhatia and Drachman [106] 1984 Hylleraas 440 224 Frolov and Efros [129,130] 1984-85 Slater 400 607.2 Hu [107] 1985 Hylleraas 500 628 Hu [108] 1987 Hylleraas 1102 658.03 Korobov et al. [131] 1987 Elliptic 2084 659.68 Szalewicz et al. [109] 1987 Hylleraas 3063 660.04a Kamimura et al. [Ill] 1988 Jacobian 2662 660.104 (30) Alexander and Monkhorst [110] 1988 Slater 2000 660.1721 Haywood et al. [84] 1991 Slater 2600 660.17786 (10) a Taken from Ref. [85], which is the value normalized to the set of masses given in Table 1.1. 'Published value shifted by 0.0252 meV to account for the different masses used. Because the loosely bound (dpt)u states resemble a pt atom and a weakly bound deuteron orbiting around it, the corrections AEJ°l and A E " * o m are similar and partly cancel each other, hence the corresponding corrections Aejv are suppressed by a factor of about ~ 102 [132]. These corrections (without hyperfine effects) amount to about a —30 meV shift (towards deeper binding) in the dpt binding energy e n [4]. Hyperfine structure of the pt and dpt further introduces corrections for F — 0 of +35.2 meV (S = 0), and for F = 1 o f -8.3 meV (S = 0), -14.9 meV (S = 1), and -8.5 meV (S = 2), where F is the total spin of pt and S, that of dpt. 2.2 Resonant molecular formation In this section we shall discuss the theory of resonant muonic molecular formation. Fo-cusing on the pt + D2 case, which is the subject of this thesis, the standard theory based on the Vesman model is treated first with a particular emphasis on the work by Chapter 2. Theoretical aspects of pCF 41 Faifman et al.[70-72,133,134], followed by the extensions to the standard theory. 2.2.1 The standard Vesman model As mentioned earlier, resonant molecular formation refers to the process: {pt)F + [D2\VitKi -> \(dpt)Sjvdee)Vj<Kj, (2.24) where the kinetic energy of pt and the energy released upon formation of dpt is absorbed in the rotational (K) and vibrational (v) excitation of the molecular complex [(dpt)dee]*, a hydrogen like molecule with dpt playing the role of one of the nuclei. The process is resonant in nature because of the discrete spectra in the final states, corresponding mainly to the ro-vibrational levels (uf,Kf)7, and the collision energy has to match the resonant condition. The resonance condition can be written, when considering purely two body collisions (i.e., neglecting three-body or phonon effects): eres[pt + D2] = t^[dpt) + E„jKj[(dpt)dee] - E^Di). (2.25) Figure 2.3 gives a schematic energy level diagram for the resonant dpt formation process. Plotted on the left is the potential curve of dpt showing the "shallow" (in the muonic scale) bound state, while on the right the molecular complex energy levels are plotted. Note that due to the reduced mass difference, the energy levels of D 2 and [(dpt)dee\ are different (by 33.7 meV for the ground states according to Faifman et al. [138]). In fact it is this difference in reduced mass which introduces the dependence of the dpt formation rates on the target molecular species, i.e., pt + D2 versus pt + HD or pt -f DT. 7 There are recent studies on the [(dpt)dee] energy level splitting (of a few meV level) due to the dpt interactions with the host molecular complex including the quadrupole finite-size corrections [135-137]. This would greatly increase the number of levels in the final state spectra (hence further complicating the notation!), but it is neglected here. Due to the Doppler effect the resonance profiles in the pt lab frame are already broad (see the discussion later in this section), so presumably these splittings would not have much effect in our measurement of epithermal molecular formation, though more care may be necessary at low energy. Chapter 2. Theoretical aspects of fiCF 42 tu. + D 2 Figure 2.3: Schematic energy diagram for resonant molecular formation fit + [D2]^ r;=o,i/,=o —> [(dfit)dee]Kf,uf, from Ref. [4]. The energy released in the formation of dfit is absorbed by the ro-vibrational excitation of the host molecular complex. Under ordinary target conditions, only V{ = 0 is populated. On the other hand, the initial rotational population depends on the target preparation procedure a,s well as on temperature. At 3 K, an equilibrated target would have nearly 100% K{ = 0 population (ortho deuterium). Our targets, however, were prepared by rapidly freezing the statistically populated deuterium (67% K{ = 0 and 33% Kj = 1) from a hot palladium filter, and because the rotational relaxation is very slow in the absence of a catalyst, we expect this population will last for the entire measurement time. The initial ortho-para Chapter 2. Theoretical aspects of pCF 43 population is relevant, because angular momentum conservation requires: L + K i = J + K f , (2.26) where L is the relative angular momentum in the reaction (2.24). Since the dpt rotational angular momentum J is 1, we have Kf = K{±1 for low energy collisions with L = 0 [139]. However, at epithermal energies L > 0 becomes important, allowing various Kf states. This is one of the reasons why we expect such a high rate at epithermal energies. Formulation of the formation process Calculation of the resonant molecular formation rates is difficult since it is a rearrange-ment process involving six bodies. Several different groups have published the calcu-lations [45,70,71,127,133,139-146], but each work is often criticized by other groups, which creates for the experimentalists a situation that is confusing (and sometimes frus-trating!). For the conventional homogeneous target experiments, the resonant molecular forma-tion rate \mf as a function of the target temperature, is written as [4,147]: A m ' (T) = NJ2 f de2n\ < i\H'\f > \2f(e,T)I(e~tlf,T), (2.27) where N is the target density, < i\H'\f > is the matrix element for molecular formation, f(e, T) is the kinetic energy distribution in the center of mass for the collision at the temperature T (e.g., the Maxwellian distribution if fit is thermalized), e;/ is the resonant energy, and I(A,T) is the resonance intensity profile. The popular choice of the interaction operator has been: H' = d • E, (2.28) where d is the dipole operator of dfit and E is the electric field at the center of mass of the dpt due both to the spectator nucleus and electrons in the molecular complex. Chapter 2. Theoretical aspects of pCF 44 The importance of including the field of the electrons was pointed out by Cohen and Martin [142] and independently by Menshikov and Faifman [143]; it screens the nuclear field, substantially reducing the matrix elements. The use of the dipole operator in H' was criticized by Petrov et al [144-146] and by Scrinzi [148], who each proposed alternative forms of the operators, but the accuracies of simplifications adopted were in turn questioned by Faifman et al. [71,134]. Faifman and his colleagues recently included the quadrupole corrections in the interaction operator in Ref. [71], and those are what we used in our analysis of the experimental data. We note that the perturbative formulation used by Faifman as well as by Petrov and by Scrinzi were criticized by Lane [141] and Wallenius [45,46]. Lane [141] also criticized the formulation of earlier work by Cohen and Leon [14]. In fact, Wallenius in his thesis goes as far as calling Menshikov and Faifman's justification for the Born approximation "nonsense." Wallenius concludes, however, that different "working formulae" in the literature actually end up to be equivalent (presumably due to cancellation of what he calls mistakes), and would yield the same cross sections, if no further approximations were made [46]. Thus theorists seem to agree, though maybe for different reasons, at least on the form of the starting formula they use to calculate the matrix elements. For actual evaluations, however, further approximations are necessary in the interaction operators, such as the multipole expansion as mentioned above, or in the wave functions as discussed next. Wave functions The initial and final wave functions \P,-,\I>/ in the molecular formation matrix element caculation are usually approximated as: *i = <(r)*%Ki{T1,r2,p2)J»-', (2.29) Chapter 2. Theoretical aspects of pCF 45 where ^^(r) is the pt ground state wave function with r being the interparticle vector from t to p, and ^ ^ - . ( r i , r 2 , p?) is the initial D 2 wave function with internuclear vector p2 and electron vectors r x , r 2 . The factor e t p r describes the relative motion of the centre of mass of the pt and D 2 . The final wave function is written: tf, = # ( r , R ) ^ ( r ^ r , , , ) , (2.30) where tpff(r, R) is the (dpt)lx wave function, and ^ ^ A ^ (ri> r 2 , p) is the final state molec-ular complex wave function where dpt is replaced by a fictitious particle of mass and charge equal to the sum of dpt and located at the centre of mass of dpt. Following Menshikov [149], most authors use an analytical approximation of ipf^(r, R) with its asymptotic form valid in the limit of infinite pt + d separation, since the main contribution to the matrix element integral comes from the region where the pt+d separa-tion is large. The asymptotic wave function is characterized by a normalization constant C [149,150]. Recently Kino et al. [114] obtained a new value of C using Kamimura's variational wave function (Section 2.1.4, page 36), calculated with the Gaussian basis functions in the Jacobi co-ordinate. As we have discussed, Kamimura's wave function treats the break up channel into pt + d directly, hence it gives an accurate description of asymptotic behaviour, while overcoming the difficulty of the Gaussian basis ("fast dumping") by using a sufficiently large number of basis functions. The formation rates are proportional to the square of the constant C, and its new value decreased the pre-dicted formation rates by 14% compared to an earlier calculation in Ref. [149] using the Adiabatic Representation, but increased them by 33% compared to Ref. [150] using a variational wave function with the Slater-type basis functions [110]. It is interesting to note that the Slater-type basis method, which was used so successfully in the energy level calculations (see Table 2.1), gives a less accurate description (assuming Kino et al. are Chapter 2. Theoretical aspects of JJLCF 46 correct) of the asymptotic wave function compared even to the Adiabatic Representa-tion method, perhaps illustrating the difficulty in achieving a unified description of the non-adiabatic three body problem. In our analysis, the new value of C by Kino et al. is used. Decay of the molecular complex Once the [(dfit)dee] complex is formed, it can either stabilize itself, which leads to fusion, or go through a back decay [151] into the entrance channel fit + D2: The back decay width T^K ,K, is determined by the same matrix element as the forma-tion, though it should be noted that, in general, v[K[ ^ V{K{ (see footnote8). lower state, (3) Auger deexcitation of (dfit)n, and (4) collisional deexciation of the com-plex due to interaction with the surrounding environment. Because of the centrifugal barrier, fusion from J = 1 states is relatively slow (< 108 s - 1 ; see Ref. [3] for a recent summary), and Lane estimates the radiative decay of dfit to be even slower (< 106 s _ 1 [151]). Hence A/ is dominated by the Auger deexcitation rate, at least at low densities. This rate has been calculated by several authors with an increasing degrees of sophistica-tion, most recently by Armour et al. [152], who took into account the molecular nature of 8 T h i s possibility of what I call the "resonant excitation" of a D2 molecule by fit is not apparent in the papers of Faifman et al. [71,133,134]. In fact, we point out that for kinematic reasons, this occurs only for Vf > 4 for most Kf (for all Kf, if the Kf distribution is relaxed to low temperature equilibrium). [(dfit)sJudee]1/jK} (2.31) (2.32) Chapter 2. Theoretical aspects of pCF 47 the host complex, and estimated the rates for (J, v) = (1,1) —> (0,1) to be in the range (6.9-10.3) x l O 1 1 s - 1 , depending on the model used. The collisional rotational de-exciation of the complex was first calculated by Ostro-vskii and Ustimov [153], and later by Padial, Cohen and Walker [154]. The latter, who claim better accuracy but still neglect the ro-vibrational transitions in the target (i.e., the surrounding) molecule, found the rates substantially smaller than the former (by a factor of two to ten depending on the transition). To our knowledge, there is no accurate calculation of collisional vibrational quenching, except a rough estimate by Lane of ~ 107 s~l at room temperature [151,155], which is much slower than other processes. However, he claims this rate increases drastically with target temperature (for equilibrated targets) and increasing v [155], rising to the order of 1010 s _ 1 at 2000 K, therefore it may compete with Auger decay for the molecular formation at epithermal energies with high Vf (see footnote9). In Faifman's calculations used in our analysis [71,72], the rotational relaxation rate of ^KfK'} — 10 1 3^ s _ 1 , with (j) being the target density in units of LHD (liquid hydrogen density), and the effective fusion rate of A/ = 1.27 x 1012 s _ 1 [156], were used. The latter is the sum of the dipole E l Auger transition from the (J,v) — (1,1) state to (0,1), (2,0), or (0,0) with each rate being 11.4, 1.3 and 0.02 (xlO 1 1 s _ 1), respectively. Total and effective formation rates For our analysis, it is convenient to express the formation rates as a function of the fit laboratory (lab) energy, as opposed to the target temperature given in Eq. 2.27. 9 Padial , Cohen and Walker promised to extend their calculations to vibrational transitions [154], and we await their results eagerly. Chapter 2. Theoretical aspects of fiCF 48 Specifying the initial and final states, we then have FS (2.33) v;SK, where (2.34) A 5 F = 27rN7(El;t\eres)Aif\<i\H'\f> (2.35) uiK'i,ujKj where is the initial rotational population distribution, which for equilibrated targets is the standard Boltzmann distribution, and A{/ is a coefficient which depends on initial and final quantum numbers. The factor "f(E, tres) is the Doppler broadening profile due to target motion and recoil, whose exact form is derived in Ref. [133], but for E1^ » kT can be approximated [157] as a Gaussian distribution with the width of: In Eq. 2.35, a 6 function resonance profile was assumed (the classical Vesman model), but even with that, the formation rate in the lab frame has a distribution with non-zero width due to the above-mentioned Doppler broadening. At epithermal energies, the transition formation matrix elements become very large, which means both the formation rate and back decay width are large, resulting in a significant probability for dfit not fusing but returning to the entrance channel fit + 732- The effective formation rate is a renormalized rate taking into account the fusion probability, as defined in Ref. [134]: 1 0 W e sometimes refer to these rates as the total formation rates, as opposed to the effective rates described below. (2.36) (2.37) Chapter 2. Theoretical aspects of pCF 49 where K&f) = l +VSF > (2-38) ^ffh'j = E rf^- f l / , A-;. (2.39) For a high density (^ > 0.1) target such as ours, Faifman assumes complete rotational relaxation of the Kj levels, hence dropping the Kf dependence, •Aj* = E ^ f ^ V (2-4°) Wff = ^ - ^ , (2.41) r"f = E ^ / ) 1 ^ ' (2-42) where UJB(KJ) is the Boltzmann distribution of the Kj states11. The effective fusion probability for the molecular complex is defined as [133]: w F = ( 2 .43) where A ^ t is the rate for non-resonant formation, which does not back-decay. For XF t >> ^Tfiti WF can be written: E K'MKK, WF = , F , (2.44) vj,S,Kf which can be understood as the average of fusion probabilities from each state weighted by the formation rate of that state. Figure 2.4 shows the molecular formation rates XF^t(E) and effective fusion probabil-ities WF for Ki = 0 (ortho) and Ki = 1 (para) cases, calculated by Faifman et al. [70-72] 1 1 In Ref. [71,134], the dependence on 5 in the fusion probabilities W?F(Kf) (Eq. 2.38) and W?F (Eq. 2.41) is not explicitly indicated, but they do indeed depend on S. Chapter 2. Theoretical aspects of fiCF 50 1.5 2.0 0.0 0.5 /at Lab energy (eV) Figure 2.4: Formation rates A f^ for pt+D2 —> [(«?ju£)c?ee]* (top) and the fusion probabil i ty W F (bo t tom) , calculated by Faifman [70-72] for 3 K. Also shown in dashed lines are the effective rate \ F ^ t ~ . The rates are normalized to l iquid hydrogen density. for a 3 K target. Also shown are the effective rates \F^t m WFXF)it. Note that A ^ t < 10 7 s _ 1 is invisible in the scale plotted. Chapter 2. Theoretical aspects of fiCF 51 2.2.2 Subthreshold resonances Rather constant behaviour of the formation rate \mf(T) is observed by experiments, especially at low temperature [62,65], together with an unexpected density dependence of the formation rate [63]. This led theorists to consider extensions of the classical Vesman mechanism of resonant molecular formation which assumed an isolated two-body collision and the 8 function resonance profile, as adopted in Refs. [71,72,133]. Recall that the Vesman model was very successful in the dfid case [12]. In the dfit case, however, strong resonance levels are expected to exist just below the fit + d threshold, e.g., K{ -> Kf = 0 1 at -14.0 meV, 0 -> 2 at -4.3 meV, and 1 -»• 2 at -11.7 meV for F = 0 [4]. These transitions have large matrix elements because of the strong overlap of the wave functions. Two main mechanisms to access these subthreshold (i.e., negative energy) resonances are: (1) intrinsic resonance width due to the finite lifetime of the molecular complex (mainly the Auger decay width), and (2) three-body or many-body collisions in which the other body (or bodies) absorbs the excess energy. The mechanism (1) is density independent, but (2) depends on the surrounding environment. There have been many attempts to treat the subthreshold resonances [144,158-163], however, complete understanding has not yet been achieved. For example, the use of the Breit-Wigner profile adopted in Ref. [158] was criticized in Ref. [160,161] at least for the high density situation. Despite these criticisms, Petrov's calculation for the formation rate for Aj^° at low temperature and low density [145,146] seems to agree with the value suggested by the PSI measurement (130 ± 20 fis) [22]. Armour recently proposed a new approach beyond the Born approximation [164,165], which also predicts non-zero width resonance profiles, but its application is still limited [121] and comparison with experiment is not yet possible. Chapter 2. Theoretical aspects of fiCF 52 We note, however, that for epithermal molecular formation, the resonant widths con-sidered in these models are smaller than the Doppler broadening; thus the effects are negligible. 2.2.3 Condensed matter effects An important advance was the observation of a striking condensed matter effect in solid D 2 [77,166], where we measured an unexpectedly high dfid formation rate. This stimu-lated the theoretical efforts in fiCF to be extended to solid state physics. In addition to the study of thermalization processes [167], investigation of molecular formation in solids has been started by several authors [167-170]. Condensed matter effects, important mainly at collision energies comparable to or below the Debye temperature of hydrogen (JCQB ~ 10 meV), include the reduced mass effect which shifts the resonance energies in the lab frame, and phonon assisted resonant molecular formation. The latter is similar to the three-body collision mentioned in the previous section, but in this case it is one or more phonons which carry away the excess energy. Fukushima made the first and so far only calculation of dfit formation in solid hydro-gen [168], but he considered only metallic hydrogen targets at extremely high pressure in order to avoid the quantum crystal nature (involving large non-harmonic lattice vibra-tions) of solid hydrogen at zero pressure. Therefore its applicability to our experiments is questionable even at an order of magnitude level. Menshikov and Filchenkov claimed that our measured dfid formation rate [77,166] could be explained by the phonon as-sisted resonant formation alone, but again because of the lack of proper treatment of the phonon spectrum, the reliability of their calculation is rather unclear 1 2. Adamczak 1 2 Note that their conclusion contradicts at least partly Adamczak's slow thermalization model [167], so they cannot both be right. Chapter 2. Theoretical aspects of pCF 53 has recently reported the first realistic calculations of resonant molecular formation in a solid for the dpd case [170], extending his work on muonic atom thermalization in solid targets [167]. Unfortunately, there are no realistic calculations for dpt formation in the solid state available to date. We further note that the role of condensed matter effects in the ro-vibrational relaxation of the molecular complex (perhaps involving rotons or vibrons), which affects the fusion probability WF, is an open question. Finally, let us note that the use of renormalized effective rates XF t (Eq. 2.37-2.42), which takes into account the fusion probability, needs very careful consideration as to its applicability. In the case of epithermal molecular formation, the use of the renormalized rates in the Monte Carlo calculations, as was done in Refs. [27,82] would result in a significant overestimate of the fusion yield [171]. We shall give a detailed discussion on the use of effective rates in Appendix B (Section B.l) . I shall come back to some of the theoretical details when we discuss our results. In the chapters that follow, we shall see what we can contribute to the understanding of the rich physics described in this chapter, involving resonant molecular formation as well as the muonic few body problem. We shall start in the next chapter with a description of our experimental apparatus, which makes these measurements possible. Chapter 3 Experimental Apparatus The experiment described here has been conducted at the M20B channel of TRIUMF (Canada's national meson facility) located on the University of British Columbia campus in Vancouver, Canada. A novel target system for films of solid hydrogen isotopes and other gases was developed, and detectors which take advantage of the thin film targets have been utilized. In this chapter, we describe the muon beam, the cryogenic target system including target characterization, and detection systems. 3.1 Muon beam TRIUMF is one of the two remaining meson factories in the world, providing intense continuous beams of protons, pions and muons. The TRIUMF accelerator is a six-sector focusing cyclotron, which accelerates H~ ions. By stripping the two electrons with a thin foil, protons are extracted at energies between 183 and 520 MeV with an intensity up to 140 fiA. The primary proton beam usually has an RF time structure of 23 MHz 1 . The cyclotron and the beam lines are shown in Fig. 3.1. Two main beam lines are BL4, primarily for proton experiments, and BL1, mainly for pion and muon experiments. In addition, BL2A will be used for the ISAC radioactive nuclear beam facility now under construction. xIt is, however, possible to change this structure. For a measurement of pion lifetime, for example, the beam cycle was reduced by a factor of five [172]. 54 Chapter 3. Experimental Apparatus Figure 3.1: Schematic site drawing of T R I U M F facilities and beam lines. Chapter 3. Experimental Apparatus 56 The muon beam at the M20B channel is obtained from primary protons striking the meson production target 1AT2, typically water cooled beryllium 10 cm thickness and 5 mm x 15 mm cross sectional area (Fig. 3.2). A cloud of pions produced near the surface of 1AT2 decays into muons and neutrinos. For our experiment, negative muons of momentum about 27 MeV/c were selected by the momentum analyzing magnet (Bl). The electron separator, a crossed electric and magnetic field, selected muons, improving the muon to electron ratio in the beam by more than two orders of magnitude. Final focusing of the beam was done by a set of quadrupole magnets, resulting in a beam size of about 30 mm (FWHM) with a rate of order 5 x 103 s_1. Focus 1AT2 Target Figure 3.2: Layout of the M20B muon channel. Chapter 3. Experimental Apparatus 57 3.2 Cryogenic target system A cryogenic target system for solid hydrogen thin films, developed and improved over several years at TRIUMF, was used in our Muonic Hydrogen research program. It is this target system which allows us to perform unique experiments including the one described in this thesis. i The TRIUMF target system is capable of making two separate solid layers of hydrogen isotopes (protium 1 H 2 , deuterium, tritium and their mixtures), and other gases such as neon, which face each other. The thicknesses range from a few jUg-cm - 2 (~1 /.tm for protium) to a few mg-cm - 2, and can be controlled to better than 1% relative accuracy. The system consisted roughly of a gas mixing system, a gas deposition system, a cryogenic system, a vacuum system and a tritium safety system. The details of the target system are given in Refs. [173,174]. A general description of each part is given below. 3.2.1 Gas handling and mixing Hydrogen isotope mixtures and other gases were prepared by a gas handling system, and introduced to the experimental vacuum space (EVS). The topology of the EVS and gas handling systems is shown in Fig. 3.3. Highly isotopically pure protium, generated by electro-decomposition of deuterium depleted water, was passed through a palladium filter heated to a few hundred degrees Celsius, providing protium gas with an impurity level of a few ppm or less, necessary be-cause of possible muon transfer to heavier nuclei. Similarly, research grade deuterium gas (purity > 99.99%) was passed through another palladium filter to remove non-hydrogenic impurities. A small amount of protium contamination in deuterium could be tolerated, Chapter 3. Experimental Apparatus 58 beam line IOEVS n -Experimental vacuum space U-bed 1 •torage device Mixing Waite volume •04 Figure 3.3: Topology of important parts of the experimental vacuum and the gas handling systems. The gases were purified and mixed in the gas handling system on the right, then transferred to the experimental vacuum system where they were condensed. After measurements were completed, tritiated targets were pumped into a uranium storage bed, while subsequent pumping was provided by a turbo molecular pump (TMP), backed by a spiral pump (SBP) and a bellows pump (MDP). Important valves (VAs) were air-controlled, and various gauges (CGs and IG) monitored pressures. since muon transfer to protons from other nuclei is kinematically forbidden at the ener-gies at which we are working. A total inventory of 200 Ci (22.5 mg) of tritium gas2 was supplied by Ontario Hydro, and stored in a 5 g uranium tritium getter bed. By heating the tritium bed to the temperature 200-400° C, the gas was released into a buffer volume. 21 c m 3 of tritium at 1 bar and 273 K contains 2.38 Ci of activity. Chapter 3. Experimental Apparatus 59 When a mixture of gases was used, its isotopic concentrations were determined by the volume of the buffer and its pressure. The gases were mixed in a volume which was connected to the experimental vacuum region via gas transfer lines. The gas mixing took place after each gas was passed through the palladium filter, hence a protium-tritium mixture, for example, is expected to be a mixture of H 2 and T 2 , but not HT molecules. Before starting a gas deposition, we waited typically for a few hours so that gas mixing was complete. Gas composition was monitored also by a residual gas analyzer (RGA), sometimes called a quadrupole mass spectrometer. 3 . 2 . 2 G a s d e p o s i t i o n In the experimental vacuum space, the gases were injected onto a cold substrate via a gas diffusion mechanism called a diffuser. The details of the diffuser are shown in Fig. 3.4. By independently diffusing gas on each side of the diffuser, the system was capable of making two films which faced each other. Different gas deposition systems have been developed to incorporate experimental demands, and to improve the performance [173,174]. For system 1, the original design, the gas was released through a diffuser made of a thin stainless steel foil perforated with many holes (~0.2 mm diameter). System 2 used the same diffuser foil, but as a result of gas line modifications to incorporate tritium compatibility, it had gas inlet tubing with a higher volume, which acted as an unwanted buffer volume as we shall see in section 3.3.2. System 3 is the latest version with the gas inlet lines replaced with low volume, high conductance tubing, and a sintered metal diffuser (2 /mi porosity) employed to ensure microscopic homogeneity of deposition. For the present work, systems 2 and 3 were used for Run Series I and II, respectively. The target substrates were made of 50 / i m thick gold foil, and cooled to about 3 K Chapter 3. Experimental Apparatus 60 with a helium flow cryostat. The cryostat, designed by a company specializing in custom refrigerators3, was based on a standard unit4. The helium flow line was extended to suppress the thermal conductivity between the mounting flange and the cold stage, and was surrounded by a buffer space (Tritium Barrier Space, TBS) for safety purposes as discussed below. The heat shield was cooled below 100 K, and the diffuser kept below 150 K to re-duce radiation heating of the target. Two flexible copper braids (detail K in Fig. 3.4) connected the diffuser and the radiation shield to provide a flexible thermal conduction path, allowing the diffuser to move vertically. The temperatures of different parts of the cryogenic system were monitored. Constant current silicon diode sensors permitted wide ranges of temperature readings at heat shields and diffuser. The coldest part of the cryostat, on the other hand, was measured by carbon glass resistors which provided a sensitivity necessary for fine control for helium flow optimization and depositions. 3.2.3 Experimental vacuum space Figure 3.5 illustrates the central part of experimental vacuum space (EVS). The diffuser (detail L) moved vertically into the space between the upstream (US) and downstream (DS) target supports foils (detail F), and was retracted during the muon beam measure-ments. The muon beam, defined by a 250 / m i plastic scintillator (TI), entered from the front port through a thin (25 / im) stainless steel window, which isolated the high-vacuum target region from the lower vacuum of the beam line and muon production target area. This also prevented tritium contamination of the rest of the beam lines. The beam then went through the upstream heat shield (consisting of a pair of 6 pm gold plated copper 3 Quantum Technology Co., 1370, Alpha Lake Road, Unit 15, Whistler, B . C . Canada. 4SuperTran, Janis Research Company, Inc. Wilmington M A , 01887-0696. Chapter 3. Experimental Apparatus 61 foil) and the US target support (50 pm gold foil), before stopping in the hydrogen layers. Almost all muons which penetrated the hydrogen layers stopped in the DS target support foil (also 50 pm gold), and did not reach the downstream heat shield (1.6 mm gold plated copper plate). Different kinds of detectors were placed at the remaining ports of the EVS (see Sec-tion 3.4). When the MWPCs were used, a thin side window (50 pm) was necessary to reduce multiple scattering of electrons. For neutron and gamma detection, thicker side windows could be tolerated. Silicon detectors were placed in vacuum, held on the heat shield, viewing directly the target without any window in between. This turned out to be a powerful feature of our experiment, allowing the detection of low energy charged particles with a high resolution. The distance between the US and DS gold foils (detail F) was adjustable, and was 42.5 mm for Run Series I, and 17.9 mm for Run series II. 3.2.4 Vacuum system The experimental vacuum space was pumped with a turbo-molecular pump5, backed by a spiral pump and a bellows pump, all of which were free of oil and elastomers. Most of the other vacuum components were made also with elastomer free materials, in order to reduce activation by the tritium6. The system typically achieved a 10 - 7 torr level of vacuum at room temperature, with the main residual being water vapor and hydrogen; when cooled to 3 K, cryopumping reduced the pressure to about 10 - 9 torr. Figure 3.6 illustrates the vacuum system. 5 TurboVac 340M, Leybold Vacuum Products Inc. D5000 Cologne 51, Germany. 6 T h e contamination takes place through exchange reactions of tritium with any compound containing hydrogen. Chapter 3. Experimental Apparatus 62 Figure 3.4: A view of the diffuser system showing the central diffuser support (L), gas supply lines (M), diffuser chamber (N), mechanical support (0), bellows and vertical motion guide shafts (P, Q), and diffuser cooling copper braid connected the heat sheild (K). The figure only shows one side of the diffuser; the gas can be deposited from the other side as well. Chapter 3. Experimental Apparatus 63 Figure 3.5: A perspective view of the central part of the experimental vacuum system (EVS) w i th a side window and a panel open (side panels normally hold Si detectors; see Fig. 3.18). Shown are the thermal heat shield (I), gold target support foils (F) , the retractable diffuser (L) and its guide rails (J). Chapter 3. Experimental Apparatus 64 -gate valve (VABLI) /-He inlet Figure 3.6: A pictorial view of vacuum system, roughly to scale. For detail of the experimental vacuum space, see Fig. 3.5. The description of the cryostat is given in Section 3.2.2. Chapter 3. Experimental Apparatus 65 3.2.5 Tritium safety system Of the total tritium inventory of 200 Ci, no more than about 10 Ci was present in the target vacuum at any time. This is considerably smaller than other dt fusion experiments, which typically use up to 100 kCi of tritium. Also because of low permeability at low temperature, tritium permeation through target walls and windows is less of concern in our normal operating conditions7. Thus, the use of the muonic atom emission mechanism from the cryogenic target allowed an experiment with less stringent safety requirements. Nevertheless, various precautions have been implemented in order to ensure the safety of the personnel and to avoid incidents. The entire target system was contained in safety enclosures, from which the air was continuously exhausted to a roof vent through a fan at a rate of 50 m 3/min. This ensured that tritium was diluted with air, and exhausted from the enclosures in case of any tritium leak from the system. Necessary access to the inside of the enclosures, such as valve operations, were done through glove panels. The tritium level was monitored with tritium monitors (TM) at different places, i.e., in the exhaust stack, in the room air in the experimental area, and at specific places in the enclosures during particular operations. See Fig. 3.7 for the schematic representation of the safety enclosure and the positions of the tritium monitors. The helium flow line for the cryostat was doubly isolated from the EVS and tritium target region with an intermediate space (Tritium Barrier Space, or TBS) in between, reducing the possibility of contaminating the helium recovery system8. The TE1S was evacuated and its pressure monitored, separately from the main vacuum system. The readings of the tritium monitors and the target vacuum ionization gauge were 7 Tr i t i um permeation through target walls and windows can be significant sometimes even at room temperature [175]. 8 A t T R I U M F , helium gas is collected for recycling after use Chapter 3. Experimental Apparatus 66 Air Flow Monitor o o o o o o o o o o o o o t t t VACUUM SYSTEM Tritium Supply o o o o o o o o o o o t t OAS HANDLING SYSTEM o Air Inlet Air Flow Glove Panel Figure 3.7: Schematic representation of the safety enclosure, showing position of tritium air monitors (TM), air inlets, glove panels, and the air flow direction. Chapter 3. Experimental Apparatus 67 taken into a programmable logic controller to provide an interlock system which con-trolled important pneumatic valves and tritium supply systems. It not only prevented operational errors by not allowing certain operations under particular conditions, but also automatically implemented basic protective measures in case of emergency. In addition, various alarm signals, such as target temperature, ventilation air flow and air pressure to the pneumatic valves, were sent to the experimental counting room to give early warning of unusual conditions. 3.2.6 Operation The amount of gas injected into the system was measured in units of Torr-litre (abbre-viated T-l), where one T-l corresponds to the number of molecules in one litre of gas at a pressure of 1 Torr and ambient temperature (~295 K). This unit was operationally convenient since the number of molecules can be compared, independent of the isotopic composition. Depositing targets required some experience in fine handling of the metering valve which controlled the gas release onto the cold foil. The target was deposited typically at a rate of the order of a few (T-l) • s - 1 or less in the steady condition, the limit imposed by the cryostat cooling power, as well as by the requirement to avoid significant intermolecular interactions which would lead to heat conduction from the relatively hot (~100 K) diffuser to the cold plate (~3 K). Close monitoring of the temperature and the pressure and quick response to any condition change during possibly several hours of target deposition was necessary in order to avoid losing the target. The cryogenic system consumed less than 3 //hour of liquid helium in normal con-ditions, hence 6 days of beam experiments were possible with a 500 / helium dewar, conveniently fitting the beam schedule at TRIUMF where beam stops once a week for Chapter 3. Experimental Apparatus 68 accelerator maintenance. Since changing the dewar requires warming up of the system to nearly room temperature, it would cost many hours of beam time, if it could not have been done during the maintenance day. The tritium-containing target film, once measurements were finished, was removed by stopping the cryostat helium flow, and the evaporated gas was mostly captured by the recovery getter bed, similar to the supply one in function, but significantly larger in volume. After the pressure went down to a few mTorr level, the remaining gas was pumped via the turbo pump into a 100 / waste volume to provide a closed cycle pumping system. After beam experiments were finished at the end of several weeks of running, the content of the waste volume was very slowly released into the air exhaust, to be diluted greatly with the air, so that the monitored tritium level would always remain much below the allowed limit. Modifications and changes to the target system, such as installation of the calibration setup described below, were performed in a separate enclosure, where personnel safety was ensured. The weekly urine samples of the operators working in a possibly contami-nated environment were tested by the TRIUMF Safety Group during the period of the operations, but no sign of the contamination has been found to date. To prepare for a potential contaminating incident, a dehydrating substance was stored in the experimen-tal counting room during experimental periods in order to facilitate timely discharge of the radioactivity. 3.3 Target character izat ion In our experiments using thin solidified gas films as targets, characterization of the target films is often important in the analysis of the experimental data. For cross section measurements, in particular, the accurate knowledge of the thickness and uniformity of Chapter 3. Experimental Apparatus 69 the target films is essential, since the uncertainty in the thickness directly propagates to the final results. This section gives a description of the thickness calibration method and the results under various conditions, together with comparisons to Monte Carlo simulations of the gas deposition mechanism. 3.3.1 Characterization method Due to the spatial and cryogenic limitations, conventional methods for thin film thickness measurements, such as optical interferometry, cannot be used with our system. Thickness measurements of condensed gases have been reported by several authors. S0rensen et al. used a quartz crystal oscillator to measure solid hydrogen film thick-nesses [176]. However, the method suffered from a severe non-linearity, which was at-tributed to the low density of hydrogen. Rutherford Backscattering (RBS) [177] which was used, for example, by Chu et al. [178] to measure the thickness of solid argon, oxygen, and CO2, is kinematically impossible for protium targets. We have used the energy loss of a particles traversing the film to measure the thick-ness [179-181]. Uniformity was determined by measuring the thickness at different posi-tions with an array of sources. Characterization Setup Figure 3.8 shows a schematic view of the experimental setup for the target characteri-zation . A linear array of five alpha spot sources was custom-manufactured9 by electro-deposition of 2 4 1 Am onto a gold-plated oxygen-free copper plate. The spot sources had nominal diameters of 3 mm, a center-to-center separation of 10 mm, and were covered by a thin gold layer (~200 /jg-cm-2) for safety and ease of handling. 