Ph.D. thesis

THE NPDGAMMA EXPERIMENTTHE WEAK INTERACTION BETWEENNUCLEONS ANDPARITY VIOLATION IN COLD NEUTRONCAPTUREMichael GerickeSubmitted to the faculty of the University Graduate Schoolin partial fulfillment of the requirementsfor the degreeDoctor of Philosophyin the Department of PhysicsIndiana UniversityNovember, 2004ii

Acceptance goes here ...i

This thesis is dedicated to my family.To my parents and sisters, for believing in me. To my parents in law, fortheir encouragement and support. To my precious daughter, Nadia, forbrightening my days. Never forget that knowledge is the only form of powerthat can not be taken away by anyone. Most of all, this thesis is dedicated tomy wife, Tatiana, for her monumental contribution to my success, for herdevoted love, and for her unwavering support throughout all of this.This is your Ph.D. Tatiana.Thank you, from the bottom of my heart. I love you all more than I canexpress.Michaeliii

I would like thank my adviser, Professor W. Michael. Snow, for his supportand guidance over the past 5 years. His insightful teaching and leadershipskills have made my years as a graduate student very pleasant. Furthermore,I would like to thank the members of the symmetries team at Los AlamosNational Laboratory, for their guidance and support over the past threeyears of my residence there. In particular, I would like to thank Dr. DavidBowman, Dr. Gregory S. Mitchell, Dr. Seppo I. Penttila, and Dr. Wesley S.Wilburn, for their tireless efforts in providing me with a stimulating andhighly educational experience in experimental physics and Mr. Gilbert G.Peralta for his technical support. I would also like to thank Professor AdamSzczepaniak for providing me with many hours of help and advice on thetheory portion of my work. Finally I would like to thank my Ph.D. defensecommittee and the members of the P-23 group office at Los Alamos NationalLaboratory for their support.v

Abstract ...vii

ContentsAcceptanceiDedicationiiAcknowledgmentsivAbstractvi1 Introduction 11.1 A Brief History of Parity Violation . . . . . . . . . . . . . . . 21.2 Parity Violation between Nucleons . . . . . . . . . . . . . . . 51.3 The NPDGamma Experiment . . . . . . . . . . . . . . . . . . 102 Basic Theory and Observables 152.1 The Basic Reaction . . . . . . . . . . . . . . . . . . . . . . . . 16ix

2.2 The Two-Nucleon Wave-function . . . . . . . . . . . . . . . . 182.3 The Weak Force as a Perturbation . . . . . . . . . . . . . . . 202.3.1 The Effective Parity Violating Weak Potential . . . . . 212.4 Origin of the Measured Asymmetry . . . . . . . . . . . . . . . 242.4.1 Electro-Magnetic Interactions and the Gamma Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Pion-Nucleon Coupling in the Standard Model . . . . . . . . . 372.5.1 Contributions from Weak and Strong Interactions . . . 392.5.2 Reduction of Terms (Quark Dynamics) . . . . . . . . . 473 Experimental Setup and Component Performance 633.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2 Flight Path 12 Beam Line and Moderator . . . . . . . . . . . 663.2.1 Moderator Brightness . . . . . . . . . . . . . . . . . . . 673.2.2 Neutron Guide . . . . . . . . . . . . . . . . . . . . . . 693.2.3 Frame Overlap Chopper . . . . . . . . . . . . . . . . . 723.3 Beam Monitors . . . . . . . . . . . . . . . . . . . . . . . . . . 743.4 The 3 He Neutron Spin Filter . . . . . . . . . . . . . . . . . . . 77x

3.5 The Radio Frequency Neutron Spin Flipper . . . . . . . . . . 833.5.1 RF Spin Flip Theory . . . . . . . . . . . . . . . . . . . 853.5.2 Spin Flip Energy Dependence and Field Ramping . . . 873.5.3 Spin Flip Efficiency . . . . . . . . . . . . . . . . . . . . 893.6 Neutron Capture Targets . . . . . . . . . . . . . . . . . . . . . 1033.6.1 Cl, Al, B 4 C In, and 6 Li Targets . . . . . . . . . . . . 1063.6.2 The Hydrogen Target . . . . . . . . . . . . . . . . . . . 1093.6.3 Detector-Target Geometry . . . . . . . . . . . . . . . . 1103.7 Data Acquisition and Storage . . . . . . . . . . . . . . . . . . 1203.7.1 Sampling Scheme . . . . . . . . . . . . . . . . . . . . . 1213.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254 The NPDGamma Detector 1294.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.1.1 Review of the Scintillation Process and Physics . . . . 1304.2 Detector Design and Operational Criteria . . . . . . . . . . . . 1344.2.1 Vacuum Photodiodes . . . . . . . . . . . . . . . . . . . 142xi

4.2.2 Low Noise Preamplifier . . . . . . . . . . . . . . . . . . 1434.2.3 Mode of Operation and Systematic Effects . . . . . . . 1444.3 Detector Photoelectron Yield and CsI to VPD Gain Matching 1464.3.1 CsI Relative Photoelectron Yield . . . . . . . . . . . . 1474.3.2 VPD Relative Gain and Efficiency . . . . . . . . . . . . 1504.3.3 CsI to VPD Matching . . . . . . . . . . . . . . . . . . 1524.3.4 Combined Relative Detector Gain . . . . . . . . . . . . 1554.4 Noise Performance and Background . . . . . . . . . . . . . . . 1584.4.1 Detector Noise . . . . . . . . . . . . . . . . . . . . . . 1584.4.2 Cosmic Ray Background . . . . . . . . . . . . . . . . . 1714.5 Operation at Counting Statistics . . . . . . . . . . . . . . . . 1734.6 Instrumental Systematic Effects and False Asymmetries . . . . 1764.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1815 Data Analysis and Results 1835.1 Asymmetry Definition . . . . . . . . . . . . . . . . . . . . . . 1845.1.1 The Raw Data Asymmetry . . . . . . . . . . . . . . . . 185xii

5.1.2 Physics Asymmetry Extraction andCorrection Factors . . . . . . . . . . . . . . . . . . . . 1865.2 Data Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1955.3 The Combined AsymmetryStatistical Treatment . . . . . . . . . . . . . . . . . . . . . . . 1985.4 Asymmetry Results . . . . . . . . . . . . . . . . . . . . . . . . 2005.4.1 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 2025.5 Neutron Interaction InducedSystematic Effects . . . . . . . . . . . . . . . . . . . . . . . . . 2045.5.1 Beam Depolarization from Incoherent Scattering . . . . 2065.5.2 Example: Mott-Schwinger Scattering . . . . . . . . . . 2075.5.3 Observed Background . . . . . . . . . . . . . . . . . . 2145.6 Background in the Hydrogen Signal . . . . . . . . . . . . . . . 2166 Summary and Conclusions 221A Conventions, Conversions and other Useful Information 227A.1 Common Relations and Properties for Low Energy Neutrons . 227A.1.1 Properties of Thermal Neutrons . . . . . . . . . . . . . 228xiii

A.1.2 Equations and Relations . . . . . . . . . . . . . . . . . 228A.2 Detector Information . . . . . . . . . . . . . . . . . . . . . . . 230A.2.1 Dynode Corrections . . . . . . . . . . . . . . . . . . . . 230A.2.2 Additional Figures . . . . . . . . . . . . . . . . . . . . 234A.2.3 Tables and Miscillaneous Information . . . . . . . . . . 234B Theory Details 237B.1 Dirac Spinor Normalization . . . . . . . . . . . . . . . . . . . 237B.2 Parity Violating Weak Potential (Derivation) . . . . . . . . . . 239B.2.1 Isospin Setup . . . . . . . . . . . . . . . . . . . . . . . 239B.2.2 Calculation . . . . . . . . . . . . . . . . . . . . . . . . 241B.3 Tree Level Amplitude Integral Tables . . . . . . . . . . . . . . 248C Data Summaries 259C.1 Thin Aluminum Target . . . . . . . . . . . . . . . . . . . . . . 260C.2 Thick Aluminum Target . . . . . . . . . . . . . . . . . . . . . 284C.3 CCl 4 Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311C.4 Cu Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314xiv

C.5 In Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316C.6 B 4 C Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318xv

Chapter 1IntroductionWhen parity is violated, the results of an experiment differ from those obtainedwith an experiment that was set up as its mirror image. If an experiment studiesan interaction that conserves parity, no difference will be observed. Amongthe four fundamental forces of nature, only the weak interaction violates parity.Parity violating processes therefore provide an opportunity to directlymeasure the weak interaction within the nucleus, which is otherwise obscuredby the much larger strong interaction. To date, the weak interaction betweennucleons is not well understood and low energy neutron capture experimentsprovide a way to study parity violation in the nucleus.The NPDGamma experiment uses the capture of cold polarized neutrons onprotons to look for a spatial asymmetry in the number of gamma ray photonswhich are emitted as a result of the capture process.The observation of1

an asymmetry or the absence thereof both have important implications forthe applicability and/or correctness of the theories that are used to describethe interactions and predict their observable properties. One such theory isreferred to as the “Standard Model” of electro-weak interactions, which hasbeen very successful in predicting experimental results. Due to the small size ofthe weak interaction, the expected asymmetry for the NPDGamma experimentis only about 50 parts per billion (ppb). Asymmetry measurements at the ppblevel are notoriously difficult and many systematic effects can produce falseresults.This thesis will describe the concept and setup of the experimentand the theoretical framework it is testing. The results of a very successfulcommissioning run will be presented, showing that possible systematic effectshave been accounted for and have been measured with high precision. We havemeasured parity violation in 35 Cl and placed upper bounds on both parityviolatingand parity-conserving (n, γ) angular correlations in Al, Cu, B 4 C andIn. Al, Cu, B 4 C , and In are materials on which neutrons will be incident andmay produce false background asymmetries in the data obtained from captureon hydrogen.1.1 A Brief History of Parity ViolationIn 1934 Fermi proposed a theory describing the β decay of a nucleus, whichwas modeled after Dirac’s theory of quantum electro-dynamics [1]. He wasmotivated to do so after Pauli hypothesized the existence of the neutrino in1933 to remedy the apparent non-conservation of energy and momentum in2

β decay, which Chadwick had observed in 1914, in the continuous energyspectrum of the emitted electrons.At this point, the Lagrangian Fermi devised was a linear combination of scalarproducts of two polar vectors in which the electron field and 4-vector potentialhad been replaced by the single particle Dirac wave functions for the neutron,proton, the electron and the anti-neutrino.L = − G F√ {ψ p γ µ ψ n ψ e γ µ ψ νe + ψ n γ µ ψ p ψ νeγ µ ψ e } (1.1)2As it stood, the Lagrangian did not include the description of parity violatingeffects. In the years following this initial proposal, it was realized that this Lagrangianis just a specific example of a more general theory which could alsoinvolve linear combinations of scalar products including axial vectors whichprovided terms that would allow parity violation. By the 1950’s, many particleshad been discovered (e.g.the muon, pions, K-mesons and hyperons)which essentially display the same characteristics observed in β decay.By1956 enough data had been collected on the non-leptonic decays of mesons toshow that there seemed to exist two meson states with identical lifetimes, thesame mass and spin, but with decay modes of opposite parity. This apparentcontradiction was known as the τ − θ puzzle in which the decay θ → π + π 0has even parity and τ → π + π + π − has odd parity. The situation could not beresolved unless parity was violated. In 1956, as a direct result of the τ − θpuzzle, T.D. Lee and C. N. Yang suggested that parity violation is actually anintegral part of the weak interaction [2]. In 1957 parity violation was observed3

for the first time by C.S. Wu et al. [3] in the β-decay of polarized 60 Co nuclei.With the addition of parity violation in the general theory, Eq. (1.1) was nolonger necessarily complete and a new general form emerged [4]M = G F√2∑nucleons∑∫jd 3 x [ ψ p O j ψ n] [ψe O j(Cj − C ′ jγ 5)ψνe], (1.2)leaving 19 constants (O j , C j and C ′ j) to be determined from experiment. Inthe following few years, several experiments constrained the possible couplingsin Eq. (1.2). For neutron decay, the final form of the Feynman amplitude wasM = G ∫F√2d 3 xψ p γ µ (1 − λγ 5 ) ψ n ψ e γ µ (1 − γ 5 ) ψ νe . (1.3)The current-current theory of weak interactions, developed independently byFeynman and Gell-Mann and Sudarshan and Marshak in 1958 in responseto the discovery of parity violation, was a generalization of Fermi’s theorycovering all charged weak processes.In this theory all such processes aredescribed by an effective Lagrangian in which a ”universal” charged weakcurrent is coupled to its Hermitian conjugate at a single space-time pointL = − G F2 √ ( )Jλ J λ† + J λ† J λ , (1.4)2where the current J λ = j lλ + J λ consists of both leptonic and hadronic terms.The theory therefore also predicted a parity violating weak contribution to theinteraction between nucleons.4

1.2 Parity Violation between NucleonsIn 1964 the first experimental results of parity violation between nucleons innuclei were observed in cold neutron capture reactions [5]. Since then, parityviolation between nucleons has been extensively studied experimentally.Parity-violating alpha transitions , parity-violating circular γ-ray polarizations,parity-violating asymmetries from polarized nuclei and, most recently,anapole moments of nuclear ground states have been observed and measuredwith good precision.In 1980, Desplanques, Donoghue, and Holstein (DDH) published a benchmarkcalculation according to which the weak parity-violating nucleon-nucleon interactioncan be described by a meson-exchange potential involving seven weakmeson-nucleon coupling constants [6]. DDH based their calculations on theWeinberg-Salam model, using SU(6) symmetry arguments to relate NN weakmatrix elements to measured hyperon decay amplitudes, thereby bypassing theneed for detailed knowledge of nuclear wave functions. The coupling constantsand the DDH best values and reasonable ranges are listed in table 1.1The weak nucleon-nucleon force interferes with the strong force. It changesthe parity and isospin (∆I = 0, 1, 2) of the nucleon-nucleon pair and perturbativelyintroduces parity violating admixtures in nuclear wave functions. Thestudy of the hadronic weak interaction is of great relevance for low energy, nonperturbativeQCD. The properties of quark-gluon wave functions of hadronsare not well understood theoretically and have only partially been tested experimentallyat low energy. The hadronic weak couplings probe short range5

Weak Meson-Nucleon Coupling ConstantsCoupling Cabbibo Weinberg-SalamRange Best Value Range Best Valuef π 0 → 1 0.5 0 → 30 12h 0 ρ -64 → 16 -25 -81 → 30 -30h 1 ρ -0.7 → 0 -0.4 -1 → 0 -0.5h 2 ρ -58 -20 → -29 -25h 0 ω -22 → 6 -6 -27 → 15 -5h 1 ω -2 → 6 -1 -5 → -2 -3h 1′ω Not Reported Not ReportedTable 1.1: DDH “best” value and reasonable ranges for six of the sevenweak meson-nucleon coupling constants. Values are given in units of 1/g πNN(g πNN = 3.8 × 10 −8 ).correlations between quarks because the quark-quark weak interaction occurswhen the distance between quarks is ≤ 2 × 10 −3 fm.In the decades following the DDH paper, many experiments were performedto measure the coupling constants. Among others, the experiments included:• (1) ⃗p + p scattering to determine the analyzing power for longitudinallypolarized protons A z which measures a linear combination of h 0 ρ, h 1 ρ andh 2 ρ.A z = −0.028(h 0 ρ + h 1 ρ + h 2 ρ/ √ 6).For example, A z = (0.86 ± 0.35) × 10 −7 was measured in the E497 ex-6

periment at TRIUMF [7, 8].• (2) The measurement of the circular polarization of photons emitted intransitions of excited nuclei, such as 18 F.P γ = 4385f π + 1.016 × 10 −4 .This was measured in five different experiments with consistent results:P γ = (1.2 ± 3.9) × 10 −4 (see for example [9, 10, 11]).• (3) The measurement of the nuclear anapole moment in 133 Cs [12], withg p = 7.3 ± 1.2(exp.) ± 1.5(theor.) and the relationg p = 2.0 × 10 5 (28.2f π − 7.8h 0 ρ − 1.9h 1 ρ + 0.5h 2 ρ − 4.5(h 0 ω + h 1 ω)) [13].• (4) The measurement of the nuclear anapole moment in 205 Tl [14], withκ a = −0.22 ± 0.3 and the relationκ a = 1.12 × 10 6 (f π − 3.23h 0 ρ) [15].The nuclear anapole moment is produced by parity violating nuclear forceswhich create spin and magnetic moment helical structures inside the nucleus[13]. Two of the coupling constants contribute most to low energy nuclearobservables, the ∆I = 1 πN coupling f π and the ∆I = 0 ρN coupling h 0 ρ.The pion-nucleon coupling is of particular interest, because it is expected to besensitive to neutral weak currents in the standard model. If neutral weak currentsdo not contribute significantly in the nucleon-nucleon interaction, then7

f π will be small. Figure 1.1 illustrates the current situation for some measuredcoupling constants. The coupling constants are expressed in terms ofh 0 ρ + 0.6h 0 ω and DDH best values are used for those constants that have notbeen determined from measurement. The 18 F measurement isolates f π andresults indicate a small value [16]. An extraction of f π from 133 Cs, using DDHbest values indicates a large pion coupling. However, in [15] it is pointed outthat the 18 F and 133 Cs could be consistent if the DDH values are taken at thevery edge of the reasonable range. On the other hand, the 205 Tl and 133 Csresults are inconsistent, calling into question the current understanding of theanapole moment theory. Both [15] and [17] reanalyze the anapole momentresults and find that there is no consistent solution to date.Figure 1.1: DDH best values and measured weak N-N coupling constants.Graph taken from [7].8

In addition to the experimental activity, three other theoretical calculations,besides DDH, have been done. The first one, done in 1993 by Kaplan andSavage, includes strangeness changing currents [18]. The second calculationwas done by Henley et al. in 1998, using QCD sum rules [19]. The thirdcalculation was done by Meißner and Weigel in 1999 and is based on an SU(3)Skyrme model [20].The status of theoretical calculations is illustrated inFig. 1.2. The DDH reasonable range and best value is (in units of 10 −7 )f ∆I=1π ≈ 4.56 (0 − 11.4) (1.5)DDH1980Kaplan and Savage1993Henley et al.1998Meissner and Weigel1999−4 −2 0 2 4 6 8 10 12 14 161f π (10−7 )Figure 1.2: Theoretical calculations of the weak pio-nucleon coupling [6, 18,19, 20].It is apparent from the inconsistencies in the experimental results, outlinedabove, and the difficulties in the development of a coherent theoretical descriptiondue to the incomplete knowledge of strong interactions at low energy,9

that a series of experiments with interpretable results is needed to establish acomplete picture of the hadronic weak interaction. This includes the need foran unambiguous measurement of f 1 π.In general, the parity violating asymmetry in a measurement can be expressedas an expansion in terms of the coupling constants,A = a 1 πf 1 π + a 0 ρh 0 ρ + a 1 ρh 1 ρ + a 2 ρh 1 ρ + a 0 ωh 0 ω + a 1 ωh 1 ω,in which the coefficients are calculated from theory. A measurement of theasymmetry from large nuclei is hard to interpret due to nuclear structure effects.On the other hand, for experiments with few-nucleon systems, the coefficientscan often be calculated directly and with small uncertainties. NPDGammais a simple two body experiment which measures an asymmetry with a simplerelation to the weak coupling constant and therefore provides a theoreticallyclean interpretation of the result.1.3 The NPDGamma ExperimentNPDGamma, currently under commissioning at the Los Alamos Neutron ScienceCenter (LANSCE), is the first experiment designed for the new pulsedcold neutron beam line, flight path 12, at LANSCE. NPDGamma will measurethe parity-violating up-down asymmetry, A γ in the reaction ⃗n + p → d + γ .The asymmetry is observed in the angular distribution of 2.2 MeV γ-rays with10

espect to the neutron spin direction.dσdΩ ∝ 14π (1 + A γ cos θ s,γ ) . (1.6)Because of the low energy of the deuteron bound state, the asymmetry measuredin this reaction is dominated by the exchange of the lowest mass mesonsand mostly isolates f 1 π.A γ = −0.107f 1 π − 0.001h 1 ρ − 0.004h 1 ω (1.7)The asymmetry is thus, to a very good approximation, directly proportionalto the weak pion-nucleon coupling constant and NPDGamma therefore determinesf π [21, 22, 23]. The error on the asymmetry due to neglecting the h 1 ρand h 1 ω couplings is about 1% and the error from strong interactions is about2% [24]. The asymmetry has a predicted size of 5 × 10 −8 [6] and the goal ofthe NPDGamma collaboration is to measure it to 10% of this value.Very recently, new theoretical work has been completed to reformulate thetheory of the hadronic weak interaction using effective field theory and chiralperturbation theory [25] and to recalculate the parity violating asymmetry in⃗n + p → d + γ , using modern two-body strong interaction models [26]. Thelong-term goal is to calculate the weak coupling constants directly from theStandard Model.The experiment is performed at the LANSCE spallation source because brightpulsed spallation neutron sources possess high instantaneous neutron fluxes11

and lend themselves to the measurement of neutron energy by time of flight.The time of flight information provides control of systematic effects, since theparity-violating observables are independent of neutron energy in this regime,whereas systematic effects depend strongly on neutron energy.This allowsmeasurements of very small observables to be made reliably. New high precisionfundamental neutron physics experiments can be designed to take advantageof these features [27] and accurately measure the coupling constants andprovide a complete picture of the hadronic weak interaction. One such classof measurements consists of searches for parity violation in polarized neutroncapture on light nuclei [28, 29, 30].The remainder of this thesis consists of 5 chapters. Chapter 2 describes indetail the derivation of the weak poin-nucleon potential in the effective fieldtheory framework, the extraction of the asymmetry A γ and its relation to fπ,1as well as the link to the standard model weak interaction. The possibilityfor inclusion of additional interactions due to the strong interaction and theresulting expansion of dynamical terms in the S-matrix expansion is also explained.This portion of the calculation is meant to illustrate how the couplingconstant can be calculated from the standart model as a starting point, butit is not completed to the level of actually calculating a coupling constant.However, a series of corresponding tree-level integrals is calculated, which canbe used in a numerical calculation, together with appropriate nuclear wavefunctions.Chapter 3 describes the setup and performance of the experimental apparatus.Chapter 4 provides a detailed account of the work done on the NPDGamma12

detector array its properties and performance. Chapter 5 discusses the performedasymmetry measurements, the analysis procedure, systematic effectsand the results obtained during the 2004 commissioning run. Chapter 6 is asummary of the work and the results.13

Chapter 2Basic Theory and ObservablesAt low energies, nucleons and mesons are generally considered to be the effectivedegrees of freedom in the theory. Strong interactions (QCD) at lowenergies are poorly understood and the weak interaction provides a way toprobe QCD in the long range, strongly-interacting limit via short-range W ± ,Z 0 exchange. The measurement of the weak πNN coupling will provide a testfor the effective theory as well as information on low energy QCD.Incorporating the weak interaction into the NN interaction has historicallybeen achieved by replacing one of the strong NN vertexes by a weak vertex,which adds to the strong parity conserving (PC) potential a parity nonconserving(PNC) potential. However, this approach implies a specific modelfor the interaction.A deeper understanding of the interactions at the weak vertex must be gainedby looking at a quark-model ( standard model ) picture, where W ± , Z 0 gauge15

osons decay into the exchanged mesons.Although such a model was attemptedby DDH in 1980, using the then state-of-the-art bag model, the rangeof possible weak NN couplings was still very broad, as discussed in section 1.2.The pion-nucleon coupling determined in this experiment pertains to the interactionat the weak vertex. The weak πNN coupling f π has been calculatedseveral times using different models (see Sec. 1.2)2.1 The Basic ReactionThe basic reaction of interest here is the thermal capture of spin polarizedneutrons on the protons in a liquid para-hydrogen target. The capture resultsin the formation of a deuteron, and in each such process a single 2.2MeVγ-photon is emitted from the nucleus.⃗n + p → d + γ (2.1)Parity is violated if the nuclear spin and the photon momentum parallel antiparallelcross-sections are different; i.e. nature shows preference for a particularhandedness. The reason for this is apparent from Eqn. 1.6, because the correlationbetween neutron spin and photon momentum 〈⃗s n · ⃗k γ 〉 is odd undera parity transformation; ⃗ k γ changes sign, but ⃗s n does not. Parity violation isobserved via the cross-sectional asymmetry of the gamma rays emitted in thecapture process. The cross-section is obtained from the transition amplitudeof the electro-magnetic part of the Hamiltonian between initial (capture) and16

final (bound) two nucleon states.Figure 2.1: Level diagram for the n-p system. Parity violation arises frommixing of the P and S wave states and interference of the E1 transitions.A γ cannot come from J=0 capture states, so it must come from 3 S 1 ↔ 3 P 1transitions.The primary process is the strong interaction induced PC transition betweenthe singlet and triplet S-wave states: 1 S 0 , 3 S 1 . The weak interaction then introducesa small PNC admixture of P-wave states. The primary PC transitionamplitude results from the 1 S 0 scattering (capture) initial state with a smallercontribution from the 3 S 1 scattering initial state, which appears here, becausea state with a polarized neutron requires a coherent superposition of these twoinitial states. The weak interaction mixes parity-odd partial waves with theS-waves. Only L=1, J=0,1 states are considered here, because higher partialwaves would correspond to higher multi-pole terms in H EMwhich producenegligible contributions to the cross-section. The final deuteron bound stateis 3 S 1 , but may also have small components of parity-odd admixed P-waves,17

introduced by the weak interaction. We neglect the D-wave of the deuteronground state. Figure 2.1 shows the level diagram for this situation.2.2 The Two-Nucleon Wave-functionThe initial and final states for the complete reaction are the two-nucleon statesdescribed in this section plus the γ photon vacuum in the initial state and thesingle photon in the final state.|ψ Ji 〉 ≡ |i〉|0〉 = ∑W Ni1 N i2(k i1 , k i2 )|N i1 N i2 〉|0〉N i1 N i2(2.2)|ψ Jf 〉 ≡ |f〉|γ〉 = ∑W Nf1 N f2(k f1 , k f2 )|N f1 N f2 〉|q, µ〉N f1 N f2(2.3)The nucleon wave-function can be written as an isospinor⎛W N =w ⎞p⎝ ⎠ (2.4)w nwhere each of the “charge states” w i , i = p, n is a spinor and is defined by theparticle’s momentum, spin and relative orbital angular momentum. The initialand final states can be written as a superposition of basis states, of specificcombinations of the momentum, angular momentum and isospin quantumnumbers of the nucleon pair.In the language of second quantization, thistakes the form18

| ⃗ k, J, M J , I, M I 〉 = ∑=W N1 N 2(k 1, k 2)|N 1N 2〉N 1 N 2∑∫ d 3 k 1 d 3 k 2W LSλ 1 t 1 ,λ 2 t 2(2π) 6 JI ( ⃗ k 1 λ 1 t 1 , ⃗ k 2 λ 2 t 2 )a † ⃗ k1a † λ 1⃗ k2|0〉λ 2(2.5)〈N 1N 2| ⃗ k, J, M J , I, M I 〉 = W LSJI ( ⃗ k 1 λ 1 t 1 , ⃗ k 2 λ 2 t 2 ) (2.6)Using momentum conservation and coupling the relative orbital angular momentumand spin as well as the isospin of the two nucleons into the totalangular momentum J and isospin I produces the wave function of the twonucleonsystemW LSJI ( ⃗ k 1 λ 1 t 1 , ⃗ k 2 λ 2 t 2 ) = (2π) 3 δ 3 ( ⃗ P − ⃗ k 1 − ⃗ k 2 )W LSJI ( ⃗ k 1 − ⃗ k 2 , λ 1 , t 1 , λ 2 , t 2 )= WJI LS (| ⃗ k 1 − ⃗ k 2 |)[Y M LL (θ, φ) ⊗ [η(m S1 ) ⊗ η(m S2 )] S ] M JJ×[ξ(m I1 ) ⊗ ξ(m I2 )] M II(2.7)The exact form of the momentum part to the wave-function depends on themodel chosen to describe the strong interaction between the nucleons. It isthe largest source of uncertainty in any final prediction of the amplitude inthe reaction 2.1.19

2.3 The Weak Force as a PerturbationAn order of magnitude estimate of the size of the weak contribution to thepion-nucleon coupling can be made from the ratio of the weak boson to pionpropagators.g 2 wg 2 πNN· m2 πM 2 w≈ 10 −6 (2.8)This estimate is modified by other factors, of order unity, stemming from thecomplete evaluation of all matrix elements in the process amplitude. Clearly, it is valid to introduce the weak interaction as a perturbation to the stronginteraction and treat it in first order perturbation theory.The Hamiltonian can then be split into a strong PC term plus a weak PNCpotentialH = H s + V P NC (2.9)In this way, the weak portion of the Hamiltonian mixes states of higher orderpartial waves (P-waves) which violate parity with the S-waves from the stronginteraction. That is,|ψ〉 = |ψ 0 〉 + a|ψ 1 〉 (2.10)where the mixing coefficient a isa = 〈ψ 1|V P NC |ψ 0 〉∆E(2.11)20

2.3.1 The Effective Parity Violating Weak PotentialIn the effective field theory context, V P NC is a product of both the strong andweak Hamiltonians and can be expressed in terms of these to any order in aperturbation expansion. A possible path to determine V P NCis to take theT-matrix elements of the Hamiltonian between final and initial nucleon pairstates. Here, T = V + V GT is the Lippmann-Schwinger equationG =1E − H o ± iɛis the appropriate Green’s function and H = H o + H P C + H P NC = H o + V .Solving for T and expanding in powers of GV leads toT = V + V GV + V GV GV + ...So that actually calculating the T-matrix element between the desired initialand final states gives〈f|T |i〉 = 〈f|(H P C + H P NC )|i〉+1〈f|(H P C + H P NC )E − H o ± iɛ (H P C + H P NC )|i〉+ ...The first term vanishes, because H P C and H P NC both contain pion creationand annihilation operators, whereas the initial and final nucleon states do not.In term two and higher terms even in V the pion operators can be contracted21

to produce the propagator. Cutting the expansion off after the first order termproduces〈f|T |i〉 = 〈f|(H P C1E − H oH P NC + H P NC1E − H oH P C )|i〉+ 〈f|(H P C1E − H oH P C + H P NC1E − H oH P NC )|i〉The term quadratic in H P C does not contribute to the parity violating amplitudeand its contribution to the PC (strong) term is in second order. Theterm quadratic in H P NC is negligible in size. The first order contribution tothe PNC weak potential is therefore given by〈f|V P NC |i〉 = 〈f|H P C1E − H oH P NC |i〉 + 〈f|H P NC1E − H oH P C |i〉 (2.12)which simply says that the order of operation of H P NC and H P C lead toindistinguishable processes whose amplitudes must add.H P C and H P NC are established using the standard field theory approach andare given byH P C = ig πNN∫d 3 xψ i (x)γ 5 ψ j (x)(⃗τ · ⃗φ(x)) (2.13)H P NC = f ∫π√2d 3 x ′ ψ i (x ′ )ψ j (x ′ )(⃗τ × ⃗ φ(x ′ )) 3 (2.14)The field equations are the standard solutions to the Dirac and Klein Gordonequations and the Dirac spinors are treated in the static (ω = E i − E j = 0)non-relativistic limit. In other words, they are expanded to order (1/M) with22

ω = 0. [31]ψ i (x) = ∑ λ i∫d 3 k i(2π) 3 (a k i ,λ iu(k i , λ i )e −ik·x + b † k i ,λ iv(k i , λ i )e ik·x ) (2.15)∫φ t (x) =d 3 k(2π) 3√ 2ω k(c k,−t e −ik·x + c † k,t eik·x ) (2.16)u i (k i , λ i )γ 5 u j (k j , λ j ) → χ † (λ i ) ⃗σ · (⃗ k j − ⃗ k i )χ(λ j ) (2.17)2Mu i (k i , λ i )u j (k j , λ j ) → χ † (λ i ) E + M2M [1 − (⃗σ · ⃗k i )(⃗σ · ⃗k j )(E + M) 2 ]χ(λ j ) (2.18)The index in the pion field runs over all isospin projections (t = −1, 0, +1).So that a φ − corresponds to the creation of a π − or the annihilation of a π + .Using all this in equation 2.12 produces the result〈f|V P NC |i〉 =∑λ f1 t f1 ,λ f2 t f2×W † LSJI∑∫λ i1 t i1 ,λ i2 t i2d 3 k f1 d 3 k f2(2π) 6 ∫ d 3 k i1 d 3 k i2(2π) 6 {( ⃗ k f1 λ f1 t f1 , ⃗ k f2 λ f2 t f2 )Vπ∆I=1 (k π )WJI LS ( ⃗ k i1 λ i1 t i1 , ⃗ k i2 λ i2 t i2 )}with the Fourier transform of the PNC one pion-exchange potential beingequal toV ∆I=1π (r π ) =∫d 3 k π(2π) V ∆I=13 π (k π )e i⃗ k π·⃗r= ig √πNNf π[⃗τ 1 × ⃗τ 2 ] z [⃗σ 1 + ⃗σ 2 ] · [⃗p, e−mr32M 4πr ] . (2.19)23

Where r = |⃗x 1 −⃗x 2 | , ⃗p = ⃗ k 1 − ⃗ k 2 , M is the nucleon mass and m the pion mass.The detailed derivation of Vπ∆I=1 (k π ) is shown in appendix B.2.The weak pion nucleon coupling constant is obtained from a measurement ofthe combined couplings in 2.19.2.4 Origin of the Measured AsymmetryThe connection between the measured asymmetry and the weak coupling constantis made by looking at the electromagnetic transition the nucleus makesduring the neutron-proton capture. The resulting gamma cross-section carriesthe signature of the parity violation via a difference in the cross-sections of thegamma emission relative to the neutron spin up or spin down.(See Figs. 2.2and 2.3) That is, the incoming neutron beam is spin-polarized, but the rateat which gammas are emitted up or down with respect to the neutron-spindirection is different if parity is in fact violated.dσdΩA γ =(⃗ S · ⃗k > 0) − dσ (⃗ S · ⃗k < 0)dΩdσ(⃗ S · ⃗k > 0) + dσ (⃗ S · ⃗k < 0)dΩ dΩ(2.20)wheredσdΩ ∝ 14π (1 + A γcos(θ ⃗n, ⃗ k)).It is evident that, given the form of the cross-section Eqn. 1.6, Eqn. 2.20 willisolate the asymmetry, but it is less obvious, how this cross-section is derived.24

Figure 2.2: Depiction of the neutron capture process and the angular dependenceof the outgoing γ-ray . θ is the angle between the direction of neutronpolarization and the γ-ray momentum. The parity operation changes the signof ⃗s n · ⃗k γ .25

+Spin Flip n + p d + γ+Figure 2.3: The spin flip of the neutron is identical to a parity transformationon the system. Aside from detecting γ-rays in the up and down direction, thespin flip is used to detect the asymmetry.2.4.1 Electro-Magnetic Interactions and the Gamma Cross-SectionAccording to Fermi’s golden rule, we have for electromagnetic transitionsω f←i = 2π|〈ψ Jf |H EM |ψ Ji 〉| 2 ρ f (2.21)where ρ f is the final density of states.Before going into the details of calculating the transition matrix elements, wehave to establish what the initial and final states will be and we can do thatfrom symmetry considerations alone. The initial and final states are knownfrom the reaction (eqn. 2.1). However, the information needed lies within the26

angular momentum and isospin structure of the states.The initial state represents the neutron-proton continuum. Observing Fermistatistics,which we must do since we are implicitly treating neutrons andprotons as identical particles when we employ isospin symmetry, the stronginteraction, which introduces the predominant S-waves states, leads to thefollowing possible configurations for the initial state:L = 0, S = 0 ⇒ | 1 S 0 , I = 1〉L = 0, S = 1 ⇒ | 3 S 1 , I = 0〉And the weak interaction perturbatively mixes in the configurationsL = 1, S = 0 ⇒ | 1 P 1 , I = 0〉L = 1, S = 1 ⇒ | 3 P 0 , I = 1〉⇒ | 3 P 1 , I = 1〉The final state represents the deuteron. Ignoring the tensor term, we obtainthe following configurationsL = 0, S = 1 ⇒ | 3 S 1 , I = 0〉L = 1, S = 0 ⇒ | 1 P 1 , I = 0〉L = 1, S = 1 ⇒ | 3 P 0 , I = 1〉27

⇒ | 3 P 1 , I = 1〉We ignore the | 3 P 2 , I = 1〉 configurations, keeping only the lowest order partialwaves consistent with parity violation. Grouping the states together accordingto total angular momentum J, we get from eqn. 2.10|Ψ Ji =0〉 = | 1 S 0 , I = 1〉 + a 0 | 3 P 0 , I = 1〉|Ψ Ji =1〉 = | 3 S 1 , I = 0〉 + a 0 | 1 P 1 , I = 0〉 + a 1 | 3 P 1 , I = 1〉|Ψ Jf =1〉 = | 3 S 1 , I = 0〉 + a 0 | 1 P 1 , I = 0〉 + a 1 | 3 P 1 , I = 1〉(2.22)The subscript on the admixture amplitudes a indicates the isospin change Iacross the amplitudes.From this, the list of possible matrix elements is〈 3 S 1 , I = 0|H EM | 1 S 0 , I = 1〉〈 3 S 1 , I = 0|H EM | 3 S 1 , I = 0〉a 0 〈 3 S 1 , I = 0|H EM | 3 P 0 , I = 1〉a 0 〈 1 P 1 , I = 0|H EM | 1 S 0 , I = 1〉a 1 〈 3 P 1 , I = 1|H EM | 1 S 0 , I = 1〉a 0 〈 3 S 1 , I = 0|H EM | 1 P 1 , I = 0〉a 1 〈 3 S 1 , I = 0|H EM | 3 P 1 , I = 1〉28

a 0 〈 1 P 1 , I = 0|H EM | 3 S 1 , I = 0〉a 1 〈 3 P 1 , I = 1|H EM | 3 S 1 , I = 0〉(2.23)To go any further, we have to insert the actual hamiltonian and establish theproper parity, isospin and angular momentum selection rules.The usual expression for the electromagnetic interaction hamiltonian is∫H int = −d 3 r ⃗j(⃗r) · ⃗A(⃗r, t) . (2.24)Where the quantized radiation field (using normalization over a box of volumeV) has the form⃗A(⃗r, t) = ∑µ=±1∫d 3 k√ 2ωk V [e ⃗ kµd ⃗kµ e ik·r + e † ⃗ kµd † ⃗ kµe −ik·r ] (2.25)The initial and final states are those given in 2.22. The nucleon fields in thecurrent density of equation 2.24 interact with the nucleons in the initial andfinal states, the details of which have to be supplied by a particular nuclearmodel. The photon field (eqn. 2.25) interacts with the photon in the finalstate.This part of the matrix element will provide information about theparity and angular momentum selection rules which will isolate the allowedstates from those given in 2.23.29

Then, together with the density of statesρ f = V k2(2π) 3 dΩ k,the transition probability becomesω f←i =k ∣ ∫ ∣∣∣〈f|8π 2d 3 r⃗j(⃗r) · e ⃗kµ e −ik·r |i〉 ∣∣2dΩk (2.26)Using a multipole expansion of the plane wave, the amplitudes can be split upinto electric and magnetic multipole moments. [32] [33]∫d 3 r⃗j(⃗r) · e ⃗kµ e −ik·r = −µ ∑ l√2π(2l + 1)i l ∫d 3 r⃗j(⃗r) · A l µ(⃗r) (2.27)whereA l µ(⃗r) = ( M l µ(⃗r) − µE l µ(⃗r) ) (2.28)The parity transformation properties of the multipoles are (−1) l and (−1) l+1for the electric and magnetic operators respectively. Since we are consideringonly the lowest multipoles consistent with parity violation (l=1) the two states(taken together) in the matrix elements involving the electric dipole operatorhave to be odd under parity while those involving the magnetic dipole operatormust be even in order for them not to vanish.In addition to that, it canbe shown that for self-conjugate nuclei (N=Z), the electro-magnetic dipoleoperator vanishes between states of equal isospin. Here, the spatial averageof the dipole operator for a nucleus with mass number A is given by E1 =30

∑ Ai=1r(j)I 3 (j) [33]. The matrix element of the dipole operator between isospineigenstates then yields 〈N, Z, I f , I f,3 |E1|N, Z, I i , I i,3 〉 ∝ 1 2 (Z − N) if I f= I iand I f,3 = I i,3[34]. Under these considerations, the list of possible amplitudesin 2.23 is reduced to1) 〈 3 S 1 , I = 0|M1| 1 S 0 , I = 1〉2) 〈 3 S 1 , I = 0|M1| 3 S 1 , I = 0〉3) a 0 〈 3 S 1 , I = 0|E1| 3 P 0 , I = 1〉4) a 0 〈 1 P 1 , I = 0|E1| 1 S 0 , I = 1〉5) a 1 〈 3 S 1 , I = 0|E1| 3 P 1 , I = 1〉6) a 1 〈 3 P 1 , I = 1|E1| 3 S 1 , I = 0〉where,∫M1 =d 3 r⃗j(⃗r) · M l µ(⃗r)etc...As will be discussed in detail below, the amplitudes containing initial stateswith J = 0 do not contribute to the asymmetry because their contribution tothe cross-section are spherically symmetric. In addition, the third and fourthamplitudes contribute little to the spherically symmetric PC amplitudes involvingthe M1 operators because they contribute to second order.Theyare thus negligible. Also, the second amplitude vanishes, because of the orthogonalitybetween the initial and final states after operating with the staticmagnetic dipole operator (µ p − µ n )(⃗σ p −⃗σ n ). The resulting list of contributing31

matrix elements isP C 〈 3 S 1 , I = 0|M1| 1 S 0 , I = 1〉P V a 1 〈 3 S 1 , I = 0|E1| 3 P 1 , I = 1〉P V a 1 〈 3 P 1 , I = 1|E1| 3 S 1 , I = 0〉.(2.29)There is another way in which amplitudes 3 and 4 above can be seen to vanish.From the subscript on the mixing amplitude a I , one can see that, for the case ofI=0, the parity violating pion exchange potential has been evaluated betweenstates of equal isospin. However, the potential (2.19) transforms as an isovector,which means that isospin has to change by one from the initial to thefinal state. On these grounds, the amplitudes multiplied by a 0 must vanish.In the derivation of the multi-pole expansion used above, it is assumed that thephoton is emitted along the z direction [32] [33]. Arbitrary photon directionscan be obtained by rotating the plane wave (2.25) through two of the threeEuler angles, using the proper rotation matrix.This then corresponds torotating the multipoles.A l µ(⃗r) −→ ∑µ ′ =±1Dµµ l ′(−φ, −θ, 0)Al µ ′(⃗r) (2.30)The rotation does not affect the quantum numbers in the initial and finalnucleon states. So one can extend the sum over µ ′ to the entire matrix element32

and move the rotation matrix outside.〈J f M Jf |A l ∑µ(⃗r)|J i M Ji 〉 −→ 〈J f M Jf | Dµµ l ′(−φ, −θ, 0)Al µ ′(⃗r)|J iM Ji 〉∑=µ ′ =±1µ ′ =±1D l µµ ′(−φ, −θ, 0)〈J fM Jf |A l µ ′(⃗r)|J iM Ji 〉So, using the Wigner-Eckart theorem [35], one can separate the angular dependenceof the amplitudes from the more complicated remainder which dependson the nuclear model used.−→=∑µ ′ =±1∑µ ′ =±1D l µµ ′(−φ, −θ, 0)〈J fM Jf |A l µ ′(⃗r)|J iM Ji 〉D l µµ ′(−φ, −θ, 0)(−1)l−J i+J f√2l + 1〈J i , M Ji , l, µ ′ |J f , M Jf 〉〈J f ||Al||J i 〉The evaluation of the Clebsch-Gordan coefficients dictates the selection rulesinvolving the coupling of angular momenta between the nuclear initial andfinal states and the angular momentum that is carried off by the emittedphoton.That is, the photon angular momentum can take values given by|J i − J f | ≤ l ≤ J i + J f and the helicity of the photon must obey µ ′ = M f − M i .To describe a polarized neutron beam interacting with an unpolarized target,one has to take the coherent superposition of the spin triplet and singlet statesof the two nucleon system. For example a spin-up polarized neutron beam33

incident on an unpolarized target can be described by the superposition|i〉 = | 3 S 1 , M s = 1, M l = 0, M J = 1〉 + | 3 S 1 , M s = 0, M l = 0, M J = 0〉+| 1 S 0 , M s = 0, M l = 0, M J = 0〉 + O(l = 1) + .....The P-wave initial state can then mix in states following the same scheme inthe spin portion but with M J= M l + M s = 0, ±1 and higher. As alreadydetermined, the singlet state only contributes to the PC amplitude and allothers contribute to the PV amplitudes. So, as far as the evaluation of theClebsch-Gordan coefficients is concerned, the parity violating amplitudes mayhave all (M J = 0, ±1) projections in the initial and final states. This is alsothe case when the neutron beam is polarized spin-down.Finally, together with the properties of the Wigner d-functions (see for example[36]), the angular dependence can be determined. For the PC amplitude onefinds for right circular polarized photons (µ = 1)∑µ ′ =±1Dµµ 1 −iµ(−1)√ 1−0+1〈0, 0, 1, µ ′ |1, M ′ Jf 〉〈1||M1||0〉3= √ −i 〈1||M1||0〉 ( )D1,1 1 + D1,−11 3= −i √3〈1||M1||0〉.34

