ABSTRACT. (V A) and the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix is unitary. Measuring the

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1 ABSTRACT PATTIE JR, ROBERT WAYNE. A Precision Measurement of the Neutron β Asymmetry using Ultracold Neutrons. (Under the direction of Albert R. Young.) In the standard model of particle physics the weak force has a purely vector axial-vector structure (V A) and the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix is unitary. Measuring the decay parameters of the neutron, namely its lifetime τ n and any of the angular correlation parameters, provides a way to test these assumptions. Precise measurements of τ n and λ G A /G V, the ratio of the axial-vector to vector coupling constants, could be combined to determine V ud, the largest element of the first row of the CKM matrix, which can be used to test the unitarity of the quark mixing matrix. The neutron-spin electron-momentum correlation parameter, A, is to leading order determined by λ and has the highest statistical sensitivity to λ relative to the other correlation parameters. The previous and most precise measurement of A by Perkeo II is discrepant with the previous world average by 4σ, where all previous measurements were performed with cold neutron beams. The UCNA collaboration designed and constructed an ultra-cold (UCN) neutron production facility at the Los Alamos Neutron Science Center (LANSCE) to exploit the different systematics associated with UCN to provide a complementary high precision measurement of A o. In the following we present the experimental setup, analysis techniques, and results of the first UCN based measurement of the neutron β-asymmetry parameter and the continuation of that experiment to higher precision. An initial proof-of-principle measurement, completed in 27 and published in 29, achieved a 4.4% precision on A o. In 28/29 a set of measurements were carried out to study the scattering systematics which were published in 21 and resulted in a relative uncertainty of σ A = 1.3%. We also present limits on the tensor coupling coefficient C T /C A derived from β-decay data from neutron and + + systems.

2 Copyright 212 by Robert Wayne Pattie Jr All Rights Reserved

3 A Precision Measurement of the Neutron β Asymmetry using Ultracold Neutrons by Robert Wayne Pattie Jr A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Physics Raleigh, North Carolina 212 APPROVED BY: Paul Huffman Harald Ade Chueng R. Ji Albert R. Young Chair of Advisory Committee

4 DEDICATION To my parents and family, who supported me while I discovered which direction to go. ii

5 BIOGRAPHY Robert W. Pattie Jr. was born in the city of Manassas, VA on August 11, 1979 to Robert and Cheryl Pattie. He attended George C. Round Elementary School, Grace E. Metz Middle School, and Osbourn High School, graduating in June of During this time the author played several sports at the club and varsity level including: soccer, swimming, wrestling, basketball, football, and baseball. After high school he attended Northern Virginia Community College for three semesters before transferring to Surry Community College in Dobson, NC, so that he could continue playing baseball at the collegiate level. In the summer of 1999 the author enlisted in the United States Marine Corps Select Reserve 9-day reservist program and entered the 13 week boot camp at Paris Island, SC the following summer. The day after graduating from boot camp as a Private First Class (PFC) the author checked into Virginia Polytechnic Institute and State University (Virginia Tech) where he majored in physics. The next six years (2-26) were spent attached to B Co. 4th Combat Engineer Battalion, Roanoke, VA as a engineer equipment operator (MOS-1345). During that period, the author completed his degree in physics with a minor in mathematics at Virginia Tech in 23, while working in the ultracold neutron guide production lab of Bruce Vogelaar. Working in the guide fabrication lab exposed the author to the UCNA and introduced him to Albert Young, who would ultimately convince the author to enter the Ph.D. program at North Carolina State University in Raleigh, NC. Graduate school provided him the opportunity to work at labs in France, The Netherlands, and the United States on projects related to UCN transport and beta-decay studies using nuclei and neutrons. In 25 (January-October), the author was deployed in support of Regimental Combat Team-2 under Colonel Taylor for Operation Iraqi Freedom III and was stationed at Al Asad Airbase. The author was honorably discharged from the Marine Corps in the summer of 26 at the rank of sergeant. His graduate work was completed in the fall of 212. iii

6 ACKNOWLEDGEMENTS To begin with I have to thank my parents, Robert Sr. and Cheryl, for providing me with an open environment to explore all of my interests. Although it took a few years for me to find my motivation for academic pursuits, they unfailingly backed by wanderings. And by extension my sisters, Michele, Leanne, and Staci, who influenced me to go against expectations by introducing me to rock music at a very young age. Being the only fifth grader listening to Guns-n-Roses, Led Zeppelin, and The Grateful Dead builds a decent amount of self-confidence. I must also acknowledge my friends: Wes Dautrich, Jarrett Galeano, Chuck Whitesell, and Tim Zacherl, who taught me at a young age, in their own unique ways, that being yourself is far more important than fitting in. During the academic process I ve had two mentors, Bruce Vogelaar and Albert Young, whose open discussions on physics and life were always helpful and interesting. There are a number of students, graduate and undergraduate, in and out of the physics department who have kept me sane during this process by helping with work in the lab or office or helping forget work with a pint at Mitch s. To all of them, thanks. And to the students who started this nine year journey with me, Adam Holley, Chris O Shaughnessy, and Grant Palmquist it s been fun. I m grateful to Albert and Leah Broussard for putting up with me working on their project to measure the 19 Ne lifetime in The Netherlands, which allowed me to experience living in different country for several months a years while working on a completely different project. And finally thanks to the many post docs, professors, students, and Los Alamos staff members in the UCNA collaboration who have made this thesis possible by your hard work. Thank you for working on the experiment and thank you for the countless conversations we have shared over the years. iv

7 ocrmarg4em TABLE OF CONTENTS List of Tables ix List of Figures xi Chapter 1 Introduction Neutron Decay in the Standard Model The Neutron The Weak Interaction and the Standard Model The Cabibbo-Kobayashi-Maskawa Matrix Observables in Neutron β-decay Most Recent Measurements Perkeo I PNPI Yerozolimskii ILL TPC Perkeo II Current Status of V ud Improvements Using Ultracold Neutrons Chapter 2 Experimental Overview Introduction The Ultra-Cold Neutron Source Ortho-Para Deuterium Scattering He 3 Neutron Detectors UCN Transport General UCN Transport From the Source to the PPM The AFP Magnet and Spin Flipping Round to Square Transition The Superconducting Solenoidal Spectrometer Decay Region and Trap β Detectors Cosmic Ray Veto Detectors Data Acquisition Trigger Generation Run Control Chapter 3 General Systematics of an Asymmetry Measurement Introduction Neutron β Decay Physics Generator Derivation Verification of Decay Observables v

8 3.2.3 Electron Energy Spectrum Proton Energy Spectrum Angular Correlations Energy Calibration Calibration to the Kellog Electron Gun Calibration and Linearity Differences Finite Detector Resolution Uncertainty in Calibration due to Finite Line Sources Summary of the Bias due to Energy Calibration Scattering Systematics A PENELOPE3 simulation of the UCNA Geometry Backscattering Edge Effects Angular Acceptance Applying Monte Carlo Scattering Corrections Optimization of the Energy Analysis Window Detector Response Functions Off Axis Radial Field Expansion Non-uniform Field Effects Backgrounds Time Dependent Backgrounds Neutron Generated Backgrounds Experimental Determination of Neutron Generated Backgrounds Instrumental Asymmetry Due to Loading Efficiency Neutron Polarization Chapter 4 Analysis of the 27 Asymmetry Data Introduction Data collection Analysis Approach Definition of Software Cuts Timing and Beam Cuts Muon Veto Cuts Wire Chamber Cuts Summary of Software Veto Cuts Energy Calibration Method Application to the in situ 113 Sn Calibration Data Post Run Calibrations with Sn, Bi, and Sr Backscattering Energy Reconstruction E vis E recon Parameterization of E(E) Background Subtraction Ambient Backgrounds Residual Backgrounds vi

9 4.9 Polarization Measurements The Raw Super Ratio Asymmetry Systematic Corrections and Uncertainties Energy Dependence of the Asymmetry Asymmetry from the Reconstructed Energy Forward Application of the Response Matrix A Coefficient from the Forward Method Conclusions and Results of the 27 Run Cycle Chapter 5 Analysis of the Multiple Geometry Asymmetry Data Introduction Data collection Data Taking Cycle Clock Blinding Data Collection Problems Different Geometries Software Cut Definitions Timing Cuts Detector Calibration Energy Reconstruction Detector Efficiency and Triggering Functions Background Subtraction Backscattering Analysis Detector Alignment Measurable Backscatter Event Fractions Event Type Energy Spectra Scattering and Acceptance Corrections Other Systematics and Corrections Polarization Radial Dependence Determination of A Comparison of the Extracted Asymmetry Asymmetry Grouping Results Summary of Systematic Corrections and Uncertainties Final Results Chapter 6 Analysis of Tensor and Scaler Couplings Introduction Derivation of Impact of the Fierz Term on the Neutron Decay Rate Comparison of the Measured and Predicted Decay Rates Limit on C T /C A Derived from a Direct Calculation b Dependence of λ Application to the Difference of Rates Method Application to the Direct Calculation vii

10 6.4 Experimental Weighting Conclusion Chapter 7 Conclusions and Future Work Conclusions Future work References Appendices Appendix A Derivation of Off-Axis Magnetic Field Appendix B Statistical Tests of Asymmetry Data Groupings B.1 Generating the Rates B.2 Calculating the Asymmetry B.2.1 The Simple (Sum Over Difference) Asymmetry B.2.2 Pulse Pair Super-Ratios B.2.3 Higher Order Super-Ratios B.3 Results Appendix C Additional Measured Distributions from the 28/29 Dataset C Energy Distributions viii

11 LIST OF TABLES Table 1.1 Possible interaction terms in the weak interaction Table 1.2 Summary of previous measurements of the β-asymmetry A Table 2.1 Summary of foil geometries Table 2.2 List of electronics used in the data acquisition system. NIM electronics used by the UCN monitors and muon vetoes are omitted Table 3.1 Energy and relative intensities for conversion and Auger electrons from the calibration sources used in UCNA Table 3.2 Bias due to energy calibration and detector response function differences are summarized for several reasonable choices of analysis energy windows Table 3.3 Simulation scattering parameters Table 3.4 Summary of a γ background simulation of the UCNA spectrometer Table 3.5 Summary of event Type fractions versus foil thickness Table 3.6 Summary of the scattering corrections 2 and 3 for various foil geometries Table 3.7 Neutron capture γ s from 63 Cu and 65 Cu Table 3.8 Event triggers generated by Cu(n,γ) interactions in a simulation Table 3.9 Residual rates after No MWPC cut - MWPC cut subtraction method for each geometry for each detector and spin state in the analysis energy window 2 6 kev Table 4.1 The number and percent of events surviving after data reduction are summarized. Event totals listed are prior to background subtraction Table 4.2 Definitions of l 1 and l 2 in Eq. 4.6 for each quadrant, where R is the radius of the scintillator Table 4.3 PE production from east and west detectors during the 113 Sn calibration runs from 26 and Table 4.4 Backscattering fraction for each detector, (E-East, W-West), and spin state compared with PENELOPE3 predictions Table 4.5 Parameters for the linear reconstruction method for the detectable event classes. 152 Table 4.6 Analysis of the excess count rate passed the endpoint Table 4.7 Fractional systematic corrections and uncertainties to A are listed and grouped by source. Only backscattering and acceptance effects have corrections, all other effects contribute only the systematic uncertainty Table 4.8 Corrections and results for each analysis choice Table 5.1 Thickness of the foils used in the runs are summarized below Table 5.2 Reconstruction parameters for geometries A-D Table 5.3 The average decay rates, after veto cuts, background subtraction, a r < 45 mm position cut and kev energy cut Table 5.4 Rotation matrix parameters and resulting asymmetries Table 5.5 Measured backscatter fractions for 28/ ix

12 Table 5.6 Summary of Type 2 and Type 3 event fractions Table 5.7 Backscattering and acceptance corrections for each geometry and relevant analysis choices Table 5.8 Table of final uncertainties Table 6.1 Weighting and phase space factors from superallowed decays Table 6.2 List of the terms in the error with ascribed error and input values Table 6.3 Solutions to C T /C A Table 6.4 Experimental results for the determination of λ from measurements of A with the energy windows and sensitivity to a Fierz parameter Table B.1 Results of the rate generator Table B.2 Results of extracted asymmetries x

13 LIST OF FIGURES Figure 1.1 The Feynman diagram for neutron β-decay Figure 1.2 Ft values for super allowed + + decays Figure 1.3 PDG ideogram of A Figure 1.4 Current Status of V ud Figure 2.1 Schematic of area b and the UCNA experiment Figure 2.2 Cross sectional view of the LANSCE UCN source Figure 2.3 UCN helium-three detector Figure 2.4 Maximum integrated UCN density as a function of Fermi potential of the storage vessel Figure 2.5 Cartoon schematic of the switcher box Figure 2.6 Schematic of the soleniodal super conducting spectrometer (SCS) Figure 2.7 Source insertion systems that have been used between 27-present Figure 2.8 UCNA β-detector schematic Figure 2.9 The configuration of the MWPC wire chamber planes Figure 2.1 Schematic of the cosmic veto array Figure 2.11 Muon veto tubes Figure 2.12 An engineering drawing of the backing veto assembly Figure 2.13 Schematic of signal routing and trigger generation in the UCNA DAQ Figure 2.14 TDC common stop diagram Figure 3.1 The difference between the non-relativistic expression and the expansion of Eq Figure 3.2 Variation of the electron energy spectrum due to a nonzero Fierz term Figure 3.3 Proton energy spectrum Figure 3.4 Monte Carlo generated asymmetries Figure 3.5 The photo-electron calibration as measured using the electron gun at Cal Tech. 59 Figure 3.6 The fractional difference in the energy spectra of detector 1 and 2 and the deviation from the ideal asymmetry Figure 3.7 Bias in the asymmetry due to gain mismatch Figure 3.8 Example response functions for several energies Figure 3.9 Bias to the asymmetry is shown for several scenarios of differences in detector resolution and efficiency Figure 3.1 The deviation of the of calibrated energy due to nonlinearity Figure 3.11 Effect of spectral distortions on the asymmetry Figure 3.12 A 2-D slice view of the detector package Figure 3.13 The probability of detection for an event with emission angle θ Figure 3.14 Schematic of backscattering trajectories Figure 3.15 Type I timing distributions Type 2/3 events separation probability based on trigger side MWPC energy deposition Figure 3.17 MWPC energy spectra for Type 2/3 events Figure 3.16 xi

14 Figure 3.18 The angular and energy dependence of the event classes for a typical UCNA geometry Figure 3.19 Backscatter corrections for the geometries relevant to the analysis of the datasets Figure 3.2 Radial dependence of the simulated asymmetry Figure 3.21 Average value of cosθ for simulated events in geometry C vs reconstructed energy. 93 Figure corrections for each geometry vs. energy Figure 3.23 The energy dependence of systematic and statistical uncertainties are shown in the panels on the left and on the right the statistical and combined uncertainties are presented for a range E low E hi Figure 3.24 The Yacout semi-empirical response function model is shown in terms of its constituent parts Figure 3.25 Fits to a simulated response function are shown for energy up to 8 kev in 1 kev intervals Figure 3.26 Fit parameters from the simulated response functions are shown as a function of incident energy Figure 3.27 Results of response function analysis compared to simulation Figure 3.28 The response functions for 113 Sn and 27 Bi for the east and west detector packages are fit to the semi-empirical model Figure Sn and 27 Bi simulated and measured response functions Figure 3.3 Bias on the asymmetry due to scaling the response function fit parameters Figure 3.31 An analytical model for the on axis value for B z is fit to the original AMI field map provided with the spectrometer Figure 3.32 Field map for Figure 3.33 Time varying backgrounds Figure 3.34 The bias in the asymmetry is shown as a function of the background scale parameter ρ Figure 3.35 Bias on the asymmetry due to several scenarios of n-generated backgrounds Figure 3.36 (n,γ) lines from naturally abundant copper isotopes Figure 3.37 Angular and energy distributions for Compton generated electrons Figure 3.38 Beta-block experiment schematic Figure 3.39 Geometry A. Residual background from the MWPC - No MWPC cut Figure 3.4 Bias on the asymmetry due to differences in the spin flipper on and off polarization.127 Figure 4.1 Beam related triggers Figure 4.2 Backing veto ADC and TDC spectra Figure 4.3 Muon tube TAC spectra and east scintillator QADC spectra Figure 4.4 Coincidence rates in the muon veto arrays for the east and west detectors array. 134 Figure 4.5 Position of background events in the energy analysis window Figure 4.6 MWPC multiplicities are shown for the x and y cathode planes in both detectors. 136 Figure 4.7 Definitions of the path length and opening angle Figure 4.8 A comparison of the toy model for light attenuation in the scintillator and the measured response in run Figure 4.9 Schematic of the gain monitoring system xii

15 Figure 4.1 Typical GMS spectrum from the east NaI detector Figure 4.11 Visible energy summed over spin-flipper states for each detector Figure 4.12 Source position and ADC spectra for run Figure 4.13 Source peak ADC locations vs. x position in the spectrometer Figure 4.14 Calibration sources versus position after corrections Figure 4.15 Peak location vs. expected E vis for all calibration positions Figure 4.16 The east versus west TDC correlation plot Figure 4.17 Comparison between the linear reconstruction and the true energy spectrum for each event type Figure 4.18 Reconstructed energy spectra with backgrounds and Monte Carlo predicted spectra.153 Figure 4.19 The residual rate passed the endpoint for the east and west detector averaged over spin states Figure 4.2 Raw integral asymmetry for each pulse-pair Figure 4.21 Averaged signal and background spectra are compared to expectation from Monte Carlo Figure 5.1 TDC Shifts due to electronics problems Figure 5.2 Missing TDC header issues Figure 5.3 The positions r w (x,y) for Type and Type 1 events are shown in panels a and b, respectively Figure 5.4 Event type specific energy spectra are compared to PENELOPE Figure 5.5 Event type anode energy spectra for geometry D Figure 5.6 Simulated MWPC spectra for Type 2/3 separation Figure 5.7 The probability of being a Type 2 events versus trigger side MWPC energy deposition is shown from a simulation of geometry A Figure 5.8 The fraction of deposited energy in the primary detector, E prim /E tot, for the Type 1 backscatter events for each geometry and detector is compared to simulation. 194 Figure 5.9 Visible energy vs timing for Type 1 backscattering for each geometry is compared to Monte Carlo Figure 5.1 The ratio of SCS monitor rate to the total measured β-decay rate is shown for each run in the dataset Figure 5.11 The radial dependence of the asymmetry is show for 8 kev Figure 5.12 Octet asymmetries from the parallel analyses of the blinded data are presented for each geometry Figure 5.13 Raw asymmetries are shown for the octet, quartet, and super-ratio run-groupings Figure 5.14 for each geometry Raw asymmetries are shown without correction for each geometry and analysis choice compared the Monte Carlo predictions Figure 5.15 Corrections, 2 and 3, are applied to A raw for each geometry, sequentially Figure 5.16 The bin-by-bin asymmetry averaged over all geometries Figure 6.1 The one σ error bands are plot for C S /C V and and C T /C A Figure 6.2 λ vs. C T /C A Figure 6.3 The impact of a b-term on λ xiii

16 Figure 6.4 The one σ error bands are plot for C S /C V and and C T /C A Figure B.1 Shown above is R/σ for the thrown rates and the average rate in each detector for each spin state. These histograms are fit to a Gaussian to ensure that the initial rates are normally distributed. For all cases the σ of the fit is Figure B.2 Residuals of the Bone-head and super-ratio analysis normalized by 1σ Figure B.3 Residuals of the sum and product quartet analysis normalized by 1σ Figure B.4 Residuals of the sum and product octet analysis normalized by 1σ Figure B.5 χ 2 contours for the Bone-head, pulse-pair, quartet, and octet analysis method after subtracting out the minimum value. Overlaying the curves for the analysis methods shows no difference ( Bone-head to others is 1 7 ) in the 68% confidence internval Figure B.6 Simulated data was grouped in sets of 1 and the χ 2 for each set is plotted (black) and shown versus the theoretical χ 2 distribution (red) of 99 degrees of freedom Figure C.1 Figure C.2 Figure C.3 Figure C.4 Event type specific energy spectra are compared to PENELOPE3. Data from the west detector during geom A with the spin-flipper off is displayed as the black circles, with the simulation shown as the red dashed line. Monte Carlo spectra are normalized to the integrated rate of the rate of the specific event type Event type specific energy spectra are compared to PENELOPE3. Data from the west detector during geom B with the spin-flipper off is displayed as the black circles, with the simulation shown as the red dashed line. Monte Carlo spectra are normalized to the integrated rate of the rate of the specific event type Event type specific energy spectra are compared to PENELOPE3. Data from the west detector during geom C with the spin-flipper off is displayed as the black circles, with the simulation shown as the red dashed line. Monte Carlo spectra are normalized to the integrated rate of the rate of the specific event type Event type specific energy spectra are compared to PENELOPE3. Data from the west detector during geom D with the spin-flipper off is displayed as the black circles, with the simulation shown as the red dashed line. Monte Carlo spectra are normalized to the integrated rate of the rate of the specific event type xiv

17 Chapter 1 Introduction 1.1 Neutron Decay in the Standard Model The Neutron From its discovery by Chadwick in 1932 [1], studies of the fundamental properties of the neutron and tests of the discrete symmetries of nature have pushed the understanding the electro-weak force to more precise and deeper levels. The neutron transitions to a proton with the emission of an electron and anti-neutrino, as shown at the quark level in Figure 1.1, the continuous energy spectrum of β-decay gave the first evidence for the existence of the neutrino [2]. Parity violation (P), discovered by Wu [3] in the 6 Co system, can be tested to high precision in the absence of large nuclear structure corrections using free neutrons via measurements of the angular correlation in β-decay [4 13]. The existence of time reversal (T) violating interactions has been tested by measuring the triple correlation parameter, D, using the neutron [14 16]. Charge conjugation and parity reversal (CP) violation was first suggested by Gell-Mann [17], measured in the Kaon system [18], and is one of the Sakharov conditions for the matter antimatter asymmetry in the universe [19], is predicted by the standard model (SM) and many of its extensions and can generate a extremely small, < 1 31 e-cm, non-zero electric dipole moment (EDM) in the neutron. Although the SM prediction for the magnitude of the EDM is below current experimental sensitivity by several orders of magnitude, precision experiments such as measures of the EDM have been 1

18 p = udu e W n = udd ν e Figure 1.1: The Feynman Diagram for the neutron β -decay, using the convention that the arrow indicates the flow of particle number and not momentum. The transition from down to up quark is mediated by a virtual W vector-boson which decays into an electron and anti-neutrino. able to constrain many proposed extensions to the SM. The decay rate of the neutron plays a key role in the nucleo-synthesis in the early universe, setting the proton-neutron ratio and determining the 4 He abundance as well (current uncertainty in the lifetime determines the uncertainty in the 4 He abundance), has been measured numerous times with several large changes in the accepted value as experimental technology evolves. Currently there is a 4-6σ discrepancy in the value measured by the most recent and most precise measurements using ultracold neutrons compared to previous results based on cold neutron beam experiments. Additionally, the neutron can be used to test baryon number non-conversation, B, through n n transitions. The current limit from free neutrons experiments is τ n n > s [2]. Even though this is an incomplete list of the experimental exploits of the neutron, which does not mention any of its applications to the solid-state and material/surface science, one can see that such a simple nuclear system has a broad range of applications in nuclear and particle physics. At present, a full review of the status of symmetry tests of the SM using the neutron, as well as nuclear systems, is available by Severijns and Naviliat-Cuncic [21] The Weak Interaction and the Standard Model In the modern theory of forces, the transfer of momentum between particles occurs via the exchange of a virtual boson: a photon for the electromagnetic force, a W ± or Z vector boson for the weak force, a gluon for the strong force, and a graviton for gravity. Early formulations of β-decay by Fermi described the it as a point vector-type interaction analogous to photon emission of a charge particle [22]. 2

19 This description could easily be generalized to accommodate the other Lorentz invariant operators, O j = 1,γ µ,σ µλ,γ µ5,γ 5, where j = S,V,T,A,P, which allowed for both J =, scalar (S) and vector (V), interactions and J =,±1, axial-vector (A) and tensor (T), exchanges; the pseudo-scalar (P) vanishes in the non-relativistic limit and is therefore neglected in this treatment. Fermi s theory of β-decay was largely successful when applied to low energy phenomena; however, since the predicted scattering crosssection were proportional to E 2, it became clear that the calculations above 3 GeV were divergent in a non- renormalizable way. This issue can be addressed by including a massive vector-boson propagator D w = g µν q µ q ν /m 2 W q 2 m 2 W. (1.1) However, this results in the scattering-matrix elements growing logarithmically with respect to the energy and still diverging. In the modern description, the weak interaction is unified with the electromagnetic interaction and naturally produces charged and neutral weak currents, W ± and Z bosons. It is renormalizable through application of local gauge transformations, spontaneous symmetry breaking, and dimensional regularization, which is briefly covered in a number of texts [23 25]. Derivation of β-decay Amplitude Neutron β-decay, n p + e + ν e shown in Figure 1.1, is a semi-leptonic process that at the quark level is the change of a down quark into an up quark resulting in the emission of an electron and an electron anti-neutrino via a W exchange boson. To calculate the relevant features of β-decay it is common to work under the impulse approximation, which treats all nucleons in the nucleus as free and independent, eliminating complications from many body effects. The matrix element describing neutron beta-decay is T f i = g2 8 V ud Ψ u γ µ (1 γ 5 )Ψ d gµν k µ k ν /mw 2 k 2 mw 2 Ψ e γ ν (1 γ 5 )Ψ ν, (1.2) where Ψ u,d,e,ν are the particle wave functions, V ud is the first element of the CKM matrix which will be discussed in section 1.1.3, γ µ and γ 5 are Dirac matrices, g µν is the Minkowski metric, m W is the mass of the W exchange vector boson, k is the momentum transfer, and g is the weak coupling constant. From 3

20 Table 1.1: Possible interaction terms in the weak interaction. Coupling Constant Definition Class Vector g V 1 st Weak Magnetism f M = µ p µ n 1 st Induced Scalar f S 2 nd Axial-Vector g A 1 st Induced Tensor f T 2 nd Induced Pseudo-Scaler f P 1 st the Feynman rules for the weak interaction the factor of g 2 /8 is from the vertices and g µν k µ k ν /m 2 W k 2 m 2 W, is the propagator for the W. Momentum transfer in β-decay is normally a few MeV/c and the mass of the W is 9 GeV/c 2 ; so for k << m W the propagator reduces to g µν /m 2 W and g2 /8m 2 W = G F/ 2, where G F is the Fermi coupling constant and is precisely determined in muon decay with an uncertainty of 5 ppb [26]. T f i = G F 2 V ud Ψ u γ µ (1 γ 5 )Ψ d Ψ e γ µ (1 γ 5 )Ψ ν, (1.3) Eq. 1.3 can be written in terms of J h µ and J lµ, the hadronic and leptonic currents, respectively: T f i = G F 2 V ud J h µj lµ. We can write the currents as the difference of a vector term and an axial-vector term J h µ V µ A µ and J lµ v µ a µ V µ = i Ψ p [ gv (k 2 )γ µ + f M(k 2 ) 2m p σ µν k ν + i f S (k 2 )k µ ] Ψn, A µ = i Ψ p [ ga (k 2 )γ µ γ 5 + f T (k 2 ) 2m p σ µν k ν γ 5 + i f P (k 2 )k µ γ 5 ] Ψn, (1.4) where m p is the proton mass, m e is the electron mass, and k is the four-momentum transfer. To describe 4

21 the internal structure of the neutron the form factors listed in Table 1.1 are introduced. These form factors represent functions with arbitrary dependence on k 2 and whose value is taken from the k 2 limit, i.e. g V = g V (). In the SM second class currents do not contribute and the pseudo-scalar term is suppressed by proton mass and can also be neglected. Whether a form factor is first or second class is determined by its behavior under a G-parity transformation, G = Ce iπt 2, a charge conjugation operation, and an isospin rotation of π about the I 2 axis. Taking this into account the interaction Lagrangian to leading order : L int = G ( F 2 2 V ud Ψ P γ µ (1 + λγ 5 ) + µ p µ n σ µν k ν 2m p ) Ψ n Ψ e γ µ (1 γ 5 )Ψ ν, (1.5) where λ g A /g V = G A /G V since g A and g V are related to G A and G V by G V = V ud g V (k 2 )G F, (1.6) G A = V ud g A (k 2 )G F. Inputs to Eq. 1.5 are the SM parameters G F, g A, g V, V ud, and the magnetic moments of the neutron and proton µ n and µ p. The conserved vector current hypothesis (CVC) asserts that the value of g V is unchanged by strong interactions and is unity. The strength of the axial vector current is affected by nuclear screening and its value changes in the nuclear medium. Efforts have been made to calculate these effects using lattice QCD, but the uncertainty of such calculations are 1-3% [21], leaving g A and V ud to be determined experimentally. An interesting aspect of Eq. 1.5 is the vector minus axial-vector structure of the interaction and its behavior under parity transformation. Vectors have an eigenvalue of -1 under parity operation, since parity is a simply a reflection of the vector about the origin. However axial-vectors remain unchanged by parity. A simple example of this is angular momentum, L = r p, where r is the position vector and p is the momentum vector, both vectors transform with -1 eigenvalue leaving L unchanged. The result is that the structure of the weak interaction maximally violates parity conservation since, it is built on equal parts that behave differently under such a transformation. 5