9Isotope Products Laboratories, 1800 N. Keystone St., Burbank, C A , USA. Chapter 3. Experimental Apparatus 70 to Cryostat, ~3K -+20mm - Omm --20mm Side View Front View Figure 3.8: Schematic view of the target characterization setup. The upstream target support foil (detail F in Fig. 3.5) was replaced by a gold plated copper plate onto which 2 4 1 Am sources were implanted (shown as Cold Plate in this figure). A Si detector was mounted on top of the diffuser for this measurement to directly view the target film (normally Si detectors were mounted in the side panels. See Fig. 3.18). Chapter 3. Experimental Apparatus 71 As in the beam experiments, the target support plate, enclosed in an evacuated cham-ber, was cooled to approximately 3 K, and hydrogen gas solidified onto it when introduced through the diffuser. All of the different gas deposition systems described in Section 3.2.2 were tested for the calibration measurements, including System 1 not actually used in the experiments for this thesis, but the results are given here for completeness. The adjustable distance between the diffuser surface and the target plate was set to about 14 mm for System 1 and 2, and about 8 mm for System 3 for the present calibration measurements. Alpha particles penetrating the hydrogen film were detected by a passivated, im-planted planar silicon detector10 of active thickness 150 pm and area 600 mm2. The detector was mounted on the diffuser frame which was part of a mechanism that allowed the diffuser to be inserted and retracted (Fig. 3.8). The detector thus moved vertically to allow a measurement of the thickness at different positions by detecting the a particles from each of the five spot sources. A collimating device which consisted of an array of small holes (diameter ~1 mm) restricted the angular path of the a particles to accept alphas from only one spot source at a time. The signal from the detector was recorded with a standard spectroscopy system consisting of a charge sensitive pre-amplifier, linear amplifier, and analog-to-digital converter, together with a CAMAC/VAXstation data ac-quisition system, a system similar to the one described in Section 3.4.4. The energy scale of the alpha detection system was calibrated using a separate 2 4 1 Am source. Calibration was frequently required since temperature variations in the detector could cause signifi-cant shifts in the gain. The system achieved a typical resolution of 0.4% (FWHM) at a detector temperature near 100 K. The profile of alpha counts versus the vertical position of the collimated detector for a bare target (i.e. no solid hydrogen layer) is shown in Fig. 3.9. The plot confirms that we detected a particles from only one source spot at a time. 10Canberra, model FD/S-600-29-150-RM. Chapter 3. Experimental Apparatus 72 3 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 4 0 Detec tor Posi t ion in m m Figure 3.9: Counts versus vertical position of the silicon detector with respect to the center. Each peak corresponds to one of the five source spots. The detector is collimated such that it sees only one spot at a time. Thickness Determination Figure 3.10 shows an example of the energy spectra of a particles penetrating hydrogen films with different amounts of gas injected, namely 0, 150, and 300 T-l. The shift of the peaks to lower energy with increased injected gas is clearly visible. Note also the peak broadening (due mainly to straggling) and the asymmetric peak shape which is due in part to the energy loss in the protective gold layer on the source. We determine the mean energy value < E > from the centroid of the energy distribution f(E) in the spectrum Chapter 3. Experimental Apparatus 73 100 w 600 zs o o ID 400 N o 200 0 - r 2 300 T-l 150 T-l 3 4 a Energy [MeV_ 0 T-l 5 Figure 3.10: Alpha particle energy spectra for the central source spot with different thicknesses of hydrogen film, where the peaks are normalized to the same number of counts. The numbers above each peak indicate the amount of hydrogen gas injected. via <E>=J<E>zL  <E>+£ f(E)EdE I (3.1) f(E)dE <E>-t where e is a finite cutoff value. The use of the centroid in the analysis achieves sufficient accuracy while avoiding the difficulties in fitting the irregular peak shapes, which vary depending on the source spot and target thickness. In the approximation that the angular dispersion of a particles is avoided due to the use of a collimator, the thickness of the target T can be obtained from R(< E >), the alpha range as a function of energy: T = R(<Einit >) — R(<Efin>), (3.2) Chapter 3. Experimental Apparatus 74 where < Einn > is the initial energy of the a particles and < Efin >, the energy after traversing the target. For our analysis, the stopping power or the range in the solid state of hydrogen for a particles in the energy range of ~2-5 MeV was needed, but no experimental data is available for solid hydrogen at these energies. The detailed discussion of the effect of physical phase on stopping power for heavy charged particles, which is reported for keV projectiles in hydrogen [182] and in nitrogen [183], as well as for MeV ions in organic and other materials [184-186], can be found in Ref. [179] (see also reviews [187-189]). After a critical survey of the literature [179], a recent compilation for gaseous hydrogen by the International Commission of Radiation Units and Measurements (ICRU) [190] was used for the conversion. Uncertaint ies The systematic uncertainties considered here include knowledge of stopping power, the effect of energy cuts, and energy calibration of the detector. The uncertainty in the stopping power is difficult to estimate, since neither experimental data nor satisfactory theory is available for our case. The ICRU table [190] claims an accuracy of ~l -2% at 4 MeV and ~2-5% at 1 MeV for gaseous hydrogen. Ziegler et al., in another commonly used compilation [191], give an estimate of stop-ping power of solids for which there are no data available, by interpolating (or, for hydrogen, extrapolating) the data from other elements under certain assumptions. They quote a 5% average accuracy for a particle stopping powers in solids, which is simply the average of deviations taken from the collections of experimental data in the literature for many elements. The two tables ([190] and [191]) agree with each other within 3% at ~ 2 MeV, and the thicknesses derived using both tables agree within 2% (the differences in stopping powers Chapter 3. Experimental Apparatus 75 partly cancel with one another upon integration over energy). We shall conservatively quote 5% of the derived thickness as the uncertainty due to the stopping power, including a possible physical phase effect. This was the limit on our accuracy in most cases. The effect of the finite cutoff values (e in Eq. 3.1) in determining the centroid of the energy spectra was investigated by changing the cuts and its uncertainty was found to be small compared to that from the stopping powers in most cases. The uncertainty in the detector energy calibration is estimated to be less than 2 keV, except for a few measurements during which a temperature variation in the silicon detector resulted in a gain shift. These systematic uncertainties, as well as the statistical uncertainties, were added in quadrature to obtain the total uncertainty in the thickness. Except for very thin layers, whose thicknesses had to be determined from a small difference in the initial and final alpha energies, the knowledge of the stopping power dominated the uncertainty. 3.3.2 F i l m thickness and uniformity-Linearity of film deposition The linearity of the deposition was tested by measuring the thickness of films made with different amounts of gas input under different conditions. The data shown in Fig. 3.11 were taken at the central source position (0 mm) for films made by diffuser system 1. Some of the films (40, 300 T-l) were made by depositing gas on top of existing films, whereas others (20, 150, 400 T-l) were made with a single deposition. The solid line represents a weighted least-squares fit to the data (plotted with error bars 1 1). The slope represents the conversion factor from gas input to the film thickness at this source position (0 mm). A possible non-linear component in deposition was investigated by allowing a 1 1 Error bars presented in this section represent total uncertainties including that from stopping power, unless otherwise stated. Chapter 3. Experimental Apparatus 76 Gas input [T-i] Figure 3.11: Test of the linearity of deposition. The solid line represents a least-squares fit to the data points plotted with error bars. quadratic term in the fitting function. The maximum allowable non-linear contribution, when extrapolated to a 1000 T-l target, was similar in size to the uncertainty in the thickness due to the stopping power. Similarly, the data from other source positions on the plate, and with diffuser systems 2 and 3, showed good linearity for these moderate thicknesses. The extrapolation of the results to very thin films requires some caution; if a small amount of the gas remained in the gas transfer tubes, it would be lost from the layer, giving a small offset in gas deposition. This gas loss is negligible for thick layers, but can be important for thin layers. The effect was examined for systems 2 and 3 by comparing two series of measurements; (1) thick films made by a single deposition of a large amount of input gas, in which the gas loss is negligible, and (2) thin films made by sequential depositions of small amounts of gas, where the gas loss from each deposition, if it exists, Chapter 3. Experimental Apparatus 77 is multiplied to give a measurable effect after several depositions. Due to the small energy loss, measurement of very thin films was difficult, and we have assumed that the same cutoff value, e in Eq. 3.1, can be used to determine both the initial and the final energies (this is not generally valid in the thick film measurement due to the peak broadening). Provided that the same value is used for both initial and final spectra, the choice of e did not affect the resulting thickness values. Figures 12(a) and 12(b) show the results of the comparisons for systems 2 and 3 respectively. For system 2 (Fig. 12(a)), sequential deposition of small amounts of gas (measurement (2), filled squares) resulted in a smaller thickness per unit gas input than thick films made with a single deposition (measurement (1), open squares), indicating that when making very thin films a non-negligible amount of the gas remains in the gas transfer tubes without being deposited. On the other hand, System 3 (Fig. 12(b)), which was in fact designed to remove the effect, shows no evidence of such gas loss. Thickness Profile Shown in Fig. 3.13 are the thickness profiles from different deposition systems. An asymmetric non-uniformity with respect to the center can be observed. The error bars do not include the uncertainty from the stopping power, since it is not relevant when considering relative uniformity. One may assume that the thickness of a film at a particular position depends only on the relative distance from the diffuser, which is the case if, for example, the molecules stick onto the cold plate at the first contact as suggested from the measurements described in Section 3.3.2. A series of measurements were made using system 3 with films that were deposited with the diffuser displaced by 0 mm, +5 mm, +10 mm and -20 mm from the nominal Chapter 3. Experimental Apparatus 78 1500 Gas input [T-l] (a) System 2 1500 Gas input [T-l] (b) System 3 Figure 3.12: Sequential deposition of very thin films shown as filled squares is compared to the thick film deposition shown as open squares for different systems: (a) system 2, where the difference in slopes between the two series of depositions indicates the gas loss effect, and (b) system 3, showing no evidence for such an effect. Inserted boxes in the figures illustrate details at small thicknesses. Chapter 3. Experimental Apparatus 79 A 7 3 E o 3. 2 in in CD c o 0 $ j — * — S y s t e m 1 ------ S y s t e m 2 — Q — S y s t e m 3 -30 - 2 0 - 1 0 0 10 S o u r c e pos i t i on [ m m 2 0 3 0 Figure 3.13: Thickness profiles for different diffuser systems. Thickness per unit gas input is plotted against the source positions. Because these are relative measurements, the error bars do not include the uncertainty from the stopping power. standard position, the positive direction being upward. In Fig. 3.14, the resulting thick-nesses are plotted against diffuser coordinates, i.e., the relative vertical distance from the center of the diffuser, unlike Fig. 3.13 which was plotted against the distance from the center of the source plate. The data from different films, thus plotted, are consistent, indicating the translational invariance of the film profile under the diffuser displacement. One exception is the filled star point at distance 0 mm in the diffuser coordinates, which represents the thickness at the center of the diffuser for a film deposited with the diffuser retracted 20 mm downwards. At this position the diffuser, which has an active diame-ter of 60 mm, was aligned with the bottom source spot (see Fig. 3.8) and a significant fraction of emitted gas molecules missed the cold plate, hence the molecules may bounce Chapter 3. Experimental Apparatus 80 A 7 3 CD C o Ic 2 0 D i f fuse r pos i t i on O 0 mm 0 +5 mm • +10 mm * —20 mm - 4 5 - 3 0 - 1 5 0 15 3 0 4 5 V e r t i c a l d i s tance f r o m the d i f f u s e r c e n t e r [ m m ] Figure 3.14: Fi lm thicknesses for system 3 are plotted against the vertical distance from the diffuser center (diffuser coordinates). The error bars do not include the uncertainty from the stopping power. See text for the detail. around inside the vacuum system, eventually sticking nearer to the bottom rather than the top of the cold plate. Nevertheless, validity of the translational invariance indicated above suggests that it is possible to measure a thickness at an arbitrary point along the vertical axis with the present technique, despite the fact that a particles are emitted only from discrete source spots. Figure 3.14 can thus be considered as a good representation of the thickness profile of the film made with system 3. Based on these results, the cali-bration factors (thickness per unit gas input) for system 3 at each position with respect to the diffuser center, including the 5% uncertainty from the stopping power, are given in Table 3.1. When two or more measurements exist for the same position, an average was taken, except that some points from the diffuser position of -20 mm, which may be Chapter 3. Experimental Apparatus 81 Table 3.1: Protium film thicknesses at various distance from the center of the film for system 3. The distance between the diffuser surface and cold plate surface was 8 mm. Position with respect to film center (mm) Thickness per unit gas input Gug-cm-MT./)- 1) 40 0.21 ±0 .08 30 1.52 ± 0 . 1 1 20 2.92 ±0 .17 15 3.10 ±0 .18 10 3.21 ±0 .16 5 3.33 ±0 .19 0 3.46 ±0 .17 -5 3.64 ±0 .18 -10 3.76 ±0 .19 -15 3.92 ±0 .21 -20 3.77 ±0 .19 -25 3.06 ±0 .17 -30 1.68 ±0 .09 in error as discussed, were not included in the final average. It should be noted that, in general, the calibration factor depends on the distance between the diffuser surface and the cold plate surface which, in the present case, was 8 mm. For system 1, which had a 14 mm separation between the diffuser and cold plate, the calibration factors at the standard vertical diffuser position are given in Table 3.2. The values for system 2 are similar and agree with system 1 within the uncertainty (see Fig. 3.13). Thicknesses of deuterium and tritium in units of / /g-cm - 2 are factors of 2 and 3, respectively, larger than for a protium film with the same number of molecules due to the isotopic mass difference. The measurement of a deuterium film, when corrected by this factor, showed good agreement with protium films. Chapter 3. Experimental Apparatus 82 Table 3.2: Thickness of the protium film for system 1, deposited with standard diffuser position. Source position (mm) Thickness per unit gas input (^g-cm-MT-O- 1) 20 2.37 ±0.13 10 2.94 ±0.15 0 3.32 ± 0.17 -10 3.60 ±0.19 -20 3.31 ±0.17 Other Measurements Measurements were made with films deposited under different conditions to see the effects on thickness and uniformity. No deviation was found for films made with and without pumping the target vacuum during deposition within the relative uncertainty of about 1%. Reducing the deposition rate by an order of magnitude also did not noticeably affect either thickness or uniformity. As described in Section 3.2, the gas deposition system is capable of making a second film on a separate cold surface through the opposite side of the diffuser. The appara-tus was designed to minimize unwanted cross deposition from one side to the other by shielding with cold surfaces. By intentionally releasing a large amount of gas through the opposite side, cross deposition on the spot source target was checked. The measurement, made with similar assumptions on the cutoff value e to the measurement of very thin films described in Section 3.3.2, indicated that less than one part in a thousand of the injected gas arrived at the central spot. This result, together with the fact that pumping the system did not affect deposition, suggests that the gas molecules are likely to stick to the cold surface on first contact, which would explain the observed translational invariance of film profiles (Section 3.3.2). Chapter 3. Experimental Apparatus 83 Table 3.3: Thickness calibration factors for a neon film at source positions -20 mm, 0 mm and 20 mm. Source position Thickness per unit gas input (mm) (^g-cm-MT-/)- 1) 20 24.7 ± 1.7 0 32.5 ±2.0 -20 31.5 ± 1.9 Beam experiments using the same solid hydrogen target sometimes lasts for a few days, so it is important to check the effect of target aging. The results of measurements, made 8 hours apart, of the same film were consistent with each other, giving cin up-per limit in thickness variation AT/T < 0.5% over 8 hours. In the analysis, a similar assumption in the cutoff value was made. Films of other gases can also be deposited with the target system (for example, neon films have been used in the experiments by our group [192,193]). Thickness measure-ments of neon films were made using two different gas inputs at three of the five source positions. The stopping power in gaseous neon taken from the ICRU tables [190] was used to convert energy loss to thickness. The same 5% uncertainty in stopping power was assumed. Calibration factors for the three spots are given in Table 3.3. M o n t e Car lo Simulat ions of Gas Deposi t ion In order to better understand the mechanism of gas deposition, Monte Carlo simulations were performed with the following assumptions: (1) molecules are emitted uniformly from the gas diffuser surface, (2) the molecules stick to the cold surface at the position of first contact, and (3) there is no interaction between the molecules. The last assump-tion is justified since the requirement to keep the target film temperature cold demands Chapter 3. Experimental Apparatus 84 an insignificant intermolecular interaction, minimizing heat conduction from the warmer diffuser to the target plate. Pressure must be maintained low enough during deposi-tion to keep the mean free path of the emitted molecules comparable to or larger than the diffuser-to-plate separation. Three different models for the angular distribution of molecules emitted from the diffuser surface were used in simulations: (a) the (unrealistic) forward emission model assuming 9 = 0, where 9 is the angle with respect to the normal to the diffuser surface, (b) the isotropic emission model assuming that the number of molecules dw emitted into unit solid angle dil = sm.9d9d(f> is constant, independent of 9, i.e. dw/d(cos 9) = const., and (c) the cos# emission model using a diffusion-like angular distribution dw/d(cos 9) ~ cos 9. The diffuser diameter of 60 mm and the distance of 8 mm between the diffuser and cold plate surface were used in simulations to compare with the measurements for system 3. There are no free parameters in the simulations other than the emission angular distribution. Figure 3.15 compares the simulation results with the film thickness profile for system 3 as deduced above, which is plotted with error bars that now include the uncertainty from the stopping power. The agreement with the cos 9 model (c) is quite good, except for the asymmetry in the shape. The asymmetry, which obviously cannot be reproduced with our simulations, is un-likely to be due to an effect of gravity on the target, since the relative non-uniformity remains constant over a wide range of the thickness. The shape may be partly explained by the fact that the gas is introduced from the bottom of the diffuser system, hence gas molecules may have a larger probability of diffusing out at the bottom rather than the top of the diffuser, indicating a breakdown of assumption (1). Similar simulations with the diffuser-to-plate distance taken to be 14 mm were com-pared to data for system 1. The cos 9 model best describes the data, but comparison indicates a slightly more forward peaked emission angular distribution. It should be recalled, however, that system 1 had a different perforation structure from system 3. Chapter 3. Experimental Apparatus 85 A 7 3 CP CO ill C D c C J 2 0 1 i 1 — - 4 5 - 3 0 - 1 5 0 15 3 0 4 5 V e r t i c a l d i s tance f r o m the d i f f u s e r c e n t e r [ m m ] Figure 3.15: The data, with error bars which include the uncertainty from the stopping power, denote the averaged thickness profile for system 3 plotted against the relative vertical distance from the diffuser center. The histograms are the the simulated thick-ness profiles from Monte Carlo calculations with different assumptions for the angular distribution of molecules emitted from the diffuser surface. Reasonable success in the simulation with the cos 6 model gives us some confidence in scaling the thickness as a function of distance between the diffuser surface and the target support foil surface. Figure 3.16 shows a comparison of the simulated thickness profiles with different distances. Effective Thickness The results given in Tables 3.1, 3.2, and 3.3 should be used with caution when the films are used as a target for beam experiments. Since the present calibration measures Chapter 3. Experimental Apparatus 86 15 _ 0 15 Pos i t ion [ m m ] Figure 3.16: Monte Carlo simulations comparing different distances between the diffuser surface and the cold foil surface. the profile only in the vertical dimension, the profile in the horizontal dimension has to be assumed when estimating an average target thickness. For a non-uniform layer, the average thickness depends on the width and profile of the beam which stops in the target. The angular divergence of the beam also contributes to the effective thickness. Thus the method of the averaging depends on the type of measurement. For example, when the target is used to stop particle beams from an accelerator as in Ref. [194], a Gaussian distribution with a certain width may be justified. However, when muonic hydrogen atoms emitted from a solid hydrogen layer are stopped in a deuterium film, as in Ref. [83], the atomic beam is divergent with an angular distribution close to dw/d(cos 9) ~ cos#, and depends on scattering cross sections. In this case the detailed averaging has Chapter 3. Experimental Apparatus 87 to be done with a Monte Carlo simulation which includes differential scattering cross sections. This will be done in a following chapter. 3.3.3 Summary of target characterization Deposited films of solid hydrogen isotopes and neon have been characterized via the energy loss of a particles. The method can be applied to a relatively wide range of film thicknesses, e.g. for protium, from ~1 /fg-cm - 2 to ~1 mg-cm - 2 . The accuracy of the measurement is limited by the uncertainty in the stopping powers, but the relative accuracy can reach better than 1%. The uniformity can be measured by sampling the thickness at different positions with an array of alpha sources. Furthermore, it was possible to determine the thickness at an arbitrary vertical position by measuring films deposited with different diffuser positions. We note that muon catalyzed fusion reactions could also be used as a mono-energetic alpha source for the thickness measurement [179] (see also [195]). As for our target system, the linearity of deposition was confirmed, with the exception of very thin film deposition with system 2. A n asymmetric non-uniformity was observed in all films, while deposition conditions (such as pumping of the high vacuum region surrounding the target, or change in gas deposition rate) did not affect the thickness or uniformity. Deposition from one side of the diffuser did not contaminate the cold plate on the other side to a limit of one part in a thousand, enabling experiments using two separated solid hydrogen targets described in this thesis. No evidence was seen for the change in the film thickness over time. The comparison with Monte Carlo simulations indicates that the angular distribution of gas emission is close to a cos 9 model. The thickness calibration as well as film profile for the target is presented in Tables 3.1, 3.2, and 3.3. Chapter 3. Experimental Apparatus 88 3.4 Detection System 3.4.1 Overview As illustrated in Sec 1.2, muonic processes in hydrogen isotopes produce a rich variety of complex reactions, leading to various kinds of radiations. Obviously, detecting as many different types of radiation as possible is advantageous in identifying and understanding the processes. The TRIUMF system was designed to allow this versatility. Figure 3.17 illustrates a top view of a detector arrangement used in Run Series 1. Muons entering the system were detected by a thin plastic muon beam counter (TI). On the beam right (i.e., on the right hand side when facing the same direction as the beam flux) was placed a set of MWPCs, which tracked trajectories of muon-decay electrons, enabling measurement of the positions and the time of muon decay. A silicon detector and a neutron detector detected fusion products, while a germanium detector monitored target impurities via muonic X-rays. Si detecter was placed in vacuum, and held on the side of the heat shield, viewing the targets without any window in bewteen. Electron, Neutron and Ge detectors were placed outside the safety enclosure (Section 3.2.5), which contained the vacuum system. For Run Series 2 (Fig. 3.18), the distance between the two gold foils was reduced to 17.9 mm from 42.5 mm in order to enhance the number of lower energy fit atoms surviving to reach the downstream reaction layer. Because of the smaller drift distance, the MWPCs, which had a position resolution of several mm, were not useful and they were replaced by a silicon detector and neutron detector to gain better efficiency for fusion product detection. Chapter 3. Experimental Apparatus 89 Figure 3.17: Schematic top view of the detector arrangement for Run Series 1. A MWPC system reconstructed the paths of the muon decay electrons, whose time was measured with plastic scintillators, and the energy with a Nal crystal. A silicon detector and a neu-tron detector detected fusion products, while a Ge detector monitored target impurities via muonic X-rays. See also Fig. 3.18. Chapter 3. Experimental Apparatus 90 shielding vacuum system T1 muon detector ~\j Ge detector Si detector Heat Shield 3 K target neutron detector electron detectors Figure 3.18: Top view of the detector arrangement for Run Series 2. Compared to Run Series 1, the target spacing was reduced and the MWPCs were replaced by another Si detector and neutron detector. Chapter 3. Experimental Apparatus 91 Detector signals were processed via a N I M / C A M A C electronics system. The signals were converted w i th Analog-to-Digital Converters (ADCs) or Time-to-Digi ta l Convert-ers (TDCs) modules and read into a workstation (DEC Vax Station) w i th a T R I U M F developed VDACS data acquisition system [196]. The VDACS uses a PDP-11 micropro-cessor (CES 2180 Starburst, hereafter called a Starburst) as a front-end-processor which handles the data acquisition according to a procedure specified by the user using a high level language called T W O T R A N [197]. The data were logged onto a magnetic tcipe on an event by event basis for off-line analysis, while an analysis program M O L L I [198] was used for on-line analysis to provide a rapid diagnosis of the experiment. 3 . 4 . 2 T r i g g e r The data acquisition system was "triggered" to read in the data, when appropriate con-ditions were met. This decision making was done in a hardware logic circuit i l lustrated in Fig. 3.19. Our basic strategy for triggering can be summarized as: 1. Provide a gate for ~ lOps after muon arrival, which is sufficiently long compared to the muon l i fetime. 2. Inhib i t the triggering if a muon had been already accepted and the gate was opened (BUSY), or the previous data was being processed by the computer (INH). 3. Identi fy a pile up, i f another muon arrived after the gate was opened. 4. Collect the signals f rom all of the detectors, i f any one of the detectors fired during the gate. 5. Clear the T D C and A D C buffers, i f no detector fired. 6. Record various scaler values for data normalization and diagnosis. Chapter 3. Experimental Apparatus 92 Tl IF EEVGF PUG BF P L E U P ^ T1H1 CALIBRATEy1^ EB EEVG TBI L COM TBIF EVG GATE STOP $ to CMU WCINH •\lh WC TRP -TT MONITOR EI EVENA TRGn DELX GATE STOP TRCF DELX-GATE STOP to to C / N H y HINH EVTR DELl. GATE Q to CPINH at GATE La E V G F OUT/-Figure 3.19: Schematic diagram for the trigger electronics, illustrating the key com-ponents. Pulses from the beam counter T l , beam busy signal BUSY, and computer busy INH were combined to provided an event gate EVG, which allowed individual sub-detector systems to accept events and give a sub-trigger TRGn. The circuit dia-grams are for illustrative purpose only, and not all of the details are given". "Electronics diagrams shown in this section are taken from or based on those in Ref. [199], for which I thank Paul Knowles. Chapter 3. Experimental Apparatus 93 The timing diagram is given in Fig. 3.20. A muon, defined by a sufficient energy deposit in TI, opened a pile up gate (PUG), which supplied a BUSY signal for 10 ps. The BU SY gate, together with the general inhibit signal (INH) due to computer delay12, was used to ensure (2.) above. Only when no muon had arrived in the preivous 10 ps, and the computer was ready, was the coincidence TI • B • I satisfied, opening the event gate (EVG) as well as giving a common start to TDCs. The EVG was sent to all the detector signal processing logic allowing them to accept events. If any of n sub-detection systems described below gave a trigger DETn during EVG, a trigger gate TRGn was opened, which was subsequently closed with the end of event gate pulse (EEVG). The master event trigger EVTR was provided as a coincidence of TRG and EEVG, requesting the Starburst to collect the ADC and TDC values. On the other hand, when there was no detector giving a trigger, EVCL was given from EVCL = EEVG • TRG, clearing the ADCs and TDCs so that they could accept new values in the following event. Once the master trigger was given, the Starburst provided a computer busy signal (CINH) in a CAMAC output register while processing the signal (~ 1 ms). However, since it took a few hundred microseconds for CINH to turn on, an inhibit had to be provided by hardware (HINH) to prevent a pile up during this time. Furthermore, an extended inhibit (IEX) was provided while clearing the ADCs and TDCs, whose duration was determined empirically by monitoring the number of lost events13. If more than one muon arrived during the gating period, the time correlation between the muon stop and the detected reaction would be lost, hence the event had to be discarded. The rejection of pile up events, which amounted to about 5% of total events at our typical beam rate of 5000 s _ 1 , was achieved in software using a pile-up bit pattern 1 2 W h e n M W P C s were used, INH was also activated by a high voltage supply failure WCINH. 1 3 T h i s was derived from comparing the number of triggers going into the Starburst (EVTR), and the number it actually accepted (STAR, a scaler counting Starburs events). Chapter 3. Experimental Apparatus 94 T1 B -10 lis EVG EB EEVG DETn TRGn EVTR HINH CINH IEX ~200 fis -1 ms -150 us Figure 3.20: Schematic t iming diagram for trigger electronics. Lower lines indicate "on," and higher lines "off," corresponding to 0 V and -0.8 V , respectively, in fast-negative N I M logic standard. The horizontal axis represents approximate t ime flow. The second pulse in T I illustrates a pile up event, which extends B, and delays EB. The E V C L pulse, not shown in the figure, was given when there was no trigger, and was generated at the same t im ing as E V T R . • given by the PUG module. Also in the case of pile up, the module extended its BUSY signal for 10 ps f rom the t ime of the second muon entrance, which in tu rn delayed the-end-of-busy pulse (EB) (see dotted lines in the figure). Chapter 3. Experimental Apparatus 95 An important scaler GMU (good muons) was derived as a coincidence between EB and EEVG (end of event gate). Since EEVG always occurred 10 ps after EVG was turned on, regardless of whether another muon came in or not, the coincidence was satisfied only if there was no pile up. Recalling that EVG(= Tl • B • I) takes into account the computer dead time, GMU provided the number of incident muons which satisfied all the trigger conditions except detector triggering, hence is to be used for absolute normalization. Various scalers including GMU were recorded for normalization and system diagnosis purposes. Their values were read into the Starburst typically every 5 s, independent of the muon trigger. Occasionally, some of the scalers were lost due to module malfunction, but there was enough redundancy to recover important scalers. For example, an alternative way of deriving the number of "good" incident muons is: m = n . J . 7 x I ^ x - ! ™ L , (3.3) T l MON-1 V ' where MON is a scaler for a pulser signal14, hence MON • I/MON is the measure of live time fraction. The factor, (7/1 • TJ • 7/7/1) x MON/(MON • 7) gives the fraction of events which do not have pile up f(pile), and GMU is derived by Tl • B • I x f(pile) A further diagnosis and normalization tool was provided by a trigger called 1/N, which was activated every 1024 hits in Tl scintillator, regardless of its deposited energy (i.e., including beam electron hits), and independent of the sub-detector triggers. The latter feature of this trigger allowed yet another way of checking the GMU scaler. 3.4.3 Multi-wire proportional chambers Multi-Wire Proportional Chambers (MWPCs) provided information of the position of muon decay electron hits. The electronics diagram for the MWPC imaging system is 4 For the purpose here, this can be any signal uncorrelated to the muon trigger. Chapter 3. Experimental Apparatus 96 given in Fig. 3.21. When an electron passed through the chamber, it ionized a gas mixture of argon, isobutane, freon, and methylal vapor, which filled the chamber. The ionized electrons drifted toward the closest anode wire(s), and near the anode where the electric field became very strong, electrons were increased from the consecutive series of ionization of the gas molecules, leading to a multiplication process. This process induced a current in the nearest cathode wires in each of two planes in orthogonal directions (y and z in our case). Each wire in a plane was connected to a delay line, and the signal reached the end of the line with a time delay (with respect to the electron hit, measured by EMI and EM2) that was proportional to the distance of the hit position from the edge of the plane, or the end of the delay line. The delay time was measured at both ends of the delay line, and the difference between them was proportional to the distance from the center of the chamber. For instance, for z wires, the distance from the center Zwc is given by ZWc(mm) = (ZL - ZH) x dispZ + offsetZ (3.4) where ZL and ZH are the delay times at the beam upstream and downstream ends of the delay line, respectively, and dispZ is a dispersion constant, determined from calibrations using an electron source, where the physical displacement of the source by a known distance was mapped to the shift in the delay time. The parameter, offsetZ was adjusted such that the center of the distribution of electron hits corresponded to zero in the z scale. The position in the y-dimension was measured in an identical manner. With three sets of MWPCs providing hit positions, the trajectories of muon decay electrons were determined from a least-squares fit to three points on a straight line. The fitted line was extrapolated back to a perpendicular plane bisecting the target, providing positional information of the muon at the time of muon decay. The energy of electrons was measured by a large Nal crystal (MINA). The cuts in Chapter 3. Experimental Apparatus 97 CnZL EM 1 EM I 2 EM2 W L 3 EUEVG CnF Figure 3.21: Electronics diagram for the Multi-Wire Proportional Chamber imaging system. Only one of three identical chambers is shown. The time of the electron is measured by a pair of plastic scintillators, while its energy is given by the Nal crystal MINA. MINA energy improved position resolution of the imaging system, by favoring higher energy electrons which scatter through smaller angles in window materials. More details of the data treatment will be given in Sec. 7.1. 3.4.4 S i l i c o n d e t e c t o r A silicon detector, placed in vacuum, detected low energy charged particles such as 3.5 MeV a particles from dt fusion and 3 MeV protons from dd fusion. For Run Series 1, Chapter 3. Experimental Apparatus 98 Figure 3.22: Electronics diagram for a silicon detector. The detector output was pre-amplified to give a fast timing signal and a slow energy signal. The timing filter amplifier, with the use of an external delay, optimized the pulse shape which gave a sub-trigger SITR if in coincidence with EVG, and opened the ADC gate for the ampli-fied energy signal. In Run Series 2, where two silicon detectors were used, both functioned in an identical manner. A 4-input 8000-channel ADC was shared between silicon detec-tor^) and a germanium detector. a passivated, ion-implanted planar silicon (PIPS) detector15 of active thickness 150 pm and area 600 mm 2 was operated at a bias voltage of 30 V. For Run Series 2, two similar PIPS detectors16, but of active thickness 300 pm and area 2000 mm 2 (diameter 50 mm) were used. Silicon detectors, such as PIPS detectors, and more commonly used silicon surface barrier (SSB) detectors, are in principle reverse-biased diodes. Charged particles slowing down in a carrier-depleted region of the silicon crystal creates electron-hole pairs, which are swept toward and collected at the electrodes by an applied bias field. PIPS detectors offer advantages over SSB detectors, such as thinner yet more robust entrance windows, and lower leakage currents, which improves energy resolution. Figure 3.22 shows the electronics diagram for silicon detectors. Pre-amplifiers were placed immediately next to the feed-through connectors of the vacuum system, in order to reduce the total capacitance. Amplification of energy and timing output of the preamps 1 5 Canberra, model FD/S-600-29-150-RM 1 6 Canberra, model FD/CY-2000-37-300-RM Chapter 3. Experimental Apparatus 99 was done with linear amplifiers and timing filter amplifiers, respectively, before the signals were sent to the counting room. Considerable efforts were made to improve energy resolution in the tritium environ-ment. This included shielding the pre-amplifier and the cables leading to it with a copper foil and copper braid material respectively, grounding the detectors and electronics sys-tem to the target vacuum chamber, and using a noise-filtered power supply. Mounted on the cryogenic thermal shields, the detectors were kept at about 90 K, which reduced the leakage currents and thereby improved the energy resolution. For Series 2 the horizontal dimension of the detectors was collimated to 13.9 mm to view only the inner sides of the gold foils, where the reactions of interest took place. Fur-thermore the collimation restricted the angular path of the alphas and protons entering the detectors, preventing the ones with grazing angles which suffer large energy loss in the layer. This reduced variations in the detection efficiencies when layers with different thicknesses were used. The energy scale of the silicon detectors was calibrated with an 2 4 1 Am source. 3.4.5 Neutron detector Detection of neutrons is among the standard techniques in conventional studies of fiCF, since they easily exit from the target containers which typically consist of thick walls. Detectors using the liquid scintillator NE213 were used with pulse shape discrimi-nation (PSD) provided by hardware modules (Link System PSD-5010 and PSD-5020). Taking advantage of the difference in the liquid scintillator light output time-structure between neutrons and gammas, the PSD module could discriminate the two at the trigger level. The electronics diagram for neutron detection is given in Fig. 3.23. A pair of plastic Chapter 3. Experimental Apparatus 100 E I N I G N I — TIME —1 LINK PSD # 5 0 10 PSD N l EVENT Y n P R E -T T L - S C A L E R - N / M ADC GATE NJ TRG COM DELA. GATE (9 0 DEAD T / M E GATE J STOP _ J L / N AMP PSD SHIFT Figure 3.23: Electronics diagram for a neutron detector. Charged particle anti-coincidence was required for the neutron/gamma detection. The pulse shape discrim-inator L I N K 5010 (5020 for the second detector) provided a trigger level discrimination of neutrons from gammas, though the latter were also recorded, pre-scaled by factor of 30. For the sub-trigger N1TRG to fire, the event has be in coincidence with E V G . scintillators placed in front of the detector was used in anti-coincidence to reject charged particles. Together with neutron triggers, gammas were recorded but were pre-scaled by a factor of 30. Chapter 3. Experimental Apparatus 101 Figure 3.24: Electronics diagram for the germanium detector, similar to the silicon de-tector circuit except the charged particle veto was used here. Both detectors shared the same A D C . 3.4.6 Germanium X-ray detector Detection of muonic X-rays emitted upon muon transfer from hydrogen isotopes to heav-ier elements was used mainly for two purposes in our experiments. One was to detect arrivals of muonic hydrogen at a specific spatial position, while the other was to measure the level of impurities in the target layers. For the work of this thesis, the latter was the main use, and because of the anomalously long diffusing path length of muonic tritium in protium, the detection of the transfer X-rays provided a sensitive measure of impurities such as nitrogen and neon in the layer. Examples of the former use of muonic X-rays are given in Ref. [193,200,201]. Figure 3.24 shows the circuit for the germanium detector with a pair of plastic scin-tillators used for charged particle anti-coincidence. The detector was placed in a lead collimator to reduce neutron induced background [202]. Chapter 3. Experimental Apparatus 102 Figure 3.25: Electronics diagram for. the electron detection by plastic scintillators and the dynode output of the neutron detectors. Only scintillators in front of the neutron detector are shown here, but all other ones were also connetced to the logical OR module to give an electron trigger DELE. The neutron dynode output provided not only the timing of the electron, but also its energy. 