Whereas, for the PV amplitudes, one finds∑µ ′ =±1Dµµ 1 i(−1)√ 1−1+1〈1, M ′ Ji , 1, µ ′ |1, M Jf 〉〈1||E1||1〉3= √ −2i √ 〈1||E1||1〉 ( )D1,−1 1 − D1,11 3 2= −√ 2i√3〈1||E1||0〉 (− cos θ) .So using these results in the expression for the transition probability givesit angular dependence. For a neutron beam polarized along the z-axis, θ isthe angle between the neutron polarization and the momentum vector of theoutgoing photon.ω f←i = k (1 − 2 √ 2 |〈E1〉| )4π |〈M1〉| cos θ |〈M1〉| 2dΩ kwhich gives the differential cross-sectiondσdΩ ∝ 1 (1 − 2 √ 2 〈E1〉 )4π 〈M1〉 cos θ(2.31)where, in keeping with the conventions, the asymmetry has a negative valueA γ ≡ −2 √ 2 〈E1〉〈M1〉 . (2.32)and〈E1〉 = a 1∫d 3 r{〈 3 P 1 ||⃗j(⃗r) · E 1 µ(⃗r)|| 3 S 1 〉 + 〈 3 S 1 ||⃗j(⃗r) · E 1 µ(⃗r)|| 3 P 1 〉}35

〈M1〉 =∫d 3 r{〈 3 S 1 ||⃗j(⃗r) · M 1 µ(⃗r)|| 1 S 0 〉}(2.33)More sophisticated calculations show that effects such as isospin violation,relativistic effects in the electro-magnetic and meson exchange currents, aswell as the inclusion of the deuteron D-state produce negligible corrections tothis relation.The final set of non-vanishing amplitudes that exhibit parity violation (2.29)are all in the ∆I = 1 channel. It is for this reason that the parity violatingweak pion-nucleon coupling constant in equation 2.19 refers to the isovector reactionchannels only. The relation between the asymmetry and the coupling isproduced through the connection of the potential (2.19), the mixing amplitude(2.11) , and the expression for the asymmetry above (2.32). The calculationof the matrix elements in 2.32 has been carried out by several people and theresult leads to the relation[6]A γ ≡ −2 〈E1〉〈M1〉≃−0.107f ∆I=1π(2.34)From an experimental point of view, equation 2.20 is most useful. It describesdirectly, the use of the cross-section results from the experiment, to extractthe asymmetry.36

2.5 Pion-Nucleon Coupling in the StandardModelBeyond its use in the effective theory for nucleon-nucleon weak interactions,discussed in the previous sections, another physics implication of a measurementof the weak pion-nucleon coupling constant is that the results can, inprinciple, be compared with those predicted from calculations based on theStandard Model of quarks. The weak interactions between quarks and leptonsare well understood and tested. Purely leptonic and semi-leptonic processeshave been studied extensively and the results mostly agree well with thosepredicted from the Standard Model.On the other hand, the weak interactions between hadrons (non-leptonic) arepoorly understood, mostly because of the role strong interactions play in determiningthe nuclear wave-functions. Evaluating matrix elements in whichquarks are considered the fundamental degrees of freedom requires, in theconstituent quark model, the description of the nucleon wave-function as athree-body system of quarks. There is no unique and exact way, to date, tomodel the interaction between the quarks and produce a definitive form forthe wave-function.Various nuclear models exist, which will aid in the approximation of wavefunctions(for example the MIT bag model). The weak pion-nucleon couplingconstant predicted from calculations involving those approximations will carry37

their uncertainties and vary from one model to the next. Experimental knowledgeof the coupling can aid in establishing the validity of models.Low energy interactions between nucleons are described through the exchangeof the lowest mass mesons. The interaction between the quarks of the pion andthose of the nucleons at these low energies are poorly understood. The strongforce is predicted to exhibit large coupling at long ranges, but the exact valueof the coupling or nature of strong interaction at low energies is unknown.Extensive calculations have been carried out, using the Standard Model, topredict ranges and “best values” for the weak pion-nucleon coupling constant[6, 37] (See section 2.3.1). The calculations involve neutral current-currentN → Nπ matrix elements (See figure 2.4), but they do not include possibledynamic contributions from strong interactions.The weak interaction between quarks is mediated via W ± and Z o boson exchange.In light of the large boson masses, the quarks are expected to be closetogether when interacting weakly, so that quark-quark spin and spatial correlationsbecome important. Thus, measureing the N-N weak interaction at lowenergy can provide important insight into the significance of these correlationsbetween quarks and aid in the development of nucleon wavefunction modelswhich incorporate these effects.The focus of the present section is a calculation that will include strong interactionsat the weak vertex in the form of quark-anti-quark pair creationand annihilation in addition to the exchange of a weak vector boson. Because38

MNNMNNFigure 2.4: Two examples of nucleon-meson dynamics treated in DDHthe pion represents the lowest energy quark-anti-quark bound state excitationproduced by the strong interaction, this type of calculation, coupled with thenewest models for nuclear wave-functions, has the potential to tell us somethingabout strong interactions at low energies.2.5.1 Contributions from Weak and Strong InteractionsAs determined in the previous sections, only isovector channels contribute tothe measured coupling. In the Standard model the (strong) isospin structureis determined through quark flavor, with up and down flavors producing adoublet and any other flavor being a singlet. Starting with the currents con-39

tributed by the weak and strong interactions and deciding on the order inperturbation to which to evaluate the S-matrix will, together with the usualisospin coupling mechanism, determine the matrix elements that contribute.The quark portion of the weak lagrangian of the Standard Model isL INTW = − g2 √ ( )Jµ†C W µ + J c µ W µ† g − J24 cos θNZ µ µ (2.35)wwhereJ µ†C = ψ u γ µ (1 − γ 5 )ψ d cos θ c + ψ u γ µ (1 − γ 5 )ψ s sin θ c−ψ c γ µ (1 − γ 5 )ψ d sin θ c + ψ c γ µ (1 − γ 5 )ψ s cos θ cJ µ C = ψ d γ µ (1 − γ 5 )ψ u cos θ c + ψ s γ µ (1 − γ 5 )ψ u sin θ c−ψ d γ µ (1 − γ 5 )ψ c sin θ c + ψ s γ µ (1 − γ 5 )ψ c cos θ candJ µ N = ψ u γ µ (1 − 8 3 sin2 θ w − γ 5 )ψ u + ψ c γ µ (1 − 8 3 sin2 θ w − γ 5 )ψ c−ψ d γ µ (1 − 4 3 sin2 θ w − γ 5 )ψ d − ψ s γ µ (1 − 4 3 sin2 θ w − γ 5 )ψ s .From the point of view of flavor, the neutral current is hermitian. The strong40

lagrangian used here is written down as∫L INTS =dt y ψ q (x)Γ ρ ψ q (y)f(⃗x, ⃗y)δ(t x − t y ) . (2.36)In the end only those terms consistent with quark-anti-quark pair creationand annihilation are kept in the calculation. However, to keep the Lorentzcovariance and make it clear how to evaluate contractions into propagatorsthe lagrangian is written containing all terms. The spatial distribution of thequark-antiquark pair is approximated by f(⃗x, ⃗y) = exp(−b(|⃗x − ⃗y|) 2 ) with (b)setting the scale for the typical distance between quarks at low energy. TheDirac structure is to be determined later, but Γ ρ can not be γ 0 , because thecurrent ψ q (x)Γ ρ ψ q (y) will vanish in that case.S-Matrix FormalismThe S-matrix is equal to∞∑ (−i) n ∫S =n!n=0∫dt 1 · · ·dt n T {H I (t 1 ) · · · H I (t n )} (2.37)Since the interaction lagrangians 2.35 and 2.36 contain no derivatives in thefields, the Interaction picture hamiltonian is the negative of the lagrangianin 2.37. This gives the same form of the S-matrix used in QED. Because ofthe unitary transformation relation between the Heisenberg and Interactionpicture operators the use of Heisenberg operators in the S-matrix will havethe same result when taken between stationary (Heisenberg picture) states41

and, in addition, the operators in the Interaction picture satisfy the samecommutation relations and equations of motion as do those in the Heisenbergpicture. As a result, we can use the same plane wave expansions in the fieldsand the usual propagators.ψ (+)q (x) = ∑ λψ (−)q (x) = ∑ λψ (−)q (x) = ∑ λ∫∫∫∫ψ (+)q (x) = ∑ λψ q (x) = ψ q(+)ψ q (x) = ψ (+)qd 3 k(2π) 3 a q( ⃗ k, λ)u( ⃗ k, λ)e −ikµxµd 3 k(2π) 3 b† q( ⃗ k, λ)v( ⃗ k, λ)e ikµxµd 3 k(2π) 3 a† q( ⃗ k, λ)u( ⃗ k, λ)e ikµxµd 3 k(2π) 3 b q( ⃗ k, λ)v( ⃗ k, λ)e −ikµxµ(x) + ψ (−) (x)q(x) + ψ (−)q (x)(2.38)Contractions between the vector boson fields lead to the usual propagators∫〈0|T {W µ (x)W ν† (y)}|0〉 = i≡∫id 4 k −g µν + kµ k νm 2 W(2π) 4 k 2 − m 2 W + iɛ e−ikµ(x−y)µd 4 k(2π) 4 Dµν F (k, m W )e −ikµ(x−y)µand likewise for the Z boson.The idea is to build on the previous current-current interaction used in [37]and [6] and introduce the strong interaction via H I = H I S + H I W . To second42

order in the weak hamiltonian H I W , we haveH I (t 1 )H I (t 2 )H I (t 3 ) = HS(t I 1 )HS(t I 2 )HS(t I 3 ) + HS(t I 1 )HS(t I 2 )HW I (t 3 )+ HS(t I 1 )HW I (t 2 )HS(t I 3 ) + HS(t I 1 )HW I (t 2 )HW I (t 3 )+ HW I (t 1 )HS(t I 2 )HS(t I 3 ) + HW I (t 1 )HS(t I 2 )HW I (t 3 )+ HW I (t 1 )HW I (t 2 )HS(t I 3 ) + HW I (t 1 )HW I (t 2 )HW I (t 3 )The only terms that contribute are numbers 4, 6 and 7.So the S-matrixexpansion readsS = (−i)36∫dt 1∫dt 2∫dt 3 T {H I S(t 1 )H I W (t 2 )H I W (t 3 ) + H I W (t 1 )H I S(t 2 )H I W (t 3 )+ H I W (t 1 )H I W (t 2 )H I S(t 3 )}Since the integrals run over all possible values in the variables, this can bewritten simply asS = i 2∫dt 1∫dt 2∫dt 3 T {H I W (t 1 )H I W (t 2 )H I S(t 3 )} (2.39)The weak and strong hamiltonians areH I W (x) =∫( gd 3 x2 √ 2)( )Jµ†c W µ + J c µ W µ† g + J µ4 cos θNZ µw(2.40)43

∫HS(x, I y) = −∫d 3 x∫d 3 ydt y J ρ Sf(⃗x, ⃗y)δ(t x − t y ) (2.41)At this point it is most convenient to use isospin symmetry to establish which ofthe possible current-current-current contributions will contribute to the overallprocess amplitude N → Nπ in the isovector channel.Isospin StructureThe currents from the strong (flavor conserving) interaction are always ∆I =0. Making sure to use only those terms that correspond to quark-anti-quarkpair creation and annihilation takes care of that.J ρ S = ψ + u Γ ρ ψ + u + ψ − u Γ ρ ψ − u+ ψ + d Γ ρ ψ + d + ψ− d Γ ρ ψ − d+ 2(ψ + s Γ ρ ψ + s + ψ − s Γ ρ ψ − s )(2.42)As a result, the isospin structure of the S-matrix elements is determined bythe weak currents alone. The charged currents,ψ u γ µ (1 − γ 5 )ψ d cos θ candψ d γ µ (1 − γ 5 )ψ u cos θ c44

transform as ∆I = 1, whileψ s γ µ (1 − γ 5 )ψ u sin θ candψ u γ µ (1 − γ 5 )ψ s sin θ ctransform as ∆I = 1 2 .The symmetric product of two ∆I = 1 charged currents transforms as ∆I =0, 2, while the antisymmetric product transforms as ∆I = 1. In the currentformalism, the antisymmetric product does not emerge (it would vanish underisospin rotations). Therefore the only ∆I = 1 contributions from the chargedcurrent are produced in the combination of two ∆I = 1 2currents which, therefore,must include strange quarks. The models used here, in the descriptionof nuclear wave functions in the constituent quark model, include only up anddown valance quarks. This then requires a contraction between the strangequarks in the weak and strong currents, producing loop diagrams from thestrange quark see in the nucleon.To highlight their transformation properties under isospin, the different neutralcurrent terms can be rewritten in the following form:J µ N = Ψ q γ µ (1 + 2 sin 2 θ w − γ 5 )τ 3 Ψ q− 2 3 sin2 θ w Ψ q γ µ IΨ q + ψ s γ µ (1 − 4 3 sin2 θ w − γ 5 )ψ s(2.43)45

where τ 3 is the third Pauli matrix, I is the identity matrix and⎛Ψ q =ψ ⎞u⎝ ⎠ .ψ dThe first current in 2.43 transforms as ∆I = 1, while the other two transformas ∆I = 0. As before, the symmetric product of two ∆I = 1 neutral currentstransforms as ∆I = 0, 2, while the antisymmetric product transforms as ∆I =1 and vanishes.So, keeping in mind that we can not have any mixing between neutral andcharged currents, the fact that all terms multiplied by sin θ c ≃ 0.22 are Cabbibosuppressed, and that, given our initial and final state, we have to have aneven number of anti-particle creation and annihilation operators in the matrixelements, the tree-level contributions to the ∆I = 1, ∆S = 0 amplitude are:J µ†N (x 1 )J ν N(x 2 )J ρ S(x 3 ) = − 2 3 sin2 θ w Ψ q (x 1 )γ µ (1 + 2 sin 2 θ w − γ 5 )τ 3 Ψ q (x 1 )×Ψ q (x 2 )γ ν IΨ q (x 2 )×Ψ q (x 3 )Γ ρ Ψ q (x 3 )(2.44)Adding the neutral and charged current terms involving the strange quarkswill introduce loop diagrams on top of the tree-level diagrams above. Theseadditional terms are not included in this work, but may pose a significantcontribution.46

2.5.2 Reduction of Terms (Quark Dynamics)With the selection of terms entering into the reaction under the proper isospinchannel, the calculation of the amplitudes amounts to the evaluation of equation2.39.Explicitly, we haveS = i ∫d 4 x 1 · · d 4 x 4 T {J µ†N (x 1 )J ν2N(x 2 )(2π)δ(t x3 − t x4 )JS(x ρ 3 , x 4 )}f(⃗x 3 , ⃗x 4 )( )= ig2 1∫16 3 tan2 θ w d 4 x 1 d 4 x 2 d 4 x 3 d 4 x 4 T {∑Ψ q (x 1 )Λ µ τ 3 Ψ q (x 1 ) ∑Ψ q (x 2 )γ ν IΨ q (x 2 )q=u,d× ∑q=u,dq=u,dΨ q (x 3 )Γ ρ Ψ q (x 4 )}δ(t 3 − t 4 )f(⃗x 3 , ⃗x 4 )D µνF (x 1 − x 2 , M Z )(2.45)where Λ µ ≡ γ µ (1 + 2 sin 2 θ w − γ 5 ).The time-ordered product is evaluated by taking the normal-ordered productwith all possible contractions between field operators . For tree-level processeswe only take the terms with zero or one contraction between fermion operators.Using the quark field equations 2.38, evaluation of 2.45 produces a series of dynamicalconfigurations where quarks with specific flavors combine to producethe Feynman amplitudes allowed under the isospin selection rules. Contractionsbetween the particle and antiparticle creation and annihilation operatorsin the S matrix terms and those in the nuclear wave function produce thequark dynamics for the entire process (See section 2.5.2).47

Nuclear Wavefunctions in the Constituent Quark ModelThe initial and final states in the reaction N → Nπ include the nucleon andpion wave functions which consist, in the constituent quark model, of threeand two quarks of the first generation respectively.|N〉 ≡ ∑ ijkΨ N (ijk)|q i , q j , q k 〉= ∑ ijk∫d 3 k i d 3 k j d 3 k k(2π) 9 Ψ N ( ⃗ k i · ·; λ i · ·; f i · ·; c i · ·)a † ⃗ ki λ ia † ⃗ kj λ ja † ⃗ kk λ k|0〉|π〉 ≡ ∑ rsΨ π (rs)|q r , q s 〉= ∑ rs∫d 3 k r d 3 k s(2π) 6 Ψ N ( ⃗ k r , ⃗ k s ; λ r , λ s ; f r , f s ; c r , c s )a † ⃗ krλ rb † ⃗ ksλ s|0〉(2.46)Here ∑ ijk is short-hand for ∑ λ i ,λ j ,λ k∑f i ,f j ,f k∑c i ,c j ,c kand λ, f, c stand for spin,flavor, and color degrees of freedom.The exact form of the wave functions in 2.46 depends on the specific nuclearmodel used, but the important decision at this stage is to choose their symmetryunder particle (quark) exchange. The symmetric behavior for the nucleonwave function is governed by the choice for a particular form of its spin-flavorpart. If the wave function is antisymmetric under the exchange of any two ofthe three quarks, then the spin-flavor portion is a linear combination of mixedsymmetric and mixed antisymmetric functions consisting of all possible spinand flavor combinations of the three quarks. The pion wave functions is alwaysantisymmetric under exchange of the quark-antiquark pair labels.48

Most models assume that the nucleon wave functions are antisymmetric onlywith respect to exchange of the two quark with the same flavor but not withrespect to exchange any of those two with the third. Since going from thismodel to one in which the wave function is antisymmetric under the exchangeof any pair of quark labels only produces a further reduction in distinct dynamicalcombinations, the more general calculation takes the antisymmetryto be with respect to exchange of equal flavor quarks. The results obtainedhere will be based on that assumption. To be precise, we take the followingsymmetry under particle permutation:Ψ N (ijk) = −Ψ N (jik)Ψ π (rs) = −Ψ π (sr).f i = f jMatrix Elements (Evaluation of Terms)The task at hand is the evaluation of the matrix elements of 2.45 between thestates described in 2.46. A reasonably detailed outline of the calculation forone of the matrix elements will be given without showing specific steps thatcould be considered as common. For the remaining matrix elements the resultsfollow from the same treatment.Before proceeding, a few more remarks about the strong interaction portionin 2.45 will clarify what is being done. The problematic feature of the strongterm is that one wants to characterize it as a process that has a finite space-time49

structure. That is, it happens over a finite time interval and within a finitevolume. Quark confinement would then require that the probability of thecreation or annihilation of a quark-antiquark pair decreases with an increase inthe distance over which the interaction is taking place. The difficulty with themethod used here is that one cannot time order a hamiltonian which explicitlycarries this information. In the current case, the delta function in time that wasplaced in the hamiltonian simply says that the probability for this interactionis zero, unless it is instantaneous, but it has a spatial distribution instead ofbeing a point interaction. This space-time structure could be described if onewere to include exchange particles (gluons), but as already mentioned, for lowenergy processes the strong coupling constant to be included in such a setupis not well known.Given the quark-flavor degrees of freedom in the initial and final nucleon statesand the isospin selection rules laid out in section 2.5.1, there are 32 possibletree-level terms arising from internal contractions between weak and strongcurrents, in equation 2.45 as well as 8 additional terms arising when thereare no internal contractions. Due to the structure of the strong interaction(eqn. 2.36), the first 32 terms must be considered separately when consideringthe internal contractions. Since we are only considering quark-anti-quark pairs,the contractions do not lead to the full, Lorentz covariant fermion propagatorsbut rather to their non-covariant quark and anti-quark counter parts. Thereis, in the end, no problem with this, since non relativistic reductions willeventually be made when evaluating the Feynman amplitudes. However, oneneeds to keep track of the sign difference in the propagators when evaluating50

the integrals. From equation 2.45, we writeig 216( 13 tan2 θ w) ∫d 4 x 1 · · d 4 x 4 M f(⃗x 3 , ⃗x 4 )δ(t 3 − t 4 ). (2.47)Where the possible internal structure is given byM = : Ψ (−)f 1(x 1 )Λ µ τ 3 Ψ (+)f 2(x 1 )iD µν (x 1 − x 2 )Ψ (−)f 3(x 2 )γ ν Ψ (+)f 4(x 2 )Ψ (−)f 5(x 3 )} {{ } Γρ Ψ (−)f 6(x 4 ) :+ : Ψ (−)f 1(x 1 )Λ µ τ 3 Ψ (−)f 2(x 1 )iD µν (x 1 − x 2 )Ψ (−)f 3(x 2 )γ ν Ψ (−)f 4(x 2 )Ψ (+)f 5(x 3 )} {{ } Γρ Ψ (+)f 6(x 4 ) :+ : Ψ (−)f 1(x 1 )Λ µ τ 3 Ψ (+)f 2(x 1 )Ψ (−)f 5(x 3 )} {{ } Γρ Ψ (−)f 6(x 4 )iD µν (x 1 − x 2 )Ψ (−)f 3(x 2 )γ ν Ψ (+)f 4(x 2 ) :+ : Ψ (−)f 1(x 1 )Λ µ τ 3 Ψ (−)f 2(x 1 )Ψ (+)f 5(x 3 )} {{ } Γρ Ψ (+)f 6(x 4 )iD µν (x 1 − x 2 )Ψ (−)f 3(x 2 )γ ν Ψ (−)f 4(x 2 ) :+ : Ψ (−)f 1(x 1 )Λ µ τ 3 Ψ (+)f 2(x 1 )iD µν (x 1 − x 2 )Ψ (−)f 5(x 3 )Γ ρ Ψ (−)f 6(x 4 )Ψ (+)f 3(x 2 )} {{ } γν Ψ (−)f 4(x 2 ) :+ : Ψ (−)f 1+ : Ψ (−)f 5+ : Ψ (−)f 5(x 1 )Λ µ τ 3 Ψ (−)f 2(x 1 )iD µν (x 1 − x 2 )Ψ (−)f 5(x 3 )Γ ρ Ψ (−)f 6(x 4 )Ψ (+)f 1(x 1 )} {{ } Λµ τ 3 Ψ (+)f 2(x 3 )Γ ρ Ψ (−)f 6(x 4 )Ψ (+)f 1(x 1 )} {{ } Λµ τ 3 Ψ (−)f 2f 2(x 3 )Γ ρ Ψ (−)f 6(x 4 )Ψ (+)f 3(x 2 )} {{ } γν Ψ (+)f 4(x 2 ) :(x 1 )iD µν (x 1 − x 2 )Ψ (−)f 3(x 2 )γ ν Ψ (−)f 4(x 2 ) :(x 1 )iD µν (x 1 − x 2 )Ψ (−)f 3(x 2 )γ ν Ψ (+)f 4(x 2 ) :+ : Ψ (−)f 1(x 1 )Λ µ τ 3 Ψ (+) (x 1 )iD µν (x 1 − x 2 )Ψ (−)f 3(x 2 )γ ν Ψ (+) (x 2 )Ψ (−)f 5(x 3 )Γ ρ Ψ (−)f 6(x 4 ) :f 4For the first term, the substitution of the quark fields 2.38 and integrationover the space-time coordinates produces the result⎡4∏C w⎣ ∑ ∫n=1 λ n⎤d 3 ∫k n ⎦(2π) 12d 4 P(2π) 4 (2π)4 δ 4 (k 1 − k 2 + k 3 − P )M(2π)δ(E k4 + E P ) ×{a † ⃗ k1 ,u a† ⃗ k3 ,u b† ⃗ k4 ,u a ⃗ k2 ,u + a† ⃗ k1 ,u a† ⃗ k3 ,d b† ⃗ k4 ,d a ⃗ k2 ,u − (u ⇀↽ d)}.51

WhereM = u( ⃗ k 1 , λ 1 )Λ µ u( ⃗ k 2 , λ 2 )iD µν (k 1 −k 2 )u( ⃗ k 3 , λ 3 )γ ν iS(P )Γ ρ v( ⃗ k 4 , λ 4 ) ˜f(− ⃗ P , − ⃗ k 4 )andC w = ig216( 13 tan2 θ w).Here, use has been made of the standard Fourier transformation∫dt 3∫dt 4 e iE P t 3e iE k 4t 4δ(t 3 − t 4 ) = (2π)δ(E k4 + E P )as well as∫d 3 x 4∫d 3 x 3 e i ⃗ P ·⃗x 3e i⃗ k 4·⃗x 4f(⃗x 3 , ⃗x 4 ) = ˜f( ⃗ P , ⃗ k 4 ).Now, momentum conservation in the strong vertex implies that˜f( ⃗ P , ⃗ k 4 ) = (2π) 3 δ 3 ( ⃗ P + ⃗ k 4 ) ˜f(| ⃗ P − ⃗ k 4 |).So that the matrix element becomes⎡⎤4∏C w⎣ ∑ ∫d 3 k n ⎦ (2π) 4 δ 4 (kn=1 λ n(2π) 12 1 − k 2 + k 3 + k 4 )M ×{a†⃗ k1 ,u a† ⃗ k3 ,u b† ⃗ a k4 ,u ⃗ + k2 ,u a† ⃗ k1 ,u a† ⃗ k3 ,d b† ⃗ a k4 ,d ⃗ − (u k2⇀↽ d) } .,u(2.48)With a Feynman amplitude given byM = u( ⃗ k 1 , λ 1 )Λ µ u( ⃗ k 2 , λ 2 )iD µν (k 1 − k 2 )u( ⃗ k 3 , λ 3 )γ ν iS(− ⃗ k 4 )Γ ρ v( ⃗ k 4 , λ 4 ) ˜f(2| ⃗ k 4 |).The corresponding Feynman graph is shown in Fig. 2.5.52

ρΓ~f (2|k 4|)q,k 4+S (-k 4)q ,k 3DµνZ(k 1-k 2)q’ ,k 2q’ ,k 1Figure 2.5: Feynman diagram for a typical amplitude in S-matrix expansion.D zµν is the neutral vector boson propagator and ˜f is the pion momentumdistribution. The large blue circle represents the strong vertex.Using the initial and final states (eqns. 2.46), the portion of the amplitudepertaining to the second quantization portion takes the form〈0|a i a j a k a π b π a † 1a † 3b † 4a 2 a † l a† ma † n|0〉Where {1, 2, 3, 4}, {l, m, n}, {i, j, k}, {π, π} denote the internal, initial and finalnucleon, as well as pion degrees of freedom respectively.Carrying out thecontractions between second quantization operators produces nineteen distinctdynamical combinations which can be split into 3 groups:1.2δ π,4 δ π,1 {2δ l,2 (δ k,3 δ j,m δ i,n − δ j,3 (δ i,n δ k,m−δ i,m δ k,n )) + δ n,2 δ i,m (δ k,3 δ j,l − 2δ j,3 δ k,l ))} (2.49)53

2.2δ π,4 δ π,3 {2δ l,2 (δ j,1 (δ i,n δ k,m − δ i,m δ k,n ) − δ k,1 δ j,m δ i,n )−δ n,2 δ i,m (δ k,1 δ j,l − 2δ j,1 δ k,l ))} (2.50)3.4δ π,4 δ l,2 δ j,1 {δ i,3 (δ k,n δ π,m − δ k,m δ π,n ) − δ k,3 (δ i,n δ π,m − δ i,m δ π,n )}+4δ π,4 δ l,2 δ j,3 δ k,1 (δ i,n δ π,m − δ i,m δ π,n ) (2.51)4δ π,4 δ n,2 δ π,l {δ k,1 δ j,3 (δ i,m + δ j,1 (δ i,3 δ k,m − δ k,3 δ i,m )} (2.52)The corresponding diagrams are shown in Figs. 2.6 and 2.7.πd(u)πd(u)πu(d)πd(u)Z propagatorStrong Vertexd(u)d(u)u(d)lmnjkiu(d)d(u)u(d)d(u)d(u)u(d)lmnijkd(u)d(u)u(d)Figure 2.6: Left : Diagram for the five dymanical from Eqn. 2.49Right: Diagram for the five dymanical from Eqn. 2.5054

ππd(u)u(d)ππd(u)u(d)d(u)md(u) ljku(d)d(u)d(u)u(d)lniju(d)u(d)u(d)niu(d)d(u)mkd(u)Figure 2.7: Left : Diagram for the five dymanical from Eqn. 2.51Right: Diagram for the five dymanical from Eqn. 2.5255

As already indicated in the diagrams, only three of the nineteen terms survive.This situation emerges as a result of the symmetry properties of the nucleonand pion wavefunctions, which was laid out in section 2.5.2, and the particularreaction under investigation (i.e. the fact that we are looking at the neutralcurrent isovector component of pion-nucleon coupling). The second group, forexample, does not contribute at all, because, for neutral currents, these termsgive rise only to Nπ o coupling.The diagrams show that, among those terms that do contribute, all isospinallowed internal combinations of flavor degrees of freedom are present. Thus,looking back at the matrix element (eqn 2.48), if we writef 1 f 3 f 4 f 2 = 1) uuuu2) uddu3) − duud4) − dddd(2.53)we can see that the diagram pertaining to group one and the first diagramin group three both include flavor combinations 2 and 3, while the seconddiagram in group three includes combinations 1 and 4.These three terms can now be combined with the wavefunctions in the initialand final states. The final result of the analytical portion of the caluclation isthen expressed in terms of a nine dimensional integral and sum over a set ofinitial and final state degrees of freedom which define the state of the nucleons56

and the pion before and after the reaction. Correspondingly, for the the firstdiagram we have (−4δ π,4 δ π,1 δ l,2 δ j,3 δ i,n δ k,m )〈N f , π|S 1 1|N i 〉 = − ig2 tan 2 θ w12∑ ∑∫ijk lππd 3 k i d 3 k k d 3 k π(2π) 9 (2π)δ(E f − E i )⊗ (2π) 3 δ 3 ( ⃗ P Nf + ⃗ P π − ⃗ P Ni )M ˜f(2| ⃗ k π |)⊗ Ψ Ni ( ⃗ P Ni − ⃗ k i − ⃗ k k , ⃗ k k , ⃗ k i ; l, k, i)⊗ Ψ ∗ N f( ⃗ k i , ⃗ P Nf − ⃗ k i − ⃗ k k , ⃗ k k ; i, j, k)⊗ Ψ ∗ π( ⃗ P π − ⃗ k π , ⃗ k π ; π, π)where, taking the low energy limit for the neutral boson,M =−1M 2 Z(k 2 π − m 2 ) u( ⃗ P π − ⃗ k π , λ π )Λ µ u( ⃗ P Ni − ⃗ k i − ⃗ k k , λ l )g µν⊗ u( ⃗ P Nf − ⃗ k i − ⃗ k k , λ j )γ ν (−̸ k π + m)Γ ρ v( ⃗ k π , λ π )and (f π = f k = f l = d(u), f π = u(d), f i = f j = u(d)).For the second diagram we find (4δ π,4 δ l,2 δ j,3 δ k,1 δ i,n δ π,m )〈N f , π|S 2 1|N i 〉 =ig 2 tan 2 θ w12∑ ∑∫ijk lππd 3 k i d 3 k k d 3 k π(2π) 9 (2π)δ(E f − E i )⊗ (2π) 3 δ 3 ( ⃗ P Nf + ⃗ P π − ⃗ P Ni )M ˜f(2| ⃗ k π |)⊗ Ψ Ni ( P ⃗ Nf − ⃗ k i + ⃗ k π , P ⃗ π − ⃗ k π , ⃗ k i ; l, π, i)⊗ Ψ ∗ N f( ⃗ k i , P ⃗ Nf − ⃗ k i − ⃗ k k , ⃗ k k ; i, j, k)⊗ Ψ ∗ π( P ⃗ π − ⃗ k π , ⃗ k π ; π, π)57

withM =−1M 2 Z(k 2 π − m 2 ) u(⃗ k k , λ k )Λ µ u( ⃗ P Nf − ⃗ k i + ⃗ k π , λ l )g µν⊗ u( ⃗ P Nf − ⃗ k i − ⃗ k k , λ j )γ ν (−̸ k π + m)Γ ρ v( ⃗ k π , λ π )and (f π = f k = f l = d(u), f π = u(d), f i = f j = u(d)).And for the third diagram we find (−4δ π,4 δ n,2 δ j,1 δ k,m δ i,3 δ π,l )〈N f , π|S 2 1|N i 〉 =ig 2 tan 2 θ w12∑ ∑∫ijk nππd 3 k i d 3 k k d 3 k π(2π) 9 (2π)δ(E f − E i )⊗ (2π) 3 δ 3 ( ⃗ P Nf + ⃗ P π − ⃗ P Ni )M ˜f(2| ⃗ k π |)⊗ Ψ Ni ( ⃗ P π − ⃗ k π , ⃗ k k , ⃗ P Nf − ⃗ k k + ⃗ k π ; π, k, n)⊗ Ψ ∗ N f( ⃗ k i , ⃗ P Nf − ⃗ k i − ⃗ k k , ⃗ k k ; i, j, k)⊗ Ψ ∗ π( ⃗ P π − ⃗ k π , ⃗ k π ; π, π)whithM =−1M 2 Z(k 2 π − m 2 ) u( ⃗ P Nf − ⃗ k i − ⃗ k k , λ j )Λ µ u( ⃗ P Nf − ⃗ k i + ⃗ k π , λ n )g µν⊗ u( ⃗ k i , λ i )γ ν (−̸ k π + m)Γ ρ v( ⃗ k π , λ π )58

and (f π = f k = d(u), f π = u(d), f i = f j = f n = u(d)).The calculation of the remaining terms proceeds in the same fashion as describedhere. Solving all terms in 2.47 produces 30 integrals having two possibleflavor combinations each. In addition to the two contributing diagramsalready shown they give rise to 8 more; 6 from the terms with one internalcontraction and 2 from the last term. The last term in 2.47 produces a differenttype of Feynman amplitude, corresponding to the disconnected strong andweak vertexes. This only manifests itself in the number possible dynamicalcombinations between internal and external degrees of freedom. The diagramsare shown in figures 2.8 and 2.9.Figure 2.8: Diagrams depicting configurations involving internal contractionsFigure 2.9: Diagrams depicting configurations without internal contractionsThe results for all integrals can best be summarized by writing down a genericintegral describing all possibilities and then writing the momenta in tables.59

For example, table 2.1 summarizes the results written out explicitly abovefor the first term in equation 2.47. Additional tables, summarizing all termstreated here, are listed in appendix B.3. The pion and nucleon wave functionsin the generic integral are expressed explicitly in terms of the momenta andflavor. The remaining degrees of freedom (spin and color) are expressed in thegeneric variables {ξ 1 · · · ξ 8 }.Collecting all terms together, we write the generic integral as〈N f , π|S|N i 〉 = (2π)δ(E f − E i )(2π) 3 δ 3 ( ⃗ P Nf + ⃗ P π − ⃗ P Ni )⊗ αC w∫ d 3 k i d 3 k k d 3 k π(2π) 9 M ˜f(2|⃗q 9 |)⊗ Ψ Ni (⃗q 1 , f 1 , ξ 1 ; ⃗q 2 , f 2 , ξ 2 ; ⃗q 3 , f 3 , ξ 3 )⊗ Ψ ∗ N f(⃗q 4 , f 4 , ξ 4 ; ⃗q 5 , f 5 , ξ 5 ; ⃗q 6 , f 6 , ξ 6 )⊗ Ψ ∗ π(⃗q 7 , f 7 , ξ 7 ; ⃗q 8 , f 8 , ξ 8 )(2.54)where, as before,C w = ig216( 13 tan2 θ w),Λ µ ≡ γ µ (1 + 2 sin 2 θ w − γ 5 ),andΓ ρ = I.Reading the values for ⃗q i , ξ i and f i from the tables and substituting theminto 2.54 then reproduces each element in equation 2.48. In the tables, α isthe constant with magnitude equal to that arrived at while carrying out the60

external contractions and with a sign being the result of both the contractionsas well as from the signs introduced by the isospin selection rules; c1 and c2denote the two different flavor combinations that arise for each integral (Seeeqn. 2.53, for example).61

Table 2.1: S-Matrix Term 1Integration over ⃗ k i , ⃗ k k and ⃗ k π , Summation over i,j,k,(l or n),π,πIntegral 1 Integral 2 Integral 3c1 c2 c1 c2 c1 c2α -4 +4 +4 -4 -4 +4i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i1 PNi⃗ − ⃗ k i − ⃗ k k , l u d PNf⃗ − ⃗ k i + ⃗ k π , l u d Pπ ⃗ − ⃗ k π , π u d2 ⃗ kk , k u d Pπ ⃗ − ⃗ k π , π u d ⃗ kk , k u d3 ⃗ ki , i d u ⃗ ki , i d u PNf⃗ − ⃗ k k + ⃗ k π , n d u4 ⃗ ki , i d u ⃗ ki , i d u ⃗ ki , i d u5 PNf⃗ − ⃗ k i − ⃗ k k , j d u PNf⃗ − ⃗ k i − ⃗ k k , j d u PNf⃗ − ⃗ k i − ⃗ k k , j d u6 ⃗ kk , k u d ⃗ kk , k u d ⃗ kk , k u d7 Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π u d8 ⃗ kπ , π d u ⃗ kπ , π d u ⃗ kπ , π d u9 ⃗ kπ ⃗ kπ ⃗ kπ−1M1MZ 2 (q2 8 −m2 ) 7, λ 7 )Λ µ u(⃗q 1 , λ 1 )g µν u(⃗q 5 , λ 5 )γ ν (−̸ q 8+ m)Γ ρ v(⃗q 8 , λ 8 )−1M2MZ 2 (q2 8 −m2 ) 6, λ 6 )Λ µ u(⃗q 1 , λ 1 )g µν u(⃗q 5 , λ 5 )γ ν (−̸ q 8+ m)Γ ρ v(⃗q 8 , λ 8 )−1M3MZ 2 (q2 8 −m2 ) 5, λ 5 )Λ µ u(⃗q 3 , λ 3 )g µν u(⃗q 4 , λ 4 )γ ν (−̸ q 8+ m)Γ ρ v(⃗q 8 , λ 8 )62

Chapter 3Experimental Setup andComponent Performance3.1 IntroductionThe small size of the asymmetry imposes stringent requirements on the performanceof the beam line and apparatus. It is necessary to achieve high countingstatistics while at the same time suppressing any systematic errors below thestatistical limit.The experiment makes use of an intense cold neutron beam at LANSCE[38]. The beam is pulsed at 20 Hz and transversely polarized by transmissionthrough a polarized 3 He cell. A radio frequency spin flipper is used toreverse the neutron spin direction on a pulse-by-pulse basis. The neutrons arecaptured in a 20 l liquid para-hydrogen target. The 2.2 MeV γ-rays from the63

n-p capture reaction are detected by an array of 48 CsI(Tl) detectors. Theentire apparatus is located in a homogeneous 10 G magnetic field to maintainthe neutron spin downstream of the polarizer and to suppress Stern-Gerlachsteering of the neutrons. Three 3 He ion chambers are used to monitor beamintensity, measure beam polarization and transmission and monitor the orthopararatio in the liquid hydrogen target. A schematic of the beam line andexperimental setup is shown in Fig.3.1Figure 3.1: Schematic of the ⃗n + p → d + γ experimental setup.The remainder of this chapter describes the components used in the experimentand the measurements performed to verify that they operate according to theirdesign. The components described in this chapter include the beam line, thebeam chopper and monitors, the neutron spin filter and spin flipper, and thevarious targets used during the 2004 commissioning run as well as the liquidhydrogen target to be used in the production data runs. The detector arrayis discussed in chapter 4 and data acquisition and storage are discussed in64

¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¨¢¢¢¢¢¢¤¤¤¤¤¤¥¦¥¦¥¦¥¦¥¦¥¦¥¦¥¦section 3.7. The detailed actual layout of the experimental components in thecave during the 2004 commissioning run is shown in Fig. 3.2Guide§¡§§¡§§¡§§¡§Borated Poly Shielding¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡ ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡DetectorArray3HeAnalyzerCell¡§¡§¡§¡§¡§¡§§¡§¡M13HeOven§¡§§¡§§¡§§¡§§¡§§¡§§¡§§¡§§¡§§¡§©¡©©¡©¡¡¡¡¡¡M2£¡££¡£RFSF¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡M 3£¡£©¡©¡¡£¡£©¡©©¡©£¡£¡£¡£¡©¡©©¡©¡©¡©¡¡©¡©¡©¡©¡©¡©¡©¡©¡©¡©©¡©¡©¡©¡©¡©¡©¡©¡©¡©¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡7.2 25.8 13.7 32.5 66 11.6 95Figure 3.2: 2004 in cave component layout. All distances are given in cm. Allhatched regions are ∼ 2.5 cm thick 7% borated poly. The borated poly and2 mm thick sheets of 6 Li loaded plastic (not shown) were used for low energyneutron shielding. The monitors are ∼ 5 cm thick. The beam collimationaround M2 was usually either ∼ 5 or ∼ 10 cm depending on the measurement.The analyzer cell had a diameter of ∼ 2.5 cm and collimation was ∼ 1.5 cmin front of it and ∼ 1.6 cm behind it.65