22 1.1.3 The Cabibbo-Kobayashi-Maskawa Matrix Weak couplings between quark generations are not one to one, instead they interact as an admixture of quark mass states via a 3x3 unitary rotation matrix, d s b = ˆV d s b (1.7) where the d, s, and b denote the mass eigenstates and the primed quarks are the weak eigenstates. The idea of mixing between the mass states was first purposed by Cabibbo to maintain the universality of the weak interaction. It brought predictions of the decay rates of Σ n + e + ν and Λ p + e + ν into agreement with data and resolve the discrepancy between the muon lifetime and 14 O ft value [27]. Kobayashi and Maskawa extended the original idea, incorporating all three quark generations. The Cabibbo-Kobayashi-Maskawa (CKM) matrix has many parameterization, although only three are commonly used, the Wolfenstein, KM, and standard. In the standard parameterization the CKM matrix is ˆV = c 12 c 13 s 12 c 13 s 13 e iδ s 12 c 23 c 12 s 23 s 13 e iδ c 12 c 23 s 12 s 23 s 13 e iδ s 23 c 13, (1.8) s 12 s 23 c 12 c 23 s 13 e iδ c 12 s 23 s 12 c 23 s 13 e iδ c 23 c 13 where c i j = cosθ i j, s i j = sinθ i j, and θ 12 = θ C 13 o, the Cabibbo angle. Thus the CKM matrix is defined by four free parameters, three angles and an arbitrary phase, and is normally written as V q1 q 2, where q 1 and q 2 are the quarks that are coupled by that element of the matrix ˆV = V ud V us V ub V cd V cs V cb V td V ts V tb =.97427(15).22534(65).351( ).2252(765).97344(16).412( 11 5 ).867( ).44(11 5 ) (21 46 ). (1.9) Values are taken from the July 212 PDG [26] and are based on a global fit of the Wolfenstein parameters to the available data assuming unitarity as a constraint on the three quark generation. The unitary condi- 6

23 tion, i V i j V ik = δ jk and j V i j V k j = δ ik, implies that our knowledge of the number of quark generations is complete. Determination of V ud Using Fermi Decays Tests of CKM unitarity involving the β-decay of nuclei and free neutrons focus on the first row, V ud Vud +V usvus +V ub Vub = 1, of the CKM matrix, which is dominated by the first element V ud as it is very close to unity. Numerous systems exist in which V ud can be measured, including super allowed + + Fermi decays, neutron β decay, and π decay. A full review of these methods is given by Severijns [28]. The most precise measurements of V ud come from the Ft values of super allowed + + nuclear beta decays and a complete summary of these measurements is given by Hardy and Towner [29 31]. Due to the pure Fermi nature of transitions, the only input to the decay rate is the vector coupling strength. Since G V is provided by muon decay, one can extract a value of V ud directly from the + + transition rate: Ft + + ft(1 + δ R)(1 + δ C ) = K 2V 2 ud G2 V (1 + V R ) (1.1) where f is the statistical rate function and the partial half-life t = t ( 1/2 1 + ε ) BR β +, (1.11) includes the branching ratio, BR, and the electron capture probability ε/β + as corrections to the half-life, t 1/2. Transition-dependent radiative corrections δ R and R and isospin symmetry breaking correction δ C have been calculated within the SM by many groups and show good agreement between the different calculation methods [29]. However, these calculations are among the leading systematic uncertainties (in fact, the uncertainty in V ud from + + transitions is dominated by the box-diagram radiative correction recently re-analyzed by Marciano and Sirlin [32]). Combining the F t values measured in eight systems, 14 O, 26 Al m, 34 Cl, 38 K m, 42 Sc, 46 V, 5 Mn, and 54 Co, V ud is determined to be V ud =.97425(22). Consistency of the F t values for pure Fermi transitions, shown in Fig. 1.2, suggests that the strength of 7

24 Ft values Ft(s) Z of Daughter Nucleus Figure 1.2: Ft values for super allowed + + decays. The red band shows the average extracted from all available data, F t = ±.79. the vector current is not perturbed by strong nuclear effects as predicted by CVC. Based on the agreement presented in Figure 1.2 across a broad range of atomic numbers, χ 2 /ν =.28 and since the Ft value is proportional to G 2 V, Towner and Hardy conclude that this verifies CVC to better than Determination of V ud from Neutron Decays Mixed Gamow-Teller / Fermi transitions in neutron beta-decay, for example, require an additional measurement to determine V ud, since there are both vector and axial vector couplings. The neutron lifetime is given as τ n = K/ln2 G 2 F V 2 ud (1 + V R )(1 + 3λ 2 ) f n (1 + δ R ), (1.12) where f n (1+δ R ) = (2) is the phase space factor, K/( hc) 6 = 2π 3 hln2/(m e c 2 ) 5 = (11) 1 1 GeV 4 s, and δ R and V R are again transition-dependent radiative corrections. From Eq. 6.4 it is clear that the lifetime depends on both V ud and g a, since g v is assumed by CVC to be unity. As we will show in Section mixed decays provide many observables, which are to leading order proportional to λ and may be used along with a measured of τ n to extract V ud. Because the theoretical and experimental systematic uncertainties differ greatly from those of + + measurements, these decays provide a valuable complementary determination of V ud. The current values from neutron decay 8

25 is V ud =.9746(19) [21]. Determination of V us and V ub V us is determined by decay rate measurements of the semi-leptonic decays of charged and neutral kaons which have been preformed by several experiments: for example BNL-E865, KTeV, NA48, KLOE, and ISTRA+. The results of these experiments are combined by the PDG to give V us f + () =.21673(46). (1.13) Calculations of f +, the semi-leptonic decay form factor at q 2 =, have been carried out using chiral perturbation and lattice QCD. The PDG and FlaviaNet have adopted the lattice value of f + =.9644 ±.49, resulting in V us =.2247(12). The reader is referred to reviews by Severijns [21] and Hardy [31] and the references found therein, for thorough discussion of these measurements and calculations. The final element in the unitarity sum, V ub, is determined from decays of B-mesons which suffer from large experimental and theoretical systematic uncertainties. Measurements of both inclusive, B X u l ν, and exclusive, B πl ν, decay channels can be used along with input from lattice or light front QCD to extract a combined value of V ub = (3.89 ±.44) 1 3 [33]. The magnitude of V ub makes it almost negligible in the unitarity sum Observables in Neutron β-decay In addition to lifetime, τ n, there are numerous spin-spin, spin-momentum, and momentum-momentum correlations that can be measured experimentally for determination of g A and V ud. Expressions for these observables were first worked out by Jackson, Wylde, and Treiman [34] for polarized and unpolarized nuclei and for measurements including the electron momentum and polarization and the neutrino momentum, which is determined by observation of the recoil nuclei. This discussion will be confined to the case of polarized neutrons, where the spin of the outgoing particles is not measured, however correlations between the electron spin, momentum, and the neutron spin have been measured precisely and 9

26 have come under recent theoretical interest as possible low energy tests of super symmetric corrections to the SM [35]. In the case of polarized neutrons the probability distribution function for the momenta of the emitted particles can be written as, dγ( p e, p ν ) = F(E e)p e E e (Q E e ) 2 de e dω e dω ν (2π) 5 1 { 2 ξ 1 + a p e p ν + b m e + ˆσ n E e E ν E e ( A p e + B p ν + D p )} e p ν, (1.14) E e E ν E e E ν where F(E e ) is the Fermi function which accounts for final state interactions, σ n is the neutron spin direction, E e and p e are the electron energy and momentum, E ν and p ν are the anti-neutrino energy, and a, b, A, B,and D are the correlation coefficients. The constant ξ is represented, in the most general sense, by a combination of all the coupling constants and the Fermi and Gamow-Teller matrix elements as, ξ = M F 2 ( C S 2 + C V 2 + C S 2 + C V 2 )+ M GT 2 ( C T 2 + C A 2 + C T 2 + C A 2 ), (1.15) which, reduces to ξ = (1 + 3λ 2 ), for neutron decay in the standard model to leading order. C i and C i (i = S,V,T,A,P) give the relative strength of the coupling in the lepton current. Note that, in the standard model, all fundamental coupling constants are zero except C V = C V and C A = C A. To leading order a, A, B, and D are given in terms of λ, the ratio of G A /G V, as a = 1 λ λ 2, (1.16a) [ ( 1 α 2 CS +C ) ( )] S b = ± 1 + 3λ 2 Re + 3λ 2 CT +C T Re, (1.16b) C V A = 2 λ 2 + Re(λ) λ 2, (1.16c) B = 2 λ 2 Re(λ) λ 2, (1.16d) D = 2 Im(λ) λ 2, (1.16e) C A 1

27 where in the SM λ is assumed to be real, time reverse symmetry is conserved, and D =. However, final state effects will produce a non-zero D term. If scalar (C S,C S ) and tensor (C T,C T ) contributions to the weak interaction are zero as predicted in the SM,then the Fierz interference term, b, is also zero (α is eq is the fine structure constant). Therefore a, A, and B are the only viable correlations to measure to determine λ. In both a and B, the electron and proton momentum must be measured simultaneously, making for a more difficult detection scheme, since the maximum proton energy is only 752 ev. The β asymmetry parameter, A, requires a high degree of polarization and precise determination of the electron momentum. In addition to a simpler detection scheme A has the greatest sensitivity to λ: δλ δa = 3.3, (1.17) δλ δa = 2.4, δλ δb = 13.6, making it the ideal candidate for such a measurement. Integrating Eq over the neutrino variables and assuming the SM values for b = and D = the decay rate is dγ(e e,θ e ) de e dcosθ S(E)(1 + Aβ e P cosθ), (1.18) where all the energy dependent terms are collected in S(E), P is the neutron polarization, and θ is the angle between the electron momentum and the neutron spin. By setting up a detection system designed to collect the electrons emitted to the hemispheres defined by cosθ/ cosθ = ±1, one can define a count rate asymmetry of A = 2 β e N 1 N 2 N 1 + N 2, (1.19) where N i is the count rate in either hemisphere. A measurement of this fashion would find a value of A which is independent of energy. However, a sufficiently high precision measurement of the β -asymmetry would be sensitive to higher order terms due to recoil order corrections of the form A(E) = A o C(E) 11

28 where we define C(E) = 1 + a o (λ, f 2, g 2 ) + a 1 (λ, f 2, g 2 )E + a 1 (λ, f 2, g 2, f 3 )/E. (1.2) The form factors f 1 () g v, f 2 f 2 ()/ f 1 () (weak magnetism), f 3 f 3 ()/ f 1 () (induced scalar), and g 2 g 2 ()/ f 1 () (induced tensor) are defined using the conventions of Holstein [25], and Gardner and Zhang [36], and ( ) ε a 1 (λ, f 2, g 2, f 3 ) = Rx(1 + 3λ 2 2λ(1 + λ + 2 f 2 ) + 2(λ g 2 f 3 ), (1.21a) ) ( ) R a o (λ, f 2, g 2 ) = 3λ(1 λ)(1 + 3λ 2 (1 + λ + 2( f 2 + g 2 ))(3λ 2 + 2λ 1), (1.21b) ) ( Rx a 1 (λ, f 2, g 2 ) = 3λ(1 λ)(1 + 3λ 2 (1 + λ + 2 f 2 )(1 5λ 9λ 2 3λ 3 ) (1.21c) ) + 4 ) 3 g 2(1 + λ + 3λ 2 + 3λ 3 ), where ε = (m e /m n ) 2, x = E/Q, and R = Q/m n. In the standard model f 3 = g 2 = and f 2 = (κ p κ n )/2 = , which can identified as the weak-magnetism form factor. Eq. 1.2 is normally included in a fit of asymmetry data with all of the input parameters fixed or integrated over the energy range to provide a bulk correction to the extracted asymmetry, amounting to a 1.5% increase of the measured asymmetry from A. 1.2 Most Recent Measurements Asymmetry experiments with free neutrons have been performed with cold neutrons since 1975 [41]. Systematics common to all previous asymmetry measurements are polarization and neutron generated backgrounds, both of which are, in some respect, related to the cold beam nature of the experiment. In beam experiments, the flux of neutrons through the decay region is 1 1 s 1 and only 1 5 % of this flux decaying in the acceptance region of the experiment, whereas the remainder of the flux has the potential to capture on materials and make an irreducible background. Over the years the ability to 12

29 Prior to UCNA 21 Prior to UCNA 21 With UCNA 21 With UCNA 21 UCNA Perkeo II Ill TPC Yerozolimski Perkeo Ill TPC Yerozolimski Perkeo UCNA Perkeo II A (a) PDG 21 values A (b) Including the 212 Perkeo II result. Figure 1.3: If the central value of a measurement is set as the mean of a Gaussian whose width is equal to the combined uncertainty of the measurement, σ, and height set as 1/σ 2, then summing the Gaussians from several experimental results will generate the relative probability distribution for the measured parameters. This practice is commonly used to by the Particle Data Group to display the combined results from a number of experiments [26]. Panel (a) shows such a curve for the current values used by the PDG to determine the world average of the asymmetry parameter, A, before (red) and after (black) the 21 UCNA publication and panel (b) includes the most recent Perkeo II results. 13

30 Table 1.2: Summary of previous measurements of the β-asymmetry A. Measurements not included in the particle data group s world average are noted. In cases where multiple measurements were made with the same apparatus, such as Perkeo II and UCNA, the reference citing the most precise value is listed. The 212 Perkeo II result is too recent to be included in the PDG world average. Mostovoi is not included in the PDG average of the β-asymmetry, however their value of λ is included. Year Group Neutron Source A Error λ 1986 Perkeo I [4] Cold n Beam.1146(19) 1.7% (5) 1997 PNPI-97 [11] Reactor n.1135(14) 1.2% (38) 1997 ILL TPC [9] Cold n Beam.116(15) 1.3% 1.266(4) 21 Mostovoi* [37] Inferred.1168(17) 1.5% (46)(7) 22 Perkeo II [8] Cold n Beam.1189(7).6% 1.274(3) 21 UCNA [13] UCN.1197( ) 1.3% (41 45 ) 26 PDG [33] Average.1176(11).94% 1.271(25) Not Included in PDG Average 212 Perkeo II [38] Cold n Beam.11951(5).42% (13) 29 Pattie [12] UCN.1138(45) 4.5% 1.26(12) 1995 Schreckenbach [39] Cold n Beam.116(9)(11) 1.% 1.266(4) 1991 Erozolimskii [1] Reactor n.1116(14) 1.3% (36) 1979 Erozolimskii [4] Reactor n.114(5) 4.8% 1.261(12) 1975 Krohn [41] Cold n beam.113(6) 5.3% 1.258(15) polarize a beam of cold neutrons has steadily improved, from 79% in Krohn s measurement which used a SternGerlach magnet [41], to the use of super mirror polarizers which first achieved 97-98% in Perkeo I [4] to 99.85% in the most recent Perkeo II [38]. A summary of the results of these measurements, shown in Table 1.2 and in Figure 1.3, is dominated by the final results of Perkeo II which achieved.42% precision, after combining several datasets the most precise of which was.48% Perkeo I The first iteration of the Perkeo experiment was performed in 1986 and used cold neutrons from the High Flux Reactor (HFR) at the ILL. Electrons emitted from β decay were detected in a magnetic spectrometer positioned coaxial to the cold beam with the detector mounted above the beam line, moving them out of line of sight of the neutron beam and limiting the solid angle for capture gammas. The electrons from neutron β -decay in the spectrometer were guided to the detectors via a tailored magnetic field. While this 14

31 arrangement reduced neutron generated backgrounds in the detector, it introduced a 5% correction due to mirroring of electrons emitted outside of the maximum field region 1. Other leading systematics include a 97% polarization and a 1. % correction from neutron generated backgrounds. Perkeo I achieved a final result of A =.1146(19) PNPI Yerozolimskii One of the few modern experiments not performed at the ILL s HFR, Yerozolimskii [1,11] used a vertical cold neutron beam from a liquid hydrogen source in the PNPI reactor and measured the coincidence of electrons and protons from polarized neutron decay. A high frequency magnetic coil was used to reverse the neutron spins entering the detection region every 2-3 s and was on for both spins states. This reduced the impact of spin flipper inefficiency, which instead appeared as a reduction in the polarization. The largest correction in this experiment, which resulted in a reanalysis of the data six years after the initial publication, was the spatial dependence of the neutron beam polarization. For the initial analysis a polarization averaged over the beam acceptance of P =.7867(7) was used to obtain a result of A =.1116(14). After determining that energy dependent interaction with air and aluminum in the neutron flight path distorted the beam polarization measurement, the polarization was further reduced by 1.65(15)%, which yielded A =.1135(14) ILL TPC One of the most unique detection schemes among the asymmetry measurements, the ILL-TPC group [9, 39] used a helium filled time projection chamber (TPC) to observe the decay of polarized neutrons at the PN7 cold neutron beam position at the HFR. The experiment used a system of bent super-mirrors to transversely polarize neutrons, and achieved a polarization of P =.981(3). Polarization was maintained by a 1 mt holding field over the 2 m transit to the spectrometer, which consisted of a TPC and scintillation detectors. A current sheet style spin-flipper, with spin-flipper efficiency of 99.(2)%, was used to alternate the neutron spin direction between measurement cycles. Ionization tracks measured in coincidence with 1 Electrons emitted toward the interior of the spectrometer from up or down stream of the tailored field will experience an increasing magnetic field and potentially magnetically mirror. Resulting in the event s direction incorrectly counted. 15

32 triggers from, 5 mm thick, plastic scintillators provided information on the energy and direction of electrons from the decaying neutrons. Leading systematic uncertainties in this experiment were due to neutron induced backgrounds (.8%) and from a correction to the angular acceptance of the detector (.4%). The angular acceptance of the scintillators increased the measured value of cosθ =.85(3) from 1/2, requiring a large, 26% correction to the obtain the final result of A =.116(15) Perkeo II To improve on the results of Perkeo I, a second spectrometer was constructed with the axis of the solenoidal magnetic field perpendicular to the neutron beam [5,6,8,42]. This orientation all but eliminated the large correction due to magnetic mirroring, with a.24% effect remaining due to the beam center passing slightly off center of the field maximum. The installation of baffling to shield from neutron capture gamma radiation brought the correction from irreducible backgrounds down to <.2%. Improved polarization and spin-flipper technology reduced the uncertainty in polarization to.2%. Results from Perkeo II were first published in 1997 and the final data set, published in 212, achieved a precision of.42% and a final result of A =.11951(5). This result was more precise than previous values published by Perkeo I by a factor of more than 2 and are in 4σ disagreement with the world average before Current Status of V ud The ratio of the axial-vector to vector coupling, λ = g A /g V, can be determined from the zeroth order form of the β-asymmetry A from the relationship given in Eq. 1.16c or more explicitly as λ = 1 1 (3A + 2)A (3A + 2) (1.22) 16

33 UCNA-1 Perkeo II.976 Unitarity V ud τ τ n,serb n,pdg ILL-TPC λ pdg PNPI-97 Perkeo I λ = G /G V A Figure 1.4: Extracted values for λ = g A /g V from the β -asymmetry measurements included in the PDG average, represented by the solid vertical lines with the with average shown as the vertical hashed area, are overlain against τ n to show the bounds of V ud from the neutron data set. These results can be compared to what is determined in the + + data, the red horizontal band, and unitarity constraints. The final Perkeo II result is shown as red dashed lines and the most precise τ n measurement from Serborov, τ n,serb is shown by the blue band. 17

34 with the uncertainty in λ given as σ λ = ( ) 5 + 3A A 3A 2 σ A (2 + 3A ) 2 1 2A 3A 2 σ λ λ σ A 4A. (1.23) Combining the results summarized in Table 1.2 with measurements of τ n, a value of V ud can be calculated as a cross check of the superallowed Fermi decays. In Figure 1.4 the bounds for A from all measurements currently included in the PDG averaged are shown in addition to bounds from the lifetime measurements [33]. The intersection for the λ bounds and the τ n bounds give the range of V ud predicted by those measurements. The limits from the superallowed decays and unitarity constraints are included to check consistency. The unitarity limits are derived by setting V us and V ub to the global averages shown in Eq. 1.9 and assuming unitarity to extract V ud [33] from, Vud 2 = 1 V us 2 Vub 2. From this analysis, it is clear that the average obtained from including the previous measurements of A does not agree with the super allowed decays or the unitarity argument, however if one only considers the results of Perkeo II then the situation seems to be resolved. When one considers only the most precise results of τ n = 878.5(7)(3) [43] and A [38] the limits on V ud derived from the three analyses are in very good agreement. At the current precision, 1.4%, the UCNA experiment also finds a larger value of λ compared to previous experiments in agreement with Perkeo II. 1.3 Improvements Using Ultracold Neutrons Neutrons that are cooled to low enough temperatures, typically< 3 mk, can be contained in material, gravitational, and magnetic bottles and are defined as ultracold neutrons (UCN). This fact has been exploited by numerous τ n and nedm measurements to eliminate many of the systematics associated with beam experiments. In the same way, using UCN for a measurement of the β-asymmetry introduces an entirely different set of polarization and background systematic errors as opposed the traditional beam experiments. UCN densities delivered to experiments are typically < 1 UCN/cc, meaning the relevant 18

35 flux of neutrons are five orders of magnitude smaller than cold beam. In addition, UCN are trapped in the decay region, thus a much higher fraction, (1/44) in the case of UCNA, of the flux can contribute to the measured decay rate. Because of their low capture-rate per bounce and high specific activity, neutron-generated backgrounds are greatly reduced. UCN experience a 6 nev/t potential in a magnetic field through the µ n B interaction, therefore fields greater than 6 T present an impenetrable barrier to one spin state. Using a sufficiently large magnetic field barrier should, in principle, give a 1% polarized sample of UCN. UCNA was designed to exploit these two fundamental differences between cold and ultracold neutron experiments and to further refine the techniques of previous experiments [44, 45]. UCN are polarized by a 7 T magnet solenoidal with a tailored gradient where an adiabatic fast passage spin flipper operates. A solenoidal 1 T spectrometer is loaded with either spin state of highly polarized UCN. To increase the poor signal to noise ratios seen with bare scintillators due to ambient gamma backgrounds, edge effects due to scattering on beam aperture and position dependent detector response, position sensitive multi-wire gas proportional counters were paired with the thin plastic scintillators. Low pressure, low-z gas counters have a very low gamma counting efficiency, so that a coincidence requirement between the scintillation and gas counter reduces ambient backgrounds by a factor of 5. In the following chapters the general scope of an asymmetry analysis along with the simulation required to gain understanding of scattering corrections, analysis of the first measurement of the β-asymmetry using UCN, and the analysis of a follow-up experiment designed to test the scattering predictions of our Monte Carlo will be discussed with emphasis on the data related to electron detection in the spectrometer. Documentation of the UCN transport and polarimetry analysis can be found in a thesis by A.T. Holley [46]. 19

36 Chapter 2 Experimental Overview 2.1 Introduction The UCNA experiment consists of three experimental subsystems: the ultra-cold neutron solid deuterium (SD2) source, UCN transport and spin conditioning, and the superconducting soleniodal spectrometer (SCS), laid out as shown in Figure 2.1. UCN created in the spallation SD2 source are extracted via stainless steel guides to the first polarizing magnet, The Prepolarizer Magnet (PPM). The µ B potential provided by the 7 T field in PPM pulls one spin state through a vacuum separation foil that, for safety reasons, isolates the source vacuum from the experimental vacuum. Downstream of the PPM, electropolished copper guides and one DLC coated quartz guide are used to maintain the polarization of the UCN. A second 7 T superconducting magnet, The AFP Magnet (AFP), fully polarizes the input flux of UCN to the SCS and the tailored gradient in the 1 T downstream end of the AFP allows for the use of an adiabatic fast passage spin flipper to alter the spin state of UCN [47]. A density of UCN polarized, with spins in the direction defined in the AFP magnet, is built up in the central 1 T field of the SCS and the decay products, constrained by the field, spiral to identical detector packages at either end of the SCS, thus creating an effective 2 2π acceptance. The event rates parallel and anti-parallel 1 to the neutron spin, σ, are then used to extract the asymmetry parameter, A. Details of the experimental setup of UCNA 1 Throughout the text the terms parallel and anti-parallel will refer to emission direction of the electrons from β-decay. We define the directions ˆσ p e > as parallel to the neutron spin, ˆσ, and ˆσ p e < as anti-parallel. 2

37 East West Round to Square Transition AFP Magnet Fe Foil Monitor Switcher Monitor Gate Valve Switcher PPM Guide to Test Port Superconducting Spectrometer Biological Shielding UCN Source Gate Valve UCN Monitor Proton Beam Figure 2.1: The layout of UCN guides, UCN monitor detectors, AFP and PPM magnets, and the superconducting soleniodal spectrometer (SCS) is shown not to scale. Not shown is a fourth UCN monitor directly under the center of the SCS. Through the 29 run cycle, the switcher to the test port was not installed. Instead a smooth, continuous 6 bent guide mated to the input to the switcher prior to the AFP magnet. The east and west ends of the SCS is noted, as it will be used to describe detectors packages throughout the text. will discussed in this chapter, a general discussion of analysis techniques and systematic corrections will be presented in Chapter 3, and specific analysis will be shown in Chapters 4 and The Ultra-Cold Neutron Source If one imagines the process of cooling thermal neutron as similar mass billiard ball scattering, neutrons on hydrogen atoms, then each collision can result, in a range of resultant neutron energies, from almost no energy loss to a completely stopped neutron. Through many successive collisions, the neutrons will lose energy until their kinetic energy is comparable to the thermal energy of the hydrogen atoms. At this point the energy exchanged scattering on the hydrogen atoms averages to zero and scattering on the much 21

38 more massive nuclei can be viewed as purely elastic. To further reduce the temperature of the neutron population there must exist another channel for energy exchange. Now we need to consider that neutrons at this low energy begin to act as quantum objects whose wave function no longer sees scattering as a point to point process but indeed samples many scattering sites. Through this scattering it is possible to excite vibrational quanta in the solid lattice, phonons, which can continue to carry away energy from the neutrons. If the temperature of the solid is reduced to where all such excitation are suppressed then it is possible for the neutron to lose energy through phonon exchange with the lattice. The Los Alamos Neutron Science Center s (LANSCE) linear accelerator delivers 8 MeV protons which are focused onto a tungsten target, with an average beam current of a few µa, creating approximately 2 spallation neutrons per proton which can be cooled to the temperature of ultracold-neutrons, < 4 K, via scattering in cold moderators and solid deuterium (SD2) [48 51]. Directly above the W target is a layer of cold polyethylene beads (the moderator) surrounding a disk of solid deuterium, at < 8 K. Some neutrons in the low energy tail of the energy distribution coming from spallation will lose enough energy in the moderator so that they become cold neutrons (CN), which then have a chance to be converted to UCN via a phonon exchange in the SD2. Since the spallation neutrons are emitted in 4π the target is surrounded by layers of beryllium, graphite, plastic, and iron(steel) to reflect a fraction of the flux back into the moderator and SD2, Fig Successfully down-converted UCN, with perpendicular kinetic energy below the Fermi potential of the material bottle, are trapped in the guide system and flow similarly to an ideal gas out of the source to the experimental area. UCN have a relatively short lifetime in the SD2 converter of the UCNA source, τ 25 ms at 5 K [51, 52], due to thermal up-scattering, nuclear absorption on hydrogen and deuterium, and spin-flip up-scattering on para-deuterium, therefore minimizing the time they are exposed to the source is essential for increasing the density down stream. To accomplish this, the beam is operated in pulse mode and a 58 Ni-plated valve opens in time with the beam bursts. The pulse structure of the beam and relative timing of the flapper has changed slightly from year to year, due to changes in the source and changes to allowable instantaneous current of the proton beam, in most cases the valve is opened.1 s prior to a burst and closes after.4 s with a.1 s opening / closing time. For the first UCNA asymmetry 22