3.4.7 Electron detection system Plastic scintillators, in addition to providing a charged particle veto for neutron and photon detection, were used to detect the electrons with an independent trigger, recording the electron time spectrum (Fig 3.25). Since statistics for the electron measurement were not a problem, the trigger was pre-scaled by factor of 4, in order to reduce the load on the data acquisition system. This circuit also provided a tool called delayed electron (Del) coincidence for reduction of muon capture related background, where one demands that the muon-decay electron be detected after the reactions of interest, hence ensuring the muon was alive, (i.e., had not decayed nor been captured by) then. In case of nuclear muon capture, the muon was converted to its neutrino, and no electron would be observed in the delayed time Chapter 3. Experimental Apparatus 103 window, which was typically from 0.2 to 5 fis after the reaction. For the Del concidence, the master trigger was always given by the detector of interest, therefore no pre-scaling of the signal was necessary. The neutron detectors could also be used for electron detection by utilizing a fast signal from the dynode of the photo-multiplier. A separate trigger, called telescope or simply Tel trigger, was implemented which required a triple coincidence of two plastic scintillators and the dynode signal, with the latter energy also recorded. Chapter 4 Experimental Runs Muonic hydrogen research at T R I U M F has evolved for over a decade, making continual progress in improving the measurements and understanding the physics. Of nearly a dozen series of experimental runs performed up to now by the Collaboration, some of which have or will have resulted in Master's, as well as Ph.D. theses [30,179,199,203], the runs analyzed for this thesis are the first two with tritium used in the target, which are summarized in Table 4.1. Run (date) Au foil spacing Gas diffusera Si Detector Active area Thickness MWPC imaging Main goals Physics Series I (Nov-Dec 93) 39.0 mm System 2 600 mm 2 150 yes Commission jit emission Series II (Jul-Aug 94) 17.9 mm System 3 2x2000 mm'2 300 fim (2x69.5 mm 2) 6 no Production TOF fusion "See Section 3.2.2 the for description of diffusers. 6 Each detector had a C u collimator with an opening of 13.9 mmx50 mm. Table 4.1: Comparison of Run Series 4.1 Run Series I Run Series I was performed for three weeks in November and December 1993. After many months of work spent on target system modification to incorporate tritium com-patibility, observing a clear peak built up within minutes after starting the first dt fusion 104 Chapter 4. Experimental Runs 105 ID Run Target Upstream GMU (xlO 6) 1-1 1526 SET1 (1000 T-l H 2 with 0.1% T 2) 9.179 1527 SET1 47.377 1528 SET1 77.682 1529 SET1 102.109 subtotal 1526-29 304.094 1-2 1536 SET1 0 3 T-l 43.726 1-3 1538 SET1 0 6 (3+3) T-l 13.471 1539 SET1 0 6 (3+3) T-l 34.918 1540 SET1 0 6 (3+3) T-l 39.038 1541 SET1 0 6 (3+3) T-l 88.066 subtotal 1536-41 219.219 1-4 1542 SET1 © 12 (3+3+6) T-l 12.533 1543 SET1 0 12 (3+3+6) T-l 9.027 1544 SET1 © 12 (3+3+6) T-l 24.896 1545 SET1 0 12 (3+3+6) T-l 49.753 subtotal 1542-45 96.209 1-5 1546 SET1 0 24 (3+3+6+12) T-l 21.302 1-6 1547 SET1 0 36 (3+3+6+12+12) T-l 22.423 1-7 1555 SET3 (1000 T-l H 2 with 0.3% T 2) 6.671 1556 SET3 33.033 subtotal 1555-56 39.704 1-8 1572 1000 T-l H 2 26.844 Table 4.2: Run summary for Series 1. For thin deuterium overlayers shown with 0, the nominal gas input values are given, but the actually thickness has some uncertainties (see Section 3.3.2). GMU, the number of "good muons," is derived from Eq. 3.3. Data was collected typically at the rate of 3.5 to 4.0xl0 3 GMU/s, i.e., 12-15xl06 GMU/hour. Chapter 4. Experimental Runs 106 run was quite exciting indeed. Run Series I was a commissioning run for the new tritium target system, and much of the time was devoted to testing of various systems and establishing safe procedures of target handling. It was the first time that we observed an a particle from muon catalyzed dt fusion with high resolution. In addition, the emission of muonic tritium into vacuum was observed for the first time. For this thesis, MWPC imaging data was analyzed to study muonic tritium emis-sion, on which the atomic beam method is based. Shown in Table 4.2 is a summary of runs analyzed from this Series. The standard emission target, denoted SET1 (or SET3) consisted of a mixture of 1000 T-l pure protium with 0.1% (or 0.3%) of tritium. An ad-ditional layer of thin deuterium, called an overlayer, was sometimes deposited on top of the emission layer. Its thickness for this Series, however, had some uncertainty due to the effects of gas remaining in the transfer tube, as described in Section 3.3.2 (see Fig. 12(a)). Other than the thin overlayers, the average conversion factor for T-l is 3.2 pg-cm~2 per T-l. With a TI beam counter rate of ~ 5 x 103/s GMU events were accumulated at the rate of 10-12xlO6/hour. The scaler GMU (Fig. 3.19) suffered an occasional malfunction during this run, therefore GMU derived from Eq. 3.3 was used for normalization. 4.2 Run Series II Run Series II was our production run, where the measurement of dpt molecular for-mation via time of flight was attempted. Improvements in the target system had been made after Series I. The gas diffuser was replaced with the one with sintered metal to ensure microscopic uniformity of the target film. The gas transfer tubing was modified to remove the problem of gas remaining in the transfer tube which had caused uncertainties in thin layer thicknesses. Target support foil spacing was reduced from 39 mm to 17.9 Chapter 4. Experimental Runs 107 ID Run Target Target GMU Upstream Downstream (xlO 6 ) II-1 1625 SET1 (1000 T-Z H 2 with 0.1% T 2 ) 20 T-Z 51.882 II-2 1630 SET1 Q 3 T-l 20 T-Z 17.250 1631 SET1 6 3 T-Z 20 T-Z 31.073 subtotal 1530-31 48.323 II-3 1635 SET1 <£ 6 (3+3) T-l 20 T-Z 50.392 II-4 1640 SET1 6 14 (3+3+8) T-Z 20 T-Z 33.483 II-5 1650 1000 T- ' H 2 - 41.391 1654 1000 T- ' H 2 - 43.501 subtotal 1530-31 84.892 II-6 1657 SET1 - 24.136 1658 SET1 - 38.082 subtotal 1530-31 62.218 II-7 1663 SET1 e 14 T-l - 39.791 1664 SET1 6 14 T-Z - 20.926 subtotal 1663-64 60.717 II-8 1666 SET1 6 14 T-Z 500 T-Z H 2 46.745 1667 SET1 8 14 T-l 500 T-Z H 2 8.073 subtotal 1666-67 54.818 II-9 1671 SET1 ® 14 T-Z 500 T-Z H2( B 3 T-Z 57.561 1673 SET1 ® 14 T-l 500 T-Z H 2( B 3 T-Z 46.514 1674 SET1 ® 14 T-Z 500 T-Z H 2( B 3 T-Z 32.627 1675 SET1 ® 14 T-Z 500 T-Z H 2( B 3 T-Z 57.345 1681 SET1 ® 14 T-l 500 T-Z H 2( B 3 T-Z 38.143 1682 SET1 ® 14 T-/ 500 T-Z H 2( B 3 T-Z 81.813 1683 SET1 ffi 14 T-l 500 T-Z H2( B 3 T-Z 54.574 1684 SET1 e 14 T-Z 500 T-Z H2( B 3 T-Z 41.567* 1685 SET1 ® 14 T-Z 500 T-Z H2( B 3 T-Z 29.243* 1686 SET1 ® 14 T-Z 500 T-Z H2( B 3 T-Z 30.421* subtotal 1671-86 469.808* 11-10 1688 SET1 e 14 T-Z 500 T-Z H2( B 6 (3+3) T-Z 50.370* 11-11 1690 SET1 (B 14 T-Z 500 T-Z H2( B 20 (3+3+8) T-Z 54.109* Table 4.3: Run summary for Series 2 (part 1) Chapter 4. Experimental Runs 108 ID Run US Target DS Target GMU (xlO 6) 11-12 1708 SET2 (1000 T-l H 2 with 0.2 % T 2 ) - 46.441 11-13 1709 SET2 © 14 T-l - 52.875 1711 SET2 © 14 T-l - 11.266 1712 SET2 © 14 T-l - 40.259 1713 SET2 © 14 T-l - 14.359 1714 SET2 © 14 T-l - 48.087 subtotal 1708--14 166.846 11-14 1719 SET2 © 14 T-l 3 T-l 58.499 1723 SET2 © 14 T-l 3 T-l 51.871 1728 SET2 © 14 T-l 3 T-l 37.135 1729 SET2 © 14 T-l 3 T-l 38.114 1730 SET2 © 14 T-l 3 T-l 10.839 subtotal 1719--30 196.458 11-15 1741 SET2 © 14 T-l 23 (3+20) T-l 51.847 1742 SET2 © 14 T-l 23 (3+20) T-l 7.385 subtotal 1741--42 59.232 11-16 1693 SET0.5 £ 20 T-l - 22.602 1694 SET0.5 ( 33 20 T-l - 70.664 subtotal 1694-95 93.266 Table 4.4: Run summary for Series 2 (part 2) Chapter 4. Experimental Runs 109 mm, in order to increase at the downstream reaction layer the number of fit for energies 0.5-2 eV, at which the largest resonances were predicted. The MWPC imaging system, which would not have been as effective due to the smaller foil spacing, was replaced by another silicon detector. The z-dimension (beam direction) of both of 2000 mm 2 silicon detectors were collimated to 13.9 mm, to view only the inner side of the Au foils, where target films exist, with reasonable (i.e., non-grazing) angles. Shown in Tables 4.3 and 4.4 is the run summary for Series 2. The standard emis-sion targets with tritium concentration 0.1% (SET1) and 0.2% (SET2) were used with deuterium overlayers shown with © in the table. The GMUs were taken from the GMU scalers, but the runs with (*) in the table (Run 1684-90) had a problem with an ADC affecting silicon detector normalization, which will be corrected in Section 8.1.2. Apart from the data for TOF measurements of molecular formation, test data for new experiments were taken during this run, which resulted in new proposals. These include precision measurement of muonic atom scattering cross sections and direct measurement of the fiCF efficiency-limiting sticking probability. 4.3 Histogramming and analysis tools Data collected during the experimental runs, which were recorded on tape on an event-by-event basis, were analyzed using several different histogramming and analysis pack-ages. A TRIUMF program, MOLLI [198] with the FIOWA histogramming package [204], which was used on the TRIUMF Data Anaysis Vax cluster, was the main mode of the histogram collection. CERN packages PAW and HBOOK, featuring NTUPLE analysis, were used on DEC Alpha Unix servers, when correlations of many different variables were important. Graphic drawing and data manipulation including fitting were done with REPLAY [205] and PHYSICA [206] both TRIUMF programs, as well as PAW, on Chapter, 4. Experimental Runs both V A X and UNIX (Alpha/OSF, Pentium/Linux) platforms. Chapter 5 Monte Carlo simulation codes In order to gain quantitative understanding of the spectra observed by our experiments, comparison with detailed Monte Carlo simulations is essential. Several simulation codes have been developed by different authors. In particular, the prime tool for the quan-titative analysis for the present thesis is SMC (Super Monte Carlo) developed by Tom Huber at Gustavus Adolphus College in Minnesota, USA. In the following section, SMC is described in detail, followed by a brief discussion of other simulation codes used in this thesis. 5.1 SMC Monte Carlo code Developed over several years by Huber, SMC simulates muonic processes in full three-dimensional geometry with energy-dependent cross sections and rates as inputs. The code also simulates the MWPC imaging process, and was a prime tool for quantitative analysis for this thesis. This section explains the physical processes modelled in the simulation. 5.1.1 Muonic atom generation The simulation was started by creation of a muonic atom in the emission layer. The spatial distributions, both in the beam direction z and in the radial direction r, of the muonic atom generation could be controlled by the input file. A uniform distribution 111 Chapter 5. Monte Carlo simulation codes 112 in z and a Gaussian distribution with a flat top in r were used as nominal input. For determination of the radial beam distribution, see Section 6.1. The fraction of muonic protium pp to muonic tritium pt was proportional to that of protium to tritium in the target1. The initial energy of the muonic atom was taken to be a Gaussian distribution with mean 1 eV and standard deviation 1 eV. Hyperfine states of muonic atoms were populated with the statistical weights, i.e., 25% F=0 and 75% F=l for both pp and pt, which are composites of two spin one half particles. 5.1.2 Reaction selection Once a muonic atom is created, various reactions can take place, which include muon decay, elastic scattering, inelastic scattering, muon transfer, muonic molecular formation followed by fusion or back decay, and muon capture (after transfer to a heavy element). Some of the theoretical input for density dependent reactions is given in terms of cross sections, but it is convenient to convert them to rates. For the process i: A; = G{ • Vrei • Hi, (5.1) where vre\ is the relative velocity between the projectile and the target, and n,- is the num-ber density of the relevant target species. Inputs for other reactions are given by rates, some such as molecular formation, dependent on the density and/or collision velocity, while others such as muon decay independent of the above. The simulation was done in an energy dependent manner, and for a given state of the projectile with a particular energy, the probability for ith. reaction Vi, out of m possible 1 Deviations of the capture ratio from the concentration ratio were reported for pionic hydrogen and deuterium (for a review, see [207]), but this does not affect our simulations, because of very small tritium concentrations used in the measurements. Chapter 5. Monte Carlo simulation codes 113 different reactions, is given by: * = -sr*-, (5-2) where A,- is the reduced rate for each reaction, given by Eq. 5.1 for cross section inputs, or by Ai = d • (/> • Xi, (5.3) for input by rates, where C,- is the relative concentration of relevant species, <j> = n/n0 is the target number density in the unit of liquid hydrogen atomic density n0 = 4.25 • 1022 c m - 3 , and Aj is the rate normalized to no- The processes such as muon decciy and nuclear muon capture are independent of the target concentration d. For muon decay, for example, A,- is X0Qk, where A 0 is the free muon decay rate and Qk the Huff factor2 in the &th element. 5 .1 .3 Time and position evolution The time between reactions r r is given by log(RAN) (5.4) where RAN is a random number between 0 and 1. This follows from the requirement that d (exp Tr Ttot dr const. (5-5) where Ttot — \ ^ L J A i ) 1 S ^ e ^°^ a ^ r e a c t i ° n time constant. 2See Section 6.4 for the description of the Huff factor. Chapter 5. Monte Carlo simulation codes 114 The position X of the reaction is calculated from X = via\, x rr. If X lays outside the boundary of the current medium, the projectile is advanced to the edge of the new medium, from which a new random number is generated for Eq. 5.4. 5.1.4 Muonic processes In this section, we discuss the details of physical processes modelled in the simulation. Elastic scattering ppF + p -> fipF + p, F = 0,1 pdF + d^ pdF + d, F = §, | HPF + t -> ppF + p, F = 0,1 (5.6) H P f + d^np F + d, F = o,i HPf + t -> fiPF + t, F = O,I lidF + p —> fj,dF + p, F 1 3 2 ' 2 lxdF + t -> iidF + t, F=\,\ iit F + P ntF + P, F = O,I fitF + d/itF + d, F = 0,1 (5.7) Elastic scattering cross sections for the symmetric (Eq. 5.6) and asymmetric (Eq. 5.7) cases were taken from Bracci et al. [16], and Chiccoli et al. [17], respectively. These are known as "Nuclear Atlas cross sections," since the calculations assume scattering with the bare nucleus, as opposed to the atom or molecule. For the symmetric case, differential cross sections published by Melezhik and Wozniak [18] were used to give the final angular Chapter 5. Monte Carlo simulation codes 115 distributions. Similar differential cross sections for asymmetric collisions have not been published yet, but were calculated by Wozniak [208] and used as inputs to the simulation. There are more recent calculations which take into account the molecular effects [23] as well as solid state effects [167]. However, not all these cross sections are available in differential forms as of the writing of this thesis. Because we are interested in the transport of muonic atoms, it is essential to use differential cross sections, and therefore we chose to use the above "nuclear" cross sections which are available in differential form. For energies above about 0.2 eV, the molecular and solid state effects are small, therefore for the resonant molecular formation at pt energies of 0.5 - 2 eV, the use of nuclear cross sections should be a good approximation. At lower energies, however, solid state processes become increasingly important. Spin flip ppF=0 + ppF=1 + p -0.1820 eV ppF=1 + p^ ppF=0 + p +0.1820 eV pdF=° + d^ pdF=1 + d -0.0485 eV pdF=1 +d-+ pdF=0 + d +0.0485 eV ptF=0 + t -> ptF=1 + t -0.2373 eV ptF=1 + t -> ptF=0 + t +0.2373 eV (5.8) Spin flip cross sections are also taken from the Nuclear Atlas by Bracci et al. [16] with differential cross sections provided by Melezhik and Wozniak [18]. Spin flip in asymmetric collisions such as pt + p requires relativistic interactions, hence is much suppressed. Cohen calculated those cross sections and found that they are smaller by 6 to 8 orders of magnitude compared to their elastic scattering counterparts [21]. Therefore Chapter 5. Monte Carlo simulation codes 116 we do not consider the asymmetric spin flip in our simulations. Charge exchange pp + d -> fid + p + 134.709 eV pp + t-+ pt + p+ 182.751 eV pd + t -> pt + d + 48.042 eV (5.9) pp + Z -+ pZ + p pd + Z -+ pZ + d pt -\- Z —> pZ + t (5.10) For isotopic charge exchange, or muon transfer reactions (5.9), differential nuclear cross sections similarly calculated by Chiccoli et al. [17] and Wozniak [209] are used, except for pp + t —>• pt + p, where the rate measured by our group, \ p t = 5.86 x 109 s _ 1 (energy independent) [83], was used as a nominal input. Although the experimental Apt, obtained in similar conditions to this thesis, appeared more reliable than the theory, simulations with the theoretical A p i were also performed, wherever possible, to check the systematic effects. Muon transfer to heavy elements takes place when muonic hydrogen reaches target materials such as gold. The rate for these reactions is taken to be a high value (10 1 2 s _ 1 ) to ensure rapid transfer in the simulation. Nonresonant molecular formation pp + p^ppp pd + p —> ppd pt + p ppt (5.11) Chapter 5. Monte Carlo simulation codes 117 Since Faifman's predictions of nonresonant molecular formation rates [51] depend only weakly on the energies, a constant rate from Ref. [51] was used: A p / X j = 5.6 X 106 s _ 1 for p/id and XPflt = 6.5 x 106 s _ 1 for pfit formation. For pfip formation, on the other hand, the measured rate by our group A p w = 3.2 x 106 s _ 1 [83] was used in the simulations. Resonant molecular formation fitF + D°rth° -> [(dfit)dee], F = 0,1 (5.12) fitF + Dp2ara [[dfit)dee], F = 0,l (5.13) As discussed in detail in earlier chapters, these are indeed the key rates which we wish to test by our measurements. The nominal rates for the resonant formation were taken from the work of Faifman and his colleagues (Refs. [70-72,133]), calculated for isolated target molecules at 3 K with the quadrupole interaction included, but correlations among target molecules not taken into account. These rates are calculated separately for different fit hyperfine states, and target molecule rotational states (ortho and para). Since our D 2 layers were made by rapid freezing of warm gas, it is expected that the ortho-para ratio is statistical (2 to 1), and is so assumed in the simulations. The predicted rates by Fcdfman are given Fig. 2.4. A peak value for F = 1 in ortho D 2 for example is 1.16 x 10 1 0 s _ 1 with a resonance energy of 0.45 eV. Fusion and back decay dfit^n + a + n.QMeV (5.14) {(dfit)dee] -*fitF + D2, F = 0,1 (5.15) Once the molecule is formed, fusion can take place rapidly releasing the energy. If the formation took place resonantly, it can break up through a back decay. The branching Chapter 5. Monte Carlo simulation codes 118 ratio for fusion (W) versus back decay (1 — W) is given by Faifman et al. [70-72,133]). The hyperfine state of fit after the back decay was kept the same as before the molecular formation. The outgoing energy of pt was one of the free parameters in the simulation. 5.1.5 Muon decay and imaging The decay rate of a muon in muonic hydrogen is taken to be the same as that of a free muon (A0 = 0.455 x 106 s _ 1), ignoring the small nuclear capture rate (A^2 = 531 ± 33 s _ 1 [210]). When muon decay is selected as the reaction, an imaging routine is called to simulate the MWPC response. Two MWPC planes (instead of three as in the actual experiment), as well as the cop-per thermal heat shield and stainless steel vacuum window, were defined in the program. The electron was generated isotropically (within a cone containing the MWPC planes) from the muon decay position, and passed through the shield and the window, where mul-tiple scattering deflected the electron angle according to a Gaussian distribution with a width given by 9 = 1 3 '^ e V'V^/X~o[l + 0.038 \n(x/X0)} (5.16) where p, f3c are the momentum and velocity of the electron, and X/XQ is the medium thickness in radiation lengths [211]. The electron energy spectrum J-dec{E) is approxi-mated by 1 exp f J E ~ 5 3 n i f 0 < £ < 5 3 M e V V2%adec \ 2<jjec J (517) 0 otherwise with a dec = 50/(2v/21ri2) MeV (normalization arbitrary). This is half of a Gaussian with FWHM of 50 MeV, displaced by 53 MeV, which is sufficiently close to the real spectrum for our purposes. When the electron track intersected with the ith MWPC at [a;,-, y,-, Z{], Chapter 5. Monte Carlo simulation codes 119 the position was smeared in y-z plane by a Gaussian with a distribution of standard deviation ,o^c, characterizing the finite resolution of MWPCs. The new positions were fitted with a straight line, and extrapolated back to the perpendicular plane bisecting the target, similar to the way the real data was analyzed. 5.2 Other simulation codes 5.2.1 Full muonic processes MCKIN, developed by Valeri Markushin3 of Russian Research Center, Moscow, was used in the early stages of the experiment, providing guidance for planning the measurements. As well, a program developed for TRIUMF Experiment 742 by Jan Wozniak of Insti-tute of Physics and Nuclear Techniques, Cracow, Poland, was used to test some of the systematics of the measurements such as solid state effects in muonic atom scattering. 5.2.2 Specific processes In addition to the above codes for full simulation of muonic processes, various smaller programs were used to study specific parts of the experiment. A GEANT-based code was developed by Marshall and used to understand the energy deposited in the liquid scintillators by the muon-decay electrons. A charged-particle energy loss calculation program APEC-97[212] featuring a package PEPPER[213], which I originally developed for the planning of the muon-alpha sticking experiment, was used to study the energy spectra of silicon detectors as well as for determining detector solid angles and energy cut efficiencies. The detail of these codes will be discussed in the analysis sections. 3 N o w at Paul Scherrer Institute, Switzerland. Chapter 6 Analysis I - Absolute normalization 6.1 Effective target thickness Because of the non-uniformity of the target observed in Section 3.3, the average layer thickness depends on the width and profile of the beam which stops in the target. Also recall that we have measured only the profile in the Y (vertical) dimension, hence the horizontal profile has to be assumed. The effective thickness can be defined via where T,- is the thickness at the i i h measured spot, and W{ the weighting factor. A weighted root-mean-square deviation of thickness is defined via This is a quantitative measure of the non-uniformity and is useful when optimizing the vertical position of the diffuser for deposition. Figure 6.1 illustrates the dependence of the effective thickness on the beam parame-ters. The average was calculated assuming rotational symmetry of the thickness profile, and weighted with two different beam parameterizations (i) Gaussian beam, and (ii) a flat top Gaussian, for which the radial intensity at the distance r, f(r) is defined by: WiTi (6.1) (6.2) (6.3) 120 Chapter 6. Analysis I - Absolute normalization 121 4.0 3.5 a. in C D c > 3.0 h 2.5 h £ 2.0 (a) FG: no R c u t - o f f (b) FG: R<30 mm (c) FG: R<35 mm (d) G: no R c u t - o f f (e) G: R<30 mm (f) G: R<35 mm 10 20 30 40 Beam width parameter a [mm] 50 1.2 h 1.0 ? 0.8 .2 0.6 I 0.4 cu :B 0.2 C J C D h 0.0 (b) (d) (e) 10 20 30 40 Beam width parameter a [mm] 50 Figure 6.1: Dependence of the effective thickness (above) and the effective deviation (be-low) on the beam parameters and the radial cut-off values, (a) (b) (c): Flat-top Gaussian beam (FG, defined in Eq. 6.3) with Rflat = a and F W H M S = 2.355 x ag = a/2, (d) (e) (f): Gaussian beam (G) with F W H M S = a. The points are slightly shifted horizontally for visual ease. The error bars represent the 5% uncertainty from the stopping power. Chapter 6. Analysis I - Absolute normalization 122 Imaged Y posi t ion [ m m ] Figure 6.2: The Y-distribution of decay electrons in the upstream layer imaged by the M W P C system plotted with error bars is compared with the Monte Carlo simulations, in the histograms, assuming Gaussian distributions of F W H M 20, 25, and 30 mm (from inside out) as the initial muon beam distribution in the XY plane. The figure also shows the dependence on the radial cut-off values which reflect the physical limit of the beam radius. For the upstream layer, the cut-off value can be considered to be 32.5 mm which is the radius of the thin gold target support frame (beyond this radius the muon would have to go through more than a millimeter of copper). As for the downstream layer, which is directly facing the upstream layer, there is no collimation of the pt beam due to the target support frame, hence an R cut-off of 35 mm in the X direction (from the external rectangular size of the gold plated copper frame) and slightly larger in the Y dimension can be expected. As can be seen from Fig. 6.1, the average thickness depends on beam profile, especially if the beam width is large. Since the knowledge of the thickness is very important, all the available information Chapter 6. Analysis I - Absolute normalization 123 Imaged Y posi t ion [ m m ] Figure 6.3: Similar to Fig. 6.2 except simulations are with the initial muon distribution being Gaussian, with flat top. The histograms, from inside out, show the beam distribu-tion of 808, 10©10, 12012, 15015, where first number is the flat top radius Rfiat and the second is the full width at half maximum of the Gaussian part F W H M 3 = 2.355 x ag (see Eq. 6.3). needs to be combined to determine the effective thickness. We attempted this in the following manner: (1) first parameterize the muon beam distribution (in the XY plane) from the image of decay electrons obtained by the MWPC system, (2) then use that as an input to the Monte Carlo simulation to calculate the distribution of the [d beam reaching the downstream layer, and (3) finally take a weighted average for each of the upstream and downstream layers, using the assumed beam profiles. Note that depending on the thickness of the upstream moderator, the fit beam profile at the downstream layer, hence the effective thickness, could be different. The knowledge of the beam profile is also necessary for the determination of the silicon Chapter 6. Analysis I - Absolute normalization 124 US beam effective thickness VI si RJlat (mm) F W H M , (mm) (^g-cm"2 (T-l)'1) (%) 0 20 3.461 2.355 0 25 3.373 2.392 10 10 3.493 2.336 12 12 3.449 2.394 15 15 3.281 2.462 Table 6.1: The effective upstream (US) layer thickness with different beam parameters. Also shown is the corresponding acceptance fis;, for the silicon detector. detector acceptance, which is tabulated in Tables 6.1-6.3, together with the effective thicknesses, but will be discussed in the following section. Figures 6.2 and 6.3 compare the experimental data and simulations, with different input parameters, of the Y distribution of the decay electrons image in the upstream layer. The data were obtained via the MWPC imaging system, while the Monte Carlo code, SMC, simulated the imaging process in the detector with the initial muon beam stopping distribution and the wire chamber resolutions as input parameters. In Fig. 6.2, Gaussian distributions with varying FWHM were assumed for the initial beam distribution in the XY plane, while flat-top Gaussian distributions, defined by Eq. 6.3, with varying Rpat and o~g were used for Fig. 6.3. The Gaussian beam of FWHM 20 ~ 25 mm, and the flat-top Gaussian with the flat top radius Rpat of 10 ~ 12 mm and Gaussian F W H M 3 X of 10 ~ 12 mm seem to reproduce the experimental data rather well. The resulting effective thicknesses are summarized in Table 6.1. The variation in the thicknesses with these values of beam parameters is less than 3%. The wire chamber resolution, cr™c, of 1 mm is used for this analysis, but variation of the wire chamber resolutions between 0.2 mm and 4 mm did not affect this conclusion. 1 The subscript g is given here to stress it is the width only of the Gaussian part of the beam, as opposed to that of the entire beam. Chapter 6. Analysis I - Absolute normalization 125 Ln t ry into downs t ream layer 6 0 0 CO c o o 4 0 0 h 2 0 0 5 0 0 -100 \-0 - 4 0 - 2 0 0 20 4 0 m m Figure 6.4: Example of the simulated radial profile of the /it beam entering the down-stream layer (line plotted with error bars), and its parameterization using a flat-top Gaussian function (histogram). The profile of the pt atomic beam reaching the downstream layer generally depends on, but differs from, that of muon beam stopping in the upstream layer. The SMC, with all the physics in it, was used to simulate the former, using the latter as SMC input. The resulting profiles were parameterized similarly to the upstream case. Figure 6.4 illustrates an example of the simulated radial profiles of the pt beam reaching the downstream layer (plotted with error bars), for which the input to the simulation of a flat-top Gaussian beam, with Rpat = 12 mm and F W H M a = 12 mm, was assumed for the upstream beam profile. Shown as a histogram is a parameterization of that profile using a flat-top Gaussian function. The flat radius of 4 mm and Gaussian FWHM of 28 mm give a reasonable x 2 per degree of freedom (DOF) of 1.05. If a Gaussian Chapter 6. Analysis I - Absolute normalization 126 distribution is assumed, a FWHM of 32.4 mm with x 2 /DOF of 1.80 was obtained in the fit. Table 6.2 summarizes the effective thickness for the downstream layer with the dif-ferent US beam distribution. For the US beam of 10010 mm, a single Gaussian function fits the simulated /it distribution entering the DS target reasonably well, but for larger US widths, the fitted FWHM values depend on the region of the fit (e.g. [-29;29] mm vs. [-35;35] mm), and a flat-top Gaussian function gave better x 2 a s w e l l a s a more stable fit. The different DS beam parameters for the same US parameter in the table reflect variations due to the fitting region size. The deuterium overlayer of 48.3 (~ 14 T-l) was assumed for the simulations. As discussed in Section 3.3.2 in Chapter 3 (see Fig. 3.16 in page 86), the layer thickness depends on the distance between the diffuser surface and the target foil surface according to our film deposition simulations. Although the distance for the upstream layer was similar to the calibration measurements in Section 3.3 (~ 8 mm), that for the downstream foil was closer, i.e. ~ 2.8 mm. Therefore the calibration factors have to scaled by a factor 1.106±0.034. This factor was determined by comparing the areas between the simulations for the foil distance 8 mm and 2.8 mm in Fig. 3.16. About 3% uncertainty in the scaling is due to the choice of the interval within which the ratio was taken; [-15, 15] mm and [-30, 30] mm are the two extreme intervals considered, and the average between them was used as the scaling factor. In general, the fit beam profile at the downstream layer depends on the thickness of the overlayer in the upstream layer. This is partly due to the more forward peaked angular divergence of fit going through the overlayer, and also because the moderated fit with larger angles are less likely to survive to reach the downstream layer due to the longer flight path. This effect was investigated in Table 6.3 with the US beam of 10010 mm assumed, and found to have about 6% effect in the effective thickness. Chapter 6. Analysis I - Absolute normalization 127 US b earn (mm) DS b earn (mm) Effective thickness Rflat F W H M 3 Rflat F W H M 5 (^g-cm-2 (T-0" 1) (%). 10 10 0 28.8 3.60 2.44 12 12 0 32.1 3.50 2.47 0 33.8 3.45 2.48 4 28 3.53 2.46 15 15 0 42.1 3.25 2.50 0 37.2 3.36 2.53 7.5 30 3.36 2.51 0 20 0 28.0 3.63 2.43 0 25 0 31.8 3.51 2.47 Table 6.2: The effective thickness and the silicon detector acceptance (Qsi) for the down-stream (DS) layer, evaluated with different parameterizations of the /it beam profile at DS, which in turn were simulated using different US profiles as input to the Monte Carlo. The upstream moderating overlayer of 14 T-l is assumed for the simulation. See the text for the details. The summary of effective thicknesses and Si solid acceptance will be given in Table 6.5. US overlayer DS b earn (mm) Effective thickness Si thickness (T-l) Rflat F W H M a (^g-cm-2 (T-Z)-1) (%) 0 0 35.8 3.40 2.49 3 0 32.1 3.50 2.47 6 0 30.6 3.55 2.46 14 0 28.7 3.60 2.44 Table 6.3: The effective thickness of the downstream (DS) layer with different upstream overlayer thickness, and the corresponding silicon detector acceptance. The US beam parameter of the flat-top Gaussian 10©10 mm was used as input for simulations. 6.2 Silicon detector acceptance The silicon detector acceptance depends both on the geometry and the distribution of the beam (which determines the distribution of the particle source). The Monte Carlo method was used to determine the acceptance. We define our Si acceptance flsi as the Chapter 6. Analysis I - Absolute normalization 128 Source Uncertainty Relative change in Si Acceptance beam X position shift ±5 mm +2.7 % beam Y position shift ±5 mm -0.9 % Si distance —0.5 mm +2.1 % Si distance +0.5 mm -1.9 % Si Z position shift — 1 mm -2.4 % Si Z position shift + 1 mm + 1.7 % collimator width -0.2 mm -1.4% collimator width +0.2 mm + 1.5 % Total uncertainty +4.1 % -3.5 % Table 6.4: Variations in Si acceptance due to uncertainties in the geometry. average of two silicon detectors (Sil and Si2), as this quantity is much less sensitive to the beam position shift as we shall see. The position of the silicon detectors is rather well defined, since they are mounted on the thermal shield box made of gold plated copper. A copper collimator sheet (aperture of 13.9+0.2 mm x 50+0.2 mm at room temperature) in front of the detector, further defined the acceptance. Thermal contraction, for both the thermal shield box and the collimator sheet, is taken into account using the values given in Ref. [214] and is estimated to give a combined relative correction of about 1.3% to the Si acceptance. Uncertainties due to the possible variations in the geometry are summarized in Table 6.4, where the total uncertainty is given as a quadratic sum of the all the uncertainties. Note that shifting the beam with respect to the target centre in the X direction always increases the solid angle (the two detectors are symmetric to the beam on the X axis), while shifting in the Y direction always reduces it. The variations of the silicon detector acceptance due to the beam distribution and its parameterizations were already given in Tables 6.1 and 6.2, for the upstream layers and the downstream layers, respectively. Chapter 6. Analysis I - Absolute normalization 129 Target Effective thickness (/tg-cm - 2 (T-Q-1) ClSl (%) DS (0 T-l US) DS (3 T-l US) DS (6 T-l US) DS (14 T-l US) US 3.43 ±0.18 3.34 ±0.22 3.44 ± 0.23 3.49 ±0.23 3.54 ±0.23 q 7c+o.ioo .0 I <J_o.086 r 19+0.105 . a i z , _ 0 0 9 0 Table 6.5: Summary of effective average thickness and Si acceptance for all the target arrangements used in this thesis. Except for the case of the US beam of size 15015 mm which appears inconsistent with the electron imaging data (Fig. 6.3), the variations of the Si acceptance due to the beam parameters are found to be less than 1%. Given in Table 6.3 are the changes in the Si acceptance due to varying the upstream overlayer thicknesses, in which about 2% relative difference is found between 14 T-l overlayer and no overlayer. Table 6.5 summarizes the final values and their uncertainties for the effective thickness and Si acceptance for all the target arrangements used in this thesis. The uncertainties for the thickness include the ones due to the beam parameterization, the scaling factor for the diffuser distance (DS only) and the stopping power, which were added quadratically. Uncertainties for flsi are due to the geometry and the beam parameterization, also added in quadrature. Chapter 6. Analysis I - Absolute normalization 130 6.3 Muon stopping fraction determination In order to obtain the normalization, we need to know the fraction of muons which stop in the hydrogen target with respect to the number of entering muons counted by the beam counter T I . Since our layer is rather thin, some of the muons stop in the Au target foils or possibly in the Cu thermal shield. We attempted to determine this fraction by two different methods, both of which use the decay electron time spectrum: (i) the amplitude ratio of different lifetime components in the electron time spectrum (Section 6.4: amplitude ratio method), and (ii) an absolute measurement of the number of electrons decaying with the hydrogen lifetime, with the detection efficiency determined using fusion signals in the silicon detectors (Section 6.5: absolute amplitude method). These methods are largely independent, but we initially observed an apparent discrepancy of some 20% between the two methods. This is why we go into great detail for the stopping fraction determination in the sections that follow. 6.4 Amplitude ratio method The first method takes advantage of the fact that the lifetime of the muon is different in hydrogen and gold, being about 2.2 fis and 70 ns, respectively. The electron time spectrum is measured with a pair of electron scintillation counters surrounding the target chamber (En l , En2, and Ege), and with NE213 neutron detectors (NI and N2) with their fast dynode output (Tell and Tel2) in coincidence with the scintillator pair hit in front of them. The time of the first hit among E n l , En2, or Ege (first E), as well as the first hit of Tel l and Tel2 (first Tel), are also histogrammed, which if we accept only one muon at a time, and if the effect of background is small, should equal to the sum of E n l , Eii2 and Ege, or Tel l and Tel2 respectively. Chapter 6. Analysis I - Absolute normalization 131 The recorded time spectra were fitted to exponential functions with a constant back-ground2. The general form of the fitting function, taking into account capture and other muon losses, as well as the finite size of the time bin, is derived in Eq. 6.4 in Appendix C: M exp - A f c (ti - I; At — exp 1 - A f c ( U + -At + bkgd, (6.4) where for kth element [k = 1... M) , e& is the electron detection efficiency, Qk the Huff factor, the initial muon population, and A^ the muon disappearance rate in element fc, with A 0 being that in free space. The above T^%t is a function of a discrete time variable ti with interval At. Let us consider the specific case of M = 2 (two exponentials), where we assume all observed muon decay takes place in one of the two materials. Also we take k = 1 for a heavy element such as gold present in the system, and k = 2 for hydrogen3. We define the fraction of muon stopping in hydrogen SH, hence N° = N°(l — SH) and = N°Sa-in actual fits, the time histogram is normalized to GMU (number of "good muons"), and thus the fit function is defined to give, together with the decay rates A^, the normalized amplitudes Ak = e^C^A^/GMU. Hence, the stopping fraction can be determined from the ratio A2 ?AR _ N$ _ t2Qi OAK _ _ ^Wi. (n r\ e i Q i e2Q2 the superscript "ARn indicating the stopping fraction being derived from the amplitude 2 T h e validity of assuming a constant background will be discussed later. 3 A fit with M = 3 was also performed to investigate a possible existence of the copper component in the time spectrum, with two of the three lifetimes fixed to those of A u and C u from Ref. [215], but this did not affect the derived hydrogen stopping fraction. Fixing of two lifetimes was necessary to avoid a fitting problem due to a strong correlation between them. Chapter 6. Analysis I - Absolute normalization 132 ratio. We define a reduced stopping fraction as, ^AR _ A2 S« = AJQTTT,' (6'6) which would equal SJJR, if we assume (a) Q = 1 for hydrogen, and further (b) ex = e2. The assumption (a) is valid since the binding energy of muonic hydrogen (~ 2.5 keV) is much smaller than the average electron energy (~ 35 MeV), but (b) needs careful consideration, which I shall revisit later. 6.4.1 Electron scintillator measurements Shown in Fig. 6.5 is an example of the electron time spectrum and fit with two exponential functions with a constant background. Plotted with error bars are the first electrons detected by electron pair scintillators for a target of 1000 T-l pure hydrogen, while the solid curve is a fit in the time interval of [0.02, 6] /is. The resulting fit amplitudes ( A i / Q i , A 2 , and bkgd), normalized to G M U , and the lifetimes (1/Ai and 1/A 2) are given in Table 6.6, for the fit from each detector (Ege, E n l , and En2) as well as that from the first hit in the three detectors (1st). Also shown is the reduced muon stopping fraction in hydrogen (SH ) derived from A2/(Ai/Qi + A 2 ) with the Huff factor for A u of Qi = 0.85. The data of runs 1650 and 1654, taken with the same conditions, were analyzed both separately and together. Fits are mostly of satisfactory quality, but in some cases we observed rather low con-fidence levels. This could be in part due to beam related background which is correlated to the 23 MHz (43 ns) cyclotron R F cycle. This periodic background, when fitted with a straight line, would increase the total x2, but should not affect the fit parameters for the signals, since it is the average over the relevant time scale that affects the fit results4. 