3.2 Flight Path 12 Beam Line and ModeratorThe NPDGamma experiment is located on flight path 12 at the Manuel LujanJr. Neutron Scattering Center at LANSCE. The statistical limit that can bereached in the NPDGamma experiment at LANSCE is limited by the availablecold neutron flux. In a spallation neutron source, the amount of neutron fluxdepends on the proton current, the energy incident on the spallation target,the moderator performance (brightness) and the neutron guide performance.The LANSCE linear accelerator delivers 800 MeV protons to a proton storagering, which compresses the beam to 250 ns wide pulses at the base, at a rateof 20 Hz. The protons from the storage ring are incident on a split Tungstentarget and the resulting spallation neutrons are cooled by and backscatteredfrom a cold H 2 moderator with a surface area of 12 × 12 cm 2 . Flightpath 12 (FP12) views the H 2 moderator which incorporates a new configuration.The first brightness measurement for this moderator has been done bythe NPDGamma collaboration in January 2003. The cold neutrons from themoderator are directed through FP12 by an m = 3 supermirror guide systemwhose performance was measured for a portion of the guide in January 2003and again, for the entire guide length during the 2004 commissioning run.Figure 3.3 illustrates the flight path and experiment cave. The distance betweenthe moderator and target is about 22 meters. The flight path 12 beamline is ≃ 19.6 m long and consists of 4 m of in pile guide, a 2 m long shutter, a66

eam chopper and ≃ 13 meters of neutron guide. The pulsed neutron sourceallows us to know the neutron time of flight or energy and the installed beamchopper allows us to select a range of neutron energies (see below).Figure 3.3: FP12 beam line, with in-pile guide, beam shutter, beam chopper,13 m of guide and experimental cave.3.2.1 Moderator BrightnessFor the moderator brightness measurement, neutrons have to be detected directlyfrom the moderator, without reflection from the neutron guide surfaces.67

£¤ ££££¤¤¤¤£¤¤¦ ¥¥¥¦¦¦¥¥¥¥¦¦¦¦¥BiologicalShield¡¢¡¢¡¢¢¡¢¡¢¡ ¢¡¢¡¢¡ ¢¡¢¡¢¡ ¢¡¢¡¢¡ ¢¡¢¡¢ £Neutron GuideCollimatorsDetectorColdH 2Moderator9.3 m 2.6 m 2.6 mFigure 3.4:To measure the direct neutrons a two-pinhole collimator system (neutron camera)was set up. The upstream collimator defined the magnification of theviewed moderator surface area, while the downstream collimator restrictedthe active detector area to reduce the count rate to levels suitable for pulsecounting. The neutron detector consisted of a 4 mm diameter (collimated toan active area of 2 mm) and 2 mm thick circular 6 Li-loaded glass scintillator.In the detector, the scintillation light produced by each product in the reaction6 Li(n, α) 3 H is processed with a photomultiplier tube. The setup is shown inFig.( 3.4).Figure 3.5 shows the measured brightness with a maximum of 1.25 × 10 8n/s/cm 2 /sr/meV/µA for neutrons with an energy of 3.3 meV. A detaileddescription of our measurement of the FP12 moderator brightness and performanceof the first 4 meters of neutron guide is given in [38].68

Figure 3.5: Measured FP12 moderator brightness, averaged over an area of0.93 cm 2 centered on the moderator surface.3.2.2 Neutron GuideThe neutron guide effectively moves the neutron source (moderator) to thefront of the guide using total internal reflection for a range of angles up tothe critical angle. Typical guides are made of 58 Ni and allow for perpendicularneutron velocities up to 7 m/s (Fig. 3.6). Flight path 12 uses a m=3 guide witha 9.5×9.5 cm 2 cross-sectional area. The guide is coated with hundreds of layersof 58 Ni and 47 Ti. It allows neutrons with 3 times the normal perpendicularvelocity to be transmitted, resulting in a ∼ 20% increase in neutron flux.The maximum glancing angle at which total internal reflection still occurs isgiven byθ c = mθ c ( 58 Ni),where θ c ( 58 Ni) is the critical glancing angle of the usual nickel coating [39].The effective potential of 58 Ni for slow neutrons is about 335 neV. This gives69

ModeratorGuideExperimentFigure 3.6: Illustration of the neutron guide mechanism. Low energy neutronsare guided to the experiment with minimal loss.a maximum perpendicular neutron velocity of 7 m/s that a slow neutron mayhave and still be reflected from a surface with this coating. Figure 3.7 showsthe reflectivity measured by the manufacturer for a 50 cm long section of theguide used on FP12.Figure 3.7: Measured reflectivity as a function of the glancing angle m =θ c /θ c ( 58 Ni) for 4.6 meV neutrons.The guide performance was measured by mounting the downstream collimatorand detector on a scanner and moving them to different positions on a 12 ×12 cm 2 grid perpendicular to the beam direction in 2 mm steps. To study70

the guide performance we looked at neutrons with an energy of 10 meV whichwere either coming directly from the moderator or were reflected once or twicefrom the guide surface.Figure 3.8 shows the beam profiles resulting from the up-down and left-rightdetector scans done for the first 9 meters of installed guide(see Fig. 3.5). Fordetector positions up to ∼ ±1 cm from the nominal center along either axis thedetected neutrons come directly from the moderator without reflection. Forpositions between ∼ ±1 and ∼ ±3 the the detected neutrons were reflectedonce and neutrons detected between ∼ ±3 and ∼ ±4 were reflected twice. Theapparent asymmetry about the origin in the up-down scan is due to a smallmisalignment between the moderator and guide [38].Figure 3.9 shows the beam profiles for the entire length of the guide, measuredduring the 2004 commissioning run. As the detector is moved further off thebeam axis, neutrons must have larger glancing angles and undergo a largernumber of reflections to be detected. Each peak in the profiles of Fig. 3.9corresponds to a certain number of reflections for the detected neutron. Themaximum number of reflections for a 3 meV neutron and with this collimationis six.The beam profile measurements were then used to calculate the average reflectivityper neutron reflection, for the entire guide [40]. The results are shownin Fig. 3.10, which should be compared with Fig. 3.7. According to Fig. 3.10the guide performance meets the specifications.71

Figure 3.8: Beam profile from the 2003 guide performance measurement.3.2.3 Frame Overlap ChopperThe frame overlap chopper is shown if Fig.( 3.11). It incorporates two bladeswhich rotate independently at up to 1200 rpm. The chopper is located 9.38 mfrom the surface of the moderator. Since the flight path is about 21 m andthe pulse period is 50 ms the slowest neutrons that reach the end of the guidein each pulse have an energy of about 1 meV. The blades are coated with athick layer of Gd 2 O 3 which was determined to be black for neutron energiesup to 30 meV. This is used to block the slow neutrons at the tail end of the72

Figure 3.9: Beam profile from the 2004 commissioning run.time-of-flight spectrum when either one or both of the blades cover the beamopening. The diameter of the blades is 1.024 m and each blade covers 4.38 radof a full circle. The chopper does not stop fast neutrons so shielding is requiredto reduce the background seen in the experimental apparatus.The ability to select only part of the neutron spectrum is an important toolto control systematic errors since it provides the ability to effectively polarize,spin-flip and capture the neutrons (see Sections 3.5 and 3.6). It prevents theoverlapping of very slow neutrons from a previous pulse with the faster onesfrom the following pulse. At a distance of about 22 m from the moderator,73

Figure 3.10: FP12 guide reflectivity per neutron reflection for 3 meV neutrons.the time required to fully open or close the beam aperture is ∼ 4 ms. Duringthe 2004 commissioning run, the chopper rotation was phased with the beampulses such that it began opening at each pulse onset (0 ms), was fully open4 ms after pulse onset, began closing about 30 ms after pulse onset and wascompletely closed about 4 ms later (Fig. 3.12). This allowed us to take beamoff(pedestal) data for ∼ 6 ms at the end of each neutron pulse, which isneeded for pedestal and background studies. The chopper feedback loop keptthe chopper in phase with the beam pulse to 30 µs.3.3 Beam MonitorsThe experiment uses three parallel plate ion chambers with a 12 × 12 cm 2active area as beam monitors. The chambers consists of an outer Aluminum74

Figure 3.11: Schematic of the frame overlap chopper.1 mm thick entrance and exit windows. Each chamber contains 3 internal 0.5mm thick Aluminum electrodes. The housing is grounded and the two outerelectrodes are supplied with −5 kV. The current signal is extracted from thecentral electrode (Fig. 3.13).The first (upstream) monitor is located immediately after the neutron guideexit. The second monitor is located downstream of the 3 He polarizer to allow insitu absolute beam polarization measurements to be made. The third monitoris located downstream of the target and detector array. It was used during the2004 commissioning run to study the spin flipper efficiency and will be usedto monitor neutron depolarization in the hydrogen target.The monitors are held at 1 atm pressure, filled with 50% He and 50% N 2 .75

Figure 3.12: Normalized signal from the first beam monitor downstream ofthe guide exit. The red curve shows the signal obtained from a run where thechopper was parked open. The black curve corresponds to a run taken withthe chopper running.The first two (thin) monitors contain mostly 4 He with small fractions (< 3%)of 3 He absorbing about 3% of the beam at 10 meV. The back monitor contains50% of 3 He and absorbs about 40% of the beam at 10 meV that hasnot captured in the target. The monitor signal is produced via the reaction⃗n + 3 He → 3 H + p + 0.765 MeV where most of the kinetic energy contributesto the ionization in the chamber. The ionized atoms (mostly N 2 ) are collectedat the central electrode. The current signal is converted to a voltage with apreamplifier almost identical to the ones used for the CsI(Tl) detector array(see Chapter 4, Section 4.2.2). The amplified monitor 1 voltage signal for twoneutron pulses in a typical run is shown in Fig. 3.14The fast neutron portion of the spectrum is not detected in the beam monitorsdue to the small 3 He absorption cross-section at that energy and the small 3 He76

Beam−5 kV0.5 mm Al electrodesCurrentSignal3 HeN 21 mm Al HousingFigure 3.13: Schematic of the beam monitor.thickness in the first monitor.3.4 The 3 He Neutron Spin FilterAfter exiting the neutron guide, the neutrons are spin filtered by passingthrough a cell containing polarized 3 He.3 He spin filters have a number ofdesirable features. They have large acceptance angles, can be operated in amagnetic holding field of 10 G with a field gradient of less than 1 mG/cm andneutron capture on 3 He does not create a γ-ray background [41]. The 3 Hepolarization and therefore the neutron polarization can be reversed withoutchanging the direction of the holding field, by adiabatic fast passage of the3 He spin. The neutron polarization can be measured with 2 − 3% accruracyand without introducing large magnetic field gradients.77

Figure 3.14: Monitor 1 signal out of the preamplifier for two neutron beampulses. Data is taken for 40 ms in each pulse. Also visible are the aluminumBragg edges from windows in the path of the neutron beam.Figure 3.15 illustrates the neutron polarization process. A small admixture ofRb vapor in the cell is polarized using circularly polarized laser light, opticallypumping the Rb D1 line [42]. The 3 He atoms are polarized through the hyperfineinteraction in which the Rb electron spin is transfered to the 3 He nucleus.Neutrons are then spin filtered by passing through the 3 He cell 1 . The crosssection for n − 3 He singlet state absorption is much larger than for the tripletstate. The capture cross-section for unpolarized neutrons is σ a = (v o /v)σ o ,where σ o = 5327 b at v o = 2200 m/s (26 meV). The cross-section for captureof neutrons with spin parallel to the 3 He nuclear spin is 4 orders of magnitudesmaller than it is for neutrons with spin anti-parallel to the 3 He nuclear spin.So neutrons with spin anti-parallel to the 3 He nuclear spin are absorbed while1 The 3 He cell used during the 2004 run-cycle was developed at NIST and was named”BooBoo”.78

Polarized LaserLightHelmholtz Coils3 He CellUnpolarizedNeutronsPolarizedNeutronsFigure 3.15: Schematic of the 3 He spin filter setup. 3 He nuclei are polarizedthrough spin exchange with the Rb valance electron which is polarized viaoptical pumping of the Rb D1 line.those with spin parallel are transmitted. The neutron transmission is given byT n = T 0 cosh (σ a nlP He ),where P He is the 3 He polarization, (n) is the number density of P He atomsin the cell and (l) is the length of the cell. T 0 = exp(−σ a nl) is the neutrontransmission through an unpolarized cell. The neutron polarization can beobtained fromP n = T + − T −T + + T −= tanh (σ a nlP He ) , (3.1)where T + = exp(−σ a nl(1−P He )) and T − = exp(−σ a nl(1+P He )) are the transmissionsof neutrons with spin parallel and anti-parallel to the 3 He spin. If thetime of flight (energy) of the neutrons is known, then the neutron polarizationcan be determined directly from the neutron transmission measurements [42]( ) 2√ T0P n = 1 − . (3.2)T n79

Figure 3.16: Neutron transmission data for an unpolarized 3 He spin filter celland a polarized cell.For NPDGamma, the figure of merit is the product between the neutron transmissionand the square of the neutron polarization P 2 nT n . The neutron transmissionincreases as a function of energy, whereas the neutron polarizationdecreases as a function of energy. In the analysis of the data taken duringthe 2004 commissioning run the neutron polarization was calculated for eachrun using eqn. 3.2.The neutron transmission through the unpolarized cellwas calculated once using the measured time of flight (tof) spectra from beammonitors 1 and 2 (see Section 3.3) at the beginning of the run cycle, hereT = M2/M1.The transmission for the polarized cell was measured for every pulse in a runand averaged to produce a single tof transmission spectrum for the run. Theresults for a typical run are shown in Fig. 3.16. The transmissions are thenused in eqn. 3.2 to calculate the neutron polarization, shown in Fig. 3.17.The transmission spectrum obtained for each run was fitted with tanh (σ a nlP He ),using a thickness of 4.84 bar · cm, which was measured during the commis-80

Figure 3.17: Plot of beam polarization as a function of neutron time of flight.sioning run [43].For each run the 3 He polarization was extracted as a fitparameter.Figure 3.18: Plot of the 3 He polarization as a function of run number. Thetime period over which these runs were taken corresponds to about 5 days.Figure 3.18 shows the the 3 He polarization for some 1000 analyzed runs. The3 He polarization varied between 42% and 48% which produced a 40% to 80%neutron polarization in the tof range corresponding to energies between 2-15 meV. The level of neutron polarization is one of the four parameters which81

introduce corrections to the measured (raw) asymmetry and must be includedin the calculation of the final physics asymmetry (see Chapter 5).82

3.5 The Radio Frequency Neutron Spin FlipperIn this experiment the asymmetry is measured continuously since the signalsfrom opposite detectors in a pair are measured simultaneously for each spinstate (See Section 5.1).However, the efficiency of the γ-ray detectors willchange slowly due to a number of effects including temperature and crystalactivation and the detector gains cannot be matched to an accuracy that wouldallow an asymmetry measurement to be made for each individual neutron pulse(see Section 4.3.4). In addition, an asymmetry measurement cannot be madeto the required level of accuracy by simply measuring the signal in a givendetector for one spin state and the corresponding signal in the same detector forthe opposing spin state, made some time later after the neutron spin has beenreversed using the neutron spin filter. This possibility is precluded because ofpulse-to-pulse fluctuations in the beam current. Both situations lead to falseasymmetries.The primary technique for reducing false asymmetries generated by gain nonuniformities,slow efficiency changes and beam fluctuations is fast neutronspin reversal. This allows asymmetry measurements to be made for opposingdetectors, removing sensitivity to beam fluctuations, and for consecutive pulseswith different spin states, removing the sensitivity to detector gain differences,drifts, and fluctuations. The asymmetries can then be measured very close83

together in time, before significant drift occurs.By carefully choosing thesequence of spin reversal, the effects of drifts up to second order can be furtherreduced (See Section 5.1).To achieve this fast neutron spin reversal, theexperiment employs a radio frequency neutron spin flipper (RFSF).The spin flipper has to satisfy the following important criteria:1. The spin flipper must rotate the neutron spin efficiently for all energiesbetween 1 meV and 100 meV.2. It must be fast enough to be turned on and off in less than 5 ms.3. It must work properly in a homogeneous field and introduce no magneticfield gradients.4. It must not introduce any AC fields to the rest of the experiment or allowelectronic signals to couple to any other component in the experiment.The RFSF is a 30 cm diameter and 30 cm long solenoid enclosed in a 40 cmdiameter, 40 cm long aluminum housing (Fig. 3.19). It operates according tothe principles of NMR, using a 30 kHz magnetic field with an amplitude of afew G and is mounted partially inside the detector array. The neutron spindirection is reversed when the RFSF is on and is unaffected when it is off. Toturn the RFSF off, the current drawn by the coil is switched to a dummy loadconsisting of a resistor circuit designed to have the same impedance as thecoil. This keeps the load on the main power circuit constant and minimizespickup of the spin flipper on-off switching in other circuits.84

Figure 3.19: Illustration of the RF spin flipper. The wire is wound around anylon cylinder. The coil is enclosed in an aluminum housing.3.5.1 RF Spin Flip TheoryDue to the 10 G holding field ( B ⃗ o ) in the experiment, the magnetic momentof the neutrons precesses about the vertical axis, perpendicular to the beamdirection. The spin flipper operates by producing an RF field oscillating alongthe beam, at a frequency that is matched with the Larmor frequency of theneutron in the holding field ω L = γB ⃗ o . The RF field will introduce a constantfield in the neutron’s frame of reference rotating at ω L and make the neutronmagnetic moment precess about this constant field, effectively rotating it aboutthe beam axis. To see this, consider the rate of change of a magnetic momentin the presence of a time dependent field H ⃗d ⃗ Mdt= γ ⃗ M × ⃗ H85

and the transformation to a rotating framed ⃗ M ′dt= d M ⃗dt − ⃗ω (2π × M ⃗ = γM ⃗ × ⃗H −⃗ω )2πγ.So if we choose ⃗ω = 2πγH ⃗ then in the rotating frame M ⃗ is constant and inthe lab frame it precesses at the Larmor frequency. One can then drive thespin flipper coil such that the magnetic field along the beam axis oscillates atthe Larmor frequency. Figure 3.20 illustrates this process. According to thiswe have⃗B rf (t) = ∣ ∣ ∣Brf ⃗ ∣∣ ẑ cos(ωL t) ,which can be written as⃗B rf (t) =∣ ⃗ B rf∣ ∣∣2(ẑ cos(ω L t) + ŷ cos(ω L t) + ẑ cos(2ω L t) − ŷ cos(2ω L t))= ⃗ B + rf + ⃗ B − rf . (3.3)So the neutron sees a static field ⃗ B + rf = | ⃗ B rf |2(ẑ cos(ω L t) + ŷ cos(ω L t)) in itsframe of reference because the field appears to be moving with the neutronprecession about the static field ⃗ B o . The magnetic moment of the neutron nowprecesses about ⃗ B + rfuntil it exits the spin flipper. The degree of precessionis determined by the amplitude | ⃗ B rf | and the amount of time for which thefield is applied. The anti rotating field ⃗ B − rf = | ⃗ B rf |2(ẑ cos(2ω L t) − ŷ cos(2ω L t))is seen in the neutron’s frame of reference as rotating at twice the Larmor frequencyand is therefore off resonance. This introduces a “jitter” effect makingthe magnetic moment vector move back and forth, but without significantly86

xB o= 10 x Gµ = γ < s >Beam− ω tω tz(−)B rfy(+)B rfB rf (t)= B rf zcos(ωt )Figure 3.20: Illustration of the spin rotation mechanism.changing the overall rotation of the neutron spin.3.5.2 Spin Flip Energy Dependence and Field RampingAs mentioned above, how far the neutron magnetic moment precesses aboutthe static field introduced by the spin flipper depends on the field amplitudeand the time period for which it is applied. Because of the finite length of thespin flipper, the latter parameter is determined by the velocity or energy ofthe neutron. The precession angle is given by θ = 2πγ| ⃗ B + rf |∆t = πγ| ⃗ B rf |∆tand to rotate the neutron spin by π about the beam axis a field with thatamplitude must be applied for a time ∆t = 1γ| B ⃗ . The length of the spinrf |flipper (L) and the velocity of the neutron v(t) then determine the amountof time the neutron actually sees the field ∆t =87Lv(t)= Lt , where (d) is thed

distance between the moderator and the spin flipper entrance and (t) is thetime of flight (tof), the time required for the neutron with a given energy totraverse this distance.So in order to rotate the spin for each neutron energy in the beam, the fieldamplitude must vary as| ⃗ B rf | = dγLt .However, in practice the amplitude | B ⃗ rf | is not constant spatially inside thespin flipper, since the normal component of the field must vanish at its endcaps.As a result, it is actually the integral of the amplitude along the length of thespin flipper that must vary with the neutron energy∫| ⃗ B rf |dz = dγLt .The ramping is achieved with two arbitrary waveform generators and a feedbackdifference circuit as shown in Fig. 3.21. The first waveform generator(AVG) in Fig. 3.21 provides the ramping envelope signal which is fed to thedifference amplifier. The difference amplifier computes the difference betweenthe input from the first waveform generator and the rectified ∼ 29 kHz feedbacksignal picked off the coil driving current. If the difference is positive thenthe difference amplifier output will increase the amplitude of the ∼ 29 kHzsinusoidal wave generated by the second waveform generator. If the differenceis negative then the ∼ 29 kHz signal amplitude will be lowered. The amplified,ramped signal is fed through a capacitor designed to match and cancelthe imaginary component of the coil impedance to reduce it to the winding88

AVGRampEnvelopeDifferenceAmpAVGFeedbackRectifierRampedSignalCoilGain AmpFigure 3.21: Ramping Circuit Schematic for the RFSF.resistance indicated by the resistor inseries with the coil.Both the voltage and the current of the ramped signal were monitored withthe data acquisition (see Section 3.7). The signals were sampled at 62.5 kHzover a period of 40 ms, followed by a 10 ms brake for each pulse. The currentsignal for two pulses is shown in Fig. 3.22.3.5.3 Spin Flip EfficiencySince both the neutron spin filter and the analyzer cell are used in this measurement,the spin flipper efficiency is extracted from the ratio of the neutrontransmission through both the spin filter and analyzer in pulses for which thespin flipper was on and pulses for which it was off. Figure 3.23 illustrates theconcept behind the RFSF efficiency measurement.89

Figure 3.22: Spin flipper current signal picked off from the transformer asshown in figure 3.21. The first pulse corresponds to the driving current beingsupplied to the spin flipper coil, while the current in the second pulse is fed tothe dummy load.As depicted in the figure and shown in section 3.4, the spin filter is not aperfect device and the neutron beam is only partially polarized. The numberof neutrons in either spin state after the spin filter is given byN 1,± = N o2 e−npσlp(1∓Pp) . (3.4)The number of neutrons in either spin state after the analyzer is given byN 2,± = N 1,± e −naσla(1∓Pa)N ′ 2,± = N ′ 1,±e −naσla(1∓Pa) (3.5)for spin flipper on and off respectively.90

N o = N o+ + N o−PolN 1 = N 1+ + N 1−RFSFOnN 1 = N 1+ + N 1−AnlN 2 = N 2+ + N 2−BeamRFSFN o = N o+ + N o−PolN 1 = N 1+ + N 1−OnN 1 = N 1+ + N 1−AnlN 2 = N 2+ + N 2−RFSFN o = N o+ + N o−PolN 1 = N 1+ + N 1−OffN 1 = N 1+ + N 1−AnlN 2 = N 2+ + N 2−RFSFN o = N o+ + N o−PolN 1 = N 1+ + N 1−OffN 1 = N 1+ + N 1−AnlN 2 = N 2+ + N 2−Figure 3.23: Conceptual depiction of spin filtering, spin flipping and polarizationanalyzing. The red arrows indicate the 3 He polarization state and theblack arrows indicate the neutron polarization. N o is the initial number ofneutrons out of the guide, N 1 is the number of neutrons out of the spin filter,N ′ 1 is the number of neutrons out of the spin flipper when it is on and N 2 , N ′ 2are the number of neutrons out of the analyzer when the spin flipper is off andon respectively. The (+,-) indicate spin up and spin down neutron polarizationrespectively.The spin flip efficiency (ε) is defined such thatN ′ 1,+ = (1 − ε)N 1,+ + εN 1,−N ′ 1,− = (1 − ε)N 1,− + εN 1,+ , (3.6)where 0 ≤ ε ≤ 1.The spin flip efficiency was measured using the signals seen in monitor 3 locatedbehind the polarized 3 He analyzer cell. The signal was normalized tomonitor 1. Figure 3.2 shows the location of monitor 3 and the analyzer cell.As indicated in Fig. 3.23, for a given 3 He polarization direction the number91

of neutrons out off the analyzer cell is different for spin flipper on and offstates. For the efficiency measurement the spin flipper was operated with the8-step sequence used for asymmetry measurements (see Section 5.1) and thecorresponding signal for eight consecutive pulses seen in monitor 3 is shown inFig. 3.24.32.5Signal (V)21.510.500 50 100 150 200 250 300 350 400Time (ms)Figure 3.24: Monitor 3 output of 8 consecutive neutron pulses. Data weretaken with a polarized beam and analyzer cell located in front of the monitor.The neutron spin was flipped using the spin flipper, following the pattern↑↓↓↑↓↑↑↓ for every sequence of 8 pulses.By looking at the current signals for RFSF on and off, as shown in Fig. 3.22,and observing the relative signal amplitudes in monitors 1 and 3, one findsthat the analyzer and polarizer cells were polarized anti-parallel with respectto each other. Keeping this in mind and using eqns. 3.4 and3.5, one findsthe expression for the number of neutrons going into monitor 3 with the spin92

flipper off to beN 2 (off) = N o e −σ(nplp+nala) cosh(n p l p σP p − n a l a σP a ) . (3.7)For a pulse with the spin flipper on, the number of neutrons into monitor 3 isN 2 (on) = N o e −σ(nplp+nala) [(1 − ε) cosh(n p l p σP p − n a l a σP a )+ε cosh(n p l p σP p + n a l a σP a )] . (3.8)Taking the ratio of spin flipper on to spin flipper off signals in monitor 3, wearrive atR ≡ N (2(on)N 2 (off) = 1 − ε + ε cosh(n )pl p σP p + n a l a σP a )cosh(n p l p σP p − n a l a σP a )(3.9)orWhereε = 1 − R1 − κ . (3.10)κ = cosh(n pl p σP p + n a l a σP a )cosh(n p l p σP p − n a l a σP a ) . (3.11)If the thickness and polarization of both the spin filter and the analyzer areknown, then the efficiency can be extracted for each energy separately or froma one parameter fit to 1 − R for all energies, assuming that the efficiency isa constant. The thickness of the spin filter was measured during the 2004commissioning run to be 4.84 bar · cm (see Section 3.4). The thickness of theanalyzer cell was also measured, however, it was later discovered that the cell93

glass contained 10 B which has a high neutron absorption cross-section and atransmission measurement with an empty glass cell was not performed, as ithad been done for the spin filter.The measured transmission for the analyzer therefore followsT anl = e −n He l He σ He −nglgσg−n T l T σ scatand the transmission through the glass was estimated with a Monte Carlomodel, using the geometry shown in Fig. 3.2.)5Neutron Captures (out of 3× 107000 Glass In650060005500500045004000350030002500Glass Out10 12 14 16 18 20Neutron Time of Flight (ms)Figure 3.25: MCNP simulation of the number of neutrons captured in monitor3 with and without an empty glass cell in front. The composition of the cellis identical to the the one used in the spin flipper efficiency runs.The composition of the glass is 62% SiO 2 , 17% AlO 2 , 8% CaO, 7% MgO,5% B 2 O 3 and 1% NaO 2 . For the analyzer thickness measurement, data runswere taken with the unpolarized 3 He filled cell in the beam and with the cellremoved. The Monte Carlo simulation was set up in the same way. It was94

un for all time bins in the tof range from 12 to 30 ms. One set of data weretaken with an empty cell in the beam and one set without the cell. The celland monitor were shielded and the beam was collimated in the same manneras was done in the commissioning run (see Fig. 3.2). The results are shownin Fig. 3.25. In the MC simulation, the removal of the glass cell left a vacuumrather than air. The analyzer cell window thickness was measured on thebench, using γ-ray transmission, and was found to be ∼ 0.75 mm. So the totalglass thickness seen by the neutrons is ∼ 1.5 mm. The density of the glasswas taken to be 2.5 g/cc. Both, the simulation data as well as the data takenduring the commissioning run were fitted to( ) Yinln = a + bt .Y outWhere Y in (Y out ) is the yield in monitor 3 for cell in(out) data. Here, the yield istaken to be the number of captured neutrons in monitor 3 for the simulationor the voltage signal from monitor 3 for the data runs.The overall factora = −n Tl Tσ scatthen determines the degree of neutron scattering from the cell,whereas b = −n g l g σ g ′ and b = −n Hel Heσ He− n g l g σ g ′ determine the amount ofneutron capture from the glass alone, as estimated by the simulation and theamount of neutron capture from the glass and 3 He together, as measured in thecommissioning. Here σ ′ t = σ o t/52 = σ o / √ E n = σ with σ o = 2.7168×10 −24 m 2and v n = 438 √ E (see Section A.1.2) and t is the neutron time of flight. Themoderator to monitor 3 distance is ∼ 22.7 m.Corresponding to the monitor signal tof range with the best counting statistics,as seen in the commissioning data, the cell-in-cell-out ratio for both the95

1-19×10Glass In/out Ratio-18×10-17×10-16×10-15×10-14×1010 12 14 16 18 20Neutron Time of Flight (ms)Figure 3.26: Exponential fit to Y in /Y out obtained from a Monte Carlo simulation(see text) in the tof range of 12 to 20 ms. The fit is used to determinethe thickness of the analyzer cell glass.simulation as well as the the run data was fitted between 10 and 20 ms. Thefit to the simulation result is shown in Fig. 3.26 and the fit to the run data isshown in Fig. 3.27. In the fit a and b were allowed to vary and the results are:1. For the simulation dataa = −0.110 ± 0.02b = −0.026 ± 0.001with χ 2 /dof = 1.56 and 23 degrees of freedom.2. For the run dataa = −0.026 ± 0.00496

= −0.099 ± 0.003with χ 2 /dof = 0.24 and 23 degrees of freedom.-14×10Cell In / Cell Out M3 Ratio-13×10-12×1010 12 14 16 18 20Neutron Tim of Flight (ms)Figure 3.27: Exponential fit to Y in /Y out (see text) as obtained from the monitordata in the tof range of 10 to 20 ms. The fit is used to determine the thicknessof the 3 He filled analyzer cellFrom this we find that n Hel Heσ ′ = 0.073 ± 0.003 ms −1 and with σ ′ =He He5.24 × 10 −26 m 2 atom −1 ms −1 , the 3 He thickness in the analyzer cell is (1.39 ±0.05)×10 24 atoms m −2 or 5.2±0.2 bar cm. Using bench tests, performed independentlyto measure the analyzer glass thickness, the analyzer 3 He thicknesswas found to be ∼ 1.42 × 10 24 atoms m −2 with an error of ∼ 5% [44].Using this result, the analyzer polarization can be found from the ratio of analyzertransmissions (Eqn. 3.7) in runs where both the polarizer and analyzercell are polarized to the transmission for runs where both cells are unpolarizedT o = N o exp(−n Tl Tσ scat) exp(−n g l g σ gt ′ − n p l p σ ′ t − n Heal a σ ′ t). For the trans-He97

mission with a polarized beam, only the spin flipper off pulses can be used toextract the analyzer polarization. The monitor 3 signals are always normalizedto the monitor 1 signals. The polarization is then found by plotting the ratioof transmissions vs. time of flight and fitting the curve withT polT unp= cosh(n a l a σ ′ He tP a − n p l p σ ′ He tP p) .Where n a l a , n p l p are the analyzer and polarizer thicknesses respectively andP a , P p are the corresponding 3 He polarizations. The analyzer polarization isthe only unknown parameter and is found from the fit on a run by run basis.The fit for a typical run is shown in Fig. 3.28 where P a = 0.62 ± 0.04 with aχ 2 /dof = 0.03 and 21 degrees of freedom.Transmission Ratio ( Pol / UnPol )1.11.091.081.071.061.051.041.031.0210 11 12 13 14 15 16 17 18 19Neutron Time of Flight (ms)Figure 3.28: Ratio of polarized to unpolarized analyzer transmission with afit to extract the analyzer polarization (see text). The Monitor 3 output wasnormalized to monitor 1.The information obtained for the polarizer and analyzer can then be used in98

eqn. 3.10 to calculate the spin flip efficiency.The 1 − R data points for atypical run are shown in Fig. 3.29, together with the ɛ(1 − κ) fit in which theefficiency ɛ is the only parameter. In this way, a single efficiency is extractedfor each run and for the entire time of flight range.To obtain the complete efficiency information for the spin flipper, several measurementshad to be performed. To verify that the spin flipper coil current wasset to its optimum value, the efficiency was measured for various amplitudesettings, as adjusted at the first waveform generator (see Fig. 3.21). Here aseparate data run was taken for each amplitude setting and this portion ofthe measurement is hereafter referred to as the spin flipper amplitude scan.A second set of efficiency measurements had to be performed while varyingthe magnetic holding field, since it determines the resonance properties of theneutron spin.Here a separate run was taken for each readjustment of theholding field, changing it in 1% steps up and down, relative to its nominalsetting of 19.165 A coil current. This set of measurements is referred to as thefield scan.Each of these measurements were taken with the analyzer cell and monitor 3located on the beam axis and due to the tight collimation of the analyzer cell,the efficiency measured for these runs is defined only for a small portion of thebeam, at its center. To obtain more complete information about the spin flipefficiency over the entire beam cross-section, the amplitude scan was repeatedfor two off-axis analyzer locations; one in the vertical direction and one in thehorizontal direction.99

1 - T(RFSF on)/T(RFSF Off)-0.2-0.4-0.6-0.8-1-1.2-1.4-1.6-1.810 12 14 16 18 20Neutron Time of Flight (ms)Figure 3.29: This plot shows 1−R data points and a ɛ(1−κ) fit (see Text). Thespin flip efficiency ɛ is the only parameter in the fit. From the fit, ɛ = 0.97±0.06at an amplitude of 850 mV. Error bars include systematic errors.The results for the on-axis the amplitude scan are shown in Fig.3.30. Theamplitude was varied between 400 mV and 1200 mV and the peak efficiencyis achieved at an amplitude setting of 780 mV. The error on the efficiency isabout 5%, as shown in the summary of table 3.1. The results for the on-axisfield scan are shown in Fig. 3.31 and summarized in table 3.2. The horizontaloff-axis amplitude scan yielded a maximum efficiency of ∼ 91% at an amplitudeof ∼ 747 mV and the vertical off-axis scan yielded a maximum of ∼ 99% at∼ 765 mV. Both off-axis scans were taken with the analyzer and monitor 3located 3.8 cm from the nominal beam center, in either direction. The resultsobtained here are in good agreement with predictions obtained from analyticalcalculations, taking into account the properties of the spin flipper and holdingfield [24].The measured 3 He polarization in the spin filter and analyzer and therefore100

1Spin Flip Efficiency0.80.60.40.20400 500 600 700 800 900 1000 11001200RFSF AWG Amplitude (mV)Figure 3.30: On-axis spin Flipper current amplitude scan. The amplitude isadjusted in the first arbitrary waveform generator. The efficiency is extractedfrom a parabolic fit to the data points. The peak efficiency is achieved for anamplitude setting of 780 mV.also the spin flip efficiency depend strongly on the application of the correctpedestal and background signals. A detailed study of these effects has not yetbeen completed and a simple subtraction of different background and pedestalruns, even if they were obtained one after the other, during the test run, canlead to polarization and efficiency results that are different by several percent.The difficult part of the pedestal and background determination is that thecorresponding signals are buried within the real signal obtained from neutroncapture on 3 He in the monitors. The background is caused by processesincluding neutron capture on Aluminum, Boron, Lithium and other materialsthat may be encountered by the beam. For the monitors, it is difficultor impossible to measure these effects within the given experimental setup.101

Run SF Pol. (frac.) Anl. Pol. (frac.) Amp (mV) Efficiency5084 0.42 ± 0.01 0.64 ± 0.04 850 0.973 ± 0.055086 0.42 ± 0.01 0.64 ± 0.04 900 0.943 ± 0.055088 0.42 ± 0.01 0.63 ± 0.04 800 0.988 ± 0.055090 0.42 ± 0.01 0.63 ± 0.04 825 0.982 ± 0.055092 0.42 ± 0.01 0.63 ± 0.04 750 0.986 ± 0.055094 0.42 ± 0.01 0.63 ± 0.04 700 0.968 ± 0.055097 0.42 ± 0.01 0.63 ± 0.04 400 0.520 ± 0.035098 0.42 ± 0.01 0.62 ± 0.04 500 0.734 ± 0.045099 0.42 ± 0.01 0.62 ± 0.04 600 0.883 ± 0.045101 0.42 ± 0.01 0.62 ± 0.04 1000 0.833 ± 0.045103 0.42 ± 0.01 0.62 ± 0.04 1200 0.420 ± 0.02Table 3.1: On-axis amplitude Scan Spin Flipper Efficiency Runs.Therefore, these effects have to dealt with by subtracting pedestals and backgroundswhich are obtained from suitable models. The development of thesemodels is involved and will be done elsewhere. Until better knowledge of thepedestals and backgrounds is available, a conservative guess on the spin flipefficiency error is about 10%, with a peak efficiency of about 97% at the settingsof 850 mV coil current amplitude and 19.165 A holding field current.These settings were used for the asymmetry measurement performed duringthe 2004 commissioning run.Run SF Pol. (frac.) Anl. Pol. (frac.) ∆A (%) Efficiency5063 0.42 ± 0.01 0.65 ± 0.04 0 0.997 ± 0.055065 0.42 ± 0.01 0.65 ± 0.04 -1 0.980 ± 0.055066 0.42 ± 0.01 0.65 ± 0.04 -2 0.928 ± 0.055068 0.42 ± 0.01 0.65 ± 0.04 -3 0.842 ± 0.045074 0.42 ± 0.01 0.64 ± 0.04 1 0.980 ± 0.055076 0.42 ± 0.01 0.64 ± 0.04 2 0.929 ± 0.055078 0.41 ± 0.01 0.64 ± 0.04 3 0.843 ± 0.04Table 3.2: Holding Field Scan Spin Flipper Efficiency Runs.102

1.11Spin Flip Efficiency0.90.80.70.60.5-3 -2 -1 0 1 2 3Holding Field Coil Current (% change)Figure 3.31: On-axis holding field coil current scan. The coil current was adjustedto be 19.165 A at the center point (0%) and was varied in 1% incrementsbetween 18.59 A and 19.74 A. The efficiency is extracted from a parabolic fitto the data points. The peak efficiency is achieved for a coil current setting of19.160 A.Over the time of flight range for which this analysis was done (10 to 20 ms), thespin flip efficiency was observed to be constant. From the efficiency measurementruns, the relaxation time of the analyzer polarization was determined tobe 150 ± 50 min.3.6 Neutron Capture TargetsThis section describes the targets used for asymmetry measurements duringthe 2004 commissioning run and the liquid hydrogen target to be used forthe ⃗n + p → d + γ production runs. The analysis procedures and asymmetry103

esults are discussed in chapter 5. The materials studied using neutron captureduring the 2004 commissioning run include Al, Cu, Cl, In, 6 Li loaded plastic,and B 4 C. The specific composition of these materials is shown in appendix A.To determine the parity violating asymmetry in neutron-proton capture tothe proposed accuracy, any possible false asymmetry from neutron captureon other materials must be measured. These asymmetries form a backgroundwhich introduces a shift in the measured ⃗n + p → d + γasymmetry if theyare non-zero and, at the very least, produce a dilution of the asymmetry, evenif they are zero. The degree of shift or dilution in the asymmetry is proportionalto the size of the background signal, relative to the signal of interest.We refer to these false asymmetries as neutron capture related or induced systematiceffects. There are also instrumental systematic effects which arise dueto changing equipment properties which may be correlated with the neutronspin. Some of the possible instrumental systematic effects have been brieflymentioned in the beginning of section 3.5 and those related to the detectorarray in particular are discussed in section 4.2.3. A more detailed discussionof systematic effects related to neutron capture and scattering is provided insection 5.5. It is difficult to model or calculate the level of parity-violation inthese targets and to establish an upper level of their contribution it must bemeasured.For a target in run, the total detector yield sampled by the ADCs in the dataacquisition (DAQ) includes both background and the signal from neutron captureon the target Y = Y s + Y b . For target out or background runs, the totaldetector yield is denoted by Y = Y b . The total, yield weighted, target in asym-104

metry is A s+b (Y s + Y b ), whereas the yield weighted, background asymmetryis A b (Y b ). The asymmetry from neutron capture on the target alone and thecorresponding error will then beA s = A s+b(Y s + Y b ) − A b Y bY s= A s+b(1 + Y bY s)− A bY bY s. (3.12)(σ As = √ 1 + Y ) 2bσA 2 Y s+b+ Y b2σA 2 b. (3.13)sY 2sThe same procedure can be used to account for beam-off false asymmetriesfrom electronic noise and instrumental systematic effects due to magnetic fieldvariations. However, as shown in section. 4.6, these asymmetries are zero andhave been measured with an error that will be negligible compared to thestatistical error obtained when the beam is on. The desired accuracy on the⃗n + p → d + γ asymmetry and the rate ratio weighted error of the backgroundthen determine to what accuracy one must measure the A s+b with the hydrogentarget installed.In addition to the need to measure false asymmetries in the background, Cl,6 Li and B 4 C were used for system diagnostic purposes. B 4 C was used toestablish that the detector array is operating at the counting statistics limit(Sec 4.5). Cl was used to verify that asymmetries can be accurately measuredwith this setup and that the data is correctly analyzed, while 6 Li was used forbackground and shielding studies. The γ asymmetry in Cl has been measuredpreviously [45] and the results are used as a benchmark in this experiment. Inaddition, Cl was used to determine the direction of beam polarization after the105

neutrons exit the 3 He spin filter, because the spin filter setup did not includethe possibility to measure the direction of 3 He polarization.3.6.1 Cl, Al, B 4 C In, and 6 Li TargetsThe size of the targets, including the hydrogen target, is chosen to satisfya number of criteria. The target must be large enough to stop most of theneutron beam by capture. This means that the target must have a diameterequal to or larger than the beam cross section ( 10 cm) and be thick (long)enough to produce a capture γ signal that is large compared to noise andbackground and allows for maximum possible counting statistics.For the2004 commissioning run the targets were always located such that they werecentered inside the detector array in all three directions.Figure 3.32: Container for the liquid CCl 4 target material. The target materialvolume is ∼ 33 cm 3 . The container is made of Teflon and is closed at the topwith a Teflon lid and Teflon screws.The CCl 4 target is shown in Fig. 3.32. It is designed to slide into the spin106

flipper, so that it can be used for diagnostic purposes when the hydrogentarget is installed. The housing is made of Teflon and the liquid CCl 4 islocated inside a ∼ 33 cm 3 rectangular volume. The CCl 4 liquid is 99.9% pure,with less than 0.01% water content. More detailed drawings of the targets areshown in figures A.1 and A.2, in appendix A.The aluminum target is shown in Fig. 3.33. It is made entirely from aluminum,except for the screws, which are made from brass. The target consists of aframe with slotted braces along the length of the target. Aluminum sheets arethen inserted into the slots and form the main volume of the target.Figure 3.33: Al target holder and target sheets. The thickness of the targetcan be adjusted by adding or removing sheets.Each aluminum sheet is a ∼ 1 mm thick square with a 3.75 cm side length. The107

total target thickness was ∼ 5 cm during the first half of the commissioningrun. For the second half, the number of slots was doubled and the numberof sheets tripled (each slot may hold 2 sheets) to increase the capture signal.The arrangement of the target into sheets, rather than having a solid target,allowed for the reduction of γ attenuation in the target.The total length(including gaps) of the target was 30 cm. The gap between slots was ∼ 5 mmfor the thin target and was reduced to about 2 mm for the thick target. Thecopper target was also composed of sheets which were mounted in the samealuminum frame. Background runs with the empty frame were conducted aswell and the background is taken into account in the final determination ofthe copper asymmetry.The boron target consists of a 1 cm thick 15 cm by 15 cm sheet of sinteredB 4 Cglued to an aluminum holder consisting of a simple (thin) aluminumsheet. The target was oriented with the B 4 Csheet facing upstream. Thesame geometry was used for the 6 Li target, but with a ∼ 2 mm thick sheet ofenriched 6 Li loaded plastic glued to an aluminum mount. The Indium targetis approximately 12 mm thick and of a somewhat irregular shape in the beamleft-right and beam up-down directions, covering a circular cross-sectional areawith a radius of ∼ 3 cm at the center of the beam.The target positions along the beam, relative to the detector array, are shownin Fig 3.36. In the beam left and right directions the detector array and targetsare centered on the beam guide axis.108