39 measurement in 27 the beam structure was a 3 pulse burst every 17 s [12]. During the run cycle the beam was operated as a burst every five seconds where each burst was comprised of five 625 µsec long pulses uniformly space over a.2 s interval, the instantaneous and average current of the beam was : 1 ma, in each pulse, 15 µa, over the burst, and 5.8 µa averaged over the 5 s period [13]. Instantaneous UCN densities of 6(12) UCN/µC/cc and 1(2) UCN/µC/cc were achieved in 29 and 21 respectively inside the source which translates to 48(1) UCN/cc in 29 and 79(16) UCN/cc in 21 at the shield wall, and 2.(4) UCN/cc in the UCNA spectrometer. Specifics of the current source and the prototype source are found in studies by Saunders et al [49, 53] Ortho-Para Deuterium Scattering During the initial design phase of the SD2 source a lot of consideration was given to possible UCN loss mechanisms during residency in the SD2 crystal. It was shown by Liu and others [51, 52] that a leading source of UCN upscattering is due to spin-flip interactions with para-deuterium, reducing the lifetime in natural deuterium to 4.6 ms. Room temperature D 2 exists in a 2:1 mixture of the ortho (even J) and para (odd J) spin states, where J is the rotational quantum number. At the temperatures relevant for the SD2 source, T < 1K only J = (ortho) and J = 1(para) are available. UCN can be up-scattered from para-deuterium by a J = 1 spin relaxation channel that transfers 7.5 mev to the UCN. Therefore, to minimize the para-deuterium fraction present in SD2 source and thus increase the achievable UCN densities, an apparatus for para to ortho conversion and monitoring of the ortho-para fraction was designed, built, and installed by Liu [54]. In normal source operation the para-deuterium fraction is below 3%, which is measured periodically by an on site rotational Raman-spectrometer. HD contamination of the deuterium can also be monitored by the Raman-spectrometer and is typically 1 3. Throughout the run cycles, anecdotal evidence was collected showing that para-deuterium is converted to ortho-deuterium during exposure to the proton beam. While a careful study to correlate the amount of time, protons on target, and reduction of para-fraction was not performed the general trend was reported in internal collaboration documents by Rios [55]. 23

40 To Experiment Tungsten Target 1 m UCN Flapper Valve SD2 Cold Poly Moderator 3K Proton Beam Be Flux Trap Figure 2.2: Cross sectional view of the LANSCE UCN source. Drawing is not to scale and in the actual experiment the proton beam path and neutron exit guide are at a right angle to each other He 3 Neutron Detectors UCN are detected by four He 3 gas counters positioned throughout the experiment as shown in Figure 2.1: just prior to the gate valve, on the drain outlet of the switcher, below a magnetized iron foil on the downstream end of the AFP magnet, and below the center of the spectrometer. Each detector housing is machined from Al and assembled in three pieces, main body, lid, and electrical and gas feed through assembly and vacuum sealed with Viton o-rings, shown in Fig. 2.3a [56]. Neutrons are accelerated by dropping them 1 m (1 m = 1 nev for UCN in a gravitational field) to increase the transmission through the.25 cm 661 T6 Al entrance window to the detector. Monte Carlo estimates put the transmission at roughly 5%. Neutrons capture on the He 3 through the He 3 (n,p)h 3 reaction resulting in the emission of a triton and a proton with a Q = 764 kev. To maximize the probability of interaction and capture of the ionization, therefore detection efficiency, the detectors are filled with a mixture of 2 mbar of He 3 and 1 bar of CF 4 [56]. He 3 has less stopping power than the B 1 F 3 used in earlier versions of this detector, because of this CF 4 was added to increase the stopping power to limit the range of charged particles to 48 mm. 24

41 several mean free paths thick in order to optimize the detector efficiency and the mean free path needs to be larger than the charged particle range to minimize the wall effects. We have designed a detector, shown in Fig. 1, which is approximately planar. The detector incorporates two cathodes held at ground, the entrance window and a copper circuit board. At the approximate center is an anode plane which is held at high voltage. The active volume is determined by the distance between the two cathode planes, 5 cm. Grid planes, spaced 2.5 mm from the anode, ensure uniform fields in the anode region even though tension. The grid planes were of similar construction with 2 mm spaced 75 mm gold-plated copper-clad aluminum wire also at 5 g m of tension. For detecting UCN, one must use low absorption materials for the entrance windows. We used aluminum alloy windows (661 T6) which are strong, robust and provide relatively low losses. The window in the detector presents both a potential barrier and introduces UCN losses due to absorption and scattering. With the Anode plane the entrance window is held Grid planes by a flange, and is bowed by the gas pressure. The anode planes were constructed on a FR-8 frame with 2 mm gold-plated tungsten wires spaced by 2 mm andentrance wound window at 5 g of tension. The wires were epoxied to the plane and soldered to the copper tabs. The ends of the wire plane were terminated with 75 mm beryllium copper guard wires wound at 1 g m of tension. The grid planes were of similar construction with 2 mm spaced 75 mm gold-plated copper-clad aluminum wire also at 5 g m of tension. For detecting UCN, one must use low absorption materials for the entrance windows. Figure We used 2.3: aluminum alloy windows (661 T6) which are strong, robust and provide relatively low losses. The window in the detector presents both a potential barrier and introduces UCN losses due to absorption and scattering. With the Grid planes Anode plane Fig. 1. Schematic view of the detector assembly. (a) Construction schematic of the detector. Fig. 1. Schematic view of the detector assembly. in the.e+ histogram. The typical resolution was 3% for the five detectorsthat 2 were 4 tested Energy (kev) In an earlier work, we used detectors of a similar construction Fig. 2. Pulse height spectrum obtained with a UCN beam. The detector is operating in ion with V A=26 chamber and V C=39 mode. In the laboratory this mode of running V. The spectra shows a peak at the 3 He(n,p) 3 H Q-value as provides well as edgessimilar at the energies resolution of both reaction and products highduegain to wall stability, effects, but was much indicated by arrows. more susceptible to both electronic and microphonic noise. Maintaining good resolution in detectors with gas gain required Count rate (arb) Counts/bin (arb) E+5 44.E+5 3.E E+5 1.E+5.E+ T 3 H Thermal Tp neutrons UCN He pressure (mbar) (b) He Energy 3 (n,p)h (kev) 3 spectrum. Fig. 3. Relative neutron efficiency for a moderated 252Cf source (dashed line) and for UCN (solid line) as a function of 3 He partial pressure. The solid curve through the UCN data is a calculation of the efficiency. Q UCN X1 Fig. 2. Pulse height spectrum obtained with a UCN beam. The detector is operating with V A =26 and V C =39 V. The spectra shows a peak at the 3 He(n,p) 3 H Q-value as well as edges at the energies of both reaction products due to wall effects, indicated by arrows. An exploded view of the UCN 3 He is shown in panel (a). A typical pulse height spectrum from the UCN monitors is shown with the three major spectral features label: Q, both proton and triton are counted, T p only the proton is counted, and T3 He only the triton is counted [56]. Count rate (arb) 8 Charge is collected by an anode plane consisting of 2 µm gold-plated tungsten wires mounted 6 on FR-8 frames with.49 N of tension, spaced every 2 mm, biased at 26 V. Grid planes of similar 4 construction as the anode with 75 µm gold-plated, copper-clad Al wires are mounted above and below Thermal neutrons the anode with 2.5 mmentrance spacing window to ensure the 2uniformity of the electric field, operated at 39 V. Cathode UCN planes, held at ground, surround the anode and grid plane assembly and define a 5 cm thick 5 mm 2 active volume. Signals are processed with an Ortec 142PC preamplifier and read by a CAEN 785 peak-sensing 3He pressure (mbar) ADC. When operating in proportional mode, three spectral features can be identified, a large peak at the Fig. 3. Relative neutron efficiency for a moderated 252Cf source (dashed line) and for UCN (solid line) as a function of 3 He partial pressure. The solid curve through the UCN data is a calculation of the efficiency. full Q value of the He 3 (n,p)h 3 corresponding to both the triton and proton ionizing in the active region, and two lower energy edges due to either reaction product capturing on the wall, shown on the right in Fig 2.3b [56]. 2.3 UCN Transport General UCN Transport The relatively large wavelength of UCN, 4 nm, allows the neutron to sample a large number scattering centers in lattice structure of a solid. This collective scattering potential can be generalized as the effective Fermi potential, V F = 2π h2 m n Na (2.1) 25

42 Maximum UCN Density vs. Material Potential 4 35 ) 3 UCN density (cm V F (nev) Figure 2.4: Maximum integrated UCN density as a function of Fermi potential of the storage vessel. Vertical line so the V Fermi for stainless-steel, PLD-DLC, and 58 Ni. where m n is the mass of the neutron, N is the number density of nuclei in the target, and a is the coherent scattering length [57, 58]. UCN with perpendicular kinetic energy, defined as E = m n v 2 /2 where perpendicular is the surface normal of the scattering material, less than the Fermi potential will undergo total external reflections. Materials such as Ni 58, Cu, and stainless steel have a Fermi potential of hundreds of nev, with V Ni58 F = 34 nev being the greatest, therefore the choice of materials used to construct the guide system out of the source will define maximum energy neutrons which can be transported and the overall fraction of neutrons trapped. A significant amount has be done by the UCNA collaboration to design and construct guides with high Fermi potential, low depolarization per bounce and low loss per bounce as detail in thesis by Mammei [59], Makela [6], and Holley [46] From the Source to the PPM After creation in the source UCN must escape into the guide system and traverse 15 m of guide tubes, two polarizing magnets, a vacuum foil, a switch box, and round guide to rectangular guide transition before getting into the spectrometer where a small fraction will decay prior to being captured, up scattered, or lost in the cracks between the guides. Such is horribly tumultuous time in the life of a UCN, Figure 2.1 shows the guide system between the source and the spectrometer. Surface roughness, material potential, gaps between guides, vacuum quality, and input velocity are a few of the numerous variables that effect 26

43 the transmission of UCN. Successfully converted UCN first ascend one meter from the flapper to a horizontal 4 in guide which leads out of the shield package through two 45 chicanes to prevent a direct path out of the source. This reduces the cold and thermal neutron and gamma flux in the experimental hall. The guides leading away form the source are another part of the experiment that has gone through many changes over the years. Initially, 23-25, the guides were diamond like carbon (DLC) coated quartz guides produced at the Virginia Tech coating facility [6]. It was determined that the DLC coating was insufficiently thick, <1 nm, and unable to contain the UCN produced in the source. The DLC-quartz tubes were replaced with uncoated electro-polished stainless steel guides, 25-present. At the exit of the biological shield stack, a high-vacuum gate valve can isolate the source from the downstream components of the experiment to shut off the flow of UCN, allowing background measurements to be taken in spectrometer without turning off the proton beam. In this way beam related backgrounds can be measured in the absence of UCN in the spectrometer. A 3 He neutron detector monitors the UCN density upstream of the gate valve through a 1/8 in diameter hole in the bottom of the guide, details on the operation of the UCN monitors are reported in Morris, et al [56]. Measuring the rate upstream of the gate valve has two main uses. When the valve is closed it directly monitors the density in the source, low rates indicate that either the source or the proton beam or both are not operating optimally. With the valve open the rate is used to normalized the rates measured downstream to remove the beam and source performance, so that the transmission of UCN can be addressed. After the gate valve is the first of two solenoidal magnets called the pre-polarizer magnet (PPM). The PPM is an American Magnetics Incorporated superconducting 7 T magnet that is used to accelerate one spin state of UCN through a thin,.5 mm, Zr vacuum safety window. Because of dangers associated with mixing deuterium and air the source vacuum must be separated from the experimental vacuum, preventing a compromise of the experimental vacuum from affecting the source vacuum. Without the energy gained in the µ B interaction a large fraction of UCN would not be able to penetrate the Zr window. Operating the magnet at 6 T, the transmission is nearly 1% for the spin state that sees a potential well when approaching the field, thus the overall transmission is 5%. Analyzing the transmission through the 27

44 PPM as a function of the magnetic field magnitude, with and without Zr window installed, gives us information about the velocity distribution of UCN exiting the shield stack allowing for benchmarks of the neutron transport Monte Carlo. Results of these studies have been reported on by A. T. Holley [46]. Downstream of the PPM all guides are electro polished copper aside from a PLD-DLC quartz guide in the AFP magnet The AFP Magnet and Spin Flipping At the exit of the PPM the guide system bends 6 to the north to head to the polarizing/spin flipping magnet, the AFP. The AFP is so-named for the adiabatic fast passage spin flipper that is resident in the tailored 1 T end of the magnet. The upstream end of the AFP is held at 7 T to polarize in the input of flux of UCN. This polarizing field reduces to 1 T at the north end of the magnet where the gradient is tailored to.1 gauss/cm so that the radio frequency (rf) spin flipper can operate at peak efficiency. In order for the rf radiation to penetrate the guide system a DLC coated quartz guide is used instead of copper. When the AFP spin flipper is operating, UCN see a 6 Gauss magnetic field oscillating at 29.8 MHz or roughly the Larmor frequency of the neutron in a 1 T field. Variations in the field gradient require that the operating frequency and input power to the spin flipper be optimized every time the magnets are cycled and periodically checked during the run cycles. The neutrons spin follows the field in a frame co-rotating with the RF-field as it transverses the AFP region and exits after a π-flip of its spin, providing a way to inject both spin states into the spectrometer. Full details of the conception, design, and testing of the UCNA AFP spin flipper and AFP spin flippers in general can be found in A. T. Holley, et al. and references found therein [46, 47]. A guide switching chamber shown in Figure 2.5 installed upstream of AFP magnet allows UCN to either pass from the PPM through the AFP to the SCS or drain from the SCS into a UCN monitor, the switcher detector. The switcher chamber contains two guide segments, mounted on a linear translation stage actuated by a Bimba pneumatic piston, which can be positioned to mate the upstream guide from the PPM to the input guide of the AFP magnet or mate the AFP guide to a descending guide that connects to a UCN counter. After β counting, the UCN population of the spectrometer and guide system can be 28

45 Switcher Vacuum Chamber To AFP To PPM Guide Cart Pnuematic Actuator Switcher Detector Figure 2.5: The switcher vacuum chamber is illustrated showing the input and output guides to the AFP and PPM and the alternate output guide down to the UCN monitor. Note that in reality the ends of the input and output guides are cut at 45 therefore the translation stage and rail system runs at 45 relative in the PPM and AFP guides to properly mate with the guides. The translation stage is manipulated by linear pneumatic actuator through a vacuum feed-through. drained into this switcher detector. Immediately downstream of the AFP magnet the UCN density is measured by a 3 He detector that is position below a magnetized thin iron foil, this detector is refer to at the iron foil monitor. The iron foil acts as a spin analyzer, preventing neutrons whose spin has been flipped to pass through to the detector. This detector was used only to confirm the spin-flipper was properly tuned. An in situ analysis of the spin content of the UCN population in the spectrometer is preformed for every β counting cycle by stopping the flow of neutrons through the gate valve, changing the state of the switcher, and then monitoring UCN exiting from the SCS in the switcher detector [46,47] Round to Square Transition Getting into the spectrometer requires that the neutrons transition from 3 in. diameter round guides to a 1 in. 3 in. rectangular guide. The entrance guide to the spectrometer is rectangular in cross section, designed to fit between the split coil design of the magnet and maximize the UCN input flux 29

46 while minimizing any field inhomogeneities that arise from having a gap in the coils of the solenoid. The rectangular guide is made of four pieces of electro polished copper that are fastened together with non-magnetic screws. The side coupling to the decay trap was shaped to match the outer radius of the trap, minimizing gaps. Where the round to square guide transition occurs a copper plate is sandwiched between them with the internal dimensions of the rectangular guide removed so that UCN not entering the rectangular guide may be reflected for second attempt to slightly increase the density. A polyethylene coupler holds the copper plate, rectangular, and round guides in place. 2.4 The Superconducting Solenoidal Spectrometer The Superconducting Solenoidal Spectrometer (SCS) is a custom American Magnetics Incorporated (AMI) split coil warm bore super conducting solenoidal magnet with cryostat built by Meyer Tools and Manufacturing, Inc. The cryostat contains a 16 L liquid helium (LHe) reservoir and liquid nitrogen (LN2) volume. It has been determined that by forcing the cold return helium vapor through the nitrogen volume, instead of holding LN2 in this reservoir, the boil off rate is reduced to 1 l/hr, minimizing the consumption of LHe [61]. The cryostat is coupled to a Koch Model 163 helium liquefier via a closed looped to recaptured and re-liquefy boil off LHe. LHe produced by the 163 is stored in a 4 l Dewar where it can be transfered via cryogenic transfer lines to either the UCN source, the SCS, the AFP, or the PPM, while the boil off gas is pushed to a 24-psi 23 l liquid equivalent tank on the high pressure side of the 163 s compressors. During normal operation the SCS is topped off every 12 hours to keep the level in the LHe reservoir above 8%. On axis the central field, as originally built and measured, is a uniform 1 T with a tailored expansion to.6 T at 2.1 m from center and held constant for 1 cm or the length of the gas counter and scintillator. Recent measurements of the on-axis field have shown variations up to.1 T in the central region (see Figure 3.32). The internal warm bore is 34 cm in diameter and 4.5 m long with an internal rail system that guides the decay trap into position. Three 7 cm 4 cm rectangular ports in the center allows access for the UCN input square guide (south side), a calibration source insertion arm (north side), and a guide to a UCN He 3 monitor for real time monitoring of the UCN density in the spectrometer (below). 3

47 To achieve the required ±5 1 5 field homogeneity and allow for a central break in the solenoid coils AMI designed a proprietary tuned multi-coil system consisting of 32 5 cm diameter, 7 cm wide coils operated by a single persistence switch and 28 shim coils controlled via a LabVIEW interface program to the magnet power supply controller. Full details of the SCS can be found in Plaster et al. [61] and a thesis by Yuan [45] Decay Region and Trap UCN are contained in three 1. m long, 12 cm diameter right cylindrical sections of electro polished copper. A thin, 15 nm coating of PLD-DLC on the surface of the copper increases the Fermi potential to 23 nev, increasing the storage times in the bottle to 3 s [59, 6]. Thin, 7 nm, mylar foils coated on the interior face with 3 nm of beryllium cap the ends of the decay trap creating a material bottle to confine the UCN to the central 3 m of the SCS, where the magnetic field is a nominally uniform 1 T. The original design of UCNA had an open decay trap, but low UCN production and/or transport issues led to the necessity of capped ends to increase the density in the bottle. A polyethylene mount holds the foils to the end of the trap and extends 2.5 mm into the inner diameter of the decay trap acting a collimator stopping electrons produced close the decay trap walls. Edge effects due to scattering on the collimator will be discussed later in Chapter 3 and are all but eliminated by the position cuts. The three sections of the decay trap are clamped together with polyethylene clamp rings to reduce the gaps such that a gap gauge could not be inserted between the guides. A.3175 cm hole at the bottom of the center section allows the density of UCN in the decay trap to be monitored. A stainless steel 4 cm diameter guide mates to the bottom of the decay trap and guides the UCN to a 3 He gas counter, mounted 2. m below the center of the SCS (see Sec ). Between a few iterations of a source calibration device was mounted opposite the square guide entrance to the SCS, shown in Figure 2.7. Prior to 28, a source was mounted on a 1.5 m arm which was attached to an Huntington Tilt-A-Port VF-179 vacuum flange that allowed the arm to be articulated such that the source could be moved in and out of the detector acceptance on the east side of decay trap. This design proved to be hard to manipulate and gave asymmetric calibration, since the east 31

48 Figure 2.6: Schematic of the soleniodal super conducting spectrometer (SCS), showing the input guides through the AFP magnetic in the decay trap. Magnetic field lines depict the expansion from 1 T to.6 T in the region between the decay trap and the detectors. detector saw the source directly and the west saw the source through the two decay trap foils. Another problem was that only one source could be mounted making in situ linearity testing impossible. After the 27 β-asymmetry measurements a series of runs to measure the linearity and position response of the detector were performed by inserting a multi-source mount in the UCN entrance port on the south side of the SCS, with 113 Sn, 27 Bi, and 9 Sr calibration sources. An updated calibration system, mounted in the same location as the Tilt-A-Port, can extend a source holder, containing up to three sources, into the center of the decay trap through a 2.54 cm diameter hole in the side directly opposite the square guide entrance. The source holder can be retracted through an airlock and mounted with a PLD DLC coated Delryn plug that closes the hole during running. Its in important to note that the plug and the calibration sources can be swapped without breaking the vacuum integrity of the SCS, preventing the need to power down the β-detectors and remove the fill gas from the MWPC s, which would make the calibration unreliable. 32

49 28-present Load-lock calibration source holder 27 In situ calibration source holder E-5 Sn-113, Bi-27, Ce-139 Sn-113, Bi-27, Sr-85 Sn-113 West Det. East Det. UCN entrance port 27 post-run calibration source insertion system Figure 2.7: Source insertion systems that have been used between 27-present are depicted and explained in the text β Detectors Detector packages consisting of a plastic scintillator, multi-wire gas proportional counter, and most of the supporting electronics and gas handling system are mounted on detector carts and attached to either end (east/west) of the SCS with vacuum fidelity maintained by a 16 inch Conflat flange and viton o-ring, shown in Fig 2.8. The MWPC provides event location in the x,y-plane and is relatively insensitive to incident gamma radiation, enabling 99.5% of the ambient background to be vetoed by requiring a coincidence between same side scintillator and MWPC in software. Event energy is measured by four photomultiplier-tubes coupled by twelve adiabatic light guides that are edge-coupled to a 3.5 mm thick 15 cm diameter plastic scintillator. Combination of this detection system and distance from the UCN source allows for experimental signal to noise ratios of 1:8 across the energy range of interest for UCNA analysis. The design, development, and commissioning of the detector carts is the subject of a previous thesis [45] and several publications [61,62], the relevant details of each subsystem will be summarized in the following sections. The Multi-Wire Proportional Counters Position sensitive multi-wire proportional counters used in UCNA, shown in Fig. 2.8, provide the ability to remove ambient gamma backgrounds through software cuts and reconstruct the event positions to within 2. mm, eliminating edge effects from scattering on the collimator in the decay trap. The wire 33

50 ARTICLE IN PRESS 59 B. Plaster et al. / Nuclear Instruments and Methods in Physics Research A 595 (28) Fig. 2. Schematic diagram of the MWPC and plastic scintillator detector package. Figure 2.8: Detector schematic for [61]. and a 24-psi medium pressure storage ambient temperature where the MWPC/scintillator is mounted). Finally, the ano ballast tank with a 23 liquid liters equivalent volume. plane was strung with 64 1-mm diameter gold-plated tungs The power supply chamber for the is main housedcoil, in aan1.5 AMI inch Model OD Al 122PS- cylindrical can wires, with and open thefaces, cathode where planes, custom separated built frames from the anode plane 42 (2 A/12 V), functions as a slave to an AMI Model 42 Power 1 mm, were strung with 64 5-mm diameter gold-pla holding thin, 6 to 25 µm thickness, aluminized mylar foils are mounted. 2 denier Kevlar yarn reinforces Supply Programmer unit. The full energized field strength of 1. T aluminum wires. The wire spacing on both the anode and requires a current of the124. foil that A. The separates powerthe supply 1 Torr for the neo-pentane shim coils, gas volume two cathode of the MWPC planes fromis the 2.54 several mm, micro-torr yielding an active area a bipolar Kepco Model BOP 2-1M (2 V= 1 A), also functions as a slave vacuum to a second of theami SCS. Model The strands 42 are Power glued Supply onto the foil in the frame 1.-T with region, vacuum providing epoxy in full 5 coverage mm spacing. of the 1-cm diame 16:3 16:3cm 2. This active area maps to a 12:6 12:6cm 2 squ Programmer unit. During magnet operations, the shim coils decay trap. are energized one-by-one Reinforcing tothe their foilsoptimized reduces thecurrent bowingvalues. due to the pressure All ofdifference the anodefrom wires 3 cm areatconnected 1 Torr to via.6conductor tracks A LabView-based program installed on a PC functions as the the anode wireplane support frame; thus, the anode signal tha cm for 6 µm thick foils and it was found that a 3.5 µm thick foil was be able to withstand a pressure interface between the user and the Model 42 Power Supply read-out is the summation of the signals on the 64 individ Programmer units. difference of up to 2 Torr. Diffusion through the 3.5 µm wires. foil was Withbeyond the wires the acceptable on one cathode limits, however, plane oriented verticall The MWPC/plastic scintillator electron detector packages, providing information on the horizontal coordinate (defined described in detailestablishing below, areamounted minimumto entrance the two window ends thickness of the ofbe 6 µ the [45]. x-coordinate), and the wires on the other cathode pla solenoidal magnet via 41.9-cm conflat flanges. The central axis of oriented horizontally, providing information on the vertic the spectrometer is located Limiting 2.74backscattering m above the and experimental high detection floor. efficiency coordinate were the (y-coordinate), main requirements theconsidered ðx; yþ position when for each event reconstructed from the center-of-gravity of the two catho the choice of fill gas was investigated. It was determined that the low atomic number, Z, to reduce planes signals Multiwire proportional chambers backscatter probability and high molecular density (high electron Note that density) the 64 to increase wires ondetection each ofefficiency the two cathode planes read-out in groups of four. This four-wire-grouping reduced A schematic diagram of neo-pentane illustrating (C the main components of the 5 H 12 ) made it an ideal candidate. Neo-pentane number of (2,2-dimethylpropane) electronics channels was (thereby chosen reducing over the cost), b MWPC/plastic scintillator detector package is shown in Fig. 2. As did not, as shown later, degrade the resulting position resoluti noted already, the MWPCs normal pentane are relatively because insensitive it is less reactive to gamma-ray and has a higher as the vapor typical pressure. four-wire-group multiplicity is 3 4 for backgrounds, offer a low threshold for the detection of backscattering events, andcharge permit is collected reconstruction threeof wire theplanes, transverse 1 anode and 2 cathodes, mounted in the center of the MWPC kev electrons from a 113 Sn conversion-electron source. 4 ðx; yþ coordinates of the b-decay events. Although described in with the anode in the middle and the cathodes symmetrically detail in Ref. [12], for completeness we briefly review some of the 3.3. Plastic spaced scintillator by 1.5 cm. detectors The wire planes consist MWPCs most important of 64 wires features mounted here. on a.92 inch thick FR4 epoxy glass frames with an active region of 15 cm 15 First, to suppress missed backscattering events (again, those Scintillator events depositing no cm: energy anode wires aboveare threshold 1µm gold in the plated MWPC), tungsten the with a 2.5The mmplastic spacingscintillator and the cathode detector wires (or, arehereafter, 5µm the b-scinti entrance window separating the MWPC fill gas from the spectrometer vacuum was designed to be as thin as possible. Second, because of this thin entrance window requirement, the fill gas pressure was required to be as low as possible. The chosen fill gas, C 5 H 12 (2,2-Dimethylpropane, or neopentane ), a low-z heavy 34 hydrocarbon, was shown to yield sufficient gain at a pressure of 1 Torr. The minimum window thickness for a 15-cm diameter window (equal to the diameter of the plastic scintillator, discussed below) shown to support a 1 Torr difference with a minimal leak rate (from pinholes) was 6 mm of aluminized Mylar, 3 reinforced by Kevlar fiber. Third, the MWPC mechanical size was tor ) is a 15-cm diameter, 3.5-mm thick disk of Eljen Technolo EJ-24 scintillator [23]. The EJ-24 wavelength of maxim emission is 48 nm. This wavelength couples well to a typi bialkali photocathode s wavelength sensitivity, and is also su ciently long for efficient optical transmission through a UVT li guide. Other desirable features of the EJ-24 scintillator wh motivated its choice include a high light output (68% anthracene), long attenuation length (1.6 m), fast rise ti (.7 ns), narrow pulse width (2.2 ns FWHM), and an index refraction (1.58) well-matched to that of UVT light guide (1.49 The 15-cm diameter in the.6-t field-expansion detec