4 T h e time scales for the R F cycle and the gold signal are rather similar, but the background amplitude is several orders of magnitude smaller, so the effect should be negligible. Chapter 6. Analysis I - Absolute normalization 133 10~6 1 1 1 1 1 1 1 0 1 2 3 4 5 6 Figure 6.5: A n example of electron t ime spectrum and fit w i th two exponential functions w i th a constant background. Plotted wi th error bars are the first electrons detected by electron pair scintil lators, while the solid curve is a fit in the t ime interval of [0.02, 6] ps. Our fitted short l i fet ime 1/Ai is reasonably consistent w i th 74.3±1.5 ns for gold given by Suzuki, Measday and Roalsvig [215], but the long lifetime I/A2 is somewhat shorter than 2.195 ps for hydrogen [215]. In the single hi t mode 5 , i f the detection efficiency is high, there could be a distort ion in the recorded accidental background, since the detector is more likely active at earlier t ime. But the individual detector fits and the fit to the first hi t among them ("1st") are consistent w i th each other, indicating that the effect of accidental background in distort ing the fit is negligible ("1st" can be considered as one single-hit detector w i th an efficiency three times as large as a single detector, hence if an efficiency dependent 5 I n this mode, the detector can accept only one event per incident muon, as opposed to the multi hit mode in which multiple events per incident muon can be recorded. Chapter 6. Analysis I - Absolute normalization 134 Run Det. ( io- 2 ) A 2 (IO"2) bkgd (IO"4) 1/Ai (ns) 1/A 2 (ps) (%) X2 /dof(cl) 1650 Ege E n l En2 5.06(13) 4.86(13) 4.86(12) 1.760(9) 1.734(9) 1.563(9) 2.4(4) 2.3(4) 2.9(4) 76.9(20) 76.0(21) 79.0(21) 2.030(23) 2.027(24) 2.041(26) 25.81(50) 26.28(52) 24.34(48) 0.99 (71%) 1.02 (18%) 1.05 (3.8%) 1st 14.71(22) 5.061(17) 6.9(7) 77.1(12) 2.029(15) 25.59(30) 1.09 (.03%) 1654 Ege E n l En2 4.72(12) 4.71(12) 4.95(13) 1.770(10) 1.722(9) 1.584(9) 2.5(4) 2.1(4) 3.0(4) 80.7(22) 77.8(21) 74.9(20) 2.003(23) 2.043(24) 2.012(24) 27.28(51) 26.75(51) 24.26(48) 1.01 (32%) 0.99 (69%) 1.04 (7.0%) 1st 14.30(21) 5.081(16) 6.8(7) 77.6(12) 2.017(13) 26.22(29) 1.02 (26%) 1650 + 1654 Ege E n l En2 4.88(9) 4.77(9) 4.90(9) 1.765(7) 1.728(7) 1.574(7) 2.4(3) 2.2(3) 2.9(3) 78.8(15) 76.9(15) 76.9(15) 2.016(16) 2.035(17) 2.025(19) 26.56(36) 26.52(37) 24.30(36) 0.99 (62%) 1.04 (8.6%) 1.13 (.01%) 1st 14.50(16) 5.071(12) 6.8(5) 77.4(9) 2.023(10) 25.91(21) 1.09 (.04%) Table 6.6: The result of a fit to the electron time spectrum with two exponential functions with a constant background, and the muon stopping fraction to hydrogen (SH ) derived from the amplitude ratio, A2/(A1/Q1 + A2) where Q i = 0.85 is the Huff factor for Au . The fit amplitudes ( A i / Q i , A 2 , bkgd) are normalized to G M U . Fits to the spectrum from each detector (Ege, E n l , En2) as well as that for the first hit in the three detectors (1st) are listed. effect was important in the fit, it would show up as the difference between the 1st and other detectors). In fact, fits with explicitly non-constant backgrounds were tried, but the stopping fraction was found rather insensitive to the background model. The dependence of the derived stopping fraction SH was tested by changing the fit region from [0.02;6] ps to [0.02; 9.5] ^ s. While the H 2 component lifetime was slightly increased (~ 2%), the variation in stopping fraction was negligible. We have also fitted the later time t > 1 ps, at which the gold signal is negligible, to a single exponential and a background. In these fits, in which 100 ns bin size was used (averaging out the R F structure), a lifetime of 2.11(1) ps was obtained with a confidence level of 8%, a lifetime closer to, yet still significantly smaller than, the literature value for muons in hydrogen. An exponential background was also tried, giving 1 /A 2 = 2.07(3) ps Chapter 6. Analysis I - Absolute normalization 135 St0 (ns) ST (%) (SSH )rel X2/dof cl (%) - 4 27.07 (23) 4.4% 1.09 0.053 - 2 26.45 (22) 2.1% 1.09 0.045 - 0 25.91 (21) 0 % 1.09 0.043 +2 25.38 (21) -2.0% 1.09 0.041 +4 24.85 (20) -4 .1% 1.09 0.039 Table 6.7: Sensitivity of the reduced stopping fraction SH to the shift of time zero (6to) for the first electron time spectrum for the sum of runs 1650 and 1654. (SSH )rei is the relative shift with respect to the 8t0 = 0 value. The variation in \2/dof is smaller than the last digit shown. with a background lifetime of 0.0829 /is with a confidence level of 11% 6. The amplitude for the hydrogen component thus derived is somewhat smaller than the ones in Table 6.6, but by no more than 3%. Alternatively, fits were tried with the lifetime fixed to 2.195 ^s and with a constant background (varying the amplitude) or an exponential background (varying the lifetime and the amplitude). Neither background gave a satisfactory fit, suggesting that the deviation of the lifetime is unlikely due to a trivial error in modeling the background, thus pointing to the existence of one or more muon loss mechanisms. This could include a small amount of non-hydrogenic contamination in the target, which would not affect the extraction of the stopping fraction, as long as it is well approximated with an exponential function. Emission of muonic protium from the solid layer, recently observed by our collaboration for the first time [216], could be partly responsible for the discrepancy also. According to Wozniak [217], several percent of muonic protium atoms may be emitted back to the gold foil, whereby the muon transfers and is captured by the gold nucleus. 6 One could conceive the background to have a form exp(—e\t), where e is the detection efficiency and 1/A the signal lifetime, if the background is an accidental one. In fact, 0.0829/A2 = 0.172, derived from the above, is close to the combined efficiency for the 1st electron obtained in Table 6.17 in page 157. Chapter 6. Analysis I - Absolute normalization 136 Det. (IO" 2) A 2 (IO"3) bkgd ( io - 8 ) 1/Ai (ns) 1/A2 (ps) (%) X2 /dof(cl) Tel l Tel2 1.43(3) 1.48(3) 8.19(2) 7.21(2) 2.3(2) 5.4(2) 79.1(16) 78.5(16) 2.092(6) 2.089(7) 36.34(38) 32.81(36) 1.02 (18%) 1.07 (.04%) 1st 2.90(4) 15.4(3) 7.6(3) 78.8(11) 2.089(4) 34.66(34) 1.06 (.3%) Table 6.8: The results of fit to the electron telescope time spectrum for the sum of Runs 1650 and 1654, to be compared with the scintillator fit in Table 6.6. The fits for Telescope 1 (Tell), located on the beam left, and Telescope 2 (Tel2) downstream, together with first hit of the two (1st) are listed. The sensitivity of SH to the shift of the time zero definition (6t0) was also investi-gated. An example for the first electron spectrum is given in Table 6.7. The fit to the individual detectors as well as to individual runs gives similar trends. The error St0 = ± 4 ns is a rather conservative estimate of the shift, and it is more likely to be less than ± 2 ns. The variation in the amplitude for the hydrogen component in this time scale is, AR nevertheless, completely negligible, hence the change in SH comes from that in the gold component. In any case, this alone cannot explain the 20% difference between method 1 and method 2, as will be discussed later. AR While SH for Ege and E n l are consistent with each other, there is a systematic difference between En2 and the others, with En2 being relatively lower by nearly 10%. This discrepancy will be addressed later in relation to the acceptance difference for the upstream and downstream targets. We note that agreement between Ege and E n l , which were symmetrically located across the target (beam-left and beam-right, respectively), exclude the possible effects of polarized muon precession about the vertical axis. 6.4.2 Electron telescope measurements Chapter 6. Analysis I - Absolute normalization 137 The liquid NE213 scintillation detectors, normally used for fusion neutron detection, can be used as charged particle detectors by reading out a fast dynode signal. We define a telescope event as the event in which the dynode fires in coincidence with the pair of plastic scintillators in front of it. Telescope 1 (Tell) is in the beam right position (as is Neutron 1 or N l ) , while Telescope 2 (Tel2) was in the downstream position; both used an independent data acquisition trigger. Given in Table 6.8 are the results of fits to the telescope spectra and derived reduced stopping fractions SH . The fits were performed.in the time region of [0.02; 9.5] ps, and the time spectra were obtained with a "nominal" energy cut on the dynode output, which was 200 ch< Edy <1200 ch. Fits in this section are done to the sum of runs 1650 and 1654. As can be observed in a comparison of Tables 6.6 and 6.8, the stopping fractions given by the telescopes were systematically higher than those from the electron scintillators. We shall look into possible effects due to the difference in the energy sensitivity of the detectors in the following section. Energy dependence of the stopping fraction We plot in the top of Fig. 6.6 the telescope energy spectra, e.g., the neutron detector dynode output, recorded in coincidence with the corresponding scintillator pair. The time cut of t > 0.02 ps was applied to remove the prompt beam signal (which would otherwise slightly increase counts at energies less than 200 ch). With the minimum ionizing energy loss dE/dx ~ 2 MeV/(g/cm 2 ) for scintillator materials, and our detector depth being about 10 cm (4 inches), the minimum ionizing electrons deposit energy of order 20 MeV. The muon decay electrons (in free space) range from 0 to about 53 MeV in energy, hence some would stop in the detector, while others would go through. Thus, the peak in Fig. 6.6 (above) near channel 700 should be due to the minimum ionizing electrons corresponding to some 20 MeV. Detailed G E A N T calculations indeed confirm Chapter 6. Analysis I - Absolute normalization 138 35000 Dynode energy (ch) < ' 0 200 400 600 800 1000 1200 Dynode energy (ch) Figure 6.6: (Top) The energy of the dynode output for the telescope events. The peak near 700 is due to minimum ionizing electrons going through the detector, depositing nearly 20 MeV. (Bottom) Energy dependence of SH (=A 2 / (Ai / (5 i + A2)), the muon stopping fraction in hydrogen obtained from the ratio of amplitudes, in a two-exponential lifetime fit to the telescope time spectrum. The width of the histogram bins roughly corresponds to the intervals of the energy cuts to which each fit was performed. Chapter 6. Analysis I - Absolute normalization 139 this picture [218]. The bottom figure in Fig. 6.6 illustrates the reduced muon stopping fraction in AR hydrogen determined from the amplitude ratio of a two-exponential lifetime fit, SH (= A2/(A1/Qi + A2)), as a function of the energy deposit in the NE213 detector. The fits were performed with the various energy cuts, with their intervals indicated roughly by the position and width of the histogram bars. This clearly demonstrates that the stop-ping fraction determined from the amplitude ratio is dependent on the energies to which the detector is sensitive. Possible mechanisms for this energy dependence are considered AR in the following sections, and will lead to the corrections to SH . Aft 6.4.3 Corrections to SH in the amplitude ratio method In this section, we investigate the possible corrections to the reduced muon stopping AR fraction in hydrogen SH derived from the amplitude ratio, in an attempt to explain —AR the observed energy dependence of SH , as well as to resolve the discrepancy with the absolute amplitude method (Section 6.5). Charged particles from muon capture Muon capture on a proton, H~ + p -> Ufj. + n, (6.7) is an elementary process (at least at the nucleon level), but when the muon is captured on a nucleus the process is rather complex. Muon capture in a heavy nucleus often leads to the emission of particles, normally neutrons but sometimes charged particles. These have the same time constants as the decay electrons, hence if detected in either the scintillator pairs or the telescope, they would bias the determination of the stopping fraction from the amplitude ratio. Chapter 6. Analysis I - Absolute normalization 140 The detection of charged particles would modify the branching ratio of muon decay versus total muon disappearance Bk (Eq. C.4) into an effective branching ratio B,t, mp where Wmp are fractional probabilities per muon capture for emission of m charged particles. Alternatively, the effect can be incorporated as the efficiency for the electron detection tk in Eq. 6.5 £fc = efc 1 + (6.9) QkXo \ / = tkh (6.10) with \3k = e^ /efc defined as a correction factor. Since the ratio \%/Qk^o is ~ 34 for the case of k = gold, a relatively small fraction of charged particle emission can affect the stopping fraction determination, hence this effect needs to be considered in some detail. The charged particle emission following muon capture has been traditionally less studied compared to neutral emission, but there is some recent interest in relation to nucleon pairing in nuclei. A few reviews exist on this subject [219-221]. Following muon capture, the emission of neutrons (and gammas) dominates over that of charged particles (p, d, a etc.), as the latter requires a multi-step reaction or capture on correlated nucleons, since the process (6.7) only gives outgoing neutrons. The total probability of the charged particle emission decreases with increasing Z of the nucleus, due to the increasing Coulomb barrier [219,220], but there is some evidence from recent measurements that the yield of the highest energy protons is independent of Z [221]. Table 6.9 lists measured and calculated probabilities for charged particle emission reactions (p~,pxn), where x is the neutron multiplicity, and p includes deuterons and Chapter 6. Analysis I - Absolute normalization 141 Nucleus Probability per muon capture Wp (%) (a) (b) (c) (d) ^ C u >1.7±0.3 3.6 2.5 ^ x Ta >0.07±0.01 0.30 0.28 0.29 j° 8Pb >Q.30±0.08 0.41 0.11 0.22 Table 6.9: Measured and calculated fractional probabilities (per captured muon) Wp for inclusive charged particle emission reactions (n~,pxn). (a) experimental lower limit for observed channels, (b) estimated total inclusive probabilities derived from (a) using approximate ratio of various channel, (c) and (d) theoretical calculations, (a), (b) and (c) are from Ref. [220], and (d) from Ref. [219]. tritons as well as protons. Column (a) is the experimental lower limit from observed channels, while (b) shows the estimated total inclusive probabilities derived from mea-sured channels, corrected by an approximate regularity among various channels. Column (c) gives calculations by Lifshitz and Singer, and (d) those by Wyttenbach et al. We ignore the charged particle multiplicity in the following analysis, i.e., we let mWmp = Wp. Note that emission of a particles is suppressed by at least several m times [220]. Assuming Wp — 0.4% for gold, we obtain eU„ = 1.14 • from Eq. 6.9, which in turn gives an upward correction to the apparent hydrogen stopping fraction AR SH . The magnitude of the correction varies from 0 to 14% (relative) as a decreasing function of SH , but for SH = 25%, the corrected value will be SfjR = 27.5%, a relative shift of 10%. For copper which is also present in parts of the target system, the emission probability Wp is nearly an order of magnitude higher than gold (Table 6.9). However, the fit to decay electron data with three exponential functions, two of which were fixed to the lifetimes corresponding to Au and Cu measured by Suzuki, Measday and Roalsvig [215], indicated that the apparent Cu component cannot be more than 10% (in most cases less Chapter 6. Analysis I - Absolute normalization 142 Nucleus Authors Eth EQ (MeV) ( io - 4 ) (MeV) S Baladin et al (1979) [222] 40.5 2.07±0.15 6.4±0.1 Ca Martoff et al (1991) [223] 40 2.32±0.26 8.Oil .5 Cu Budyashov et al (1971) [224] 42.5 3.8±1.4 13.1±1.6 Cu Krane et al (1979) [225] 40 19.6±1.2 8.3±0.5 Y Cummings (1991) [221] 40 0.72±0.07 7.2±1.0 Pb Krane et al (1979) [225] 40 1.71±0.28 9.9±1.1 Table 6.10: Integrated yields of proton emission following muon capture, based on the compilation in Ref. [221]. Eth is the energy threshold above which the yield is integrated. EQ is derived from a fit of the energy spectrum to the form exp(—E/T 0). than a few percent) of gold. Furthermore, the capture rate for copper is less than half that of gold, while the Huff factor for copper is larger. A l l this suggests that the relative contribution of charged particles from copper is significantly smaller than that from gold and thus is neglected. Nevertheless, the estimate of 14% effect on eAu is a worst possible case of detecting all the charged particles produced. In reality, the emitted particles can be ranged out in the walls and other materials, hence the effect is expected to be smaller. This of course depends on the energy spectrum of the emitted charged particles, which we discuss in the following. Table 6.10, based on a compilation by Cummings [221], lists previously measured yields TFth of high energy protons following muon capture, integrated above a threshold energy Eth- According to Cummings [221], some of the older results by Budyashov et al. and by Krane et al. may be unreliable, and excluding those would suggest high energy proton yields of l - 2 x l 0 - 4 , independent of the atomic number. For the En2 counter located downstream of the target, there is substantial material Chapter 6. Analysis I - Absolute normalization 143 between the gold target support and the detector, including 1.6 mm gold-plated copper, 2.5 mm stainless steel and 3 mm Plexiglas. According to the ICRU range table [190], the copper and stainless steel alone can stop protons up to ~ 50 MeV. Therefore, the effect of charged particles should be less than of the order of 10 - 3 , which is completely negligible for the SH determination. On the other hand, the amount of material between the gold and electron counters on the sides of the target (Ege, Enl) is considerably less, in particular, for the paths which go through the silicon detector. A rough estimate based on the range table [190] suggests that protons of energies above 20 to 25 MeV may reach the second scintillator E2ge (providing a coincidence of the scintillator pair) after going through the silicon detector (300 / t m ) , a copper thermal shield foil (13 / m i ) , a stainless steel window (76 / m i ) , and the first scintillator Elge (3.2 mm) with three layers of mylar wrappings (3 x 285 / i m ) (recall that Ege = Elge • E2ge). Assuming the proton spectrum from the gold is similar to that from yttrium given by Cummings [221] and using the partial yield above 40 MeV and the exponential decay rate in Table 6.10, the yield above 20 MeV, r 2 u can be derived to be 2.3 x 10~3. This would give BAu = CAU/CAU = 1-08 (see Eq. 6.9), which in turn would give a relative 6% AR correction, if SH = 25%, to 26.5% from Eq. 6.5. However, if the charged particle did not go through a Si detector, but hit the Cu collimator around the active Si area, it would be less likely to reach the scintillators. The relative acceptance of Ege (or Enl) compared to that of Si detector, which was assumed to be unity above, is actually less than a half, hence the effect of charged particles is expected to be smaller than the above, and of a few percent level for the side counters Ege and Enl . Chapter 6. Analysis I - Absolute normalization 144 1 1 1 1 1 1 , i | i i , Free , . lron\ ... O C K : / / / \ ^ -/ // \ \ / / / \ \ -/ 7 / \ / // \ i i 1 i i i 1 i i \ 0 20 40 60 80 Energy (MeV) Figure 6.7: The energy spectra of muon decay electrons, for the muon bound in lead and iron, calculated by Huff [226]. Also plotted is the electron spectrum for the muon decay in free space (without radiative corrections), R(E) oc 16E2[3(1 — 2E) + §/>(8i? — 3)], with the Michel parameter p = 0.75, and the maximum energy of 52.8 MeV. A l l spectra are normalized to a free muon decay rate of unity. E lec t ron energy spectrum As was seen in the previous section, the effect of capture induced charged particles alone —AR cannot explain the energy dependence of SH . Indeed, for the downstream counters, it was found very unlikely to cause any measurable effect. We therefore turn our atten-tion to other effects in these two sections, i.e., electron energy spectra, and upstream-downstream acceptance difference. The energy spectrum of decay electrons from muons bound in a nucleus is known to differ from that in free space, due to effects such as the final state Coulomb interaction and reduced phase space [226-228] (the same effects are responsible for the Huff factor Chapter 6. Analysis I - Absolute normalization 145 describing the reduction in decay rates). Shown in Fig. 6.7 are the electron energy spectra calculated by Huff [226], for muons bound in lead and iron, together with that for free muon decay. As all the spectra are normalized to a free muon decay rate of unity, the integral of the spectra gives the Huff factor (Q = 0.844 for lead, 0.975 for iron). More recent and more elaborate calculations on the energy spectra [227,228] essentially agree with Huff's. Figure 6.7 illustrates that the strength of the spectra above ~ 20 MeV, energies which would give a peak in the telescope spectrum (Fig. 6.6), is significantly larger for free decay than for lead (which are similar to decay for hydrogen and for gold, respectively). This AR would cause the amplitude ratio for hydrogen SH to increase at higher energies. At lower energies, on the other hand, the spectrum is stronger for lead than for free decay, AR which would decrease SH . Thus the difference in the electron energy spectra shown in Fig. 6.7 provides an explanation, at least in part, for the energy dependence of the —AR amplitude ratio SH observed in Fig. 6.6. Relative acceptance In our stopping fraction measurements, most or all of the hydrogen was placed on the upstream target, while the muon can stop in gold at both upstream and downstream targets. The relative geometrical acceptance for the electrons from hydrogen and gold is expected to be similar for the counters at the sides (Ege, En l ) , but can be different for the downstream counters (En2). Taking this into account, the expression for the stopping fraction Eq. 6.5 should read: Chapter 6. Analysis I - Absolute normalization 146 where Fu, Fd are relative fractions of muons which stops in the upstream and downstream gold foil, respectively (Fu + Fd = 1), and KQ — Q,d/Clu is the acceptance ratio. Obviously, when Fd « 1 or ~ 1, the expression reduces to Eq. 6.5. If the beam momentum width is larger than the corresponding thickness of gold foils and the hydrogen layer, one would very roughly expect Fu ~ Fd for the beam momentum optimized for the maximum stopping in hydrogen, ACQ can be roughly estimated to be about 1.3 for En2 from the fact that target spacing between the upstream and the downstream layers is some 12% of the detector distance. Assuming a point source, this would make about a 10% upward correction to SHR (from 25% to 27.4%) for the downstream counter pair En2. Combined corrections We have looked above at the possible individual influence on the stopping fraction de-termination by charged particles following muon capture, decay electron energy spectra, and upstream-downstream acceptance variations. Here we combine all the factors.to give a general expression. When the muons stop in M different materials, located in L different locations (such as upstream foil, downstream foil and so on), the fraction of the muon which stops in the A;-th material, S£R, determined from the ratio of the amplitudes Ay for the electron spectrum lifetime fit, is Ak QAR ( \ (6.12) Chapter 6. Analysis I - Absolute normalization 147 E Ak - ( ^ (6-13) E-pkfk where JF* is the fraction of muons stopping at the location I for the specific material k (Y^iFk — 1)) a n ( i ti is the effective efficiency for electron detection from muons stopped on material k at location /, = Q*en?#7f, (6.14) with Qk the Huff factor for the material k, e the intrinsic detector efficiency, fif the detector acceptance for material k at location /, and Bk and 7* the corrections to the efficiencies due to charged particle emission and electron energy spectrum, respectively. In our specific case where we assume M = 2 and hydrogen is present only at the upstream foil (F^s = 1, F^us = 0), while the muons stop in gold both at the upstream and the downstream layer with the relative fraction FAu,FAu but nowhere else (FlA^uud = 0), we can simplify the expression to SHR = , (6.15) A A u + AH v(FAU + KF£) where 77 is the ratio of the effective efficiencies for gold and hydrogen both at the upstream location, and K that for the upstream and the downstream gold: tAu (~~\AUQAU QAU^AU >tts V ^us rus I us / /> -j n\ " _ tH ~ OHClH flH^H ' 10.10 j > M S T > * Lusrus lus „ A H C)AU RAU^,AU K = S — S S f6 17) KAu OAuQAuryAu' v " ' us us rus Jus The expressions above reduce to Eq. 6.11, if n ~ QAu (i.e., gold and hydrogen have the same efficiencies for the upstream except the Huff factor), and further to Eq. 6.6 if K ~ 1 (the same efficiencies for upstream and downstream gold) or FASU <K FASU (negligible stopping in the downstream gold). Chapter 6. Analysis I - Absolute normalization 148 We consider the case for counters at the sides (Ege, Enl) and downstream (En2) separately. For En2, if we assume ft\ = 1 (k = Au,H and / = us,ds), and fi^ 4" ~ f)^., we approximately have V{En2) ~ QAu-^-, (6.18) 7 M S < E n 2 ) ~ f^- (6-19) On the other hand, for Ege, Enl , assuming (3f^ ~ flA^ ~ f l ^ " , we have CiAu (iAu ryAu V(Ege,Enl) ~ ^ ^ - 2 ^ , (6.20) M S P « S 7 M S K(Ege,Enl) ~ 1, (6.21) where there is some uncertainty in r](Ege, Enl) due to the charged particle emission effect Pu^/Pus- m addition, there is potentially a large error in the factor due to the possible non-flatness of the gold foil which could cause significant shadowing of the electrons either from gold or hydrogen7. Thus, the use of the downstream counters En2, in which the shadowing effect will not be present and the charged particle emission effect is negligible, appears more reliable for the determination of the stopping fraction. The factors F^su, F^, 77, K in Eq. 6.15 were determined from detailed GEANT simula-tions [218] taking into account the full geometry. Simulation for the muon beam assuming a momentum of 27.0 MeV/c with Ap/p of 5.7% yielded the gold stopping upstream and downstream to be Fu — 0.48 and = 0.52. A separate series of simulations for the decay electrons using the energy spectrum for lead (Fig. 6.7) to represent that for gold, and the free muon to represent hydrogen gave the absolute detection efficiencies efif7^ (k = H, Au; / = US, DS), presented in Table 6.11. Note that only the relative efficiencies are relevant in this context. 7 T h e frame for the foil is rather thick, so the foils could be bent such that the view from the side is blocked by the foil frame. We have observed evidence for such effects in the November 1993 runs with the M W P C imaging system. Chapter 6. Analysis I - Absolute normalization 149 Efficiency (£ftf7f x IO2) US DS H Au Au Ege 6.03(8) 4.88(7) 4.90(7) Enl 6.14(8) 5.02(7) 4.93(7) En2 5.11(7) 4.06(7) 5.59(7) Table 6.11: The absolute efficiencies, efif^ f (k = H, Au; / = US, DS) for electron detection, calculated by GEANT simulations [218]. Uncertainties given are statistical only. Detector Q A u us lus 1 us lus Pus 1 Pus OAu1oAu Pds 1 Pus V AC Ege©Enl 0.85 0.813* 1.04 1 0.719 1.007* En2 0.85 0.795* 1 1 0.676 1.375* Table 6.12: The correction factors 77, AC in Eq. 6.15, and assumed quantities for the derivation by Eqs. 6.16, 6.17. The first row is the average of Ege and Enl. Values with '*' are taken from the ratio of the efficiencies calculated by a GEANT simulation given in Table 6.11. The correction factors, which take into account the differences in solid angle for upstream and downstream, electron energy spectra between gold and hydrogen, and effects of charged particle emission, can now be determined. Together with the assumed quantities, the values of 77 and K are given in Table 6.12. Note that assuming Pf™/P^ = 1, K is the direct ratio of the electron detection efficiencies QAs7dsU l^uslus • As mentioned before, n(Ege,Enl) can have relatively large uncertainties, due to possible errors in and pi\ The effect of these corrections on the stopping fraction is summarized in Table 6.13. The uncertainty in the estimated value of PAu (1.04) for Ege, Enl is assumed to be ±0.04, which is included quadratically in the uncertainties presented in the table. Possible errors in ^ds/^us f° r E§ e a n < i Enl are not included. The error for En2 is statistical only. Chapter 6. Analysis I - Absolute normalization 150 Detector s? (%) s H R (%) Ege E n l En2 26.6 (4) 26.5 (4) 24.3 (4) 23.5 (12) 23.5 (12) 23.4 (7) Table 6.13: The uncorrected stopping fraction SHN from Eq. 6.6 and the corrected one SHR from Eq.6.15 for the runs 1650, 1654. The errors on SHR for Ege, E n l include estimated 4% (relative) uncertainties due to the charge particle emission. The other errors are statistical only. AR The total of about —12% relative correction to SJJ for Ege, E n l , is dominated by the —14% correction due to the difference in the relative electron detection efficiency ^ds'lds™/^us7usU (which in turn is dominated presumably by the electron energy spectrum effect), partly offset by a +3% effect due to charged particle emission. AR For En2, the correction to SH is rather small as a result of cancellation between the — 16% relative correction, due to the difference in the energy spectrum, and the +14% correction, due to the difference in the upstream-downstream relative acceptance. The AR latter correction in SH is reasonably close to the approximate +10% given on page 146, estimated with simplified assumptions without detailed simulations. 6.5 Absolute amplitude method In this section, we shall discuss another method for the determination of the stopping fraction,' referred to earlier in this chapter as the absolute amplitude method. In this method, the same fit is performed to the decay electron time spectra as in Fig. 6.5, but we use only the amplitude information for the hydrogen component, and do not rely on the gold component in the fit. With the knowledge of the absolute electron detection efficiency, the number of muon stops in H 2 can be directly measured. Chapter 6. Analysis I - Absolute normalization 151 The absolute efficiencies for electron detection are determined by taking advantage of a condition called delayed electron coincidence, or Del-e, and delayed telescope coincidence, Del-t (together, generally called Del cuts). These cuts were implemented originally to suppress the muon-capture related background, particularly useful for detection of low energy fusion neutrons from dd fusion [166]. In the Del cut, it is demanded that the electron be observed in the scintillator (telescope) within a certain time window after a hit in a detector of interest (e.g., neutron or Si). This ensures with a high probability that the event in the detector comes from a / i C F related process, as opposed to the capture process, since in capture the muon would be converted into a neutrino, and would not decay into an electron. The suppression of background by more than two orders of magnitude in neutron detection was essential in order to overcome the poor signal-to-noise ratio in the dd fusion neutron measurements, but in the Si detectors, <frj-fusion a particle signals can be identified without Del-e (Del-t) cuts, thanks to the good resolution of the detector. This leaves us with the ability to measure with precision the efficiency of Del-e (Del-t) cuts, from which the absolute efficiency of the electron scintillators (telescopes) can be determined. The delayed electron (telescope) coincidence cut efficiency tdei is a product of the electron scintillator or telescope detection efficiency (including the solid angle) eeVte, the branching ratio for electron emission Be, and the time cut efficiency ttime (assuming the Huff factor is 1 for hydrogen), Jti where the etime intergral is over the time difference between the Si event time (tsi) and the delayed electron time (tdei), and the total electron disappearance rate A e^/ (= Ao + Ax) (6.22) (6.23) Chapter 6. Analysis I - Absolute normalization 152 is the sum of the free decay rate A 0 and muon loss rate A^ accounting for possible muon loss, as observed in the electron lifetime. The branching ratio Be is defined as Be = (6.24) It should be noted that in our earlier paper [166], an incorrect expression for the delayed electron efficiencies (Eq. (4)) was used without the factor Be. That would be correct only when there is no muon loss (i.e. X^ei = Ao), which is generally not the case either in Ref. [166] or in this thesis. For Ref. [166], however, tde\ was used only for relatively minor corrections related to background subtraction, hence the conclusion there should not be significantly affected. For our case, on the other hand, Be enters directly into the stopping fraction, as we shall see, so this distinction is rather important. The Del cut efficiencies, tdei, were determined experimentally from the ratio of the yield Y"gl for fusion a, to that for the a with a Del-e condition demanded, Ydael. The electron disappearance rate was obtained by fitting the time spectrum of the electrons detected after the fusion a signal in the Si detectors. Thus, the electron detection efficiency can be derived via: n _ (-del Bedtime = V^(\^-) [ e x p ( - A d e ; t 1 ) - e x p ( - A ^ 2 ) ] - 1 (6.25) Isgl \AdelJ The fit of decay electrons to exponential functions, discussed in the previous section, had given us the amplitude A2 corresponding to the hydrogen component (see Table 6.6). With A2 normalized to the number of incident muons (GMU), in was the case in Table 6.6, we have A2 = e e n e iV 2 0 /GMU, where N2° is the number of muons stopping in the hydrogen. Hence, the muon stopping fraction in hydrogen SHBS (= A ^ / G M U ) , using the knowledge of the absolute efficiency of electron detection eeQe obtained from the Del analysis, can Chapter 6. Analysis I - Absolute normalization 153 be given by: cABS A2 = A2 (6.26) (•del We shall discuss each factor in Eq. 6.26 in the following subsections. 6.5.1 Delayed electron lifetime The disappearance rate of delayed electrons \dei enters in the determination of stopping fraction (6.26), directly in the branching ratio Be and indirectly in the time cut efficiency eame- The rate \j_ei was determined by fitting the time tdei — tsi, i-e-, the time between the silicon a event in Sil or Si2 and the first electron (telescope) event following the Si event, with a single exponential function. In order to ensure accurate determination of Xdei, cuts were made on both energy and time of the Si events. A cut on tsi (Si time with respect to muon entrance time t0) of 0.02 < tsi < 0.5 ps was applied to select prompt fusion events from the upstream D2 moderator overlayer (where the signal-to-background ratio is most preferable), and to ensure a uniform time efficiency of the Del cuts. Since the event gate was open for a finite width (~ 10/is) after the muon entrance, the efficiency for the Del cuts for the Si events occurring late (with respect to t0) is reduced, due to the smaller time window for the detection of delayed electrons8. Two different energy cuts were applied. The nominal [2000, 3700] ch9 cut (noted as Energy cut "a" in Tables 6.14, 6.15) covered a good portion of the fusion a peak, 8Recall that any event, including the delayed electron event, can be collected only when the Event Gate is open. Thus, if for example, a Si event occurred at 9 ps, there is only about a 1 ps window for the delayed electrons to be detected to fulfill the Del cut, as opposed to the nominal Del window (t2 — ti) of ~ 5 /is. The Del time efficiency is constant as long as tsi < TEV — (^ 2 — ^l), where TEV is the event gate width. 9 Recall that 1 ch ~ 1 keV. Chapter 6. Analysis I - Absolute normalization 154 Run Energy 1st E after S i l 1st E after Si2 cut l/Xdel (PS) X 2 /dof cl (%) l/Xdei (ps) X 2 /dof cl (%) A a 2.025(31) 1.05 34 2.094(34) 1.18 12 I 2.095(56) 0.99 50 2.260(63) 1.03 41 B a 2.027(32) 1.03 40 1.966(34) 1.23 6.7 I 2.150(66) 1.09 27 2.073(49) 1.12 23 Table 6.14: The lifetime of the first electrons (1st E) after the Si signal, fitted to a single exponential with a constant background. Energy cut a represents Si energy of [2000, 3700] ch, selecting fusion a events, while the cut / is for [2000, 3000] ch, avoiding fusion events occurring at the surface. while the lower energy cut [2000, 3000] ch (noted as "/" in Tables 6.14, 6.15) avoided the fusion events which occurred near the surface of the D2 overlayer. The latter cut was implemented to test a possible systematic effect which depends on the depth of the fusion event in the layer, such as p~~ or [id escaping from the layer. The energy of the a is related to the event depth thanks to a energy loss in the layer. Tables 6.14 and 6.15 give the fitted results of \dei for Del-e and Del-t time spectra, respectively. Two different series of runs, (A) Runs 1671-831 0, and (B) Runs 1709-30 1 1 , were used for the fit. While a thick hydrogen layer (500 T-l) was present in the downstream target for Run A , there was no (Target 11-13) or only a very thin (11-14) layer in the downstream target for Run B. For the delayed electron (Table 6.14), the use of a constant background term was necessary to obtain reasonable fits. This was not the case for the delayed telescope (Table 6.15) where the background was smaller, and fits both with and without the constant term were tried to check the consistency. As shown in Tables 6.14 and 6.15, for S i l , Runs A and B give a consistent value of 1 0Target ID = II-9. "Target ID = 11-13, 11-14 Chapter 6. Analysis I - Absolute normalization 155 Run Energy 1st T after S i l 1st T after Si2 bkgd cut l/Xdei (ps) X 2 / d o f cl (%) 1/Arfe/ (ps) X 2 / d o f cl (%) A a 1.978(48) 1.20 9.7 2.102(46) 0.95 62 yes I 2.054(89) 1.21 9.0 2.113(78) 0.92 69 B a 2.034(51) 1.13 19 1.952(44) 0.99 50 I 2.141(91) 0.94 64 2.077(94) 1.18 12 A a 2.116(31) 1.38 1.0 2.096(26) 0.94 84 no I 2.102(52) 1.21 8.4 2.102(44) 0.91 73 B a 2.090(30) 1.15 16 2.013(28) 1.04 37 I 2.187(51) 0.94 65 2.103(54) 1.17 13 Table 6.15: The lifetime of the first telescope (1st T) after Si signal, fitted to a single exponential with (bkgd yes) and without (bkgd no) a constant background term. Xdei for both Del-e and Del-t, while for Si2, Run B gives a smaller value than Run A by 2 to 3 cr. If Runs A and B are averaged, however, S i l and Si2 give consistent Xdei-The averages over S i l and Si2 as well as over Runs A and B were thus taken and are presented in Table 6.16. We note the following. First, Del-e and Del-t are for the most part consistent with each other. Second, not including the constant background term increases the value of fitted Xdei- Though not shown in the tables, this holds true for the Del-e fits as well. Third, the energy cut / gives a Xdei that is 2-4 o lower than the cut a in all cases in Table 6.16. Our determination of Xdei is thus limited by systematic effects, which are possibly due to the finite thickness of our layer. Taking the average of the two extreme values in Table 6.16 we assign Xdei = 2.081 ± 0.064 p,s with the error covering the two extremes. Thus we have the time cut efficiency, etime = 82.0 ± 0.4%, and the electron branching ratio, Be = 94.7 ± 2.9%, which combine to give the factor Be • etime — 77.7 ± 2.8% for Eq. 6.26. Note that the errors are correlated. Chapter 6. Analysis I - Absolute normalization 156 Detector bkgd Energy cut a Energy cut / 1st E yes 2.028(16) 2.145(29) 1st T yes 2.017(24) 2.096(44) no 2.079(14) 2.124(25) Table 6.16: The lifetime of 1st delayed electrons, and 1st delayed telescope, averaged over S i l and Si2, as well as Run A and Run B. 6.5.2 Delayed electron cuts efficiency The efficiencies for the delayed electron and telescope cuts were determined from the ratio of the a signal yields with and without the Del cuts. From each spectrum the background was subtracted, usually using a pure H 2 target in which no fusion takes place 1 2. The a events were selected with an energy cut of [2000, 3700] ch, and a time cut of [0.02, 0.4] ps was applied to the Si signals. Columns 3 to 5 in Table 6.17 show the Del cut efficiency, Cdei for S i l , Si2, and their average. Run A is the same as that in Tables 6.14 and 6.15. Run C was similar to A but had no downstream layers (500 T-l H 2 and 3 T-l D 2 were present for Run A) . Run D had a 6 T-l upstream overlayer (as opposed to 14 T-l as in others), testing possible effects of the layer thickness and the reaction depth. Given in brackets for tdei are statistical errors. Run A had sufficient statistics, while Runs C and D, as well as Del-t cuts (Tell, Tel2) had relatively poor statistical precision. Some variations of the Del cut efficiencies are observed in Table 6.17: Run C and D have a lower e^i than A by 2 to 3 a, possibly pointing to a systematic effect due to the target conditions. The interpretation of the tdei results requires some caution; recall that in the present analysis, detection of the fusion reactions is used to tag the muon as a well-defined 1 2 F o r Run D below, the standard emission target without overlayer (target ID=II-7) was used. Chapter 6. Analysis I - Absolute normalization 157 Run Det. €del (%) SFHBS Si l Si2 Average (%) (%)• A Ege E n l En2 1st E 3.99(5) 5.05(6) 4.13(6) 13.1(1) 5.26(5) 3.87(7) 4.25(5) 13.3(1) 4.63(4) 4.46(5) 4.19(4) 13.2(1) 5.95(5) 5.74(6) 5.39(5) 17.0(1) 29.6(11) 30.1(11) 29.2(11) 29.8(11) Tell Tel2 1st T 2.35(4) 1.90(4) 4.24(6) 1.99(4) 1.98(4) 3.96(6) 2.17(3) 1.94(3) 4.10(4) 2.79(4) 2.50(4) 5.28(5) 29.1(11) 28.9(11) 29.2(11) C Ege E n l En2 1st E 3.67(13) 4.85(15) 4.07(13) 12.54(25) 5.12(11) 3.78(16) 3.93(13) 12.76(24) 4.40(9) 4.32(11) 4.00(9) 12.65(17) 5.66(11) 5.56(14) 5.15(12) 16.29(22) 31.2(13) 31.1(14) 30.6(13) 31.1(12) Tell Tel2 1st T 2.23(10) 1.78(9) 4.01(13) 2.07(10) 1.83(9) 3.89(13) 2.15(7) 1.81(6) 3.95(9) 2.77(9) 2.33(8) 5.09(12) 29.6(14) 30.9(15) 30.3(13) D Ege E n l En2 1st E 4.17(23) 4.63(25) 4.11(23) 12.86(44) 4.44(22) 3.80(27) 3.81(24) 11.93(44) 4.31(16) 4.22(18) 3.96(17) 12.40(31) 5.55(21) 5.43(23) 5.10(22) 15.96(40) 31.8(17) 31.8(18) 30.9(17) 31.8(14) Table 6.17: The Del cut efficiency, tdei, the absolute electron detection efficiency, eefle, and the muon stopping fraction, SF^BS determined from the absolute yield method. Run A is the same as that in Table 6.14, and Run C is similar to A but without the downstream 500 T-l H 2 and 3 T-l D 2 (target ID=II-7). Run D had a 6 T-l upstream overlayer (target ID=IT3) instead of 14 T-l as in others. The errors (in brackets) for tdei and eette are statistical only, while those for SF^BS include the ~ 3.6% systematic uncertainty due to Be. The values in bold face highlight the downstream detectors (En2 and Tel2) which are less suceptible to the Si solid angle effect. source for the electron, into which it decays. Detecting the fusion a, however, biases the electron source to preferentially concentrate towards the region where the acceptance for the particular Si detector is higher, e.g., the beam-left edge of the target for S i l (see Fig. 3.18, page 90). The overall efficiency of the Del cuts, which is a convolution of the a particle acceptance in the Si detector and the electron acceptance in the scintillators Chapter 6. Analysis I - Absolute normalization 158 (telescope), is thus higher in the case of the Del coincidence in the same side of the target than that in the opposite side (Fig 3.18); e.g., for S i l , tdei is higher for Ege at the beam left than E n l at the beam right, as can be seen in Table 6.17. This acceptance bias effect can be largely avoided by taking the average of S i l and Si2, yet the effect persists to increase tdei (unless the target is infinitely small), and is at the several percent level in our case. On the other hand, the downstream electron detectors (En2, Tel2), which were on the beam (Z) axis (as opposed to the perpendicular axis (X)), are less susceptible to the bias effect, since they have relatively uniform acceptance over the entire target. This can be seen in the consistency between S i l and Si2 in tdei for En2 and Tel2 in Table 6.17. En2 and Tel2, which have lower values of tdei, are thus more reliable compared to other detectors. With all the factors in Eq. 6.26 determined, the stopping fraction with the absolute yield method can now be computed and the results are given in the last column of Table 6.17. The error includes 3.6% systematic uncertainty due to the delayed electron life time, Tdei, which dominates the total error for Run A , but is comparable to the statistical error for Runs C and D. The values in bold face are for the detectors downstream (En2, Tel2), which are more reliable than other detectors, as discussed above. Among these gpABS v a m e s f o r En2 a n d Tel2, there are about ±3 .3% variations depending on the target conditions or the choice of the detector (i.e., En2 versus Tel2), which indicates a measure of further systematic effects. Note that unlike the case for the relative amplitude AR method, in which there was some 30% discrepancy in SFH between the scintillators and telescopes (see Table 6.6 in page 134 and 6.8 in page 136), the results in the two detectors are nearly consistent in the absolute yield method here. Taking the average of the two extremes of the downstream SF^BS, we quote SF^BS = (29.9 ± 1.5)%, where with the assumption that the uncertainty is dominated by the systematics, the 5% error is given Chapter 6. Analysis I - Absolute normalization 159 by a quadratic addition of the 3.6% systematic error from each measurement and 3.3% due to run variations. 