3.6.2 The Hydrogen TargetAn illustration of the liquid hydrogen target is shown in Fig. 3.34Figure 3.34:It consists of a target vessel containing the liquid hydrogen, surrounded by avacuum chamber. The hydrogen itself and the heat radiation shield, locatedaround the vessel, are cooled by two cryogenic refrigerators. In the coolingprocess, the hydrogen is converted to liquid para-hydrogen, from its usualstate of mostly ortho-hydrogen, to prevent the depolarization of the neutronspin in the target via spin-flip scattering. Beam depolarization in the varioustarget materials is briefly discussed in section 5.5.The 30 cm diameter and 30 cm long hydrogen vessel is large enough to stop109

most of the neutron beam.Monte Carlo calculations performed using thedouble differential scattering cross sections for cold neutron scattering in liquidparahydrogen [46] indicate that a target of this size will capture about 60% ofthe incident neutrons [21]. These calculations are based on the neutron energyspectrum emitted by the coupled LH 2 moderator viewed by the NPDGammabeam line [38]. The beam entrance windows in the vacuum chamber, radiationshield, and target vessel are as thin as possible to efficiently transmit theneutron beam and create minimal prompt capture radiation. The hydrogenvessel and vacuum chamber are made of aluminum, with a total entrancewindow thickness of ∼ 3.2 mm and ∼ 6 mm respectively.The hydrogenvessel exit window is ∼ 3.8 mm thick.The target vessel is constructed ofnon-magnetic materials, so that the neutron spin direction can be efficientlytransported into the target with negligible beam depolarization. The fractionof the beam that is not captured in the target and may be scattered with acomponent perpendicular to the beam direction must be absorbed efficientlyto avoid creating background ionization in the detector (see Sec.4.2). To aidin this process, sheets of ∼ 2 mm enriched 6 Li loaded plastic are attached tothe inside walls of the vacuum chamber. Indium seals are used in the windowsof the main vacuum chamber and copper is used for the heat radiation shieldand as the seal in the conflat flanges. An illustration of the target assemblytogether with the detector array is shown in Fig. 4.2.3.6.3 Detector-Target Geometry110

The geometric dependence for a point source, point detector measurement ofA γis given by the basic differential cross-section derived in chapter 2. Inthis experiment, the cosine dependence of the cross-section is referred to asthe geometry factor .This name is meant to be indicative of the complicationsintroduced by the finite size of the targets and the detectors as well as theangular and spatial dependence of the energy deposited in a given detector.Beam Upxφ91087Beam Right116θ0Beam5z1423Figure 3.35: Coordinate system setup for target and detectors. For a givenγ-ray source point (x=0, y=0, z=0), and direction (red arrow), the cosine ofthe angle that the γ-ray makes with the vertical (x-axis) is given simply bythe standard spherical coordinate direction cosine (green arrow). Detectors ina ring are numbered from 0 to 11, while the four rings are numbered 0 to 3,starting with the most upstream ring.yIn the calculation of the geometry factor as well as in the asymmetry analysis(see Sec. 4.6 and Chp. 5) the coordinate system is set up such that the x-axislies along the vertical (defined by the holding field direction) and the z-axislies along the beam direction. Figures 3.35 and 3.36 illustrate the axis choiceand show the detector numbering scheme used throughout the analysis.111

%%&&&%%%%&&&&%¥¥¦¦¦¥¥¥¥¦¦¦¦¥"""""""$$$$$$$¨¨¨¨¨¨¨a)Shielding§¡§¡¡¡¡¡¡¡¡¡¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢¡¢§¡§§¡§§¡§§¡§Target¡¡¡¡¡¡¡¡Beam ¡¡¡¡¡¡¡¡CCl 4¡¡ ¡¡ContainerDetectorsSpinFlipper§¡§§¡§©¡©¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤£¡£¡£¡£¡£¡£¡£¡£¡£¡£¡£¡£©¡©©¡©©¡©©¡©©¡©©¡©Ring Numbers3 2 1 0!¡!!¡!!¡!!¡!!¡!!¡!b))())()'(''(''(''(')())())())()'('In, B 4C,Target¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡SpinFlipper!¡!)() '(')() '(')() '(')() '('#¡##¡#¡¡¡¡¡¡¡¡¡¡¡¡'('¡¡¡¡¡¡¡¡¡¡¡¡#¡##¡##¡##¡##¡#¡ ¡¡¡ ¡c)¡¡¡¡¡¡¡¡¡¡¡¡¡¡ ¡¡¡¡¡¡¡¡¡¡¡¡¡Al TargetSpinFlipper¡¡¡¡¡¡¡¡¡¡¡¡¡¡ ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡ ¡¡¡ ¡¡¡¡¡¡¡¡¡¡¡¡¡ ¡ ¡¡ ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡ ¡ ¡¡ ¡¡¡¡¡¡¡¡¡¡¡¡¡¡ ¡¡¡¡¡¡¡¡¡ ¡¡¡ ¡¡¡ ¡Figure 3.36: Target positions with respect to the detector array, along thebeam.The finite size of the target and detector raises several questions that must beconsidered when the asymmetry is extracted from the data:• Where, in the target, will neutrons capture?• What direction does an emitted gamma ray have and which detector(s)does it intersect?• What is the gamma ray’s angle with respect to the holding field given112

the detector(s) it intersected?• How much energy will the gamma ray deposit in each detector it intersects?The capture distribution in the targets, i.e. the probability of a neutron capturingin one region of the target versus another, depends on the geometry ofthe target, the properties of the target material, i.e. how black it is for neutrons,and on the energy of the incident neutron. This probability determinesthe frequency with which γ-rays originate from a given point within the target.Or, in other words, it determines the intensity of that point in the target.The angle that an emitted γ-ray has with respect to the vertical determinesthe detector(s) it will intersect and its subsequent path length through them.This in turn determines how much energy the photon deposits in each detectorbefore it is completely absorbed or exits the detector array.The fact that a single γ-ray may deposit its energy in more than one detectorhas two effects. The first one is a small decrease in the angular resolution.The design of the detector array and, specifically, the size and number ofindividual detectors in the array was based in part on the size of this errorfor certain design considerations. This is described further in section 4.2. Thesecond effect is a dilution of the true geometry of the detector array. Here, thesituation in which a given γ-rayonly deposits part of its energy in a givendetector is essentially equal to a situation in which the detector was actuallythinner in the direction of γ-ray propagation, effectively changing the arraygeometry the γ-ray sees. This experiment uses current mode detection and,113

ecause of that, there is no way to identify events which deposit differentamounts of energy in a detector. Therefore, the average direction cosine for agiven detector must be weighted by the directional average energy depositedin it, for all possible γ-ray source points and directions.Both the 6 Li and B 4 C targets are relatively thin sheets which are largelyblack to all low energy neutrons. In these cases the geometry factor has beencalculated only on the basis of target-detector geometry and the directionalaverage of the γ-ray energy deposited in the detectors. The Cl, Al, and Cutargets, on the other hand, are not entirely black to low energy neutrons andare longer (thick) targets, as shown in Fig. 3.36. For each of these targetsa separate Monte Carlo calculation was performed to establish the captureprobability density as a function of position along their length.)-1Prob. Dens. ( cm0.080.060.040.0204035302520Neutron Tof (ms)15103025201510Capture location along beam (cm)50Figure 3.37: Monte Carlo calculation of the neutron capture probability inAluminum as a function of the capture location along the length of the targetand as a function of neutron time of flight.Figure 3.37 shows the capture probability density for the aluminum target asa function of incident neutron energy by time of flight and as a function of114

position along the target. For each neutron energy the distribution falls offexponentially with the distance of the capture location into the target. Thedecay parameter λ(t i ) is extracted for each time of flight bin t i from a fit tothe Monte Carlo results. The probability that a neutron with a given time offlight will capture in a region between δz around z is then given byh(z, δz, t i ) =∫ z+δz/2z−δz/2dz λ(t i)e −λ(t i)z1 − e −λ(t i)(30) . (3.14)To estimate the average energy deposited in each detector for each γ-ray sourcepoint and all directions, a simple separate Monte Carlo program has been written.Here, the target volume is sampled uniformly using a 5 mm grid in alldirection. At each grid point in the target volume a total of 64800 γ-ray tracksare emitted uniformly into 4π. If a γ-ray with a given direction enters a detector,its path and energy deposition is determined via the Compton scatteringformula. The scattering formula is repeatedly evaluated each time the photonhas traveled a predetermined distance (step size) until it has either depositedmore than 95% of its energy or exits the detector. For each scattering event anew γ-ray direction is randomly chosen and when the γ-ray exits a detectorits (new) direction determines if it enters another detector.So, for a given source point (⃗x) in the target and γ-ray direction (θ, φ), theenergy weight for the direction cosine, corresponding to this event is given bythe following recursive expression.f(⃗x, θ, φ) =∞∑i=1( Ei−1γ − EγiEγ0)θ(0.95 − f(⃗x, θ, φ)) (3.15)115

Where, E 0 γ is the initial γ-ray energy when it enters a detector andE i γ =E i−1γ1 + Ei−1 γ(1 − cos α)mc 2(i > 0)is the remaining photon energy after the ith scattering event, with α being thescattering angle. The series is terminated when the total amount of energylost by the γ-ray reaches 95% or when it exits the detector. The γ-ray sourceposition and directional dependence is not explicitly shown in Eqn. 3.15, butthe coordinates appear there, because whether or not a γ-rayscatters in agiven detector, the scattering angle (α) and the subsequent direction of thescattered photon all depend on these coordinates.In this fashion, the γ-rayenergy deposit and neutron capture distributionweighted average direction cosine for the jth detector is given byG j (θ, φ, E n , ⃗x) ≡∫ d 3 x ∫ π0 dθ ∫ 2π0 dφ g(θ, φ)f(⃗x, θ, φ)h(z, dz, E n )∫d3 x ∫ π0 dθ ∫ 2π0 dφf(⃗x, θ, φ)h(z, dz, E n )(3.16)where g(θ, φ) = sin θ cos φ.1G(θ,φ,E n ,x)0.50-0.5-14540353025Detector201510500540353025201510Neutron Tof (ms)Figure 3.38: Monte Carlo calculation of the detector geometry factor for thealuminum target as a function of neutron time of flight.116

Figure 3.38 shows the result of the geometry factor calculation for aluminumin the time of flight region 5-40 ms for all detectors. The cosine dependence ofthe detector geometry factor is clearly visible Figure 3.39 shows the geometryfactor for all detectors at 25 ms time of flight. The difference in the geometryfactor for detectors in different rings is a result of the size (length) of thetarget, its position with respect to the detector array (see Fig. 3.36) and thecapture distribution.10.5G(θ,φ,E n ,x)0-0.5-10 5 10 15 20 25 30 35 40 45DetectorFigure 3.39: Geometry factor for all detectors with the aluminum target at 25ms time of flight.The time of flight dependence for four up detectors which have a nominallysmall angle with respect to the vertical is shown in Fig. 3.40. Figure 3.41shows the same for four side detectors which have a nominally large angle withrespect to the vertical. For detectors in the two upstream rings the geometryfactor increases with an increase in time of flight because lower energy neutronsmostly capture in the front of the target, while higher energy neutrons travelfurther toward the end of the target before capturing. For the same reasons,detectors in the two downstream rings show a decrease in the geometry factorwith decreasing time of flight.117

G(θ,φ,E n ,x)1.0510.950.90.850.80.750.70.65Det 0, Ring 0Det 12, Ring 1Det 24, Ring 2Det 36, Ring 310 15 20 25 30 35 40Neutron Time of Flight (ms)Figure 3.40: Al target geometry factor versus neutron time of flight for 1 upperdetector in each ring.G(θ,φ,E n ,x)0.260.250.240.230.220.210.20.190.180.17Det 2, Ring 0Det 14, Ring 1Det 26, Ring 2Det 38, Ring 310 15 20 25 30 35 40Neutron Time of Flight (ms)Figure 3.41: Al target geometry factor versus neutron time of flight for 1 sidedetector in each ring.The same calculation has been done for the Cl target.The correspondingprobability density is shown in Fig. 3.42. At low neutron energy the fall offin capture probability with distance into the target is much more pronouncedthan it is in aluminum. This a result of both the high density in CCl 4andits large capture cross-section, which falls off fast for higher energy neutrons.For CCl 4the differences in geometry factor between detectors and rings isnot very different from the differences seen with the aluminum target becausemost of the capture in the aluminum target is concentrated in a small regionin the front.118

)hst1Entries 48Mean 23.22RMS 13.34-1Prob. Dens. ( cm1.81.61.41.20.8 10.60.40.204035302520Neutron Tof (ms)151050400.511.522.533.5Capture location along beam (cm)Figure 3.42: Monte Carlo calculation of the neutron capture probability inCCl 4 as a function of the capture location along the length of the target andas a function of neutron time of flight.10.5G(θ,φ,E n ,x)0-0.5-10 5 10 15 20 25 30 35 40 45DetectorFigure 3.43: Geometry factor for all detectors with the CCl 4time of flight.target at 25 msThe time of flight dependence of the geometry factor in CCl 4 , for four updetectors is shown in Fig. 3.44.For the left-right asymmetry measurement, one is interested in the direction ofphoton emission with respect to the horizontal axis, rather than the vertical.For this case, the geometry factor was determined using the same calculation,as described above, but with g(θ, φ) = sin θ sin φ replaced in Eqn. 3.16.119

G(θ,φ,E n ,x)1.0510.950.90.850.80.750.70.65Det 0, Ring 0Det 12, Ring 1Det 24, Ring 2Det 36, Ring 30 5 10 15 20 25 30 35 40Neutron Tof (ms)Figure 3.44: CCl 4 target geometry factor versus neutron time of flight for 1upper detector in each ring.3.7 Data Acquisition and StorageBecause of the small size of the parity violating signal and the presence ofthe large isotropic signal, the gamma intensities within each of the 8 neardetectors and each of the 4 corner detectors in a given ring are nearly equal.It is therefore possible to equalize the signals from near and corner detectorsby gain adjustments (discussed below) and sample only (1) the average signalin a ring, (2) the differences in each detector from the average signal in a ring.This strategy allows one to exploit the increased dynamic range to increase thegain of the system and minimize the effects of noise in the sampling electronics.Here we describe the details of how this idea for sampling the array signalswas implemented.120

3.7.1 Sampling SchemeThe preamplifier output for each detector is sampled by the NPDGamma dataacquisition. The DAQ incorporates four sum and difference amplifier boardswith 12 channels each. Each sum and difference board forms an average voltageover the 12 detectors in a given ring and each individual detector signal has itscorresponding ring average subtracted. The process is shown schematically inFig. 3.45. Here each difference amplifier contributes a gain factor of 10 and eachBessel filter (denoted by F in the schematic) contributes an additional factorof 3 to the gain. The 48 resulting difference signals and four average signalsare sampled by 16-bit ADCs.The sum and difference signals are sampledat 62.5 kHz and 50 kHz respectively. A macro pulse of data is collected bysampling for a duration of 40 ms, followed by a 10 ms break, before the nextframe of neutrons arrives. This results in 2000 difference and 2500 sum samplesfor each macro pulse. In the data stream (before the raw data are written tofile) every group of 20 difference samples and 25 sum samples is summed toproduce a final value for each of 100, 0.4 ms wide time bins. The sampled dataare transfered via fiber optic connection to a 3.5 Tbyte RAID array storagedevice. Figure 3.46 shows the 10 macro pulses of electronic pedestal (beam off)output for a typical detector, obtained using the described sampling scheme.According to Fig. 3.45 and the sampling scheme just described, the data actuallystored for each time bin are a sum of 20 difference samples for each detectorD i r = ∑ 20j=1 D i rj and a sum of 25 average samples for each ring S r = ∑ 25j=1 S rj .A ring average sample is given by S rj = 3/12 ∑ 12i=1 I i rj and a difference samplefor a given detector in the ring is given by D i rj = 30(I i rj − S rj /3). In the anal-121

I 1 rjDF1D rj2I rjDFD2rj3I rjDFD3rj12I rjDF12DrjSS o rjFS rjFigure 3.45: DAQ sum and difference amplifier schematic. The 12 individualdetector signals for the ring are denoted by I i rj. The 12 corresponding differencesignals are denoted by D i rj. The ring average signal is denoted by S rj .Here, r, j denotes the rth ring and the jth sample.ysis, the time bin average of the difference and sum signals are recombinedto produce the average detector signal for the time bin at the ADC inputI i r = 1/30 (D i r + 10 S r ), in ADC counts. Here, D i r = D i r/20 and S r = S r /25.The sum and difference scheme increases the effective dynamic range of theADCs, which are limited to ±10 V, therefore allowing for a larger gain to beapplied and staying above the bit-noise of the ADCs. The Bessel filters providehighly correlated ADC samples, filtering out high frequency components in thesignal, and the high sampling rate averages out the bit noise in the ADCs. Thechosen time bin width removes the correlation between the data points actuallyused in the calculation of asymmetries (see Section 4.4.1).122

Figure 3.46: A typical detector pedestal signal, showing 10 macro pulses with100 data points each. Each time bin is 0.4 ms wide containing one data point.The 40 ms long sampling period is followed by a 10 ms break in each pulse.Also seen are eight outliers, where the larger signal is due to an incident cosmicray depositing ∼ 100 MeV in the crystal.123

internetslow signals &beam informationvme1365 MHz Celeron128 MB10 GB diskdetector difference signals48 chan. 16−bit ADC at 50 kHzvme2365 MHz Celeron128 MB10 GB disk385 kB/s100 Mbitfiber ethernetdetector sums, beam monitors,spin state48 chan. 16−bit ADC at 50 kHzvme3365 MHz Celeron128 MB10 GB disk100 Mbitfiber ethernet385 kB/sFP12 cave wallPerl DAQ controlhazel2.4 GHz Xeon512 MB3.5 TB RAID arrayprimary data storagefiverdual 1.6 GHz Opteron2 GBremovable 185 GBhard disk backupPerl/Tk run control GUIROOT online analysisFigure 3.47: DAQ computer and storage layout. The two analysis and storagecomputers Hazel and Fiver, together with VME crate 1 are located outside theexperimental cave and are connected via fiber optic Ethernet with VME crate2 and 3 inside the cave. VME 1 records the proton beam current level anddiagnostic data, VME 2 acquires only the 48 difference signals, while VME 3acquires the four detector ring sum signals as well as the beam monitor andspin flipper signals. Fiver is the main on-line analysis computer including thedata backup storage space. VME 1 receives a signal from the accelerator tosynchronize the data acquisition with the neutron beam pulses.124

3.8 SummaryFor the ideal experiment, with 100% beam polarization, 100% spin flip efficiency,point sources and detectors, and no systematic effects resulting inbackground asymmetries or beam depolarization, the yield from a single detectoris given by Eqn. 2.31, with the intensity essentially determined by thenumber neutrons leaving the FP12 moderator. However, the actual signal inthe detectors is more closely related to the number of neutrons that captureon the target and the detector solid angle.The FP12 beam guide was installedto deliver the maximum possible number of low energy neutrons tothe experimental apparatus. All components between the guide exit and thetarget have been optimized to ensure that they can properly function whileattenuating the beam as little as possible. This includes optimization of themonitor and spin filter thicknesses, reduction of the aluminum windows onthe monitors and the spin flipper and the overall reduction of the experimentlength to reduce beam divergence.In addition to maximizing the number of neutrons, a successful asymmetrymeasurement requires a stable, polarized beam and the ability to reverse thebeam polarization without significant losses. The emitted γ-rays have to bedetected with high efficiency and with reasonably good angular resolution,which is limited by the finite size of the detectors and targets. As described inthe preceding sections, the level of beam polarization and spin flip efficiency arelimited, depending on energy of the neutrons. During the 2004 commissioningrun, the beam polarization increased from 40% to 80%, in the time of flight125

ange of 10 to 34 ms. The spin flip efficiency was seen to be constant, at ∼ 95%,over the same time of flight region, averaged over the beam cross-section. Theability to keep track of processes as a function of the neutron energy is a keyfactor in maintaining their efficiency and controlling systematic effects. Ourknowledge of the neutron energy is provided by the nature of the spallationsource and the use of the beam chopper.Each of these “real world” properties of the beam line and experimental apparatusfactor into the yield one actually sees in the detectors. Together withthe beam polarization factor, the spin flip efficiency and the geometry factor,the measured cross-section for a detector, as a function of neutron energy andthe γ-ray source location and angle isdσdΩ ∝ 14π [1 + A γg(θ, φ)P n (E n )ɛ(E n )S(E n )] h(z, E n )f(⃗x, θ, φ) (3.17)In the asymmetry analysis the beam polarization and geometry factor areapplied on a time bin basis (see Chp. 5). The geometry factor calculations describedin the previous section are done for each detector and each time bin inthe time of flight range. Then, the source location and angular dependence foreach detector can be integrated out and when solving for the physics asymmetry(see Chp. 5) the geometry factor is seen to emerge in the form of Eqn. 3.16.An additional factor which contributes to the size of the observed asymmetryand its error is the amount of depolarization of the beam due to spin flip scatteringin the various targets. The factor S(E n ) is unity for materials in whichspin flip scattering is negligible and decreases for materials where it is more126

significant. Beam depolarization due to scattering is discussed in section 5.5.127

Chapter 4The NPDGamma Detector4.1 IntroductionTo measure A γ to an accuracy of 5 × 10 −9 the experiment must detect at leasta few ×10 17 gamma-rays from ⃗n + p → d + γ capture with high efficiency.The average rate of gammas deposited in the detectors for any reasonablerun-time is therefore high, and the instantaneous rates at a pulsed neutronsource are, of course, even higher than for a CW source. Because of thesehigh rates and for a number of other reasons discussed below, the detectorarray uses accurate current mode gamma detection. Current mode detectionis performed by converting the scintillation light from CsI(Tl) detectors tocurrent signals using vacuum photo diodes (VPD), and the photocurrents areconverted to voltages and amplified by low-noise solid-state electronics.Another stringent constraint for the detector system is the detection and elim-129

ination of any instrumental systematic effects inducing false asymmetries associatedwith imperfections in the detector or data acquisition system. Theseeffects must be periodically in the course of the experiment. It is thereforeessential to perform these measurements in a short time, compared to the runtime of the experiment. The time required for these measurements is determinedby the time required to average the electronic noise. For the currentmode detection to be effective the electrical noise in the detector system mustbe much smaller than the beam-on shot noise.A series of measurements have been performed both on individual detectorsand their components as well as on the detector array as a whole, in conjunctionwith the DAQ. The results show that the detectors meet all requirementsdescribed above. The remainder of this paper describes the measurements indetail, including the setup, the procedures used and the results found.4.1.1 Review of the Scintillation Process and PhysicsThe following is a short review of the underlying physical processes in sourcesof gamma-rays and their interaction with matter, as they relate to the testsdescribed herein. A description of the setup, the procedures used in the varioustests and the various results are given in the main sections (See table ofcontents).In the interaction with matter, gamma-rays may transfer all or part of theirenergy to electrons within the material. How much of their energy they giveto the electrons depends on the process of interaction. The three main pro-130

cesses involved in the interaction of gamma-rays with matter are photo-electricabsorption, Compton scattering, and pair production [47]. In photo-electricabsorption an incoming gamma-ray with the right energy will be absorbed byan atom in the material, in the process of which an electron is ejected. Theenergy of the electron is equal to that of the gamma-ray minus the bindingenergy of the electron in the atom E e − = hν − E b . However, since the bindingenergy of the electron is on the order of eV while the energy of the photon willbe on the order of at least keV the electron essentially has the energy of thegamma-ray. The probability of gamma-ray absorption depends on the atomicnumber of the nuclei in the material and is approximately given byτ ∼ = const × ZnE 3.5γWhere n varies between 4 and 5 [47]. This is the primary reason why high Zmaterials such as lead are used for gamma-ray shielding.The ejected electronwill move through the material, exciting any atom in its path through collisions.Each atom will then de-excite by giving off a photon with an energythat depends on the specific atomic transitions in the material. For CsI(Tl)the emitted scintillation light has a wavelength centered around 550nm (Seetable A.2, section A.2.3). The number of atoms that are excited in this waydepends directly on the kinetic energy of the electron and since the electronreceived all of its energy from the gamma-ray, the number of photons emittedfrom the material is a direct measure of the gamma-ray energy. The pulsecounting method used (see below) to measure the number of photoelectronsper MeV relies on the accurate determination of the gamma peak position for131

the isotopes used.The other common process is Compton scattering. In this case, the incidentgamma-ray is not absorbed but scatters off a large number of electrons inthe material. Each recoiling electron excites its host atom which then alsode-excites by emitting a photon with a frequency that is characteristic of thematerial. However, in this case the energy given off to the electron may varyanywhere from zero to a large fraction of the gamma-ray energy. The energyof the scattered gamma-ray is given byhν s =hν1 + hνm oc(1 − cos(θ))2(4.1)Where m o c 2 is the rest-mass of the electron. Because of the many possibleenergies Compton scattered electrons may take on, the spectrum of any sourcedue to this process will be washed out over a range up to the Compton edge,at which point Compton scattering decreases and photo absorption increases.The majority of scintillation light produced by the 2.23 MeV γ-ray, releasedin the p(n, γ)d reaction, will be due to the Compton process. In isotopes withclosely spaced photo-peaks the Compton edge contributes to the spread seen inthe photo-peak and distorts the resulting efficiency measurement. The sourcesused for the efficiency measurements described here are either low enough inenergy to not have an appreciable Compton effect ( 241 Am) or have a photopeakthat is well separated from the Compton edge ( 137 Cs).The excited electrons in the scintillation material will decay with a half-life of132

about a µs. Because the time it takes for the electron that was emitted in theprocess following the gamma-ray interaction to migrate through the materialis much shorter than that, all the excited states are essentially formed at onceand a large number of visible light photons will be emitted at nearly the sametime with a spread determined by the above half-life.At high gamma-ray energies, well above twice the rest-mass energy of an electron,the process of pair-production becomes probable. In this case, the photonis replaced by a positron-electron pair which will annihilate after slowingdown, emitting two 0.511MeV gamma-rays. Due to the low count rate fromthis process (as compared to photo-electric absorption) the peaks created frompairFigure 4.1: Relative contribution to scintillation from three major types ofgamma-ray interaction.production can not be used to determine the CsI efficiency within a suitabletime frame.133

4.2 Detector Design and Operational CriteriaThe detector array consists of 48 CsI(Tl) cubes arranged in a cylindrical patternin 4 rings of 12 detectors each around a cylindrical 20 l liquid hydrogentarget (Fig. 4.2). In addition to the conditions set on the detector array by theneed to preserve statistical accuracy and suppress systematic effects (see Section4.2.3), the array was also designed to satisfy criteria of sufficient spatialand angular resolution, high efficiency, and large solid angle coverage. Herewe discuss some of the reasoning behind certain design choices and describethe specific properties of our array.To measure the asymmetry, a small (5 × 10 −8 ) parity-odd component must bedetected in the presence of an intense isotropic (parity-even) gamma signal.The parity-odd component of the signal is proportional to cos θ, where θ isthe angle between the direction of neutron polarization and the momentumvector of the emitted γ-ray. As long as the change in cos θ over a detectorelement is small, the finite size of the detector elements will not reduce thestatistical accuracy of the experiment. The error in the measured asymmetrydue to spatial resolution, for N detected gammas, can be estimated as follows:For an array of only two detectors, one covering each hemisphere, the yieldsof the upper (Y U ) and lower (Y D ) detectors areY U =Y D =∫ 2π0∫ 2π0∫ π/2 1dφ0 4π (1 + A γ cos θ) sin θ dθ = 1 2 + A γ4∫ π 1dφπ/2 4π (1 + A γ cos θ) sin θ dθ = 1 2 − A γ4 .134

So that (ccf. 2.20) from the measured (raw) asymmetry, we findA raw ± σ A√ rawN= Y U − Y D= A γY U + Y D 2and therefore the error in A γ for this case is ɛ Aγ = 2/ √ N.For an infinitely fine grained array the error can be calculated using a χ 2minimization. We then haveχ 2 = ∑ i(Yi − Ω iN4π (1 + A γ cos θ i ) ) 2σ 2 Y i,where Y i and Ω i are the signal yield and solid angle for the ith detector and,from Poisson statistics, σ 2 Y i= Y i . The error on A γ can then be calculated fromɛ −2A γ= 1 ∂ 2 χ 22 ∂A 2= N 4π∑Ωi cos 2 θ i .Which, in the limit of infinitely many detectors, approachesɛ −2A γ⇒ N 4π∫ 2π0∫ πdφ cos 2 θ sin θdθ = N03 .So the error in A γ for an array with infinitely many detectors is ɛ Aγ= √ 3/ √ N. Since the error is a slowly-varying function of the degree of segmentation,there is no pressing need for the detectors to be finely segmented, and thelateral dimensions can be chosen with regard to other criteria.A segmented detector with elements large enough to fully contain the gammaenergy reduces the noise per detector from fluctuations in the fraction of135

gamma energy shared among different detectors and simplifies the identificationof the gamma emission angle, which as noted above need not be determinedwith high precision. Gamma cross sections in high Z materials reacha minimum around2 MeV energy and the corresponding mean free path( 5.5 cm for 2.2 MeV γ-rays in CsI) sets the scale for the dimensions of thedetector elements. The size of the individual detectors is 152×152×152 mm 3 .With these dimensions each crystal absorbs about 84% of the energy for a 2.2MeV γ-ray incident at the center of the front face. MCNP 1 [48] and EGS42 [49] calculations have shown that about 3% of the energy is backscatteredfrom the front face of the crystal, 11% leaks out through the rear face, and2% leaves through the 4 remaining sides. This reduces the cross talk betweendetector elements to a level that is small enough to allow the measurement ofthe asymmetry to the proposed accuracy. The main effect of cross talk is asmall loss in angular resolution. A 20% increase in thickness in the directionof the incident γ-ray increases the amount of energy absorption by only 4%.The overall size of the array, on the other hand, is constrained by the size of thesource (target), which depends on the diameter of the neutron beam ( 10 cm)and the mean free path of a cold neutron in the liquid hydrogen target (14 cmfor a 2 meV neutron) (see Sec. 3.6.2). The neutrons that are not capturedin the liquid hydrogen are absorbed in a thin (2 mm) plastic material loadedwith 6 Li to prevent activation of the CsI by neutron capture. Gammas are1 MCNP is a trademark of the Regents of the University of California, Los Alamos NationalLaboratory, http://laws.lanl.gov/x5/MCNP/index.html2 The EGS code system and its various tools and utilities are copyrighted jointlyby Stanford University and National Research Council of Canada. All rights reserved.http://www.slac.stanford.edu/egs/136

Figure 4.2: NPDGamma detector array and target assembly. The array surroundsa 20 l liquid hydrogen target. There are 48 detectors grouped into fourrings of 12 detectors each and arranged in a cylindrical pattern around thehydrogen target.transmitted through the low Z of the plastic and the aluminum target vesselwith high efficiency. Since neutron absorption in 6 Li is dominated by chargedparticle emission as opposed to gamma emission, background gammas whichcan dilute the signal from n-p capture are suppressed. For design purposeswe can therefore choose to concentrate on the signal from n-p capture eventsrather than background events.The detector array is arranged in cylindrical rings to surround most of thehydrogen target and allow the neutron beam to enter and exit without activatingthe CsI. It is important to detect the majority of the photons emittedtransverse to the neutron beam, since the neutrons are transversely polarizedand gammas emitted along the neutron polarization direction contribute most137

to the parity-odd component of the asymmetry. Photons emitted along thebeam direction therefore contribute little to the asymmetry, and the questionbecomes how many rings need to be included. Monte Carlo calculationshave show that the error in the asymmetry as a function of the number ofrings along the neutron beam axis reaches 87% of its asymptotic value for 4rings [21]. Together with the individual detector dimensions mentioned abovethis geometry covers a solid angle of nearly 3π.With the detector size chosen to be large compared to the gamma mean freepath, the predicted peak gamma rate into a single detector in the experimentis estimated to be 100 MHz, based on moderator brightness measurements [38]and Monte Carlo calculations. At this rate pulse counting is impractical forCsI given the decay time of the scintillation light pulses (1 µs [50, 51]). Foran array composed of CsI detectors, the high photon rates must be handledusing current mode gamma detection.The crystals were manufactured and encased in the housing by Bicron 3 . Eachdetector module consists of two rectangular pieces of optically coupled Thalliumdoped Cesium Iodide crystals. The slightly hygroscopic CsI(Tl) crystalsare wrapped in PTFE Teflon, a diffuse reflector, and hermetically sealed ina 1.0 mm thick Aluminum housing.The Optics program [52] was used tostudy which reflector and crystal surface treatment to use, in order to obtainthe maximum light output and best overall uniformity for the given detectorgeometry. We found that diffuse reflection produced the best results. Thecrystals are coupled to a 76 mm diameter K+ glass window at the top of3 Bicron, Saint-Gobian Industrial Ceramics, Inc , www.bicron.com138

the housing assembly to facilitate the detection of the scintillation light bya vacuum photodiode (VPD) during standard operation. The detectors areindividually mounted on the array to minimize potential stress and plasticdeformation of the crystals.Figure 4.3: Illustration of an individual detector. Each detector consists oftwo coupled CsI(Tl) scintillators, a VPD and a preamplifier stage.139

CsI(Tl) was chosen because of its high density (4.53 g/cm 3 ), large Z, high lightyield and its relatively low cost. For the alkali iodides, thallium activation isrequired to achieve a high light output. For CsI(Tl) the emitted scintillationlight has a wavelength centered around 540 nm which is well matched withthe absorption characteristics of the type of VPD used in this experiment.The light yield of CsI(Tl) is 54000 photons per MeV with a wavelength ofλ max = 540 nm at maximum emission and a scintillation efficiency of 12% [47].The current collected from the VPD anode is amplified by a low noise solidstateamplifier.A cylindrical aluminum housing for the VPD and preamplifier is mounted ontop of the CsI crystal assembly.The housing is designed to be light-tight,to minimize noise contributions from capacitive coupling between electroniccomponents on the preamplifier board and to shield the assembly from outsidefields such as those produced by the radio frequency spin flipper. To avoidground loops the detector housing is grounded via the signal cable shield onlyand the detectors are individually mounted to a stand which allows the electricalisolation between detectors. Each detector comes equipped with two lightemitting diodes (LED), one in each crystal half. The LEDs are used duringbeam off detector diagnostic tests (see Section 4.6).Radiation damage will decrease the self-transparency of the crystals, resultingin a decrease in detected light. CsI(Tl) has been found to be rather radiationhard up to doses of more than 500 Gy [53, 25, 54, 55, 56], with the precisethreshold for significant radiation damage in the crystal dependent on crystalimpurities as well as the radiation damage rate. This radiation dose is approx-140

Figure 4.4: CsI Detector Physical Specifications.141

imately the dose that the detectors will receive over the course of the entireexperiment (a few thousand hours of running), and the corresponding damagerate is small compared to those that have caused significant radiation damagein CsI(Tl) detectors in the past.4.2.1 Vacuum PhotodiodesTo convert the scintillation light to a current the detectors employ 76 mmS-20 Hamamatsu 4 vacuum photodiodes rather than photomultiplier tubes(PMT). The decision to use VPDs was based on the fact that photomultipliersare very sensitive to magnetic fields. A 1 G field leaking into the PMT canchange its gain by 100%. The experiment uses magnetic fields to control theneutron spin direction and any field leaking into the detectors may producelarge gain changes. On the other hand, the sensitivity of a vacuum photodiodeto magnetic fields, is only about 1 × 10 −4 /G in a 10 G DC field or 1 × 10 −5 /G 2for 10 G AC field (see Section 4.3.2) [45].This particular type of vacuum-photodiode was chosen for the low photocathodesheet resistivity. The low sheet resistivity of the S-20 photocathode reducesthe degree of gain non-linearities across its surface.With an S-20 cathode, theVPD has a quantum efficiency of ≈ 10% at the CsI maximum emission wavelength.A bias of 90 Volts is applied across the VPD via two 45 V batterieslocated on top of the VPD and preamplifier housing (Fig. 4.3). This removesthe necessity for an external supply to be connected directly to the VPDs which4 Hamamatsu Corporation, 360 Foothill Road, P.O. Box 6910, Bridgewater, N.J. 08807-0910, USA, www.hamamatsu.com142

Figure 4.5: Properties of S-20 (500K) photocathodes usedcould cause ground loops and introduce additional noise in the VPDs. Thebatteries 5 have a capacity of 140 mAh, the average beam-on current drawnfrom the VPDs is about 30 nA and the VPDs can therefore nominally be runfor 10 7 hours continuously. So the time before exchange should be limited bythe battery’s five year shelf life.4.2.2 Low Noise PreamplifierThe VPD current in each detector is converted to a voltage signal by a threestage low-noise current-to-voltage preamplifier [57]. The first stage uses anop amp-based current-to-voltage amplifier with a gain of 5 × 10 7 (Fig. 4.6).The second amplifier stage serves as an inverter with a nominal gain of −2.15.The resistor in the second stage is used to adjust relative detector gains (seeSection 3.7 and 4.3.4). The third stage serves as a low impedance line driver.5 Eveready Industries India, LTD. , www.evereadyindustries.com143

C 1anodecathode90V+−−+R 1Rg−+R2R3C3+−Figure 4.6: The detector preamplifier consists of a decoupled power supply,two 45 V batteries to provide the bias across the VPD and three operationalamplifier stages.The detector preamplifier has been designed to operate at noise levels close tothe theoretical limits set by Johnson noise, so that the time required to measurethe asymmetry to the level of 5×10 −9 is dominated by the collection of countingstatistics rather than by the need to average electronic noise (see Section 4.5)[58]. Various filter stages and ground isolation between the preamp power andsignal circuit have been implemented to satisfy these requirements.4.2.3 Mode of Operation and Systematic EffectsAchieving the desired accuracy in measuring the parity violating gamma asymmetrydepends on good counting statistics (Section 4.5) with comparativelysmall errors from other sources, such as electronic noise and systematic effects.During beam-off measurements, the time required to determine any systematiceffect is governed by the noise in the preamplifier and the rest of the DAQ.Since these effects need to be studied periodically during the experiment, itis essential to perform the beam-off measurements quickly compared to thetime required to collect counting statistics.To satisfy these requirements,144

the detector array must have a high photoelectron yield, a low sensitivity toexternal radioactivity and electromagnetic effects and have very good noiseperformance.The photoelectron yield enters into the calculation of the average photo currentseen at the detector preamplifier output as well as the shot-noise seen at theVPD cathode. The time needed to measure an asymmetry to a given accuracyis proportional to the inverse of the average photo-current. In a current modemeasurement, if the detector has a high photoelectron yield, counting statisticsare manifest in the form of shot noise due to the fluctuations in the numberof gammas entering the detector. The corresponding expected RMS width isgiven by [59, 60]σ Ishot =√2qI√f B , (4.2)where q is the amount of charge created at the photo cathode per detectedgamma-ray, I is the average photo-current per detector and f B is the frequencybandwidth, set by the filtering in the data acquisition system (DAQ).Since the experiment intends to determine an asymmetry with a precisionof 5 × 10 −9 , any systematic effect resulting in a false asymmetry has to bemeasured to at least this level of accuracy in a short period of time. The measurementof such a small quantity requires the careful evaluation and analysisof any possible systematic effects. For the detector array the two most seriouspotential instrumental systematic effects may be caused by a radio frequencyspin flipper (RFSF) [45], which is used to reverse the spin of the neutrons (seeSection 3.5).145