51 aluminum with 2.5 mm spacing. The cathode planes are oriented at 9 to each other to define an x and y pickup. Charge collection on the anode is ganged into a Multi Channel Systems PA33 amplifier module with its main purpose to monitor energy deposition in the gas for event selection and possible energy reconstruction. The cathode wires are ganged into groups of four consecutive wires and read out as 16 channels per plane, which are also amplified by PA33 amplifier and mounted on a Multi Channel Systems CPA16. Operating in this mode the position of events can be reconstructed by either assuming a simple charge-weighted average of the wire positions X ev = 16 i=1 Q ix i, (2.2) 16 i=1 Q i where Q i is the charge recorded in the wire group and x i is the center position of the wire group or the charge cloud can assumed to be modeled well by a Gaussian and the measured charges can be fit and the center of the cloud backed out. In either model the spatial resolution can be shown to be < 2 mm [62]. Summing all the cathode channels has shown to be a good redundancy check for the anode signal and analysis can be performed with either or both for cuts on event identification. A -27 V bias is applied to the anodes from a BNC Wire Chamber Bias NIM module. Studies of the charge collection efficiency between the cathode planes and the mylar foils suggested that the cathodes be biased positive relative to ground, +4 V. This biasing scheme has been used during from 29 to present, prior to this the cathodes were left at ground. Only the inner 14 wire groups were instrumented on the cathode plane, as of 28 the radius of the decay trap was increased to 1 cm requiring that the outer most wires be used to cover the entire acceptance. β-scintillators A polyvinyltoluene (PVT) based plastic scintillator EJ24 disk, 3.5 mm thick and 15 cm diameter, was chosen to measure energy of the β s. Twelve edge-coupled adiabatic Lucite light guides transport scintillation photons to four photomultiplier tubes (PMT) located in the fringe of the SCS field. Layers of Mu-metal shielding and bucking coils are used to compensate for residual fields [61]. The thickness of 35

52 Cathode Planes Y-Plane X-Plane Anode Plane Figure 2.9: The configuration of the MWPC wire chamber planes. 36

53 Table 2.1: Thicknesses of the decay trap foils and wire chamber windows are summarized for the data collection. In all cases the foils were Mylar and an additional 3 nm of Be was coated on one side of the decay foils. Year Geometry Decay Trap (µm) MWPC (µm) A B C D.7 6 the plastic scintillator was optimized such that the minimal ionizing peak was close to the endpoint of neutron decay and that exponential tail of the γ background decayed at low ADC channel number, below the region of interest of extraction of the asymmetry. Studies of the detector response performed with the electron gun at Cal Tech [45] found the photo-electron production to be 34 NPE/MeV, greater than three times the light output of the Perkeo II detectors [5]. Subsequent analysis of the performance of the scintillators shows a 2% resolution at 34 kev, the measured 113 Sn peak, and 23% averaged over the β spectrum. Foil Geometry Summary As part of the initial commissioning and subsequent assessment of scattering corrections the foil thickness on the decay trap and MWPC changed several times. Table 2.1 summarizes the foil geometries used between December 27 and January Cosmic Ray Veto Detectors Top and Side Vetos Los Alamos National Lab is at an altitude of roughly 75 ft. increasing the cosmic radiation background by about a factor of ten relative to sea level. An efficient cosmic ray veto for the beta detector packages was therefore required. UCNA uses a composite system of three detectors to cover the top, sides, and ends of the spectrometer. Originally the top and side detectors were one inch thick, 3.65 m long, 1 m 37

54 West Muon Detector Array Gas Counters East Muon Detectro Array Large Area Plastic Scintillator SCS SCS Unistrut Support Gas Counters Figure 2.1: Cosmic veto arrays for the east and west detector are shown schematically from end on point of view. The large area plastic scintillator on the east side is monitored by three PMT s, via three edge-coupled light-guides, which are only mounted on the far end of the scintillator away from the center of the spectrometer. Not depicted on either side is the backing veto, which is a 2.54 cm thick cm diameter plastic scintillator disk coaxial with the SCS bore. wide plastic scintillator paddles coupled to three pmt s via light guides and optical grease. One such paddle is installed above the east side of the spectrometer, but it was determined that these modules were expensive, heavy, and hard to install and reinstall when the detector carts needed to be moved. In a parallel effort the Mu-Rad group at LANSCE developed sealed drift tube detector panels for muon radiography. The size, shape, efficiency, low cost, and ease of installation made them a prime candidate for an alternative to a large area plastic scintillator muon veto system. Each panel consists of eight 2 inch OD (.89 cm wall thickness) 3.65 m long Al tubes filled with a 5/5 Ar and ethane mixture at 8 mbar. End caps are orbitally welded in place and a central feed through of standard.32 Swagelok fittings are used for gas filling and holding the anode wire, a 2 µm Cu wire tensioned with 5 g and biased at 2.1 kv, shown in Figure Tubes are packed side by side in a mounting bracket and the measured efficiency of 97% per panel is equal to the geometric coverage provided by the tubes, implying the inefficiency is due solely to the gaps between tubes and the wall thickness. Adding a second panel of tubes will increase the efficiency to nearly 1%. The construction and testing of the muon drift tubes are detailed by Rios, et al [63]. Five panels mounted on a Unistrut cage, shown in the right panel of Figure 2.11, were used to cover 38

55 the west side of the spectrometer with a few loose tubes added in between panels to cover the gaps. The cage, commonly refer to as the dog house, was designed to pivot about a sliding hinge located near the center of the spectrometer for easy access to the west detector mounting flange. On the east side of the spectrometer one of the previously mentioned scintillator paddles is mounted on a Unistrut support structure above the SCS and two panels of drift tubes are mounted vertically to either side with a small, 6 inch, gap between the top paddle and the tube panels. The solid angle allowed by this gap has been shown to have minimal effect on the efficiency due to both the cos 2 θ dependency of the muon flux and the straight line redundancy of the bottom of the opposite sides drift tube panel, i.e. the muon would have to enter through the gap and have a large angle scatter to miss the opposite side s detector. Signals from the tube arrays are ganged into a single channel for each side extracted via a capacitive coupling (1 nf) high voltage pickoff box and shaped in a time filtering amplifier. A logic pulse generated by the amplified signal is delayed and used as the stop single for a Time-to-Analog Converter (TAC), which is started by the global trigger of β detectors (trigger logic and a compete description of the DAQ will be given in the following section). The analog voltage output of the TAC is read by a peak sensing analog to digital converter (PADC), CAEN V785, showing a peak corresponding to a veto tube β scintillator coincidence. Output from the three PMTs monitoring the east top plastic scintillator are amplified and read out by a CAEN V792 charge integrating ADC (QADC) which is read out during a global trigger event. During a global trigger a logic signal from a leading edge discriminator monitoring the PMT output is read by a time-to-digital converter (TDC) operating in common stop mode, providing a T between an above threshold event in the paddle with the source of the global trigger. Software cuts on the veto PMT signals in the QADC and TDC are used determine a coincidence. A full discussion of the cut efficiencies, and the impact of the cuts on analysis will be discussed in Chapter 4. Backing Veto The horizontal cosmic ray muon flux is strongly suppressed by it characteristic cos 2 θ dependence, where θ is relative to vertical, but it is not negligible; therefore, a 7.75 inch diameter 1 inch thick plastic scintillator backing veto and plastic mounting bracket was designed at N.C. State. Since the backing 39

56 pumps that are backfilled with argon and pumped again. They are leak checked and filled with a 5/5 mixture of argon and ethane gas at 8 mbar (approximately the ambient atmospheric pressure in Los Alamos). Filling is done through a Swagelok tee fitting at one end of the drift tube. Finally, the filling tube is mechanically crimped to seal the gas in the tube. After optimizing the technique the average assembly time for a tube dropped to about 15 min. At this time the dominant cost in building these detectors was the cost of materials. 4. Performance Fig. 2 shows the muon veto system installed around the spectrometer. Amplified signals are 1 mv for cosmic rays. The thresholds are set at 3 mv with the noise levels around 1 mv. Over the four years of operation, no degradation in signal amplitude or increase in noise has been observed. Neutron drift Swagelok fittings PEEK tube Lower fittings BrassTube Fig. 2. Photograph of the muon veto detector covering one side of the UCNA experiment. Fig. 1. Cut-away view of one end of a drift tube muon detector. Figure 2.11: Left Panel: The Swagelok vacuum feed-through and anode wired holder of the muon veto tube. Right Panel: An array of muon veto tube packages on the west UCNA spectrometer. [63] veto was designed after the β-detector carts were built and installed the available space behind the thin scintillator and between its light guides was the major constraint for the size of the backing veto and extracting the light to a PMT. Four 1 mm deep 4 mm wide troughs were cut into the sides of the backing veto scintillator in which scintillating fiber is wrapped in a close packed fashion so that the fiber lays flat in the trough. Saint Gobain & Plastic, Inc. BCF-91A wavelength shifting scintillating fiber was selected to capture the UV light output of the scintillator and shift it to 494 nm, which is near the response peak, 42(5) nm, of the Hamamatsu R555 PMT. A single continuous four meter fiber, the 1/e length of the BCF-91A is 3.5m, was wrapped several times in each trough and epoxied in place with Nye Optical Products Nyogel OCK-451 coupling gel. The eight ends of the fibers were bundled together and fixed in the mount plate and optically coupled using the Nyogel to the photo cathode of the PMT. The scintillator was sealed in an opaque material to prevent light leakage to the main detector scintillator or light guides. In the initial design the PMT would be mounted to the bulkhead just behind the β-detector in the approximately.6 T magnetic field and 1 Torr nitrogen environment. The low pressure nitrogen volume equalize pressure behind the MWPC and to prevent degradation of the bare scintillators and light guide assembly. The R555 was selected for its small size and ability to operate stably in magnetic fields up to one Tesla. Unfortunately the Pashen Curve for nitrogen, which relates the breakdown voltage to gas pressure and gap distance, is at a minimum at 1 Torr and the first test run of the backing veto resulted in arcing and failure. The PMT mounting bracket shown in Figure 2.12 was removed and replaced with a 4

57 Figure 2.12: Schematic of the backing veto mount, scintillator, and PMT assembly is shown as designed. As noted in the text the PMT assembly had to be moved out of the 1 torr N 2 volume, therefore the PMT as pictured is replaced with a flexible light guide to the vacuum flange. bracket to hold a flexible commercially available fiber bundle light guide, commonly used for podium lighting, to transport the light from the scintillating fiber out to a window on the rear flange of the detector housing. A 1 m straight light guide is coupled externally to the window with optical grease on one end and the PMT at the other end. Signals from the PMT, operating at 2.1 kv, are amplified by a Phillips Scientific 776 PMT constant x1 Amplifier and read out by the CAEN V792 QADC. Like the east top veto, the backing veto signals are discriminated and read by the common stop TDC. Software cuts on the ADC and TDC spectra can be used to eliminate cosmic backgrounds and possibly identify high energy exotic events, like Compton generated electrons from the opposite detector. 2.5 Data Acquisition The UCNA collaboration started development on a MIDAS-based data acquisition system with NIM, CAMAC, and VME hardware components in the early 2 s, refinement of both hardware and software continues to the present day [45, 64]. A summary of the hardware modules used for signal logic and analog to digital conversion of energy and timing input is provided in Table 2.2. MIDAS is a modular 41

58 Table 2.2: List of electronics used in the data acquisition system. NIM electronics used by the UCN monitors and muon vetoes are omitted. Module Description VME Units CAEN V792 AA 32-ch., 12-bit, integrating multi-event QADC CAEN V755 AA 32-ch., 12-bit, multi-event TDC, 12ns range 3 x CAEN V ch., 12-bit,multi-event peak sensing PADC CAEN V ch., programmable leading edge discriminator CAEN V83 AA 32-ch., 32-bit, multi-event Scaler CAEN V82 AA 32-ch., 32-bit, Scaler CAEN V486 8-ch., Gate and Delay Generator CAEN V462 2-ch., Gate Generator CAEN V495 Programmable Pulse logic unit (PLU) CAEN V538 8-ch., NIM/ECL level converter SIS11/31 PCI/VME VME controller interface 25 MBytes /s SIS36 32-bit, multi-event input register NIM Units LeCroy 428F Quad Linear Fan in/out LeCroy 429A Quad Logic Fan in/out LeCroy 623B Octal Discriminator LeCroy 365AL 2-ch., 4-input Logic Unit LeCroy 222 Gate Generator P/S 7 NIM/ECL level converter 16-Ch. C/C++ based program which communicates with the VME acquisition cards when triggers are generated and collects and packages the data in PAW or ROOT ntuples and histograms for off-line analysis [65, 66] Trigger Generation Event records are generated from all ADC, TDC, and scaler VME modules when the global trigger condition is satisfied in the absence of the BUSY LOGIC being true. Signals from the PMT s are split by a LeCroy 428F linear fan in fan out module with the outputs routed to the CAEN V792 AA charge integrating QADC and a LeCroy 612AM x1 PMT amplifier. Logic signals are generated by leading edge discrimination of the amplified PMT signals by a CAEN V895; the threshold for each each is set at the two photo-electronic level. If two of the PMT s on either side pass the leading edge discrimination a main trigger is generated by the CAEN V495 pulse logic unit (PLU). Another PLU OR s together all 42

59 PMT Linear FiFo QADC 1x Amp Disc. Scaler PLU 2-Fold UCN Monitors GMS Triggers PLU 2-Fold From Other Detector Busy Logic Main trigger TDC Read Gates to Adc s and Tdc s Gate Generator Figure 2.13: PMT signals are routed through a Linear FiFo into a CAEN V792 QADC and a leading edge discriminator logic circuit for global trigger generation. The dashed lines correspond to the PMT from the opposite side of the spectrometer. 2-Fold triggers from both detector packages are OR ed together with triggers from UCN monitors and the GMS to create the main trigger. Busy logic signals, created during through the conversion time of the VME ADC and TDC modules, can veto any main trigger to prevent re-triggering. possible sources of main trigger generation from the β -scintillators, UCN monitors, and the GMS system and generates a global trigger. On a global trigger event records on event VME module are created. To prevent retriggering during the conversion time of the ADC and TDC module a busy logic circuit combines the busy output of all modules and vetoes any triggers generated [45]. The source of trigger generation is recorded by the SIS36 for offline event identification. Common Stop TDC The timing difference from a global trigger and any subsequent triggers are measured by a CAEN V channel TDC operating in common stop mode. In common stop mode each channel of the TDC is started independently and they are all stopped by a common signal. By delaying the global trigger and using it as the common stop signal, the timing difference between the global trigger and any other trigger can be determined up to the delay length. Timing distributions generated in this way will appear reversed with the longer T s at smaller TDC values and the short T s close to the self-timing peak. The self 43

60 Individual Pmt Triggers Individual Channel Triggers Trigger Logic Unit TDC Channel Start Stop 1 ns Delay Global Trigger Out Figure 2.14: The TDC s common stop timing works by passing a copy of all individual triggers to start the timer for that channel, assuming the global trigger criterion has been met for the DAQ. A copy of the global trigger is delayed by approximately 1 ns, using cable delay, and is connected to the common stop input. This stops the timer for every channel 1 ns after the generation of a global trigger. Using the TDC in this manner produces a spectrum which is opposite to expectation, triggers generated closer in time to the global trigger will have larger TDC values while ones with longer delays will occur closer to the delayed stop and have smaller TDC values. timing peak appears at TDC values corresponding to the delay time of the global trigger, since copies of all the triggers are sent to start the TDC and to the global trigger logic, the channel that triggered the DAQ can start and stop its own TDC channel. Because the main use of the TDC is to reconstruct the initial direction for events where both beta-scintillators are triggered, delaying much longer 1 ns (note the minimum transit time for a beta-generated coincidence event is about 14 ns) would not be of much use. Setting the delay to 1 ns allows for 97.5% of these events to be properly identified. 44

61 2.5.2 Run Control Operation of the DAQ, gate valve, switcher valve, and spin flipper were automated by a LabVIEW program developed by R. Rios for the UCNA experiment. Via a National Instruments BNC 211 controller, TTL logic pulses were used to change the state of the gate valve, switcher valve, and spin flipper state for the signal, depolarization and background measurements. After November of 28 the LabVIEW program was also able to start and stop MIDAS-DAQ runs in-line with the cycling of the valves. 45

62 Chapter 3 General Systematics of an Asymmetry Measurement 3.1 Introduction The ultimate precision which will be obtained by UCNA will be determined by understanding and the ability to reduce all relevant systematic uncertainties inherent in the experiment. Most sources of bias, such as cut efficiency, detector linearity, and collimator effects, can be directly measured or at the least upper limits can be empirically placed on the size of the bias they induce. An analytical framework can be constructed to include the effect of many of these systematics in a calculation of the asymmetry. Others must be estimated using Monte Carlo simulations of the electron scattering in the spectrometer or neutron transport through the guide system. Bias due to charged particle scattering will be assessed using the PENELOPE3 (v23) simulation package for geometries relevant to the run period as well as a few addition geometries of interest. Observables upon which to benchmark these Monte Carlo estimates must be identified to determine the validity of the scattering corrections and size of their uncertainty. 46

63 3.2 Neutron β Decay Physics Generator Derivation i To correctly model the systematic effects of interest, an accurate neutron event generator was needed which could reproduce the energy and angular distributions of all the decay products. Starting from the decay probability of the free neutron d 6 [ Γ( p e, p νe ) d 3 p e d 3 = C 1 F(Z,E) p νe 1 + a p e p νe + b m e + σ E e E νe E e ( A p e + B p ν e E e E νe )] δ 4( ) p n p e p νe (3.1) where C 1 is a constant, a, b, A, B are the β-decay angular correlation coefficients, F(Z,E) is the Fermi function, and p i (i = e, ν e ) are the electron and anti-neutrino momenta. The T-odd triple correlation parameter D has been omitted, since it is zero in the standard model and not an observable in the scope of the UCNA experiment. The correlation coefficients can be represented to leading order in terms of the ratio of the axial-vector and vector coupling constants, λ = G A /G V 1.271(25): a = 1 λ 2.13(4) 1 + 3λ 2 (3.2a) 2λ(1 + λ) A = 1 + 3λ (11) (3.2b) λ(λ 1) B =.987(3) 1 + 3λ 2 (3.2c) and the Fierz interference term, b, depends on the existence of tensor or scalar contributions to the weak interaction and is therefore zero in the SM will be discussed in a later section and in Chapter 6, values for these parameters are taken from the PDG [26]. The Fermi function F(E e,z), which distorts the electron energy spectrum due to Coulomb effects on final state wave function of the electron, can be represented as the ratio of perturbed to unperturbed wave functions Ψ e (r)/ψ e (r) 2 for a point nucleus, Ψ e (r) 2 Ψ e (r) = 2(1 + κ )(2p e r) 2(1 κ) e πn Γ(κ + in) 2 Γ(1 + 2κ ) 2, (3.3) 47

64 F NR E,Z F E,Z Figure 3.1: The difference between the non-relativistic expression and the expansion of Eq. 3.4 e where n = ±Ze 2 /v e with the ± depending on whether it is β ± -decay, Γ(x) is the standard Gamma function, and κ = (1 Z 2 α 2 ) 1/2. An approximation of Eq. 3.3 can be made to consider a finite sized nucleus of radius R, F(E e,z) = n= a n (αz) n, (3.4) where this expansion is taken to third order by Wilkinson [67] a = 1, a 1 = πe/p, a 2 = 11 4 γ E ln(2pr) π2 (E/p) 2, a 3 = π(e/p)[ 11 4 γ E ln(2pr)], (3.5) where γ E is Euler s constant At electron energies above 1 kev the error between this expansion and a direct calculation of the Dirac equation, which takes realistic charge distributions and nuclear screening effects into account, is below.1%. However, below 1 kev the error quickly increases to 3%. We can exploit the fact that the non-relativistic solution to this scenario is known to estimate the 48

65 Fermi function at low energy, F NR (E e,z) = 2πE e α, (3.6) p e (e 2πEeα pe 1) where α is the fine structure constant [67]. For the analysis of the electron spectrum and event generation in the simulation the expansion of F(E e,z) will used. Using conservation of momentum the neutrino variables can be replaced with proton momentum variables, [ ( dγ = C 1 F(Z,E) 1 a p2 e + p e p p cosθ ep E e (m n E e E p ) + bm e + σ A p )] e p e + p p B E e E e m n E e E p d 3 p e d 3 p p δ ( m n E e E p p e + p p ), (3.7) where we have used p νe = ( p e + p p ) and E νe = p e + p p. Evaluating the remaining delta function by integration over the magnitude of the proton s momentum we find p ± p (p e,θ ep ) = µ ± S 2α (3.8) where we have following kinematic relations µ = 4 p e cosθ ep (d 2 + m 2 p p 2 e) d = m n E e α = 4(d 2 p 2 2cos 2 θ ep ) S = µ 2 4αh h = 4(m 2 n m 2 p)(e e E ecrit )(H E e ) H = 1 2 (m p + m n + m2 e m n +m p ) E ecrit = 1 2 (m n m p + m2 e m n m p ) and the angle between the electron and proton momentum is defined as cosθ ep = sinθ e sinθ p cos(φ e φ p ) cosθ e cosθ p (3.9) θ e,θ p are measured from the spin of the neutron and φ is the azimuthal angle, which is in agreement with [68]. Since the proton momentum is determined by solving a quadratic we have two roots which 49

66 will contribute to the distribution. Summing over the roots δ(p p x i ) δ( f (p p )) = i f (x i ) f (p p ) = m n p 2 p + m 2 p E e p p + p e f (p p ) = p p E p p p + p e cosθ ep m m E e p 2 p + m 2 p (3.1) where x i are the roots of p p. Using the above substitutions for the proton energy and changing d 3 p p = p 2 pd p p dω p we get the distribution function in terms of the electron energy and the momentum directions of both the proton and the electron. W( p e, p p ) = 1 a p2 e + p e p ± p cosθ ep E e (m n E e E p ± ) + bm e + A p e cosθ e B p e cosθ e + p ± p cosθ p E e E e m n E e E p ± (3.11) ( W( pe d 5, p p )p +2 p δ o (p p f + ) Γ(E e,ω e,ω p ) = C 1 F(Z,E) f + + W( p e, p p )p 2 p δ o (p p f ) )E f e Ee 2 m 2 e de e dω p dω e (3.12) While we now have the distribution in the form we sought, neither root is globally invalid and some care must be taken when dealing with these roots and the effect on the distribution function. If the electron is emitted with E e < E ecrit 235keV the minus root falls outside the limits of integration and may be neglected. The majority of the decays result with an electron being emitted with energy greater than this threshold resulting in both roots corresponding to kinematically valid proton momentum. Throwing a random number into the relative weights of plus/minus term in the distribution function we select the proton energy for that event. For a given electron energy, E e > E ecrit, and θ ep there will be two values of p p which will be valid and both will contribute a term to the distribution function. The weight of each 5

67 term, given by: W ± W(p e, p ± p ) = W(p e, p + p ) +W(p e, p p ) (3.13) will define regions ( : W + : W + +W = 1). Randomly generating a number between :1 will select either root and determine the proton energy for this event. Finally the standard Monte Carlo rejection method is applied which results in the keeping these values or selecting new values for E e,ω e,ω p Verification of Decay Observables Integration over selected subsets of variables allows one to construct an expression for experimentally observable quantities, such as angular correlations and asymmetries with respect to the spin axis of the parent neutron. With this generalized distribution function observables can generated by setting logical flags to determine the if the neutron population is polarized, the degree of polarization, the measured decay products, and the value of λ Electron Energy Spectrum The shape of the electron spectra has a slight dependence on the Fierz interference term, b. The proton kinetic energy amounts to a 1 7 correction to the maximum available energy and will be neglected for now. Recoil order corrections will be hidden in the definition of the Fermi function to account for the nonzero energy of the proton, w(e e ) = F(Z,E e )p e E e (m n m p E e ) 2, (3.14) dγ(e e ) = w(e e )(1 + b m e E e )de. (3.15) In the standard model b is effectively zero thus any deviation of Eq from Eq will be due in part to a nonzero Fierz term, which corresponds to nonzero scalar or tensor couplings in the weak interaction 51

68 from Severijns review [28] we have the approximate expression, b γ 1 + ρ 2 [Re(C S +C S ) + ρ 2 Re( C T +C T )] C V C A (3.16a) γ = 1 α 2 Z 2 (3.16b) ρ = C AM GT C V M F (3.16c) where M GT and M F are the Gamow-Teller and Fermi matrix elements and C A, C V, C S, C S, C T, and C T are the axial, vector, scalar, and tensor coupling constants. Analysis by Hardy and Towner [31] give limits of.44 < C S +C S C V <.44, (3.17) for ratio of the scalar to vector couplings and measurements of 17 In [28] are used to set limits on the ratio of the tensor to axial-vector couplings of :.34 < C T +C T C A <.5. (3.18) Using these limits and the SM value of ρ = 3λ we find.2616 < b <.484 at the 9.% confidence level. We see from Figure 3.2 the difference of the electron energy spectrum from the SM with b as it varies within the stated limits. The discrepancy is < therefore only high precision measurements of the electron energy will have the ability to detect a variation due the Fierz term from the spectral measure alone, if it is non-zero. This also shows that the sensitivity to the Fierz is a factor of 2 greater at low energies. Since a non-zero b alters the total energy spectrum it will also modify the measured β-asymmetry as A m = A /(1 + b ), where b is the average value of bm e /E e over the analysis energy window, (See Chapter 6 for a full discussion). A possible b-term could have as large as a.3% impact on the measured asymmetry, assuming b = ±.4. We would expect b to much smaller than the maximal limits stated and the effect on the asymmetry is <.1%. 52

69 Β b e A b e Figure 3.2: (Left).1% contours of the changed in the energy spectrum, 1 Γ(E e,b)/γ(e e,), due to the existence of a Fierz term. (Right) The change in the energy spectrum is propagated through to a calculated of the β -asymmetry. The contours shown are for: b =.2 dotted-dashed line, b =.1 dotted line, b =.1 dashed line, and b =.2 solid line Proton Energy Spectrum Using the formulation from Glück [68, 69] to express the proton energy spectrum we have, w p (E p ) = m n G 2 wξ 4π 3 [W P(E emax,e p ) W P (E emin,e p )], (3.19a) W P (E e,e p ) = 1 2 (1 + a)e2 e (m n m p 2 3 E e) am n E e (E pmax E p ) + bm e E e (m n m p 1 2 E e), (3.19b) where E pmax is the maximum proton kinetic energy ( 752 ev) and E emin/max is the maximum and minimum kinematically allow electron energy for a given proton momentum, E pmax = (m n m p ) 2 m 2 e 2m n, (3.2a) E emin/max = 1 2 [m n m p ± p p + m 2 e m n m p ± p p ]. (3.2b) 53

70 Dalitz Plot for Electron and Proton energies Normalized Probability (a) Proton Energy Spectra. Electron Kinetic Energy MeV Proton Kinetic Energy MeV (b) Proton Electron Dalitz Plot Figure 3.3: Panel (a) shows the proton energy spectra corresponding to; (red) λ =, (green) λ = 1.267, and (blue) λ = 1 generated via Monte Carlo rejection methods using Eq and compared to the analytical expression Eq. 3.19b. Panel (b) shows the Dalitz plot of the proton and electron energy defining the allow range of energy with respect to each other. The shape of the spectrum contains a terms which are dependent on the electron-neutrino correlation parameter, a, and the Fierz interference parameter, b. Comparing the variation of Eq. 3.19b with the prediction from the event generator for different values of λ will verify that the dependence on the correlation parameter is captured by the simulation, shown in Figure 3.3. Additionally, the energy and proton energy can be tallied for each event to show that correlation between their energies obeys the limits set by Eq. 3.2b Angular Correlations There are numerous correlations and asymmetries that can be defined as shown in [68], of primary importance to us will be the beta- asymmetry, proton-asymmetry, electron-neutrino correlation, and the ratio of the proton and electron asymmetries. The electron and proton asymmetries will be defined as the difference of the number of particles emitted with ˆp ˆσ > and ˆp ˆσ <, where p is the particle momentum and σ is the neutron spin vector, α = N N N + N (3.21) 54

71 For these asymmetry measurements the neutron spin will be fixed to the z-axis and the momenta of the decay products will be measured relative to that axis. β Asymmetry, A Electrons emitted have an energy dependent angular distribution with respect to the neutron s spin, d 2 Γ(E e,θ e ) de e dcosθ e = 1 + A P βcosθ (3.22) where P is the average polarization of the neutrons, and A is the β asymmetry coefficient. Corrections to the β-asymmetry due to Coulomb interactions, weak magnetism, and G V G A interference are outlined by Wilkinson [67] and change the raw measured value of A.1173, A = A{1 + A µm (A 1 Q + A 2 E e + A 3 /E e )}, (3.23) where Q = m n m p and the terms in the expansion are A µm = λ + µ m n λ(1 λ)(1 + 3λ 2 ), A 1 = λ λ 1 3, A 2 = λ 3 3λ λ + 1 3, A 3 = 2λ 2 (1 λ), (3.24) where µ = µ p µ n is the difference in the magnetic moments of the proton and neutron. Extracting the asymmetry from the Monte Carlo data is done by fitting A(E e ) = 1 2 A β e (3.25) and using A from equation 3.23 as the fit parameter. Since this Monte Carlo is ideal in that; we know precisely the energy of the decay products we can fit over the entire energy range. We will see in a 55