6.6 Discussion of stopping fraction 6.6.1 The discrepancy As we have seen in the previous sections, the results of two methods for stopping fraction determination are in disagreement. For the downstream detector En2, we have: SHR = 23A ± 0.7% (Relative Amplitude Method) SHBS = 29.9 ± 1.5% (Absolute Amplitude Method), where the disagreement is about 25% (at the 4 u level). Although the absolute amplitude method has about twice as large a quoted error, it is not necessarily less reliable than the relative amplitude method. The advantages of the former include that it: (1) does not rely on the Au component fit in electron spectrum, which can be rather difficult due to the fast time slope, (2) is much less sensitive to the time zero, to, position, (3) does not require the corrections via Monte Carlo simulations for the electron detection threshold and decay energy spectrum effects, as well as the relative (upstream versus downstream) solid angle effect, and (4) is not susceptible to the effects of particle emission after muon capture. On the other hand the disadvantages of the absolute amplitude method include: (1) lack of a complete understanding of the muon loss mechanism in a thin layer, which might affect the results via the branching ratio factor, Be, (2) possible bias due to the different acceptance functions for electron detection in scintillators (telescope) and a detection in Si. It should be stressed, however, that these effects have been estimated rather reliably to be less than a few percent by using our data, and are thus reflected in our quoted Chapter 6. Analysis I - Absolute normalization 160 errors for this method. 6.6.2 A solution In an attempt to resolve this discrepancy, we revisit the relative amplitudes in the tele-scope measurements. As discussed in Section 6.4.2 (p. 136), the telescope data gave some 30% higher value of the reduced stopping fraction SH , compared to the electron scintillator data. We suspect that this might be due the energy sensitivity of the detec-tors to the stopping fraction, as the telescope events required a certain energy deposit in the liquid scintillator. This was experimentally demonstrated in Fig. 6.6 (page 138), and qualitatively explained to be due in part to the muon decay electron energy spectra (p. 144). Here we postulate the possibility of muon-capture related processes still not fully accounted for by various corrections which we have made in Section 6.4.3. One such example is gamma induced reactions in the target. We make use of the minimum ionizing peak (~ 15 — 20 MeV) in the telescope energy spectra, and make a comparison with the G E A N T simulations of full electronic processes, to extract the relative amplitude. The possible advantages of using the energy cuts at this peak, as opposed to no energy cuts, include: (a) energy deposited from a muon-capture related process, if it exists, should be less important at high energies, and (b) with the lack of an energy calibration for the Tel spectra, the minimum ionizing peak itself can provide a calibration. Nevertheless, in order to account for some ambiguity 1 3 in the exact cut positions corresponding to the experimental energy cut at the peak (590<E<790 ch for both Tell and Tel2; see Fig. 6.6), we applied two different sets of cuts in the Monte 1 3 I n addition to lack of independent energy calibration, we note that the simulation does not explicitly include the resolution of the detector, though the simulated physics processes in the detector partly account for it. Chapter 6. Analysis I - Absolute normalization 161 10 15 20 Energy (MeV) 25 30 e!2 800 10 15 20 Energy (MeV) 25 30 Figure 6.8: Energy spectra for Tell (top) and Tel2 (bottom), obtained by G E A N T simulations. Vertical lines indicate the cuts used to determine the relative electron de-tection efficiencies, corresponding to the cuts in the experimental data (590<E<790ch in Fig. 6.6). Narrower cuts (enl, cn2) and wider cuts (cwl, cw2) were tried to check the systematic effects. Chapter 6. Analysis I - Absolute normalization 162 Detector QAu Au/riH H us lus 1 us lus RAulQH rus 1 rus V K Tell 0.600(9)(9)* 1 I 0.510(8)(8) 1.059(18)(5)* Tel2 0.578(12)(7)* 1 I 0.491(10)(6) 1.188(25)(4)* Table 6.18: The correction factors n, K in Eq. 6.15, and assumed quantities for the derivation by Eqs. 6.16, 6.17. Values with '*' are taken from the ratio of the efficiencies calculated by G E A N T simulation with the cuts given in Fig 6.8. The first parentheses are the Monte Carlo statistical errors, while the second indicate systematic uncertainties due to the different cuts. Carlo spectra. Shown in Fig. 6.8 are the simulated spectra and applied cuts. With these cuts, we determined the relative efficiencies of electron detection necessary to derive the stopping fraction, as given in Table 6.18. The first parentheses in the tables are the Monte Carlo statistical errors, while the second indicate systematic uncertainties due to the different cuts in the Monte Carlo energy spectra. As can be seen, the resulting relative efficiencies are not very sensitive to the choice of the cuts. Note that the charged particle effect ratio fi^s/Pus °f 1 w a s u s e d for Tell (bold face in the table), as opposed to 1.04 in the scintillator measurement, because of the higher energy cuts applied here. The stopping fractions SHR determined with the telescope thus are given in Table 6.19, together with uncorrected values. Energy cuts (%) Tell Tel2 ~~qAR CAR D H nominal min. ion. peak min. ion. peak 36.3(4) 42.4(8) 31.3(14) 32.8(4) 39.9(7) 29.6(14) Table 6.19: The uncorrected stopping fraction SH and the corrected one SHR using the minimum ionizing peak of the telescope detectors in the relative amplitude method. Uncorrected values with nominal energy cuts (200<E<1200 ch) are also given for com-parison. Chapter 6. Analysis I - Absolute normalization 163 The results for SHR from the telescope are in good agreement with the absolute amplitude method, while in disagreement with the relative method with the scintillators. The stopping fractions determined from various methods are summarized in Table 6.20. Also listed is the result of a G E A N T based beam stopping simulation performed by Marshall assuming the peak momentum of 27.0 M e V / c with Ap/p = 5.7% (these values were carefully tuned to match the relative hydrogen stopping fractions for a range of beam momentum). Our results suggest that the relative amplitude method, using electron scintillators, may be unreliable for a stopping fraction determination, even with elaborate corrections applied as in the preceding sections, due perhaps to further capture related background such as nuclear gammas which are not accounted for. In fact, a preliminary study by Art Olin using an evaporation model for particle emission following nuclear excitation from muon capture suggests this may be the case. Excluding the scintillator relative amplitude values, we obtain consistent results on the stopping fraction. The absolute amplitude method with its advantages discussed above, appears to be more reliable, hence I shall use the value SH = 29.9 ± 1.5% for the present work. Method Detector SH (%) Comment Relative amplitude En2 23.4(7) fi capture related background? Relative amplitude (min. ionizing peak) Tel l , 2 30.5(14) Energy cut ~ 16-20 MeV Absolute amplitude En2, Tel2 29.9(15) No M C correction necessary Beam simulation (GEANT) 31.1 p = 27.0 MeV/c , Ap/p = 5.7% Table 6.20: Summary of stopping fractions determined via various methods. The absolute amplitude result (bold face) will be used for this work. Finally, I point out that since many of the previous experiments in the field used the simple relative amplitude method, often without any efficiency corrections, their results may require some re-interpretation, particularly when muon stopping in non-hydrogenic Chapter 6. Analysis I - Absolute normalization materials is significant, as in a low density gas targ Chapter 7 Analysis II - Emission of muonic tritium In this chapter, we present the analysis of Run Series I, in which the emission of muonic tritium is studied. The first section deals with the treatment of the M W P C imaging data, and in the second section we present a quantitative analysis using detailed Monte Carlo calculations. 7.1 M W P C imaging data 7.1.1 Cuts As described in Section 3.4.3, the M W P C system determines the position of the electron hit from the delay times, the time it takes the signal to reach both ends of the delay line (see Eq. 3.4). In addition to a hardware discrimination (via pre-amplifier threshold) against noise in the delay signal lines, the following off-line software cut procedure was applied in order to ensure the quality of imaging. Since the delay lines have a fixed length, the sum of the delay times is expected to be constant, i.e., for the Z-direction, SUMZ = (ZL + ZH) x dispZ = Const., where ZL and ZH are the delay time measured at beam upstream and downstream ends of the delay lines, respectively, and the sum is multiplied by the dispersion constant, dispZ with a length dimension for convenience. In reality, however, a small but finite amount of time is required between the ionizing electron and the current being induced in the 165 Chapter 7. Analysis II - Emission of muonic tritium 166 cathode wires, mainly due to the drift time of electrons to the anode, which creates a distribution in SUMZ (and SUMY = (YL + YH) x dispY) reflecting the proximity of the initial hit to the anode. There is a strong correlation between distributions of SUMZ and SUMY, since they are due to the same physical process for both z and y wires, i.e., electron drift to the anode (e.g. if ionization occurs very close to the anode, both SUMZ and SUMY are small due to a small drift time). Because of this correlation, the plot of SUMZ vs. SUMY for good events occu-pies only a small area of two dimensional space (a long rectangular box oriented in a diagonal direction), so making independent cuts on each parameter SUMZ and SUMY (corresponding to accepting a large square box on the plane) is rather inefficient in terms of noise elimination. We therefore performed a linear transformation of the coordinate system, introducing new uncorrelated variables, SUMSUM = SUMZ + SUMY and DIFSUM = SUMZ - SUMY. With these coordinates, applying the cuts becomes straightforward. DIFSUM, the difference in the sums, typically had a smaller disper-sion compared to that of SUMZ or SUMY and SUMSUM larger. Therefore a small interval for DIFSUM cut (±8 or ±10 mm for this analysis) and a large interval for SUMSUM cut (±50 mm), cover the entire parameter regions for good events, while efficiently eliminating the noise. Good hits in each of three M W P C s , thus passing the cut, were fitted to a straight line with the least-squares method. Cuts in the chi-square (< 4) and the fit-residual for the second of three chambers (< 10 mm) for both z and y variables, were applied to ensure a reasonable fit. For this analysis, all the above cut values described in this section were taken to be relatively non-restrictive, the largest effect being about 20% rejection due to the chi-square cut, in order to prevent biasing of the data set and to allow for maximum statistics. Chapter 7. Analysis II - Emission of muonic tritium 167 Z (mm) Figure 7.1: The time and extrapolated position in the z-direction of the muon decay. The contour level spacing is log-arithmic. Emission from targets of (a) H 2 with 0.1% T 2 (H/T) , and (b) pure H 2 , both with thickness 3.2 mg-cm - 2 , are compared. Shown in (c) is the H / T target with a thin overlayer of D 2 . 7.1.2 Muonic tritium emission spectra The trajectories of muon-decay electrons, reconstructed from the least-squares fit of three wire chamber positions, were extrapolated back to a perpendicular plane bisecting the target, providing an estimate of the position of the muons at the time of decay. The time of electrons was measured by the fast output signal of plastic scintillators. Recalling that the number of muon decays in unit time is proportional to the muon population (—dN^/dt = XNp), the detection of decay electrons provides the measurement of the muon population at a particular position and time. The evidence for p,t emission is clearly seen in Fig. 7.1. The time and the extrapolated position of muon decay in the 2-direction (along the beam) are shown in contour plots Chapter 7. Analysis II - Emission of muonic tritium 168 for the targets of (a) H 2 with 0.1% tritium, and (b) pure H 2 . The intense region near z = —20 mm in both figures is from muons decaying inside the hydrogen layer and upstream target support, whereas the events near z — 20 mm are muons stopping in the bare downstream gold foil where they disappear quickly via nuclear capture. While the pure H 2 target (b) shows very few events in the vacuum region between two gold foils (around z = [—10,10] mm), a strong signal is observed in this region for the H / T target (a), indicating emission of the muonic system into vacuum. The emitted muonic system, when allowed to collide with a separate D 2 layer, produced dt fusion providing unambiguous identification as fit. The information on the velocity distribution of the emitted pt can be obtained from the correlation between the time and position of muon decay. Since the time for emission is relatively short (~100 ns), the slope z/t in the plot roughly corresponds to the longitu-dinal component of the pt velocity. Shown in Fig. 7.1 (c) is another measurement, where the standard H / T target was covered by an additional very thin layer of D 2 (overlayer). A significant shift in average slope, hence velocity, is observed. Figure 7.2 compares a series of measurements with varying overlayer thickness of D 2 as well as with no overlayer, where the time spectra of muon decay in the vacuum region (z = [—10,10] mm) are plotted. The events were normalized to G M U , and the pure H 2 run (Fig. 2 (b)) was used for background subtraction. The change in the mean of the time distribution is clear, demonstrating that we can control the velocity of the emitted pt, an important feature for optimizing the pt beam energy for the time-of-flight measurements. Further quantitative analysis requires a comparison with Monte Carlo calculations, which will be discussed in the following section (7.2). Chapter 7. Analysis II - Emission of muonic tritium 169 CO I o CO - M c CD > CL) " D CL) N 73 E i _ o 14 /xg-cm -2 28 /xg-cm -2 60 /xg-cm -2 2 3 Time (/xs) Figure 7.2: Time spectra of muon decay in the region z=[-10,10] mm with varying thick-ness of D 2 overlayer. The thicknesses given are approximate. Chapter 7. Analysis II - Emission of muonic tritium 170 7.2 Measurement of the Ramsauer-Townsend effect In this section, we shall quantitatively analyze the muonic tritium emission spectrum obtained in Section 7.1, by comparing with Monte Carlo calculations based on theoretical cross sections. This will constitute the first spectroscopic measurement of the Ramsauer-Townsend effect in the muonic system. 7.2.1 Monte Carlo parameter determination In order to make a comparison with data, some parameters in the SMC code need to be determined, including (a) muon beam distribution, and (b) wire chamber position resolution. This was achieved using a pure H 2 layer. We have already discussed (a) in the context of determining the effective thicknesses (Section 6.1), where the beam distribution was determined from the M W P C image of the target region in the Y direction. For the present analysis, a flat-top Gaussian beam distribution with a flat top radius R/iat of 10 mm and Gaussian F W H M 3 of 10 mm (noted 10©10 mm) was used. Because the target is extended in the Y direction, the Y image is rather insensitive to the wire chamber resolution parameter a^0'. We therefore used the Z image to determine cAJc and assumed that o^c had a comparable value. Note that we need not know a^0 too precisely because the Y distribution is already broad and the Y image is not aifected much by the choice of o^c'. We have performed iterative fits of the Monte Carlo calculations to the data to find the best value for oYC (Fig. 7.3). Plotted with error bars in Fig. 7.3 (a) is a Z image of M W P C system for a pure H 2 target1, from which one expects no muonic atom emission except at very low energies [216]. A time cut of t — t0 > 2/J,S (t0 being muon entrance time) is applied to eliminate the background from muon stopping in heavy elements in target ID 1-8 in Table 4.2 in page 105. Chapter 7. Analysis II - Emission of muonic tritium 171 the target frames2. The data was fitted in the interval shown in the figure with a Monte Carlo assuming o^0' = 1.3 mm with a beam parameter of 10 © 10 mm (histogram). The normalization factor being the only free parameter, the fit gave x 2 / D O F of 1.62 (10.3% confidence level). Figure 7.3 (b) shows the result of an iterative fit, varying crz for two different beam parameters3. Thus we conclude that the best value for the M W P C resolution parameters is 1.3 ± 0.1 mm with only a small dependence of the beam parameters. 7.2.2 The Ramsauer-Townsend minimum energy With these Monte Carlo parameters fixed, we can now compare the time of flight spec-trum of the emitted muonic tritium with Monte Carlo simulations to test theoretical cross sections based on few-body theory. We shall focus on the measurement of the position of the Ramsauer-Townsend minimum ER, since this is important both as a source of the pt beam and as a test of the quantum three-body calculations. The Nuclear Atlas cross sections [16,17] for the muonic processes were used as a nominal input to the Monte Carlo. Molecular and condensed matter effects [167,170,216] are expected to play negligible roles in the transport properties of muonic atoms at the energies above a few eV. Iterative calculations were performed by multiplying the energy scale of the fit + p elastic scattering cross section by a constant factor n, thus shifting the RT minimum as ER —• KER. The resulting simulated time spectrum is fitted to the experimental data with one free parameter (relative normalization), and x2 w a s calculated for each value of K. For both the data and the M C , a longitudinal spatial cut of z = [—10,10] mm was applied to select the vacuum region. 2 Recall that muons bound in the atomic states of heavy elements disappear quickly (within a few 100 ns) due to a high rate of nuclear muon capture via the semi-leptonic weak interaction. 3See Section 6.1 for the notation of beam parameters. Chapter 7. Analysis II - Emission of muonic tritium 172 MWPC image -30 -20 -10 0 Z (mm) 10 20 (a) Z image ofthe M W P C system for pure H 2 target (error bars), fitted by a Monte Carlo assuming a™c = 1.3 mm (histogram). (b) Total chi-squared in the fit (D0F=9) with varying M W P C resolution parameter < T ^ c . Dependence on the beam width pa-rameters also shown. Figure 7.3: Determination of M W P C resolution parameters. Chapter 7. Analysis II - Emission of muonic tritium 173 o o 3 h o x 2 I 1 -1 fit region A fit region B 2 3 4 time /u.s in 1 o CD u - 2 - 3 0 1 2 3 4 5 6 time yus Figure 7.4: Above: Plotted with error bars is a background-subtracted time spectrum of pt emitted from the standard emission target (protium with 0.1% tritium). A spatial cut of z — [—10,10] mm was applied. The histogram shows a fit to a Monte Carlo calculation, with an energy scaling factor K = 1.1. Two different intervals for the fit are also shown. Below: residuals from the fit in the region A. Chapter 7. Analysis II - Emission of muonic tritium 174 Figure 7.4 shows an example of such a fit (top) and its residuals (bottom). Plotted with error bars is the fit time spectrum from the standard emission target4 (1000 T-l H 2 with 0.1% T 2 ) from which a background run 5 of pure 1000 T-l H 2 , normalized to G M U , was subtracted, while the histogram shows a simulated time spectrum assuming the energy scaling factor of re = 1.1, which turned out to give the best x 2 /DG"F G f 1.06 for fit interval t = [0,6] fis (region A) with DOF = 59. Plotted in the bottom of Fig. 7.4 is the fit residual, i.e. the difference between the Monte Carlo and the data, normalized to one standard deviation. Illustrated in Fig. 7.5 is the global trend of total x 2 versus the energy scaling factor re, while Fig. 7.6 shows the details near the minimum. The horizontal axis is plotted against the inverse of square root of re to reflect our sensitivity to the time of flight, rather than the energy6. Dependences of the various parameters were investigated in detail for the potential systematic effects. Figs. 7.5 and 7.6 show some examples of such investigations including: 1. Muon transfer rate from protons to tritons, Xpi, as input to the simulation. Our standard input uses an experimental rate measured by our collaboration, Xpt = 5.86 x 109 s - 1 [83] for this process, but fits using the theoretical energy-dependent value of Xpt from the Nuclear Atlas cross section [17] were also tried. 2. Time interval of the fit region. Region A: t = [0,6] fis, Region B: t = [0.1,3] fis (see Fig. 7.4). If our fit model is correct, the fit results should be independent of fit region except to add a constant value in total x2i which was confirmed in Figs. 7.5, 7.6. 3. M W P C resolution parameter o^0 ,o~YG • Our nominal value 1.3 mm was varied 4Target ID = 1-1 in Table 4.2 in page 105. 5Target ID = 1-8. 6Recall that t = / x ^JM^JIE, where t is the time of flight over the drift distance /. Chapter 7. Analysis II - Emission of muonic tritium 175 4 0 0 CM * 2 0 0 3 5 0 -3 0 0 -2 5 0 h 100 -150 -50 -0 0.6 0.8 1.0 . - 1 / 2 1.2 1.4 fC Figure 7.5: Total x2 versus n-1/2 in the fit of pt emission data with Monte Carlo calcula-tions, where AC is a scaling parameter shifting the Ramsauer-Townsend energy minimum. The inverse of square root n is plotted to reflect our sensitivity to the time of flight, rather than the energy. Effects of the muon transfer rate from protons to tritons, Xpt, and the time interval of fit regions are also shown. A p i(exp)= 5.86 x 109 s - 1 is taken from Ref. [83], and A p i(th) from Ref. [17]. Fit region A is the time interval t = [0,6] ps with D 0 F = 59, and region B , t = [0.1,3] ps with DOF= 29 (see Fig. 7.4). from 1.0 mm to 1.5 mm (only some selections are shown in Fig. 7.6.) 4. Beam width parameters characterizing a flat top Gaussian distribution in the XY plane. 10©10 mm (nominal) and 15©15 mm were plotted. 5. Muon beam stopping distribution in the z direction of the emission layer. a: Gaussian with the peak at the surface of the layer with the standard deviation width of half the layer thickness. (3: Gaussian with the peak at the centre of the layer with the standard deviation Chapter 7. Analysis II - Emission of muonic tritium 176 90 8 0 -7 0 -60 -5 0 4 0 3 0 h 2 0 0 .80 T -B (71.3, 10©10, • — CT1.3, 100 10, 0- - CT1.3, 10© 10, •- - cr1.3, 10© 10, 0-1.3, 10910, JT1.3, 100 10, ff 1.25,15© 15, ffl.25,15© 15, cr 1.0, 15© 15, O-1.0, 15© 15, ffl.O. 10© 10, ff1.0, 10© 10, - A - A--O--•--X •-+ (A) (B) (A) « (B) a (A) P (B) p (A) (B) (A) (B) (A) (B) / / / / / / • • " s -0.85 0 .90 K -1/2 0.95 1.00 Figure 7.6: Total x2 versus the energy scaling parameter A C - 1 / 2 near the minimum region with the dependence of the fit to various systematic effects. The plot legend indicates, from the left to right, M W P C resolution parameter (see item 3 in the text on page 175), beam width parameters (item 4), the fit time region (item 2), beam stopping distribution in Z direction (item 5, no symbol indicates uniform stopping in Z). width of 0.4 times layer thickness. No symbol: uniform Z stopping distribution. For the nominal values of parameters, we obtained the best fit with K = 1.1 (re - 1/ 2 ~ 0.95 in the figures). The value of K shifted between 1.05 to 1.15 depending on the parameters (see Fig 7.6). In addition, we estimate that the uncertainty in the drift distance scale can give rise to a shift in K of order ±0.05. With the two major systematic errors (due to the parameters (l)-(5) and the distance scale) added in quadrature, and the statistical errors much smaller, as can be observed in Fig. 7.6, our measurement indicates Chapter 7. Analysis II - Emission of muonic tritium 177 a scaling factor of K = 1.10 ± 0.07, i.e., the Ramsauer-Townsend minimum energy of EeRxp = 13.6 ± 1.0 eV (cf. the theortecial minimum, Eg = 12A eV). The results of a similar analysis for a measurement using an emission target with a tritium concentration of 0.3%7 were consistent with K = 1.1. In conclusion, we have reported in this chapter, (1) the first observation of fit in vacuum, (2) the first quantitative spectroscopic evidence of the Ramsauer-Townsend effect in an exotic system, confirming the theoretical RT minimum energy in the fit + p elastic scattering at the 10% level, an accuracy sufficent for our goal of molecular formation rate measurements. 7Target ID = 1-7. Chapter 8 Analysis III - Molecular formation 8.1 Si detector data Detection of the alpha particles provided the signature of dpt molecule formation. From the data, information on two somewhat independent aspects can be extracted, namely, time spectra and absolute yields. As is the case in many experiments, the absolute measurement of the data was more difficult compared to the time spectrum measurement. In this section, the procedure for the analysis of the a particle data from Run Series 2 is presented. 8.1.1 Si detector energy calibration The energy scales of the two silicon detectors were calibrated with 2 4 1 A m sources, which were attached to the diffuser and hence could be removed during the muon beam mea-surements. Figure 8.1 illustrates the calibration spectra for one of the silicon detector (Si2) taken with the source. Measurement with no target present in the system is shown in Fig. 1(a) which had a full width half maximum resolution of about 40 keV. This is partly due to the resolution of the source itself, and the detector resolution may be better. The spectrum in Fig. 1(b), on the other hand, was taken with 2 T-/, or about 6.8 Ci of trit ium 1 present (Target 11-14 in Table 4.4 in page 108). The effect of tritium caused the peak width to 1 T h e specific activity of tritium is 2.58 C i ' / c m 3 at STP, hence 1 Tl corresponds to about 3.4 Ci. 178 Chapter 8. Analysis III - Molecular formation 179 4 0 0 0 5100 5200 5300 5400 5500 5600 5700 ADC channel (a) With no target in the system (b) With about 6.8 Ci of tritium present in the target Figure 8.1: Silicon (Si2) energy spectra taken with 2 4 2 A m source (peak energy 5.486 MeV). One A D C channel is close to 1 keV. S i l had similar spectra. Chapter 8. Analysis III - Molecular formation 180 5 5 0 0 5 4 9 0 5 4 8 0 ? 5 4 7 0 Q 5 4 6 0 5 4 5 0 5 4 4 0 5 4 3 0 0 2 4 6 8 10 12 14 Cal ibrat ion ID number Figure 8.2: Centroid values of A D C for 2 4 1 A m peak for different calibration runs, showing the stability of the system. be about 65 keV, due to the (3 decay background. Since energy spectroscopy is not the purpose of our experiments, the absolute energy scale was not very crucial. However, as we found in the thickness measurements in Section 3.3, the change in the detector temperature could cause significant variation in the energy gain, hence frequent calibration was performed to ensure the relative stability of the energy scale. In addition, the fusion signal itself, whose energy is well known, could provide the calibration information. Figure 8.2 shows the position of the centroid of the peak corresponding to 2 4 1 A m for different calibration runs. The centroid value was determined by Eq. 3.1 (page 73) in a similar manner to the thickness measurements described in Section 3.3 with the lower and upper cut off values 5300 ch and 5600 ch, respectively. The error bars in the figure Chapter 8. Analysis III - Molecular formation 181 indicate statistical uncertainties in the centroid. Changing the lower cut off value to 5200 ch caused less than 3 ch change in the centroid, indicating the systematic uncertainty in the calibration. The average of the measured centroid values is 5486.9 ch (4.7 ch) for S i l and 5473.5 ch (3.4 ch) for Si2, respectively with the values in parentheses being the standard deviation of 14 measurements. These should be compared to the weighted average of 3 alpha lines of 2 4 1 Am, 5491 keV (note that the peak energy for the strongest line is 5486 keV). Assuming there is no significant offset, which was confirmed by a test using a pulse generator, the calibration is very close to 1 ch = 1 keV for both detectors. 8.1.2 A D C dead time correction Various off-line systematic checks of scaler values, as well as comparison of the fusion yield amongst different runs were done to make sure that the system was functioning properly. During one such check, it was discovered that some runs had fewer counts in the Si spectra compared to other runs. After some investigation, it was realized that a change made to the germanium detector circuit had caused the problem in the Si detectors, and a method was devised to correct for the loss. The runs in questions are Runs 1683 - 1690 taken with Target II-9 (Table 4.3, page 107). A few initial observations included: • During Runs 1683 - 1690, both fusion signal and background counts were reduced by the same factor, indicating a normalization problem rather than target related real physical effects. • There were very many zeros in the silicon energy spectra for Runs 1683 -1690, which went unnoticed during the measurements, since on-line spectra had a cut to remove zeros. Chapter 8. Analysis III - Molecular formation 182 • These zero energy events had a similar time structure as the other non-zero events. • Other detector yields appeared unaffected. Attention then was drawn to a circuit modification which took place during Run 1683. Before this run, a germanium detector had been disabled simply by unplugging the fast timing signal from the circuit, because its high trigger rate was causing a significant dead time for the data acquisition. (Removing the germanium detector from the trigger re-duced the dead time from about 30% to 20%). In the middle of Run 1683, the circuit was modified with intention of allowing the germanium data to be recorded when other sub-detector systems fired while keeping the germanium sub-trigger out of the master trigger so that there would be no trigger if only the germanium fired. This was implemented by reviving the germanium sub-detector circuit, but with its sub-trigger (c.f. T R G n in Fig. 3.19 in page 92) not connected into logical OR for the master trigger ( T R G F in Fig. 3.19). The problem was, as it turned out, the spectroscopic A D C (AD413A, Ortec), which was shared among the germanium detector and two silicon detectors, and which was being "blocked" when only the germanium sub-trigger fired, but nothing else. AD413A, unlike other camac ADCs and TDCs, had no "fast clear" function which would ensure in the event of no master trigger, the A D C / T D C buffer was cleared by a fast pulse signal in order to be ready for the next event. Of course, when there was a master trigger, the A D C buffer was cleared (by a C A M A C command) after the data was read into the computer. Even when no sub-trigger but germanium was activated, however, the A D C gate2 was opened by the sub-trigger, hence the germanium energy data was still accepted by the A D C . The problem is that AD413A worked by design in such a way that when the master gate closes, no pulses were accepted until a C L E A R command was given, even 2Here I am referring to the master gate for the A D C ; the use of a gate for individual channels was redundant in our case. Chapter 8. Analysis III - Molecular formation 183 •if the master gate opened again. Therefore, if the germanium fired without any other triggers, and then either one of the Si detectors triggered on the following event, the Si signal was not accepted by the A D C , hence only zero was recorded. Yields of Si events were thus affected by the change in the germanium circuit. In order to obtain a correct normalization, we derived a procedure to determine the ratio of "good" A D C events to "bad" ones. We first define the following conditions in order to identify the different events: 1. Firing of Si sub-trigger (Sil or Si2) 2. Recorded Si energy = 0 3. Recorded Ge energy > 0 (i.e., not 0) 4. Recorded Ge time is an over flow We used a combination of above conditions to select the events. Several different types of events can give a similar signature, and they are summarized in Table 8.1. Type Event ( i - 1 ) Event i Condition A D C Ge Si Ge Si blocking I yes no no yes 1-2-3-4 yes II yes no yes yes 1-2-3-4 yes III no no no L L D 1-2-3-4 no IV no no yes L L D 1 - 2 - 3 - 4 no Table 8.1: Comparison of different types of events and their signatures. Event (i — 1) is the event preceding a silicon detector event (Event i). L L D stands for low level discrim-ination. The conditions are defined in the text. From the discussion above, the conditions 1 and 2 select obvious candidates for the "blocked" events. These candidate events could have zero silicon energy for two reasons; [a] the A D C blocking which is our concern (Event type I, II in Table 8.1), or [b] a real Chapter 8. Analysis III - Molecular formation 184 silicon event, but with its energy lower than the low level discrimination 3 (LLD) of the A D C (Event type III, IV) 4 . Ignoring for the moment the small chance of coincidence between an L L D event and germanium trigger in Event i (Type IV in Table 8.1), the above [a] and [b] can be distinguished by Condition 3. That is, if the event was due to blocking (case [a]), there should be non-zero germanium energy in the A D C which came from the previous Ge event (Type I, II). On the other hand, if it was due to L L D (case [b]), there is no germanium energy information left over from the previous event, hence the germanium energy is zero (Type III), unless it is a Ge-LLD coincidence event (Type IV). The condition 4 provides information as to whether there was a germanium hit in Event i. If there was no germanium trigger, then the T D C clock would not be stopped and an overflow would be recorded. This could be used to discriminate between event type I and II, as well as between III and IV. However, in the end only Condition (1-2-3) was used to identify the blocking events. This condition inevitably included event type IV as well, which was not a blocking event, but the analysis of normal runs suggested that this type of event occurs in less than 0.5% of the total Si events. The alternative Condition (1-2-3-4) would have been much worse, since it would miss selecting event type II which has a higher probability than type IV. The problem here was that when there was Ge hit in Event i, there is no way of knowing where a Ge hit also occurred (and blocked the ADC) in Event (i - 1). A n independent check of the A D C problem was conducted using a quite different effect. It took advantage of an unwanted background for our main measurements, namely, the signal from the scattered beam muon directly stopping in the Si detectors5. When 3 L L D was specified by a C A M A C command, and set a threshold, below which energy only zero was recorded in the A D C . This could save read out time. 4 F o r our purpose, an event where the silicon detector was triggered due to noise on the logic signal line can be included in [b]. 5 T h i s background is removed in future runs by placing a thin Cu foil to shield the Si from the beam. Chapter 8. Analysis III - Molecular formation 185 looking at the fusion signals, this background was reduced to a negligible level by rejecting the signals at prompt times, but here we used the intensity of this prompt beam peak to check to see if the silicon detector system was properly functioning. When the A D C was blocked, not only fusion events but also the beam stopping background events would be lost. Hence, by normalizing the beam background intensity to G M U , we could effectively measure the A D C live-to-dead time ratio. The limitation was that we had to assume that the beam was stable; a small change in the position of the beam could change the number of muons that could go through a tiny gap in the cryostat to reach the Si detectors. The A D C dead times derived from this method and the above method using scalars agree reasonably well, giving us some confidence in correcting for the A D C blocking events. 8.1.3 Energy spectrum features Before we go into the detailed quantitative analysis, it may be intructive to take a look at gross qualitative features of our data. Therefore we shall spend this section and the next for that purpose. Characteristics of the Si energy spectra, for different time cuts, are shown in Figs. 8.3 and 8.4. Figure 8.3 is for our standard time-of-flight measurement arrangement, i.e., 1000 T-l of H 2 with 0.1 % tritium and 14 T- l of D 2 overlayer in the upstream target, plus 3 T-l D 2 downstream (target ID = II-9 in Table 4.3 in page 107). Three spectra are plotted with the time cuts of (1) t > 0, (2) t > 0.02 ps, and (3) t > 1.5 ps. The histogram (1) shows a signal from direct beam muon stops in the Si detector, in addition to a broad fusion signal near 3.5 MeV, predominantly from the US moderating overlayer, and a strong low energy background peak. The beam signal is very prompt in time, and a 20 ns delay cut eliminates this very efficiently (histogram (2)). The histogram (3) with a time cut of t > 1.5 ps, on the other hand, shows a much narrower fusion peak mostly Chapter 8. Analysis III - Molecular formation 186 GO 0 2000 4000 6000 8000 channel Figure 8.3: Energy spectrum of Si detector (Sil) for standard T O F target with different time cuts, showing important characteristics as indicated in the figure. The energy scale is close to 1 ch = 1 keV. Si2 has a similar energy spectrum. coming from the DS reaction layer. The US fusion takes place typically in the first few 100 ns, while fusion in DS occurs after at travelled a separation distance of 18 mm, hence appropriate time cuts can separate these two events. Furthermore, the US D 2 layer had 14 T-l thickness while the DS only 3 T-/, resulting in the difference in the peak width. There is indication that we are seeing 3 MeV protons from dpd fusion, which contributes to the background. This will be treated in detail in Section 8.2.4. Comparison of Fig. 8.3 with Fig. 8.4, the latter for pure H 2 target with no D 2 , confirms that the peaks near 3000 - 3500 ch in the former come from fusion in the D 2 layers. On the other hand, the peak near 2 MeV persists in Fig. 8.4 (also in the bare target runs), and in fact the peak energy shifted as beam momentum was changed, providing the Chapter 8. Analysis III - Molecular formation 187 channel Figure 8.4: Energy spectrum of Si detector (Sil) for the pure H 2 target with different time cuts. The fusion peak is absent, while the muon beam peak persists. evidence that this peak is due to muons stopping in the detector. Other sources of background include: (a) muon decay electrons, (b) charged particle (proton, deuteron etc.) emission following nuclear muon capture on heavy elements, (c) muon induced nuclear break up, and (d) scattered beam electrons. Also the neutrals, such as neutrons and gammas following muon capture, muonic X rays, or bremsstrahlung from tritium j3 decay, could cause background signals in the Si detectors, but probabilities for these are expected to be small since the detectors had a thickness of only 300 pm. Among other possible background processes for targets containing tritium are conversion muons from muon catalyzed pt fusion: ppt —>4 He + p (19 MeV), and muon capture on Si from emitted pt hitting Si detectors. Chapter 8. Analysis III - Molecular formation 188 8.1.4 Time spectrum features 0 2 4 6 8 10 pis Figure 8.5: Time spectrum of Si detector (Sil) for standard T O F target with different energy cuts. Gross features of Si time spectra, with different energy cuts, are illustrated in Figs. 8.5 and 8.6. Shown in Fig. 8.5 is the S i l time spectrum for the standard time-of-flight arrangement, whereas that for pure H2 is given in Fig. 8.6. A l l the histograms have a sharp spike at time zero, which, at least in part, comes from direct muon stops in the Si detector. The low energy part (E < 2000 ch) of the spectra, which we saw was dominated by a large background signal (Figs. 8.3, 8.4), has two exponential components, a fast one with the order of 100 ns and a slow one about 2 pis. This is consistent with muon disappearance rates in heavy elements and hydrogen, respectively, suggesting the signals in this energy region come from muon decay electrons and charged particles from muon capture. Conversion muons from ppit fusion (19 MeV) could also contribute to the Chapter 8. Analysis III - Molecular formation 189 0 2 4 6 8 10 pis Figure 8.6: Time spectrum of Si detector (Sil) for pure H 2 target with different energy cuts. The fusion peak is absent, while muon beam peak persists. long lifetime. The time spectrum with an energy cut 2001 < E < 4000 ch in Fig. 8.5 exhibits fusion time signals; exponentially decaying in early time (t < Ip s) is fusion from the upstream target, while events in ~ 2 — 4ps are mostly due to fusion from pt flying across the drift distance to reach the downstream layer (though the signal is not so clear from the figure due to the unrestrictive energy cut). Whereas these are obviously absent from the same energy region in Fig. 8.6, com-parison between the two figures of the higher energy part (4001 < E < 8000 ch) of the spectra indicates excess events in Fig. 8.5. These events, unlikely due to dpt fusion since the maximum a energy is about 3.5 MeV and the probability of pile up is very small, are attributed to emitted pt reaching the Si detector where the muon is transferred to Si and Chapter 8. Analysis III - Molecular formation 190 then captured, emitting charged products. In fact, the signal in this region is enhanced when there is no moderating overlayer because of the higher yield of pt emission into vacuum. 