During normal (beam-on) operation, when gammas from neutron capture createa large signal in the array, any magnetic fields leaking into the VPDs canproduce a systematic effect through a multiplication of the overall detectorgain. We call such an effect a multiplicative systematic error. In addition, anyelectronic pickup could add a false signal on top of the real signal. We callsuch an effect an additive systematic error. If these signals are correlated withthe spin state of the neutrons, through the spin flipper, this could lead to falseasymmetries.4.3 Detector Photoelectron Yield and CsI toVPD Gain MatchingThere are important reasons for making the overall efficiency of the elementsof the detector array as uniform as possible. Equalizing the overall efficienciesthrough relative gain matching between detectors prevents saturation of thedifference signal channels in the ADC (see Section 3.7) and allows an expansionof the dynamic range as discussed earlier. Furthermore, uniform detector efficienciesmake the observed parity-odd up-down gamma asymmetry signal lesssensitive to potential neutron spin-dependent crosstalk from parity-conservingleft-right asymmetries which are known to be present at small levels in theinteraction of the neutrons with hydrogen and in the gamma angular distribution[61]. Unfortunately it was not practical to obtain all the individualcomponents of the detector with sufficiently uniform properties to ensure thisby design. For these reasons great care has been taken to characterize the rel-146

evant properties of all of the individual components for each detector so thatthey can be individually matched to minimize variations in the overall gain ofeach detector/VPD/preamp combination. After hardware matching is optimized,final adjustments can be made in amplifier gains and also in software.This section describes these measurements.To establish the properties and performance of the individual detector components,a variety of measurements were performed prior to their assembly. Thephotoelectron yield of the CsI scintillators and the efficiency of the VPDs weremeasured independently and the results were used to match them and obtaina reasonably uniform relative gain between all CsI-VPD detector modules.After assembly of the detectors, a current mode measurement was performedto establish the combined detector gain and to refine the gain matching usingthe resistors in the second preamplifier stage.4.3.1 CsI Relative Photoelectron YieldThe primary photo-peaks of two radioactive sources, 241 Am (0.4 MBq, E γ =0.06MeV) and 137 Cs (0.3 MBq, E γ = 0.67MeV) and standard pulse countingmethods were applied to determine the number of photoelectrons per MeVfrom the RMS width of their respective peaks. This procedure relies on theassumption that the photo-peak widths (σ p= FWHM/2.35) data are dueprimarily to the fluctuations in the number of photoelectrons made at thephotocathode and subsequent dynodes of the PMT (shot noise), as well asintrinsic properties of the crystal.Contributions to the peak width due to147

electronic noise were combined with those due to crystal intrinsic propertiesinto a single width σ int .Crystal¡ ¡¡¡¡¡ ¢¡¢¡¢¡¢¡¢¡¢¢¡¢¡¢¡¢¡¢¡¢¢¡¢¡¢¡¢¡¢¡¢¢¡¢¡¢¡¢¡¢¡¢£¡£¡£¡£¡£¡£¡££¡£¡£¡£¡£¡£¡££¡£¡£¡£¡£¡£¡££¡£¡£¡£¡£¡£¡££¡£¡£¡£¡£¡£¡£¡ ¤¡¤¡¤¡¤¡¤¡¤¡¤¤¡¤¡¤¡¤¡¤¡¤¡¤¤¡¤¡¤¡¤¡¤¡¤¡¤¤¡¤¡¤¡¤¡¤¡¤¡¤ ¥¡¥¡¥¡¥ ¦¡¦¡¦¥¡¥¡¥¡¥ ¦¡¦¡¦¥¡¥¡¥¡¥ ¦¡¦¡¦§¡§¡§¡§ ¨¡¨¡¨¤¡¤¡¤¡¤¡¤¡¤¡¤ ¢¡¢¡¢¡¢¡¢¡¢ ¡ Collimator(Pb, 5 cm thick)Source¨¡¨¡¨ §¡§¡§¡§Figure 4.7: Setup for detector photoelectron yield measurement. The gammaraysources were centered such that both scintillator crystals are illuminatedequally. The sources were collimated to avoid smaller energy deposits in partiallycontained events.The gamma sources were centered on one side about 6.8 cm from the detectorhousing to illuminate both halves of the crystal equally (Fig 4.7). The γ-rayswere collimated down to ∼ 0.2 sr, using a 5 cm thick lead shield. The sourcewas mounted on a reproducible mount, to ensure that the relative sourcedetectorposition was always the same. A 127 mm Hamamatsu R1513 PMTwas optically coupled to the detector window using BCS 260 optical couplinggrease. To study the effects of optical coupling quality on the overall detectorefficiency two randomly chosen detectors were later used in conjunction withtwo other coupling methods, and the results were compared to those obtainedhere (see Section 4.3.3).To extract the photoelectron yield, a plot of the two relative peak variancesΣ 2 = σ 2 p/p 2 versus inverse peak-energy was made. A linear fit was made148

etween the two points, usingΣ 2 = a 1 E + σ2 int.Here p is the zero-offset corrected peak mean. The slope of the linea = 1 N (1 + 1δ − 1 )was extracted from the fit and determines the number of photoelectrons perMeVN = a −1 δδ − 1 ≃ 1.37a .Here, δ is the Poisson distributed gain in the number of electrons produced atthe PMT cathode which, for a 10 stage Hamamatsu R1513 PMT operated at1.6 kV, is ∼ 3.7. The overall gain for the PMT used is 5 × 10 5 at 1.6 kV. Thefactor1δ−1≃ 0.37 emerges due to the fluctuations in the number of electronsmade at each dynode stage of the PMT, which contribute to the overall RMSwidth in the photopeaks (see Appendix A). Corresponding to the activities ofthe two sources, the errors from counting statistics for the time counted are∼ 0.1% for ( 241 Am) and ∼ 0.07% for ( 137 Cs). The overall error on the resultsis dominated by the quality of the least-squares fit.Possible gain non uniformities from the two crystal halves were searched for byseparately exposing each half to the 137 Cs source. Measurements were takenwith the source on the left side, right side and the center of the scintillator,where the center is defined as seen in Fig. 4.7. In the worst case, the gainvaries by about 7% from one crystal half to the other. The photoelectron yield149

measurements produced an average of 1300 photoelectrons per MeV with anoverall variation of ± 20% between detectors. Figure 4.8 shows the results for48 detectors in the order they were taken. The error in the values is about5%, mostly due to the fitting procedure.17001600CsI rel. eff.Error 5%1500pe/MeV14001300120011001000900CsI Crystals (time ordered)Figure 4.8: CsI detector efficiency. The average measured efficiency is 1300 ±260 photoelectrons (pe) per MeV. The efficiencies are shown in the order theywere measured.4.3.2 VPD Relative Gain and EfficiencyThe response of the VPDs to the CsI scintillation is different from the responseseen using a tungsten lamp, which was used by Hamamatsu Photonics to calibratethe VPDs. Therefore, it was decided that the VPD relative efficiencieshad to be studied using the CsI scintillation light. The VPD efficiencies weremeasured using capture gammas from a neutron beam at KEK and relativeefficiency measurements were performed at Los Alamos, using the LED’s inthe detectors (see Section 4.3.2). At the pulsed epithermal neutron beam lineat KEK, Cd and In targets were used to convert neutrons to gamma-rays150

through radiative neutron capture at the Cd cutoff and at the In 1.46 eV and9.1 eV resonances. One of the CsI crystals and its preamplifier was installed,together with each tested VPD, next to the target, using proper neutron andgamma shielding. In this way, relative efficiencies of 57 VPDs were determinedwith an error of 6%. Comparisons of normalized VPD efficiencies at differentneutron energies show a very high correlation, whereas the correlation betweenefficiencies measured with neutrons and those measured with a tungsten lampis very poor [62].In a separate measurement the LEDs in each detector were used to establishthe relative VPD efficiency again and verify the quality of the CsI-VPD gainmatching. To measure the relative VPD efficiency, a single CsI crystal wascoupled, in turn, with each VPD, using vacuum grease as coupling compound(see Section 4.3.3). A 2 V, 100 Hz square wave was applied to both LEDs in thedetector. The current drawn by the LEDs was constant at 18.5 mA throughoutthe measurement for 48 VPDs. The preamplifier output was monitored witha scope and with a precision volt meter. The relative gains differ by up to afactor of 2.5. The results are shown in Fig. 4.10. A conservative estimate onthe error in the efficiency is 7%, based on the fluctuations seen in the outputof the precision volt meter.These measurements are compared with those done at KEK. Figure 4.11 showsthat the results of the two independent measurements agree to within errors.The differences are most likely due to the change in optical coupling whenswitching the VPDs.151

Power S.V s (Pre−amp)100ΩScope100ΩPulse Gen.Figure 4.9: VPD and LED Test Setup87Pre-amp out (Volts)6543VPD (time ordered)Figure 4.10: VPD relative efficiency results using LEDs. The efficiencies areshown in the order they were measured.4.3.3 CsI to VPD MatchingAs already mentioned, the CsI and VPD efficiencies are matched to reducethe overall gain variations in the detector array. However, fluctuations in thequality of the VPD-CsI optical coupling also affect the overall gain. A pair ofdetectors was tested with three optical coupling compounds:1. BC 630 Bicron optical coupling grease,152

1.110.90.8LA0.70.6VPD rel. Eff.Fit0.5KEK error 5%0.4LA error 6%0.4 0.5 0.6 0.7 0.8 0.9 1 1.1KEKFigure 4.11: A comparison of VPD relative efficiencies, measured with neutronsat KEK and with LEDs (see text).2. Dow Corning Sylgard 184 Silicone Elastomer (cookies) and3. high vacuum grease (translucent).The Sylgard elastomer was used to make ”cookies”, circular 76 mm diameterdisks, about 3 mm thick. The initially liquid compound was pumped to removeair and cast into a mold standing on its side to make the surfaces as flatas possible. Unfortunately the compound was too hard and small non uniformitieson the VPD window or on the casting surfaces prevented the elastomerfrom producing a significant increase in coupling quality.The best resultsfor the Sylgard elastomer are achieved by pouring it and allowing it to set inplace, between scintillator and VPD. However, this option was not consideredbecause the detector array configuration makes it difficult to exchange entireCsI-VPD assemblies in-situ. The best efficiency results were obtained usingoptical coupling grease, which increased the gain by a factor of two over nocoupling compound. However, BC 630 was disqualified due to its low viscosityand the vertical orientation of some of the VPD-CsI boundaries in the array.153

Since high vacuum grease is much more viscous and gave 90% of the gain ofBC 630, it was chosen for the coupling.Figure 4.12 shows the gain ordered efficiency measurement results for theVPDs, the CsI crystals and their product. The curve showing the productof the efficiencies provides an upper limit on how well the gain of the detectorscan be matched in hardware without further adjustments. The actual relativegain shifts are expected to differ from this prediction, due to variations in opticalcoupling quality among the detectors and the variations in scintillationlight response of the crystals. The CsI crystals were matched with the VPDsaccording to the efficiency results presented in Fig. 4.12. This selection servedas a starting point from which to conduct additional efficiency measurementsand further improve the gain via the adjustment of the feedback resistors inthe detector preamplifier.10.9Relative Gain0.80.7CsI measured rel. gainVPD measured rel. gainCsI*VPD theoretical rel.gain0.60.5CsI Crystals and VPDs (gain ordered)Figure 4.12: CsI and VPD gain matching. The CsI relative gain is shownin increasing order and the VPD relative gain is shown in decreasing order.Their product shows the theoretical overall efficiency spread after matchingthe VPDs and scintillators in this way.154

At this stage, all feedback resistance values in the preamplifiers (Fig. 4.6) wereidentical for all detectors. The overall variation observed in the efficiency of thecompletely assembled detectors determine the change in feedback resistancenecessary to adjust the gain in any given detector.4.3.4 Combined Relative Detector GainTo compare the predicted values of the relative gain between detectors, theassembled detectors were tested with a 137 Cs source, intense enough to producea current mode output. The corresponding preamplifier output was fedthrough a low-pass filter and monitored with a precision voltmeter. The sourcewas located at the outside surface of the detectors and centered on one side.A source with the given activity (see Section 4.3.1) and in the given configuration(fractional solid angle ≃ 0.4) is expected to deposit approximately6.54 × 10 4 MeV/s into the detector. With the measured average CsI photoelectronyield of ∼ 1300 pe/MeV one expects to measure an output on theorder of a few millivolts.The low-pass filter was adjusted to have a time constant of 15 seconds tostabilize the voltage. Three measurements were taken for each detector, onewithout source, one with source in place, and again without the source. Themeasurements were performed quickly to avoid fluctuations with long timeconstants (∼ minutes). The mean voltage out of the preamplifier was 2.4 mV.Figure 4.13 shows the normalized difference between the signal with sourceand the average of the two signals taken without the source, versus detector155

serial number. The overall spread in gains is still ∼ 40%, but the relative gainsshifted a bit, as expected.1Rel. Gain (Normalized)0.80.60.40.2Current Mode Gain TestCsI-VPD Matching Prediction0 5 10 15 20 25 30 35 40 45DetectorFigure 4.13: Detector relative gain measured in current mode, as comparedto the theoretical prediction. The gains were measured on the bench underidentical conditions for each detector.Resistors ranging from 1.50 kΩ to 2.80 kΩ were then installed in the secondamplifier stage to reduce the gain variations. After adjusting the preamplifierresistors and assembling the detectors into the array stand (Fig. 4.2 for thefinal configuration), an additional set of data was taken to verify the finalrelative detector gains (before using the gain modules).This measurementwas done, using a rotating 2 MBq 137 Cs source and a lock in-amplifier. Thesource was located at the center of the ring corresponding to the detector thatwas tested. The reference phase for the lock in-amplifier was generated by aninfrared emitter-receiver mounted onto the shaft of the rotating source. Themeasured normalized relative gains per detector are shown in Fig. 4.14. Thelock in-amplifier and the rotating source were used to filter out noise and otherfluctuations to obtain a cleaner measurement of the detector gains.The corner detectors cover a solid angle that is ∼ 20% smaller than it is for156

1.2Current Mode Gain Test (Lock-In Technique)Rel. Gain (Normalized)1.110.90.8Side DetectorsCorner Detectors0.70.6CsI Crystals (odered by ring and position)Figure 4.14: After assembly of the detector array the relative gain was againmeasured using a rotating gamma source located at the center of each detectorring. The corner detectors are shadowed by the detectors above an below themand therefore cover a smaller solid angle.the side detectors (Fig 4.2 and Fig 5.1). The gain in the corner detectors wasthen matched with the rest of the array by further adjusting their feedbackresistors.This level of hardware gain matching is sufficient to prevent thedifference channels from saturating.The final precision of gain matching in the array will be limited by the qualityof optical coupling as well as any time dependent gain fluctuations in thedetectors. The next stage of gain matching will be implemented using custombuilt adjustable gain VME modules in the DAQ stream and will be performedduring the commissioning run, in conjunction with a neutron beam and signalsfrom neutron capture on a target. This work will be described elsewhere.157

4.4 Noise Performance and BackgroundIn this section we describe studies of long term fluctuations in detector pedestalsand electronic noise due to radioactive and electromagnetic background. Weshow that cosmic ray background in the detector array is understood and hasno effect on the measured asymmetries. We discuss measurements performedto verify that false asymmetries due to electronic pickup and magnetic fieldinduced gain changes in the VPDs are negligible.These measurements required the acquisition of data over long periods of timeand under conditions that are as close as possible to those encountered whenthe experiment is running. Accordingly, the entire array and spin flipper wereassembled in their final configuration and data have been taken with the sameDAQ setup to be used in the final experiment.4.4.1 Detector NoiseIn a current mode experiment, the accuracy of the measurement is governedby the rate and quality of sampling of a signal that may be viewed as a continuousstring of values of a random variable. The randomness and the spread(RMS width) of the samples determine how many samples one must take toachieve a certain level of accuracy in the measurement, while the sampling ratedetermines how long that will take and whether or not one has in fact measureda representative subset of the signal. As the sampling rate is increased,158

a larger fraction of the width in the signal is due to the correlation betweensamples and the observed error will be larger than expected from 1/ √ N countingstatistics. Thus, for a given number of samples taken, oversampling leadsto loss in statistical information. Undersampling, on the other hand, will leadto information loss in the noise since high frequency fluctuations are aliasedinto lower frequencies.The detector preamplifier (see section 4.2.2) and the rest of the DAQ wasdesigned under the requirement that there be no substantial additional noisecontribution beyond the expected Johnson noise from the resistors in the firstpreamplifier stage and the intrinsic noise of the operational amplifiers used [57].To verify that this constraint has been met, it is important to consider thenoise behavior one would expect based on the circuit design.Thermal or Johnson NoiseThe thermally excited motion of the free electrons in a conductor producesrandom voltage pulses on a very short time scale.At any given instant, alarge number of these pulses contribute to produce a particular voltage dropacross this conductor. The values measured are random and, from the centrallimit theorem, are expected to fluctuate according to a Gaussian distribution,around some DC voltage offset. Because of the short duration of the individualpulses making up the measured voltage drop, the spectral density (powerspectrum) is approximately flat [59] (S VJ (f) ≃ S VJ (0)).The amplifier stages in the detector preamplifier are referred to as fixed param-159

eter linear systems; i.e. the output of the system depends on the parameter(s)in the same way as the input to the system and such a system could be describedby a set of linear differential equations. Most basic electronic circuitsfall under this category (See for example [59]). Each such system has associatedwith it a system function H(iω) which relates the input (X (iω)) to theoutput (Y(iω)) of the systemY(iω) = H(iω)X (iω).Here the input and output is expressed as the Fourier transform of the relevanttime dependent signals.For the relation between the input and output power spectra, we also haveS y (ω) = |H(iω)| 2 S x (ω).In the case of a noisy resistor we have a system function which relates outputcurrent noise to the input voltage noise; i.e current fluctuations seen due torandom variations in voltage drop across the resistor.One can use severaldifferent equivalent circuits to describe a noisy resistor. One such system is anoiseless resistor connected in series with a regular inductor, across a voltagesource (V J ) [59]. Since the variance of a signal due to a stationary randomprocess is equal to the intensity of its time varying component, the powerdissipated in the model circuit can be expressed as〈I 2 〉 =∫ ∞−∞160S I (f)df.

And for the equivalent RL circuitL2 〈I2 〉 = k BT2 .The appropriate system function for this circuit isH(iω) =1√R 2 + (iωL) 2 .So one finds thatandS IJ (f) =〈I 2 〉 =∫ ∞−∞S VJ (f)R 2 + 4π 2 f 2 L 2S VJ (f)dfR 2 + 4π 2 f 2 L 2 . (4.3)Then, since we normally only deal with positive frequencies and H(iω) is aneven function, we have∫ ∞0S VJ (f)dfR 2 + 4π 2 f 2 L 2 = S V J(0)4RL . (4.4)and the noise due to thermal excitation in a resistor is predicted to beS VJ (0) = 4k B T R. (4.5)161

Expected Noise PerformanceTo establish the actual RMS width one expects to see in the voltage signalat the output of the preamplifier, one has to understand the origin of all noisecontributions, their propagation through the DAQ and their final processingin the data stream. This includes determining a suitable sampling rate andthe correct bandwidth for the predicted noise levels, set by the filtering in theDAQ.For the purposes of noise analysis, the most important component of thepreamplifier circuit (Fig 4.6) is the first stage connected to the VPD anode.This stage incorporates a 50 MΩ feedback resistor (R 1 ) which is expected tocompletely dominate the noise. Subsequent resistors in the preamplifier andDAQ are smaller by three orders of magnitude and their noise contribution isthus negligible. However, each additional amplifier and filter stage will have amultiplicative effect on the noise generated in the first stage.The thermal noise spectral density in the output of the first amplifier stage,is predicted to be[ ] V2S VJ (f) = 4k B T R 1 . (4.6)HzHere k B is Boltzmann’s constant and T is temperature.A small addition to this noise density comes from the op-amp intrinsic noise.The total, Spice model 6 predicted, RMS width in the current noise density forbeam-off and LED off measurements is ≃ 19 fA/ √ Hz [57].6 EECS Department of the University of California at Berkeley,http://bwrc.eecs.berkeley.edu/Classes/IcBook/SPICE/162

In the DAQ system, the signal from the preamplifier is processed by the sumand difference amplifiers, where it is fed through a six-pole Bessel filter withan average 38 µs time constant (see Section 3.7, Fig. 3.45). The Bessel filterremoves higher frequency noise components present in the preamplifier output[57]. This filtering has to take place before the digitization. The 16-bitADCs do not have sufficient resolution to retain the full noise informationat those frequencies and would cause aliasing of these components into lowerfrequencies. Further averaging in the data stream, after digitization, can notremove this noise without destroying the information content of the signal.Figure 4.15: Auto-correlation function for a six pole Bessel filter. The graphshows the correlation between samples taken with a given time difference (τ)between them. Here, τ = kt o (t o = 1 f s) is the integrated time between thekth sample and the onset of the sampling interval (P ), over which samplesare taken. At a sampling rate of either 50 kHz (t o = 0.02 ms) or 62.5 kHz(t o = 0.016 ms) the samples are ∼ 55% correlated. The correlation is zero forτ > 0.1 ms.To find the expected RMS width at the output of the Bessel filter, one cancalculate the corresponding auto-correlation function for the system [59, 63].The effect of the Bessel filter can be calculated using the corresponding six-pole163

amplitude response function [64, 65]|H(iω)| =K o|B 6 (iω)| , (4.7)where B 6 (ω) is a 6th order Bessel polynomial and K o is chosen such that thedc gain of the filter is unity. The correlation between time samples can thenbe calculated viaR IJ (τ) =∫ ∞0|H(iω)| 2S VJ (f) e 2πifτ df = K o∫ ∞04k B T R 1 e 2πifτ df|B 6 (i2πf)| 2 , (4.8)whereas the variance seen in an individual sample taken from the Bessel filteris given byσ 2 k = R IJ (τ = 0) =∫ ∞04k B T R 1 df|B 6 (i2πf)| 2 . (4.9)The graph shown in Fig. 4.15 is the result of a numerical integration of Eq. 4.8,for a range of intervals (τ) between samples.Due to the Bessel filter, the values sampled by the ADCs are highly correlatedat the sampling frequencies of 50 kHz (t o= 0.02 ms) or 62.5 kHz(t o = 0.016 ms) used in the DAQ, indicating that any high frequency componentsin the preamplifier output are filtered out. As a result of this oversampling,the variance for an individual time bin will be dominated by thecorrelation between samples.Given the sampling scheme described in Section 3.7, the total variance for a164

time bin in the sum and difference signal is given by [59]σ 2 (V ) = σ k 2N + 2 t oP(N∑1 − k t )oR(k t o ). (4.10)k=1PHere, the correlation R(τ = k t o ) is given by Eq.(4.8), τ is the time differencebetween the initial sample and any subsequent sample taken during thesampling interval (P ) and N is the total number of samples taken during thesampling interval.In the calculation of the relevant statistic (mean, standard deviation, etc ...), ifwe consider an individual time bin measurement as the fundamental randomvariable then T = 0.4 ms, N = 20 or 25, and R(τ) will be the correlationbetween all ADC samples within that time bin.On the other hand, if weintegrate over an entire macro pulse and consider that value to be the fundamentalquantity then T = 40 ms and R(τ) will be the correlation betweenall ADC samples within that macro pulse and N = 2000 or 2500. Accordingto Fig. 4.15 any average value obtained for one time bin is essentially statisticallyindependent from the value obtained for any other time bin. Takingeither a time bin or an entire macro pulse as the fundamental quantity is thereforeequivalent, provided that the measured signals have a distribution thatis independent of time over the chosen interval. For beam-on measurementsthe signals observed in the detector array have a time dependence, becausethe neutron flux from the spallation source varies with neutron energy. Thepolarization of the neutrons as well as the capture distribution in the targetare energy (time) dependent as well. So in the measurement of an asymmetryfrom radiative neutron capture at a pulsed neutron source, it is necessary165

to select the time bin interval as the fundamental statistical quantity. The0.4 ms time bin width provides an energy resolution small enough to suppressthe corresponding errors below the statistical limit [45].Noise Propagation through the Sum and Difference AmplifiersIf we start with the expected power spectrum seen in an individual detector,we can propagate the signal through the sum and difference circuit to establishwhat we see at its output. The recombination of data in the data stream andits effect on the noise is also studied.From chapter 3.7 (Fig. 3.45) the relevant signals are:1. The ith detector signal I i rj in the rth ring,2. the rth sum signal S rj = 3 ∑ 1212 i=1 Irj i in the rth ring and3. the ith difference signal D i rj = 30 ( I i rj − 1 3 S rj)in the rth ring.Under the assumption that the noise levels are the same for all detectors, thecorresponding variances are1. σ I irj,2. σ 2 S rj≃ 912 σ2 I i rjand3. σ 2 D i rj≃ (30) 2 ( 1112 )σ2 I i rjin the rth ring.166

Since the samples taken from different detectors are statistically independent(the detectors and the DAQ channels are electrically isolated), the autocorrelationfor the sum signal isR Srj (τ) = 〈S rj (t j )S ∗ rj(t j + τ)〉=9(12) 2 12∑i=1〈Irj(t i j )Irj(t i∗j + τ)〉≃ 912 R I i rj (τ) (4.11)So the noise seen in the sum signal is approximately equal to 3/ √ 12 the noiseseen in any individual detector of that ring, provided that the noise in eachdetector is about the same.For the difference signal, we haveR D irj(τ) = 〈D i rj(t j )D i∗rj(t j + τ)〉= (30) 2 〈I i rj(t j )I i∗rj(t j + τ)〉+ (30)29 〈S rj(t j )S ∗ rj(t j + τ)〉− (30)23 〈Ii rj(t j )S ∗ rj(t j + τ)〉− (30)23 〈S rj(t j )I i∗rj(t j + τ)〉Which, using equation 4.11, givesR Di (τ) = (30) 2 (R Ii (τ) + R S (τ) − 212 R I i(τ))167

≃ (30) 2 1112 R I i(τ)(4.12)The sum and difference signals are sampled by the ADCs, after going throughthe Bessel filters. From eqn. 4.10, the RMS widths seen in a time bin, beforerecombination of the detector signals in the analysis, areσ 2 D i rj=σ 2 Drji20 + 2 t oP(20∑j=11 − k t oP)R D irj(j t o ) (4.13)andσ 2 S rj= σ2 S rj25 + 2 t oP(25∑j=11 − k t oP)R Srj (j t o ). (4.14)After recombining the sum and difference signals, we have I i r = 1/30 (D i r +10 S r ), and the corresponding variance in the recombined detector signal isσ 2 I i rj= 1)(σ 2 + (10) 2 σ 2(30) 2 D i Srjrj+ 20〈D i rjS rj 〉 . (4.15)The covariance 〈D i rjS rj 〉 vanishes, in the limit that the signals in each detectorare the same. Taking a time bin as the sampling interval, the first term inEqns.(4.13,4.14) is found from a numerical evaluation of the integral in Eq.(4.9)with σk/20 2 = 0.28 σ3dB 2 and σk/25 2 = 0.22 σ3dB. 2 The second term evaluates to0.30 σ3dB 2 for N = 20 and 0.35 σ3dB 2 for N = 25, where σ3dB 2 = 4k B T R 1 f 3dB andf 3dB is set by the Bessel filter time constants. Putting everything together, theRMS width in the recombined detector signal is expected to be approximately168

equal toσ Iirj≃√4k B T R 1 f 3dB (0.6) . (4.16)So the RMS width is reduced by a factor of √ 0.6 as compared to the thewidth expected from a simple RC low-pass filter with a bandwidth of f 3dB .The reduction factor is a direct result of the Bessel filter characteristics andthe sampling rate and one finds that the sum and difference amplifiers have noeffect on the noise as seen in the data stream. According to this calculation,the variance in the recombined detector signal is the same that one wouldexpect to see if the detector output was directly filtered with the Bessel filtersand sampled by the ADCs.Using the known values of the preamplifier resistors and the expected noisedensity from the first amplifier stage Eq.(4.16) gives an expected RMS widthat the preamplifier output of ≃ 0.1 mV. For samples taken with τ > 0.05 ms,the second term in equation 4.10 evaluates to less than 1 µV and increases byless than 1% when 25 samples are taken as is done for the sum channels. Gainfactors multiplying the RMS width after the preamplifier are known and thewidth observed in the data can be related back to the noise at the preamplifieroutput. Other contributions to the noise introduced by the sum and differenceamplifiers are negligible compared to the noise of the preamplifier.Noise Measurement Results169

The RMS width of the noise is expected to be different for each detector due tothe different feedback resistance values in each preamplifier (see section 4.3.4).The noise was measured with the data acquisition and the data were averagedover an 8 minute period. The noise seen in the detectors is shown in Fig. 4.16.Figure 4.16: Calculated and measured noise levels for all 48 detectors. A 3sigma cut was placed on the samples in the calculation of the RMS noise,to filter most of the cosmic background. The measured noise levels includecontributions from any activation within the CsI crystals as well as any radiationand electro-magnetic backgrounds found in the general area where themeasurement was performed.Detector pedestals and noise were monitored over a period of 60 hours. Thepedestals were seen to drift by about 1 mV on average and the noise RMSwidth stayed the same (Fig. 4.17).The measured noise levels shown in Fig. 4.16 also include contributions fromdark currents and cosmic ray background (See section 4.4.2), after a 3 sigmacut to filter large cosmic ray signals. These contributions to the noise are notaccounted for, in the noise expected from the calculation above.The measured RMS width in Fig. 4.17, on the other hand, includes all contri-170

0.0220.02180.0216Volts0.02140.02120.0210.02080 50 100 150 200 250 300 350 400 450Run (8.3 minutes each)Figure 4.17: Long-term pedestal and noise vs. run (time) for a typical detector.Each run is 8.3 minutes long. The center band is the mean run pedestal. Thedata points above and below the pedestal mean indicate the noise RMS width.butions to the noise and is observed to be a factor of 3 higher, on average, thanthe estimated noise level. These noise levels determine the time required toperform beam-off measurements of false asymmetries. During beam-on measurements,the expected shot noise RMS width is approximately 28 mV, afactor of 70 larger than the largest noise components for beam-off measurements.4.4.2 Cosmic Ray BackgroundThe detector signals seen during a pedestal run exhibit frequent sample outliersmany standard deviations above the mean (see Fig. 3.46 and Fig. 4.18). Theseare due to cosmic rays incident on the detector array. For an 8-minute longrun, the number of outliers above 6 sigma, one expects to see from noise inthe entire array, is about 24. The observed number is approximately 4 orders171

of magnitude higher.Figure 4.18: Electronic pedestal histogram for a typical detector. The fit(dashed line) shows an RMS width of ∼ 2.3 × 10 −4 V. The data were takenover an 8.3 minute long period.Considering the surface area of the detector array and its location, the numberof cosmic muons that are expected to enter the array, per run, is on the orderof a few times 10 5 .This corresponds to detection of cosmics in less than0.5% of the time bin samples taken in a given sampling period and a rateof about 7 Hz in a single detector. Several measurements were performed toestablish that the observed outliers do, in fact, correspond to cosmic radiation.These measurements used filtering techniques similar to those used on the L3detector at CERN [66] and the SND detector at the Budker Institute forNuclear Physics [67].One of the measurements incorporated the use of cosmic ray event coincidencesbetween a pair of detectors, one below the other. Two additional scintillatorpaddles were installed above and below the pair to trigger only those muonstraversing the entire detector. For such events, the expected energy deposition172

454035csmc_peak_2Entries 99Mean 7.307RMS 3.6833025201510500 2 4 6 8 10 12 14 16 18 20 22mVFigure 4.19: Histogram for muons traversing an entire detector. The detectorpedestal has been subtracted. The narrow peak (dashed line) is due to thoseparticles entering the detector normal to the crystal surface. The broaderbase (solid black line) emerges due to particles that enter the crystal at aslight angle with respect to its surface normal. These particles have a slightlylonger or shorter path length, therefore depositing different amounts of energy.The mean of the narrow peak is ≃ 6.94 ± 0.09 mV.is about 100 MeV. Figure 4.19 shows the result, as obtained for the upperdetector of a the pair.According to the measured CsI photoelectron yieldand the known detector gains, an instantaneous energy deposition of 100 MeVshould produce a signal, in one time bin, of approximately 7 mV above thepedestal mean.4.5 Operation at Counting StatisticsTo establish that the array operates at counting statistics, it is desirable toconduct a neutron capture experiment in which a single neutron capture producesat most one outgoing gamma ray with a known energy. The reaction10 B(n,α) 7 Li produces an excited 7 Li nucleus with a branching ratio of 94%.173

The excited nucleus decays, emitting a 0.48 MeV gamma ray, 100% of thetime. As a result, the width in the data taken with a B 4 Ctarget can becompared to the width expected from neutron counting statistics.The boron target used in this measurement is described in section 3.6.1.The B 4 Ctarget is black for cold neutrons. It was located in the center of thedetector array. The beam was collimated down to a 10 cm diameter crosssection,so that the target completely covered the beam. Three consecutivemeasurements were made. Three 500 s long pedestal runs, without beam, one500 s data run with beam incident on the target and one 500 s backgroundrun with beam but without the target installed.Because of the single gamma released in the 10 B(n,α) 7 Li capture reaction andEq.(4.2) a direct comparison can be made between the RMS width expectedfrom counting statistics and the RMS width seen in the data stream. Severalfactors contribute to the overall width seen in the data stream.Contributionsfrom electronic noise and signals from activation decay in the variousmaterials are measured during beam-off, pedestal runs. Contributions fromradiative capture and scattering on materials other than the intended targetare measured by performing background runs. Both of these contributions canbe removed by subtracting their RMS widths in quadrature from the overallwidth. In addition, any fluctuations in the magnitude of the signal over thecourse of the measurement will also contribute to the width. To remove thosecontributions, the data and background runs were normalized to the beamcurrent and separate variances were calculated for each group of 20 macropulses and then averaged. For all of these calculations a single time bin, cor-174

esponding to the peak signal (E = 11 meV), was used in each macro pulse.600500Electronic NoisensestatsEntries 10000Mean-2.934e-06RMS 0.0005333400300Counting Statistics2001000-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02VoltsFigure 4.20: Counting statistics analysis results for a typical detector. TheRMS width due to counting statistics is compared to the width seen frompedestal runs (electronic noise). The fit to the beam on, target in data histogramshows an RMS width of 6.1 ± 0.04 mV.Figure 4.20 shows the RMS width for a typical detector, as seen at the preamplifieroutput.For this detector, using Eq.(4.2), the measured values for qand I and the gain of the preamplifier circuit, the RMS width expected atthe preamplifier output is 5.7 ± 0.3 mV. The error on the expected width isdominated by the accuracy to which we know the efficiency (number of photoelectronsper MeV) of the detector. Additional contributions to the width,that are not currently accounted for, include fluctuations in the amount ofenergy a gamma-ray actually deposits in the detector (currently, it is assumedthat each photon deposits its full energy) and background due to neutronsthat are scattered off the target and subsequently capture in other surroundingmaterials. An effort to carefully model these effects, using Monte-Carlomethods is currently underway.Using the known proton current (126 µA) and an appropriate Monte-Carlomodel to simulate the transmission of the neutrons through the guide and theexperiment, the number of neutrons expected to capture on the B 4 C target,175

for the peak energy, is about 7.5 × 10 6 n/ms/pulse. Considering the gammaattenuation in the target as well as the solid angle for the detector, the numberof photons into a detector, per time bin is ≃ 4.6×10 4 , so that the relative widthis1 √N≃ 0.5%. Using the corresponding voltage (∼ 1.2 V), one expects to seeout of the preamplifier, the RMS width expected due to neutron countingstatistics is ≃ 5.7 ± 0.6 mV. The error is due to the accuracy to which weknow the detector efficiency (∼ 5%) and the estimate on the accuracy in thenumber of neutrons capturing in the target (∼ 5%).4.6 Instrumental Systematic Effects and FalseAsymmetriesIn this section, we describe measurements performed to study systematic effectsthat may introduce false asymmetries as well as the time required tomeasure these effects (see Section 4.2.3).The first measurement describes the tests for the sensitivity of the VPDs to ACand DC magnetic fields. The second measurement was done to establish thetime required to average electronic noise down to the desired accuracy in theasymmetry. Here we essentially measured the asymmetry due to electronicnoise, which is expected to be zero.This was done without operating anyequipment other than the detector array itself and the DAQ. For the thirdmeasurement the RF spin flipper was operated and data were taken withoutany signal going into the detector array. This was done to search for an additive176

effect, in which an asymmetry may be induced as a result of an addition tothe signal in the VPDs, due to spin flipper correlated electronic pickup. Thefourth measurement looked for a multiplicative effect, a spin flipper correlatedgain change in the VPDs, due to any spin flipper magnetic field leakage. Thiseffect can only be seen by having a signal (light) going into the VPD, which wasaccomplished by using the LEDs in each detector so that the VPD photodiodecurrent was approximately equal to that produced by the scintillation lightexpected during beam-on measurements.VPD Magnetic Field SensitivityVPD gain can change due to magnetic fields interacting with the photoelectrons.This nonlinear effect increases with increasing current between cathodeand anode. As already mentioned in section 4.2.3, if this effect is large, smallfluctuations in the field could cause pulse-to-pulse variations in the detectorsignal and therefore produce false asymmetries. The sensitivity of the VPDsto magnetic fields was measured using both dc and ac fields.An unshielded VPD connected to a preamplifier and a green LED were placedinto a light-tight box which was located in a magnetic field up to 10 G . Theoutput of the preamplifier was monitored with a lock-in amplifier. The VPDwas tested in a 10 G dc field used in the experiment to control the neutronpolarization and suppress Stern-Gerlach steering.The LED was pulsed at90 Hz and produced a 100 mV peak-to-peak signal with various offsets up to1.0 V at the preamplifier output. The tests were performed with the VPD177

in parallel with and perpendicular to the magnetic field direction as well aswith the VPD rotated around its axis of symmetry. In each configuration, thechange in gain for a dc field is only about 1 × 10 −4 /G. No gain dependenceon the voltage offset was observed.For the ac measurement, the LED was held at a constant voltage provided bya battery. The magnetic field was varied according to B sin ωt with B ≃ 15 G.The lock-in amplifier was used to measure first and second order changes inthe gain and was synchronized to the field frequency.To first and secondorder the gain changes were 2 × 10 −5 /G and 1 × 10 −5 /G 2 respectively [45].The Aluminum housing normally placed around the VPD further reduces thecoupling and for a pulse-to-pulse fluctuation of a few mG in the holding fieldthe gain change in the VPD is negligible.False Asymmetry ResultsIn calculating a false asymmetry, one asymmetry was calculated for each timebin and over any valid sequence of eight consecutive macro pulses (see Section3.7) with the correct neutron spin state pattern, as described in section 5.1.For the LED-on tests the asymmetries are calculated the same way as for thebeam on data. For the beam-off, LED-off data the only difference is that thedenominator of Eqn.( 5.2) is set by the expected rate of γ-rays when the beamis on, as well as gain and sampling factors in the DAQ. The expected ADCcount in a single detector during beam on measurements is ≃ 5 × 10 3 . Thisnumber can be estimated from the measured neutron flux and Monte Carlo178

predictions (see section 4.1) [38]. The denominator value used in the asymmetrycalculation is d = 4.8 × 10 7 . Both for the LED-on and LED-off, falseasymmetry measurements, the beam polarization and spin flip efficiency areset to unity.Many data runs contribute to the asymmetry measurements and for each detectorpair in the array a combined mean and standard deviation were calculatedfrom all runs. A total error-weighted average asymmetry is then calculatedfor the entire array. A 6 sigma cut was placed on the data to remove most ofthe cosmic background. For the spin flipper on and off (no LED) asymmetrymeasurements, the time required to achieve a certain accuracy in the asymmetriesis limited by the RMS width in the signal. As described above, theRMS width is set by the Johnson noise Eq.(4.10) and additional noise fromthe detector preamplifier [57] as well as cosmic ray and other background.Without LEDs, the asymmetry can be measured down to the 5 × 10 −9 levelin 3 hours. In 5 hours, the additive false asymmetry with the spin flipper wasmeasured toA noise = (−1 ± 3) × 10 −9 .The individual detector pair asymmetries from the spin flipper runs are shownin Fig. 4.21. The noise observed in the detectors, with the spin flipper running,did not significantly change from the levels shown in Fig. 4.16. If no cuts areapplied to remove the cosmic background, the time required to measure thenoise asymmetry to the above accuracy increases by about a factor of two.With the LEDs turned on, the RMS width of the detectors signals is dominated179

Graph×100.60.50.40.3-7A noise0.20.10-0.1-0.20 5 10 15 20 25Detector pairFigure 4.21: Measured additive false asymmetries with the spin flipper on andLEDs off.by shot noise at the photocathode.If the shot noise is characterized by asingle electron, then the expected average noise density is ≃ 95 fA/ √ Hz at anaverage current of 28 nA out of the VPDs. The multiplicative false asymmetryfor the 24 detector pairs with LED signal is shown in Fig. 4.22.Graph×100.60.50.40.3-6A LED0.20.10-0.1-0.20 5 10 15 20 25Detector PairFigure 4.22: Measured multiplicative false asymmetries with the spin flipperand LEDs on.In 11 hours, the noise asymmetry with the LED signal, for the combined array,180

was measured toA LED = (−8 ± 5) × 10 −9 .The noise in the detector preamplifier, as expected from calculation, predicts arun time estimate of 1 hour to measure the beam-off, LED-off noise asymmetryto 5 × 10 −9 . From the estimate of the shot noise for LED-on measurements,the run time should be 5 hours for an average photo-current of 28 nA . Theperformed measurements show that the required run-times are actually 2 to3 times larger than predicted. However, the average beam-off noise in thepreamplifier, as seen in Fig. 4.16, is ≃ 32 fA/ √ Hz, about 70% higher than expectedand the runtime required with this noise level corresponds to ≃ 3 hours.The same was seen to be true for the LED-on (beam-on) measurement, wherethe noise observed was about 50% larger than expected from the above shotnoise estimate.4.7 SummaryThe NPDGamma CsI(Tl) detector array has been designed and built to operatein current mode and at low noise without introducing instrumental systematiceffects at the 10 −9 level. The prerequisite for a successful current modemeasurement is the suppression of noise levels much below the statistical limit.The noise in the preamplifier due to thermal fluctuations in the circuit componentswas measured to be smaller, by a factor of 70, than the expectedshot noise during measurements with a neutron beam. Since, in current mode181

detection, counting statistics appear as shot noise at the VPD photo-cathode,the accuracy of the asymmetry measurement will therefore be determined bycounting statistics.Any systematic false asymmetries arising due to the operation of the array andspin flipper must be suppressed to have an effect below 5 × 10 −9 . The beamoff,LED-off additive false asymmetry due to spin flipper correlated electronicpickup as well as the LED-on multiplicative false asymmetry due to spin flippercorrelated gain changes in the VPD were measured to this level of accuracywithin a few hours and were consistent with zero.182

Chapter 5Data Analysis and ResultsThe asymmetry measurements performed in March and April of 2004 includedthe following targets: Al, CCl 4 , B 4 C , Cu, and In. In addition, measurementswithout any target were taken to determine the background asymmetry fromexperimental apparatus and shielding alone. As explained in section 3.6, allof these targets are made from materials which can be encountered by theneutron beam during production data taking with the liquid hydrogen targetand may produce a false background asymmetry or dilute the accuracy of thefinal A γvalue. In addition to looking for an up-down asymmetry A γ , thedata was also analyzed for the presence of a possible left-right asymmetryA γ,LR [68]. In the case of a slight left-right misalignment or rotation aboutthe beam, of the detector array, a left-right asymmetry may “leak” into theup-down asymmetry and produce a false asymmetry.183

This chapter describes, in detail, the analysis procedures used and the asymmetryresults obtained, followed by a discussion of the impact on the A γmeasurementin ⃗n + p → d + γ .5.1 Asymmetry DefinitionGiven the vertical polarization direction of the neutrons in the NPDGammaexperiment, the parity violating asymmetry is essentially seen in a differenceof the number of γ-rays going up and down. For the ideal case, described bythe theoretical framework in section 2.4, the γ-ray cross section is proportionalto Y = 1 + A γcos θ, where θ is the angle between the neutron polarizationand the momentum of the emitted photon. A third term is introduced if a leftrightasymmetry exists Y = 1 + A γ cos θ + A γ,LR sin θ. However, as discussedin section 3.8, the basic expression for the γ-rayyield is modified due tolimitations in the properties of the experimental apparatus and interaction ofneutrons with elements other than hydrogen. For the current experimentalsetup, the expression for the yield in a given detector is given by Eqn. 3.17Y d ≡ V d4π [1 + A γg d (θ, φ)P n (E n )ɛ(E n )S(E n )] h(z, E n )f d (⃗x, θ, φ) . (5.1)Where V d is the average signal out of the preamplifier for detector d whichincludes all factors such as overall gain and efficiency (see Chp. 4).The aim of the analysis procedure described here is to measure a single, totalasymmetry for the entire detector array and from all runs. However, because of184

the dependence of beam polarization, spin flip scattering and geometry factoron neutron energy, asymmetries must be calculated first for each time bin andover any valid sequence of eight consecutive macro pulses (see Section 3.7)with the correct neutron spin state pattern, before combining them into atotal asymmetry.A valid 8-step sequence of spin states is defined as ↑↓↓↑↓↑↑↓. The spin flipper isturned on and off on a pulse by pulse basis to produce this sequence of neutronspin states. This particular pattern was chosen to suppress first and secondorder detector gain drifts within the sequence. The necessity for fast neutronspin reversal, used to cancel detector gain non uniformities and slow efficiencychanges, was discussed in section 3.5. In addition, the section also discussesthe need to measure asymmetries simultaneously as a difference between an upand down detector pair in a ring, in order to reduce the effect of pulse-to-pulsebeam fluctuations.5.1.1 The Raw Data AsymmetryA pair of detectors is defined as shown in Fig. 5.1. As shown in Eqn. 2.20,the measured (raw) asymmetry for each detector pair can be extracted byforming a ratio of differences over sums. Then, taking into account the 8-stepsequence, if we let Y d,↑ (t i ) and Y d,↓ (t i ) denote the sum of all four up detectorsignals with the corresponding spin states in a spin sequence and likewise forthe down detector in the pair, the asymmetry for a detector pair and time bin185

can be expressed asA raw,p (t i ) = Y A p,↑(t i ) − Y Bp,↑(t i ) − Y Ap,↓(t i ) + Y Bp,↓(t i )Y Ap,↑(t i ) + Y Bp,↑(t i ) + Y Ap,↓(t i ) + Y Bp,↓(t i ) . (5.2)The average signal in a given detector (V d ) in Eqn. 5.1 may vary from onepulse to the next due to fluctuations in neutron flux. Since the beam polarizationis reversed on a pulse-to-pulse basis this can lead to false asymmetries orasymmetry shifts, if the fluctuations are correlated with the direction of neutronbeam polarization, but will, at least, produce a dilution of the asymmetryerror. In addition, because of efficiency and gain differences, the signal alsovaries between detectors in a pair. To show these effects explicitly, the averagedetector signal is defined to be proportional to the product of the overall detectorgain and the neutron flux as a function of spin state V d = g d f ↑↓ ξ, whereξ is a constant that depends on the target material.As mentioned before, the use of fast spin reversal and simultaneous up-down/ left-right asymmetry measurements reduces these effects. The next sectionshows how A γis extracted from the raw asymmetry and how these systematiceffects enter into the asymmetry and at what level they are likely to contribute.5.1.2 Physics Asymmetry Extraction andCorrection FactorsAside from the correction factors due to beam polarization, spin flip efficiencyand detector-target geometry, which are included in Eqn. 5.1, there are addi-186