72 Thrown β-asymmetry Entries χ / ndf / 76 A.1179 ±.1 Asymmetry Proton Asymmetry hep Entries 13 Mean RMS Initial Energy [kev] (a) Beta asymmetry Energy (kev) (b) Proton Asymmetry Figure 3.4: Monte Carlo generated asymmetries for the electron and proton in terms of the electron kinetic energy. The asymmetry is define as difference over the sum or the counts rates parallel versus anti-parallel to the neutron spin. later discussion that if the energy resolution is imperfect or that the noise or statistics dominate the low or high energy ranges the fit must be restricted to a smaller energy range. The results of PERKEO II were obtained by applying a similar fit in the 2-7 kev range. Another experimental concern is the acceptance of the detector geometry, if there is a restriction in the angular distribution the factor of 1 2 in equation 3.25 will be altered since the integration over the positive and negative hemisphere is not complete. Proton Asymmetry Analogous to the electron asymmetry one can also measure the rate that protons are emitted relative to the neutron spin. A simple discussion of the proton asymmetry can be found in [68] while a more analytical description is presented in [69]. The ratio of the proton and electron asymmetries gives, to leading order, a polarization independent way in which to extract λ [7, 71]. 56

73 3.3 Energy Calibration The detector calibration in UCNA is a multi-step process in which the first operation is to remove temporal and spatial gain fluctuations from the PMT s before using simulated and measured calibration points to translate ADC values into energies. Since the mechanisms for controlling or correcting hardware gain changed from year to year, the discussion of this step in the calibration will be left to later chapters which specifically address the 27 and analyses. Here we begin by assuming the ADC values have been converted to what we have termed the visible energy, E vis, a quantity defined by the Monte Carlo which is proportional to the number of photoelectrons produced in the scintillator PMTs, which is assumed to be proportional to the energy deposition in the plastic scintillator after quenching and deadlayer effects have been taken into account. The reason for not calibrating the detectors to the true average energy of line conversion sources like 113 Sn or 27 Bi is that we will eventually need to correlate energy dependent scattering corrections predicted by the simulation to the measured asymmetry, thus we need a compatible energy scale between the two. In the following sections a discussion of how the simulated quenching was calibrated and how possible sources of calibration uncertainty propagates to the final systematic uncertainty of the asymmetry will be presented Calibration to the Kellog Electron Gun Light output along the path of an electron in the scintillator is related to the energy loss by Birks law dl dx = S de dx de 1 + k B dx (3.26) where the SdE/dx gives the number of excited molecules per unit path length, k B de/dx is the number of damaged molecules along the path length multiplied by the probability that a damage molecule will capture an exciton quenching the fluorescence, and dl/dx is the differential light output per unit path length [72]. For high energy electrons de/dx is small and Eq reduces to dl/dx S de/dx, in this regime the light output is linear with energy loss. For low energy electrons or heavier particles, like alpha particles, the energy loss density is much larger, increasing the density of damaged molecules which do 57

74 not fluoresce and in turn quenches the total light output [73]. The total light generated by an event must propagate out of the scintillator via the edge-coupled bent light guides to the photo-cathodes where a number of photo electrons (NPE) are created proportional to the incident number of photons and the quantum efficiency of the PMT. Therefore, a realistic simulation must use Birks law to translate the raw energy deposition in the scintillator to the NPE recorded by the detector electronics. S, k B, and the dead layer were measured using the Kellog electron gun over an energy range of 2-13 kev [45]. The measurement of the Suzuno detector response was performed with a pencil beam of electrons normally incident on the plastic scintillator. Because of the light guide geometry of the Suzuno detector the scintillator could not be directly coupled to the vacuum system. The vacuum of the electron gun was sealed with a 6 µm thick mylar foil with the scintillator positioned 1.2 cm away. Eq is numerically integrated along the path of each electron to determine the number of PE generated, with the value of de/dx extracted from a fit to the NIST ESTAR database [74] using the functional form ae b, a = 117(3), b = 729(5), S =.425, and k B =.2 ( their measured values). The deadlayer thickness was included in a fit to the light production as a function of the energy of incident electrons, L(E), by transforming Eq to E E SdE L(E) =, (3.27) de 1 + k B dx where the upper limit is reduced by E due to the dead layer. Using the ae b form of de/dx and integrating to a deadlayer thickness of t the upper limit of Eq becomes E E = ( E 1 b (1 b)at ) 1/(1 b). (3.28) Combining this result with Eq the results from the electron gun test were fit to determine the dead layer thickness and quenching factor. The above quoted values of S and k B and a dead layer thickness, t = 3 µm were found in this manner in the work of [45]. Fig. 3.5 shows that calculating the number of PE from the energy loss, using the measured value 36 NPE/MeV, the number of PE is over predicted at low energies and will be under predicted at higher energies. After increasing S by 24%, Eq was 58

75 NPE Calculated by Birks Law 4 NPE Calculated by Birks Law NPE Calculated from de/dx 36 NPE/MeV Measured NPE vs E 3 NPE Beam Energy (kev) Figure 3.5: The photo-electron calibration as measured using the electron gun at Cal Tech shown in green is plotted against simulation output where the photo-electrons are estimated by converting the energy deposition to NPE via a simple calibration constant (blue) and integrating Eq over the track in the simulation (red). fit to the data with S and k B as free parameters and the resulting agreement between the fit parameters extracted from the Kellog data and the PENELOPE3 simulation was.995% Calibration and Linearity Differences We will now analyze the effect of calibration errors on the extracted asymmetry. We can assume that the calibrated energy spectrum differs slightly from theoretical or Monte Carlo expectation therefore we can describe this difference in terms of an expansion in the kinetic energy, E, E = n i= α i E i, (3.29) in the case of a simple offset α 1 = 1 and α = E. A more likely case would be for α and α 1 1 and we ll assume for now that the difference does not have quadratic or higher dependence for simplicity. It is also physically reasonable to assume that the offset and gain mismatch is different for each PMT. For the moment we ignore the detection efficiency and point response function and only consider the 59

76 calibrated spectrum of detectors 1 and 2 : dγ 1(2) (E ) = F(E i)p(e i)(e i + m e )(Q E i) 2( 1 ± 1 2 Aβ(E i) ) de = W i (E ) ( 1 ± 1 2 Aβ i(e ) ) de (3.3) where W i (E ) is the shape of the calibrated spectrum for detector i {1,2} integrated over a hemisphere. Calculating the difference over sum asymmetry, any offset or scaling in the calibration comes in at first order ( ) W 1 (E ) W 2 (E ) + A/2 W 1 (E )β 1 (E ) +W 2 (E )β 2 (E ) A m = ( ) (3.31) W 1 (E ) +W 2 (E ) + A/2 W 1 (E )β 1 (E ) W 2 (E )β 2 (E ) and the resulting effect on the asymmetry is shown in Figure 3.6b. In this analytical model one detector is assumed to have perfect calibration to the true physics distribution and the gain or energy offset of detector two is altered. A modest gain mismatch of 1-5% leads to an integrated difference in the spectrum of the same order however this propagates directly through a simple calculation of the asymmetry leading to much larger corrections especially in the analysis window of.2.6 MeV. Through either pedestal subtraction or low energy non-linearities an energy offset can enter the calibration, where we can assume that any offset on the order of the bin width, 25-5 kev for the measurements reported here, would be noticed and corrected prior to asymmetry analysis. Therefore we ll restrict the size of the offset to the maximum of half of the bin width, -2 kev. The effect on the spectrum due to an offset is minimal (<.1%) at low energies (E < 3 kev), but rapidly divergences at higher energy, see Figure 3.6(a). Since an offset does not cancel in ratio of the rates, the < 1% bias in the spectrum translates to 1.5-2% bias on the asymmetry, which is much larger than the bias due to 5% difference in the linear term. As one would expect if the sign of the offset or difference from unity gain was reversed the energy dependence of these corrections follows suit. In the end we see that the simple asymmetry, provides no protection from the bias due to calibration differences, in fact these differences are enhanced. The results of this analytical model were checked by applying these calibration parameters to the energy of events in a Monte Carlo of the experiment, the black circles overlaid on Figure 3.6(b). 6

77 E e A Α, Α1.95 Α, Α1.99 Α 1keV, Α1 1 Α 2keV, Α1 1 Α 1keV, Α1.95 Α 2keV, Α e (a) Energy spectrum difference (b) Difference in the simple asymmetry. Figure 3.6: The fractional difference in the energy spectra of detector 1 and 2 and the deviation from the ideal asymmetry is shown in panels a and b, respectively. The offset parameter is set at 1 and 2 kev and the linear scaling parameter is 99% and 99.9%. Black circles in panel b are from analysis of a Monte Carlo with a 1 kev energy offset between detectors. The legend of panel b corresponds to both plots. If the asymmetry is formulated using the super-ratio, these calibration differences become manageable with Eq becoming S = W 1(E ) ( Aβ 1(E ) ) W 2 (E ) ( Aβ 2(E ) ) W 1 (E ) ( Aβ 1(E ) ) W 2 (E ) ( Aβ 2(E ) ) ( = Aβ 1(E ) )( Aβ 2(E ) ) ( Aβ 1(E ) )( Aβ 2(E ) ) (3.32) A m = 1 S 1 + S (3.33) the residual differences come in through the energy dependence the β weighting to the asymmetry. Since β = p e /E e, any effect due to linear scaling of the energy is greatly suppressed in this ratio, Figure 3.7 shows that even a 5% gain mismatch has a maximum A of 1.2% with A.6%. Even the corrections due to offsets are suppressed by roughly a factor of 1 relative to the sum/difference method to A < 1.5% for α = 2 kev. The power of this method however, lies in the fact differences in shape of the β -spectrum are suppressed which in turn removes the divergence at higher energy that plagues the simple asymmetry method. From this analysis we should expect uncertainties or corrections below 1% due to calibration assuming we can restrict any relative offsets or a gain mismatch to the sizes presented here. 61

78 A Α, Α1.99 Α kev, Α1.95 Α 1keV, Α1 1 Α 2keV, Α1 1 Α 1keV, Α1.97 Α 1keV, Α1.97 Α 2keV,Α e Figure 3.7: The fractional difference in the extracted asymmetry from Eq for similar gain differences and energy offsets used in analysis of the simple asymmetry. Note the when the gain is reduced, α 1 <, and the constant offset is positive, α > the effect on the asymmetry is reduced Finite Detector Resolution Scintillator detectors are known to have energy resolution between 1 2% due to the statistical nature of number of scintillation photons reaching the photo-cathode, photo-electron production by these photons, and the electrons produced in the gain stages of the PMT-base. Since the resolution of each PMT in the detector package is related to the light transport to the photo-cathode and the performance of the PMT-base, it is reasonable to assume that the resolution of each detector package will not be identical. This effect can be simulated by convolving Eq with a Gaussian, R(E,H) = e (E H)2 /(2σ(E) 2 ) 2πσ(E) (3.34) where H is the measured signal for an event with true energy E, experimentally this would be an ADC value that is then calibrated to the appropriate energy scale, and σ(e) = ρe where ρ can be set by fitting the simulated calibration source spectra to the data, typically a ρ of 2.5 gives good agreement 62

79 at the 113 Sn energy of 364 kev. An analytical model was again used to investigated how this would propagate through the energy spectrum to a determination of the asymmetry. To include realistic detector behavior Eq. 3.3 is convoluted with a Gaussian and trigger efficiency function which is estimated as η(e) = tanh(a(e b)) (3.35) where a scales the slope of the efficiency s rise and b sets the energy where the efficiency is 5% nominally between 5 15 kev for the geometries of interest. In reality the trigger and point response function are dependent on angle and position, η(e,θ,x,y) and R(E,H,θ,x,y), but for the sake of simplicity we will assume that these are separable dependences which may be addressed as needed. Initially we will set a =.4 and b = 1 kev which is a good approximation of the trigger function from both simulation and data, these parameters will be adjusted to match the specific geometries as needed or efficiency histograms can be directly generated by simulation. In Figure 3.8(a), the point response function is folded to with Eq. 3.35, shown as the dotted black line and scaled to the peak of the β-spectrum, to produce a few representative line source spectra for E (15,4,8) and with Eq. 3.3, the dashed black curve, to show the expected true energy distribution of detected events. A few additional details are added to the simple Gaussian resolution function, Eq. 3.34, to more accurately model the detector: first a constant multiple scattering and backscatter tail is included via a 1 (tanh ( ) ) (E H)a E + m (3.36) where a 1 sets the size of the tail, a 2 determine how quickly the tail goes to zero as H E, and the inverse energy dependence suppresses the contribution of the tail at higher energy. Next an energy loss term can be used to modify H in Eq and Eq. 3.36, and for now we ll assume the energy loss is related to the inverse of the initial energy by some constant. From simulations of a typical UCNA geometry, the mean energy loss for 113 Sn is 25 kev. The effect of adding both Eq and an energy loss term to the response function is shown by the dashed blue curve in Figure 3.8 for 4 kev initial energy. Based on this model of the detectors, instrumental asymmetries can enter through differences in the trigger 63

80 function, resolution, and size of the multiple-scattering tail. For now we will assume that the detectors are identical and that the physical construction of the detector determines, to leading order, the size of tail, so that the constants in Eq are the same for both detectors. By a similar argument, the energy loss term is also roughly the same between detectors. This leaves the production and transport of scintillation light and the performance of the PMT s to be varied, both of which strongly influence the trigger probability and resolution. With this model we can first investigate the bias due only to resolution differences between detector 1 and 2, where σ 1 and σ 2 are the widths of the Gaussian smearing functions, the detection efficiency is set to 1 and the multiple scattering tail is turned off. Varying the ratio of σ 1 /σ 2 from.6 to 1 we can see in Figure 3.8(b) that for all cases where the this ratio is not unity the asymmetry is greatly increased at E < 1 kev and E > 65 kev. When the resolutions are equal (red line) or when the super-ratio formulation is used to calculate the asymmetry (black dashed line) this trend is reversed, decreased at E < 1 kev and slightly decreased at higher energies. As with the analysis of the calibration differences using the super-ratio suppresses the effect of varying resolutions between the detectors as shown in Figure 3.8(b) (the dashed black curve corresponds to σ 1 /σ 2 =.6). A reasonable expectation is that the variation in the resolution is below 1% for the 27 data which gives.5% bias in the asymmetry in the energy region of interest, E (2 6) kev. When this analysis is carried out on a flat, energy independent spectrum without the β weighting on the asymmetry term, the detector resolution introduces no bias to the observed asymmetry. Since the smearing of the spectrum happens in a smooth and continuous manner throughout the spectrum, there is an equal weighting to events leaving or entering a bin due to the convolution. Even with the energy dependence of the decay distribution included, there is no bias to the asymmetry if we assume the asymmetry itself is energy independent. However, when the v/c weighting is included, a small bias of.1% is induced because the convoluted energy dependence of the asymmetry does not match the expectation from theory. For energies below the peak of the β-spectrum a larger number of events are scattered to bins where they have a larger than expect v/c weighting, thereby increasing the observed asymmetry. At energies greater than the peak the larger v/c weighted part of the distribution is suppressed 64

81 by the higher statistics of events at smaller energies, which decreases the asymmetry. At low energy the rapid variation of v/c and the energy spectrum leads to a much large bias in the asymmetry than the high energy behavior. For all reasonable values of the width, ρ < 1., the induced bias is <.1%. Including a detector efficiency which is 5% at 1 kev, a tail which is normalized such that the peak to tail area ratio is 5%, and simplistic energy loss term allows us to be one step closer to analyzing a realistic detector. From this analysis we see that if the trigger function is identical the low energy, E < 2 kev, behavior of the asymmetry bias is dominated by the inefficiency and above that is unchanged. If, however, the 5% energy is varied with resolution held constant the bias toward a increased asymmetry falls more rapidly as the energy falls below 25 kev with the high energy bias unaffected. In both cases the super-ratio formulation highly suppresses these effects. From this simplified picture we can assume that the trigger function will, in part, set the lower limit of the energy window and that a small bias (<1%) above 625 kev from detector resolution is unavoidable. Geometric factors such as foil thickness will dominate where the detection efficiency reaches unity, which we will see in the analysis of the asymmetry data. Time Dependent Gain Drifts Operating conditions in the experimental area can vary due to the heat load on the electronic components or the temperature of the hall due to day/night cycling, both can effect the performance of the scintillatorlight guide-pmt system. To first order, gain drifts are eliminated by periodic calibration to sources or an internal gain monitoring system. Details of the operation and corrections from the gain monitoring systems (GMS) are left to Chapters 4 and 5 where the data collect are discussed relative to the GMS used during that run cycle Uncertainty in Calibration due to Finite Line Sources There are, unfortunately, a limited number of line sources with which to calibrated the detector in the energy range of interest. Additionally, these sources are not truly monoenergetic, in that they are internal conversion sources where the outgoing electrons can come from K, L, or, M shells which have differing 65

82 Arb. Units Η E R ij 15,H Η E R ij 4,H Η E R ij 4 de,h Η E R ij 8,H E Η E Η E E A 1 A m A o Σ 1 3. Σ Σ Σ Σ 1 1. Σ 1.1 Super Ratio Σ (a) Example β-spectrum with a point response function for several energies. (b) Bias in the asymmetry due to resolution. Figure 3.8: Panel (a) shows several example response functions for E true of 15(red line),4(blue solid line), and 8 (green) kev after convolution with Eq and Eq. 3.36, along with the detection efficiency (dotted black line), and the theoretical β-spectrum (dashed black line). The solid black curve represents the application of detection efficiency to the theoretical β -spectrum and the dashed blue curve is the result of adding an energy loss term to Eq Panel (b) shows the bias to the simple asymmetry in several scenarios of detector resolutions. The dashed black line is the expected bias for σ 1 = 2.5 if the super-ratio formalism is used A 1 Am Ao Σ 1 Σ 2.6 Σ 1 Σ 2.68 Σ 1 Σ 2.8 Σ 1 Σ 2.92 Σ 1 Σ 2.96 Σ 1 Σ E E 2 E 15 E 1 E 5 E Visible Energy kev (a) Bias in the asymmetry due to differences in resolution. (b) Bias in the asymmetry from difference in detection efficiency. Figure 3.9: Bias to the asymmetry is shown for several scenarios of differences in detector resolution and the energy that detector efficiency function is 5% in left and right panels respectively. The black line in both plots are for the worst case using the super-ratio formalism. 66

83 Table 3.1: Energy and relative intensities for conversion and Auger electrons from the calibration sources used in UCNA, with relative intensities within 1% of the most intensive line, are listed below. Energies are in units of kev. All conversion electrons except 27 Bi result from a single transition. The 56.7 kev line from 27 Bi is Auger that is visible in geometries with thinner foils and the lines from kev and kev are from separate γ emissions in the transitions of 27 Pd 7/2 to 27 Pd 1/2 after electron capture on the parent nucleus. This can result in multiple conversion electrons being emitted simultaneously and is accounted for in the event generator in simulations. 113 Sn 27 Bi 139 Ce 85 Sr 19 Cd (28.8%) 56.7 (2.9%) (17.15%) (.6%) (5.6%) (1.53%) (2.3%) (.68%) (1.14%) (.44%) (.48%) (.21%) (.11%) (7.8%) (1.84%) (.44%) binding energies. In the analysis that follows there are two main uses of measured sources: simple point calibration for ADC vs. E vis and determination of the detector response function for convolution analysis. In both cases the results obtained from the available sources must be extrapolated over the entire energy range, 1 MeV. UCNA has made use of 113 Sn, 139 Ce, 19 Cd, 27 Bi, and 85 Sr as calibration sources, where the energies and relative intensities of conversion electrons are listed in Table 3.1. A discussion of the specific use of these sources to calibrate the data is left to Chapter 4 and 5, here we will investigate the uncertainty introduced by the extrapolation process when the full set of sources or fewer are used for calibration. Point Source Calibration Let us assume that there are sufficient sources to have a calibration point every.1q, where Q is the full energy range in question. For this analysis the calibration range is -1 MeV, so the source energies are spaced every 1 kev. If the detection system has a non-linear response given by f ADC (E) = 2 i= α ie i, where we will set the offset parameter α = 1, the linear scaling to 1 ADC channel /kev, and the quadratic parameter will be allowed to vary, α 2 (.1,.2). Using this detector response, a set of 67

84 Gaussian calibration points can be simulated with a 1% uncertainty on the mean and σ = 2.5E. The question becomes: how well will a nonlinear term be characterized by a linear fit to the available calibration sources and how much will it the bias the asymmetry? And how does the energies of the calibration sources used affect this? First we assess how poorly the quadratic response of the detector is captured by a linear fit to ten sources and a collection of three source sets: 1. 2, 6, and 9 kev 2. 2, 3, and 9 kev 3. 2, 3, and 4 kev 4. 8, 9, and 1 kev. The source energies were selected to represent scenarios where 1- source energies span the energy range and are regularly spaced, 2- the energy range is spanned, but two of the sources are very close, 3- and 4- the sources do not span the range and are all at one end of the spectrum. Source energies available for the UCNA experiment are similar to case 2. The difference between the linearly predicted energy and the generated calibration points are shown in Figure 3.1 for α 2 =.1,.5,.1,.2. This analysis shows that, at best when all 1 sources are used, there is a average discrepancy of <1% across the energy range of interest. Cases where the highest and lowest energy sources are well separated the energy of the third source makes marginal difference. When the sources are closely grouped the deviation at energies below the sources diverges more rapidly than above, so in this scenario it appears having the source at the lower end of the spectrum is preferable to minimize the impact in the analysis energy range. If we now generate the measured spectra for detectors one and two, by convoluting the energy spectrum of neutron β-decay given by Eq. 3.3 with the nonlinear detector response, we can determine the bias to the asymmetry if the linear calibration of one set of sources is used. The measured ADC channels are converted to an energy based on the linear fits previously described, and the deviation from the ideal asymmetry is shown in Figure Interestingly enough, this analysis finds that a linear 68

85 / ADC meas ADC cal =.2 α 2 All 1 Sources 2, 6, and 9 kev 2, 3, and 9 kev 2, 3, and 4 kev 8, 9, and 1 kev Energy(keV) (a) 2% non-linear term. / ADC meas ADC cal =.1 α 2 All 1 Sources 2, 6, and 9 kev 2, 3, and 9 kev 2, 3, and 4 kev 8, 9, and 1 kev Energy(keV) (b) 1% non-linear term. / ADC meas ADC cal =.5 α 2 All 1 Sources 2, 6, and 9 kev 2, 3, and 9 kev 2, 3, and 4 kev 8, 9, and 1 kev Energy(keV) (c).5% non-linear term. / ADC meas ADC cal =.1 α 2 All 1 Sources 2, 6, and 9 kev 2, 3, and 9 kev 2, 3, and 4 kev 8, 9, and 1 kev Energy(keV) (d).1% non-linear term. Figure 3.1: The deviation of the of calibrated energy, as determined using a linear fit to the peaks of available sources, from the true quadratic response of the detector is shown for several scenarios of the sources and magnitude of quadratic term. For each case, the calibration curve is determined for 1 sources equally spaced at 78.2 kev and several sets of three sources. 69

86 calibration curve derived from the low energy side of the spectrum generates the smallest average bias to asymmetry. The average bias to the asymmetry, tabulated for each case in Figure 3.11 for the analysis window of kev, is below 2% for a calibration scenario similar to UCNA. It should be noted that this analysis assumes that any nonlinearity is either not corrected for or unrecognized and has the form of a positive quadratic term. Its is entirely possible that the nonlinearity can any other form and should be handled differently for each unique basis Summary of the Bias due to Energy Calibration A realistic detector would exhibit all the systematics discussed previously and any bias imparted to a measured asymmetry will be arise through correlations of these effects. To minimize the impact of such bias, the energy window used in the analysis is normally restricted. From the previous analysis a range starting between 1-3 kev and ending between 6-7 kev should provide a value of asymmetry which is as insensitive to calibration issues as one can expect. Table 3.2 summarizes corrections to the asymmetry for several scenarios of detector resolution and differences in resolutions and calibration for a few choices of analysis energy windows. In the case of a resolution difference, σ 2 is fixed at 2.5 and σ 1 is altered to achieve the listed ratios. From the analytic models presented, one should expect that the calibration effects will bias the asymmetry at or below.2% from detector resolutions and the differences in detector calibration could lead to similar corrections. 3.4 Scattering Systematics In the previous sections we have assumed that the detector s response function is independent of the angular distribution of the emitted β s, even when a multiple scattering and backscattering tail was included in the response function. To continue building a realistic model of the detector we must now introduce angle dependent effects. Scattering on the non-active materials in the spectrometer such as foils bottling the decay trap and windows of the MWPC will cause a small fraction of events to have their initial direction misidentified or systematically alter the angular distribution measured in each energy 7

87 1 - A cal /A , 6, and 9 kev, A = , 3, and 9 kev, A = , 3, and 4 kev, A =.598 8, 9, and 1 kev, A = Energy (kev) (a) 2% non-linear term. 1 - A cal /A , 6, and 9 kev, A = , 3, and 9 kev, A = , 3, and 4 kev, A =.42 8, 9, and 1 kev, A = Energy (kev) (b) 1% non-linear term A cal /A , 6, and 9 kev, A = , 3, and 9 kev, A = , 3, and 4 kev, A =.257 8, 9, and 1 kev, A = Energy (kev) (c).5% non-linear term. 1 - A cal /A , 6, and 9 kev, A = , 3, and 9 kev, A = , 3, and 4 kev, A =.33 8, 9, and 1 kev, A = Energy (kev) (d).1% non-linear term. Figure 3.11: Spectral distortions which arise from linearly calibrating a nonlinear response are propagated through to a measured asymmetry and compared to the ideal energy dependence. A simple mean bias is calculated for kev, which is similar to the energy range used in the UCNA analysis. 71

88 Table 3.2: Bias due to energy calibration and detector response function differences are summarized for several reasonable choices of analysis energy windows. Calibration Resolution Resolution Diff. Energy Range (kev) α,α 1 A σ 1 A σ 1 [%] σ 1 /σ 2 A σ 1 /σ 2 [%] - 782, , , , , , , , , , , , , , , , , , , , ,

89 bin. Monte Carlo studies of each experimental geometry using the PENELOPE3 simulation package [75] were carried to determine the corrections required to offset these effects as well as identify experimental observables to benchmark the accuracy of the simulation A PENELOPE3 simulation of the UCNA Geometry A simulation of the particle tracking in the spectrometer using the scattering calculations and geometry handling of PENELOPE3 [75] is used to assess biases that might arise. The accuracy to which the simulation package can predict the energy loss and angular deflection due to scattering will determine the uncertainty applied to any corrections to the asymmetries that are derived from the Monte Carlo. Martin, et al [44, 76, 77] find that there is as large as a 2 3% discrepancy between the predicted angular dependence of backscattering and their measurements on Si, Be, and plastic scintillator for both PENELOPE3 and GEANT4. A scale factor was used to minimize the χ 2 between the models and data, setting the size of the discrepancy, and it is found that only in the case of measuring the total normal incidence backscatter from the scintillator does GEANT4 have a lower χ 2, after scaling, than PENELOPE3. It is assumed that this is due to PENELOPE3 being a mixed algorithm, all collisions over a step where the energy loss or angular deflection exceeds a user defined limit are explicitly calculated. GEANT4 applies a condensed algorithm which averages over the effects of multiple scattering and can be susceptible to the choice in the step sized used in the calculation. Based on the these findings and the thesis of Seth Hoedl a redundant approach to simulating the scattering corrections was adopted where both GEANT4 and PENELOPE3 models were developed in parallel by separate groups. The geometry of the spectrometer and detector packages was developed using PENELOPE3 s geometry definition routine PENGEOM, which uses a quadric equation to define surfaces that can be grouped to enclose bodies. Several modifications to PENGEOM were required to accommodate a larger number of surfaces and bodies than the default settings would allow. Including the individual wires in the MWPC for each detector alone, 194 surfaces and 192 bodies for each detector, this exceeded the default body and surface limit of 2 each. Using PENGEOM to manage the geometry definition is not required in PENELOPE3 but loading a geometry definition in this way allows for the use of several routines that 73