8.2 The dfit fusion measurements In this section we shall extract, taking into account the background and various other systematic effects, the time-of-flight fusion spectra as well as the fusion yields. While the former contains the main physics we are seeking, i.e., the resonant molecular formation at epithermal energies, the latter can provide us with valuable information, as we shall see. Our fusion measurements can be divided into four distinct categories, three for the fusion yield and one for the time spectrum measurement: (a) US fusion yield with varying US layer thickness (denoted USY measurements) , (b) DS fusion yield with varying US thickness but DS thickness fixed (Moderation yield or M O D Y measurements6), (c) DS fusion yield with varying DS thickness but US thickness fixed (Time-of-flight yield or T O F Y measurements), and (d) Time spectrum of the DS fusion for various DS thickness (Time-of-flight spectrum or TOFS measurement). Note that (a) and (b), as well as (c) and (d) use the data from mostly the same run series, respectively, but they each look at different aspects of the data. Let us first discuss the background in general before going into the specifics of the each measurements. 6 T h e notation M O D is given because of its sensitivity to the pt moderation process in the US D 2 layer Chapter 8. Analysis III - Molecular formation 191 8.2.1 Background subtraction Since the Si data contains counts from non-fusion events as seen in the previous sections, subtracting appropriate background is important in the yield and spectrum determina-tion. It is one of the advantages of our multi-layer film targets that the process of interest can be controlled, i.e., "turned on" and "turned off" without much affecting other pro-cesses (e.g., by simply depositing a thin layer), hence providing rather reliable means of background subtraction. Thus our strategy generally is to use the data from the target, in which the particular processes is turned off, for the background subtraction when-ever possible (with both the signal and background data normalized to the number of muons G M U ) . For a given measurement, nonetheless, there were sometimes more than one possibility for the choice of the background runs, or no one best choice, in which case different methods were compared to see the systematic effects. Summaries of background runs used are given in Table 8.2. As we will see, in some cases further corrections were necessary to account for a residual background. Label ID Ct (%) US (T-Z) DS (T-Z) Use G M U (106) BG1 II-5 pure H2 1000 0 USY, M O D Y 84.9 BG2 II-l 0.1 1000 20 D 2 U S Y 51.9 BG3 II-6 0.1 1000 0 USY, M O D Y 62.2 BG4 11-12 0.2 1000 0 U S Y 46.4 BG5 II-7 0.1 1000014 D 2 0 M O D , T O F Y , TOFS 60.7 BG6 II-8 0.1 1000014 D 2 500 H 2 T O F Y , TOFS 54.8 BG7 11-13 0.2 1000014 D 2 0 T O F Y , TOFS 166.8 Table 8.2: Summary of runs used for background subtraction. Chapter 8. Analysis III - Molecular formation 192 8.2.2 Yield measurements Cuts Yield measurement US layer (T-l) DS layer (T-l) Time cut U S Y 3, 6, 14, (14)t, (20)* 20/0 0.02 < t < 0.4 ps M O D Y 0, 3, 6, 14 20 0.3 to 0.8 < t < 6 ps T O F Y Series A 14 3, 6, 20 1 < t < 6 ps T O F Y Series B+ 14t 3 f , 23+ 1 < t < 6 ps Table 8.3: The target combinations for fusion yield measurements. Tritium concentration of ct = 0.1% was used, except the ones marked f (ct = 0.2%) and J (ct = 0.05%). The layers with bold face were kept fixed during the measurements. Some of the US yield measurements were done with no DS layer. The time cuts for the yield measurements have been chosen to maximize the fusion signal of interest, while avoiding the background. For example, the US and DS fusions, when they coexist, can largely be separated by appropriate time cuts, since most US fusion takes place nearly promptly after the muon stop and disappears with typical time constant of ~ 100 ns (depending on the tritium concentration), while DS events occurs after pt time-of-flight across the drift distance, typically of order ps. The standard time cuts of 0.02 < t < 0.4 ps, and I < t < 6 ps were used for the U S Y and T O F Y measurements, respectively, while the M O D Y time cuts were slightly varied to maximize the signal-to-background ratio (e.g., with a thin US moderator, the pt energy is rather high, hence the time-of-flight short, which could overlap with US fusion). Table 8.3 summarizes the main conditions for the yield measurements. Together with the partial yields for given time cuts, we give the total yield with the acceptance for the time cut estimated by SMC simulations. These corrections are typically 10% or less. The comparisons with Monte Carlo calculation results, which will be discussed later, will be done using the same time cuts for both the data and M C , hence Chapter 8. Analysis III - Molecular formation 193 > o X in I O X Ui o o 1000 Si 2 i i i 2000 3000 4000 5000 1000 2000 3000 4000 5000 Energy (ch) Figure 8.7: Si energy spectra with 3 T-l US D 2 layer in the early time range. Top figures show the fusion signal (filled circles ) and a background run without US D 2 layer (open circles), while bottom figures give background subtracted spectra. the time cut efficiencies will cancel in those analyses (assuming the time distribution is predicted correctly by the MC) . The effects of the energy cuts on the yield were carefully studied by using different cuts, as well as estimating the efficiency using a dedicated Monte Carlo. These and other corrections will be discussed later. Chapter 8. Analysis III - Molecular formation 194 3.0 -0.5 1000 Si 2 ft 4-fc 2000 3000 4000 5000 1000 2000 3000 4000 5000 Energy (ch) Figure 8.8: Top: early time Si energy spectra with 14 T-l US D 2 (filled circles) and a pure H 2 background run (open circles). Bottom: background subtracted Si energy spectra. Notice the wider peak and the tail at low energy compared to the 3 T-l measurement in Fig. 8.7. USY: Upstream Yield The time cut of 0.02 < t < 0.4 was used for the US measurements to select the fusion events in the upstream layer. The first 20 ns was avoided due to the prompt beam muon background, while the long time was excluded to remove the fusion events in the downstream layer. Chapter 8. Analysis III - Molecular formation 195 For these measurements, we had two main choices of background run in most cases: B G l , a pure H 2 target, and BG2, a standard emission target with 20 T-l D 2 DS (see Table 8.2). Fortunately, thanks to a good signal-to-background ratio and the high yielding US fusion, the effect of background choice is small. Examples of the US fusion data with the background are illustrated in Figs. 8.7 and 8.8 for 3 T-l and 14 T-l cases, respectively. Notice the differences in the peak width and the extent of the lower energy tail, both due to a particle energy loss in the D 2 layer. To determine the a yield, we typically used at least two different energy cuts. The energy cut efficiency corrections were done using a dedicated Monte Carlo code, which will be discussed later. Si yield for the US fusion measurements with different backgrounds are given in Table 8.4. When the effects due to different background subtractions are larger than the statistical errors, we considered the former as a measure of the systematic uncertainty. These cases are indicated with * in the table. Runs with target ID=II-2, II-3, and II-4, were performed consecutively in identical conditions (except US thickness), hence the relative yields are more reliable than, other separate runs. MOD: Moderation Yield Similar to the US measurements, runs B G l and BG2 were used for background sub-traction for most runs. Because of the lower signal-to-background ratio, the choice of background had larger effects on the yield determination for the M O D measurements. Since B l has no tritium or deuterium, it may underestimate the background, while unmoderated pt emission from B2 may contribute to the over-subtraction of the back-ground. Some events in B2 appear to come from pt reaching Si detectors (p capture on Si produces charged products), which have time structure similar to the real data with a characteristic time-of-flight. Table 8.5 gives the yield with different backgrounds. Chapter 8. Analysis III - Molecular formation 196 US (DS) ID E cut ( x l O 3 ch) B G Yield (10 - 4 /GMU) B G avg. S i l Si2 Si avg. 3 T-l (20) II-2 2.7; 3.7 B G l BG2 0.580(15) 0.582(15) 0.572(15) 0.575(16) 0.576(11) 0.579(11) 0.577(9) 3.1; 3.7 B G l BG2 0.537(13) 0.543(13) 0.541(13) 0.545(14) 0.539(9) 0.544(10) 0.541(8) 6 T-l (20) II-3 2.0; 3.7 B G l BG2 1.64(2) 1.64(2) 1.67(2) 1.68(2) 1.66(2) 1.66(2) . 1.658(14) 2.7; 3.7 BG2 BG2 1.55(2) 1.55(2) 1.60(2) 1.61(2) 1.68(1) 1.58(2) 1.628(48)* 14 T-l (20) II-4 1.0; 3.7 B G l BG2 4.26(4) 4.28(4) 4.27(4) 4.28(4) 4.27(3) 4.28(3) 4.273(24) 2.0; 3.7 B G l BG2 3.97(4) 3.97(4) 3.99(4) 3.99(4) 3.98(3) 3.98(3) 3.980(22) 14 T-l (0) II-7 1.0; 3.7 B G l BG3 4.24(3) 4.24(3) 4.41(3) 4.48(3) 4.33(2) 4.36(2) 4.343(19) 2.0; 3.7 B G l BG3 4.01(3) 4.02(3) 4.15(3) 4.21(3) 4.08(2) 4.12(2) 4.098(17) 14 T-l (0) (H = .2% 11-13 1.0; 3.7 B G l BG3 4.55(2) 4.58(2) 4.54(2) 4.65(3) 4.55(2) 4.62(2) 4.580(35)* 2.0; 3.7 B G l BG3 4.28(2) 4.30(2) 4.28(2) 4.34(2) 4.28(1) 4.32(1) 4.300(20)* 20 T-l (0) ct = .05% 11-16 1.0; 3.7 B G l BG2 4.41(3) 4.41(3) 4.54(3) 4.60(3) 4.48(2) 4.51(2) 4.490(16)* 2.0; 3.7 B G l BG2 3.99(2) 4.00(2) 4.10(2) 4.16(2) 4.05(2) 4.08(2) 4.063(14) Table 8.4: Si yield per G M U for the US fusion measurements with different background subtraction and energy cuts. A time cut of 0.02 < t < 0.4 ps was applied to select the US events. The error values with * include systematic effects due to the different background subtraction. Tritium concentration of ct = 0.1% was used, unless otherwise stated. Similarly to the US measurements, the background subtraction effects are included in the error (noted with *), if it is larger than the statistical error. When one background run is clearly more applicable than the other, that alone was used for the yield determination, which was the case for 0 and 14 T-l measurements. Chapter 8. Analysis III - Molecular formation 197 US (ID) T cut (lis) E cut ( x l O 3 ch) B G Yield (10 - 5 /GMU) B G avg. S i l Si2 Si avg. 0 T-l (II-l) 0.3; 6 1.0; 3.7 BG1 BG3 27.3(3) 25.7(3) 26.6(3) 25.0(3) 27.0(2) 25.4(2) 25.35(20) (BG3 only) 2.0; 3.7 BG1 BG3 25.4(2) 24.5(2) 24.6(2) 23.6(2) 25.0(2) 24.1(2) 24.04(17) (BG3 only) 3 T-l (II-2) 0.5; 6 1.0; 3.7 BG1 BG3 16.0(2) 14.3(2) 16.5(2) 14.7(3) 16.3(2) 14.5(2) 15.38(88)* 2.0; 3.7 BG1 BG3 15.2(2) 14.2(2) 15.6(2) 14.6(2) 15.4(1) 14.4(1) 14.90(50)* 6 T-l (11-3) 0.6; 6 1.0; 3.7 BG1 BG3 9.58(18) 7.82(19) 9.84(18) 8.02(19) 9.71(13) 7.92(13) 8.82(90)* 2.0; 3.7 BG1 BG3 9.41(15) 8.38(15) 9.38(15) 8.34(16) 9.40(11) 8.36(11) 8.88(52)* 14 T-l (II-4) 0.8; 6 1.0; 3.7 BG1 BG5 1.81(14) 1.79(15) 2.16(14) 2.07(15) 1.99(9) 1.93(11) 1.93(11) BG5 only 2.0; 3.7 BG1 BG5 1.92(9) 1.76(10) 2.06(10) 1.91(10) 1.99(7) 1.84(7) 1.84(7) BG5 only 14 T-/ a (11-11) 0.8; 6 1.0; 3.7 BG5 BG6 3.17(16) 2.29(16) 2.98(16) 2.01(17) 3.08(11) 2.15(12) 2.15(12) BG6 only 2.0; 3.7 BG5 BG6 1.99(10) 1.96(10) 1.89(10) 1.76(10) 1.94(7) 1.86(7) 1.86(7) BG6 only "With 500 TI H 2 layer at DS under 20 Tl D 2 layer. Table 8.5: Si yield per G M U for the M O D measurements (DS fusion with varying US moderator thickness) with different background subtractions and energy cuts. The error values with * include a systematic effect due to the different background subtraction. When one B G method is clearly more applicable than the other, we quote that wilue in the final column. Tritium concentration of ct = 0.1% was used. T O F Y : T i m e - o f - f l i g h t Y i e l d Two separate series of T O F measurements were performed with tritium concentration ct = 0.1% (Series A) and ct = 0.2% (Series B). Due to the small counting rates, the T O F measurements, especially the ones with thin DS layers, were most difficult among the fusion measurements. We give in Table 8.6 the Si detector yield per G M U for T O F Y measurements with different background and energy cut conditions. The information Chapter 8. Analysis III - Molecular formation 198 we obtain in this section, such as the background and the fusion yield, will be directly applicable to the time-of-flight spectrum measurements which are our main physics data. Hence we give some detailed discussion here of the background subtraction. TOF series A The measurement for a 3 T-l layer in Series A (ct = 0.1%) had an additional difficulty in the background subtraction, which was overcome by using different methods of back-ground estimation and comparing them. The most suitable background run for the ct = 0.1% series (BG6 in Table 8.2, Page 191) had rather limited statistics with about 1/9 of GMUs of the production run. Performing our standard background subtraction procedure using BG6, illustrated in Fig. 8.9, exhibited two potential systematic effects: (a) a S i l to Si2 detector ratio of up to about 30% was observed in the yield using BG6 for 3 T-l (see values with f in Table 8.6), which appear to be due mostly to asymmetry in the background BG6, not the signal, and (b) there appears to be low energy tails in the background subtracted spectra down to 2 MeV (2000 ch) (see Fig. 8.9, and Table 8.6). Using instead BG5, which does not have DS 500 T-l H 2 under the DS deuterium layer, makes the Si l /Si2 ratio consistent with 1. However, the lack of the H 2 layer in BG5 could cause an underestimate of the background, since muon decay electrons are a major source of background especially at lower energies. In fact, yields with BG5 are higher than those with BG6 in all cases, but clearly with a lower energy cut (see values with | in Table 8.6). Another possible run to be used for background subtraction is BG7, which had an emission target with ct = 0.2%, and no H 2 DS substrate layer. Despite the difference in ct, which might possibly introduce further systematic error, this run had much better statistics (about 3 times that of BG6). Chapter 8. Analysis III - Molecular formation 199 US ct E cut B G Yield ( 1 0 - 6 / G M U ) (ID) (%) ( x l O3 ch) Si l Si2 Si avg. 3 T-l 0.1 2.0; 3.7 BG5 5.74(40) 5.81(42) 5.77(29) (11-9) BG6 5.64(42) 4.29(47) 4.96(31) BG5+6+7 5.66(24) 5.61(25) 5.63(18) 2.5; 3.7 BG5 4.95(33) 5.07(34) 5.01(24) BG6 5.20(34)+ 3.67(39)* 4.43(26) BG5+6+7 4.88(21) 4.75(21) 4.82(15) 3.1; 3.7 BG5 4.04(25) 4.39(24) 4.22(17) BG6 4.35(25) 3.53(28) 3.94(19) BG5+6+7 4.12(16) 3.98(16) 4.05(11) 6 T-l 0.1 2.0; 3.7 BG5 8.83(81) 9.89(85) 9.36(59) (11-10) BG6 8.73(82) 8.38(88) 8.55(60) BG5+6+7 8.75(74) 9.67(79) 9.22(54) 2.7; 3.7 BG5 8.79(70) 9.55(72) 9.17(50) BG6 9.01(70) 8.48(74) 8.75(51) BG5+6+7 8.68(66) 9.26(68) 8.97(47) 20 T-l 0.1 1.0; 3.7 BG5 30.2(15)* 28.0(15)* 29.1(11)* (11-11) BG6 22.4(16) 18.9(16) 20.7(11) BG5+6+7 29.1(14) 26.5(14) 27.8(10) 2.0; 3.7 BG5 19.3(10) 18.8(10) 19.05(68) BG6 19.2(10) 17.3(10) 18.24(69) BG5+6+7 19.2(9) 18.6(9) 18.91(64) 2.7; 3.7 BG5 16.3(8) 16.6(8) 16.45(59) BG6 16.5(8) 15.5(9) 16.03(59) BG5+6+7 16.2(8) 16.3(8) 16.25(56) 3 T-l 0.2 2.0; 3.7 BG7 2.81(33) 3.85(36) 3.33(24) (11-14) 2.5; 3.7 BG7 3.02(29) 3.45(30) 3.25(21) 3.1; 3.7 BG7 2.86(23) 2.93(23) 2.89(16) 23 T-l 0.2 1.0; 3.7 BG7 23.6(10) 22.7(10) 23.16(73) (11-15) 2.0; 3.7 BG7 21.1(7) 21.1(8) 21.08(52) 2.7; 3.7 BG7 18.6(7) 18.9(7) 18.70(46) Table 8.6: Si yield per G M U for the T O F measurements (DS fusion with US moderator fixed at 14 T-l) with different background subtractions and energy cuts. Two separate series of runs were performed with tritium concentration ct = 0.1% and 0.2%. Chapter 8. Analysis III - Molecular formation 200 Energy (ch) Figure 8.9: Si energy spectra for the T O F yield measurement with 3 T-l DS D 2 in Series A (ct = 0.1%), with the time cut of 1 < t < 6 /is. Top figures show the fusion signal (filled circles ) and a background run B G 6 (open circles), while bottom figures give background subtracted spectra. Given this situation for the 3 T-l measurement in the ct = 0.1% series, we made several different background subtractions, and compared them to see the systematic effects. Method 1 is the standard procedure using BG6, but with a wide energy cut (2.5 < E < 3.7 MeV) to account for the possibility that the lower energy tails are due indeed to real a particle events, which gave the Si averaged yield of Ymeth\ = 4.43(26) x 1 0 ~ 6 / G M U . The energy spectra obtained with Method 1 had nonzero counts even at energies Chapter 8. Analysis III - Molecular formation 201 2.5 2.0 h 1.5 O 1.0 co 0.5 0.0 CD I 1.5 X i-o h 0.5 h cn o O o.o -0.5 S i l + ^ a * 5 5 G B * C S 3 o « j; 1 B - L O SG B-HI J ll •rfx i i $ *• if Sat!? + 4- + B - L O SG B-HI I i j i U L j i ^ i i i i i , ! 1000 2000 3000 4000 5000 1000 2000 3000 4000 5000 Energy (ch) Figure 8.10: The energy spectra for the T O F 3 T-l (ct — 0.1%) measurement with the background as sum of BG5, 6, and 7, describing Method 2. The additional correction to the yield in the signal region SG was applied for the residual background present in the bottom figure, using the average of the background region B-LO and B-HI. See text for the details. much higher than the fusion signal, where we expect an average of counts consistent with zero. In Method 1-b, taking into account the possibility that this is due to a potential error in the relative normalization (although the A D C blocking effect in Si detectors was corrected carefully in Section 8.1.2), we fitted the region between 4000 and 8000 ch with a constant base line, and subtracted that from the yield from Method 1 above to obtain Chapter 8. Analysis III - Molecular formation 202 O 1000 2000 3000 4000 5000 6000 1000 2000 3000 4000 5000 6000 Energy (ch) Figure 8.11: Background subtraction Method 2-c, where the background subtracted spec-trum obtained using BG5+6+7 (bottom in Fig. 8.10) was fitted with an exponential function to remove residual background. Ymethi-b — 4.06(27) x 1 0 - 6 . Using a line with a slope or exponential curve in the fit gave a larger base line yield (hence smaller fusion yield), but we use the value Ymeth\-b as our upper limit in calculating the average later. Method 2 used the entire background data set available (sum of BG5, BG6, and BG7) to make a first order background subtraction (see bottom in Fig. 8.10). But keeping in mind some differences in the conditions, we made an additional, second order correction from the above background. From the narrowly defined signal region (denoted SG, 3.1 < E < 3.7 MeV), the average of counts from the region below (shown as B-LO, 2.2 < E < 2.8 MeV) and above (B-HI, 3.8 < E < 4.6 MeV) were subtracted, i.e., Ymeth2 =SG - 0.5(B-LO + B-HI). With SG=4.05(11) x IO" 6 , B-LO=0.34(6) x IO" 6 , and B-HI=0.73(10) x IO" 6 , we have Ymeth2 = 3.52(12) x IO" 6 . The systematics in Method 2 were checked for the manner in which the additional Chapter 8. Analysis III - Molecular formation 203 correction was made; the background cut, B-LO, might contain the signal, which, to-gether with the narrow cut for SG, might lead to over-subtraction of background. We present two more systematic checks of Method 2, in which the second order additional correction was done in a different way, while keeping the first order the same as Method 2. Method 2-b is simply to use a wider signal region of 2.7 < E < 3.7 MeV, with B-LO 2.1 < E < 2.6 MeV and B-HI 3.8 < E < 4.3 MeV. Note that because of the low energy structure of background, we do not desire to use the spectrum much lower than 2 MeV. Thus B-LO and B-HI now each have half the energy width of SG. The detector averaged yield thus obtained with Method 2-b is: Ymeth2b = 3.65(18) x 10~6. In Method 2-c, illustrated in Fig. 8.11, we have fitted the residual background obtained in Fig. 8.10 with an exponential function. The fit was performed simultaneously in the regions B-LO (2.0 < E < 2.4 MeV) and B-HI (3.8 < E < 6.0 MeV), and are plotted with a solid line in Fig. 8.11. The yield was determined in the SG region (2.5 < E < 3.7 MeV), which was excluded from the fit, by subtracting the area under the fitted curve. We obtain Ymeth2c = 3.45(15) x 1 0 - 6 , where the error is due to the statistics in the SG counts. The Si yield for 3 T-l in T O F Series A (ct = 0.1%), using different background sub-traction methods, are summarized in Table 8.7. Energy cut width dependent corrections, i.e., the energy cut efficiency and the dd proton contribution, which will be discussed in detail in Sections 8.2.4 and 8.2.3, are applied here. As can be seen, Method 2, 2-b, 2-c, which all used combined background BG5-r-6+7 but had different methods of residual background subtraction, gave consistent results, giving some confidence at least in the residual background subtraction procedure. In comparison with Method 1, Method 1-b appears more reliable since it takes into account the possible relative normalization error by subtracting a constant base line. Note that in Chapter 8. Analysis III - Molecular formation 204 Method 1 1-b 2 2-b 2-c BG run BG6 BG6 BG5+6+7 BG5+6+7 BG5+6+7 SG cut (MeV) SG yield (1(T 6 /GMU) 2.5; 3.7 4.43(26) 2.5; 3.7 4.43(26) 3.1; 3.7 4.05(11) 2.7; 3.7 4.64(14) 2.5; 3.7 4.81(15) B-LO cut (MeV) B-HI cut (MeV) Fit/sum for ith bin B yield (1(T 6 /GMU) - 4.0; 8.0 yi = const. 0.38(6) 2.2; 2.8 3.8; 4.4 E i Vi 0.54(6) 2.1; 2.6 3.8; 4.3 0.99(11) 2.0; 2.4 3.8; 6.0 yi = ae~Xilh 1.36 TOF yield (10- 6 /GMU) 4.43(26) 4.06(27) 3.52(12) 3.65(18) 3.45(15) dd proton E cut efficiency 2.4(9)% 1 2.4(9)% 1 0 0.980(3) 2.4(9)% 1 2.4(9)% 1 T O F C O T r (10- 6 /GMU) 4.32(26) 3.96(27) 3.59(12) 3.56(18) 3.37(15) Table 8.7: Summary of the detector averaged Si yield for 3 T-l T O F measurement (c t = 0.1%) with different background subtraction methods. The energy cut dependent corrections (cut efficiency and dd proton contribution) are made. Taking the average of two bold values we obtain Y3TI = 3.51 ±0.25 x 1 0 _ 6 / G M U , without the time cut and the nitrogen contamination corrections (see text for the details). Series B , the constant term was consistent with zero, as expected. Discarding Method 1, we chose the two extreme values in (Method 1-b and Method 2-c) as representative of the deviations due to the background subtraction procedure. Since these measurements are not completely independent, determining the average and its error is somewhat tricky. Obviously, since the two values are incompatible within in the quoted error, simply taking a weighted mean and combining the error would underestimate the total uncertainty. Here we adopt a procedure used by the Particle Data Group [229,230], which scales the error bar to give x2/dof — 1. In our case, with the numbers of points and the free parameter being 2 and 1 respectively (hence dof — 1), we scale the error by a factor 1.9 and obtain Y 3 T L = (3.51 ± 0.25) x 1 0 - 6 / G M U . As for other T O F series A measurements (with ct — 0.1%), the statistics of the pro-duction runs and the standard background run (BG6) are comparable, and the accuracy is not limited by the background statistics. Also because of a better signal-to-background Chapter 8. Analysis III - Molecular formation 205 ratio, effects of the background are not as serious. Furthermore, the Si l /Si2 asymmetry in Si yields are at the 10% level for the most part. Thus, we shall use BG6 for our background subtraction. Using other backgrounds in fact gives consistent results for a reasonable range of the energy cuts (Table 8.6). TOF series B The T O F measurements with ct = 0.2% (Series B) had a good background run (BG7) with high statistics, comparable to that of the production run. Neither of the runs had a H 2 substrate under the DS D 2 . The yields with different energy cuts are given in Table 8.6. In the 3 T-l run with ct = 0.2% there is some 10% increase in the yield when the cut lower limit is extended from 3.1 MeV to 2.5 MeV (Method 3-nr and Method 3-wd). This is due to an increase in the Si2 counts (Sil remains constant). Again, we test different background subtractions. In Method 3-b, the high energy (> 4 MeV) part of the background subtracted spectra (bottom of Fig. 8.12) were fitted with a constant line as in Method 1-b above. These gave the results consistent with zero base line, indicating there is no normalization problem. The spectra was then fitted with either exponential or linear functions in the region excluding the signal (1500 < E < 2400 ch and 3800 < E < 6000 ch) to look for a potential residual background (Method 3-c). While the S i l fit was consistent with zero background, the Si2 fit found 0.50(6) x 1 0 6 / G M U background in the signal region (2500 < E < 3700). The results are summarized in Table 8.8. The shift between Method 3-wd and Method 3-c is due to the yield B (the area under the fitted curve in the signal region), which is well determined from the fit. Taking into account both possibilities that B is due to the background or due to the real signal, we take an average of two values and accept half the difference as our systematic uncertainty. Adding the statistical error in quadrature, we obtain the yield for Series B , Chapter 8. Analysis III - Molecular formation 206 2.5 2.0 h > CD o co X i o X co +-> ti O 1.5 1.0 h 0.5 0.0 1.5 1.0 0.5 h 0.0 -0.5 i Hi % - 11 ^ i ^ i i i i i i t i i i i l I* Si2 i i i i i 5J i iMiiiiiuJilii 1000 2000 3000 4000 5000 1000 2000 3000 4000 5000 Energy (ch) Figure 8.12: Si energy spectra for the T O F yield measurement with 3 T-l DS D 2 in Series B (ct = 0.2%), with the time cut of 1 < t < 6 fis. Top figures show the fusion signal (filled circles ) and a background run B G 6 (open circles), while bottom figures give background subtracted spectra. y 3 f ( = (3.05 ± 0.24) x 1 0 - 6 / G M U . Chapter 8. Analysis III - Molecular formation 207 Method 3-nr 3-wd 3-c B G run BG7 BG7 BG7 SG cut (MeV) SG yield (10 - 6 /GMU) 3.1; 3.7 2.89(16) 2.5; 3.7 3.25(21) 2.5; 3.7 3.25(21) B-LO cut (MeV) B-HI cut (MeV) B yield (1O-7GMU) - -1.5; 2.4 3.8; 6.0 0.28(9) T O F yield (10 - 6 /GMU) 2.89(16) 3.25(21) 2.97(23) dd proton E cut efficiency 0 0.980(3) 2.4(9)% 1.0 2.4(9)% 1.0 T O F c o r r (10-7GMU) 2.95(16) 3.17(21) 2.90(23) Table 8.8: The detector averaged Si yield for Series B (ct = 0.2%) 3 T-l T O F mea-surement with different energy cuts, with and without a residual background fit. The energy cut dependent corrections (cut efficiency and dd proton contamination) are included. Taking into account the systematic effect in the background we obtain Y3Tl = 3.05 ± 0.24 x IO -7GMU. 8.2.3 Systemat ic effects Sil-Si2 symmetry Generally, higher count rates were recorded in Si2 compared to S i l . This is presumably due to the higher background rates, as Si2 was geometrically slightly closer to the beam axis. However, as was seen in Tables 8.4-8.6, the background subtracted yields are mostly consistent with the Sil-Si2 symmetry, except when the signal-to-noise ratio is low as in 3 T-l T O F measurements where we observed up to 30% asymmetry. We note, however, that while the individual detector yield can be sensitive to a slight shift in the beam position, this effect is cancelled in the averaged yield to first order. For example, in the limit of a small beam size, a 5 mm shift in the beam can cause some 40% asymmetry in the individual detector yields, yet the averaged yield is affected only by a few percent. Therefore it is crucial to have two detectors particularly for the absolute Chapter 8. Analysis III - Molecular formation 208 yield determination. Time zero shifts We have observed some shift in the time zero over the course of the three week experimen-tal run. The time shift between the signal runs and background could cause some errors in background subtraction. The uncertainty introduced by the few ns shift is obviously negligible for the T O F measurements which has a time scale of ps, but its effect on the US yield was investigated by intentionally introducing the time shift upon background subtraction, and found to be about 0.5% for a shift of 4 ns, which can also be safely neglected. Si detector double peaking Both S i l and Si2 time spectra exhibited a small satellite peak for a prompt signal, which came 15 to 20 ns before the nominal time zero with an amplitude (averaged over two detectors) of about 5% relative to the prompt peak. The origin of the second peak is not clear, but if it contains real counts, the total yield would be underestimated if we only counted the events in the prompt peak. Again, it is negligible for the T O F and M O D measurements. The possible effect on the US yield was estimated assuming the fusion decay time of 100 ns to be less than 1%, hence we neglected the double peaking effect in our analysis. Cut efficiency correction Efficiencies for the time cuts can be determined by making similar cuts in the full Monte Carlo time spectra. For this purpose, we used the standard SMC input with modified low energy formation rate (the nominal input, see Section 8.3). For the comparison with Chapter 8. Analysis III - Molecular formation 209 c3 -2 I 1 1 1 i 1 i 1000 1500 2000 2500 3000 3500 4000 4500 Energy (ch, keV) Figure 8.13: Energy cut efficiency determination using a dedicated Monte Carlo code APEC-97 [212] (plotted with histograms), featuring a package P E P P E R [213] which incorporates a non-uniform fusion depth profile calculated with the full Monte Carlo SMC. Without P E P P E R , a uniform fusion distribution is assumed. US 14 T-l S i l data are plotted with error bars. The use of P E P P E R in APEC-97 does not seem to be justified. various Monte Carlo calculations with different physics input, we will be applying the same time cuts between the M C and data, hence the time cut efficiencies cancel out as long as the M C correctly describes the fusion time spectrum (not necessarily the absolute yield). For energy cut efficiencies, we have used a dedicated Monte Carlo code, APEC-97 (Alpha Particle Energy Computation ver. 1997), featuring a package P E P P E R (Profile Evaluation Package for Particle Energy and Range). A n example of the APEC-97 out-come is illustrated with and without the use of P E P P E R in Fig. 8.13. APEC-97 with Chapter 8. Analysis III - Molecular formation 210 PEPPER assumes the fusion depth profile calculated by SMC, while APEC-97 without PEPPER uses a uniform fusion profile. It is interesting to notice that the simple uniform fusion depth profile (without PEP-PER) seem to describe the observed Si spectrum better than the profile derived from a detailed full simulation of the muonic processes leading to fusion (with PEPPER). This might be due to real physics, but from these data alone we cannot exclude other possi-bilities such as non-uniform target layers (which is only included as the effective average thicknesses, see Section 6.1) or an inaccurate assumption of beam radial distribution7. Nonetheless we take the average of the cut efficiency derived from each depth profile, and take the difference as a measure of uncertainty. The derived energy cut efficiencies, together with various other corrections, will be given later in Table 8.13. Nitrogen contamination If there was a contamination of heavier elements in the emission layer, the yield of fit would be reduced due to muon transfer from the proton to the heavy element. In some of the runs, we observed muonic X rays from muonic nitrogen atomic transitions, indicating contamination of the target. From the analysis of the Lyman series X rays, the contamination level was estimated to be a few ppm level by Francoise Mulhauser [83,231]. In the constant rate, infinite medium approximation, the probability of pt production Vj^t can be expressed as: -pN (j>ct\pt ^ where CJV2 is the concentration of N 2 molecules, and \PN the muon transfer rate from a proton to a nitrogen atom. Using the proton to triton transfer rate A p i = 5.86 x 103 ps-1, and ppp formation rate A p w = 3.21 ^s _ 1 , both obtained from our earlier measurements 7Flat-top Gaussian 10 © 10 mm was assumed for the fusion radial profile in these simulations Chapter 8. Analysis III - Molecular formation 211 in solids [83], together w i th the transfer rate to nitrogen Apjv = 0.34 x 101 1 fis f rom Ref. [232] (measured in a gas), we can estimate the reduction factor for the fit production = V^JV%, which is normalized to the pure target yield V% = 0.621. Table 8.12 presents the correction factor for two series of target sets which are of relevance in our analysis. Target ID cN2 (Ref. [231]) VN r fit ' eN S E T Q - 0 .1% (a) SETc t = 0 .1% (b) I I I to I I-4 II-6 to 11-11 3-3+3;3 ppm ^-ii P P m o.62i^;js 0 . 6 1 l i o ; 2 3 0.966 ± 0.034 0.976 ± 0.024 Table 8.9: The fit production probabil i ty V^t w i th possible nitrogen contamination, and the reduction factor eN, normalized to a pure emission target V°t = 0.621. We note that the estimate of nitrogen contamination by Mulhauser gives asymmetric errors as quoted in the table, but we take the average of two extreme values of la error bars for convenience in the data treatment. In our analysis, we assume that reduction in fit emission is proportional to the factor e/v given here, neglecting the effects of the fit —> fiN transfer, which is presumably much smaller. 8.2.4 dfid fusion Protons f rom dfid fusion in the deuterium layer can cause a background in the a mea-surements depending on the energy cut (recall that the proton energy is 3 M e V ) . They can come f rom two different sources: (a) direct stopping of muons in the deuterium layer, or (b) recycled muons, i.e., the muons released after the fusion reaction (wi th the prob-abi l i ty 1 — u>s, where LOS is the sticking probabi l i ty) . For both cases, because of solid state effects and finite thickness, estimating the proton yield is diff icult, and there is yet no satisfactory theoretical model. The use of the Monte Carlo is untested in these Chapter 8. Analysis III - Molecular formation 212 conditions, and its reliability is questionable without solid state effects included in the input. Nonetheless, we first make an analytical estimate using the two node kinetics model which successfully, if accidentally, described the time evolution of dpd fusion in a bulk solid deuterium [199]. According to this kinetics approximation, with an assumption of an infinite target and ignoring dpd cycling, dpd fusion yield per muon Vd^d can be obtained as: •pkin _ f r dfid — J I HK,d + A H ) + A 0 cf>{X%d + A H ) + A 0 <j>X%d + A 0 + h , (8.2) ^ld + Ao where <f> = 1.4 is the target density in units of liquid hydrogen atomic density, fz = = | are the initial hyperfine populations, and A 0 = 0.456 ps'1 is the free muon decay rate. Using the effective formation rates from our measurements in thick solid deuterium targets [166,199], i.e., molecular formation rates from each hyperfine state, 3 1 3 j Adnd ~ 2-7 ps'1, Xd/id = 0.044 ps'1, and the spin flip rate, A22 = 34 ^ s, we obtain ^dfld ~ 0.16. The fusion branching ratio into protons Bp is a somewhat complicated function of atomic hyperfine states and molecular angular momentum states, but for our analysis here, it suffices to let Bp ~ 0.5. Hence we have the proton yield per muon in the kinetics model, Vpm Vfn = Vkdlnd • Bp ~ 0.08, (8.3) of which 30% comes from the fast (34 ps'1) component, and the rest with the slower time slope. This level of proton yield would give a non-negligible contribution in the a yield mea-surements, when the direct stops and the recycling components are combined. However, Chapter 8. Analysis III - Molecular formation 213 a proton yield at the 10% level relative to a can be excluded by comparing the expected - peak shape, simulated with an energy loss Monte Carlo program [212], with experimental energy spectra. Recall that the dd proton, with an energy of 3 MeV, has energy loss much smaller than a 3.5 MeV a, by a factor of roughly (Zp/Za)2(Mp/Ma) ~ 1/16, hence a 10% proton yield would appear as a sharp peak in the Si energy spectra, which is absent from the data. Given the inconsistency with this simple analytical approach, we pay closer attention to our data set to extract the proton contribution in the sections which follow. D i r e c t s t o p c o n t r i b u t i o n a t l a t e t i m e As we mentioned, the fusion yields in the upstream layers were measured with a time cut of 0.02 < t < 0.4 ^s, while the downstream measurements were done mainly with 1 < t < 6 ps. The proton contributions at these two time ranges, as well as those from different sources (e.g., direct vs. cycling), need to be considered separately. We first investigate the protons from direct muon stops in a D 2 layer at late time. This is important for the measurements using thick downstream layers, from which the DS direct proton contribution cannot be subtracted by using the standard background runs which had no downstream layer. In order to estimate this contribution, we looked at the yield of protons coming from the 14 T-l upstream deuterium layer when only a very thin downstream layer was present. Given in Table 8.10 is the yield in the peak at 3 MeV (2960 < E < 3040 ch) with time cut of 1 < t < 6 ps for two different series of runs. The counts just above and below the peak in the energy spectra (with the same total bin width), were subtracted to account for the background. As we will see later cycling from the US layer contributes very little at this late time, and assuming cycling from DS is negligible8 since the layer is very thin, we obtain Ypdvflate, 8 T h i s approximation introduces an error of no more than 0.5% in the final correction. Chapter 8. Analysis III - Molecular formation 214 Target Y i e l d / G M U ( x l O " 7 ) US DS ID S i l Si2 SET1©14T-/ D 2 SET2014T-/ D 2 3T-Z D 2 3T-Z D 2 II-9 11-14 1.77(45) 1.36(74) 1.81(48) 1.79(79) Table 8.10: Yp_late, the proton yield in late time (1 < t < 6 ps). the proton yield from direct stops in 17 T-l (14 US + 3 DS) of deuterium layers in the time region of 1 < t < 6 ps. Averaging the above values, we have Ypd"late = 1.68(32) x 10~7 protons per G M U . Our downstream thicknesses ranged from 6 T-l to 23 T-l, hence this yield was scaled proportionally to the thickness in order to give the direct stop contribution at late time. Cycling contribution For the cycling contribution, we take advantage of the fact that we had two independent Si detectors, and study the correlations between them. With an energy cut of 2501 < E < 3700 ch and the time cut of 0.02 < t < 0.4 ^s in one of the detectors, which selected alpha events with a high probability, we looked at the other detector. With an energy cut in the second detector of 2701 < E < 3100 ch to select proton events, we obtained the coincidence rates with two different time cuts: (a) —0.01 < (Tp — Ta) < 6 ps, and (b) 0.01 < (Tp — Ta) < 6 ps, where Tp — Ta is the time difference between the proton and the a candidate events. The yield with time cut (b) was corrected for the cut efficiency assuming the dpd fusion disappearance rate of 34 ps~x given by the hyperfine transition rate in the kinetics model [166], as well as the average time-of-flight difference between the 3 MeV proton and 3.5 MeV a, while the cut (a) required no time corrections. The conditional solid angle for the protons given an a is detected, could be different from the solid angle singles events, since the a detection biases the proton Chapter 8. Analysis III - Molecular formation 215 distribution towards the first detector (i.e., away from the proton detector). This effect was estimated using a Monte Carlo [212] taking into account the geometrical correlation of the two detectors, and found to give about 10% (relative) reduction in the second detector. Using the coincidence solid angle, £lco,-n = 2.1%, the a yield with the energy cut 2501 < E < 3700 ch, Ya = A x 10~ 3/GMU, a n d the time cut efficiency et = 0..71 for cut (a) (et = 1 for the cut (b)), we obtain the proton yield per dt fusion from cycling, V;yc as -* a 1 ' c o i n ^4 where Ycoin is the coincidence yield per GMU, and we ignored the small sticking proba-bility in dpt fusion. The resulting Vpyc from Target 11-13 are given in Table 8.11. Time cut T — T Proton yield/dt fusion (xlO 2) Sil(p) • Si2(a) Si2(p) • Sil(a) [-0.1,6] ^s [0.1,6] ps 2.6 (4) 0.9 (3) 2.3 (4) 0.9 (3) Table 8.11: Vpyc, the probability that the proton is produced after dt fusion is observed, is given for each coincidence pairs and the time cuts. The inconsistency between the results from the two time cuts can be due either to the time cut efficiency or possible presence of a prompt background (such as p decay electron). For example, if the dpd fusion disappearance rate is different in a thin layer from 34 ^s _ 1 measured in bulk solid targets, the time correction made is not appropriate. As well, with limited statistics, it is difficult to estimate the prompt background contribution. Given the uncertainty, we take the average of the two cuts, and quote an error covering two extremes of the error bars. Thus we have V^c = 1.6(9) x 10 - 2 as the contribution from cycling. In our analysis below, we assume the cycling proton yield per dt a to be independent of the D 2 layer thickness, which is sufficient for the accuracy required here. Chapter 8. Analysis III - Molecular formation 216 We note that at late time 1 < (Tp — Ta) < 6 ps, the coincidence yield is more than an order of magnitude smaller than at early times, hence the slow lifetime component of dpd (observed in bulk solid and other targets) appears nearly absent from cycled fusion in a thin layer. The value of Vpyc here should be compared with our earlier estimate with the kinetics model Vptn ~ 0.08 given by (Eq. 8.3), which is significantly higher. It is interesting to note, however, that our value is more or less consistent with the estimated fast component yield in the kinetics model (0.3 x Vpyc ~ 0.024). One possibility is that the muonic deuterium in the lower hyperfine state (F = | ) escapes from the layer before fusion takes place, since the low rate of non-resonant fusion (Az ~ 0.04 ps), which is responsible for the slow component, implies a rather long interaction length. Another possibility, of course, is that the muon escapes from the layer before stopping to form muonic deuterium [233]. Physics of the muon cycling and the pd transport in thin layers is a very interesting topic on its own, indeed. Direct stop contribution at early time The contributions from direct muon stops in the deuterium layer at early times are diffi-cult to estimate experimentally from our data set, since at such times, an overwhelming dpt signal is present in the spectrum. We therefore use the information from the cycling contribution and express the direct stop proton yield per G M U Ypd"fast as: •pcyc Yp-jast = SH-TDi^—ttsi, (8.5) Vcyc where SH = 0.299 is the muon stopping fraction in the emission layer, Tp2 = (14 T-/)/(1000 T-l) is the thickness fraction of the D 2 layer with respect to the emission layer, and r\cyc = Vpyc/Vptr is a phenomenological parameter which describes the ratio in the proton yields between dpd fusion after dt cycling and the direct stop. If we assume Chapter 8. Analysis III - Molecular formation 217 the spatial distribution of cycled muons is the same as that of direct stop muons, the Tjcyc parameter simply accounts for the muon escape from the layer before stopping to form fid. Even if we allow a value as low as rfcyc = 0.5 (i.e., up to 50% of muons escaping), the yield will be Ypdir ~ 1.8(1.0) x 10"6, which is only 0.46(26)% of the a yield. Thus the uncertainty in rfcyc is not serious for our purpose here. For thicknesses other than 14 T-l, we estimate the yield by scaling to the thickness ratio. Table 8.12 summarizes the corrections due to protons from dfid fusion. We shall apply the correction for the DS (estimated here for the time cut 1 < t < 6 /is) to the MOD measurements as well, ignoring some differences in the time cut. D 2 target Proton yield Ydfld per a (%) Cycling Direct Total US 20 T-l 14 T-l 6 T-l 3 T-l 1.64 (94) ditto ditto ditto 0.66 (37) 0.46 (26) 0.20 (11) 0.10 (6) 2.3 (10) 2.1 (10) 1.8 (9) 1.7 (9) DS 23 T-l 20 T-Z 6 T-Z 3 T-Z 1.64 (94) ditto ditto ditto 1.10 (21) 0.95 (18) 0.64 (6) 0.72 (6) 2.8 (10) 2.6 (10) 2.3 (9) 2.4 (9) Table 8.12: Summary of the correction due to protons from dfid fusion. US is with the time cut of 0.02 < t < 0.4 fis and DS with 1 < t < 6 fis. 8 .2 .5 T h e y i e l d r e s u l t s Table 8.13 summarizes our results for the yield measurements with the corrections discussed in this section and others. Those include the energy cut efficiency efut, dd proton contribution Ydtld, the time cut efficiency tTcut, and nitrogen contamination effects e .