BθBB43B5B6A 1¡¡¡¢¡¢¡¢¡¢¡¢ ¡¡ ¡¡¡¢¡¢¡¢¡¢¡¢¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡¡ ¡¡¡¢¡¢¡¢¡¢¡¢¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡Target180 oA2AA34¡¡¡¢¡¢¡¢¡¢¡¢ ¡B2£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£B 1£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡££¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£A6A5Figure 5.1: A ring of detectors and one up-down pair, as seen with beamdirection into the page. ⃗ B is the magnetic holding field defining the directionof the neutron polarization. There are six pairs per ring, starting with the firstas shown and then proceeding in clockwise direction.tional correction factors from the systematic effects just mentioned and dueto the possibility of a slight asymmetry in detector positioning. The latterwill result in slightly different angles with respect to the vertical for the twodetectors in a pair.Figure 5.2 illustrates the four geometrical possibilities that emerge for thesetup used in this experiment. In the application of the geometry factor, θis the angle between the first detector of the pair and the vertical coordinatesystem axis, rather than with respect to the direction of neutron polarization.To simplify the discussion, the situation is depicted in two dimensions, for apoint source and two point detectors forming a pair. In this case the geometryfactor consists only of the cosine and sine dependence and h and f are bothunity.187

n γcos( α)spinupdowndetectornθ=αγθγAcos( α)cos( θ)αcos( α)= cos( θ)nθnθαcos( α)= cos( θ)cos( α)cos( θ)BγγαnFigure 5.2: Simplified illustration of the geometrical dependence for the extractionof the asymmetry from the measurements for a point source and twopoint detectors forming a pair. ⃗n is the direction of neutron polarization and⃗γ is the γ-ray momentum. θ is always the angle of the first detector in thepair (A) w.r.t. the vertical (⃗x) axis (see Fig. 3.35). ⃗n and ⃗γ are unit vectors.Beam Fluctuation and Detector Gain Difference CorrectionAssume, for the moment, that the correction factors E, P n and S are unityand that the detector alignment and rotation is perfect, so that the anglesfor the detectors in the pair are identical.The case of different angles forthe detectors in a pair is briefly treated below. The extraction of the physicsasymmetry would then proceed as follows:The measured detector yield for the two detectors of a pair and for each neutronspin state can be expressed asY A,↑ (t i ) = g A f ↑ (t i )[1 + A UD (⃗n ↑ · ⃗γ A ) + A LR (⃗n ↑ × ⃗γ A ) + P A (t i )]188

Y A,↓ (t i ) = g A f ↓ (t i )[1 + A UD (⃗n ↓ · ⃗γ A ) + A LR (⃗n ↓ × ⃗γ A ) + P A (t i )]Y B,↑ (t i ) = g B f ↑ (t i )[1 + A UD (⃗n ↑ · ⃗γ B ) + A LR (⃗n ↑ × ⃗γ B ) + P B (t i )]Y B,↓ (t i ) = g B f ↓ (t i )[1 + A UD (⃗n ↓ · ⃗γ B ) + A LR (⃗n ↓ × ⃗γ B ) + P B (t i )] .(5.3)The pedestals P A (t i ), P B (t i ) are a function of neutron energy or time of flight(t i ), as explained in section. 5.5, but are assumed to be constant on a pulseto-pulsebasis for a given neutron energy. They are therefore also assumed tobe independent of the neutron spin state, which may not always be true if theinteraction which produced the background is spin dependent. This must bechecked separately in background runs.Figure. 5.2 shows the relation between the inner products for the four differentneutron spin and γ-ray momentum directions:⃗n ↓ · ⃗γ A = −⃗n ↑ · ⃗γ A⃗n ↓ · ⃗γ B = −⃗n ↑ · ⃗γ B⃗n ↑ · ⃗γ B = −⃗n ↑ · ⃗γ AThe same applies to the cross product terms for the left-right asymmetries.Then using Eqn. 5.2 together with Eqns 5.3 one findsA p raw(t i ) =(g A −g B )(f ↑ (t i )−f ↓ (t i ))(g A +g B )(f ↑ (t i )+f ↓ (t i )) + (Ap UD cos θ + A p LR sin θ)1 + (g A−g B )(f ↑ (t i )−f ↓ (t i ))(g A +g B )(f ↑ (t i )+f ↓ (t i )) (Ap UD cos θ + A p LR sin θ)(5.4)189

One can then define the beam fluctuation asymmetry,A f (t i ) ≡ f ↑(t i ) − f ↓ (t i )f ↑ (t i ) + f ↓ (t i )(5.5)and the detector gain asymmetryA p g ≡ g A − g Bg A + g B. (5.6)As seen in Scn. 4.3.4, the detector gain asymmetry is ∼ 10 −2 . The beam fluxasymmetry is established from the monitor data and is seen to be consistentwith zero. The RMS width in the beam asymmetry is ∼ 0.001 for ∼ 120 µAof beam current if each asymmetry is calculated as an integral over the time offlight and over an 8-step sequence. The flux asymmetry is then zero to within10 −3 / √ n ≃ 3σ γ , where (n) is the number of histogrammed 8-step sequences.The RMS width in the gain asymmetry is also ∼ 0.001. Thus the product of thegain and beam flux asymmetries is on the order of 10 −5 / √ n. In addition, boththe up-down and the left-right asymmetries are expected to be very small. Sothe second term in the denominator of Eqn. 5.4 is negligible. In addition, onealso has asymmetries from noise and background. So the complete measuredraw asymmetry is given byA p raw(t i ) = (A p UD + βA p UD,b ) cos θ + (Ap LR + βA p LR,b ) sin θ +A f (t i )A p g + A p noise .(5.7)Where β = Y b /Y s is the background to target-in signal yield ratio. The noise190

asymmetries were measured to be zero with a high precision (see Sec. 4.6).The effect of the background asymmetries were discussed in section 3.6 andthe results are presented later in this chapter.Combing Eqn. 5.7 with the actual geometry factor, integrated over the solidangle of a detector and the volume of the target and including all other correctionfactors in Eqn. 5.1 and solving for A γ ≡ A UD + A LR , we find the physicsasymmetry per time bin for the jth sequence and for a given detector pairA j,praw(t i ) ≡ A j,praw UD(t i ) + A j,praw LR(t i )A j,praw UD(t i ) =A j,praw LR(t i ) =((Aj,pUD + βA j,pUD,b )〈g UD h(t i)f〉 ) E(t i )P n (t i )S(t i )〈h(t i )f〉((Aj,pLR + βA j,pLR,b )〈g h(t LR i)f〉 ) E(t i )P n (t i )S(t i )〈h(t i )f〉=⇒(Aj,pUD(t i ) + βA j,pUD,b (t i) ) 〈G UD (t i )〉 + ( A j,pLR(t i ) + βA j,pLR,b (t i) ) 〈G LR (t i )〉(Aj,praw − A p gA f (t i ) − A p )noise=E(t i )P n (t i )S(t i )(5.8)Here,∫〈|g|h(t i )f〉 ≡∫ π ∫ 2πd 3 x dθ dφ g(θ, φ)f(⃗x, θ, φ)h(z, dz, t i )0 0191

and∫〈h(t i )f〉 ≡∫ π ∫ 2πd 3 x dθ dφ f(⃗x, θ, φ)h(z, dz, t i ).0 0The definition of 〈G(t i )〉 then follows from Eqn. 3.16.The situation in Eqn. 5.8 is actually significantly more complicated, since thebackground has a geometry factor that may significantly differ from the one forthe actual target. The background geometry factor may in fact be impossibleto determine, unless one had good knowledge of the source distribution. This isone of the reasons why one wants to suppress background as much as possible,aside from the obvious reasons such as counting statistics.Detector Pair Angle Difference CorrectionIf the detector array is slightly misaligned with the target or the beam, thenthe angles which the detectors of a pair make with respect to the verticalholding field axis may be different. In the presence of a left-right asymmetry,this can lead to a false up-down asymmetry and, even if there is no leftrightasymmetry, may lead to a perceived shift in the up-down asymmetry.Considering again only the simple two dimensional case, a difference in thedetector angles within a pair can be expressed as a small change in the yieldfor one of the detectors. From Eqn. 5.3 we then haveY A,↑ (t i ) = (1 + A UD cos(θ) + A LR sin(θ))Y A,↓ (t i ) = (1 − A UD cos(θ) − A LR sin(θ))Y B,↑ (t i ) = (1 + A UD cos(θ + δ) + A LR sin(θ + δ))192

Y A,↓ (t i ) = (1 − A UD cos(θ + δ) − A LR sin(θ + δ))So that the raw asymmetry is approximately equal to(A raw ≃ A UD cos(θ) − δ )2 sin(θ)=(A UD + δ 2 A LR)cos(θ) ++ A LR(sin(θ) + δ 2 cos(θ) )(A LR − δ 2 A UD)sin(θ) .The error on the up-down asymmetry due to the mixing with the left-rightasymmetry is δσ ALR /2.So if the detector array is aligned such that δ ≃20 m radians then the left-right asymmetry will contribute 1% of its error andwill shift the up-down asymmetry by 1% of its value.During the 2004 commissioning run, the detector array alignment scheme wastested but the analysis of this data has not yet been completed. The alignmentscheme attempts to use the signals in the detectors to align it with respect tothe center of the target capture distribution. Before data taking began, thearray and target were centered on the center of the beam guide and surveyedto within a few millimeter in each direction.However, the true center ofthe capture distribution may vary from this slightly. A 5 mm misalignmentof the detector array in each direction produces a 5% error and shift in theasymmetry as measured with a corner detector pair (a pair with a nominalangle θ = 45 deg at its center).For the analysis procedure described here, the geometry factor for both detec-193

tors in each pair are taken to be identical, which is a good approximation towithin a few percent, as long as the target and detector array are reasonablywell centered on the beam.Target Out Background LevelsAs indicated in Eqn. 5.8, the background due to neutron interaction withanything other than the intended target may produce false asymmetries andwill dilute the accuracy to which it is measured. As described in section 3.6,the way in which this happens may be expressed asA s = A s+b(Y s + Y b ) − A b Y bY s= A s+b (1 + β) − A b β . (5.9)σ As =√(1 + β) 2 σ 2 A s+b+ β 2 σ 2 A b. (5.10)Where Y s and Y b is the γ-ray yield from target in and target out runs respectively.This is correct, regardless of the fact that the signal from background has adifferent geometry factor. The detector-target-out geometry is important inthe extraction of the actual background asymmetry. However, an additionaldifficulty is that the background which is measured in separate (target-out)runs may in fact be different from the background with the target installed. Anexample of this is the scattering of neutrons from the target material. In thesecases, the level of background and its shape must be estimated from modelsand analytical calculations. A list of neutron interactions that may pose a194

Relative BackgroundB 4 C β ≤ 17%Al β ≤ 15%In β ≤ 11%CCl 4 β ≤ 8%Cu β ≤ 7%Table 5.1: 2004 commissioning materials and background contributions. Ineach case the maximum value is stated for the detector with the largest backgroundsignal.problem is provided in section 5.5. Some of these interactions are described inmore detail, but a full model for backgrounds is not yet in place.The target-out relative background levels for the materials measured duringthe 2004 commissioning run are listed in table 5.1.5.2 Data CutsAs discussed above, beam fluctuations may induce false asymmetries or asymmetryshifts. This effect was suppressed using two cuts. First, the protonbeam current was monitored and a given 8-step sequence of pulses was discardedwhen a large change in current occurred within that sequence, or whenthe current fell below 80 µA. Large changes in proton current over an 8-stepsequence occur, for example, when the beam goes down completely and duringramp up after the beam comes back on.Second, beam monitor 1 was used to place a cut on the data. This was donebefore any asymmetry was extracted and independent of the signals seen in195

the detectors. To do this, the signal seen for each pulse of an 8-step sequencewas integrated over the entire time of flight range, producing 8 integrated“signal intensity” values per sequence. Then, an average pulse intensity wascalculated for the entire sequence.∫ 40S M,p ≡ dt S M,p (t)0S M ≡ 1 8∑S M,p8p=1The fractional deviation of the individual pulse intensities from the average inthe sequence is then monitored.∆S M,p =∣ 1 − S ∣ ∣∣∣∣M,pS MSequences in which one or more of the pulses deviated from the average by morethan 2% were discarded. The same calculation was performed for each detectorsignal and the fractional pulse deviations for each were histogrammed over theduration of a run. This procedure is more effective in suppressing small scalefluctuations, which can not be detected by the proton current signal, becauseit is discretized at the 1 µA level. Figures 5.3 and 5.4 show the histogrammedfractional deviations for a typical detector when no cuts are applied and whenvarying levels of maximum deviations are applied.If the sequence cuts are based on the monitor signal alone, less than 10% of the196

MBC_CutMBC_Cut MBC3_CutsMBC_CutEntries Entries 1068480 Entries 10684801068480Mean Entries Mean 0.002573 Mean 10684800.0025730.002573RMS Mean RMS 0.001966 RMS 0.0025520.0019660.001966RMS 0.001972MBC_CutMBC_Cut MBC3_CutsMBC_CutEntries Entries 1068480 Entries 10684801068480Mean Entries Mean 0.002573 Mean 10684800.0025730.002573RMS Mean RMS 0.001966 RMS 0.0025730.0019660.001966RMS 0.001966MBC_CutEntries 1068480Mean 0.002552RMS 0.001972MBC_CutEntries 1068480Mean 0.002552RMS 0.001972Asymmetry Fluctuations510Number of Pulses4103102101010 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Fractional Dev. From Sequence MeanAsymmetry Fluctuations510Number of Pulses4103102101010 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16Fractional Dev. From Sequence MeanFigure 5.3: Histogrammed detector pulse fractional deviation (see text), withvarying constraints on monitor signal fluctuations. The second figure showsthe same data zoomed in. The red histogram shows the result when no cutsare applied. The black histogram shows the result of a proton current cutonly. The blue and green histograms show proton beam current and monitorcuts together with the maximum fractional deviation set to 10% and 2%respectively.data are lost due to the cut. If the cuts are based on the deviations observedin the detectors, the data losses exceed 10%. In addition, rejection of neutronpulses based on monitor data is an unbiased cut since it does not distort thedistribution of signals seen in the detectors. Cutting on the detector signal,on the other hand, is a biased cut, since it cuts into the Gaussian distributionof pulses, as seen in Fig. 5.4. For this reason, all sequence cuts were based onthe monitor and proton current fluctuations.197

MBC_CutMBC_Cut MBC3_CutsMBC_CutEntries Entries 1068480 Entries 10684801068480Mean Entries Mean 0.002573 Mean 10684800.0025730.002573RMS Mean RMS 0.001966 RMS 0.0025730.0019660.001966RMS 0.001966MBC_CutEntries 1068480Mean 0.002552RMS 0.001972Asymmetry Fluctuations510Number of Pulses4103102101010 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04Fractional Dev. From Sequence MeanFigure 5.4: Same data as in Fig. 5.3. The magenta colored histogram showsthe results when the sequences are cut for a fractional pulse deviation > 2%seen in the detectors. The dashed line is a Gaussian.5.3 The Combined AsymmetryStatistical TreatmentFor each sequence, which was not discarded due to the cuts described aboveand which had the correct 8-step pattern, a single raw asymmetry was calculatedfor each detector pair and for each time bin, according to Eqn. 5.2. Forraw asymmetry per sequence and pair, a corresponding pair physics asymmetrywas extracted according to Eqn. 5.8. In each case, the raw asymmetry wasweighted by the corresponding correction factors for the particular neutronenergy, spin flip efficiency, as well as the geometry factor and degree of spinflip scattering (see Sec. 5.5). The noise asymmetry and the product of thedetector gain and beam monitor asymmetries are negligible and were set tozero. All sequence asymmetries, extracted in this way, are then statisticallyequal and can be further combined by statistical means.To find the combined pair physics asymmetry for an 8-step sequence an error198

weighted average is formed over the time of flight range from 12 to 34 msfor which the neutron signal was appreciably higher than the pedestal (seeFig. 3.14).The beam chopper began closing at approximately 34 ms (seeSec. 3.2.3). This time of flight range corresponds to 55 time bins (30 through84) out of the 100 sampled time bins per macro pulse (see Sec. 3.7.1).A j γ,p =84∑i=3084∑i=30A j γ,p(t i )σ j2γ,p(t i )1σ j2γ,p(t i )(5.11)The sequence physics asymmetry for the entire array is then formed by combiningthe 24 pair asymmetries in a second error weighted average.A j γ =24∑p=124∑i=1A j γ,pσ j2γ,p1σ j2γ,p± √84∑i=3011σ j2γ,p(t i )(5.12)This forms a complete asymmetry result for the entire array for a single 8-step sequence. All measured sequence asymmetries for a target are then histogrammed.In addition, the asymmetries for each pair were histogrammedseparately as well. This was done to highlight the geometry dependence of theasymmetries when measured with respect to the pair angle (see Sec. 5.4).The individual time bin values had to be combined in an error weighted averagebecause of the variation in the signal level from one time bin to another.The same thing is true for the detector pair asymmetries, since gain and efficiencyvariations as well as geometry differences between detector pairs leadto different signal levels and asymmetry values.199

5.4 Asymmetry ResultsThe known asymmetry in CCl 4 was used to verify that a nonzero asymmetrycan, in fact, be measured with this experimental setup. The CCl 4 asymmetrywas also used to verify the geometrical dependence of the pair asymmetries.For this purpose, all 24 pair asymmetries, extracted from the histogrammed8-step sequence asymmetries from all data obtained with that target, wasmultiplied by its mean geometry factor and plotted versus its correspondingmean error. The resulting graph is shown in Fig. 5.5. The fit function used toextract the total array asymmetry is A UD cos θ + A LR sin θ.If one expects tosin(θ)cos(θ) + A LRA UD0.0030.0020.001-0-0.001-0.002-0.003-0.004-2x10Ring 1 Ring 2 Ring 3 Ring 4-0.0050 2 4 6 8 10 12 14 16 18 20 22Pair Angle + ring*2π (rad)Figure 5.5: CCl 4 asymmetries for each pair, plotted versus angle of the firstdetector in the pair w.r.t the positive x-axis (see Sec. 3.6.3). The total arrayasymmetry is extracted from the fit. Pairs 22 and 23 had noisy detectors atthe time this data set was taken.see a nonzero result for both the Up-Down and Left-Right asymmetries in thesame target, then this is, in fact, the only way to extract the correct valuesfor both.200

Figure 5.6 shows the asymmetry histograms for the entire array, for both thethin target and thick target. The complete asymmetry results for the targetsused in the 2004 commissioning run are listed in table 5.2.More detailedstatistical information on the data and the results are listed in appendix C.Number of 8-Step Sequences410 Thick TargetTot_AsyEntries 664484310210101Mean -2.694e-07RMS 0.0003369410 Tot_AsyThin Target310210101Entries 584570Mean1.665e-07RMS 0.0004083-0.002 -0.001 0 0.001 0.002Asymmetry-0.002 -0.001 0 0.001 0.002AsymmetryFigure 5.6: Total Up-Down aluminum asymmetries from the thick and thintargets. The Gaussian fit in each case gives (3 ± 4) × 10 −7 for the thick targetand (−2 ± 5) × 10 −7 for the thin target.A γUp-Down Left-RightAl (−0.02 ± 3) × 10 −7 (−2 ± 3) × 10 −7CCl 4 (−19 ± 2) × 10 −6 (−1 ± 2) × 10 −6B 4 C (−1 ± 2) × 10 −6 (−5 ± 3) × 10 −6Cu (−1 ± 3) × 10 −6 (0.3 ± 3) × 10 −6In (−3 ± 2) × 10 −6 (3 ± 3) × 10 −6Table 5.2: Up-Down and Left-Right asymmetries for the target materials usedduring the 2004 commissioning run. Stated errors are statistical only.201

5.4.1 ErrorsIn the extraction of the final asymmetry and its error from the data, we distinguishbetween statistical and systematic errors. A systematic error producesa signal in the detector that is coherent with the state of the neutron spin orotherwise non-random. A source of statistical error, on the other hand, producesa detector signal that is not correlated with the neutron spin directionand is random in nature; for example fluctuations in the number of detectedgamma rays due to counting statistics or drifts in amplifier offsets.The error on the physics asymmetry has the same form as A γitself,σ j γ,p(t i ) =σ j raw,p(t i )|〈G(t i )〉|E(t i )P n (t i )S(t i )and the error on the raw asymmetry is proportional to the square root of thedenominator, based on the assumption that the signal fluctuations around themean, in each time bin, are Poisson distributed.σ j raw,p(t i ) ∝1√YU,↑ (t i ) + Y D,↑ (t i ) + Y U,↓ (t i ) + Y D,↓ (t i )The error in the combined pair asymmetry is given byσ j γ,p =√84∑i=3011σ j2γ,p(t i ).The raw asymmetry error could, in principle, be calculated for each time bin,202

using Eqn. 4.2, without the need to assume a particular distribution. However,this is difficult for targets in which neutron capture reactions may producemany γ-rays with different energies due to the resulting variation in the VPDshot noise RMS width with γ-rayenergy. On the other hand, it is knownthat the number of photoelectrons produced at a photocathode is Poissondistributed (see for example [59]) and that, as laid out in Chp. 4, there is alinear relationship between the signal out of the detector preamplifier and theaverage number of γ-rays that entered the detector. In this regard, the benefitof the error weighted average is that the proportionality constant divides out.The final statistical asymmetry errors are taken from the distribution of sequencevalues σγ/N 2 = (E(A 2 γ) − E(A γ ) 2 )/N, with N histogrammed 8-stepsequence asymmetries. Together with the errors from background and noise,the statistical error in the final asymmetry is given by Eqn. 5.10. If the 8-stepsequence values had been histogrammed before applying the neutron polarizationand geometry factors and combining them in error weighted averages,their variations would have led to RMS widths and errors that would not berelated to the random statistical distribution of asymmetries, but rather tothe systematic variations just mentioned.The systematic errors enter in several ways. Any non-random effect such asthose introduced by the correction factors |〈G(t i ), 〉|, E(t i ), P n (t i ), S(t i ) areconsidered systematic errors. These enter asσ γ,Sys = A γ√ (σPnP n)+( ) ( )σE σG+ +E G( ) σS.S203

The errors on the beam polarization and spin flip efficiency were calculatedin Chp. 3, to be 4% and 10% respectively. The error on the geometry factoris estimated to be less than 1% from variations observed in the values whenvarying the step size in the Monte Carlo, simulating γ-ray interaction in thedetectors. The error on the spin flip scattering is estimated to be on the orderof a few percent. Since the systematic errors are scaled by the asymmetry, theircontribution to the overall error on the asymmetry is negligible compared tothe statistical, except for the case of the CCl 4target, which has a non-zeroasymmetry. For CCl 4 , the absolute systematic error is ∼ 2.3 × 10 −6 . So theCCl 4 Up-Down physics asymmetry and its total error is (−19 ± 3) × 10 −6 .In addition to the errors discussed here, possible false asymmetries due tosystematic effects would introduce an error in the measured asymmetry itself.Some instrumental systematic effects were discussed in other parts of this document(see Chapters 3 and 4) and shown to be negligible. Neutron interactioninduced systematic effects are discussed in the next section.5.5 Neutron Interaction InducedSystematic EffectsIt is important to isolate and study experimentally potential sources of systematicerrors. The aim of the 2004 commissioning run was, in part, to searchfor false asymmetries from activation of components in the apparatus due tothe capture of polarized neutrons and from possible false asymmetries from204

any material, other than LH 2 , used in the experiment. This section summarizesthe possible neutron interactions that may produce false asymmetries orintroduce effects that will dilute the ⃗n + p → d + γ asymmetry and its error.Mott-Schwinger scattering is treated in more detail to provide a specific exampleand background seen in the detectors and monitors is shown and brieflydiscussed.Systematic errors can be classified according to whether they are instrumentalin origin and are present whether or not neutrons are being detected orarise from an interaction of the neutron spin other than the directional γ-asymmetry in the ⃗n + p → d + γ reaction. One possible example has alreadybeen mentioned, the parity-allowed left-right asymmetry ⃗s n · ( ⃗ k n × ⃗ k γ ).Physical effects that arise from an interaction of the neutron spin anywherein the experiment can be arrived at by considering all contributing scalarcombinations of the dynamical variables in the experiment.In this experiment, the dynamical variables of interest are the neutron momentum⃗ k n and spin ⃗s n and the γ-ray momentum ⃗ k γ and helicity ⃗s γ . In orderto produce a false asymmetry, an interaction must occur after the spin is reversedby the RF spin flipper, otherwise the effect of the interaction would beaveraged out by the eight-step reversal sequence.The interaction must involve the inner product between the neutron spin andsome other vector consisting of vectors and scalars from the initial and finalstates. At least one quantity from the final state must deposit its energy in thedetector. Most of the the amplitudes which satisfy these criteria are listed in205

table 5.3. The size for false asymmetries due to any of these interactions haspreviously been estimated [21] and some of the results are shown in table 5.4.5.5.1 Beam Depolarization from Incoherent ScatteringOne of the correction factors in the extraction of the physics asymmetry fromthe data is provided to take into account the possible depolarization of neutronsin the beam, as they move through the target. This happens via incoherent,spin flip, scattering of the neutrons from the nuclei in the target.A first,rough, estimate of the size of the effect can be arrived at by comparing thesize of the incoherent and total cross-section for a given material. Table 5.5lists the relative size of the incoherent cross-sections to the total cross-sectionsfor the target materials used during the 2004 commissioning run and the correspondingaverage correction factor 〈S(t i )〉.The only targets in which the correction factor was applied were CCl 4 and Cu.The amount of spin flip scattering is actually neutron energy dependent andin the asymmetry extraction, it is therefore applied on a time bin by timebin basis. Figure 5.7 shows the results of a Monte Carlo calculation for thecorrection factor as a function of time of flight. The same factor was used forboth the CCl 4and Cu target.206

0.980.960.94S(tof)0.920.90.8810 15 20 25 30Neutron Time of Flight (ms)Figure 5.7: Monte Carlo Calculation of the depolarization factor from spin flipscattering in the targets [69].5.5.2 Example: Mott-Schwinger ScatteringOne of the systematic effects that we encounter in the NPDGamma experimentis the steering of the neutron beam, either left-right or up-down. Given thefinite accuracy to which the detector can be aligned with the vertical andhorizontal, if the beam moves left-right and this movement is correlated withthe neutron spin, then this effect would produce a false asymmetry.One possible interaction which could induce such a left-right beam shift whichis correlated with the neutron spin is introduced by the interaction of a neutronwith the effective magnetic field as it moves through the Coulomb field of anucleus. The field the neutron sees is given by B ⃗ = ⃗v × E, ⃗ where ⃗v is theneutron velocity and E ⃗ is the Coulomb field of the nucleus. This field exerts aforce on the magnetic moment of the neutron. The energy of this interactionis ⃗µ · ⃗B = ⃗µ · (⃗v × E ⃗ ) . Thus, a neutron which is polarized in the spin updirection will be steered to the right, if it passes the nucleus on the right orto the left, if it passes the nucleus on the left and visa-versa if it is polarized207

spin down.This type of interaction is called Mott-Schwinger scattering [70]. This is aspin-orbit interaction and the Hamiltonian associated with it may be writtenasH ′ = b m( 1er)∂V e⃗L ·∂r⃗S ≡ − 1 m V (r)⃗ L · ⃗S (5.13)where m is the reduced mass of the neutron-hydrogen system and b = g Nα¯hc/Mc 2≃ −2.9 × 10 −3 fm. M is the neutron mass and g N= −1.91316 is the neutronmagnetic moment in units of e¯h/2Mc, .We are interested in the analyzing power of a given target element (parahydrogenand the other materials found in the beam) for a polarized beam ofneutrons, where the direction of polarization is perpendicular to the scatteringplane.A MS =dσ− dσdΩ + dΩ −dσ+ dσdΩ + dΩ −(5.14)Where θ is the angle between the incident ⃗p i and final ⃗p fmomentum of ascattered neutron.The differential cross-sections can be calculated from the S-matrix, which, inturn, is calculated from the scattering amplitudes found in the asymptotic formof the scattering solution to the Schrödinger equation. In the most generalcase, the scattering potential in the Schrödinger equation must contain thespin independent strong interaction, the spin dependent strong interactionand the spin dependent electromagnetic interaction described by the potentialin Eqn. 5.13. For the low energy of the neutrons in this experiment, the spin-208

dependent strong interaction is negligible for scattering from a single atom,but must be taken into account for scattering from molecular compounds, suchas para-hydrogen (H 2 ) and others [71]. The electromagnetic spin-orbit termis a perturbation to the spin independent strong interaction and the potentialthat enters the Schrödinger equation consists of two parts U(r) = V s (r) + H ′ .For a spinless target, the asymmetry in Eqn. 5.14 may be expressed in termsof the scattering amplitudes for spin-independent scattering g(θ) and spindependent scattering h(θ). These are related to the S-matrix as S = g(θ) +ih(θ)⃗σ · ˆn [72], where ˆn = (− sin φ, cos φ, 0) is normal to the scattering planeand points in the direction of ⃗p i ×⃗p f , and ⃗σ is the spin of the incident neutron.The scattering amplitudes areg(θ) = 12ikh(θ) = − sin θik∑(2j + 1)(e 2iδ lj− 1)P l (cos θ)jl∑(e 2iδ l+− e 2iδ dl−)ld cos θ P l(cos θ)and the differential cross-section isdσ= |g(θ)| 2 + |h(θ)| 2 + i (g ∗ (θ)h(θ) − g(θ)h ∗ (θ)) ( PdΩ ⃗ o = ±ˆn) · ˆn±for a transversely polarized beam.In terms of the scattering amplitudes, the analyzing power becomesA MS(θ) = −2Im[g(θ)h∗ (θ)]|g(θ)| 2 + |h(θ)| 2 (5.15)209

In general, the analyzing power then has to be computed for all interactions,as stated above.The spin independent terms only contribute to g(θ) andthe spin dependent terms only contribute to h(θ). It can be seen from h(θ)that interactions between the incident neutron and the target, for which thedifferential cross-section is different for the partial waves (j, l) = (l + 1/2, l)and (j, l) = (l − 1/2, l) will produce a non-vanishing analyzing power. For lowenergy neutrons the phase shifts δ l+ and δ l− are small and the spin dependentscattering amplitude can be written ash(θ) = − 2 sin θk∑d(δ l+ − δ l− )ld cos θ P l(cos θ) .The phase shifts are then expressed in terms of the scattering potential andthe radial eigenfunctions of the Schrödinger equation:sin δ lj = sin δ NNlj− 2mk¯h 2∫ ∞0r 2 drj l (kr)∆V jl u (+)jl (kr)where⎜⎝∆V jl ≡ 〈jl|H ′ |jl〉 = − 1 m V (r) ⎛l, j = l + 1 2−(l + 1), j = l − 1 2⎞⎟⎠and sin δ NNljis the phase shift due to scattering from the strong interactionV s (r). It may be shown that for low energy particles, scattering from a sphericallysymmetric potential, the evaluation of h(θ) for the electromagnetic contributiongives [71]h em (θ) = 2 cot( θ2) (r 2 dV b(r)dr)( ) θ ∫ ∞− 8π cot r 2 j 0 (qr)drρ b (r) , (5.16)2r=00+210

where ρ b (r) is the charge density corresponding to V b (r) ≡ bV e e(r). The lowerlimit of the integration (0+) is meant to indicate that the origin is excluded.For scattering from an atom, the strong interaction only contributes to g(θ).To estimate this contribution one can use the experimentally know scatteringlength. For the hydrogen atom the amplitude, when averaged over the spin ofthe proton, evaluates to〈g(θ)〉 = 6.28 + i 4(kσin4π + 649k )(fm)and〈|g(θ)| 2 〉 = 162 (fm 2 ) .The experimentally known singlet and triplet scattering lengths are a( 1 S 0 ) =−23.748 fm and a( 3 S 1 ) = 5.423 fm. The inelastic portion corresponding tokσ in4πis ignored in the second term, because it only becomes comparable to theelastic part for neutrons with energies of about 3 × 10 −4 meV or smaller.The classical calculation, done by Schwinger [70], ignores the screening effectdue to the electron cloud. In this case Eqn. 5.16 evaluates to h em (θ) =2b cot(θ/2) and the analyzing power is( θA MS(θ) = −b cot2) kσin4π+ 649k324.Contributions from g em (θ) in the numerator and |h(θ)| 2 in the denominatorare negligible to first order.211

With screening one findsh em (θ) = 2b cot( θ2) (ka 0sin(( 2θ 2 + ka0 sin2)) ( )) 2θ(21 + ( ka 0sin ( )) ) 2 2, (5.17)θ2where a 0 is the Bohr radius. For the energies in this experiment (0 to 15meV) the momenta of interest are 0 ≤ ka 0≤ 1.6 and in this limit the amplitudereduces to h em (θ) ≃ b(ka 0) 2 θ. In general, the analyzing power involvingEqn. 5.17 can be calculated numerically.Mott-Schwinger Scattering from Para-HydrogenIn contrast to the case of the hydrogen atom, for para-hydrogen the value ofthe spin dependent scattering amplitude now has two contributions h(θ) =h em (θ) + h s (θ). The only difference in h em (θ) between the atomic case andthe para-hydrogen case is the charge density in Eqn. 5.16. A suitable chargedensity can be derived from wavefunctions provided in [73].The detailedderivation of the scattering amplitudes is show in [71]. Only the results arestated here.The electromagnetic contribution to the spin dependent scattering amplitudeis⎛⎞( θh em (θ) = −8πb cot b2)j 0(ka)⎜2π ⎝ 1 − 1( ) 2 ⎟(1 + ∆) 1 + k2 a 2 ⎠04212

+ 8πb cot( θ2)( √ )K a2 a 4 + k2 a 200 b∆( ) (2πK 2a2 1 + k2 a 2 0a 0 4)(1 + ∆),(5.18)where 2a ≃ 0.74Ȧ is the assumed, rigid, distance between the two protons.∆ is the overlap integral [73]∫∆ =(u(|⃗r 1− ⃗a|)u(|⃗r 2+ ⃗a|)) d 3 randu(r) = √ 1 e −r/a 0 .πa20The spin dependent strong force contributesh s (θ) = − 16mλ (LS 2(ka) 2 )θµ 3 3, (5.19)where µ = 760MeV/¯hc is the range of the strong spin-orbit force, which isset by the mass of the ρ-meson, the dominant piece for this interaction, andλ LS= 1577MeV arises from the ⃗ L · ⃗S term. For the strong central force, onefindsandg s (θ) =|g s (θ)| 2 = 4( a( 1 S 0 ) + 3a( 3 S 1 )4) 2j 2 (ka) (5.20)04k ( a( 1 S 0 ) + 3a( 3 ) 2S 1 ) ∫ 2kadx sin2 (x)2(ka) 2 40 x. (5.21)Using these results one can use numerical methods to calculate the analyzingpower for para-hydrogen.213

5.5.3 Observed BackgroundOne of the uses of the chopper (see Sec 3.2.3) is that it allows one to usethe time of flight region following the chopper cut-off (Fig. 3.12) to study thedetector and beam monitor backgrounds. This is useful for studying backgroundsfrom signals with some time delay. As is the case, for example, inthe capture of neutrons on some material and the subsequent beta decay ofthe resulting unstable nuclei. Figure 5.8 shows the last 3.6 ms of the time offlight spectrum for a typical detector and run. The fit shows a decay rate of150 µsec, which would convert to a half-life of about 5 ms. However, there isno material in the beam that would produce unstable nuclei with this half-life.0.225Datab+cxFit = a+eSignal (V)0.220.215a = 0.178 ± 0.002b = 2.5 ± 0.25c = -0.15 ± 0.0082χdof=6.360.210.20536 36.5 37 37.5 38 38.5 39 39.5 40Neutron Time of Flight (ms)Figure 5.8: The last 3.6 ms of the time of flight spectrum in a detector.The other possibility would be that the observed decay is a result of the delayin time a neutron spends diffusing around in the target before capture occurs,but this time constant is on the order of a few microseconds. In the absenceof any possible neutron interaction that would induce this phenomenon, the214

only other possible explanation is that it is introduced in the electronics. Onepossibility is the preamplifier slew rate. This is the rate at which the amplifieroutput signal can change and may be on the order of a few milliseconds. Thisexplanation seems especially appealing, because the decay is observed after arapid decrease in signal amplitude (the chopper cut-off). This can be testedon the bench, but has not yet been done.Figure 5.9 shows the monitor background after the chopper cut-off. In thiscase the situation is more complicated since there seems to be some neutronleakage, as suggested by the appearance of a secondary (much smaller) pulsewith a time of flight dependence.0.010.0098Signal (V)0.00960.00940.00920.00934 35 36 37 38 39 40Neutron Time of Flight (ms)Figure 5.9: The last 6 ms of the time of flight spectrum in the second beammonitor.Both the detector and monitor background signals require further study. Thisis true for our basic understanding of the processes within the experiment ingeneral, but good knowledge of the background in the monitors is required tocalculate the spin flipper efficiency and beam polarization to a higher accuracy.For the measurement of the asymmetry, however, the knowledge of the back-215

ground is not crucial as long as it is independent of the neutron spin, becausethe pedestals get subtracted out in the 8-step sequence. The estimated smallsize of the false asymmetry due to the various interactions listed in table 5.3suggests that the backgrounds should be largely independent of spin, at theaccuracy to which the asymmetries are calculated here.5.6 Background in the Hydrogen SignalDuring the production data runs with the hydrogen target, the signals seenfrom Al, Cu, In, and B 4 Cbecome background to the hydrogen signal. Todetermine the level of background from each of these materials and assess theaccuracy to which their asymmetries have to be measured, an estimate has tobe made about the relative yield expected for the target-in and target-out runs.Table A.1 in section A.2.3 lists the results of a Monte Carlo simulation for theexpected neutron flux out of the guide and on the target, with attenuation dueto the spin filter and 10 cm beam collimation taken into account. Figure. 5.10shows the neutron flux out of the guide from the Monte Carlo, versus neutronenergy and time of flight [69].Figure 5.11 shows the signal amplitude from a run with the thin aluminumtarget (see Sec 3.6.1). The thin aluminum target was about 5 times thickerthan the amount of aluminum that will be in the beam with the hydrogentarget installed. In Fig. 5.12, the expected signal in one detector, with theγ-ray rate obtained from the Monte Carlo, is compared with the aluminumsignal scaled by a factor of 0.2 for thickness.216

66FP12 Monte Carlo neutron flux out of the guideneutrons/ms/pulsex 10302520151050 10 20 30 40 50 60energy (meV)FP12 Monte Carlo neutron flux out of the guideneutrons/ms/pulsex 10302520151055 10 15 20 25 30 35 40time of flight (ms)Figure 5.10: Monte Carlo Calculation of the neutron flux out of the guide at100 µA proton current.If both the aluminum signal and the H 2 Monte Carlo signal are integrated between10 and 34 ms time of flight, then the background to signal for aluminumis ∼ 6%. The same method is used for the other materials and the results arelisted in table 5.6. The thickness of the material we expect to have in thebeam, as a fraction of the thickness actually used during the commissioningrun is also shown.217

1.41.2γ-signal (V)10.80.60.40.200 5 10 15 20 25 30 35 40Neutron Time of Flight (ms)Figure 5.11: Detector 0 (first detector, first ring) averaged signal from onethin aluminum target run. Error bars are statistical (RMS width) only. Therun was averaged over 1 × 10 4 macro-pulses.Reaction Observable/ Effectinteraction1 ⃗n + p → ⃗n + p np elastic scattering left-right beam shift2 ⃗n + p → n + p np inelastic scattering beam depolarization3 ⃗n + p → d + γ np capture⃗s n · ⃗k γup-down asymmetry⃗s n · ( ⃗ k n × ⃗ k γ )left-right asymmetry⃗s n · ⃗s γγ circular polarization4 ⃗n → p + e − + ¯ν neutron beta decay⃗s n · ⃗k e −asymmetry from betas⃗s n · ⃗s γγ circular polarization5 ⃗n + p → t + γ np capture⃗s n · ⃗k γup-down asymmetry⃗s n · ( ⃗ k n × ⃗ k γ )left-right asymmetry⃗s n · ⃗s γγ circular polarization6 ⃗n + X → ⃗n + X Mott-Schwinger scattering⃗s n · (⃗v n × E ⃗ X )left-right beam shift7 Stern-Gerlach steering⃗s n · ∇Bup-down beam shift8 spin rotation⃗s n · ⃗k nup-down asymmetryTable 5.3: Neutron interaction induced systematic effects in ⃗n + p → d + γ.218

Observable/Estimated falseinteractionasymmetry upp. lim.np elastic scattering≤ 2 × 10 −10left-right asymmetry ⃗s n · ( ⃗ k n × ⃗ k γ ) ≤ 1 × 10 −10neutron beta decay (betas)≤ 7 × 10 −13γ circular polarization (Bremsstrahlung) ≤ 3 × 10 −11beta decay in pol. Nuclei≤ 6 × 10 −10Mott-Schwinger scattering ≤ 1 × 10 −10 at 2 meVStern-Gerlach steering≤ 1 × 10 −10Table 5.4: Estimated neutron interaction induced false asymmetries in⃗n + p → d + γ.2σ inc3σ tot〈S(t i )〉Al 3 × 10 −3 1Cu 2 × 10 −2 0.95CCl 4 7 × 10 −2 0.95In 2 × 10 −3 1B 4 C 5 × 10 −4 1Table 5.5: Spin flip scattering estimate and asymmetry correction factors forthe 2004 commissioning targets.Background in Hydrogent H2 /t c LevelB 4 C 2% β ≤ 0.7%Al 20% β ≤ 6%In 0.6% β ≤ 0.4%Cu 2% β ≤ 2%Table 5.6: Expected background levels in the hydrogen signal, from target materialsused during the 2004 commissioning run. t H2 is the material thicknessexpected with the hydrogen data and t c is the thickness used in the run.219

5Al BackgroundSignalMC H 24γ-Signal (V)32100 5 10 15 20 25 30 35 40Neutron Time of FlightFigure 5.12: A Monte Carlo calculation of the signal in one detector as expectedwith the H 2 target installed, compared with the aluminum backgroundas measured during the 2004 commissioning run. Data are scaled to accountfor actual Al material thickness in the beam with the H 2 target. Both signalscorrespond to a 100 µA proton beam.220