90 track boundaries and the relative location of the particle in the geometry. A schematic of the detector geometry is shown in Figure Cross-section tables are built using PENELOPE3 s MATERIAL routine which uses data adapted from NISTIR [74] to compile the information for elements 1-92 as well as 28 compounds and mixtures that have empirical scattering tables. Materials not found in the NISTIR database can be generated by specifying the elemental composition either by mass or stoichiometrically, and then the MATERIAL routine creates a scattering table that is the weighted average of the cross-sections of the constituent elements. This routine also allows for the mass density of the material to be altered which is relevant for modeling the low pressure gas in the MWPC, between the MWPC and scintillator, and the residual vacuum in the spectrometer. Material dependent parameters W cc, W cr, C 1, and C 2 govern when a step is treated using the detailed or condensed calculation. C 1 is the upper limit on the average angular deflection, 1 cosθ, due to multiple scattering and C 2 limits the fractional energy loss. Both parameters have a suggested value of.5 and may be as large as.2. W cc and W cr set the cutoff energy, in ev, for hard inelastic collisions and bremsstrahlung emissions. The final parameter for each material is the maximum allowed step size. PENELOPE3 samples the step size based on the mean free path in the material, which may be longer than the object s thickness in the simulation creating the possibility of not interacting in a material, therefore interactions can be enforced by setting ds max to a value smaller than the simulated thickness. This does not induce artificial collisions, it forces the simulation to sample the step to a scattering site multiple times during a transit of material. It is suggested that ds max be set so that a minimum of 1 steps are required to transverse a material. Parameters and materials used in the simulation of UCNA are listed in Table 3.3. PENELOPE3 UCNA Geometry - The Decay Trap The decay trap is modeled using a 3 m long, 6. mm ID copper cylinder centered at the origin and coaxial with the z-axis. The ends of the decay trap are enclosed by thin,.7-12 µm, mylar foils with 3 nm of Be coated on the inward face and a 2.3 cm long plastic collimator external to the trap extends 1.5 mm in to the ID of the decay region. Physically the decay trap is comprised of three 1 m tubes held 74

91 Table 3.3: Scattering parameters of each material used in the PENELOPE3 simulation of UCNA are summarized. The material identifier used in the simulation is give by ID, NISTIR database identifier DB Id used in the generation of the scattering table, and W cc,w cr, C 1,and C 2 are the scattering parameters which are defined in the text. ds max denotes the maximum step length in centimeters in a given material. Material ID DB Id ρ (g/cc) W cc (ev) W cr (ev) C 1 C 2 ds max (cm) Neo-Pentane Plastic scintillator Mylar SiO Al N 2 (1torr) Air(1 5 torr) Be Glass Si together by plastic clamping mechanisms and the central tube has three holes in the tube wall for the inlet of the square 2. cm x 6. cm guide, a small.3175 cm aperture for neutron density monitoring, and source insertion port (usually closed with a removable section of guide) which is 2.5 cm in diameter. These gaps and holes are neglected for the purpose of electron transport simulation. Any possible effect due to scattering near the walls of the decay will be dominated by the plastic collimator which holds the decay trap foils in place. PENELOPE3 UCNA Geometry - The Detectors The detector package geometry is represented as shown in Figure 3.12 with Al used as the external housing material shown in red. Entrance and exit window foils to the MWPC have thicknesses ranging from -25 micron depending on the geometry simulated. The entrance window is modeled on a segment of a sphere with r=28.65 cm offset from the center by 1 cm to model bowing due to the pressure differential in the 1 torr neo-pentane environment of the MWPC and the roughly micro-torr vacuum in the spectrometer. The bowing effect was measured during the construction phase of the MWPC by Ito, et al [62]. A 1 cm, 1 torr N 2 filled gap is left between the exit window and the front face of the plastic scintillator, the pressure differential in the nitrogen region and the MWPC is controlled to < 5 75

92 Figure 3.12: A 2-D slice view of the detector package from the PENELOPE3 geometry view packaged. Included but difficult to see are the thin, 6-25 micron, Mylar entrance and exit window foils to the MWPC. Not included in the picture, due the limited number of bodies the viewer can handle, are the wires of the anode and cathode planes. torr so no bowing effects are included in the rear window. Based on the measurement of Yuan [45], the first 3 µm of plastic is considered the dead layer of the scintillator and is considered a separate body by PENELOPE3. All empty space is treated as a vacuum of 1 5 torr Dry Air which was found to be superior to setting the material to with respect to finding the boundary crossing into the thin foils. To crudely simulate the charge collection in the MWPC, the gas region is divided into 1 mm thick slices centered on each cathode wire that extend from the entrance or exit window to the anode plane and run the full length of the wire. The front cathode wires run along the x-axis and the back cathode plane runs along the y-axis. Thus a similar charge weighting procedure that is used in the data analysis can be used to determine the MWPC position coordinates. Grouping the planes in sets of four should be equivalent to the actual detector, however charge drift and collection are not included in the simulation. Therefore, the simulation predicts lower multiplicities than are measured. 76

93 Simulation Operation Scattering calculations carried out by PENELOPE3 are integrated in a FORTRAN77-based simulation code to handle particle tracking through the magnetic field and energy loss tallying. The simulation tracks particles from generation to either absorption in a material or an exit from the experimental volume using a 4 th order Runge-Kutta numerical integrator with 5 th order error checking taken from Numerical Recipes: Fortran77 [78]. The bi-cubic spline interpolating routines used to determine the components of the magnetic field from a measured field map are also taken from Numerical Recipes. PENELOPE3 s JUMP subroutine is use to determine the length of step and the STEP subroutine is integrated into the Runge-Kutta integration so that boundary crossing can be checked during each translational step. Initial kinematic parameters of the electron and proton, location of the decay, east/west trigger time, and the energy deposited in each body in the geometry are collected and stored in a PAW ntuple for each event. Several derived quantities are also stored in the output ntuple: MWPC multiplicities, MWPC reported position, location and energy of event if aborted, abort code, event type to identify calibration sources, and the angle and energy of backscatter events leaving the detector. Step-by-step output can be implemented to diagnose event tracking issues, but for a simulation of 1 6 events or greater, tracking each event requires too much space. To limit simulation time, trajectories are tracked for 15 s of computational time, after this limit the event is aborted with its position and energy recorded. Two other scenarios will lead to an event being aborted: events escaping the spectrometer volume and unphysical trajectories. Electrons in simulations of neutron beta-decay never escape the confines of the spectrometer, however it is possible for electrons with E >1 MeV, as would be generated in simulations of 21 Bi and Compton electrons from the high energy photons in the Cu(n,γ) simulations, to punch through the plastic scintillator and continue on. These events are aborted once z > 235 cm or r > 15 cm. On rare occasions, a frequency of < per event, one of the kinematic variables is returned from the trajectory integrator as NAN requiring the event be aborted. Corrections to the extracted asymmetry derived from the simulation must account for these aborted events which could unphysically alter cosθ reported in the analysis. In all simulations used to determine the scattering corrections, the deviation, abort = 1 cosθ NA /(1/2), was shown to be less than 1 4, where cosθ NA is determined 77

94 from all non-aborted events. One could imagine the case where the computational time limit per event discriminates against low-energy, high-pitch angle events, artificially increasing the average angle of simulated events and biasing the results Backscattering In the idealized description of a detector observing the decay of a polarized neutron that we have used in previous sections to investigate the effects associated with calibration there has been an assumption that each detector has an angular acceptance of 2π. However, this must be generalized to accommodate the reality of backscattering from materials in the spectrometer. Thus if we describe the emission probability of an electron as d 3 Γ(E e,θ e,φ e ) de e dω e = W(E e ) ( 1 + Aβ(E e ) P cosθ ), (3.37) where W(E e ) contains the allowed shape of the β spectrum and all energy dependent corrections and P is the mean polarization of the neutron population which we will assume to be 1, then for an ideal detector we assume that the count rate in a detector is given as dn de e = W(E e ) 1 1 ( 1 + Aβ(Ee )cosθ ) U(±cosθ)dcosθ. (3.38) Here U(x) represents the unit step function which is unity when its argument is positive. In this case the angular acceptance is unity in the hemisphere w.r.t the neutron spin nearest to the detector and zero for the opposite direction. To generalize the measured angular distribution, the step function is replaced with an angular efficiency which has a more complex relationship with the emission angle, η(cosθ), shown in Figure 3.13 integrated over all energies. The angular acceptance can be modeled with the four-parameter expression: ( η(cosθ) a 1 tanh(a2 (cosθ a 3 )) + a 4 ) (3.39) which captures most of the functional dependence. The main source of the large detection inefficiency, shown in Figure 3.13, of 2%, is due to including the low end of the energy spectrum in this analysis, 78

95 E < 2 kev. Calculating the asymmetry using a convolution of Eq and Eq and comparing the results to the angular acceptance for a geometry similar to 29-C,.7 µm trap foils and 6. µm MWPC windows, we find that the measured asymmetry will be reduced by.45% over the energy range of 2-7 kev. A conservative estimate on the bias generated by unrecognized backscatter is given be assuming we can separate Eq N 1 = + N 2 = + EQ E 1 EQ 1 E W(E)(1 + Aβ cosθ)(1 ε 1B (E,cosθ))d(cosθ)dE W(E)(1 + Aβ cosθ)ε 2B (E,cosθ)d(cosθ)dE = N 1 (1 ε 1B ) + N 2 ε 2B (3.4) EQ 1 E EQ E 1 W(E)(1 + Aβ cosθ)(1 ε 2B (E,cosθ))d(cosθ)dE W(E)(1 + Aβ cosθ)ε 1B (E,cosθ)d(cosθ)dE = N 2 (1 ε 2B ) + N 1 ε 1B (3.41) where the probability that an event is detected on the correct side is assumed to be (1 ε) where ε 1B(2B) (E,cosθ) is the probability of an unrecognized backscatter from the opposite side, E O and E Q are the lower and upper energy limits of the analysis window, and N represent the average decay rate in that hemisphere. After integrating over energy and angles, ε 1B(2B) (E,cosθ) ε 1B(2B), which can be taken as the total backscattering fraction. Calculating an asymmetry from the rates described in Eq gives N 1 N 2 N 1 + N 2 = N 1 (1 ε 1B ) + N 2 ε 2B N 2 (1 ε 2B ) N 1 ε 1B N 1 (1 ε 1B ) + N 2 ε 2B + N 2 (1 ε 2B ) + N 1 ε 1B = N 1 N 2 2( N 1 ε 1B N 2 ε 2B ) N 1 + N 2 = A m 2 N 1 ε 1B N 2 ε 2B N 1 + N 2, (3.42) where A m is the measured asymmetry that includes all the energy dependent effects discussed in previous 79

96 sections. In the case where ε 1B = ε 2B = ε this results reduces to N 1 N 2 N 1 + N 2 = A m (1 2ε). (3.43) Another special case of interest is where ε 1B = N 2 ε 2B / N 1 which eliminates the bias from backscattering. We should expect that in reality neither of these cases are valid because of the cosθ weighting the angular distribution incident on the detectors is not equal, therefore we can assume that the backscattering fractions will be different and most likely not fulfill the special case where the bias is offset. In the end what is important for a high precision is not only to reduce the total amount of backscattering, but also a reduction of the uncertainty placed on the Monte Carlo predictions. If we assume that Eq can represent the average side-dependent missed backscatter correction, then to have the uncertainties in the asymmetry not be dominated by the scattering correction, we have the requirement on the uncertainty on ε of σ ε ε < 1 σ A 2ε A, (3.44) as shown by the Hoedl [44]. If the ε factor is roughly.5% then to achieve the goal of σa/a <.2%, the requirement on the backscatter correction uncertainty is σε/ε < 2%. With these simplified ideas of the angular dependence of backscattering and how the amount of backscattering propagates to a bias on the asymmetry, we can investigate simulated cases relevant to UCNA and similar experiments. Experiment Specific Backscattering Backscattering in the UCNA spectrometer occurs in several very different scenarios all related to the location of dead and active material of the detectors and decay trap. There are three identifiable types of events in the UCNA geometry: 1. Only one scintillator and one MWPC are above threshold on the same side. (Type,IV, V) 2. Both detector package s scintillators trigger. (Type I) 3. Both MWPC s are above threshold and one scintillator triggers. (Type II, III). 8

97 hd.9 Probability of Detection vs. Emission Angle hd Entries Probability of Detection Cosθ Figure 3.13: The probability of detection for an event with emission angle θ the detector parallel to the neutron spin, detector 1, is plotted versus cosθ. Detection criterion requires either energy deposition in detector 1 to be greater than zero and zero in detector 2 or energy deposited in both and the trigger time of detector 1 to be less than detector 2. The simulation is representative of geometry C of the 29 data set having 6 µm MWPC foils and.7µm decay trap foils. Type Scintillator Type 1 Type 2 Type 3 Type 4 Type 5 Decay Trap Foils MWPC Figure 3.14: Schematic of backscattering trajectories. Trajectories which pass through detector region are assumed to have triggered that detector. Arrow heads denote where the trajectory ends and also assume the current region is triggered. Not shown here events which are lost in transit in foils or MWPC gas during the initial approach to a detector or during backscatter. These event contribute to a reduction in the detection efficiency and systematic uncertainties in the energy reconstruction of each event class. 81

98 Each type of event is shown schematically in Figure The first class of events combines the Type, IV, and V events from simulations (correct events, missed backscatter from detector windows, and missed backscatter from decay trap windows). Experimentally there is now way to separate the correct events and the Type IV/V events. Therefore these events are assigned to the scintillator trigger side and energy calibration and reconstruction is based on simulation results which group these three classes together. For events where both scintillators are triggered, the relative timing between the triggers can be used to identify the first detector hit if the transit time is < 1 ns. Because the digitization time of the ADC is 6 µs and the enforced dead time is 12 µs, events with transit times longer than 1 ns do not appear as new events. This leads to a slight correction to the asymmetry due to the incomplete energy reconstruction. Events which trigger one scintillator and both MWPC s (Type II/III) can have their initial direction determined on a statistical basis using input from Monte Carlo, vetoed all together, or assigned to the scintillator trigger side. The impact to the asymmetry for each of these choices is determined from Monte Carlo and benchmarked versus a set of experimental observables. Type IV and V are the missed backscatter or misidentified events which trigger the opposite detector w.r.t to their initial momentum direction and leave no trace in the other detector. They are separated into distinct classes due to their very different nature. Scattering from the decay trap windows happens in the high field region of the spectrometer. More importantly, at the same field magnitude as the neutron decay point within the decay trap, meaning that electrons of all pitch angles interact with the foils including grazing incidence. Type 4 event scatter from the front face of the MWPC in the.6 T region where the maximum angle of incidence is 56 due to adiabatic transport of the electrons in the expanding field, thus suppressing the total backscattering fraction. Events which are backscattered from the detector also have a high probability of being mirror by the increasing field. Type I Backscatters Events which trigger both β-detector packages with a time of flight between detectors of less than 1 ns are labeled as Type I events. With both detectors triggering, the timing information allows the first detector hit to be determined and events to be properly assigned to a side. A small fraction, 2.5%, of 82

99 these events have TOF longer than 1 ns and will be viewed by the analysis as a Type 2/3 event. The probability of both detectors being triggered increases with energy until plateauing between 3-4 kev, due to the energy loss in the foils. Triggering both detectors requires that an electron transit three sets of foils and have sufficient energy when it reaches the second scintillator to satisfy the 2-PMT trigger criterion. Without the foils or the magnetic field pinch, we should expect that the backscattering fraction from the plastic to be roughly 6%. Including the effects of the field reduces this by about half. Analysis of various foil geometries shows that there is a very strong correlation between foil thickness, either in the MWPC or in the decay trap, and Type 1 event fraction. The results of these simulations are tabulated in Table 3.5. Observables from Type 1 events including: initial and secondary detector energy spectra, time of flight, time of flight versus mean energy, total energy, and total fraction can all be extracted from the data and used to benchmark the performance of the Monte Carlo. Based on the work of Martin et al. [76, 77] which measured the backscattering from a bare scintillator, we have confidence that predictions from PENELOPE3 should be accurate to better than 2%. In Figure 3.15(a) the TDC spectrum illustrates how the Type 1 events are identified. The large peak at t > 13 ns is the self-timing peak described in Chapter 2. It should be noted that in common stop timing, t cs is related to time of flight, t TOF, by taking the difference of t cs and the location of the self timing peak, 13 ns in the TDC spectrum shown. Therefore t cs of 12 ns represent a very short t TOF. From this timing spectrum we can investigate correlations such as the energy deposited in the first and second detector versus TOF, as well as the relation to the initial energy, shown in Figure 3.15(b). The difference in t TOF due to the energy of backscattering electrons is on the order of 1 ns, 26 ns at 1 kev and 15 ns at 8 kev for a direct trajectory, therefore the long tail of the timing distribution is related to the angle w.r.t to the field. It is interesting to note that after 2 ns, t cs < 11 ns, events tend to deposit an equal amount of energy in both detectors, a trend which appears to be independent of the foil thickness. The amount of energy lost varies by < 2% ( A-142 kev, B-146 kev, and C - 128keV) from geometry to geometry. The average initial energy tends toward higher values as the foil thickness is increased, as expected. Comparing simulation of the geometries used in the data, where the total foils thickness that a Type 1 would pass through if its trajectory is as 83

100 htdc Entries Mean RMS 64.6 Entries h3_pfy h2_pfy χ Entries 2 Entries / ndf / 65 Prob Mean Mean 2.667e Mean p Mean y y 1.66e e+5 RMS ± 1. RMS p1 RMS RMS y y e e+5 ±.84 p e+5 ± 2.898e+3 p ± 36. Counts Backscatters Time (ns) (a) Type 1 common stop TDC timing distribution. Energy Deposited in the Second Detector (ev) t vs. Energy variables of Type 1 backscatters Second Detector First Detector Initial Energy Common Stop Timging (ns) (b) Time of flight vs. average energy. Figure 3.15: Panel (a) shows a typical TDC spectrum from the 27 dataset, where the narrow peak between ns is the self-timing peak and timing cut used to identify backscatters is shown in red. In panel (b) the correlation between the TOF and the averaged detected and initial energy is depicted. shown in Figure 3.14 is: A µm, B µm, C - 39 µm the average energy deposited in the initial detector are : A kev, B kev, C - 39 kev. To make a connection with previous experiments such as Perkeo I and II [5, 6] all of the events that they classify as recognizable backscatters [42] are considered Type I events in UCNA. Sources of False Type 1 Events It is possible for background events to mimic the timing signal and appear as a backscatter event. Muons transiting the spectrometer that trigger both scintillators will typically be vetoed, thus do not contribute to this type of background. It is possible for a background photon to Compton scatter in the plastic scintillator, knocking the electron free and triggering the detector. If the liberated electron has sufficient energy it may leave the first detector package and follow the field lines to the opposite detector. A simulation was performed to check the size of the contribution and energy spectrum from Compton scattering induced backscatter events. In the simulation, γ-ray s with the energies and relative intensities measured on the floor of area B [79] were incident on the back of the scintillator. It should be noted that the study by Tipton [79] was performed with the accelerator off, thus any prompt γ component or long 84

101 Table 3.4: Summary of a γ background simulation of the UCNA spectrometer Description Event Fraction Compton β s Triggering MWPC and Scintillator 2.38(5)% Appearing as Type I Events (-2MeV).22(5)% Total Measured γ rate (GV Closed) 45.2 Hz Measured Rate of Type I Events (GV Closed) (-2MeV).571 Hz lived activation from the beam is not included in those results and therefore not included in the input of the simulation. The results of the simulation are summarized in Table 3.4. Type II/III Backscatters The β detection package does not generate a timing pulse or a trigger to the DAQ for signals in the wire chamber above threshold, any charge collected on the wires during a global trigger is read by the peak sensing ADC. This information will allow us to see events which trigger one scintillator and have a signal above background in both MWPC s, but not the ability to separate which side was hit first. In the case where the β scatters from the dead layer of the scintillator or the rear MWPC window and triggers the scintillator on the opposite side (Type II), only one transit of the trigger side MWPC is made. However, for β s which scatter from the scintillator after having triggered it, but with insufficient energy to pass through both detector foils on the opposite side (Type III), there are two transits through the trigger side MWPC. Example trajectories are shown in Figure By placing a cut on the trigger side detector at 4.1 kev the Type II and Type III events can be separated with 8% efficiency based on input form Monte Carlo. Events with E MWPC > 4.1 kev would be assigned to the trigger side and E MWPC < 4.1 kev the opposite side. It is also possible to use the energy deposited in the MWPC of the trigger side scintillator to construct a probability distribution function from the Monte Carlo that allows for a probabilistic assignment, Figure This allows for three possible analysis scenarios: A - Reconstruct all Type II/III events to belong to the trigger side detector. B - Use the statistical reconstruction based on trigger side MWPC ADC cut. C - Veto all Type II/III to remove ambiguous events. 85

102 Scaling Type 2/3 Events An issue that will arise is how to apply the correction due to the missed backscatter fraction if we know predicted Type 2/3 fractions is up to 3% lower than measured. One can think of the Type 3 events as the low energy tail of the Type 1 events which are related to the simulation s ability to describe bulk scattering in the plastic. Therefore the scaling applied to the Type 3 events should be similar to any scaling used to correct the Type 1 events. Type 2 events, however, scatter on the foils, gas, or wires of the MWPC making this class more related to the missed backscatters. The simplest method is to scale the entire 2/3 population to match the benchmark backscattering measurements which will most likely not properly address the weighting of the Type 2/3 populations. In the top panel of Figure 3.16 the energy deposition in the wire chamber is shown with the 2/3 s summed and broken out separately, where the total spectrum is given by f 2/3 = a 2 f 2 + a 3 f 3, a i being the scaling factor applied to each event Type. Performing a least squares analysis can determine the weighting of the Type 2/3 s and give a more accurate assessment of the missed backscatter corrections. In this method, the Type 2 scaling will be applied to all missed backscatter event classes for the determination of the bias to the asymmetry for each geometry. Another option that was considered to determine the relative ratio between the Type 2 and 3 events relies on using the MWPC spectra from the Type 1 events. A typical Type 1 event will transit the first detector s MWPC twice and the final detector s MWPC once in much the same manner as a Type 3 event. Therefore the MWPC deposited energy spectrum of the primary detector for Type 1 event should be similar to the trigger side spectrum of Type 3 event and the secondary detector of the Type 1 s can be compared to the trigger side of the Type 2 events. This method would provide model independent spectra for one and two transits of the MWPC which could be scaled to match the Type 2/3 joint MWPC spectrum to determine the scaling factors. Unfortunately, Type 1 events have significantly higher energy when transversing the wire chamber, therefore the energy deposited in the gas is on average 1 kev less per pass is shown in Figure

103 East Anode Signal for Type 2/3 Counts East Anode Signal for Type 2/3 East Anode Signal hmwpc_23 for Type 2 Entries 1189 East Anode Mean Signal for Type West RMS Anode Signal for Type 4.182/3 West Anode Signal for Type 2 West Anode Signal for Type Energy (kev) Probability of being a Type MWPC Energy (kev) Figure 3.16: Type 2/3 events separation probability based on trigger side MWPC energy deposition / N total ) Normalized Counts (N bin Type 1, first htype12 htype3 htype11 htype2 detector Type Entries 1, Entries second detector Mean Mean Mean Type 2 RMS RMS RMS Type MWPC Energy (kev) Figure 3.17: MWPC energy deposition spectra for Type 2 and 3 are compared to the spectra from the primary and secondary sides in the Type 1 events. The energy spectra are fit to a Landau distribution to determine the most probable value which shows that Type 1 events lose 1 kev less energy per transit. 87

104 Missed Backscatters In the worst possible case, undetected backscatter events have an isotropic angular distribution and contribute a correction to the asymmetry as calculated in Eq where the ε s are just the missed backscatter fraction. The 2.5 µm decay trap foils located in the uniform 1 T field region backscatter 5% of events which would result in a correction of 1%. Luckily, as shown in Figure 3.18, the angular distribution of event backscattering from the foils is sharply peaked at larger angles where the cos θ weighting suppresses the asymmetry. Therefore, as the probability for scattering increases the inherent asymmetry is decreasing and if electrons emitted in both direction scatter equally from the decay trap foils the correction to the asymmetry is greatly reduced. From simulations, the probability for backscattering from the decay trap foils and triggering the wrong detector is sharply peaked at grazing incidence, with 95% of these events having an initial pitch angle w.r.t. the neutron spin of θ > 8. Because of this, the resulting correction to the asymmetry due to only Type V backscatters is < 1% for most cases, Table 3.6. It is also possible for the β s to scatter from the front MWPC window, 25 µm of Mylar and trigger the opposite detector, which we define as Type IV events. This is suppressed in two ways by the expansion of the magnetic field from 1 T to.6 T at the detector face. As the strength of the field decreases, the flux trapped by the electron s orbit must remain constant, thus the electron s transverse momentum decreases thereby reducing the possible angle of incidence on the detector window to a maximum value of 56 and in turn reducing the probability of backscattering. Also, electrons leaving the detectors heading into an increasing B field will undergo magnetic mirroring if θ > sin 1 B /B max, reflecting these events back into the detector Edge Effects Scattering from the collimator at the ends of the decay trap will select electrons emitted with lower pitch angles. This will increase the asymmetry measured at larger radii. The maximum gyration orbit in the 1 T field is 7 mm at the total Q of neutron β decay. The decay trap is 6 mm in radius, and the collimator extends another 1.5 mm into opening of the decay trap, therefore edge effects should become noticeable at 51.5 mm. Results from simulation are shown in Fig.3.2. By defining a radial cut, r < 5 mm, edge 88

105 True Energy and Direction Probability Type I Backscatters Type II Backscatters h1 Entries Type Mean III I Backscatters.421 Type RMS IV Backscatters.268 Type V Backscatters Counts/1keV Probability Type Prob Type 1 Prob Type 23 Prob Type 45 Prob Energy(keV) Cos θ (a) Probability vs. cosθ Energy (kev) (b) Probability vs. Energy Figure 3.18: The angular and energy dependence of the event classes for a typical UCNA geometry. Each probability distribution gives the event fraction in that bin. Panel (a) shows that Type s I-IV have similar dependence on the emission angle, however Type V events are strongly peaked around 9. Table 3.5: Summary of event Type fractions for various configurations of foil thicknesses for an analysis energy range of kev. This energy range is representative of the analysis window used in the UCNA for the 28/28 dataset. In the case of a thickness decay trap and wire chamber foils, the MWPC gas and wires are still present, therefore scattering from the wires, gas, or scintillator dead layer lead to the Type 2/3 events. MWPC Foils (µm) Trap Foils (µm) Type Type 1 Type 2/3 Type 4 Type (2) 2.86(4).95(2).64(5) 6.44(5) (2) 3.9(4) 1.11(2).61(5) 5.2(5) (2) 3.44(4) 1.22(2).52(5) 2.65(3) (2) 3.47(4) 1.2(2).52(5) 2.29(3) (2) 3.53(4) 1.26(2).47(5) n/a (2) 4.7(4).69(2).17(3) 5.15(5) (2) 4.53(4).89(2).15(2) 2.32(3) (2) 4.63(4).92(2).13(2) 1.45(2) (2) 4.65(4).94(2).9(2) n/a (2) 4.28(4).63(2).7(2) 5.21(4) (2) 4.95(5).82(2).4(2) 1.42(2) 94.3(2) 5.2(5).82(2). 89

106 /A o ) (1 - A i i A i Geometry A 2 2, 2,1 2,2 /A o ) (1 - A A i i i Geometry B 2 2, 2,1 2, Energy (kev) Energy (kev) /A o ) (1 - A A i Geometry C 2 2, 2,1 2,2 /A o ) (1 - A A i Geometry 7 2 2, 2,1 2, Energy (kev) Energy (kev) Figure 3.19: Backscatter corrections for the geometries relevant to the analysis of the data are presented as a function of the reconstructed energy, averaged in 25 kev bins. 9