^ The dt fusion yield Y^t is normalized to stopped muons using the stopping fraction Chapter 8. Analysis III - Molecular formation 218 0.299(15) and the target thickness dependent Si solid angle (Table 6.5, p. 129). That is, where Ysi is the a yield in the Si detector per G M U (averaged over S i l and Si2), SH is the stopping fraction, flsi is Si solid angle per detector, and e; are the various corrections such as the cut efficiencies and N 2 contamination. The final uncertainties quoted in the table include quadratically added errors for these corrections. The results with different energy cuts are mostly consistent with each other9. In order to avoid the occasional presence of low energy background, bold-faced values with narrower energy cuts (when more than one is available) are chosen as our final results. Figure 8.14 plots our final yield results for the US, M O D , and T O F measurements. The comparison of the data taken with different conditions, e.g. with different ct, re-quires some caution. For a quantitative understanding, the comparison with Monte Carlo calculations is essential, and will be dealt with in the sections that follow. 9 In a very crude statistical estimate, 6 cases out of 16 data point have the deviation between two energy cuts which is larger than the uncorrelated error (i.e., without normalization errors). This is perfectly consistent with statistical fluctuation. Chapter 8. Analysis III - Molecular formation 219 US (DS) ID (ct) E cut (10 3 ch) Si yield ( 1 0 - 5 / G M U ) e E tcut (%%) tcut e N Yit (10~3/n stop) 3 TI (20) II-2 2.7 3.1 3.7 3.7 5.77(9) 5.53(8) 1.0 .980(3) 1.7(9) n /a .890 .966(34) 9.45(72) 9.24(70) 6 T-l (20) II-3 2.0 2.7 , 3.7 , 3.7 16.58(14) 16.28(48) 1.0 .975(5) 1.8(9) .886 .966(34) 27.3(20) 27.5(22) 14 T-l (20) II-4 1.0 2.0 ; 3.7 ; 3.7 42.73(24) 39.80(22) .970(4) .920(10) 2.1(10) .884 .966(34) 72.6(54) 71.3(54) 14 T-l (0) II-7 1.0 2.0 ; 3.7 ; 3.7 43.43(19) 40.98(17) .970(4) .920(10) 2.1(10) .884 .976(24) 73.8(55) 73.4(55) 14 T-l (0) 11-13 .2% 1.0 2.0 ; 3.7 ; 3.7 45.80(35) 43.00(20) .970(4) .920(10) 2.1(10) .884 1.0 75.2(50) 74.4(50) 20 T-Z (0) 11-16 .05% 1.0 2.0 ; 3.7 ; 3.7 44.90(16) 40.63(14) .914(16) .828(29) 2.3(10) .884 1.0 78.2(53) 78.2(58) US (T cut) ID E cut (IO 3 ch) Si yield ( 1 0 - 5 / G M U ) ^cut ydfid (%%) tcut e N Yit (10~3/H stop) 0 T Z (-3; 6) II-1 1 2 3.7 3.7 25.35(20) 24.04(17) .954(18) .903(35) 2.6(10) 1.0 .966(34) 36.6(28) 36.7(31) 3 T Z (•5; 6) II-2 1 2 3.7 3.7 15.38(88) 14.90(50) .954(18) .903(35) 2.6(10) 1.0 .966(34) 22.5(22) 23.0(21) 6 T-l (.6; 6) II-3 1 2 3.7 3.7 8.82(90) 8.88(52) .954(18) .903(35) 2.6(10) .988 .966(34) 13.1(22) 13.9(21) 14 T-Z (-8; 6) II-4 1 2 3.7 3.7 1.93(11) 1.84(7) .954(18) .903(35) 2.6(10) .998 .966(34) 2.85(27) 2.87(26) 14 T-Z (.8; 6) 11-11 1 2 3.7 3.7 2.15(12) 1.86(7) .954(18) .903(35) 2.6(10) .998 .976(24) 3.14(29) 2.87(25) DS (ct) ID E cut (IO 3 ch) Si yield ( 1 0 - 6 / G M U ) 'cut (£/<*) tcut e N Final Yie ld (lQ-4/fi stop) 3 T-Z (.1%) II-9 3.51(25)xl0- b (see Table 8.7) .969 .976(24) 5.04(50) 6 T-Z (.1%) 11-10 2.0; 3.7 2.7; 3.7 8.55(60) 8.75(51) 1.0 .975(4) 2.3(9) .962 .976(24) 12.4(12) 13.0(12) 20 T-Z (.1%) 11-11 1.0; 3.7 2.0; 3.7 20.7(11) 18.24(69) .954(18) .903(35) 2.6(10) .948 .976(24) 31.9(29) 29.7(26) 3 T-Z (.2%) 11-14 3.05(24)xl0~ e (see Table 8.8) .971 1.0 4.27(54) 23 T-Z (.2% ) 11-15 1.0;3.7 2.0; 3.7 23.16(73) 21.08(52) .936(29) .876(52) 2.8(10) .948 1.0 34.6(27) 33.7(31) Table 8.13: The fusion yields Ydt per stopped fi in the US (top), MOD (middle) and TOF (bottom) measurements, with corrections due to the energy and the time cuts efut, e^ut, the effects of dd proton YpdtJ,d, and of N 2 contamination e^. The Si solid angles from Table 6.5 and the stopping fraction of 0.299(15) are used. Bold-faced values are chosen as our final results. Chapter 8. Analysis III - Molecular formation 220 US • 11-2,3,4 o II-7 • ct=0.2% X ct=0.05% 50 100 U S thickness (/ig-cm~2) 150 1 1 1 " I MOD i i i DS 140 //gem" 2 i • 11-1,2,3,4 x 11-11 i i i 1 — B — 1 1 1 1 20 40 60 80 100 U S thickness (/xg-cm-8) 120 TOF US 96 /ig-cm - 2 + • ct=0.1% • ct=0.2% 50 100 150 200 D S thickness (/tg-cm -2) Figure 8.14: Fusion yield per stopped muon for the US (top), MOD (middle), and TOF (bottom) measurements. The error bars include the absolute normalization error. Chapter 8. Analysis III - Molecular formation 221 8.2.6 Time-of-flight spectra In this section, we shall discuss our time-of-flight spectrum (TOFS) measurements. Ta-ble 8.14 summarizes the runs for T O F time spectra measurements. In all of these runs, we had a 14 T-l D 2 layer US to slow down the pt beam emitted from the emission layer. Recall that the RT minimum is around 10-15 eV in lab pt energy, which is too high for the resonant molecular formation energies that we are interested in. While Series A had a 500 T-Z H 2 substrate under the D 2 reaction layer DS, in Series B the deuterium was deposited directly on the Au foil. The H 2 substrate was initially used with the intention of reducing a background (especially in the neutron detectors) from muon capture on Au, which comes from pt passing through the thin reaction layer and reaching the Au foil, hence having the time structure similar to that of a real signal. But as we saw in the yield measurements, the H 2 causes significant background due to muon decay, and as far as the background with the characteristic time-of-flight is concerned, it is dominated by the pt directly hitting the Si detectors. Furthermore, it was learned in the course of the analysis that the H 2 substrate creates some ambiguity in the interpretation of the data due to possible pt re-emission. TOF spect. ID US DS DS Energy cut BG (T-l) substrate (T-0 (MeV) Run Method Series A II-9 14 H 2 3 3.1; 3.7 6+7+8 2 (ct = 0.1%) 11-10 (500 T-l) 6 2.7; 3.7 6 standard 11-11 20 2.0; 3.7 6 standard Series B 11-14 14 A u foil 3 3.1; 3.7 7 3-nr (ct = 0.2%) 11-15 (50 pm) 23 2.0; 3.7 7 standard Table 8.14: Summary of the runs for the T O F time measurements. See Tables 8.7, 8.8 for the background subtraction methods. Much of the details about background and various other corrections were already Chapter 8. Analysis III - Molecular formation 222 Si 1 ii + Si2 2 4 6 8 2 4 6 8 Time (/us) Figure 8.15: The TOF run with 3 T-l DS for Series A (ct = 0.1%) in filled circles, together with background run BG6 in open circles, both with an energy cut of 3101 < E < 3700 ch (top). Background subtracted time-of-flight fusion spectrum (bottom). given in the previous sections. It should be recalled that the background includes events from muon decay electrons, muon capture on Si, delayed dt fusion from US, and dd fusion protons (if the energy window extends lower than 3 MeV). The complexity in the background processes makes it virtually impossible to predict the time structure, and makes it unreliable to use a simple analytical function, hence we rely on bin-by-bin subtraction, which at the cost of statistical precision allows us better control of the Chapter 8. Analysis III - Molecular formation 223 2 4 6 8 2 4 6 8 Time (lis) Figure 8.16: The T O F run with 3 T-l DS for Series B (ct = 0.2%) in filled circles, together with background run BG7 in open circles, both with an energy cut of 3101 < E < 3700 ch (top). Background subtracted time-of-flight fusion spectrum (bottom). systematics. We show some examples of the fusion time-of-flight spectra in Figs. 8.15 and 8.16. Figure 8.15 shows the T O F spectrum for Series A (ct — 0.1%) 3 T-l measurement, together with the background data from BG6, both with the energy cut of 3.1 < E < 3.7 (see Table 8.2 for the background run information), while Figure 8.16 is for the 3 T-l measurement in Series B (ct = 0.2%). In the top figures, the non-exponential peak in the Chapter 8. Analysis III - Molecular formation 224 fusion time spectra is noticeable at around 2-4 /is, which, together with the lack of such a peak in the runs without DS reaction layers, indicate the fusion is indeed taking placing in the DS layer after a traversing the drift space. The fact that for the thin DS layer measurement, the energy width of these delayed events was so narrow in Fig. 8.9-8.12 in the previous section, corroborates that fusion is occurring at the downstream layer in which the a particle suffers less energy loss. 8.3 Monte Carlo analysis In this section, we make use of the Monte Carlo simulations to understand our data and extract the physics. The details of our Monte Carlo code SMC has been already given in Section 5.1. The physics input into the simulations is mostly taken from the Nuclear Atlas [16,17] for scattering cross sections, and Faifman et al. for the resonant [70-72,133] and nonresonant [51] molecular formation rates, with exceptions that (a) the muon transfer rate in pp —> pt, Xpt = 5.86 x 109 s _ 1 , and ppp formation rate (nonresonant) A P A i p = 3.2 x 106 s _ 1 were taken from our measurements using the same data set [83], and (b) the low energy behavior of Faifman's resonant dpt formation rate was modified to reflect the recent measurement [22] and theory [146]; the rate in fitF=0 + D^^0 —* [(dfit)dee] at E^ < 15 meV was set to 130 / i s - 1 . This set of physics input parameters will be referred to as the nominal input to the simulations. Our experimental data will confront these physics cross sections and rates. 8.3.1 Fusion yield analysis Before we test our main results on time-of-flight fusion spectra in the next section, let us first discuss the fusion yield measurements, which give us some confidence in our model as well as an indication of any new effects. Let us start with the analysis of the US yield Chapter 8. Analysis III - Molecular formation 225 .07 3. .06 h .00 0 2 4 6 8 10 12 14 US thickness (T-l) Figure 8.17: Monte Carlo simulations for US fusion yield (lines) compared to the exper-imental yield (open squares). MC(1) shows the nominal predictions with the uncertain-ties due the layer thickness indicated with the dotted lines. In MC(3), a constant rate \ c d t = 5 ps is added to the nominal dpt formation rates, whereas MC(2) is with the original dpt formation from Ref. [70-72,133] without the low energy modification. measurements. Figure 8.17 shows a comparison of the upstream fusion yields with the different Monte Carlo calculations. Only data from the same series of runs (ID=II-2, II-3, and II-4) are plotted to avoid a possible influence of the target conditions. Denoted MC(1) in the figure is our nominal physics input discussed above, with the possible variations in the yield due the layer thickness uncertainties, which are shown with dotted lines. Separate Monte Carlo runs with varied thickness inputs were performed for the latter. MC(2) is using US yield Chapter 8. Analysis III - Molecular formation 226 the original dpt formation from Ref. [70-72,133] without the low energy modification, while for MC(3) an energy independent rate \ c d t = 5 ps (also independent of F) was added to the nominal dpt formation rates. Our results are in rough agreement with the nominal M C , but using the original model of Faifman worsens the agreements at larger layer thicknesses. Many Monte Carlo calculations were performed in order to find a better agreement, varying the input such as formation rates and scattering cross sections and the transfer rates, but no simple scaling of any of these parameters allowed perfect agreement with the data. A significant improvement is observed, however, when the constant rate \° d t is added. This is a phenomenological parameter motivated primarily by reproducing the experi-mental data, and its interpretation requires careful consideration, which we shall later come back to. The dependence of the fusion yield in thick layers on \ c d t suggests the importance of, and our sensitivity to, the low energy processes in the solid state of hy-drogen, but fortunately for our measurements of the epithermal molecular formation in thin layers, the low energy effects are expected to be rather small as we shall see. In our nominal model, \ d ^ . ° at low energy is already set to a considerably higher value than in the original Faifman's model, hence the main effect in the difference be-tween MC(2) and MC(3) comes from the increase in the triplet formation rate. Indeed, increasing the singlet rate alone to Xd^t° =4000 ps-1, while keeping the triplet rate at Faifman's value, does not change the fusion yield. M O D yield In the M O D yield analysis, before comparing with the M C yield, we used a simple attenuation model to analyze our data, which gave us crude yet useful insight into the physics involved. In the one dimensional approximation, the yield Y of particles after going through a Chapter 8. Analysis III - Molecular formation 227 medium of thickness d can be expressed as: y = A o e x p ( - ^ - ) , (8.7) \L>a,ttJ where A0 is the number of the original particle, d medium thickness, and the LaU, the attenuation interaction length. The attenuation interaction is defined here as a process which absorbs or deflects the particle. Fitting our data using Eq. 8.7 gave us a rather good fit with x2/dof = 1.67 (con-fidence level 19%, dof = 2), which is somewhat surprising, considering the degree of approximation introduced 1 0. Nonetheless, with some caution in its interpretation, we can extract a phenomenological parameter, the attenuation interaction length in pt + d collisions L^j. = 37.7 ± (1.2) j a ± (2.0)^,^ /"g - cm - 2 , where the first error is given by the fit, and the second error is due to the uncertainty in the layer thickness (the latter was obtained from different fits in which the thickness was varied). The error in the experi-mental yield used in the fit included only the relative uncertainties, since normalization errors such as muon stopping fraction and Si solid angle cancel in the thickness depen-dence. The value of IfJJt can be converted to the effective attenuation interaction cross section oeJJ = 4.4 x I O - 2 0 c m - 2 , using a = l/(nL) with n the number density. This can be compared to the pt + d total elastic scattering cross section [17] ~ 20-50 x I O - 2 0 c m - 2 at pt lab energies of 1-20 eV, near which pt emission is distributed. The difference in the two cross sections can be in part understood in that it takes more than one collision to attenuate the pt, especially given the scattering angular distribution, which we will shortly come to. We now turn to the comparisons with the SMC simulations. Encouraged by its success in fitting the experimental data, we apply the attenuation model also to characterize the 1 0 T h e detection of pt using the fusion signal is assumed to be constant in this model. This is not precisely correct due to the energy dependence of the molecular formation rates. In addition, the effect of muon decay is obviously not included in the model. The fraction of the pt reaching DS before muon decay changes if the energy of the transmitted pt changes. Chapter 8. Analysis III - Molecular formation 228 3. 0 20 40 60 80 100 120 US thickness (/xg-cm-2) Figure 8.18: Experimental data (EXP) and Monte Carlo results for the MOD yield measurements plotted on a log scale. MC(a) used the nominal input which had the ut + d scattering angular distribution from Ref. [17] which is forward peaked, while MC(b) used an isotropic distribution dcrfJ-t+d(6)/d(cos 6) = const. The solid line is a fit of the MC(a) to an expression A0 exp(—x/LeJJ) with the dashed lines indicating variations in the the slope due to the fit uncertainty and to thickness uncertainty, while the dash-dotted line is a fit to MC(b), indicating the importance of the p-wave scattering in the pit + d interaction. Both fits are normalized to the experimental point at thickness 0. MC results. Figure 8.18 illustrates our Monte Carlo analysis. Plotted with error bars are experimental data, while crosses indicate the yield from the MC. The lines are fits to the MC yield from which we extract the effective attenuation lengths as above. Shown in the solid line (MC(a)) is a fit to the MC results with the nominal input, with dotted lines indicating the variations in the slope due to the fit uncertainty and the thickness errors, where we obtained - k ^ ( a ) = 34.9 ± (2.4)/t< ± (2A)th%ck /zg-cm~2. This is consistent at the 10% level with the experimental value extracted above. Indeed, the same conclusion can Chapter 8. Analysis III - Molecular formation 229 Lid (/^g'0 1 1 1 2) ^ c r i i + /d(cos0) comments This experiment 37.7 ± 1.2 ± 2.0 MC(a) nominal 34.9 ± 2.4 ± 2.1 from Ref. [17] p-wave dominant MC(b) isotropic 16.9 ± 2.3 ± 2.1 Constant s-wave assumed Table 8.15: Summary of the attenuation analysis using the M O D yield data be made from a direct comparison of the Monte Carlo and the data without the use of the intermediate approximation of the attenuation model. On the other hand, given in the dot-dashed line is a fit to the M C assuming an isotropic angular distribution in fit + d elastic scattering with the total cross section kept the same (MC(b)), where we obtained - £ ^ ( 6 ) = 16.9 ± ( 2 . 3 ) ± (2.1)^*: / ig-cm - 2 . The Monte Carlo with an isotropic angular distribution is in disagreement with our data. In Table 8.15 we present the summary of our attenuation analysis. We shall defer the discussion of the implications of these measurements to Chapter 9, but for now it suffices to state that a reasonable agreement of our L1SJ with the M C gives us some confidence in our model of fit moderation in a deuterium layer. 8.3.2 Time-of-flight analysis In this section, we shall extract two important physics parameters, the rate for the resonant molecular formation and its resonant energies. We show in Fig. 8.19 the results of our Monte Carlo calculations for a 3 T-l DS layer using the nominal input to illustrate the features of our T O F measurements. Plotted are simulated two dimensional scatter-plots of the fusion time and the energy at which dfit formation takes place. The direct events (top) refer to a process in which fit forms dfit directly, before being scattered by D 2 in the DS layer, while the total events (bottom) include also indirect processes where the fit first loses energy in collision with another D 2 molecule in the DS layer. The Chapter 8. Analysis III - Molecular formation 230 0.0 0.5 1.0 1.5 2.0 2.5 Formation energy (eV) Figure 8.19: The simulated correlations between the fusion time and the energy at which dpt molecular formation takes place for the TOF measurements using 3 T-l layer. Chapter 8. Analysis III - Molecular formation 231 0.0 0.5 1.0 1.5 2.0 2.5 Formation energy (eV) Figure 8.20: The simulated time vs. energy for total events in a thick (20 T-Z) layer, where the correlation is weaker than the thin layer. direct process shows a strong correlation between the fusion time and formation energy exhibiting individual resonance structure as shown on the projection on the time axis, whereas for the total events the correlation is obscured by the indirect processes. The role of the indirect processes is more prominent, hence the correlations further weakened, in thick DS layer measurements as shown Fig. 8.20, in which only the total events are plotted (the direct process is nearly independent of layer thickness in these examples). In addition, the thick layer measurements are sensitive to the low energy process as also seen in the US measurement, thus there are relatively large theoretical uncertainties due to solid state effects (notice that a considerable amount of fusion is taking place at very low energies in Fig. 8.20). This is why we focused our measurements and analysis on thin layers. It should be stressed, however, that despite the presence of the indirect process, a significant correlation is preserved between the fusion time and formation energy in the Chapter 8. Analysis III - Molecular formation 232 thin layer measurements. In extracting physical quantities, formation rate and resonance energy, our approach is to perform iterative fits to the data using Monte Carlo calculations with varied input parameters. We varied the formation rate X^t a n d the resonant energies E(dpt) by scaling —*• S\Xdnt (8-8) E(dfit) — • SEE(dfit), (8.9) where S\ is taken as energy-independent, and SE, rate-independent. The fusion proba-bility W was kept at Faifman's value during scaling. We made full use of our accurate absolute fusion yield determination. In each fit (i.e., fit to each Monte Carlo result for a particular physics scaling input), the x.2 was minimized by varying the overall normalization by the Monte Carlo spectrum by a factor af. But since our M C yields are already normalized, taking into account factors such as the number of incident muons, the muon stopping fraction, and the Si solid angle,' together with experimental corrections due to nitrogen contamination (if present), dpd proton contribution, the energy cut efficiency, and systematic effect in the background subtraction, all of which have been discussed in great detail in this thesis, we expect af = 1 in an ideal case. Thus constraining af to its uncertainty Saf (which is the relative error in our overall normalization), we define our x2 as: N , h e x p — afh^c \ 2 ( l — a f \ 2 X2 = fSv(^) 2 + («WC)V + \ ~ S a f ) ' ( 8 " 1 0 ) where he*v, hf40 are experimental and simulated counts in bin i, with Sh6^, 6hfIC be-ing the respective uncertainties. Thus the likelihood function is maximized, hence x2 minimized, when the shape and the yield match between experiment and Monte Carlo. Chapter 8. Analysis III - Molecular formation 233 1 2 3 4 5 6 /us Figure 8.21: Fit of the calculated M C spectrum (upper histogram) to the experimental data (error bars) for 3 T-l in Series A , giving x2/dof = 0.964 (confidence level 54.6%). The contribution from the direct process is also plotted (lower histogram). Both the data and M C results are normalized to G M U . Fit results We show in Fig. 8.21 a fit to the 3 T-l data from Series A (cf = 0.1%) with the nominal physics input, where we obtained x2/dof = 0.964 and confidence level 54.6%, with the normalization factor ctf = 1.001(31). The fit is very good, perhaps somewhat accidentally. But the situation is less perfect in the case for the 3 T-l in series B (ct = 0.2%) shown in Fig. 8.22, in which we obtained x2/dof = 1.47 and C L 1.67% with af = 0.78(4). The result of the formation rate scaling is presented in Fig. 8.23, where the total x2 Chapter 8. Analysis III - Molecular formation 234 1 2 3 4 5 6 fLLS Figure 8.22: Fit of the calculated M C spectrum to the experimental data for 3 T-Z in Series B , giving x2/dof = 1.47 (confidence level 1.67%). is plotted against the log of the formation scaling parameter S\. The log scaling of the horizontal axis was chosen to give relatively symmetrical shape of the resulting curve, compared to linear scaling. More physically motivated scalings (such as (1 — exp(—S\)) were tried, but none of them symmetrized the curves for both Series A and Series B si-multaneously. The points near the minimum (indicated by filled squares) were fitted with a quadratic function to obtain the best fit value and its error was estimated from finding the S\ in which x 2 is increased by 1 from the minimum. The dotted line in the figure indicates reduced X 2 (i.e., x2/dof) of one. The results for two separate measurements (Series A and B) are given in Table 8.16. They are in an apparent disagreement by two standard deviations, indicating the existence of an unanticipated systematic uncertainty, Chapter 8. Analysis III - Molecular formation 235 which we shall discuss below. The results for the resonant energy scaling are shown in Fig. 8.24. The horizontal axis is a linear scale here due to a reasonably symmetric distribution of the data points. A quadratic fit similar to above was performed to obtain the best fit, SE, and its error. Table 8.17 summarizes the energy scaling measurement. The results for two series of runs are in agreement within one standard deviation for the resonant energy measurements. Note that Series A had a higher sensitivity for both the rate and the energy measurements due to better statistics. We note that, for both formation rate scaling and resonance energy scaling measure-ments, there are some uncertainties in the determination of the best fit value and its error due to the non quadratic shapes of the x 2 curves. But these effects are rekitively small compared to other uncertainties involved in the measurements. Measurement Formation rate scaling Sx Series A Series B 0.88+^ 0 . 5 5 ^ 2 Table 8.16: The results of x 2 fit for dpt formation rate. Measurement Resonance energy scal-ing SE Series A 0.928 ± 0.040 Series B 0.994 ± 0.087 Table 8.17: The results of x 2 fit for dpt formation resonance energy. Chapter 8. Analysis III - Molecular formation 236 180 160 140 N-w 120 OS o 100 80 -60 -40 -10 1 • i / — 4 (A) c =0.1% , / / / / • \ ^ . . . . .XYd° f= l 10" 101 180 160 140 CM X 120 ce • J 100 o E -80 60 40 (B) ct=0.2% x7dof=l _i L 10 -1 10° 10" Figure 8.23: Total x2 versus dpt formation rate scaling factor S\, for Series A and Series B. Chapter 8. Analysis III - Molecular formation 237 200 i—{ 1 1 1 1 r - r — \ / \ (A) ct=0.1% / • \ / 150 I- ^ ' \ / N I \ / X \ , <S \ / "g 100 - x / H \ P \ / \ / 50 h w . X 2 /dof=l — i 1 i i i i i 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Resonant energy scaling, S 200 150 h 50 h i 1 1 1 r (B) ct=0.2 % J L X 2/dof=l 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Resonant energy scaling, S Figure 8.24: Total x2 versus the scaling factor of the resonant energy for dfit formation factor SE, for Series A and Series B. Chapter 8. Analysis III - Molecular formation 238 8.3.3 Monte Carlo uncertainties Many of the systematic effects in our Monte Carlo modelling were tested by running the M C calculations with the parameters in question varied in the input. In some cases we estimate the uncertainties without the Monte Carlo, relying on, for example, an analytical approximation. We treat these Monte Carlo errors separately from our measurement errors given in the previous section. In most of these tests, we ignore the difference in spectrum shape and consider only the changes in the fusion yield due to the input variations, which is usually a good approximation. Table 8.18: Estimated effects of various systematic uncertainties on the T O F fusion yield. Values denoted * were either ignored or included in other errors. M C error source Estimated AY/Y (%) Layer thickness 10.3 Muon beam size 2.2 Nonuniform fi stopping in layer 3.3 US-DS spacing 4.8 RT minimum energy 2.0 Low energy processes (formation, thermalization, backdecay energy) 9.0 M C error total 15.2 Exp fusion yield error source Estimated AY/Y (%) Muons stopping fraction 5.0 Si solid angle 4.2 Time zero shift < 0.5* Si double peaking < 1.0* N 2 contamination 2.5 (0) Proton from dfid 0.9* Energy cuts < 0.5* Background subtraction 6.4 (4.0) S i l (Si2) Exp error total 9.5 (7.7) S i l (Si2) statistical error 3.4 (6.6) Chapter 8. Analysis III - Molecular formation 239 Our estimate of the Monte Carlo uncertainties in the time-of-flight fusion yield are given in Table 8.18. By performing fits with intentionally varied Monte Carlo amplitudes, we found that 15.2% error in the fusion yield translates into a relative 8.6% uncertainty in the formation rate strength parameter S\. Also given in Table 8.18 are the statistical and the experimental uncertainties in the fusion yield which have been extensively discussed and already included in the fit. Regarding the resonance energy measurements, our major uncertainty is expected to come from the US-DS target foil spacing, and (US) layer thickness, whose effects in energy we estimate to be about 6% each 1 1, resulting in about an 8.5% error. Note that the energy measurement is not very sensitive to small variations in the fusion yield. 8.3.4 Formation rate and resonance energy results We combine the values of scaling parameters and uncertainties to give the final results on the formation rate and the resonance energy for dpt muonic molecules. For the formation rate measurements, the best S\ from Series A and B disagree by more than the error bars as given in Table 8.16. This discrepancy is curious. The difference in the run conditions, between the two series, in addition to the tritium concentration difference (i.e., ct = 0.1% vs. 0.2%), was the presence of a 500 T-l H 2 substrate under the thin D 2 layer for Series A . Comparisons of two runs for other observables such as the US fusion yields and the pt transfer time suggest that there is no problem in the pt production in the emission layer. There is also no evidence from our run record that there was a problem in target preparation for either run. One possible effect is that due to the presence of the thick H 2 substrate in Series 1 1 T h e former was estimated by doing similar \ 2 minimization with fits from varied M C geometry, but with much reduced number of fit points, while the latter was estimated by observing the shift in mean time-of-flight with one fixed SE • Chapter 8. Analysis III - Molecular formation 240 A , pt which passed through a thin D 2 layer may thermalize in H 2 , and may in turn be re-emitted back into the D 2 layer to form dpt and fuse. In fact this effect was already taken into account in our analysis; using our Monte Carlo we estimated that the re-emission would increase the fusion yield by a factor of 1.05 ± 0.05, where the uncertainty is estimated taking into account the lack of knowledge of formation rates as well as scattering cross section at very low energies. However, estimating these effects is difficult without proper solid state cross sections, and in light of our recent observation of pp emission from a pure H 2 layer [216], it may still be underestimated. Note, however, that due to a high ppt formation rate, low energy pt emission from H 2 should be somewhat suppressed compared to pp. Another possibility is the presence of some unaccounted errors in our values of Xpt and Xpiip (the pt transfer rate and ppp formation rate, respectively), which were measured with our solid targets [83]. There is no strong reason to doubt those values except perhaps that pp emission just mentioned was not considered at the time. Using the theoretical values from Refs. [17] and [51] would reduce the discrepancy significantly but not completely. Accepting the discrepancy as a measure of an unaccounted systematic uncertainty in our measurement, we shall increase our errors in accordance with Particle Data Group's procedure [229,230]. Thus our final result for the resonant molecular formation rate is: Sx = 0.747 ± (0.161)m e a s ± (0.086)m O { W , (8.11) where the first error is the combined average error increased by a factor of 2.1, and the second error is due to the M C model uncertainty discussed in the previous section. As for the resonance energy measurement, since the values from Series A and B agree within the error, we take the standard weighted average to obtain: SE = 0.940 ± (0.036)m e a s ± (0.085)model, (8.12) Chapter 8. Analysis III - Molecular formation 241 where similarly the first error is a combined measurement error and the second error is due to uncertainties in the M C modelling. Chapter 9 Discussion and Conclusion 9.1 Muonic tritium scattering 9.1.1 The Ramsauer-Townsend effect We have observed emission of pt from a hydrogen layer into vacuum via imaging of muon decay electrons. From the position and time of the muon decay, the information on the energy distribution of the emitted pt is obtained. Comparing the electron time spec-trum in the vacuum region with detailed Monte Carlo calculations, we determined the Ramsauer-Townsend energy minimum to be ER = 13.6 ± 1 . 0 eV, in agreement with the theoretical prediction of 12.4 eV by Chiccoli et al. using the multi-level calculations in the Adiabatic Representation of the three body Coulomb problem (the Nuclear Atlas [17]). Molecular and condensed matter effects are not included in our analysis, but their influ-ence on pt transport properties is expected to be negligible at these energies. Figure 9.1 illustrates the preferred variations (shaded band) of cross section from this measurement together with theoretical values (dashed line) [17]. Our results are consistent with the emission probability of about 15% per muon stopped in the production layer. In addition to the above M W P C measurements from Run Series I, the dpt fusion measurements from Run Series II which were performed in a quite different setup (see Table 4.1) can add some information. A preliminary result of the time spectrum of fusion at the DS thick layer with no US moderation layer is consistent with the RT scaling of 242 Chapter 9. Discussion and Conclusion 243 Lab energy (eV) Figure 9.1: The preferred variations in pt + p elastic scattering cross sections from this measurement (shown in a shaded band), together with the original theoretical cross sec-tions from Ref. [17] (dashed line). The dot-dashed line is the constant cross sections used in the comparison given in Fig. 9.2. The box is an expansion near the Ram-sauer-Townsend minimum plotted on a linear scale. 1.05 ±0.05 when all other nominal physics input, and the nominal US-DS target spacing, are assumed. Given the considerable difference in the setup and detection method, this gives us further confidence in our measurement of the RT minimum reported in this thesis. We note that preliminary results of recent measurements [234] performed by our collaboration, but using a different X-ray technique and independent analysis with a Chapter 9. Discussion and Conclusion 244 4 ^3 o X '2 r-c o C J 1 h 0 Figure 9.2: The time spectrum of fit decay in vacuum region (error bars) compared with a Monte Carlo calculation assuming no RT minimum in the pt+p cross section (histogram). An energy independent cross section of 5 x 1 0 _ 2 1 c m 2 was assumed in the simulation, which gives a similar pt yield in the vacuum region. The comparison clearly rules out the possibility of a constant cross section, establishing the existence of a minimum in pt + p cross section. separate M C , indicate a shift of the RT minimum (0.4 ± 0.05 eV, or relative 3% shift) to lower energy, opposite to the indication given here. It should be noted that not all uncertainties were included in the quoted value; for example, the error in the target spac-ing has been neglected so far. The difference between the X ray measurement and the M W P C measurement reported here probably gives a measure of unexpected systematic uncertainties. We stress, however, that for our goal of molecular formation rate mea-surements, the confirmation of the theoretical RT minimum energy at the 10% level is sufficient, in comparison with uncertainties in other processes1. Finally, we present in Fig. 9.2 a comparison of our M W P C emission data with a Monte 1 The energy scaling method in Ref. [234] is slightly different from ours described here, but this difference alone would not resolve the apparent discrepancy between the two results. Chapter 9. Discussion and Conclusion 245 Carlo calculation assuming no RT minimum. The simulation using an energy independent cross section of 5 x I O - 2 1 cm 2 gives a similar yield of pt emitted in vacuum, but its time-of-flight distribution is very different. Thus our measurements provide direct evidence for the existence of a deep minimum in the pt + p cross section. Note that in diffusion type measurements, which are mainly sensitive to the integrated diffusion length, it would be more difficult to rule out the possibility of an energy-independent cross section. 9.1.2 /it + d scattering From the measurements of the DS fusion yield with varying US overlayer thickness (MOD measurements), we have extracted the effective attenuation interaction length LeJJ, which agrees at the 10% level with the value given by the Monte Carlo using the pt + d elastic cross section from Ref. [17]: LeJJ{exp) = 37.7 ± (1.2) / i t ± (2.0) t W c f c ^g • cm" 2 (9.1) LeJJ{mc) = 34.9 ± ( 2 . 4 ) / 4 i ± ( 2 . 1 ) ^ pg • cm" 2 (9.2) where the first error is due to the fit and the second due to the uncertainty in the US layer thickness. The attenuation length is determined mainly by the pt + d elastic scattering process, thus the agreement between Eqs. 9.1 and 9.2 suggest confirmation of the pt + d scattering cross section of Chiccoli [17] at the 10% level. This is the first quantitative measurement of the pt + d cross section to our knowledge, as conventional cycling measurements in D / T mixtures are not directly sensitive to this process. We note that dependence of our results on the value of dpt formation rate is rather weak; a change of by a factor of 2 produces variations in LeJJ similar to or smaller than those due to the thickness uncertainty. Our data could alternatively be used to extract information on the scattering angular distribution. A M C calculation assuming isotropic scattering with the same total cross Chapter 9. Discussion and Conclusion 246 section as Ref. [17] resulted in the effective attenuation length: LafJ(mc,iso) = 1 6 . 9 ± ( 2 . 3 ) / r t ± ( 2 . 1 ) t W d b pg-cm~2, (9.3) which is in clear disagreement with our experiment. The pt + d elastic cross sections are in fact predicted to have large p-wave contributions, whose angular distribution is strongly peaked forward [17]. This is due to the existence of the loosely bound dpt molecular state with an angular momentum of J — 1, the very state responsible for the resonant molecular formation. The energy loss of a particle in the lab frame in an elastic collision is uniquely determined by the scattering angle 6. A scattering to large angles will reduce the energy substantially (depending on the masses involved), while the particles do not suffer much energy loss for forward angle scattering. Hence, the forward peaked angular distribution results in a smaller attenuation, compared to the isotropic one for the same total cross section. In order to reproduce our experimental attenuation length L\H using the isotropic angular distribution, the total cross section needs to be half of the predicted value, which seems rather unlikely (the experimental absolute fusion yield at US thickness zero is reasonably well reproduced by either model). Thus, our results given in Table 8.15 confirm the importance of the p-wave contribution in pt + d scattering. Furthermore, the suggestion of the large p-wave contribution in turn provides indirect yet intriguing experimental evidence for the existence of the J = 1 state near the pt + d threshold. Note that while for the dpd molecule there is some evidence for its J state from the fusion branching ratio of the two channels (it is predicted to be sensitive to J according to jR-matrix calculations [235]), there has been little direct experimental information of the J state of the loosely bound dpt. Finally, our analysis presented here suggests that the measurement by Strasser et al. [195], who assumed an isotropic angular distribution in the simulation of the pd Chapter 9. Discussion and Conclusion 247 deceleration in deuterium, may require re-interpretation. Since there is a significant p-wave contribution also in the pd + d interaction, neglecting its angular distribution is not justified, as demonstrated in our analysis. Therefore, their conclusion, which stressed the importance of molecular effects, should be taken with caution. 9.2 Condensed matter and subthreshold effects Despite the considerable success of our analysis in epithermal energy scattering above a few eV, low energy processes are complicated by the solid state effects on both pt slowing and dpt formation. Our fusion yield dependence on layer thickness is inconsistent with that predicted by the standard Faifman model, or even with our nominal model where the low energy formation rate for pt(F = {y)-\-D2wtho is set to 130-/US - 1 . An improved yet still not perfect agreement has been achieved only after inclusion of a small constant term \ c d t ~ 5 ps~l for dpt formation. The value of Xcdflt is sensitive to the detail of the pt deceleration process, hence it should be taken as a model-dependent phenomenological parameter at this stage. Regardless of its rate, however, there is an indication that a nonzero value for F = 1 formation rates plays an important role in our measurements. The nonresonant dpt formation rate is predicted to be very small at low temperature (< 0.5 ps-1), and does not appear to explain our observation regardless of the thermalization model. Recalling that there is some evidence for subthreshold resonances for F = 0, the same might be possible for F = 1, although for the latter the resonance energies Er are expected to be more negative (Er ~ — 50 meV). However, if one assumed the Breit-Wigner resonance profile as a zeroth approximation (despite the fact that this form is criticized for high densities [160]), the subthreshold formation rate Xsub falls off as Chapter 9. Discussion and Conclusion 248 \Er\~5/2 [141], and if one takes into account the experimental evidence for F = 0 that ^sub ~ 300 / ts - 1 with Er ~ —10 meV, it is not completely implausible to have a few /xs - 1 for F = 1 at low temperature. We note that the possibility of a nonzero F = 1 rate has not previously been ruled out experimentally either, since in the previous D / T mixture experiments at low tem-peratures, F = 1 rates were usually assumed to be zero in the fit (e.g. Ref. [36]). At any rate, small rates for F = 1 would be difficult to measure in the cycling experiments due to fast pt spin flip. Thus, our measurements in multilayers may offer a unique sensitivity to the resonance profile at large detuning energy, if theoretical uncertainties due to solid effects can be removed. To our knowledge, there are no realistic calculations of \sub for F = 1, and we urge theorists to extend their calculations to F = 1. Calculations for pt scattering processes and molecular formation in solid hydrogen are in progress [170]. We eagerly await these new results. Fortunately, the solid state effects did not overwhelm our measurements using very thin layers. 