Chapter 6Summary and ConclusionsThe aim of this thesis is three fold. It provides a thorough explanantion ofthe NPDGamma experiment and the measurement it intends to make.Itexplaines, in some detail, the theory behind the measurement and is meantto highlight the relation to important questions such as QCD properties atlow energy and the predictions of the Standard Model regarding the weakinteraction between nucleons. And it reports on the measurements performedduring the 2004 commissioning run, the results and their significance for the⃗n + p → d + γ measurement.Based on the 1980, DDH paper the weak parity-violating nucleon-nucleoninteraction can be described by a meson-exchange potential involving sevenweak meson-nucleon coupling constants [6]. The weak interaction changes theparity and isospin (∆I = 0, 1, 2) of the nucleon-nucleon pair and perturbativelyintroduces parity violating admixtures in nuclear wave functions. The221

study of the hadronic weak interaction is of great relevance for low energy,non-perturbative QCD. The hadronic weak couplings probe short range correlationsbetween quarks because the quark-quark weak interaction occurs whenthe distance between quarks is ≤ 2 × 10 −3 fm. The NPDGamma experimentdetermines the long range pion-nucleon coupling f π by measuring the directionalγ-ray asymmetry in the capture of cold polarized neutrons on protons inthe reaction ⃗n+p → d+γ. The asymmetry has a predicted size of 5×10 −8 [6].The the pion portion of the weak meson-nucleon exchange potential was derivedin chapter 2. The electro-magnetic transitions in ⃗n + p → d + γ reactionare evaluated to the point of reduced matrix elements, to extract the angularform of its differential cross-section and show the relationship between theweak pion exchange potential and the measured asymmetry. A connection ismade, between the effective field theory and the standard model, by evaluatingthe S-matrix elements of the standard model weak hamiltonian to secondorder. In this context, a set of tree-level integrals are derived to show the possibilityof additional dynamical terms from strong interaction effects at lowenergy by including quark-antiquark pair creation and annihilation within theweak vertex. With a suitable model for the the nuclear wavefunction theseintegrals can be evaluated numerically to calculate f π directly. Such modelscan include short range correlations between valance quarks.As shown in chapter 3, each component in the experiment, except the hydrogentarget, was commissioned during the 2004 run cycle and each componentperformed as designed. The neutron moderator brightness was measured tohave a maximum of 1.25 × 10 8 n/s/cm 2 /sr/meV/µA and the neutron guide222

performance was verified to operate as designed. The neutron beam polarizationwas measured to be between from 40% to 80%, in the time of flight rangeof 10 to 34 ms and the spin flip efficiency was measured the be 95% over thesame time of flight range. Chapter 4 provides a detailed report of the detectorarray design and performance with the main concerns including the detectorefficiency, noise performance, counting statistics performance, and instrumentalsystematic effects inducing false asymmetries. It was shown there that theaverage detector photoelectron yield is per MeV of γ-ray energy is 1300. Itwas also shown that the average detector noise RMS width is 0.15 mV, whichis 50% larger than the RMS width expected from the design criteria, but smallenough to make the noise RMS width negligible to counting statistics. Measurementswith beam on a B 4 C target demonstrated that the detector arrayoperates at counting statistics, where a measured RMS width of 6.1±0.04 mVcompares to a predicted 1 √NRMS width of ≃ 5.7 ± 0.6 mV.To establish the level of false asymmetries that may be present for the hydrogenproduction runs, several measurements were performed. Possible false asymmetriesdue to instrumental systematic effects involving the detector arrayand spin flipper were measured to be zero at the 5 × 10 −9 level. Asymmetriesdue to beam fluctuations were measured using the beam monitors. The beamasymmetry enters into the main data asymmetry as a product with the detectorpair gain asymmetry (see Sec. 5.1.2) with a combined RMS width of10 −5 . False asymmetries due to neutron capture on materials other than hydrogenwere measured for Al, Cu, In and B 4 C. These asymmetries were foundto be consistent with zero. The CCl 4asymmetry was used as a diagnostic223

tool. The 35 Cl asymmetries obtained from the CCl 4 measurements were previouslymeasured by this collaboration [45, 74] and an Up-Down asymmetry of(−21.2±1.7)×10 −6 was previously found by M. Avenier and collaborators [75].The asymmetries and their RMS widths are listed in table 6.1.Asymmetries and RMS widthUp-Down Left-Right RMS width(typ.)Al (−0.02 ± 3) × 10 −7 (−2 ± 3) × 10 −7 1.2 × 10 −3CCl 4 (−19 ± 2) × 10 −6 (−1 ± 2) × 10 −6 1.0 × 10 −3B 4 C (−1 ± 2) × 10 −6 (−5 ± 3) × 10 −6 0.7 × 10 −3Cu (−1 ± 3) × 10 −6 (0.3 ± 3) × 10 −6 1.0 × 10 −3In (−3 ± 2) × 10 −6 (3 ± 3) × 10 −6 0.4 × 10 −3Noise (add.) (2 ± 5) × 10 −9 (−7 ± 5) × 10 −9 2.0 × 10 −6Noise (mult.) (3 ± 7) × 10 −9 (−9 ± 7) × 10 −9 0.2 × 10 −3Beam*Gain 0 0 1.0 × 10 −5Table 6.1: Up-Down and Left-Right asymmetries for the target materials usedduring the 2004 commissioning run. Stated errors are statistical only. TheRMS widths are taken from histograms with single 8-step sequence asymmetriesfor a detector pair as individual entries.As shown in section 5.6, based on the signals obtained from the commissioningtargets, the total background due to these materials in the hydrogen signal isless than 10%. When calculating the asymmetry contributions from elementsother than hydrogen, one can take credit for the lower rates listed in table 5.6.Together with the asymmetries listed in table 6.1, the final asymmetry contributionfor each target material is listed in table 6.2These results show that the experimental apparatus, including all componentsexcept for the liquid hydrogen target, are operating as designed and meetall criteria needed to perform a successful measurement of the weak parity-224

Background Asymmetries in HydrogenBackground Level UD AsymmetryB 4 C ≤ 0.7% (−0.7 ± 1.4) × 10 −8Al ≤ 6% (−0.01 ± 1.8) × 10 −8In ≤ 0.4% (−1.2 ± 0.8) × 10 −8Cu ≤ 2% (−2.0 ± 6.0) × 10 −8Table 6.2: Expected background asymmetries in the hydrogen signal, fromtarget materials used during the 2004 commissioning run.violating γ asymmetry with an accuracy of 5 × 10 −9 in the neutron capturereaction ⃗n+p → d+γ. The accuracy in the background measurements is goodenough to perform a 5 × 10 −8 measurement of the ⃗n + p → d + γ asymmetryin the first production run cycle and can be improved by collecting more dataon these targets. The longest run time was required for aluminum (7 dayscontinuous) and a 5 × 10 −9 error on it would require ∼ 90 additional days ofcontinuous running at the same rate.225

Appendix AConventions, Conversions andother Useful InformationA.1 Common Relations and Properties for LowEnergy Neutrons227

A.1.1Properties of Thermal NeutronsMassM n = 1.674928(1) × 10 −27 kgM n = 940 MeV/c 2Radiusr = 0.7 fmSpin s = 1 2Magnetic MomentCompton WavelengthdeBroglie WavelengthGroup Velocityµ = −9.6491783(18) × 10 −27 J/Tλ C = 1.319695(20) × 10 −15 mλ B = 1.8 × 10 −10 mv = 2200 m/sβ decay Lifetimeτ = 887(2) sCoherence Length ∆ c = 12δk = 10−8 mEnergyE = 26 meVMomentumk = 3.68484 × 10 −24 kg m/sk = 6.969 keVMomentum Width δk = ¯h2δxElectric Polarizability α = 12.0 × 10 −4 fm 3A.1.2Equations and RelationsNatural Units228

¯h = c = 1¯hc = 197 MeV fm(A.1)(A.2)Energy:E = ¯h2 k 22M nE = k B T(A.3)(A.4)Momentum:√k = ¯h 2EM n= √ 2E 940= 43.4 √ E [Mev/c] [E] = MeV(A.5)Wavelength:Compton:λ C = h mc(A.6)229

de Broglie:λ B = h mv = h k=h43.4 √ E≃ √ 29 [fm] E[E] = MeV(A.7)Velocity:v =43.4 √ E h2π 197 × 10 −15 M n[m/s][E] = MeV, [M n ] = kgv ≃ 438 √ E [m/s] [E] = meV(A.8)A.2 Detector InformationA.2.1Dynode CorrectionsThe additional RMS width seen in the signal from a PMT due to its noiseis not only a result of the shot noise at the photocathode but also containscontributions from each of its additional dynode. The corrections from thedynodes are derived from Poisson statistics.230

The number of photoelectrons emitted at the photocathode follows the Poissondistribution〈¯n〉 = E(n) =∞∑n=0n e−δ δ nn!= δ〈¯n 2 〉 =∞∑E(n 2 ) = n 2 e−δ δ nn=0n!=∞∑(n + n(n − 1)) e−δ δ nn=0n!=∞∑δ + n(n − 1) e−δ δ n−1 δn=0n[(n − 1)!]=∞∑δ + δ (n − 1) e−δ δ n−1n=0(n − 1)!=∞∑δ + δ n e−δ δ nn=0n!= δ + δ 2So that the variance is σ 2 = E(n 2 ) − (E(n)) 2 = δ and the relative uncertaintyisσ(E(n)) = 1 √δ.However, this is only taking into account the distribution at the photocathode.Each additional dynode stage in the PMT contributes more to the overallobserved width. At the first dynode we may have n = 1...∞ possible numberof electrons incident on it for a particular event. Each of these events maymake ∑ ni=0 k i new electrons which are Poisson distributed themselves. So,231

with this one findsE(n ′ ) ==⎛∞∑ n∑⎝n=0∞∑n=0= δ 2 .∞∑i=0 k i =0nδP (n)⎞e −δ δ k ik i⎠ P (n)k i !Where P (n) = e −δ δ n /n!. On also findsE(n ′2 ) ==⎛∞∑ n∑⎝n=0∞∑n=0i=0 j=0⎛n∑ ∞∑⎝∞∑k i =0 k j =0k i k je −δ δ k ik i !(nδ(δ + 1) + n(n − 1)δ2 ) P (n)= δ 2 + δ 3 + δ 4 .⎞⎞e −δ δ k j⎠⎠ P (n)k j !Then, the variance one would see from the first dynode is given byσ 2(E(n)) 2 = 1 δ + 1 δ 2 .If one expands this analysis to more stages, then the variance seen from theNth stage will beσ 2(E(n)) 2 = 1 δ + 1 δ 2 + 1 δ 3 · · · + 1δ N+1 ≃ 1δ − 1 .The Hamamatsu R1513 PMT used for the detector tests (see Chapter 4) has10 stages and its gain is about 5 × 10 5 at 1.6 kV bias. So for 10 stages, the232

average number of photoelectrons (δ) produced at the photocathode is givenby δ 10 = 5 × 10 5 or δ ≃ 3.7.233

A.2.2Additional FiguresFigure A.1: CCl 4 target front view.Figure A.2: CCl 4 target side view.A.2.3Tables and Miscillaneous Information234

E n tof n/meV/ n/ms/ n pol. n trn. P 2 ∗ T FOM γ’s/ms/(meV) (ms) pulse pulse pulse75.00 5.78 0.08 2.05 0.181 0.677 0.022 0.05 0.01450.00 7.08 0.19 2.67 0.221 0.623 0.030 0.08 0.01745.00 7.46 0.25 3.02 0.232 0.608 0.033 0.10 0.01940.00 7.91 0.33 3.33 0.246 0.591 0.036 0.12 0.02035.00 8.46 0.48 3.94 0.262 0.571 0.039 0.15 0.02330.00 9.13 0.70 4.62 0.282 0.547 0.044 0.20 0.02625.00 10.01 1.33 6.67 0.307 0.519 0.049 0.33 0.03620.00 11.19 3.04 10.86 0.341 0.483 0.056 0.61 0.05515.00 12.92 6.89 16.01 0.388 0.436 0.066 1.05 0.07314.00 13.37 8.41 17.60 0.401 0.425 0.068 1.20 0.07813.00 13.88 9.75 18.27 0.414 0.413 0.071 1.29 0.07812.00 14.44 11.29 18.76 0.429 0.400 0.073 1.38 0.07811.50 14.75 12.34 19.24 0.437 0.393 0.075 1.44 0.07911.00 15.09 13.29 19.38 0.445 0.385 0.076 1.48 0.07810.20 15.67 14.81 19.29 0.460 0.373 0.079 1.52 0.07510.00 15.82 15.09 19.08 0.464 0.370 0.080 1.52 0.0739.50 16.23 16.16 18.91 0.474 0.362 0.081 1.54 0.0718.50 17.16 19.44 19.25 0.496 0.343 0.085 1.63 0.0698.00 17.69 21.50 19.45 0.509 0.334 0.086 1.68 0.0677.20 18.65 25.75 19.88 0.531 0.317 0.089 1.78 0.0666.50 19.62 29.07 19.25 0.553 0.301 0.092 1.77 0.0604.10 24.71 48.08 15.96 0.655 0.232 0.099 1.59 0.0384.00 25.02 49.38 15.79 0.661 0.228 0.100 1.57 0.0372.90 29.38 63.19 12.47 0.732 0.185 0.099 1.23 0.0242.30 32.99 67.21 9.37 0.781 0.156 0.095 0.89 0.0152.10 34.53 68.57 8.34 0.799 0.146 0.093 0.78 0.0132.00 35.38 67.91 7.68 0.808 0.140 0.092 0.70 0.0111.60 39.55 65.19 5.27 0.850 0.117 0.084 0.44 0.0061.00 50.03 56.99 2.28 0.920 0.075 0.063 0.14 0.002Table A.1: Monte Carlo calculation of neutron flux on the target and γ-ray rate out of the target at 100 µA. The figure of merit (FOM) isP 2 T × n/ms/pulse. Rates are stated in units of 10 −6 [69].235

Table A.2: Cesium IodideProperties for common dopantCsI(Tl) CsI(Na) CsI(undoped)Density [g/cm 3 ] 4.51 4.51 4.51Melting Point [K] 894 894 894Thermal expansioncoefficient [K −1 ] 54e −6 49e −6 49e −6Cleavage plane none none noneHardness (Mho) 2 2 2Hygroscopic slightly yes slightlyWavelength of emissionmaximum [nm] 550 420 315Lower wavelengthcutoff [nm] 320 300 260Refractive index atemission maximum 1.79 1.84 1.95Primary decay time[µs] 1 0.63 0.016Afterglow (after 6 ms)[%] 0.5-5.0 0.5-5.0 —Light yield[photons/MeV γ] 52 − 56e 3 38 − 44e 3 2e 3Photo-electron yield[%ofNaI(T l)] (γ rays) 45 85 4-6Absolute ScintillationEfficiency for Fast Electrons 11.9% 11.4% —236

Appendix BTheory DetailsB.1 Dirac Spinor NormalizationThe following definitions pertain to all fermion and boson states and fields,regardless whether they describe states and particles in the effective theory,where nucleons and mesons are the fundamental degrees of freedom, or in theStandart Model, with quarks and vector bosons as the degrees of freedom.We choose the conventions for fermion normalization in such a way that thefollowing is true:〈N( ⃗ P ′ )|N( ⃗ P )〉 ≡ (2π) 3 δ 3 ( ⃗ P ′ − ⃗ P )(B.1)and237

{ψ(x), ψ † (y)} ≡ δ 3 (x − y)(B.2)So with the fermion fields given asψ i (x) = ∑ λ i∫d 3 k i(2π) 3 (a(k i, λ i )u(k i , λ i )e −ik·x + b † (k i , λ i )v(k i , λ i )e ik·x )(B.3)the choices in B.1 and B.2 impose the normalization condition on the Diracspinors.〈N( ⃗ k i , λ i )|N( ⃗ k j , λ i )〉 = 〈0|{a(k i , λ i ), a † (k j , λ j )}|0〉=⇒{a(k i , λ i ), a † (k j , λ j )} = (2π) 3 δ 3 ( ⃗ k i − ⃗ k j )δ λi ,λ j(B.4){ψ i (x), ψ † j(y)} = ∑ λ i∫=⇒d 3 k(2π) 3 u(k i, λ i )u † (k i , λ i )e −i⃗ k i·(⃗x−⃗y)u(k i , λ i )u † (k i , λ i ) = I(B.5)238

Where I is the identity matrix.Thus, the exact form of the spinors isu(k i , λ i ) =⎛1 (E + M)λ i√⎝2E(E + M) ⃗σ · ⃗k i λ i⎞⎠(B.6)B.2 Parity Violating Weak Potential (Derivation)B.2.1Isospin SetupThe parity conserving and parity violating currents transform as iso-scalarsand iso-vectors. The fields in the currents (See equations 2.13 and 2.14) describenucleons and pions which couple via the strong and weak interactionsrespectively. The cross-product in the weak current is a result from its assumedinvariance under CP. The fields are not considered iso-spinors. The iso-spinspin dependence sits in the nucleon wavefunction. So again, the hamiltoniansareH P C = ig πNN∫d 3 xψ i (x)γ 5 ψ j (x)(⃗τ · ⃗φ(x))(B.7)and239

H P NC = F ∫π√2d 3 x ′ ψ i (x ′ )ψ j (x ′ )(⃗τ × ⃗ φ(x ′ )) 3(B.8)The three charge states of the pion form an iso-vector and in spherical notationthe fields representing the pion with the three possible charges are the usualirreducible tensor operators (ITO’s)φ + = 1 √2(φ 1 + iφ 2 )φ − = 1 √2(φ 1 − iφ 2 )φ o = φ 0(B.9)Similar ITO’s can be formed from the three iso-spin transformation matricesτ + = (τ 1 + iτ 2 )τ − = (τ 1 − iτ 2 )τ o = τ 0(B.10)240

The combinations found in the iso-scalar and iso-vector hamiltonians are thenexpressed in terms of these ITO’s⃗τ · ⃗φ = τ 1 φ 1 + τ 2 φ 2 + τ 3 φ 3= 1 √2(τ + φ − + τ − φ + ) + τ o φ o(B.11)(⃗τ × ⃗ φ) 3 = τ 1 φ 2 − τ 2 φ 1= i √2(τ + φ − − τ − φ + )(B.12)The pion fields produced here are the same as those in equation 2.16 andrepresent the creation of a π + or the annihilation of a π − with a φ + and thecreation of a π − or the annihilation of a π + with a φ − .B.2.2CalculationThe actual calculation starts with the insertion of a complete set of twonucleon,one-pion states between the pion propagator and the two hamiltoniansin equation 2.12.241

Start with the first term:λ a′1∑,λ ′ ,t ′ a a2∑∫λ a1 ,λ a2 ,t a〈f|H P C |N a1 N a2 π a 〉〈N a1 N a2 π a |d 3 k a1 d 3 k a2 d 3 k ad 3 k ′ ad 3 k ′1 ad 3 k ′2 a×(2π) 9 (2π) 91|N ′E − H a o1N a′2π a′ 〉〈N a′1N ′ aπ ′2 a |H P NC |i〉The energy E appearing in the denominator is that of the initial state (thefree nucleons). It is given byE = P 2 12M + P 2 22M .While the energy operator H o operating on the intermediate two-nucleon, onepionstate producesE o = P 2 12M + P 2 22M + ω k.Where ω k = √ k 2 + m 2 , k = p 1 − p 2 and m being the pion momentum andmass respectively.Thus, when acting on the intermediate states and integrating over the primedvariables, the denominator reduces to the pion energy∑λ ′ ,λ ′ ,t ′a a a1 2∫ d 3 k ′ ad 3 k ′1 ad 3 k ′2 a(2π) 9 × 〈N a1 N a2 π a |1|N ′E − H aN ′o1 aπ2 a′〉242

=∑λ ′ ,λ ′ ,t ′a a a1 2∫ d 3 k ′ ad 3 k ′1 ad 3 k ′2 a(2π) 9 δ aa′ω kaSo, we are left with the calculation of the matrix element:∑∫ d 3 k at a(2π) 〈f|H 3 P C|π a 〉 1 〈π a |H P NC |i〉ω kaSo, using equations B.7 and B.8, together with the pion fields and the nonrelativisticDirac spinor reductions 2.17 and 2.18, the calculation can be split upinto a spin-momentum part, an iso-spin part and a dynamical portion:∑∫ d 3 k a∑∫ d 3 k c1 d 3 k c2∑∫ d 3 k b1 d 3 ∫k b2 d 3 k π1 d 3 k π2t a(2π) 3λ c1 ,λ c2(2π) 6λ b1 ,λ b2(2π) 6 (2π) 6∫∫ξ † (f) Ω Π F(k..; λ..) d 3 xe i(kc 1 −kc 2 −kπ 1 )·x d 3 x ′ e i(k b 1−k b2 +k π2 )·x ′ ξ(i)where (See section 2.2 , eqn. 2.5)ξ(i)≡∑∫λ i1 t i1 ,λ i2 t i2Ω ≡ 〈0|a f1a f2a † c 1a c2a † b 1a b2a † i 1a † i 2|0〉d 3 k i1 d 3 k i2W LS(2π) 6 JI ( ⃗ k i1 λ i1 t i1 , ⃗ k i2 λ i2 t i2 )243

Π ≡ i 2 〈0|(τ c,+c π1 ,+ + τ c,− c π1 ,−)c † a,t a|0〉〈0|c a,ta (τ b,+ c † π 2 ,− − τ b,− c † π 2 ,+)|0〉The non-relativistic expansions (2.17 and 2.18) of the dirac spinors to orderO(1/M) lead to the Feynman amplitudeF(k..; λ..) ≡ χ † (λ c1 ) ⃗σ c · ( ⃗ k c2 − ⃗ k c1 )2Mδ λb1 λ b2χ(λ c2 ) √2ω ka ωkπ1 ω kπ2Note that the one body operator ⃗σ c · ( ⃗ k c2 − ⃗ k c1 ) = ⃗σ c · (− ⃗ k π ) is negative forpion creation at the weak vertex and absorption at the strong vertex. Theopposite is true for the second term in 2.12. The second term represents thetime reversed interaction, where the pion is created at the strong vertex andabsorbed at the weak vertex.In this case the one body operator above ispositive, but the ”propagator” (ω −1k π) will be negative. The dynamical picturefor the first term is shown in figure B.1. There are four possible combinationsof initial and final nucleon pairs, two direct terms and two exchange terms. Thetwo exchange terms vanish for this particular case, because in np interactions,this would require an iso-spin rotation in the strong vertex, but the stronginteraction conserves isospin.The dynamics and iso-spin dependence can be evaluated directly.244

a)N 1SN 1¤¡¤ £¡£¡£¤¡¤ £¡£¡£π + −¢¡¢ ¡¡ ¢¡¢N 2WN 2b)N 1S¦¡¦¡¦ ¥¡¥¡¥¦¡¦¡¦ ¥¡¥¡¥N 1π + −N 2¨¡¨¡¨ §¡§¡§N 2¨¡¨¡¨ §¡§¡§WFigure B.1: Dynamical depiction of the OPE interaction: a) Direct term. b)Exchange term.Ω = 2δ f1 c 1δ c2 i 1δ f2 b 1δ b2 i 2Π = i 2 (τ c,−τ b,+ − τ c,+ τ b,− ) 3 δ π1 ,aδ π2 ,aWhere δ i,j ≡ (2π) 3 δ 3 ( ⃗ k i − ⃗ k j )δ αi ,α jwith α representing all appropriate discretevariables.Together with all the constants, we arrive at245

−i(i)g πNN F π4 √ 2M ξ† (f) ∑λ c1 ,λ c2∫d 3 k c1 d 3 k c2(2π) 6∑∫λ b1 ,λ b2d 3 k b1 d 3 k b2(2π) 6 (2π) 3(2π) 3 δ 3 ( ⃗ k c1 − ⃗ k c2 + ⃗ k b1 − ⃗ k b2 )(τ c,− τ b,+ − τ c,+ τ b,− ) 3 χ † (λ c1 ) ⃗σ c · ⃗k π χ(λ c2 )=δ λb1 λ b2ωk 2 πδ f1 c 1δ c2 i 1δ f2 b 1δ b2 i 2ξ(f)−i(2π) 3 δ 3 ( ⃗ k f1 − ⃗ k i1 + ⃗ k f2 − ⃗ k i2 ) ×ξ † (f)χ(λ f1 ) ig πNNF π4 √ 2M (⃗τ 1 × ⃗τ 2 ) 3 ⃗σ 1 · ⃗k π1ω 2 k πχ(λ i1 )ξ(i)The calculation of the second term in 2.12 proceeds in exactly the same way.However, as already mentioned, because of the reverse order of the hamiltoniansin that term the the signs of the spin-momentum operator as well as thepropagator will be reversed. Also, the pion momentum now couples to thespin of the the second nucleon as opposed to the first one. This can be seen bysimply switching the variables [c 1 , b 1 ] and [c 2 , b 2 ] (which is synonymous withswitching the the order of the two hamiltonians) and observing that c 1 and c 2now couple to nucleon two in the overall momentum conserving delta function.The second term therefore produces−i(2π) 3 δ 3 ( ⃗ k f1 − ⃗ k i1 + ⃗ k f2 − ⃗ k i2 ) ×246

ξ † (f)χ(λ f2 ) ig πNNF π4 √ 2M (⃗τ 1 × ⃗τ 2 ) 3 ⃗σ 2 · ⃗k π1ω 2 k πχ(λ i2 )ξ(i)Together the two terms produce the pion exchange term of the parity-violatingNN potential.〈f|V P NC |i〉 = −i(2π) 3 δ 3 ( ⃗ k f1 − ⃗ k i1 + ⃗ k f2 − ⃗ k i2 )ξ † (f)V ∆I=1π ξ(i) (B.13)V ∆I=1π= ig √πNNF π[⃗τ 1 × ⃗τ 2 ] 3 [⃗σ 1 + ⃗σ 2 ] · ⃗k 1π32M ωk 2 π(B.14)The fourier transform of this relation produces the coordinate space form ofthe potential (2.19) including the Yukawa term.247

248B.3 Tree Level Amplitude Integral Tables

Table B.1: S-Matrix Term 2Integration over ⃗ k i , ⃗ k k and ⃗ k π , Summation over i,j,k,(l or n),π,πIntegral 1 Integral 2 Integral 3c1 c2 c1 c2 c1 c2α -4 +4 +4 -4 -4 +4249i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i1 PNi⃗ − ⃗ k i − ⃗ k k , l d u PNf⃗ − ⃗ k i + ⃗ k π , l d u Pπ ⃗ − ⃗ k π , π d u2 ⃗ kk , k d u Pπ ⃗ − ⃗ k π , π d u ⃗ kk , k d u3 ⃗ ki , i u d ⃗ ki , i u d PNf⃗ − ⃗ k k + ⃗ k π , n u d4 ⃗ ki , i u d ⃗ ki , i u d ⃗ ki , i u d5 PNf⃗ − ⃗ k i − ⃗ k k , j u d PNf⃗ − ⃗ k i − ⃗ k k , j u d PNf⃗ − ⃗ k i − ⃗ k k , j u d6 ⃗ kk , k d u ⃗ kk , k d u ⃗ kk , k d u7 Pπ ⃗ − ⃗ k π , π d u Pπ ⃗ − ⃗ k π , π d u Pπ ⃗ − ⃗ k π , π d u8 ⃗ kπ , π u d ⃗ kπ , π u d ⃗ kπ , π u d9 PNi⃗ − ⃗ k i − ⃗ k k PNf⃗ − ⃗ k i + ⃗ k π PNf⃗ − ⃗ k k + ⃗ k π−1M1u(⃗q MZ 2 (q2 1 −m2 ) 5, λ 5 )Λ µ v(⃗q 8 , λ 8 )g µν u(⃗q 7 , λ 7 )γ ν (̸ q 1+ m)Γ ρ u(⃗q 1 , λ 1 )−1M2u(⃗q MZ 2 (q2 1 −m2 ) 5, λ 5 )Λ µ v(⃗q 8 , λ 8 )g µν u(⃗q 6 , λ 6 )γ ν (̸ q 1+ m)Γ ρ u(⃗q 1 , λ 1 )−1M3u(⃗q MZ 2 (q2 3 −m2 ) 5, λ 5 )Λ µ v(⃗q 8 , λ 8 )g µν u(⃗q 4 , λ 4 )γ ν (̸ q 3+ m)Γ ρ u(⃗q 3 , λ 3 )

Table B.2: S-Matrix Term 3Integration over ⃗ k i , ⃗ k k and ⃗ k π , Summation over i,j,k,(l or n),π,πIntegral 1 Integral 2 Integral 3c1 c2 c1 c2 c1 c2α -4 +4 +4 -4 -4 +4250i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i1 PNi⃗ − ⃗ k i − ⃗ k k , l d u PNf⃗ − ⃗ k i + ⃗ k π , l d u Pπ ⃗ − ⃗ k π , π d u2 ⃗ kk , k d u Pπ ⃗ − ⃗ k π , π d u ⃗ kk , k d u3 ⃗ ki , i u d ⃗ ki , i u d PNf⃗ − ⃗ k k + ⃗ k π , n u d4 ⃗ ki , i u d ⃗ ki , i u d ⃗ ki , i u d5 PNf⃗ − ⃗ k i − ⃗ k k , j u d PNf⃗ − ⃗ k i − ⃗ k k , j u d PNf⃗ − ⃗ k i − ⃗ k k , j u d6 ⃗ kk , k d u ⃗ kk , k d u ⃗ kk , k d u7 Pπ ⃗ − ⃗ k π , π d u Pπ ⃗ − ⃗ k π , π d u Pπ ⃗ − ⃗ k π , π d u8 ⃗ kπ , π u d ⃗ kπ , π u d ⃗ kπ , π u d9 ⃗ kπ ⃗ kπ ⃗ kπ−1M1MZ 2 (q2 8 −m2 ) 5, λ 5 )Λ µ (−̸ q 8+ m)Γ ρ v(⃗q 8 , λ 8 )g µν u(⃗q 7 , λ 7 )γ ν u(⃗q 1 , λ 1 )−1M2MZ 2 (q2 8 −m2 ) 5, λ 5 )Λ µ (−̸ q 8+ m)Γ ρ v(⃗q 8 , λ 8 )g µν u(⃗q 6 , λ 6 )γ ν u(⃗q 1 , λ 1 )−1M3MZ 2 (q2 8 −m2 ) 5, λ 5 )Λ µ (−̸ q 8+ m)Γ ρ v(⃗q 8 , λ 8 )g µν u(⃗q 4 , λ 4 )γ ν u(⃗q 3 , λ 3 )

Table B.3: S-Matrix Term 4Integration over ⃗ k i , ⃗ k k and ⃗ k π , Summation over i,j,k,(l or n),π,πIntegral 1 Integral 2 Integral 3c1 c2 c1 c2 c1 c2α -4 +4 +4 -4 +4 -4251i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i1 PNi⃗ − ⃗ k i − ⃗ k k , l u d PNf⃗ − ⃗ k i + ⃗ k π , l u d Pπ ⃗ − ⃗ k π , π d u2 ⃗ kk , k u d Pπ ⃗ − ⃗ k π , π u d ⃗ kk , k d u3 ⃗ ki , i d u ⃗ ki , i d u PNf⃗ − ⃗ k k + ⃗ k π , n u d4 ⃗ ki , i d u ⃗ ki , i d u ⃗ ki , i u d5 PNf⃗ − ⃗ k i − ⃗ k k , j d u PNf⃗ − ⃗ k i − ⃗ k k , j d u PNf⃗ − ⃗ k i − ⃗ k k , j u d6 ⃗ kk , k u d ⃗ kk , k u d ⃗ kk , k d u7 Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π d u8 ⃗ kπ , π d u ⃗ kπ , π d u ⃗ kπ , π u d9 PNi⃗ − ⃗ k i − ⃗ k k PNf⃗ − ⃗ k i + ⃗ k π PNf⃗ − ⃗ k k + ⃗ k π−1M1u(⃗q MZ 2 (q2 1 −m2 ) 7, λ 7 )Λ µ (̸ q 1+ m)Γ ρ u(⃗q 1 , λ 1 )g µν u(⃗q 5 , λ 5 )γ ν v(⃗q 8 , λ 8 )−1M2u(⃗q MZ 2 (q2 1 −m2 ) 6, λ 6 )Λ µ (̸ q 1+ m)Γ ρ u(⃗q 1 , λ 1 )g µν u(⃗q 5 , λ 5 )γ ν v(⃗q 8 , λ 8 )−1M3u(⃗q MZ 2 (q2 3 −m2 ) 5, λ 5 )Λ µ (̸ q 3+ m)Γ ρ u(⃗q 3 , λ 3 )g µν u(⃗q 4 , λ 4 )γ ν u(⃗q 8 , λ 8 )

Table B.4: S-Matrix Term 5Integration over ⃗ k i , ⃗ k k and ⃗ k π , Summation over i,j,k,(l or n),π,πIntegral 1 Integral 2 Integral 3c1 c2 c1 c2 c1 c2α -4 +4 +4 -4 +4 -4252i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i1 PNi⃗ − ⃗ k i − ⃗ k k , l u d PNf⃗ − ⃗ k i + ⃗ k π , l u d Pπ ⃗ − ⃗ k π , π d u2 ⃗ kk , k u d Pπ ⃗ − ⃗ k π , π u d ⃗ kk , k d u3 ⃗ ki , i d u ⃗ ki , i d u PNf⃗ − ⃗ k k + ⃗ k π , n u d4 ⃗ ki , i d u ⃗ ki , i d u ⃗ ki , i u d5 PNf⃗ − ⃗ k i − ⃗ k k , j d u PNf⃗ − ⃗ k i − ⃗ k k , j d u PNf⃗ − ⃗ k i − ⃗ k k , j u d6 ⃗ kk , k u d ⃗ kk , k u d ⃗ kk , k d u7 Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π d u8 ⃗ kπ , π d u ⃗ kπ , π d u ⃗ kπ , π u d9 PNf⃗ − ⃗ k i − ⃗ k k PNf⃗ − ⃗ k i − ⃗ k k⃗ ki−1M1MZ 2 (q2 5 −m2 ) 7, λ 7 )Λ µ u(⃗q 1 , λ 1 )g µν u(⃗q 5 , λ 5 )Γ ρ (̸ q 5+ m)γ ν v(⃗q 8 , λ 8 )−1M2MZ 2 (q2 5 −m2 ) 6, λ 6 )Λ µ u(⃗q 1 , λ 1 )g µν u(⃗q 5 , λ 5 )Γ ρ (̸ q 5+ m)γ ν v(⃗q 8 , λ 8 )−1M3MZ 2 (q2 4 −m2 ) 5, λ 5 )Λ µ u(⃗q 3 , λ 3 )g µν u(⃗q 4 , λ 4 )Γ ρ (̸ q 4+ m)γ ν v(⃗q 8 , λ 8 )

Table B.5: S-Matrix Term 6Integration over ⃗ k i , ⃗ k k and ⃗ k π , Summation over i,j,k,(l or n),π,πIntegral 1 Integral 2 Integral 3c1 c2 c1 c2 c1 c2α +4 -4 -4 +4 -4 +4253i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i1 PNi⃗ − ⃗ k i − ⃗ k k , l u d PNf⃗ − ⃗ k i + ⃗ k π , l u d Pπ ⃗ − ⃗ k π , π d u2 ⃗ kk , k u d Pπ ⃗ − ⃗ k π , π u d ⃗ kk , k d u3 ⃗ ki , i d u ⃗ ki , i d u PNf⃗ − ⃗ k k + ⃗ k π , n u d4 ⃗ ki , i d u ⃗ ki , i d u ⃗ ki , i u d5 PNf⃗ − ⃗ k i − ⃗ k k , j d u PNf⃗ − ⃗ k i − ⃗ k k , j d u PNf⃗ − ⃗ k i − ⃗ k k , j u d6 ⃗ kk , k u d ⃗ kk , k u d ⃗ kk , k d u7 Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π d u8 ⃗ kπ , π d u ⃗ kπ , π d u ⃗ kπ , π u d9 Pπ ⃗ − ⃗ k π⃗ kk ⃗ ki−1M1MZ 2 (q2 7 −m2 ) 5, λ 5 )Λ µ v(⃗q 8 , λ 8 )g µν u(⃗q 7 , λ 7 )Γ ρ (̸ q 7+ m)γ ν u(⃗q 1 , λ 1 )−1M2MZ 2 (q2 6 −m2 ) 5, λ 5 )Λ µ v(⃗q 8 , λ 8 )g µν u(⃗q 6 , λ 6 )Γ ρ (̸ q 6+ m)γ ν u(⃗q 1 , λ 1 )−1M3MZ 2 (q2 4 −m2 ) 5, λ 5 )Λ µ v(⃗q 8 , λ 8 )g µν u(⃗q 4 , λ 4 )Γ ρ (̸ q 4+ m)γ ν u(⃗q 3 , λ 3 )

Table B.6: S-Matrix Term 7Integration over ⃗ k i , ⃗ k k and ⃗ k π , Summation over i,j,k,(l or n),π,πIntegral 1 Integral 2 Integral 3c1 c2 c1 c2 c1 c2α -4 +4 +4 -4 +4 -4254i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i1 PNi⃗ − ⃗ k i − ⃗ k k , l u d PNf⃗ − ⃗ k i + ⃗ k π , l u d Pπ ⃗ − ⃗ k π , π d u2 ⃗ kk , k u d Pπ ⃗ − ⃗ k π , π u d ⃗ kk , k d u3 ⃗ ki , i d u ⃗ ki , i d u PNf⃗ − ⃗ k k + ⃗ k π , n u d4 ⃗ ki , i d u ⃗ ki , i d u ⃗ ki , i u d5 PNf⃗ − ⃗ k i − ⃗ k k , j d u PNf⃗ − ⃗ k i − ⃗ k k , j d u PNf⃗ − ⃗ k i − ⃗ k k , j u d6 ⃗ kk , k u d ⃗ kk , k u d ⃗ kk , k d u7 Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π d u8 ⃗ kπ , π d u ⃗ kπ , π d u ⃗ kπ , π u d9 Pπ ⃗ − ⃗ k π⃗ kk PNf⃗ − ⃗ k k − ⃗ k i−1M1u(⃗q MZ 2 (q2 7 −m2 ) 7, λ 7 )Γ ρ (̸ q 7+ m)Λ µ u(⃗q 1 , λ 1 )g µν u(⃗q 5 , λ 5 )γ ν v(⃗q 8 , λ 8 )−1M2u(⃗q MZ 2 (q2 6 −m2 ) 6, λ 6 )Γ ρ (̸ q 6+ m)Λ µ u(⃗q 1 , λ 1 )g µν u(⃗q 5 , λ 5 )γ ν v(⃗q 8 , λ 8 )−1M3u(⃗q MZ 2 (q2 5 −m2 ) 5, λ 5 )Γ ρ (̸ q 5+ m)Λ µ u(⃗q 3 , λ 3 )g µν u(⃗q 4 , λ 4 )γ ν v(⃗q 8 , λ 8 )

Table B.7: S-Matrix Term 8Integration over ⃗ k i , ⃗ k k and ⃗ k π , Summation over i,j,k,(l or n),π,πIntegral 1 Integral 2 Integral 3c1 c2 c1 c2 c1 c2α +4 -4 -4 +4 -4 +4255i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i1 PNi⃗ − ⃗ k i − ⃗ k k , l u d PNf⃗ − ⃗ k i + ⃗ k π , l u d Pπ ⃗ − ⃗ k π , π d u2 ⃗ kk , k u d Pπ ⃗ − ⃗ k π , π u d ⃗ kk , k d u3 ⃗ ki , i d u ⃗ ki , i d u PNf⃗ − ⃗ k k + ⃗ k π , n u d4 ⃗ ki , i d u ⃗ ki , i d u ⃗ ki , i u d5 PNf⃗ − ⃗ k i − ⃗ k k , j d u PNf⃗ − ⃗ k i − ⃗ k k , j d u PNf⃗ − ⃗ k i − ⃗ k k , j u d6 ⃗ kk , k u d ⃗ kk , k u d ⃗ kk , k d u7 Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π d u8 ⃗ kπ , π d u ⃗ kπ , π d u ⃗ kπ , π u d9 PNf⃗ − ⃗ k i − ⃗ k k PNf⃗ − ⃗ k i − ⃗ k k PNf⃗ − ⃗ k k − ⃗ k i−1M1u(⃗q MZ 2 (q2 5 −m2 ) 5, λ 5 )Γ ρ (̸ q 5+ m)Λ µ v(⃗q 8 , λ 8 )g µν u(⃗q 7 , λ 7 )γ ν u(⃗q 1 , λ 1 )−1M2u(⃗q MZ 2 (q2 5 −m2 ) 5, λ 5 )Γ ρ (̸ q 5+ m)Λ µ v(⃗q 8 , λ 8 )g µν u(⃗q 6 , λ 6 )γ ν u(⃗q 1 , λ 1 )−1M3u(⃗q MZ 2 (q2 5 −m2 ) 5, λ 5 )Γ ρ (̸ q 5+ m)Λ µ v(⃗q 8 , λ 8 )g µν u(⃗q 4 , λ 4 )γ ν u(⃗q 3 , λ 3 )

Table B.8: S-Matrix Term 9Integration over ⃗ k i , ⃗ k l and ⃗ k π , Summation over i,j,k,l,(n or m),π,πIntegral 1 Integral 2 Integral 3c1 c2 c1 c2 c1 c2α +4 -4 +4 -4 -4 +4256i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i1 ⃗ kl , l u d ⃗ kl , l u d ⃗ kl , l d u2 PNi⃗ − ⃗ k i − ⃗ k l , m u d PNf⃗ − ⃗ k i + ⃗ k π , k u d PNi⃗ − ⃗ k i − ⃗ k l , m d u3 ⃗ ki , i d u Pπ ⃗ − ⃗ k π − ⃗ k l + ⃗ k i , n d u ⃗ ki , i u d4 ⃗ ki , i d u ⃗ ki , i d u ⃗ ki , i u d5 − ⃗ k π , j d u − ⃗ k π , j d u − ⃗ k π , j u d6 PNf⃗ − ⃗ k i + ⃗ k π , k u d PNf⃗ − ⃗ k i + ⃗ k π , k u d PNf⃗ − ⃗ k i + ⃗ k π , k d u7 Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π d u8 ⃗ kπ , π d u ⃗ kπ , π d u ⃗ kπ , π u d9 ⃗ kπ ⃗ kπ ⃗ kπiM1u(⃗qMZ 2 7 , λ 7 )Λ µ u(⃗q 2 , λ 2 )g µν u(⃗q 6 , λ 6 )γ ν u(⃗q 1 , λ 1 )u(⃗q 5 , λ 5 )Γ ρ v(⃗q 8 , λ 8 )iM2u(⃗qMZ 2 7 , λ 7 )Λ µ u(⃗q 1 , λ 1 )g µν u(⃗q 4 , λ 4 )γ ν u(⃗q 3 , λ 3 )u(⃗q 5 , λ 5 )Γ ρ v(⃗q 8 , λ 8 )iM3u(⃗qMZ 2 6 , λ 6 )Λ µ u(⃗q 2 , λ 2 )g µν u(⃗q 7 , λ 7 )γ ν u(⃗q 1 , λ 1 )u(⃗q 5 , λ 5 )Γ ρ v(⃗q 8 , λ 8 )