107 East Raw Counts vs. R for 2<E< RadHist.hradw Entries Mean RMS East Raw Counts vs. R for 2<E< RadHist.hrade Entries 4 Mean RMS Figure 3.2: The left panel shows the counts in 1 mm annuli for the two detectors where the red and black lines represent the detector in front and behind the neutron spin respectively. The right panel shows the asymmetries in these bins where the green markers are the true asymmetry and the blue and red markers are backscatter corrected and uncorrected measured asymmetry for reconstructed energies 2 < E < 65 kev. effects are eliminated. All simulations of UCNA geometries include the collimator and generate decay events out to the walls of the decay trap resulting in 8% of all decays being lost due to scattering in the decay trap Angular Acceptance In the 2 2π spectrometer design, all electrons are ideally mapped to a detector via the field lines regardless of their initial position or emission angle. In this ideal scenario there is no position dependent detection efficiency and cosθ = 1/2 for all energies. Deviations from the cosθ = 1/2 ideal, lead to an effective acceptance correction which is highly dependent on the spectrometer geometry. In experiments such as PERKEO II the angular acceptance at the edges of the decay region causes an enhancement in the measured asymmetry due to collimation and requires a.24% correction which is called edge effects [6]. The case of UCNA is unfortunately more subtle. As previously stated the position sensitivity allows 91

108 for the suppression of the edge effects by using a radial cutoff in software. However having dead material in various regions of the spectrometer results, effectively, in a cut on the acceptance of beta-decay events which depends both on the beta energy and pitch angle. Higher pitch angle trajectories will have an increased path length in the foils or detector gas relative to lower pitch angle events resulting in angle dependent energy loss in the dead material. This reduces the efficiency for detecting the higher pitch angle events, increasing cos θ and in turn the measured asymmetry. Additionally, the value of β derived from the reconstructed energy does not have a one-to-one correlation to the initial β, and like cosθ, the average value of β enters linearly into the weighting the asymmetry. The scattering effects which alter the angular acceptance are similar to those that cause deviation in the measured β. Therefore we will consider correcting cosθβ recon i to 1/2 β true i in one step for each energy bin i. The inclusion of the decay trap foils to increase the neutron density in the spectrometer resulted in a significant reduction of the spectrometer acceptance for high pitch angle events. In the 1 Tesla region, z < 15 cm, an electron can have any pitch angle, allowing grazing incidence scattering on the decay trap windows. This greatly increases not only energy loss and absorption in the foils but also backscattering. Monte Carlo studies were carried out to determine how much the correction due to scattering on the foils can be reduced by constructing thinner ones, but also to predict the effects of doubling and tripling the thickness so that the bias to the measured asymmetry can be used as a benchmark of the simulation. Eventually the 3 nm Be coating on the foils, which functions as the UCN reflector, will be the limiting factor for reducing scattering in the decay trap. To determine the acceptance correction from simulation an isotropic or cosθ distributions, events with an uniform energy or a spectrum governed by the β-decay shape are generated and propagated through the geometry. The original emission angle, W, multiplied by β = E(E + 2m e )/(E + m e ) in terms of the initial kinematic energy E, is tallied in each energy bin for E true, E vis, and E recon after backscatter reconstruction. If an isotropic source is used, the results can be weighted by a probability distribution function (see Eq. 3.3) when filling histograms and calculating corrections. Results of this analysis, where simulations of neutron β-decay events are propagated through the spectrometer, are presented in Figure 3.21 where the deviation of measured pitch angle is shown for geometries B 92

109 /(1/2) Cosθ 1 - B Choice A events Choice A Choice B Choice C Choice D Choice E Reconstructed Energy (kev) (a) Worst Case Cosθ /(1/2) 1 - B Choice A events Choice A Choice B Choice C Choice D Choice E Reconstructed Energy (kev) (b) Best Case Figure 3.21: The fractional difference of the average cosθ measured in each reconstructed energy bin from 1/2 is shown for geometry B (12.7 µm decay trap foils and 25 µm MWPC foils) in the left panel and geometry C (.7 µm decay trap foils, 6 µm MWPC foils) in the right panel. Simulated output is fit to a 5 th order polynomial to generate a smooth function used to correct the asymmetry data. and C for the different analysis methods and the full acceptance correction, 3, is shown in Figure 3.22 for the geometries used between It is interesting to note, in Figure 3.21, that at low energy, E < 2 kev, the measured angle is less than 1/2, at medium to high energies the average measured angle is rapidly increases. This is again, a result of the shape of the β-energy spectrum. Because the spectrum peaks at 33 kev, bins below the peak have competing contributions from higher energy events with high pitch angles, which lose a lot of energy prior to the detector, and the few forward directed events whose initial energy is similar to the limits of the bin. In this case the large statistics from the center of the spectrum dominates the low energy bins producing a reduction in the angular content of these bins. At high energy, E > 6 kev, it is impossible for the full angular range to populate a bin and contribution from higher energy events, which could loss energy to populate the bin, are rapidly diminishing, thus forward directed events are the majority of counts in these bins leading to significant corrections to cosθ. Comparing the acceptance correction for analysis method C for the four relevant experimental geometries we can see that by increasing the dead material that 3 (E) gains a steeper slope rotating about an energy slightly less than the peak of the β-spectrum (see Figure

110 / A o ) 3 Acceptance Correction ( 1 - A Geo A 8-9 Geo B 8-9 Geo C 7 Geo PENELOPE GEANT Reconstructed Energy (kev) Figure 3.22: 3 correction, which is defined as 3 = 1 β recon cosθ /(1/2β true ), for the geometries used between are plotted versus reconstructed energy events are simulated to generated the presented corrections. These corrections are for analysis choice C and are compared to similarly derived corrections based on GEANT Applying Monte Carlo Scattering Corrections Application of the scattering corrections is broken into several steps to allow for direct comparison of the corrections between the simulation groups and to allow for scaling of the scattering Typespecific correction to obtain agreement with the measured scattering fractions. Corrections summarized in Table 3.6 are fractional and defined by the following procedure. Backscattering is corrected in three steps: for each step, a specific event class is correctly assigned and compared to the previously calculated asymmetry, thus the three successive corrections, 2, 21, and 22, are for misidentified Type s, Type 1 s and then for Type 2/3 s. Summing the 2i s gives a fractional correction that takes A 1 A 1 (1 + 2 ) A 2. The final correction factor 3 is from the angular acceptance and defined as A 2 (1/2β)/ β cosθ A. In this method, a separate scale factor can be assigned to bring Monte-Carlo calculations of the backscatter fractions into agreement with "calibrating" measurements, such as the total backscatter fraction. For example, if only the Type 1 total backscatter fraction differs from the observed total Type 1 backscatter fraction, then only the 21 correction is scaled leaving the other scattering 94

111 corrections unaffected. In practice, however, the correction factor is not directly scaled, instead the simulation results are reanalyzed with the event class specific weights adjusted to match the measured rates and all corrections redetermined. Most geometries require scaling of 15 25% for the Type 2/3 events and since the Type 2/3 scattering is closely related to thin foil scattering the misidentified backscatters are scaled by the same fraction Optimization of the Energy Analysis Window In the previous sections the corrections to the asymmetry due to detector response, backscattering, and angular acceptance have been presented as a function of energy and in all cases these effects tend to impose a significantly larger bias at energies below 2 kev or greater than 65 kev. To determine an analysis energy range that minimizes the systematic uncertainty while maximizing the counting statistics governed by the shape of the β-spectrum and the signal to noise ratio, one can perform the following analysis. If the statistical uncertainty in an energy bin is σ i and σ i ( j ) gives the correlated systematic uncertainty then we can describe the total uncertainty for a choice of analysis energy window as σ stat 2 1 = E max i=e min 1/σi 2 ( E max σ sys 2 i=e = min σ i ( j )/σ 2 ) i j σ stat 2 σ tot (E min ;E max ) 2 = σ stat 2 + σ sys 2 (3.45) where the magnitude of the systematic uncertainties are weight by the statistics. By varying upper and lower energy limits for the analysis window the optimal energy range can determined. Figure 3.23 shows an example of this analysis where in the left panels the individual systematic and statistical uncertainty are shown for each energy bin and on the right the raw statistical uncertainty and total combined uncertainty. In this example analysis, we see that the rapidly increasing the systematic uncertainties at low energies lead to E min = 175 kev. 95

112 Energy (kev) A [%] A [%] A [%] A [%] Energy (kev) Energy (kev) Energy (kev) Energy Uncertainty Missed Backscatter Uncertainty Angle Effect Uncertainty Statistical Uncertainty (kev) E hi (kev) E hi E low (kev) A Combined A Statistical E low (kev) Figure 3.23: The energy dependence of systematic and statistical uncertainties are shown in the panels on the left and on the right the statistical and combined uncertainties are presented for a range E low E hi. Table 3.6: Summary of the scattering corrections 2 and 3 for various foil geometries. The energy range was restricted to kev for this analysis. MWPC (µm) Trap(µm)

113 3.5 Detector Response Functions The pulse height spectrum, N(H), measured by either detector is a convolution of the true physics spectrum, Eq. 3.1, the point response function R(E,H) which is normalized to unity, and the unnormalized detection efficiency η(e,θ). It is common to define the product of the point response function and the detection efficiency as the detector response function. Here we have explicitly acknowledged that detector response is dependent on the angular distribution via the detection efficiency. We can write the differential count rate in a detector as dn 1 (H) dh Q π { } de dθ sinθ Γ(E)F(E)R(E,H)η(E,θ)(1 βa(e)cosθ) + δ (3.46) where the energy dependence of the asymmetry is included in A(E) and δ is any residual background. In the case of a perfect detector R(E,H) would simply be a delta function, translating a given true energy to a single ADC value, and the efficiency would be separable to η(e,θ) = η 1 (E)η 2 (θ) in energy and angle where η 1 (E) = 1 and η 2 (θ) = U( π/2 ± θ) with U(θ) being the unit step function and the sign being determined by which detector we are considering. Physically, this refers to a detector with zero backscatter or angular dependence in the acceptance. Event Selection and Identification The effective 2 2π acceptance of the spectrometer adds a subtle complication to the idea of the response function. Incident radiation is both direct and backscattered from the opposite detector. Thus some care must be taken in the selection of events for building R(E,H). A composite response can be defined as R g (E,H,θ) = R dir (E,H,θ) + R obk (E,H,θ) + R ms (E,H,θ), (3.47) where the global detector response R g is defined as a sum of the direct response R dir, the observable backscatter R obk, and the missed backscatter R ms. In this notation the detection efficiency η(e,θ) is included in the response, so that R(E, H, θ) captures the angular dependence of both detection probability 97

114 and energy loss. Experimentally, one can not separate the direct response from the missed backscatter response and any measured response function becomes the sum of one and two detector responses, R g = R 1D + R 2D. The two detector response can either be in the measured energy in both detectors, a system response, or one could simply ignore the second detector using it only for particle identification. In the following analysis the calibrated visible energy will be summed for the backscattering events and assigned to a side using timing in formation where available and the scintillator trigger side in the case of Type 2/3 events. Response function will then correspond to the global detector response summed over all event classes. A Semi-empirical Model The semi-empirical model developed by Yacout [8] describes each part of the response function as with a functional form and amplitude, which are both energy dependent and can be fit to a few simulated or measured point response spectra and extrapolated across the spectrum. The response function is broken into five functional forms: R(E,H) = A 1 (E)G[E E,H,σ(E E)] + A 2 (E)G[E E,H,σ(E E)] ) + (A 3 (E) + A 4 (E)e α(e)(h E) + A 5 e β(e)(h E) F[E E,H,σ(E E)], (3.48) where E is energy of the incident particle, H is the corresponding detected pulse height, σ(e) is the energy dependent width which will be taken from previous sections as 2.5E, A i (E) are the amplitudes of the terms, α(e) and β(e) describe the tails the exponential terms, and the G and F functions are defined as G[E,H,σ(E)] = e (H E)2 2σ 2, (3.49) [ (H E) F[E, H, σ(e)] = πerfc ], (3.5) 2σ corresponding to a Gaussian and the complementary error function. The amplitudes of this model determine the contributions from: 1- full energy peak, 2- escape peak, 3- flat continuum, 4- short 98

115 Intensity arb. units Summed Spectra Full Energy Peak Escape Peak Flat Continuum Short Term Exp. Long Term Exp Figure 3.24: parts. The Yacout semi-empirical response function model is shown in terms of its constituent exponential tail, and 5- long exponential tail, shown in Figure In the original work of Yacout, this model was used to describe X-ray spectra on a SiLi detector, therefore the escape peak and flat continuum which are components of a X-ray spectra are included. We have found that these components are not required to explain the response to electrons on a scintillator, thus we set A 2 (E) = A 3 (E) =. As we will show, the multiple scattering and backscattering tail in the response function can be well described by adjusting the amplitudes and decay constants of the short and long exponential term. The purpose of the complementary error function is to cut the functional forms of each term off at the center of the full energy peak, preventing a non-zero amplitude above the A 1 term. Using this set of functions for R(E,H) we have five free energy dependent parameters that describe the response function. This model can then be fit to either the simulated or measured detector response. In the case of using simulated response functions, the line spectra can be tailored so that unique line energies are produced at set calibration points throughout the energy range. This eliminates the complicated structure of internal conversion calibration sources which have multiple lines convoluted into an inseparable response function. Line sources were simulated every 1 kev between 8 kev and the energy dependence of the parameters in Eq where determine from a fit to the resulting spectra, see Figure 3.25 and 99

116 Figure The short tail exponential term, A 4 (E), was also set to zero in this analysis. Using only the full energy peak and the long exponential tail was sufficient to describe the point response functions. The response histograms were normalized by total number events generated at that energy, so that the integral of the histogram gives the detection efficiency, shown in Figure 3.25s. The A 1 (E) term was factored out of R(E,H) to enforce the normalization, so that the integral of the fit function is equal to the histogram integral, so that detection efficiency is not altered by the fit. Instead of fitting a parameter s energy dependence to a functional form, a cubic-spline interpolation routine is used to determine parameter value between calibration points [81], see Figure It is interesting to note the behavior of ρ(e) shown in Figure Since the line spectra that are fit in this analysis are Monte Carlo generated, the width of the distribution should be constant, σ = 2.5E. However, in the fit function the width of the distribution enters in both the Gaussian representation of the full energy peak but also in the complementary error function. This allows Eq. 3.5 to smoothly cutoff as E H. Therefore as the contribution from the multiple-scattering tail to the overall shape of the response function increases, the optimal value of ρ is reduced to allow error function to cutoff more sharply. A full detailed analysis of the covariance of the fit parameters would be required to further assess any constraints on the individual parameters. Integrating Eq with R(E,H) and η(e,θ) determined from Monte Carlo will allow us to describe the measured spectrum and by extension the energy dependence of the asymmetry. Forming the simple sum over the difference asymmetry from Eq. 3.46, where the leading order form of the asymmetry, A o (λ), is separated from the energy dependent part C(E), A(E) = A o (λ)c(e,λ), [ N (H) N ] (H) N (H) + N = A o (λ) P [Γ i(e)η i (E)C i (E,λ)β i (E) cosθ i R i j (E,H)] j, (3.51) (H) j [Γ i (E)η i (E)R i j (E,H)] j where, in this notation, the contents of visible energy, H, bin j is determined by summing over the product of true energy, E, bin i with the corresponding efficiency, η i (E) and response functions R i j (E,H). In the case that R(E,H) = δ(e H) and η(e) = 1 the right hand side of Eq would simplify to A o (λ)c(h,λ) P β(h) cosθ with H = E. An example of this convolution procedure is shown in 1

117 Response for 71 kev Response for 142 kev Response for 213 kev Response for 284 kev Counts.2.15 Counts Counts Counts Energy (kev) Energy (kev) Energy (kev) Energy (kev) Response for 355 kev Response for 426 kev Response for 497 kev Response for 568 kev Counts Counts Counts Counts Energy (kev) Energy (kev) Energy (kev) Energy (kev) Response for 639 kev Response for 71 kev Response for 781 kev Response for 852 kev Counts.15 Counts.15 Counts.15 Counts Energy (kev) Energy (kev) Energy (kev) Energy (kev) Response for 923 kev Response for 994 kev Counts.1 Counts Energy (kev) Energy (kev) Figure 3.25: Fits to a simulated response function are shown for energy up to 8 kev in 1 kev intervals. The thick black line represents the full response function, the dashed line is the full energy Gaussian, and the dotted line is the long tail exponential. 11

118 E (kev) Energy (kev) 8 1 σ(e) = ρ(e) E Energy (kev) 8 1 A 1 (E) Energy (kev) 8 1 A 5 (E) Energy (kev) 8 1 α(e) Energy (kev) Efficiency η(e) Energy (kev) 8 1 Figure 3.26: Fit parameters from the simulated response functions are shown as a function of incident energy. The fit parameters from upper left are energy difference, E, between the visible and incident energy, the width parameter ρ, from σ = ρe, the amplitude of the full energy peak A 1 (E), the amplitude of the long tail exponential term A 5 (E), the exponential parameter α(e), and the detection efficiency η(e). Results of a polynomial interpolation (red line) and a cubic-spline interpolation (blue line) are shown. 12

119 West Convoluted β-spectrum Intensity Residual [%] Energy (kev) Figure 3.27: The energy spectrum generated by folding R(E,H) (black), determined from fits to Monte Carlo line sources with the theoretical β-spectrum, is compared to the results of a full simulation of UCNA for geometry C of (green). The discrepancy between these spectra is shown in red and is less than 5% between 2 6 kev. The asymmetry as a function of visible energy from the simulation (black with error bars) is compared to Eq (blue) and the theory energy dependence of the asymmetry (blue dashed) A Figure 3.27, where the predicted N (H) and N (H) spectra are compared to results from a full simulation of the same geometry, along with an asymmetry generated using Eq Determining the final value of A o or λ is accomplished by minimizing a χ 2 contour between Eq and a simulated or measured asymmetry. Assessing the uncertainty or bias in the asymmetry due to this method requires evaluating the discrepancy between the measured/simulated spectrum and the predicted spectrum and propagating the residuals through to a calculation of the asymmetry. Figure 3.27 shows that the discrepancy between the spectra, shown in red, is < 5% between 2 6 kev. When the asymmetry is calculated from both methods, we find that there is a large deviation from the theoretical shape (dashed blue line) at low energy. In fact, the convoluted asymmetry does not go to zero as E, this is because the efficiency for 13

120 detecting the low energy end of the β(e) curve is zero and the events that populate this energy range are from the multiple scattering tails of higher energy events and therefore carry a higher intrinsic asymmetry to lower energy bins. Getting these tails correct when modeling the response function will dominate the low energy shape of the spectrum and the predicted shape of the asymmetry. It is useful to point out that backscattering and angular acceptance have not been corrected for in Figure 3.27, which would, in part, resolve the low energy discrepancy in the asymmetry. A comparison between the measured and Monte Carlo derived detector response functions allow for a straightforward assessment of the semi-empirical model as well as the multiple-scattering and energy loss predictions of the simulations. The measured response can be fit to Eq. 3.48, as shown in Figure 3.28 for 113 Sn and 27 Bi after calibrating to the expected E vis, however as previously discussed these are not mono-energetic lines and therefore the convolution of the response to the inner shell lines K, L, and M leads to an artificial broadening of the full energy peak, especially in the case of 27 Bi where the K and L lines are separated by 5 kev. This exercise, does however show that the measured response is also fit well using only the A 1 (E) and A 5 (E) terms in the semi-empirical model. Using the response function fit parameters, the spectra of calibration sources can be generated as N(H) = = n i=1 ( n ) ω i R(E,H)δ(E E i ) de i=1 ω i R(E i,h), (3.52) where the sum is over the relevant conversion lines with the intensity of each line given by ω i, and δ(e E i ) is the Dirac delta function, which represents the discrete lines of the conversion source. Line energies and intensities relevant to this analysis are listed in Table 3.1. A straightforward comparison of the generated calibration source spectra to the measured spectra, shown in Figure 3.29, tells us that peak centers and widths are well reconstructed by this method, meaning that E and ρ(e) extracted from the Monte Carlo are capturing the energy dependence of scattering in the detector material. In both measured spectra there is an excess of low energy events which could be due to incomplete background subtraction, detector non-linearities not captured in the fit, or improper pedestal subtraction. Figure 3.28 shows that 14

121 113 East Sn Measured Response: East Measured Response Bi : Counts 2 15 Counts Energy (kev) Energy (kev) 113 West Sn Measured Response: West Measured Response Bi : Counts Counts Energy (kev) Energy (kev) Figure 3.28: The response functions for 113 Sn and 27 Bi for the east and west detector packages are fit to the semi-empirical model. 15

122 these excess events can be accommodated by this model in a direct fit. However we then return to the problem of artificially increasing the width of the peak. The decay parameter α(e) and the amplitude A 5 (E) of the exponential tail can be varied to improve the agreement with data, however the real test is how varying any of these fit parameters will impact the extracted asymmetry. An analysis was performed where a scaling factor γ i, between.5 3.5, was applied separately to A 5 and α and the best fit asymmetry parameter A min was extract by minimizing χ 2 between Eq and a simulated asymmetry curve. A similar χ 2 test was performed to determine the best fit values of γ by comparing the measured and folded calibration spectra. Results of the analysis show that the asymmetry is rather robust against variation of A 5, A 5 /A 5 = 3 A/A.1, see Figure 3.3. The same range for the scaling of α leads to roughly a 3% change in A. We have found that only scaling A 5 up by 32.3(4)% leads to the best agreement with the measured calibration spectra. Coincidentally this is approximately the same amount that the backscatter fractions are under predicted by the Monte Carlo. The relative change in A with respect to the scaling of the model parameters is smooth and is well fit by a quadratic, f (γ i ), where γ i is the scaling factor. Using the functional forms for the model parameters, we can estimate the uncertainty in the extracted asymmetry from f (γ i ± 2σ γ ), where σ γ is the fit errors determined from the χ 2 = ±1,2... contours from the calibration source analysis and are 1.5%. This allows us to estimate an uncertainty of A/A(γ) <.6% at the 2σ confidence level, therefore negligible at the current precision of UCNA. While reassuring, we must consider that this uncertainty is only valid if the simple scaling of A 5 (E) is sufficient to match the measure response. 3.6 Off Axis Radial Field Expansion Original maps of the spectrometer field provided by AMI the axial, B z, and radial, B ρ, components in a coarse 5 cm x 5 cm grid. To integrate trajectories through the field map, numerical interpolation was required to find the field components in all space. Initially, a bi-cubic spline interpolation routine from Fortran 77: Numerical Recipes was used, however it was found to be computationally intensive limiting the total statistical precision of the simulations [78]. Additionally, subsequent in situ measurements of the field only measured the axial component on axis, B z (,z). Therefore a method for calculating the off-axis 16

123 Intensity Intensity Energy (kev) Energy (kev) Figure 3.29: Shapes of the 113 Sn (left panel) and 27 Bi (right panel) generated by the simulated response function (red histogram) are compared to the calibration measurements from the 27 geometry (black circles) Variation of ρ(e) Variation of α(e) / A o 1 A min Scale Factor Figure 3.3: The fractional change it the asymmetry relative to A o is shown versus a scaling factor applied A 5 (E) and α, effectively altering the amplitude and shape of the exponential multiple-scattering tail. These curves are well fit by a 2 nd order polynomial which can be used to estimate in the uncertainties in the asymmetry due to these variations. 17

124 Tanh(z) fit to Measured Field Map (,z) Telsa B z Z (cm) Figure 3.31: An analytical model for the on axis value for B z is fit to the original AMI field map provided with the spectrometer. components based only on B z (,z) developed by Jackson [82] was adopted. A detailed calculation of the field components is given in Appendix A with the result that we can define : B z (r,z) = B r (r,z) = n= n= ( 1) n z 2n f (z) 2 2n (n!) 2 r2n f (z) 2 2n+1 n!(n + 1)! r2n+1 (3.53) ( 1) n+1 z 2n+1 where f (z) is the on-axis axial field B z (,z). In the case of UCNA, the on-axis field B z (,z) is found by a four parameter fit of B z (,z) = a + btanh((z + c)d) to the field map, Fig 3.31 and the partial power series expansion is taken to fourth order for use in the simulation. The small dip in the measured field for z > 226 cm is beyond the back of the detector package and is not important in the simulation. To check the accuracy of the trajectory integration, electrons of random energies and pitch angles are emitted from the front face of the wire chamber back toward the decay region. The angular distribution of electrons arriving either at the decay trap foil or back at the wire chamber window after being reflected by the increasing magnetic field is compare between the analytic and numerical methods. The mean angular deviation was statistically consistent with zero using a test simulation of 1 6 events, identical 18

125 event list, and an allowed error of 1 5 in the 4 th order Runga-Kutta trajectory integrator. Residual gas scattering was turned off in the space between the wire chamber face and the decay foil and particles were not propagated into material. These results suggest that there will be negligible differences between using the analytical model of the field and the measured map Non-uniform Field Effects Dips, ripples, and other non-uniformities in the spectrometer magnetic field can lead to trapped trajectories and magnetic mirroring in decay region. Small dips of (3 6) 1 3 T, which can trap high pitch angle electrons in the central 1 T region, are known to exist due to operational issues with the shim coils. The initial field measurements reported by Plaster [61] found the central field to be uniform to B < ±3 1 4 T, when the SCS was first installed and commissioned at LANSCE. Since then the field uniformity has decreased due to repeated magnet quenches. Representative measurements of the on axis magnetic field, B Z, are shown in Figure 3.32 for From these measurements we see that from the field maximum has fallen by.25 T, the central field dip has increased from (3 5) 1 3 T, and malfunctioning shim coils on the west side of spectrometer have widened the dip to include the majority of the western half the decay trap. The condition for the reflection of a charged particle heading into an increasing magnetic field is v (z o ) v (z o ) = B max tan 1 θ < 1, (3.54) B (z o ) where v and v are the initial parallel and perpendicular components of velocity, B is the initial magnitude of the field and B max is the maximum field strength. In 27 the ratio B max B = 1.29 giving a trapping angle of θ t > 86.92, corresponding to 5.38% of the electrons generated at the bottom of the dip. Simulations performed using GEANT4 which varied the pressure in the spectrometer from torr found the trapped electrons will scatter on residual gas molecules and exit the field dip with their directions randomized with times on the order of a millisecond. Therefore, the symmetric field dip present in 27 will conservatively create a.6% bias to the asymmetry, [83]. 19

126 Repeated quenching of the SCS magnet is thought to have caused persistence switch heaters on a few of the shim coils to fail over the years, resulting in the asymmetric non-uniformities seen in the measurements. Having an asymmetric field introduces an additional systematic correction; not only can electrons be trapped in the field dip, but because B west max < B east max, there is a set of events that can only escape from the west side of the trap. In effect increasing the missed backscatter fraction. For the majority of 28, B east max B west max = B max = 5 gauss, applying the reflection condition from Eq we find a critical angle of θ t > 89.2 or.1% of the total flux. This increase in the backscattering fraction results in a A <.1% for 28. In 29 B increases to introducing a bias of A <.2%. At the level of precision that the field is measured and included in particle tracking simulation these results are considered as a systematic uncertainty and not a true correction to the asymmetry. 3.7 Backgrounds Time Dependent Backgrounds The run cycle of UCNA is such that neutrons are loaded into the decay trap and the decay products are counted for approximately one hour. The trap is then unloaded in such a way that the polarization can be measured, either prior to the trap being loaded or immediately after the polarization measurement. The background rate is monitored for six to twelve minutes. The length of the background runs were nominally a 1:5 ratio with the signal runs. Sometimes the data collection cycle was compressed so that entire octet could be collect during a single night of beam time. In the 27 data the background were always measured after the depolarization run. The run cycle was changed in 28 and is used for all subsequent data collection to include two sets of twelve runs which alter the order of background measurements and spin state in the trap. One could imagine a scenario where the an external source could cause the background rates to drift with time, i.e. the temperature cycling in Area B increasing the dark rate in the PMT as the area warmed up. Using the Monte Carlo model described in Appendix B a background rate which varies linearly or quadratically with time can be added in to determine the bias to the asymmetry. Figure 3.33 shows a 11

127 1 Map of the Central B z (gauss) B z Z (cm) Figure 3.32: A Hall probe was used to map the on axis value of B z across the decay region at various times through each runs cycle. Representative measurements for the 27 (black), 28(red), and 29 (blue) are presented and show an increasing reduction of the magnitude of the field on the west side of the spectrometer. The reduced field strength at z > 4 cm or z < 1 cm is due to the tailored field expansion region between the decay trap and detectors. The z-axis is measured from the flange that seals the west detector approximately 1 m from the end of the decay trap. These measurements were performed with a Hall probe mounted on a cart which used the same rail system the decay trap is mounted on, therefore mapping was done without the decay trap installed. 111