9.3 Resonant molecular formation Using the emitted beam of pt from a hydrogen layer, we have measured the formation rates and resonance energy of dpt muonic molecules. The combined results of the rate and energy scaling parameters S\, SE from two separate sets of runs are: SX = 0.747 ± (0.161) r o e o. ± (0.086) w e ; (9.4) SE = 0.940 ± ( 0 . 0 3 6 ) m e a s ± ( 0 . 0 8 5 ) m o ( f c , , (9.5) where the first errors are experimental uncertainties and the second ones are M C modeling uncertainties (including target geometry). When the errors are added in quadrature, we have achieved accuracies of about 25% and 10% respectively for the formation rates and Chapter 9. Discussion and Conclusion 249 energy. Our measurement of resonant molecular formation corresponds to a peak rate2 of (8.7 ± 2.1) x 109 s"1 for the reaction quantitatively confirming for the first time the existence of the strong epithermal reso-nance. Our measurement of the resonance energy scaling corresponds to the position for the strongest the resonance peak in the reaction (9.6) of 423 ± 41 meV in the pt lab frame. If we assume the molecular spectrum of the complex [(dpt)dee] is predicted reliably, our results can be considered a first direct measurement of the loosely bound state energy level. Our accuracy of ~ 25 meV in the center of mass frame, compared to the muonic atomic energy scale [2] of = (mAj/m e)2i?3 / = ~ 5630 eV, is better than 10 ppm. Indeed it is comparable to the vacuum polarization correction in the loosely bound state energy level. Until a few years ago, the problem of muonic molecule binding energies appeared to have been completely settled at least for the dpd case. Extraction of the binding en-ergy from the temperature dependence of dpd formation rates, as analyzed by Scrinzi et al. [12], showed remarkable agreement with the theoretical prediction. However, new studies, both theoretical and experimental, seem to indicate the real situation is not as clear. For example, recent calculations by Harston et al. on the dpd finite size effects indicates that previous values used in Ref. [12] were significantly underestimated. Sim-ilar effects are suggested for the dpt case by Bakalov et al. [137]. Experimentally, new and precise measurements of dpd fusion at PSI [236] are not in complete agreement with those used in Ref. [12]. In addition, the assumption of complete pd thermalization may not be valid at low temperatures. It should also be stressed that in the dpt case, because 2 Assuming the Doppler width given by Faifman. pt(F=l) + D°2 ,ortho (9.6) Chapter 9. Discussion and Conclusion 250 of the lack of experimental observation of the temperature dependence predicted by the standard Vesman theory, there was virtually no experimental information on the dpt energy level. Thus our measurements of the resonance energy may provide an interesting new opportunity to test the calculations of dpt binding energies. Let us make a few remarks on the theoretical assumptions in our analysis. • Inclusion of explicit back decay in the simulation is essential. The use of the renormalized effective rates without an explicit resonant scattering channel would lead to a significant overestimate in the calculated fusion yield. See appendix B for a detailed discussion. • In our Monte Carlo simulations, the dependence of the formation rates X^t and fusion probability W on the quantum states S,Kf are separately averaged, i.e., the averages for A ^ t and W over the quantum states are factorized3. This is not rigorously accurate if the their dependence on 5" or Kf is large. Although this is probably not a large effect, the precise estimate of the degree of accuracy requires comparisons with more complete calculations. • Since our D 2 target is not in thermal equilibrium rotationally (i.e., U>B(KJ) =4 u>K{Kf) ) the use of Eq. 2.40-2.42 assuming the Boltzmann distribution for Kf may introduce some error, though judging from the difference in the results be-tween using Eq. 2.40-2.42 and Eq. 2.37-2.39 at 300 K , shown in Figs. 10 and 11 of Ref. [133], the effects appear rather small. The Kj distribution could also be affected if the rotational relaxation is substantially smaller than predicted by Ostrovskii and Ustimov [153], as suggested in Ref. [154]. 3See discussion ofthe factorization approximation in the appendix B. Chapter 9. Discussion and Conclusion 251 • The use of a (state independent) effective fusion rate Ay appears valid at least for its Auger decay contribution, since it is presumably independent of i>f,Kj, or S. However the value used, A/ = 1.27 x 10 1 1 s _ 1 , could be overestimated by ~ 40%, if the lowest value amongst Armour's several predictions [152] turns out to be the correct one. This would not affect low energy formation where the fusion probability is very high, but could have a significant effect in epithermal formation, where fusion and back decay branches are comparable. On the other hand, possible collisional vibrational quenching would increase Xj. • Our method of scaling the formation rate, given in Eq. 8.8, while keeping the fusion probability W fixed, is phenomenological in nature. More physically motivated scaling would be to scale the matrix elements, which change both A^* and W. By the same token, scaling of the dpt binding energy as input into the formation rate calculations, and comparisons with the resulting resonance structures (assuming scaled binding energies) may be more justified than our simple scaling in Eq. 8.9 for the resonance energy measurements, although these comparisons would require considerable input from theorists. Thus, our scaling results should be taken as a first step toward a more complete analysis. • In our analysis, the resonance width is given by Faifman et al., which is determined by the Doppler broadening due to the D 2 motion assuming a 3 K gas. In a solid, however, the Doppler broadening is expected to be larger due in part to the larger zero-point motion of a D 2 bound in the lattice, hence the resonance profile may be broader. The precise evaluation of this effect would require more theoretical work which is in progress [170]. Finally, let us present in Fig. 9.3 a comparison of our data with the M C calculations assuming a constant rate for dpt formation, a value of which was chosen to reproduce Chapter 9. Discussion and Conclusion 252 20 15 CO I * 10 o u 5 h 0 -5 Figure 9.3: Comparison of the experimental data (error bars) from Series A with the M C (histogram) assuming a constant formation rate. The comparison rules out the possibility of an energy independent formation rate, establishing the existence of a resonant structure in epithermal dfit formation. the fusion yield in the US moderator layer. The indication is quite convincing; we have confirmed the existence of epithermal resonant dfit formation in fit collisions with D 2 -9.4 Concluding remarks 9.4.1 Improvements and Future directions One of the difficulties of our time-of-flight measurements was its low event rate. A l -though our accuracy for the molecular formation rate is not dominated by the statistical uncertainty, the small size of the fusion data sample made it difficult for us to investigate Chapter 9. Discussion and Conclusion 253 thoroughly the systematic effects such as background subtraction. More data would help us to understand these effects better, and reduce the uncertainties. At the present moment, theoretical uncertainties due to solid state effects give a large contribution to our total uncertainty. It is hoped that the situation will be improved in the near future, as theoretical efforts are underway. As discussed in Appendix B , the evaluation of the back-decayed pt energy would be necessary to significantly improve the accuracy of our measurements. For better determination of the RT energy and dpt resonance energy, more precise knowledge of the target spacing distance would be required. Measurement in-situ of the distance is not trivial in our setup, and appropriate methods should be investigated. Some minor modification of our gold target support (e.g., using a flat plate rather a than very thin foil for the DS) could help to better define this distance. As a future project for our collaboration, I have spent a considerable amount of effort in the past few years, together with Peter Kammel and Glen Marshall, in developing a new method for a direct measurement of the sticking probability using the multilayer solid targets [171,179,237]. Sticking, as discussed in Chapter 1, places the most stringent limit on the fusion yield per muon, hence, has attracted much attention in this field, but the discrepancy between experiment and theory persists to date. Most previous experiments are sensitive to the final sticking, which is a combination of initial sticking and stripping, hence cannot be readily compared with theory. Taking advantage of our multilayer target, we propose to (a) experimentally separate initial sticking and stripping, and (b) unambiguously determine sticking at high density where the pCF efficiency is highest, but the discrepancy is largest. The experiment is already approved by the T R J U M F Experimental Evaluation Committee, but our situation is a little unclear due to funding difficulties. The further discussion of this experiment is beyond scope of this thesis and interested readers are referred to Refs. [171,179,237]. Chapter 9. Discussion and Conclusion 254 9.4.2 Summary In this thesis, we have reported a new approach in pCF studies, viz the time-of-flight method using an atomic beam of muonic tritium. With this new technique we have made measurements on pt scattering as well as epithermal dpt resonant formation, which have been quantitatively studied for the first time. Various experimental challenges have been overcome in order to complete the exper-iments. Technical contributions of this thesis to the field of / /CF studies include: 1. Characterization of target layer thickness and uniformity to an accuracy of up to a few tens of nanometers, and the evaluation of effective average thickness using the muon beam profile obtained from the M W P C imaging. 2. New methods for determinig the stopping fraction, such as the absolute amplitude method via delayed electron coincidence, and the relative amplitude with electron energy cuts. 3. Considerations of resonant scattering in the pCY processes with detailed expressions for scattered pt energy. The physics results of this thesis can be summarized as follows. 1. We have observed an emission of muonic tritium in vacuum via imaging of muon decay electrons. From the position and the time of muon decay, information of the pt energy was obtained, enabling us to spectroscopically establish the existence of the Ramsauer-Townsend effect in pt + p interactions. The energy of the RT minimum was measured to be 13.6 ± 1.0 eV, in fair agreement with quantum, three body calculations by Chiccoli et al. [17]. 2. Using the pt beam, we have confirmed theoretical pt + d scattering cross sec-tions [17] to the 10% level by measuring the attenuation of pt through deuterium. Chapter 9. Discussion and Conclusion 255 Comparisons with Monte Carlo simulations, assuming different scattering angular distributions, also confirmed the importance of p-wave scattering in the fit + d in-teraction, giving angular momentum information on the loosely bound state of the dpt molecule. 3. The existence of the predicted large resonance in fit + D2 collisions was directly confirmed for the first time. Our results of the resonance strength correspond to a peak rate of 8.7 ± 2.1 x 109 s _ 1 when the resonance width given by Faifman is assumed. This is more than an order of magnitude larger than room temperature rates. Our measurement of the resonance position indicates a resonance energy of 0.42 ± 0.04 eV for the F = 1 peak in ortho deuterium. 4. Assuming the theoretical [(du-t)dee] energy spectrum, our results for the resonant energy imply sensitivity to the binding energy of the loosely bound J = l,v = 1 state of the dfit molecule, with an accuracy approaching the magnitude of the rel-ativistic and QED corrections, providing potential future opportunities to directly test quantum few body calculations. 5. Indications of solid state effects have been observed in the layer thickness depen-dence of the fusion yield, but more theoretical input is needed for better understand-ing. Efforts have begun by theorists to calculate fit interactions in solid hydrogen. The data obtained here will confront any future calculations. Appendix A Abbreviations and notation A . l Abbreviations List of imporant abbreviations and relevant pages Abbreviation Explanation Page A D C Analog-to-digital converter 91 A R Adiabatic Representation in three-body Coulomb 29 problems cl Confidence level in \ 2 fit various Del Delayed electron coincidence 102 Del-e Delayed electron coincidence with electron scintillator 102, 151 Del-t Delayed electron coincidence with electron telescope 102, 151 dof Degrees of freedom in x2 fit various DS Downstream target 60 DSY (Measurement of) fusion yield in the downstream target 190 EVS Experimental vacuum space 60 F W H M 5 Full width at a half maximum of the Gaussian part of 120 flat-top Gaussian distribution Ge Germanium X-ray detector 90 G M U Scaler for the number of incident muons identified as 94 "good" muons taking into account computer dead time and pile up IA Improved Adiabatic approach 32 L H D Liquid hydrogen density 20 M C Monte Carlo simulation or code various M M C Muonic Molecular Complex [(dfit)xee\ where x = p,d,t 260 M O D (Measurement of) the downstream fusion with variable 190 upstream moderation layer thickness M O D Y Yield for M O D 190 M W P C Multi-wire proportional chamber 89 NI NE213 neutron detector 1 90 256 Appendix A. Abbreviations and notation 257 List of imporant abbreviations and relevant pages Abbreviation Explanation Page N2 NE213 neutron detector 2 90 NE213 Liquid scintillating material 99 QAU Huff factor for muon capture on gold 131 RT Ramsauer-Townsend (effect) 7 Rflat Radius of the flat part of flat-top Gaussian distribution 120 SH Fraction of the muon stopping in the upstream hydrogen 130 layer uon stopping fraction obtained from absolute ampli- 150 tude method SHR Muon stopping fraction obtained from absolute ratio 131 method AR SH Reduced stopping fraction in the amplitude ratio method 132 SE Scaling parameter for the dpt formation resonance energy 232 S\ Scaling parameter for the dpt formation rate 232 Series A T O F run series with a tritium concentration ct = 0.1% 221 Series B T O F run series with a tritium concentration ct = 0.2% 221 511 Silicon charged particle detector located on the beam-left 90 (looking along the beam direction) 512 Silicon charged particle detector located on the beam 90 right SET Standard emission target 105 t 0 Time of the muon entrance defined by T I 94 T I Beam defining plastic scintillator muon counter 88 TBS Tritium barrier space 65 T D C Time-to-digital converter 91 Tell Electron telescope 1 102, 151 Tel2 Electron telescope 2 102, 151 T4 Target thickness measured in Torr x litre 67 T O F Time of flight various T O F Y (Measurement of) yield in the time-of-flight measurement 190 TOFS (Measurement of) the time-of-flight spectrum 190 WF Effective fusion probability 47 US Upstream target. The upstream target almost always 60 had 1000 T- layer with ct = 0, 0.1, or 0.2%, so sometimes US is just referred to the overlayer USY (Measurement of) yield in the upstream overlayer target 190 A J u t dpt molecular formation rate for pt hyperfine states F 47 Appendix A. Abbreviations and notation 258 List of imporant abbreviations and relevant pages Abbreviation Explanation Page Adnt Effective dfit formation rate, renormalized for the fusion 47 probability Ad[it Phenomenological parameter for an energy and F inde- 247 pendent dfit formation rate Si detector acceptance 127 <t> Density in units of liquid hydrogen atomic number den-sity N0 = 4.25 x 10 2 2 c m " 3 17 awc M W P C resolution function parameter in the x axis 170 M W P C resolution function parameter in the y axis 170 Represents deposition of an additional layer (overlayer) on top of a emission layer 105 Appendix A. Abbreviations and notation 259 A.2 Error notation Throughout this thesis, unless otherwise noted, the uncertainty for the given quantity is presented in parentheses immediately following the mean value of the quantity, with the order of the last digit of the uncertainty equal to the order of the last digit in the mean. That is, regardless of the position of the decimal point, mmmm(ee) = mmmm ± ee, where mmmm is the mean value and ee is the uncertainty for that quantity. We show a few numerical examples below: 23.4(7) = 23.4 ± 0 . 7 23.5(12) = 23.5 ± 1 . 2 0.510(8) = 0.510 ±0.008. Appendix B Resonant scattering of fit In this appendix, we discuss the influence of resonant scattering in the modelling of the muon catalyzed fusion processes. As we have seen elsewhere in this thesis, resonant formation of dpt molecular ions in a loosely bound state of J = v = 1 occurs via formation of a metastable muonic molecular complex [(dpt)uxee\ (denoted M M C hereafter) in a collision of pt on a molecule DX, pt + DX -»• [(dpt)llXee\*, ( B . l ) where X = H,D,T and x = p,d, t. Here we shall denote the total rate for the formation of the complex as Xtot. After the formation of the complex, two competing processes take place: stabilization of the complex leading to fusion (mainly via Auger de-excitation of the dpt molecule) with the effective fusion rate A/ , and back decay [151] to pt + DX with the rate Tback- The above rates generally depend on the quantum numbers, such as ro-vibrational, spin, and hyperfine states, of initial and final states, but in the first part of this appendix we consider the averaged rates for simplicity and drop indices for quantum numbers. This simplification does not affect our conclusion. In conventional analyses of pCF, the effective formation rate , a renormalized rate taking into account the back decay probability, has been widely used, and is defined (when ignoring the indices for the quantum numbers) as: \eff = Xtot . w ^ 260 Appendix B. Resonant scattering of pt 261 where W is the branching ratio for the fusion channel -V T 1 back With the recent recognition of the importance of transport properties of muonic atoms, theoretical cross sections for muonic atom scattering have been calculated to high accuracy with sophisticated methods. Despite the vast theoretical efforts in pushing the accuracy of various scattering cross sections, including electronic screening, atomic and molecular structure, and most recently solid state effects, little detailed attention has been paid to the back decay process as a scattering mechanism of muonic atoms, except in the context of spin flip in the dpd system [34]. In early work, resonant scattering processes were neglected or treated incorrectly. For example, in the pioneering studies of Markushin [9,27,75,82], who first performed the full three-dimensional Monte Carlo calculations of muonic processes, renormalized effective formation rates were used to account for the back decay processes. On the other hand, in the theoretical analysis of the pCF kinetics in D / T mixture [238], Somov claimed that pt emitted after back decay of the muonic molecular complex has a thermal distribution with a temperature of the surrounding medium. Jeitler et al. followed this in their Monte Carlo analysis of D / T and H / D / T mixtures [22, 74]. They claim that this is justified when one assumes complete thermalization (of the translational motion) and the rotational relaxation of the M M C [74]. The purpose of this appendix is to point out the importance of the resonant scattering process resulting via backdecay of the molecular complex, particularly in the atomic beam type of experiments, as performed in this thesis. We will show first that the use of renormalized effective rates as was done in Refs. [9,27, 75,82] significantly overestimates the calculated fusion yield, and second that thermalized pt emission after back decay is not justified even if one assumes the complete M M C thermalization. Appendix B. Resonant scattering of fit 262 B . l Effective formation model B . l . l Validity Obviously, in transport calculations, neglecting one of the scattering channels leads to inaccurate results, if the contribution of that channel is significant. Therefore the back decay process has to be looked at carefully. On the other hand, in the description of fusion yield, the effective formation approximation has been used in many analyses (see for reviews Refs. [1-4]). Let us investigate the validity of this approximation in the fusion yield description. We point out that in order for the effective formation approximation to be justified in terms of describing the fusion yield, at least one of the following criteria must be met: (a) trivial conditions that the back decay probability (1 — W) ~ 0, (b) a negligible change in fit energy before and after back decay in the laboratory frame1, (c) fast (compared to the formation rate) "re-thermalization" of back-decayed fit (fid) in a thermal equilibrium condition. For example, at low temperatures the condition (a) is satisfied for dfit formation, while the condition (c) applies for dfid formation at least at high densities (one may need to be careful about this at very low density). In case of dfit formation at epithermal energies, however, none of these conditions apply, hence the back decay process cannot be neglected. In the following section, we shall consider the implications of the resonant scattering. 1 T h i s may be possible, for example, in the case of molecular formation in a solid at energies low compared to its Debye temperature, where recoilless processes analogous to the Mossbauer effect may dominate. Appendix B. Resonant scattering of pt 263 B.1.2 Epithermal Collisions Let us consider an epithermal collision between pt and a D X molecule, with a pt lab energy E^t (velocity V^). Throughout this appendix, solid state effects such as lattice binding and phonon exchange are neglected unless otherwise specified. Resonant scattering may occur via the following sequence of processes: (1) the molec-ular complex (MMC) is formed in a collision pt + DX, (2) M M C receives a recoil from the impact of pt, (3) M M C may be (partly) thermalized via collisions with the rest of the target molecules, (4) ro-vibrational states of M M C may change as a result of colli-sions with other molecules, and (5) back decay occurs leaving molecule DX in either the ground state (elastic channel) or in an excited state (inelastic). The energy of the pt after back decay is important. It can be shown that if the pt energy is the same before and after the back decay, the effective formation approximation gives the correct fusion yield (condition (b) above). However, if the pt energy changes such that it is removed from the resonant region, the effective formation approximation fails as we shall see. It should be noted that resonant structure has a narrow width for low temperature targets, therefore a small change in the pt energy is sufficient to remove it from the resonance. Even in the completely elastic case (no M M C thermalization, no M M C relaxation, and back decay via the elastic channel), the mean energy of pt in the lab frame is significantly reduced after back decay, because of recoil of the M M C in process (2) and that of D X in (5). In reality, M M C thermalization, M M C ro-vibrational relaxation, and back decay via inelastic channels, all give contributions to reducing the pt energy, hence the pt is likely to be removed from the resonance region. Appendix B. Resonant scattering of pt 264 B.1.3 Comparison with the explicit back decay model We illustrate the effect of back decay in the following simplified comparison. We consider a case in which a pt beam of resonant energy collides with a D 2 layer of varying thickness, and we wish to describe the fusion yield in the D 2 layer as a function of its thickness. We compare the two models (a) using effective formation rates renormalized for the back decay fraction as was used in Ref. [9,27,75,82], and model (b) using the total formation rate and explicit back decay. We consider only two competing processes of elastic scattering and molecular formation, the latter leading to either fusion or back decay, and neglect other possibilities such as scattering followed by molecular formation (muon decay is also neglected). Wi th these assumptions, the fusion yield as a function of layer thickness in the effective formation model (a) can be expressed as: Ya(x) = ^ [1 - exp{-(cr e / / + ascat)x}], (B.4) aeff + a scat where x ( cm - 2 ) is the thickness multiplied by number density n, and cross sections Oi and rates a, are related with A = vna, v being velocity of the pt for a target molecule at rest. With back decay explicitly included (model b), the fusion yield as a function of layer thickness is now written: Yb(x) = —Gt°f^—[1 - exp{-(<7( 0t + crscat)a;}] (B.5) °~tot + °~ scat These expressions can be expanded in series in x: Ya ~ OreffX - i(<7 e / / + ascat)x2 H (B.6) Yb ~ atotWx - ^(o-tot + o-scat)x2 H (B.7) Recalling oejj = cr t o t W, both models give the same results to leading order. Therefore, in the limit of a thin layer, model (a) can be justified for describing the fusion yield. Appendix B. Resonant scattering of pt 265 However, the next-to-leading order corrections become important at x comparable to the mean interaction length, xint = (atot + a s c a i ) _ 1 . In the limit of a thick layer (x > xint), the fusion yield is simply the coefficient in Eqs. B.4 and B.5. Comparing Eq. B.6 and B.7, it is clear that Ya > Yb, unless W = 1 (B.8) resulting in an overestimate of fusion yield with model (a). In our earlier analysis of the time-of-flight fusion experiments, the effective rate (model a) was used, and we observed that the calculated fusion yield was significantly larger than the experimental data. Since our thickness was comparable to the interaction length x t n t , the discussion given here explains the discrepancy. Detailed Monte Carlo calculations indeed show that the fusion yield is overestimated by nearly 50% in our time-of-flight fusion measurement arrangement, compared to model (b). It is interesting to note that Eq. B.5 suggests that, within model (b) and given the knowledge of crscat, one can determine both A i o i and W by the absolute measurement of the thickness dependence of fusion yield Y(x), i.e., \tot from the curvature of Y(x), and W from Y(x) at the limit of large x (via the coefficient OtotW/(atot + Cscat) )• Note that it is difficult to disentangle \tot and W in conventional measurements using a homogeneous D / T mixture. B.2 Back-decayed fit energy distribution B.2.1 M M C recoil and thermalization In this section, we shall take a more detailed look at each step in the back decay in order to estimate the energy of pt after back decay. When the muonic molecular complex is formed in the collision, it will receive recoil Appendix B. Resonant scattering of pt 266 velocity Vc = m ^ M d x ' For t n e case of pt + D2 collisions at the main resonance of E^f - 0.5 eV, Vc ~ 0.24 cm-ps'1. The final velocity of M M C is important since it affects the lab energy of back-decayed pt. Two kinematic extremes are the complete thermalization of M M C before it decays, and no thermalization at all. For either case, the maximum possible energy of back-decayed pt (for a low temperature target) is ob-tained when "elastic scattering" (i.e. no excitation of the target D2 molecule) takes place. However, even in elastic scattering, pt is decelerated in the lab frame due to the recoil of M M C and D2 as discussed above. Cross sections for the interaction of [(dpt)xee] + D2 has been calculated by Padial et al. [154,239]. Extrapolating a figure given in Ref. [239] to epithermal energies, the elastic scattering cross section CTQ appears to have a value of order 3 x 1 0 - 1 5 c m - 2 . The average time between elastic collision At ~ ( u c ^ ^ c ) - 1 ' where n is the D 2 number density, can be compared to the M M C life time, re-in the case of Elff ~ 0.5 eV, which corresponds to the largest resonance for molecular formation in pt + D2, the M M C recoil velocity is Vc ~ 0.24 cm-^s - 1 , hence at a molecular density of 1.4 iV 0 /2 (N0 = 4.25 x 10 2 2 c m - 3 , the atomic density of liquid hydrogen) corresponding to a solid at 3 K , At ~ 5x 1 0 - 1 4 s. Comparing this to TC ~ 1 0 - 1 2 [152,156], we have At > r c , (B.9) suggesting a fair number of collisions occur before the M M C decays. The average number of collisions needed to slow the M M C of E^MC t ° ^ M M C I C A N be roughly estimated, by generalizing the formula for neutron thermalization [240]: if in n = m ^ M M £ (B.10) where ^ _ j, j (MMMC — Mpx)2 ^ MMMC — Mpx (B 11) IMMMCMDX MMMC + Mux Appendix B. Resonant scattering of fit 267 For El*t ~ 0.5 eV in fit + D2, the M M C recoil energy is E^MC ~ 0.2 eV, and we have n ~ 8 for EMnMC = 0.4 meV. Obviously, the formula B.10 gives simply a crude estimate, but together with Eq. B.9, one can assume that the M M C slows down significantly before back decay occurs. B.2.2 M M C ro-vibrational relaxation Relaxation in ro-vibrational states of M M C would change the Q-value available for the back decay reaction, affecting the fit energy. According to the estimate of Lane [151], vibrational de-excitation rates appear to be a few orders of magnitude smaller than the M M C decay rate (although the former increases with the target temperature), hence we ignore its contribution here. Future calculations of the M M C vibrational relaxation would be very helpful. Rotational transitions in the M M C have been calculated by Ostrovskii and Usti-mov [153], and by Padial et al. [154] for the case of targets in thermal equilibrium. Ostrovskii and Ustimov estimate relaxation rates of order of 10 1 3 s _ 1 , while Padial et al, who claim higher accuracy, give ~ 0.3 x 10 1 3 s _ 1 at 300 K . Given the M M C decay rate of ~ 10 1 2 s _ 1 , if rotational thermalization is achieved only partially, then detailed rate equations2 should be solved to calculate the energy distribution of back-decayed fit. In our calculation below, we give the two extreme cases of complete rotational thermaliza-tion and no thermalization at all. 2 Scrinzi et al. give an example of such rate equations in Ref. [12], though for different purposes. Appendix B. Resonant scattering of fit 268 B.2.3 Resonant excitation of D X molecule We should stress here the importance of resonant excitation of the D X molecule, which has not been well considered in the literature to our knowledge: fit + WuiKi -* [{dfit)udee}* -+fit + [D2}u[K[ (B.12) The process when vi', K[ > vi, K{ leaves the target D 2 molecule in an excited state, hence the back-decayed fit energy is correspondingly reduced3. B.2.4 Back-decayed fit energy As we have seen, the energy distribution of back-decayed fit depends on the details of back decay reactions as well as M M C interactions with the surrounding environment, which include: (a) the back decay matrix elements4 < vjKj\H\v[K'j >, (b) collisional ro-vibrational relaxation of the M M C , (c) the M M C elastic scattering cross section for [(dfit)dee]* + D2, and perhaps (d) solid state effects such as emission of phonons upon back decay and recoilless M M C decay. Apart from the plausible yet still speculative effect (d), the processes (a) and (b) affect the Q-value of the back decay reaction, while •(c) changes the motion of the M M C centre of mass frame, hence affecting the fit energy in the lab frame. The complete analysis of this complex chain of processes is beyond the scope of this appendix, and we deal with two limiting cases of (b), i.e., no rotational relaxation at all, and complete rotational relaxation (no vibrational relaxation assumed in either case). We develop in the following expressions for the double differential rates for resonant scattering which depend on both outgoing and incoming fit energies in the lab frame 3 Note thabresonant deexcitation (e.g., v', K'j < Vf,Kj) is possible for high temperature targets, but is negligible at low temperature. 4 I am using the language of the perturbation theory here with some reservation, recognizing the controversy in the formulation of the M M C formation and back decay processes. Appendix B. Resonant scattering of pt 269 XResScati.F'lltEllt). Our formulation is analogous to the resonant formation rate calcula-tions by Faifman et al. [133]. We explicitly treat the back-decayed pt energy E't, paying attention to the energy balance. Thermal motion of the M M C is included to give the Doppler broadening [157] of the E't distribution. In the first limit of no relaxation at all in the M M C states, the rate for the resonant scattering rate A ^ e s 5 c a i ( E ^ 4 ; E^) can be written: "i,Ki "f,Kf,S v\,K\ AS + / j 1 v,,K,v[,K\ J dEuD(Eu)I [Elt - EltKt(vjKfS; v[K[) - Eu] (B.13) where KKfrfKfS; = M ™ D * M i i t bres(vfKf) - AEVtKt,v,Ki\ (B.14) is the pt energy from the decay channel (uj, Kf, S) —> (ul, Kl, S) with AEVtKtt<K, = - [E(ulK't) - E(uiKt)} (B.15) being the binding energy difference between the initial and final state of DX (i.e., exci-tation or de-excitation energy of DX due to the resonant scattering). Other notations used in Eq. B.13 include: • (As usual) S, F denote the spin of dpt and hyperfine state of pt, respectively. • / [A] is the resonance intensity profile for detuning A (e.g., I [A] = 8(A) in Faifman's model). • D(EU) is the Doppler broadening distribution due to the motion of M M C at the time of back decay (e.g., Gaussian distribution for thermalized M M C if E't ^> kT). • K , K , is the back decay width for the channel (uj,Kf) —*• (ul,Kl) with given S,F. Appendix B. Resonant scattering of pt 270 • AfiKi,uf,Kf(^nt) is the total formation rate for the incident pt lab energy E^t. • u(vi,K{) is the initial Vi,Ki distribution of target molecules DX. In the other limit that M M C rotational relaxation is complete, the following substi-tution should occur: KnivjKfS-MKl) — KxKx{yfK)S-v\K[) (B.17) with UB{K'J) being the Boltzmann distribution for the rotational states of M M C , hence simplifying the summation in Eq. B.13. Because the expression given in Eq. B.13 is rather complicated, we can alternatively take advantage of the already calculated formation rates \ F t and fusion probability WF to write, in the limit of no M M C rotational relaxation, that w here AResScat(Kt; ^ 0 ~  A L ( E ^ ) ' (1 " WF) • f{E'- E,t) (B.18) VJKJS v'^K'l X y dEuD(Eu)I [E'^ - E'ViKi(ufKfS; v[K[) - Eu] (B.19) is the energy distribution of back-decayed pt for a given initial energy E^t, and ySF is the branching ratio for decay to the state (y[, K-, S), given the M M C state of (vj, Kf, S), and \SF i p \ h^KfS; E „ ) = ^ (B.21) Vf Kf S Appendix B. Resonant scattering of fit 271 is the conditional probability that the M M C has the state (ui,K{,S), given that the M M C is formed with a fit of energy E^. Again, in the other limit of complete M M C rotational relaxation, we can average Kf states over the Boltzmann distribution, and let E ' 5 > « ^ - E C w ; ^ ) = E VjKSS vfS uFS 2.^ Ki,KiVS,K'}{Ent) UJK'JS together with the substitution in Eq. B.17. Note in Eq. B.18, we made an approximation by factorizing the state dependence, and this is not rigorously accurate when the state dependence of each factor is large, hence the approach given here should be taken as a first approximation. On the other hand, the first expression given in Eq. B.13 does not rely on the factorization approximation hence it is more accurate, though more complicated. In order to numerically evaluate these expressions, we need some several hundred matrix elements for M M C transitions. It should be stressed, however, these have been already calculated, for example by Faifman et al. for calculations of A ^ t , thus with their assistance we can readily estimate the energy distribution of back-decayed pt (within the approximation that the M M C is translationally thermalized, and is rotationally relaxed or not relaxed at all). We note that in case of the elastic scattering, [v'^K'f) = (vi,Ki) with translation-ally thermalized M M C , the energy of back-decayed E^ is given by the resonance en-ergy eres divided between pt and DX (convoluted with the Doppler broadening profile). Hence the claims by Somov and Jeitler [22,74,238] of thermalized pt after back decay is not physically justified. Jeitler also considered the limit of no M M C rotational relax-ation at all, but the simple expression given for the pt energy distribution for that case (£Y T = £ ^ ( _ M M A J £ Z _ ) 2 ? Eq. 4.72 in Ref. [74]) is not accurate, as the back decay into DX[y[, K[ / v, Ki) (the resonant excitation of DX), and the Doppler broadening due to Appendix B. Resonant scattering of pt 272 M M C thermal motion are neglected. B.3 Implications for our measurements Finally, we discuss the implication of resonant scattering in our atomic beam experiments. Regarding the validity of the effective model, we used detailed Monte Carlo simula-tions to compare the fusion yields in our standard time-of-flight target arrangement5 for both model (a), the effective model with the renormalized rates, and model (b), with explicit resonant scattering. Assuming that resonant scattering removes fit from the res-onance region, which is well justified from the discussion above, we observed that model (a) overestimates the fusion yield in the DS layer by nearly 50%. The use of model (b) thus resolves the inconsistency of our data with our earlier analysis using model (a), as reported in Ref. [78]. As for our sensitivity to the scattered fit energy, we have performed Monte Carlo calculations with different assumptions of the back-decayed fit energies. As mentioned, the exact evaluation of our expressions given in this appendix for the back-decajred fit energy distribution cannot be performed yet, since it requires the transition matrix el-ements which are not currently available to us. Our Monte Carlo calculations in the time-of-flight arrangement, with E't varied between 1 meV to 0.3 eV, suggest some 7% difference in the DS fusion yield. It should be noted that these variations depend also on the assumptions for fit interactions in a solid at low energies. Nonetheless, our present lack of precise knowledge of the back-decayed fit energy gives, a significant, yet not overwhelming contribution to the total uncertainty in our measurements of the resonant molecular formation rate, hence significant improvement in the accuracy of the latter would require, among others, the exact numerical evaluation of our E't expression. 5 Emission target with ct = 0.1% with a 14 T Z D 2 moderation layer in the US, and 3 T-l D 2 in the DS. Appendix B. Resonant scattering of pt 273 In summary, we have pointed out in this appendix the importance of the resonant scattering via back decay, a process which has not previously been well-considered, and (a) showed that the use of renormalized effective formation rates leads to an overes-timate of fusion yield and (b) gave detailed expressions for the.energy distribution of back-decayed pt, which is considerably different from the previously assumed thermal distribution. From the full Monte Carlo calculations, the effect of (a) is estimated to be ~50% and (b) < 10%, the latter depending on the possible solid state effects in pt thermalization and molecular formation. Appendix C Muon decay electron time spectrum fit Fits of muon decay electron time spectra to the exponential functions were performed taking into account the finite size of the time bins. For the muon stopping in M different elements, it obviously holds in the continuous approximation that ? = (ci) k where N£(t) are the numbers of muons in the element k at time t, in which the muon M disappearance rate is A*, and N^(t) = E - ^ f c ( 0 - By solving Eq. C l for 7VM, we have a k number of fi decay per unit time M dN^t) = - E A ^ ° e x p ( - A f c t ) , (C.2) dt k M where N® is the number of muons at t = 0 in the element k with ./V0 = Nk. Note k that Eq. C.2 can be decomposed into a set of M uncoupled independent equations. On the other hand, the decay electron detection rate, which we measure in our ex-periments, is dNe(t) B dN»(t) -dT = ~tkBk^r M = ^^khKexpi-Xkt), (C.3) 274 Appendix C. Muon decay electron time spectrum fit 275 where ek is the detector efficiency for decay electrons from the element k, and Bk is the branching ratio of the muon-to-electron decay to total muon disappearance B k = Qk\0Q+\c+\r  { C A ) where A 0 is the muon decay rate in free space, and, for the element k, Qk is the Huff factor, Xk the nuclear capture rate, and Aj^ is an effective rate representing other loss mechanisms such as muon transfer to heavier elements and muonic atom emission1. The Huff factor (Q < 1) takes into account the effect of muon binding to the nucleus, resulting in reduced phase space available as well as the time dilation of the muon's proper time with respect to the lab frame, both of which in turn lead to the reduction in muon decay rate, according to Huff [226]. Thus we have k M M = E ^ A o A ^ e x p t - A , * ) . (C.5) k We used Xk — Qk^o + Af + Af\ Note that because of muon loss channels it is Qk^Oi rather than A^, which is in the normalization factor in the final line of Eq. C.5. The actual experimental time spectra are histogrammed in a finite size of time bin At, hence are discrete function of ti (i = 1,2,3 . . . ) . To reflect this, Eq. C.5 is integrated over At, 5 C 0 = / Evaluating Eq. C.6, we have AN°(ti)  I ^rP-dt. (C.6) 2 At d t M n \ 1 Here we are approximating these loss processes as an exponential function with a simple "effective" rate. The real situation could be more complex. Appendix C. Muon decay electron time spectrum fit 276 exp — Afc ti -At exp - A * ( + -At (C.7) This can be rearranged as x exp [—Xkti] yexp which can be expanded in A^Af to 0(XkAt), -XhAt — exp ^ A •^XkAt (C.S) M ANe(U) QkXo 0 J2ek^N°kexV[-Xkti} x XkAt dNe{tt) dt At. (C.9) Eq. 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