128 Graph Background difference Run Time (s) Background Rate A2 A5 A7 A1 B2 B5 B7 B Run Time Figure 3.33: Time varying background rates overlaid with the data taking cycle for a A to B octet. The black hashed areas denote a signal measurement period and red hashing are background measurements. A linearly increasing background, red curve, and a quadratic background, blue curve, are set such that the maximum rate is 1% larger than the initial. The top panel shows the difference of the measured background and the average rate during the signal measurement. typical octet style run cycle where the Ai, i (2,5,7,1), denote hour long beta-decay measurements and the red hashed areas are the corresponding background measurement periods. For runs 2 and 5 the background is measured prior filling the trap with UCN, runs 7 and 1 measure the background afterward. A five minute gap after the signal run is included to account for the depolarization run. The main experimental difference in the A and B sets of runs is the spin state, whether the AFP-spin flipper is on during that run, because of the loading efficiency difference between the two spin states this will cause the signal to noise of A2 run, for example, to be different than a B2 run even with a constant background. In this model the peak decay rate is set to 1 s 1 for the spin flipper off and a 3% reduction in the rate with it on. The initial background rate is set at.5 s 1 and the relative change due to either a linear increase or at the apex of the quadratic is set to 1%. By varying the background rate in this manner it is readily apparent that the measured background rate is not equal to the average rate during a measurement 112

129 of the asymmetry, see Figure A simple calculation of the asymmetry would result in ( (R1 + B 1 ) B ) ( 1 (R2 + B 2 ) B A m = 2) ((R 1 + B 1 ) B 1 ) + ((R 2 + B 2 ) B 2 ) = (R 1 R 2 ) + ( B 1 B 2 ) (R 1 + R 2 ) + ( B 1 + B 2 ) (3.55) where R i is the event rate in detector 1 and 2, B i is the background rate during the signal measurement, B i is the measured background rate, and B i is the difference between the background rates. In the scenario where the background is only drifting linearly, the offset between the measured and actual background is constant if the background is always measured at the same time relative to the signal. However, with a quadratic dependence this is not the case. If the backgrounds are periodic on the time scale of the measurement cycle, then one can arrange the background measurements to cancel this drift. The bias in extracting a simulated asymmetry from a measurement with linear or quadratic backgrounds is shown in Figure 3.34 as a function of a scaling parameter, ρ, to the size of the background drift. At ρ = 1 that maximum drift is 1% and at ρ = 5 it is 5%. These results show that the bias to the asymmetry is roughly 1/1 th of a quadratic background drift and 2/3 for a purely linear drift. This analysis assumed that the cause of any background drift is external and is equally shared by the detectors, this does not have to be the case. Detailed analysis of the time dependent behavior of the background is required for each specific case to assess the impact to any measurement observable. It will be presented in later chapters that there is no evidence of correlated background drifts in UCNA Neutron Generated Backgrounds UCN which captured on materials in the spectrometer undergo (n,γ) reactions creating photons which may be incident on and trigger the β-detector s scintillator. If a Compton electron is created in the foils or scintillator it may have enough momentum to propagate through MWPC causing a false coincidence event and irreducible background. In the cold neutron experiments, this unsubtracted background reduces the measured asymmetry by up to 3% [4 6, 9]. A significant amount of work has been done in the latest iterations of Perkeo II and Perkeo 3 to shield the detectors from the neutron beam reducing the correction 113

130 A Linear SuperRatio Linear Drift Octet Octet, Bck After Octet, Bck Before Octet, Bck Optm Figure 3.34: The bias in the asymmetry is shown as a function of the background scale parameter ρ (ρ = 1 is a 1% drift, ρ = 5 is a 5% drift). The think line represents a super-ratio analysis with a linear drift and background measured at the beginning of the run and the dot-dashed line is the octet analysis for the same linear drift. Quadratic drifts were analyzed with an octet where the background was measured before (thick dashed), after (thick line), and alternating (dotted). to the asymmetry to less than.5%. Because the UCN density is roughly five orders of magnitude smaller in the decay region than in CN experiments, almost all UCN in the decay region contribute to the decay rate as oppose to the extremely small fraction of CN that decay in the detector acceptance. Given the high efficiency with which gamma events are vetoed by the MWPC, we expect the correction to the asymmetry in UCNA to be suppressed to well below this level. The effect on the asymmetry can be calculated by forming the super-ratio of the rates in detectors 1,2 and with spin flipper on and off, R 1(2) On(O f f ), ( ) R 1(2) On(o f f )(E) = N1(2) On(O f f ) (E) (1 ± Aβ(E)cosθ) + γ 1(2) (On)O f f (3.56) where the γ terms are added in as the residual neutron generated component of the background proportional to the total rate N 1(2) On(O f f ). When integrating over rates it is important that γ is not constant since it is proportional to the entire density of neutrons in the trap and not the decays counted in an energy interval, thus there is some energy dependent coefficient that normalizes γ to the energy window used for 114

131 integration so that the overall rate, can be factored out of the super-ration and asymmetry calculations. Using the simple, difference over sum, asymmetry : A meas (E) = R1 On (E) R2 On (E) R 1 On (E) + (3.57) R2 On (E), and the super-ratio method where the asymmetry is given as S(E) = R1 On (E)R2 O f f (E) R 2 (3.58) On (E)R1 O f f (E), A meas (E) = 1 S(E) 1 + S(E). (3.59) we can explore the size of this correction for the two analysis methods. Rearranging Eq and subtracting S o, the value of the super-ratio in the absence of any residual backgrounds is (1 A β cosθ ) 2 /(1 + A β cosθ ) 2. from both sides we can define the change in the super ratio as a function of the residual backgrounds S S o = (1 A β cosθ )2 + (1 A β cosθ )(γ 1 On + γ2 O f f ) + γ1 On γ2 O f f (1 + A β cosθ ) 2 + (1 + A β cosθ )(γ 1 O f f + γ2 On ) + γ1 O f f γ2 On (1 A β cosθ )2 (1 + A β cosθ ) 2. (3.6) Several physically reasonable scenarios exist: A - no detectable residual background, γ 1 On = γ1 O f f = γ2 On = γ2 O f f = B - both detectors measure equal residual rates, γ 1 On = γ1 O f f = γ2 On = γ2 O f f = γ C - there is an equal but spin state dependent rate in both detectors, γ 1 On = γ2 On = γ On and γ 1 O f f = γ2 O f f = γ O f f D - detectors 1 and 2 measure a different but spin independent rate, γ 1 On = γ1 O f f = γ1 and γ 2 O f f = γ2 On = γ 2 E - And any number of cases where one or both detectors measure unequal rates that are spin dependent 115

132 A Simple Asy, Γ 1 Γ 2 Simple Asy, 2Γ 1 Γ 2 Γ On 1 Γ Off 1 Γ On 2 Γ Off 2 Γ On 1 Γ Off 1 Γ, Γ On 2 Γ Off 2 2Γ Γ On 1 Γ On 2 Γ, Γ Off 1 Γ Off 2 2Γ Γ 1 On Γ Off 1 2 Γ 2 On 4 Γ Off 3 2 Figure 3.35: Evaluations of Eq. 3.6 and its propagation through to a calculation of the asymmetry for a few cases of neutron generated background rates with the simple asymmetry. Figure 3.35 shows the correction to the asymmetry for several scenarios as a function of the residual background to signal ratio over an extended range, up to 1%. In the case where the backgrounds are independent of detector and spin state regardless of analysis method, the dilution is linear with a 1:1 slope. If there is a spin dependence the correction, which is greatly increased, in the super ratio analysis is equal to the case of the difference over sum asymmetry analysis where detector 1 and 2 see different residuals. If there is a spin independent difference between the detectors the correction is suppressed in the super ratio relative the normal analysis. In the worst possible cases the residual background is different for each permutation of spin and detector dependence. To first order we should expect that the neutron generated background is proportional to the density of UCN in the trap and solid angle between the detectors and the trap, both of which should be spin independent Experimental Determination of Neutron Generated Backgrounds Three methods were used to determine the γ flux at the detectors from neutron capture : A - A 1/4 inch acrylic sheet was positioned between the decay trap and the detectors effectively stopping all electrons from neutron β-decay while allowing capture γ s through, the β-blocker method. The acrylic was positioned near and far from the decay trap allowing for an estimation of 116

133 63 Cu 65 Cu E (kev) I γ /I k E (kev) I γ /I k Table 3.7: Neutron capture γ s from 63 Cu and 65 Cu. the forward Compton component created in the plastic itself. B - Use the fact the MWPC has a high efficiency for rejecting γ events by taking the difference between background subtracted spectra built using the MWPC cut and not using it. This difference spectrum should be residual backgrounds not generated by the cosmic or ambient flux. C - Fit the background subtracted spectrum from β -decay past the end point, 1-2 MeV, and extrapolate the residual back on under the measured spectrum. This is the method used in most cold neutron experiments. To estimate the shape of the background and the scaling between the full γ flux and the backwards Compton component due to neutron capture, a Monte Carlo simulation of the capture γ s was performed with an intensity weighted event generator. Since the gamma cascades generated by the capture of neutrons on a material can be extremely complicated, in natural copper each isotope has greater than 25 levels, the simulation assumes each event creates one photon such that the resulting intensity spectrum matches the what is empirically measured and reported by the NNDC s CapGam website [84]. The decay trap is natural copper which is 67.87% and 3.6% of 63 Cu and 65 Cu respectively. a list of the highest intensity gammas is given in Table

134 h4 Entries 27 Mean RMS Geometry A : Trigger Rate n-captures Trigger Scintillator n-captures Trigger MWPC + Scintillator h3 Entries 1 Mean RMS Capture γ Lines Counts 2 1 Counts Energy (MeV) (a) (n,γ) lines from 63 Cu and 65 Cu Energy (kev) (b) Energy spectrum of triggering capture γ s. Figure 3.36: (n,γ) lines from naturally abundant copper isotopes, shown in panel (a), were generated on the walls of the UCNA decay trap and propagated through the spectrometer geometry. The energy spectrum of triggers on the scintillator are compared to events triggering both the scintillator and the wire chamber shown in panel (b), suggesting a factor of 2 suppression. Angular Distribution of Compton Electron Emission The angle an electron is emitted from an atom, φ, due to Compton scattering can be derived in terms of the incident photon s scattering angle, θ, and initial energy, hω, as cotφ = (1 + α)tan θ 2 (3.61) where α = hω/m e c 2, the photon energy in units of the electron rest mass. Similarly the electron energy, in units of its rest mass, is also given in terms of the scattering angle and photon energy E e m e c 2 = 2α α + (1 + α) 2 tan 2 φ, (3.62) showing that the electron emission energy is maximized when φ or π, with E max being 5 kev. For Compton electrons to have an impact on the measured rates after cuts, either φ > π/2 or through multiple scattering, its direction must be reversed and possess enough energy to escape the scintillator and trigger the MWPC. Simulations based on the (n,γ) lines from copper in the decay trap show that the 118

135 majority of backscattered Compton electrons that can trigger the MWPC are generated by the E γ > 7 MeV lines, which corresponds to an α of Klein and Nishina [85] have derived the probability of a photon Compton scattering into a sold angle dω from a free atomic electron dσ(θ) dω { [ = r2 o 1 ( ) cos 2 θ + α2 (1 cosθ) 2 ]} 1 + α(1 cosθ) 1 + α(1 cosθ) (3.63) where r 2 o = e 2 /m e c 2 and since we can arrange the coordinate system to have azimuthal symmetry dω = 2π sinθdθ. By exploiting the relationship between θ and φ, Eq can be integrated over some angular range (θ 1 θ 2 ) which can be transformed via Eq to give the probability of the electron scattering into the range (φ 1 φ 2 ). An expression for integrating Eq from to an angle θ 2 was derived by Davisson and Evans [86], [{ σ(α,θ 2 ) = πro 2 (4 + 1α + 8α 2 + α 3 (4 + 16α + 16α 2 + 2α 3 )cosθ 2 } + (6α + 1α 2 + α 3 )cos 2 θ 2 2α 2 cos 3 θ 2 ) (2α 2 (1 + α α cosθ 2 ) 2 ) 1 + ] (α2 2α 2) α 3 ln(1 + α α cosθ 2 ). (3.64) To determine the fraction of electrons which scattering into φ > π/2 using Eq one can calculate 1 σ(α,θ 1 )/σ(α,θ 2 ), where the θ correspond to φ = π/2 and π resulting in % of the electrons being backscattered. From the kinematic described by this scenario, it is reasonable that any momentum transferred to the electrons must be in the forward direction and since both GEANT4 and PENELOPE3 sample Eq to determine the angular distribution of Compton scattering, we should expect the simulated results to be similar. Therefore we must consider that either forward Compton electrons from foils in the spectrometer or multiple scattering of electrons generated in the scintillator are the major contributors to the (n,γ) induced coincidence rate in the detector package. Another condition to consider is the fact that in Eq the electrons which the photons scatter on are at rest. Simple simulations of 8 MeV photons incident on slabs of mylar and plastic scintillator with infinite extent in the x,y-plane and thicknesses representative of the UCNA geometry were performed with 119

136 Angular distribution of emerging electrons Energy distribution of Electrons PDF (1/deg).1 1e-5 1e-6 1e-7 1e-8 1e-9 Mylar 12 um Mylar 25. um PVT 3.5 mm Mylar.7 um Mylar 6. um PDF (1/MeV) 1e-1 1e-11 1e-12 Mylar 12 um Mylar 25. um PVT 3.5 mm Mylar.7 um Mylar 6. um 1e-1 1e angle (deg) (a) Angular distribution of Compton Electrons. 1e Energy (MeV) (b) Energy distribution Compton electrons Figure 3.37: Angular and energy distribution of Compton generated electrons upon exiting the mylar and plastic scintillator slabs with thicknesses representative of the UCNA geometry due to 8 MeV incident photons. Because of the triggering condition of the detector packages the energy of backscattered electrons are shown for the scintillator and the energy of the transmitted electrons for the foils. normal incidence and angle of incidence up to 4 for the scintillator and 1 for the mylar. The limiting angles of the incidence were determined from the solid angle from the decay trap to detector package and end cap foils. These simulations show that forward going electrons created by Compton scattering in the Mylar decay trap foils or even the plastic collimator contribute with similar probability as backscattered Compton electrons from the scintillator itself. Scaling the scattering probability by solid angle from the decay trap, which reduces the probability by a factor of 2 for materials in the detector package relative to the those in the decay trap, we see that even the.7 µ has a similar impact as the PVT. Taking into account the energy distribution of the electrons from each material, scattered in the appropriate direction to satisfy the MWPC coincidence, we find that the forward scattered events have energies up to the full photon energy, 8 MeV, would be measured as a minimal ionizing event around 6 kev because of the thickness of the PVT. Electronics emitted backward from the scintillator are from the multiple scattering of large pitch angle Compton s and therefore have an energy spectrum peaked at 2 kev. β-blocker Measurement An acrylic absorber was positioned as shown in Figure 3.38 to prevent all β decay electrons from triggering the detector. Therefore any residual signal, after background subtraction, will be due to physics 12

137 15 cm OD,.365 cm thick Acrylic Disk Decay Trap A B β-scintillator Backing Veto MWPC Detector housing ± 1 16 in. from flange Figure 3.38: An acrylic disk was placed between the decay trap and the detector in positions A and B. The disk had a 15. cm OD and a 3.65 thickness. In position A, the disk was mounted on the on a cart and pushed to within 2 inches of the decay trap and in position B the disk was taped to the front of the detector housing. Position A is measured relative to the 16 inch vacuum flange. events correlated with having neutrons in the trap. The high energy photons emitted after neutron capture can pass through the acrylic and, with a small probability, Compton scatter in the plastic scintillator creating an electron which can multiple scatter into the MWPC to create the required coincidence. The major systematic of the this method is that the acrylic disc becomes a source of forward scattered Compton electrons which happen at a much higher probability due to large solid angle and the angular dependence of Compton scattering strongly favors forward propagating electrons. Running this test in a near and far position, labeled A and B respectively, allows us to normalize out the component of the signal due to Compton scattering in the acrylic. The expected rate in this measurement is Γ = Γ (n,γ) (ε FC Ω disk + ε BC Ω detector ), (3.65) where Γ (n,γ) is the total neutron capture γ flux, ε FC and ε BC are the efficiencies for creating a Compton electron which will fulfill the coincidence criterion, and Ω disk and Ω detector are the solid angles to the acrylic disk and the scintillator detector. One can calculate the solid angle to the acrylic disk, of radius R 121

138 and z-axis position z disk, from any position on the surface of the decay trap x(r,z), Ω(z) = 2π θmax sinθdθdφ, (3.66) where θ max is the opening angle to the edge of the disk, tanθ max = Rsin( π φ sin 1 ( r sinφ R )). (3.67) (z disk z)sinφ Integrating Eq over the surface of the decay trap and taking the ratio of the solid angles at positions A and B we find µ = Ω A /Ω B = Subtracting the rates measured in positions A and B, Γ A and Γ B, after scaling Γ B by µ we finding µγ B Γ A µ 1 = Γ (n,γ) ε BC Ω Detector Γ Nbck, (3.68) where Γ Nbck is approximately the trigger rate due to neutron capture in the absence of the acrylic disk. This result is approximate, because the angular distribution of incident photons on the acrylic is slightly difference for positions A and B and therefore, the efficiency for creating a Compton electron is not identical. To account for these effect, the β-blocker experiment was simulated with the results, for positions A and B and an additional simulation with no β-blocker, summarized in Table 3.8. Data was collected with 1 hour of running with the gate valve open and 1 hour of running with the gate valve closed and the spin flipper off. Cosmic ray veto, wire chamber cuts, and beam timing cuts were applied to the signal and background runs prior to analysis. The blocking plastic in position A gives a rate of.128(17) s 1 and in position B.24(3) s 1. Based on this measured an estimate of the rate due to backwards Compton electrons created in the either the scintillator or the backing veto is.1(4) s 1 and an upper limit of.3 s 1 for the neutron generated backgrounds. This set of tests was repeated with the flipper on, but acrylic was only placed on the west side of the SCS. If we assume the east and west see the same flux then we can set a limit of as a correction to the asymmetry. 122

139 Table 3.8: Event triggers generated by Cu(n,γ) interactions in a simulation of the near and far β-block positions and without the acrylic disk in place. The fraction of event triggering the scintillator is consistent between geometries however the fraction that trigger the MPWC is dependent on the position of the disk. Position A Position B No β-blocker n-captures Simulated Scintillator Triggers 8154 ( ) 1354( ) 3916 ( ) MWPC + Scintillator Triggers 424 ( ) 147( ) 23 ( ) MWPC vs. No-MWPC A complementary measurement of the intensity of the (n,γ) background can be derived by viewing events that pass the anode cut as a background. Subtracting this effective background from all triggers, after removing events tagged by the veto detectors, should result in a spectrum comprised of events which trigger the scintillator and not the MWPC. The difference between this spectrum with and without neutrons in the decay trap will determine if there is a (n,γ) background. One issue with this method is that the MWPC inefficiency will mimic the γ signal. The rate when the MWPC cut is applied is R veto = (S veto B veto )(1 ε) where S veto and B veto are the signal and background after MWPC cuts and ε is the MWPC inefficiency. Without the MWPC cut the rate is R noveto = (S veto B veto ) + (S γ B γ ) where S γ and B γ are rates due to gammas normally cut by the MWPC. The difference of these is then R = (S γ B γ ) + ε(s veto B veto ) = Γ γ + εγ n where Γ γ is the (n,γ) flux incident on the scintillator and γ n is the neutron β decay rate. By dividing out Γ n, R/R veto = Γ γ /Γ n + ε, we see that ε adds directly to ratio of the (n,γ) rate and the neutron decay rate. Therefore this method of determining the (n,γ) background will be ultimately limited by the MWPC efficiency, ε, which was previously measured to be 99.5% using a collimated source. The efficiency in-situ could be greater as the longer path length of spiraling electrons in the magnet field leads to higher energy deposition. Translating the measured γ rate into an estimate for the (n,γ) background which biases the asymmetry requires two model-dependent inputs, ε and the fraction of backward going Compton scatter electrons generated. The most conservative estimate is derived by assuming ε = 1 and there is no reduction due to 123

140 East Off Rate (Hz) Rate Off (2-6 kev).919 ±.6 Ratio (2-6 kev).25 ±.13 Rate On (2-6 kev).857 ±.56 Ratio (2-6 kev).213 ± Energy (kev) West Off Rate (Hz).25 Rate Off (2-6 kev).811 ±.65 Ratio (2-6 kev).147 ±.12.2 Rate On (2-6 kev).567 ±.57 Ratio (2-6 kev).112 ± Energy (kev) Figure 3.39: Geometry A. Residual background from the MWPC - No MWPC cut the requirement that electrons are backscattered in the MWPC. Under these assumptions the measured value of R/R veto can be used in Eq. 3.6 to calculate the bias. 3.8 Instrumental Asymmetry Due to Loading Efficiency In addition to detector efficiency differences and nonuniform magnetic field it is possible to introduce a bias in the asymmetry if the two spin states do not equally populate the decay volume. When the AFP spin flipper is not activated, UCN enter and leave the AFP magnet with the same kinetic energy. However when the AFP spin flipper is active, UCN experience a spin flip in a 1 T field. The change in the neutrons energy is E = 2µB 12 nev, which is a significant amount of energy relative to the Fermi potential of the walls. In the transition region between the AFP and spectrometer the loss rate for the spin flipped population is increased to the point that the measured decay rate is reduced by approximately 3%. The large spin dependent loading efficiency prevents one form performing an analysis of the spectrometer 124

141 Table 3.9: Residual rates after No MWPC cut - MWPC cut subtraction method for each geometry for each detector and spin state in the analysis energy window 2 6 kev. Geometry East Off East On West Off West On A Rate.245(67) -.7(61).269(73) -.179(61) Ratio.7(15) -.5(17).75(13) -.47(17) B Rate.919(6).857(56).811(65).567(12) Ratio.25(13).213(15).147(12).112(15) C Rate -.9(64) -.172(58).14(65).96(55) Ratio -.9(22) -.127(23).67(18).66(23) data in a fashion similar to the Perkeo experiments, where each detector is viewed as its own experiment and the asymmetry is calculated for each detector using the count rates, with and without the spin flipper active, and averaging the final result between detectors. 3.9 Neutron Polarization The polarization of the UCN population in the spectrometer is a complex and fascinating subject that requires understanding spin transport through the magnetic field and wall collisions. Mechanisms for depolarization and the polarimetry of UCNA are discussed in detail by Holley [46] and will not be covered here. What is of interest here is how the degree of polarization for both spin states propagate to final calculations of the asymmetry. One of the unique features of UCNA is that measurement of the polarization is performed on a run-by-run basis, eliminating the reliance on periodic checks of the polarization and spin flipper efficiency. This simplifies the analysis, in that we only need the polarization, P, for that spin flipper state to extract the true asymmetry. If there are inefficiency in the spin flipper then P is decreased, but since the total polarization is measured for each run whether P is decreased from material interactions or inefficiency in the spin flipper is irrelevant. We can begin this analysis by assuming that the polarization in the spectrometer is different when the spin flipper is on, P on and P o f f. This is reasonable not only because of the spin flipper inefficiency, but also because of the different velocity distribution of UCN when the spin flipper is active. As noted previously, UCN experiencing a spin flipper in the 1 T field are boosted by 6 nev, therefore significantly 125

142 altering the effect of material interactions. Therefore we can summarize that the populations of UCN resident in the spectrometer in the two spin states have different attributes, one of which is polarization. The rates measured for each spin flipper state are R 1(2) o f f = W(E)(1 ± A P o f f β cosθ) R 2(1) on = ηw(e)(1 ± A P on β cosθ), (3.69) where 1 and 2 correspond to the detector and therefore the ± with their ordering reversed when the spin flipper is on, W(E) contains all the energy dependent terms, and η is the loading efficiency which reduces the UCN density in the SCS when the spin flipper is on. Calculating the super-ratio gives S = R1 onr 2 o f f R 2 onr 1 o f f = ( 1 A Po f f β cosθ )( 1 A P on β cosθ ) ( 1 + A Pon β cosθ )( ), (3.7) 1 + A P o f f β cosθ and the asymmetry is then (1 + P on P o f f )(1 + S) ± (1 + P on P o f f ) 2 (1 + S) 2 4 P on P o f f (1 S) 2 A =, (3.71) 2P on (1 S) β cosθ which we can simplify to where A = ε ( 1 ± 1 ε P o f f β cosθ 2), (3.72) ε = 2 P on P o f f 1 S (1 + P on /P o f f )(1 + S). It is then straightforward to compare the results of Eq with the ideal asymmetry for several scenarios of P on and P o f f, shown in Figure 3.4. In UCNA, we expect P o f f > 99.5% and 1 P on /P o f f <.5%, therefore any polarization corrections should be below.5%. The practical problem, which will be discussed in the following chapters, is the precision to which the polarization is measured in UCNA is not sufficient to determine a correction to the asymmetry and instead is used to set the systematic uncertainty associated with the polarization. In most cases the measured depolarization is statistically 126

143 A.6 P on P off 1.4 P on P off.99.2 P on P off.97 P on P off off Figure 3.4: The bias in the asymmetry, A, due to depolarization and a spin flipper dependent polarization difference is shown as versus P o f f. For this analysis the bias is shown for P on /P o f f = 1,.99,.97, and.95. indistinguishable from zero [46]. 127

144 Chapter 4 Analysis of the 27 Asymmetry Data 4.1 Introduction In December of 27 on the last weekend of the accelerator cycle about 8 k events were collected during 36 hours of production running along with the required background, calibration, and depolarization measurements. Resulting in the first measurement of the neutron β-asymmetry using ultracold neutrons (UCN) with a combined statistical and systematic uncertainty of 4.5% [12]. This first measurement was in many ways a proof of principle for UCNA, showing that sufficient densities of highly polarized UCN could be achieved in a material bottle and the energy dependent rate of the decay electrons measured to high precision. Analysis of the asymmetry data and predictions of the scattering systematics were performed in parallel by groups at Cal Tech and North Carolina State University with all the depolarization and neutron transport analysis done by A. T. Holley, [46]. Results of the two analysis groups agreed to better then.4%, well below statistical uncertainty. The analysis of the NCSU group will be presented in the following and one can find the results of the Cal Tech analysis summarized in an internal collaboration report by Plaster [87]. 128

145 4.2 Data collection Acquisition of asymmetry data was organized by pulse pairs which includes two sixty minute data (β on(o f f ) ) with the spin flipper in opposite states, fifteen minute background (Bkg on(o f f ) ) associated with the data run, and a depolarization measurement (D on o f f ) immediately after the β-decay data. A - β o f f D o f f on Bkg o f f B - β on D on o f f Bkg on In such pulse pairs were collected consisting of 8 k total events. After timing, fiducial, veto, and energy cuts and background subtraction 38k were used for analysis. One of the noted flaws of this acquisition scheme was the susceptibility to possible periodic drifts in the measured background rate due to the backgrounds always being measured at the same time relative to the data collection. Effects due to periodic timing varying backgrounds were determined to be below the sensitivity of this measurement if they existed at all. Every eight hours or every four pulse pairs, the detector were calibrated using a thin foil 113 Sn check source inserted at the east end of the decay trap. 4.3 Analysis Approach Analysis of the spectrometer data is comprised of six tasks : A - Determine veto cuts (timing and detector cuts) B - Extract initial calibration points from 113 Sn runs and determine linearity from post running multisource calibrations. C - First pass replay to remove non-spectrometer events, apply initial calibration and make adjustments based on GMS and event position. Also calculate physics quantities from ADC values such as position, time of flight, wire chamber multiplicities, etc... D - Extract end point calibration constants from results of first replay for each calibration period 129

146 Figure 4.1: Triggers during the first second of the 17 s beam cycle are shown with the red line representing the timing cut, at.8 s, used to eliminate beam related noise. The discontinuity between the steady state and the first.5 s is due the clock being reset by each of the 3 pulses in a beam burst, piling up the counts during the burst. E - Repeat replay of data with parameters from the Kurie plot analysis F - Extract asymmetries from the reconstructed energy spectra and the convolution of the response function with the theoretical asymmetry. 4.4 Definition of Software Cuts The data acquisition hardware of UCNA was designed to trigger all VME collection boards when any of the UCN monitor detectors trigger or a two-pmt coincidence occurred in the β -scintillator detector. This philosophy of data collection allowed for the maximum amount of information to be gathered during the run which could be sorted by software offline. Events identified by beam, veto, GMS, or neutron monitor cuts are removed from the data stream by a first pass replay code that calibrates the measured ADC values to physics times and energies. Prior to cuts, the total trigger rate of the UCNA DAQ was roughly 2 s 1, mostly coming from the GMS and UCN monitor triggers while only 1 s 1 were from β-like events; the full trigger source distribution is shown in Table