Table B.9: S-Matrix Term 9Integration over ⃗ k i , ⃗ k l and ⃗ k π , Summation over i,j,k,l,(n or m),π,πIntegral 4 Integral 5 Integral 6c1 c2 c1 c2 c1 c2α -4 +4 -4 +4 -4 +4257i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i ⃗q i , ξ i f i f i1 ⃗ kl , l u d ⃗ kl , l u d ⃗ kl , l d u2 Pπ ⃗ − ⃗ k π , π u d PNf⃗ − ⃗ k i + ⃗ k π , k u d Pπ ⃗ − ⃗ k π , π d u3 PNf⃗ − ⃗ k l + ⃗ k π , n d u Pπ ⃗ − ⃗ k π − ⃗ k l + ⃗ k i , n d u PNf⃗ − ⃗ k l + ⃗ k π , n u d4 ⃗ ki , i d u ⃗ ki , i d u − ⃗ k π , j, i u d5 − ⃗ k π , j d u − ⃗ k π , j d u ⃗ kj , j u d6 PNf⃗ − ⃗ k i + ⃗ k π , k u d PNf⃗ − ⃗ k i + ⃗ k π , k u d PNf⃗ − ⃗ k j + ⃗ k π , k d u7 Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π u d Pπ ⃗ − ⃗ k π , π d u8 ⃗ kπ , π d u ⃗ kπ , π d u ⃗ kπ , π u d9 ⃗ kπ ⃗ kπ ⃗ kπiM1u(⃗qMZ 2 4 , λ 4 )Λ µ u(⃗q 3 , λ 3 )g µν u(⃗q 6 , λ 6 )γ ν u(⃗q 1 , λ 1 )u(⃗q 5 , λ 5 )Γ ρ v(⃗q 8 , λ 8 )iM2u(⃗qMZ 2 4 , λ 4 )Λ µ u(⃗q 3 , λ 3 )g µν u(⃗q 7 , λ 7 )γ ν u(⃗q 1 , λ 1 )u(⃗q 5 , λ 5 )Γ ρ v(⃗q 8 , λ 8 )iM3u(⃗qMZ 2 6 , λ 6 )Λ µ u(⃗q 1 , λ 1 )g µν u(⃗q 5 , λ 5 )γ ν u(⃗q 3 , λ 3 )u(⃗q 4 , λ 4 )Γ ρ v(⃗q 8 , λ 8 )

Appendix CData SummariesThis appendix lists the condensed run by run data information for the varioustargets. Run by run Asymmetries are listed for Up-Down only. Beam statisticsare calculated from good 8-step sequences only.All quantities are unitlessunless specificly indicated otherwise.A γUp-Down Left-RightAl (0.02 ± 3) × 10 −7 (2 ± 3) × 10 −7CCl 4 (19 ± 2) × 10 −6 (1 ± 2) × 10 −6B 4 C (1 ± 2) × 10 −6 (5 ± 3) × 10 −6Cu (1 ± 3) × 10 −6 (−0.3 ± 3) × 10 −6In (3 ± 2) × 10 −6 (−3 ± 3) × 10 −6Table C.1: Total Up-Down and Left-Right asymmetries.259

C.1 Thin Aluminum TargetThin Target SummaryTotal Number of good runs : 502Total Number of sequences from all runs : 592852Number of good sequences : 584570Number of lost sequences from bad spin state : 473 ( 0.1% loss)Number of lost sequences from beam cuts : 7809 ( 1.3% loss)Total Number of lost sequences : 8282 ( 1.4% loss)Average Run Asymmetry Error : 1.3 × 10 −5Average Run Asymmetry Error RMS : 5.4 × 10 −6Average 3He Polarization : 0.45Average 3He Polarization RMS : 5.9 × 10 −3Average Energy Average Beam Polarization : 0.63Average Energy Average Beam Polarization RMS : 4.4 × 10 −3Average Beam Current : 128 µAAverage Beam Current RMS : 5.5 µATable C.2:260

261Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2031 126.6 0.46 0.48 0.95 1156 2 5 (+0.0 ± 1.2) × 10 −5 (+0.0 ± 1.2) × 10 −52032 126.6 0.46 0.48 0.95 1244 1 5 (+0.2 ± 1.1) × 10 −5 (+1.1 ± 8.3) × 10 −62033 126.4 0.46 0.48 0.95 1247 1 2 (−0.1 ± 1.2) × 10 −5 (+0.5 ± 6.7) × 10 −62034 126.8 0.46 0.48 0.95 1230 2 6 (+0.5 ± 1.1) × 10 −5 (+1.6 ± 5.8) × 10 −62035 126.9 0.46 0.48 0.95 1248 1 1 (−1.0 ± 1.2) × 10 −5 (−0.7 ± 5.2) × 10 −62036 127.2 0.46 0.48 0.95 1249 1 0 (−0.1 ± 1.2) × 10 −5 (−0.8 ± 4.8) × 10 −62037 126.8 0.46 0.48 0.95 1247 1 2 (−1.3 ± 1.1) × 10 −5 (−2.6 ± 4.4) × 10 −62038 126.8 0.46 0.48 0.95 1248 1 1 (+1.2 ± 1.2) × 10 −5 (−0.8 ± 4.1) × 10 −62039 126.5 0.46 0.48 0.95 1246 1 3 (+2.4 ± 1.2) × 10 −5 (+1.9 ± 3.9) × 10 −62040 126.8 0.46 0.48 0.95 1243 1 6 (+0.2 ± 1.2) × 10 −5 (+2.0 ± 3.7) × 10 −62041 126.4 0.46 0.48 0.95 1243 1 6 (+0.1 ± 1.1) × 10 −5 (+1.9 ± 3.5) × 10 −62042 126.6 0.46 0.48 0.95 1245 1 4 (−0.3 ± 1.1) × 10 −5 (+1.5 ± 3.3) × 10 −62043 126.5 0.46 0.48 0.95 1241 1 8 (+1.1 ± 1.1) × 10 −5 (+2.2 ± 3.2) × 10 −62044 126.7 0.46 0.48 0.95 710 1 539 (+2.2 ± 1.5) × 10 −5 (+3.1 ± 3.1) × 10 −62045 127.0 0.46 0.48 0.95 1246 1 3 (−0.4 ± 1.2) × 10 −5 (+2.6 ± 3.0) × 10 −62046 126.9 0.46 0.48 0.95 1244 3 3 (+0.8 ± 1.2) × 10 −5 (+2.9 ± 2.9) × 10 −62060 128.0 0.46 0.48 0.95 1178 0 72 (−0.1 ± 1.2) × 10 −5 (+2.7 ± 2.8) × 10 −62061 127.8 0.46 0.48 0.95 1247 0 3 (−1.2 ± 1.1) × 10 −5 (+1.9 ± 2.8) × 10 −62062 127.7 0.46 0.48 0.95 1249 0 1 (+1.2 ± 1.1) × 10 −5 (+2.4 ± 2.7) × 10 −62063 127.8 0.46 0.48 0.95 1247 1 2 (+0.5 ± 1.2) × 10 −5 (+2.5 ± 2.6) × 10 −6Table C.3:

262Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2064 127.8 0.46 0.48 0.95 1248 0 2 (−1.4 ± 1.1) × 10 −5 (+1.7 ± 2.5) × 10 −62065 127.7 0.46 0.48 0.95 1249 0 1 (−0.1 ± 1.1) × 10 −5 (+1.6 ± 2.5) × 10 −62085 116.6 0.47 0.49 0.95 1236 0 14 (−0.3 ± 1.2) × 10 −5 (+1.4 ± 2.4) × 10 −62086 127.8 0.47 0.48 0.95 1249 0 1 (−0.9 ± 1.1) × 10 −5 (+0.9 ± 2.4) × 10 −62087 127.3 0.47 0.48 0.95 1246 0 4 (+0.6 ± 1.2) × 10 −5 (+1.1 ± 2.3) × 10 −62088 127.0 0.47 0.48 0.95 1247 0 3 (+1.1 ± 1.1) × 10 −5 (+1.5 ± 2.3) × 10 −62089 126.4 0.47 0.48 0.95 1240 0 10 (−0.0 ± 1.2) × 10 −5 (+1.4 ± 2.2) × 10 −62090 125.9 0.47 0.48 0.95 1231 1 18 (+2.2 ± 1.2) × 10 −5 (+2.2 ± 2.2) × 10 −62091 125.6 0.47 0.48 0.95 1209 0 41 (+0.2 ± 1.1) × 10 −5 (+2.2 ± 2.2) × 10 −62092 127.4 0.47 0.48 0.95 1227 0 23 (−0.8 ± 1.1) × 10 −5 (+1.8 ± 2.1) × 10 −62094 128.7 0.47 0.48 0.95 1249 0 0 (−0.4 ± 1.1) × 10 −5 (+1.7 ± 2.1) × 10 −62095 128.5 0.47 0.48 0.95 1247 0 3 (+1.6 ± 1.1) × 10 −5 (+2.1 ± 2.0) × 10 −62096 128.5 0.47 0.48 0.95 1248 0 2 (−1.1 ± 1.2) × 10 −5 (+1.7 ± 2.0) × 10 −62097 128.6 0.47 0.48 0.95 1246 0 4 (−1.7 ± 1.1) × 10 −5 (+1.1 ± 2.0) × 10 −62098 127.8 0.47 0.48 0.95 1244 0 6 (+0.1 ± 1.1) × 10 −5 (+1.1 ± 2.0) × 10 −62099 127.3 0.47 0.48 0.95 1247 0 3 (+0.5 ± 1.1) × 10 −5 (+1.2 ± 1.9) × 10 −62100 127.4 0.47 0.48 0.95 1247 0 3 (+0.4 ± 1.1) × 10 −5 (+1.3 ± 1.9) × 10 −62101 127.5 0.47 0.48 0.95 1248 0 2 (−0.1 ± 1.1) × 10 −5 (+1.2 ± 1.9) × 10 −62102 127.5 0.47 0.48 0.95 1249 0 1 (−0.8 ± 1.1) × 10 −5 (+1.0 ± 1.8) × 10 −62103 127.2 0.47 0.48 0.95 1247 0 2 (−0.4 ± 1.1) × 10 −5 (+0.9 ± 1.8) × 10 −62104 127.1 0.47 0.48 0.95 1248 0 2 (−2.0 ± 1.1) × 10 −5 (+0.4 ± 1.8) × 10 −62106 127.2 0.46 0.48 0.95 1248 1 1 (+0.1 ± 1.1) × 10 −5 (+0.4 ± 1.8) × 10 −6Table C.4:

263Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2107 127.3 0.46 0.48 0.95 1248 1 1 (−0.2 ± 1.1) × 10 −5 (+0.3 ± 1.7) × 10 −62108 127.9 0.46 0.48 0.95 1248 1 1 (−0.7 ± 1.1) × 10 −5 (+0.1 ± 1.7) × 10 −62110 127.9 0.46 0.48 0.95 1245 1 4 (−0.3 ± 1.1) × 10 −5 (+0.1 ± 1.7) × 10 −62111 127.9 0.46 0.48 0.95 1249 1 0 (−0.8 ± 1.1) × 10 −5 (−0.1 ± 1.7) × 10 −62116 126.7 0.46 0.48 0.95 1243 1 6 (−1.0 ± 1.2) × 10 −5 (−0.3 ± 1.7) × 10 −62117 125.9 0.46 0.48 0.95 1245 0 5 (−0.8 ± 1.1) × 10 −5 (−0.5 ± 1.6) × 10 −62118 126.0 0.46 0.48 0.95 1246 0 4 (+1.8 ± 1.1) × 10 −5 (−0.1 ± 1.6) × 10 −62120 125.9 0.46 0.48 0.95 1243 1 6 (+0.9 ± 1.1) × 10 −5 (+0.1 ± 1.6) × 10 −62121 125.7 0.46 0.48 0.95 1245 2 3 (−0.5 ± 1.2) × 10 −5 (−0.0 ± 1.6) × 10 −62122 125.1 0.46 0.48 0.95 1239 1 10 (+0.3 ± 1.2) × 10 −5 (+0.0 ± 1.6) × 10 −62123 124.9 0.46 0.48 0.95 1243 2 5 (−0.4 ± 1.1) × 10 −5 (−0.1 ± 1.6) × 10 −62124 124.1 0.46 0.48 0.95 282 1 967 (−2.5 ± 2.4) × 10 −5 (−0.2 ± 1.6) × 10 −62125 124.5 0.46 0.48 0.95 1243 1 6 (−1.4 ± 1.2) × 10 −5 (−0.4 ± 1.6) × 10 −62126 122.6 0.46 0.48 0.95 1237 2 11 (+0.2 ± 1.2) × 10 −5 (−0.4 ± 1.5) × 10 −62127 125.8 0.46 0.48 0.95 1244 1 5 (−0.8 ± 1.1) × 10 −5 (−0.5 ± 1.5) × 10 −62128 126.2 0.46 0.48 0.95 1141 0 8 (−2.0 ± 1.2) × 10 −5 (−0.8 ± 1.5) × 10 −62129 127.0 0.46 0.48 0.95 617 1 7 (+2.4 ± 1.6) × 10 −5 (−0.6 ± 1.5) × 10 −62130 126.4 0.46 0.48 0.95 1246 1 3 (+1.2 ± 1.2) × 10 −5 (−0.4 ± 1.5) × 10 −62131 126.3 0.46 0.48 0.95 1245 1 4 (−2.2 ± 1.1) × 10 −5 (−0.8 ± 1.5) × 10 −62132 126.4 0.46 0.48 0.95 1239 1 10 (−3.4 ± 1.1) × 10 −5 (−1.3 ± 1.5) × 10 −62133 126.4 0.46 0.48 0.95 1244 1 5 (−1.5 ± 1.2) × 10 −5 (−1.6 ± 1.5) × 10 −62134 126.6 0.46 0.48 0.95 1243 1 6 (+0.6 ± 1.1) × 10 −5 (−1.4 ± 1.4) × 10 −6Table C.5:

264Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2135 126.4 0.46 0.48 0.95 1246 1 3 (−0.9 ± 1.1) × 10 −5 (−1.6 ± 1.4) × 10 −62137 126.9 0.46 0.48 0.95 1207 1 24 (−0.5 ± 1.2) × 10 −5 (−1.6 ± 1.4) × 10 −62138 126.0 0.46 0.48 0.95 1169 1 16 (+1.3 ± 1.2) × 10 −5 (−1.4 ± 1.4) × 10 −62139 126.0 0.46 0.48 0.95 1089 2 47 (−0.5 ± 1.2) × 10 −5 (−1.5 ± 1.4) × 10 −62140 126.1 0.46 0.48 0.95 1239 1 10 (+0.7 ± 1.2) × 10 −5 (−1.3 ± 1.4) × 10 −62141 126.4 0.46 0.48 0.95 1243 1 6 (−1.0 ± 1.2) × 10 −5 (−1.5 ± 1.4) × 10 −62142 126.7 0.46 0.48 0.95 1242 1 7 (+0.3 ± 1.1) × 10 −5 (−1.4 ± 1.4) × 10 −62143 126.7 0.46 0.48 0.95 1245 1 4 (−1.7 ± 1.2) × 10 −5 (−1.6 ± 1.4) × 10 −62144 127.0 0.46 0.48 0.95 1248 1 1 (+0.2 ± 1.1) × 10 −5 (−1.6 ± 1.4) × 10 −62145 127.6 0.46 0.48 0.95 1248 1 1 (−1.2 ± 1.1) × 10 −5 (−1.7 ± 1.3) × 10 −62146 127.3 0.46 0.48 0.95 1249 1 0 (+2.4 ± 1.1) × 10 −5 (−1.4 ± 1.3) × 10 −62147 127.2 0.46 0.48 0.95 1246 2 2 (+0.3 ± 1.2) × 10 −5 (−1.3 ± 1.3) × 10 −62148 126.8 0.46 0.48 0.95 1245 1 4 (−1.7 ± 1.1) × 10 −5 (−1.5 ± 1.3) × 10 −62149 128.1 0.46 0.48 0.95 1246 1 3 (+1.0 ± 1.2) × 10 −5 (−1.4 ± 1.3) × 10 −62150 128.2 0.46 0.48 0.95 1245 1 4 (+2.4 ± 1.1) × 10 −5 (−1.0 ± 1.3) × 10 −62151 128.2 0.46 0.48 0.95 1248 1 1 (+0.8 ± 1.1) × 10 −5 (−0.9 ± 1.3) × 10 −62153 126.8 0.46 0.48 0.95 1128 2 4 (+1.4 ± 1.2) × 10 −5 (−0.8 ± 1.3) × 10 −62154 126.7 0.46 0.48 0.95 1245 1 4 (−0.2 ± 1.2) × 10 −5 (−0.8 ± 1.3) × 10 −62155 126.6 0.46 0.48 0.95 1245 1 4 (+0.6 ± 1.2) × 10 −5 (−0.7 ± 1.3) × 10 −62156 126.4 0.46 0.48 0.95 1245 1 4 (−1.5 ± 1.1) × 10 −5 (−0.9 ± 1.3) × 10 −62157 126.2 0.46 0.48 0.95 1243 1 6 (+2.1 ± 1.1) × 10 −5 (−0.6 ± 1.3) × 10 −62158 126.8 0.46 0.48 0.95 1237 1 12 (−1.4 ± 1.1) × 10 −5 (−0.8 ± 1.2) × 10 −6Table C.6:

265Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2159 127.4 0.46 0.48 0.95 1248 1 1 (+0.4 ± 1.1) × 10 −5 (−0.7 ± 1.2) × 10 −62160 127.7 0.46 0.48 0.95 1248 1 1 (−1.7 ± 1.1) × 10 −5 (−0.9 ± 1.2) × 10 −62161 127.9 0.46 0.48 0.95 1246 1 3 (+0.9 ± 1.1) × 10 −5 (−0.8 ± 1.2) × 10 −62162 127.4 0.46 0.48 0.95 1245 1 4 (+1.5 ± 1.1) × 10 −5 (−0.6 ± 1.2) × 10 −62163 127.0 0.46 0.48 0.95 1244 1 5 (−0.8 ± 1.2) × 10 −5 (−0.7 ± 1.2) × 10 −62164 128.9 0.46 0.48 0.95 1247 1 2 (+0.2 ± 1.1) × 10 −5 (−0.7 ± 1.2) × 10 −62165 129.3 0.46 0.48 0.95 1247 1 2 (−0.2 ± 1.2) × 10 −5 (−0.7 ± 1.2) × 10 −62166 128.9 0.46 0.48 0.95 1248 1 1 (+1.8 ± 1.1) × 10 −5 (−0.5 ± 1.2) × 10 −62167 128.9 0.46 0.48 0.95 1233 1 16 (−0.4 ± 1.2) × 10 −5 (−0.5 ± 1.2) × 10 −62168 128.6 0.46 0.48 0.95 1246 1 3 (−0.7 ± 1.1) × 10 −5 (−0.6 ± 1.2) × 10 −62169 127.9 0.46 0.48 0.95 1241 1 8 (+0.0 ± 1.1) × 10 −5 (−0.6 ± 1.2) × 10 −62170 128.1 0.46 0.48 0.95 1243 2 5 (+0.0 ± 1.2) × 10 −5 (−0.6 ± 1.2) × 10 −62171 129.0 0.46 0.48 0.95 1242 1 7 (−0.1 ± 1.2) × 10 −5 (−0.6 ± 1.2) × 10 −62172 128.9 0.46 0.48 0.95 1246 1 3 (−1.4 ± 1.2) × 10 −5 (−0.7 ± 1.2) × 10 −62173 129.2 0.46 0.48 0.95 1241 1 8 (−1.7 ± 1.2) × 10 −5 (−0.9 ± 1.1) × 10 −62174 129.0 0.46 0.48 0.95 1241 1 8 (−2.5 ± 1.1) × 10 −5 (−1.1 ± 1.1) × 10 −62175 129.2 0.46 0.48 0.95 1244 1 5 (+0.6 ± 1.2) × 10 −5 (−1.0 ± 1.1) × 10 −62176 129.9 0.46 0.48 0.95 1243 1 6 (+0.7 ± 1.1) × 10 −5 (−1.0 ± 1.1) × 10 −62177 129.9 0.46 0.48 0.95 1240 1 9 (+0.6 ± 1.1) × 10 −5 (−0.9 ± 1.1) × 10 −62178 130.1 0.46 0.48 0.95 1244 1 5 (+0.4 ± 1.1) × 10 −5 (−0.8 ± 1.1) × 10 −62179 129.6 0.46 0.48 0.95 1238 2 10 (+0.3 ± 1.1) × 10 −5 (−0.8 ± 1.1) × 10 −62180 129.6 0.46 0.48 0.95 1233 1 16 (+0.6 ± 1.1) × 10 −5 (−0.7 ± 1.1) × 10 −6Table C.7:

266Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2181 129.4 0.46 0.48 0.95 1234 1 15 (+0.5 ± 1.1) × 10 −5 (−0.7 ± 1.1) × 10 −62182 129.1 0.46 0.48 0.95 1235 1 14 (−0.5 ± 1.1) × 10 −5 (−0.7 ± 1.1) × 10 −62183 129.3 0.46 0.48 0.95 1243 1 6 (+0.8 ± 1.1) × 10 −5 (−0.6 ± 1.1) × 10 −62184 128.9 0.46 0.48 0.95 1241 1 8 (+1.0 ± 1.1) × 10 −5 (−0.5 ± 1.1) × 10 −62185 128.8 0.46 0.48 0.95 1242 1 7 (−0.8 ± 1.1) × 10 −5 (−0.6 ± 1.1) × 10 −62186 129.0 0.46 0.48 0.95 1235 1 14 (+0.6 ± 1.1) × 10 −5 (−0.6 ± 1.1) × 10 −62187 129.1 0.46 0.48 0.95 1240 1 9 (+0.7 ± 1.2) × 10 −5 (−0.5 ± 1.1) × 10 −62188 128.9 0.46 0.48 0.95 391 1 3 (+2.4 ± 2.1) × 10 −5 (−0.4 ± 1.1) × 10 −62195 131.0 0.46 0.48 0.95 651 1 1 (+4.4 ± 1.6) × 10 −5 (−0.2 ± 1.1) × 10 −62196 131.1 0.46 0.48 0.95 1247 1 2 (+0.2 ± 1.2) × 10 −5 (−0.2 ± 1.1) × 10 −62197 131.0 0.46 0.48 0.95 1248 1 1 (+0.0 ± 1.1) × 10 −5 (−0.2 ± 1.1) × 10 −62198 130.9 0.46 0.48 0.95 1247 1 2 (+0.0 ± 1.1) × 10 −5 (−0.2 ± 1.1) × 10 −62199 130.9 0.46 0.48 0.95 1246 1 3 (−0.5 ± 1.1) × 10 −5 (−0.2 ± 1.1) × 10 −62200 131.1 0.46 0.48 0.95 1248 1 1 (−1.1 ± 1.1) × 10 −5 (−0.3 ± 1.0) × 10 −62201 130.9 0.46 0.48 0.95 1248 1 1 (−0.5 ± 1.1) × 10 −5 (−0.4 ± 1.0) × 10 −62202 130.8 0.46 0.48 0.95 1245 1 4 (+0.7 ± 1.1) × 10 −5 (−0.3 ± 1.0) × 10 −62203 126.0 0.46 0.48 0.95 1237 2 5 (−0.7 ± 1.2) × 10 −5 (−0.4 ± 1.0) × 10 −62204 128.4 0.46 0.48 0.95 1243 1 6 (+0.1 ± 1.1) × 10 −5 (−0.3 ± 1.0) × 10 −62205 128.4 0.46 0.48 0.95 1249 1 0 (+0.4 ± 1.1) × 10 −5 (−0.3 ± 1.0) × 10 −62206 128.5 0.46 0.48 0.95 1247 1 2 (+1.9 ± 1.1) × 10 −5 (−0.1 ± 1.0) × 10 −62207 128.6 0.46 0.48 0.95 1245 2 3 (+2.6 ± 1.2) × 10 −5 (+0.1 ± 1.0) × 10 −62208 129.0 0.46 0.48 0.95 1247 1 2 (−1.9 ± 1.1) × 10 −5 (−0.1 ± 1.0) × 10 −6Table C.8:

267Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2209 128.9 0.46 0.48 0.95 1247 1 2 (+0.5 ± 1.2) × 10 −5 (−0.0 ± 1.0) × 10 −62210 129.1 0.46 0.48 0.95 1247 1 2 (−0.6 ± 1.2) × 10 −5 (−0.1 ± 1.0) × 10 −62211 129.1 0.46 0.48 0.95 1247 1 2 (+0.1 ± 1.2) × 10 −5 (−0.1 ± 1.0) × 10 −62212 129.1 0.46 0.48 0.95 1244 1 5 (+0.1 ± 1.1) × 10 −5 (−0.8 ± 10.0) × 10 −72213 129.1 0.46 0.48 0.95 1247 1 2 (−0.1 ± 1.2) × 10 −5 (−0.8 ± 9.9) × 10 −72214 129.1 0.46 0.48 0.95 1247 1 2 (+1.1 ± 1.1) × 10 −5 (+0.0 ± 9.9) × 10 −72215 129.4 0.46 0.48 0.95 1248 0 2 (−0.8 ± 1.1) × 10 −5 (−0.6 ± 9.9) × 10 −72218 129.2 0.46 0.48 0.95 1248 0 2 (+0.8 ± 1.1) × 10 −5 (+0.0 ± 9.8) × 10 −72219 129.1 0.46 0.48 0.95 1248 0 2 (−0.2 ± 1.1) × 10 −5 (−0.1 ± 9.8) × 10 −72220 129.2 0.46 0.48 0.95 1247 0 3 (−0.1 ± 1.1) × 10 −5 (−0.2 ± 9.8) × 10 −72221 129.4 0.46 0.48 0.95 1248 1 1 (+2.4 ± 1.2) × 10 −5 (+1.6 ± 9.7) × 10 −72222 129.2 0.46 0.48 0.95 1248 0 2 (+0.3 ± 1.2) × 10 −5 (+1.8 ± 9.7) × 10 −72223 129.1 0.46 0.48 0.95 1231 2 2 (+0.0 ± 1.2) × 10 −5 (+1.8 ± 9.7) × 10 −72224 129.1 0.46 0.48 0.95 1246 2 2 (+0.6 ± 1.1) × 10 −5 (+2.2 ± 9.6) × 10 −72225 129.0 0.46 0.48 0.95 1248 1 1 (+1.2 ± 1.2) × 10 −5 (+3.1 ± 9.6) × 10 −72226 129.2 0.46 0.48 0.95 1246 1 3 (−0.7 ± 1.1) × 10 −5 (+2.6 ± 9.6) × 10 −72227 128.8 0.46 0.48 0.95 1246 2 2 (+1.4 ± 1.1) × 10 −5 (+3.6 ± 9.5) × 10 −72228 128.7 0.46 0.48 0.95 1247 1 2 (+1.7 ± 1.1) × 10 −5 (+4.7 ± 9.5) × 10 −72229 128.7 0.46 0.48 0.95 1246 1 3 (+0.6 ± 1.1) × 10 −5 (+5.1 ± 9.5) × 10 −72230 128.8 0.46 0.48 0.95 1248 1 1 (−0.2 ± 1.1) × 10 −5 (+5.0 ± 9.4) × 10 −72231 128.7 0.46 0.48 0.95 1246 1 3 (+0.0 ± 1.2) × 10 −5 (+5.0 ± 9.4) × 10 −72232 128.8 0.46 0.48 0.95 1247 1 2 (+0.0 ± 1.1) × 10 −5 (+5.0 ± 9.4) × 10 −7Table C.9:

268Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2233 129.0 0.46 0.48 0.95 1246 2 2 (−0.8 ± 1.1) × 10 −5 (+4.4 ± 9.3) × 10 −72234 128.9 0.46 0.48 0.95 1247 1 2 (+0.5 ± 1.1) × 10 −5 (+4.7 ± 9.3) × 10 −72243 128.6 0.45 0.48 0.95 1244 0 6 (+0.9 ± 1.2) × 10 −5 (+5.3 ± 9.3) × 10 −72244 128.5 0.45 0.48 0.95 1241 0 9 (−1.0 ± 1.2) × 10 −5 (+4.6 ± 9.2) × 10 −72245 128.5 0.45 0.48 0.95 1241 0 9 (−0.9 ± 1.2) × 10 −5 (+3.9 ± 9.2) × 10 −72246 128.6 0.45 0.48 0.95 1234 1 15 (+3.4 ± 1.2) × 10 −5 (+6.1 ± 9.2) × 10 −72247 128.3 0.45 0.48 0.95 624 1 625 (−1.4 ± 1.7) × 10 −5 (+5.6 ± 9.2) × 10 −72248 128.2 0.45 0.47 0.95 1243 1 6 (+1.6 ± 1.2) × 10 −5 (+6.6 ± 9.1) × 10 −72249 126.6 0.45 0.47 0.95 1232 1 17 (+1.3 ± 1.2) × 10 −5 (+7.4 ± 9.1) × 10 −72250 128.3 0.45 0.47 0.95 230 1 3 (−1.7 ± 2.8) × 10 −5 (+7.2 ± 9.1) × 10 −72251 128.6 0.45 0.47 0.95 832 1 4 (+1.0 ± 1.4) × 10 −5 (+7.6 ± 9.1) × 10 −72252 128.9 0.45 0.47 0.95 360 1 2 (−1.4 ± 2.3) × 10 −5 (+7.3 ± 9.1) × 10 −72253 128.6 0.45 0.47 0.95 1238 1 11 (+1.2 ± 1.2) × 10 −5 (+8.0 ± 9.1) × 10 −72254 128.6 0.45 0.48 0.95 1236 1 13 (−0.9 ± 1.2) × 10 −5 (+7.4 ± 9.0) × 10 −72255 128.5 0.45 0.47 0.95 1243 1 6 (+1.8 ± 1.1) × 10 −5 (+8.5 ± 9.0) × 10 −72256 128.3 0.45 0.47 0.95 1245 1 4 (−0.6 ± 1.1) × 10 −5 (+8.1 ± 9.0) × 10 −72257 128.2 0.45 0.47 0.95 1240 1 9 (−0.1 ± 1.2) × 10 −5 (+8.0 ± 9.0) × 10 −72258 128.2 0.45 0.47 0.95 1232 1 17 (+2.2 ± 1.2) × 10 −5 (+9.3 ± 8.9) × 10 −72259 128.5 0.45 0.47 0.95 1237 1 12 (+1.1 ± 1.2) × 10 −5 (+9.9 ± 8.9) × 10 −72260 128.1 0.45 0.47 0.95 1230 1 19 (+1.9 ± 1.1) × 10 −5 (+10.9 ± 8.9) × 10 −72261 128.0 0.45 0.47 0.95 1235 1 14 (−2.7 ± 1.2) × 10 −5 (+9.3 ± 8.9) × 10 −72262 127.9 0.45 0.47 0.95 1239 0 10 (−0.0 ± 1.2) × 10 −5 (+9.2 ± 8.8) × 10 −7Table C.10:

269Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2263 127.6 0.45 0.47 0.95 1234 1 15 (−1.3 ± 1.2) × 10 −5 (+8.4 ± 8.8) × 10 −72264 127.6 0.45 0.47 0.95 1236 1 13 (−2.2 ± 1.2) × 10 −5 (+7.1 ± 8.8) × 10 −72267 127.5 0.45 0.47 0.95 1232 2 16 (+0.7 ± 1.2) × 10 −5 (+7.5 ± 8.8) × 10 −72268 127.4 0.45 0.47 0.95 1228 1 21 (−1.1 ± 1.1) × 10 −5 (+6.8 ± 8.7) × 10 −72269 127.6 0.45 0.47 0.95 1245 1 4 (−0.7 ± 1.2) × 10 −5 (+6.4 ± 8.7) × 10 −72270 127.7 0.45 0.47 0.95 1245 1 4 (+0.2 ± 1.1) × 10 −5 (+6.5 ± 8.7) × 10 −72271 127.9 0.45 0.47 0.95 1244 1 5 (−0.3 ± 1.2) × 10 −5 (+6.3 ± 8.7) × 10 −72272 127.8 0.45 0.47 0.95 1248 1 1 (−0.8 ± 1.2) × 10 −5 (+5.8 ± 8.6) × 10 −72273 127.7 0.45 0.47 0.95 1248 1 1 (+1.4 ± 1.2) × 10 −5 (+6.5 ± 8.6) × 10 −72274 128.0 0.45 0.47 0.95 1247 1 2 (+0.8 ± 1.1) × 10 −5 (+6.9 ± 8.6) × 10 −72275 127.8 0.45 0.47 0.95 1246 2 2 (−1.0 ± 1.2) × 10 −5 (+6.3 ± 8.6) × 10 −72276 127.8 0.45 0.47 0.95 1241 1 8 (+1.6 ± 1.2) × 10 −5 (+7.1 ± 8.5) × 10 −72277 127.8 0.45 0.47 0.95 1247 1 2 (+1.1 ± 1.1) × 10 −5 (+7.7 ± 8.5) × 10 −72278 127.7 0.45 0.47 0.95 1242 3 5 (+1.5 ± 1.2) × 10 −5 (+8.5 ± 8.5) × 10 −72279 127.9 0.45 0.47 0.95 1242 1 7 (+0.6 ± 1.2) × 10 −5 (+8.8 ± 8.5) × 10 −72280 128.2 0.45 0.47 0.95 1242 1 7 (+1.2 ± 1.2) × 10 −5 (+9.4 ± 8.4) × 10 −72281 128.0 0.45 0.47 0.95 1244 1 5 (−1.7 ± 1.2) × 10 −5 (+8.4 ± 8.4) × 10 −72282 128.0 0.45 0.47 0.95 1244 1 5 (−0.1 ± 1.2) × 10 −5 (+8.3 ± 8.4) × 10 −72283 127.9 0.45 0.47 0.95 1240 2 8 (+0.8 ± 1.2) × 10 −5 (+8.7 ± 8.4) × 10 −72284 128.0 0.45 0.47 0.95 1245 2 3 (+1.3 ± 1.2) × 10 −5 (+9.3 ± 8.4) × 10 −72285 127.9 0.45 0.47 0.95 1246 1 3 (+0.6 ± 1.2) × 10 −5 (+9.6 ± 8.3) × 10 −72286 128.1 0.45 0.47 0.95 1246 1 3 (−1.2 ± 1.2) × 10 −5 (+9.0 ± 8.3) × 10 −7Table C.11:

270Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2287 128.1 0.45 0.47 0.95 1247 1 2 (+0.8 ± 1.1) × 10 −5 (+9.3 ± 8.3) × 10 −72288 128.0 0.45 0.47 0.95 1240 1 9 (+1.5 ± 1.2) × 10 −5 (+10.1 ± 8.3) × 10 −72289 128.2 0.45 0.47 0.95 1245 1 4 (−1.0 ± 1.2) × 10 −5 (+9.5 ± 8.3) × 10 −72290 127.9 0.45 0.47 0.95 1245 1 4 (−0.6 ± 1.1) × 10 −5 (+9.1 ± 8.2) × 10 −72291 128.1 0.45 0.47 0.95 1199 2 49 (+0.2 ± 1.2) × 10 −5 (+9.2 ± 8.2) × 10 −72301 128.4 0.45 0.47 0.95 1244 1 5 (+2.7 ± 1.1) × 10 −5 (+10.5 ± 8.2) × 10 −72302 128.4 0.45 0.47 0.95 1248 1 1 (−2.5 ± 1.2) × 10 −5 (+9.2 ± 8.2) × 10 −72303 128.5 0.45 0.47 0.95 1242 1 7 (−0.1 ± 1.1) × 10 −5 (+9.1 ± 8.1) × 10 −72304 128.5 0.45 0.47 0.95 1238 1 11 (−1.7 ± 1.2) × 10 −5 (+8.2 ± 8.1) × 10 −72305 128.6 0.45 0.47 0.95 1239 0 10 (−2.8 ± 1.1) × 10 −5 (+6.7 ± 8.1) × 10 −72306 128.1 0.45 0.47 0.95 1243 1 6 (−0.5 ± 1.2) × 10 −5 (+6.4 ± 8.1) × 10 −72307 128.0 0.45 0.47 0.95 1247 1 2 (+1.3 ± 1.1) × 10 −5 (+7.0 ± 8.1) × 10 −72308 127.7 0.45 0.47 0.95 1244 1 5 (−1.1 ± 1.1) × 10 −5 (+6.5 ± 8.0) × 10 −72310 128.1 0.45 0.47 0.95 1248 1 1 (+0.6 ± 1.2) × 10 −5 (+6.7 ± 8.0) × 10 −72311 127.9 0.45 0.47 0.95 1246 1 3 (−0.2 ± 1.2) × 10 −5 (+6.6 ± 8.0) × 10 −72312 128.1 0.45 0.47 0.95 1247 1 2 (+1.0 ± 1.2) × 10 −5 (+7.1 ± 8.0) × 10 −72313 127.9 0.45 0.47 0.95 1242 1 7 (−0.2 ± 1.2) × 10 −5 (+6.9 ± 8.0) × 10 −72314 127.8 0.45 0.47 0.95 1243 1 6 (+0.3 ± 1.2) × 10 −5 (+7.0 ± 8.0) × 10 −72315 127.7 0.45 0.47 0.95 1247 1 2 (+0.3 ± 1.2) × 10 −5 (+7.1 ± 7.9) × 10 −72316 127.7 0.45 0.47 0.95 1238 1 11 (+1.3 ± 1.1) × 10 −5 (+7.7 ± 7.9) × 10 −72317 127.8 0.45 0.47 0.95 1243 1 6 (+2.7 ± 1.1) × 10 −5 (+9.0 ± 7.9) × 10 −72319 127.6 0.45 0.47 0.95 1217 1 32 (−0.4 ± 1.1) × 10 −5 (+8.7 ± 7.9) × 10 −7Table C.12:

271Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2320 127.6 0.45 0.47 0.95 1228 1 21 (−0.8 ± 1.2) × 10 −5 (+8.3 ± 7.9) × 10 −72321 127.3 0.45 0.47 0.95 1211 1 27 (+0.3 ± 1.2) × 10 −5 (+8.4 ± 7.8) × 10 −72322 127.3 0.45 0.47 0.95 1229 1 20 (+0.2 ± 1.2) × 10 −5 (+8.5 ± 7.8) × 10 −72323 127.0 0.45 0.47 0.95 693 1 556 (+1.2 ± 1.6) × 10 −5 (+8.8 ± 7.8) × 10 −72324 127.1 0.45 0.47 0.95 1238 1 11 (−2.4 ± 1.2) × 10 −5 (+7.6 ± 7.8) × 10 −72326 127.2 0.45 0.47 0.95 1236 1 13 (−0.4 ± 1.2) × 10 −5 (+7.4 ± 7.8) × 10 −72327 126.7 0.45 0.47 0.95 1241 1 8 (+0.3 ± 1.2) × 10 −5 (+7.5 ± 7.8) × 10 −72328 127.0 0.45 0.47 0.95 1240 1 9 (+1.4 ± 1.2) × 10 −5 (+8.1 ± 7.8) × 10 −72329 127.0 0.45 0.47 0.95 1233 1 16 (−0.5 ± 1.2) × 10 −5 (+7.9 ± 7.7) × 10 −72331 126.8 0.45 0.47 0.95 1216 1 33 (+0.1 ± 1.2) × 10 −5 (+7.9 ± 7.7) × 10 −72332 127.4 0.45 0.47 0.95 124 0 1 (+0.5 ± 3.5) × 10 −5 (+7.9 ± 7.7) × 10 −72333 127.2 0.45 0.47 0.95 124 0 1 (−3.2 ± 3.7) × 10 −5 (+7.7 ± 7.7) × 10 −72334 127.0 0.45 0.47 0.95 123 0 2 (−1.1 ± 3.7) × 10 −5 (+7.7 ± 7.7) × 10 −72335 126.7 0.45 0.47 0.95 122 0 3 (−0.3 ± 3.6) × 10 −5 (+7.7 ± 7.7) × 10 −72336 126.9 0.45 0.47 0.95 121 0 4 (−6.2 ± 3.8) × 10 −5 (+7.4 ± 7.7) × 10 −72337 126.8 0.45 0.47 0.95 121 0 4 (−6.1 ± 3.5) × 10 −5 (+7.1 ± 7.7) × 10 −72338 126.9 0.45 0.47 0.95 121 0 4 (−2.6 ± 3.8) × 10 −5 (+7.0 ± 7.7) × 10 −72339 126.9 0.45 0.47 0.95 120 0 5 (−0.7 ± 3.6) × 10 −5 (+7.0 ± 7.7) × 10 −72340 127.5 0.45 0.47 0.95 119 1 5 (−4.5 ± 3.7) × 10 −5 (+6.8 ± 7.7) × 10 −72341 127.2 0.45 0.47 0.95 124 0 1 (−2.2 ± 3.7) × 10 −5 (+6.7 ± 7.7) × 10 −72342 127.3 0.45 0.47 0.95 122 0 3 (+0.3 ± 3.9) × 10 −5 (+6.7 ± 7.7) × 10 −72343 127.3 0.45 0.47 0.95 122 0 3 (−2.2 ± 3.5) × 10 −5 (+6.6 ± 7.7) × 10 −7Table C.13:

272Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2344 126.8 0.45 0.47 0.95 122 0 3 (+0.8 ± 4.0) × 10 −5 (+6.6 ± 7.7) × 10 −72345 127.2 0.45 0.47 0.95 123 0 2 (+3.3 ± 3.9) × 10 −5 (+6.8 ± 7.7) × 10 −72346 127.7 0.45 0.47 0.95 123 0 2 (−1.8 ± 3.4) × 10 −5 (+6.7 ± 7.7) × 10 −72347 127.3 0.45 0.47 0.95 125 0 0 (−1.7 ± 3.3) × 10 −5 (+6.6 ± 7.7) × 10 −72348 126.9 0.45 0.47 0.95 124 0 1 (−1.0 ± 3.1) × 10 −5 (+6.6 ± 7.7) × 10 −72349 127.1 0.45 0.47 0.95 125 0 0 (+11.0 ± 3.9) × 10 −5 (+7.0 ± 7.7) × 10 −72350 127.1 0.45 0.47 0.95 124 0 1 (+1.2 ± 3.2) × 10 −5 (+7.1 ± 7.7) × 10 −72351 127.3 0.45 0.47 0.95 125 0 0 (+7.8 ± 3.7) × 10 −5 (+7.4 ± 7.7) × 10 −72352 127.2 0.45 0.47 0.95 124 0 1 (−3.1 ± 4.1) × 10 −5 (+7.3 ± 7.7) × 10 −72353 126.6 0.45 0.47 0.95 1242 0 8 (−1.0 ± 1.2) × 10 −5 (+6.8 ± 7.7) × 10 −72354 126.7 0.45 0.47 0.95 1246 0 4 (+0.2 ± 1.2) × 10 −5 (+6.9 ± 7.6) × 10 −72355 126.9 0.45 0.47 0.95 1245 1 4 (+0.1 ± 1.2) × 10 −5 (+6.9 ± 7.6) × 10 −72356 127.0 0.45 0.47 0.95 1242 1 7 (+0.1 ± 1.2) × 10 −5 (+6.9 ± 7.6) × 10 −72357 126.9 0.45 0.47 0.95 1239 1 10 (+0.6 ± 1.2) × 10 −5 (+7.2 ± 7.6) × 10 −72358 127.4 0.45 0.47 0.95 1248 0 2 (−0.6 ± 1.2) × 10 −5 (+6.8 ± 7.6) × 10 −72360 127.3 0.45 0.47 0.95 1241 0 9 (−0.8 ± 1.2) × 10 −5 (+6.5 ± 7.6) × 10 −72361 127.6 0.45 0.47 0.95 1247 0 3 (−0.6 ± 1.2) × 10 −5 (+6.2 ± 7.6) × 10 −72362 127.8 0.45 0.47 0.95 1245 0 5 (−0.1 ± 1.1) × 10 −5 (+6.1 ± 7.5) × 10 −72363 128.4 0.45 0.47 0.95 1249 0 1 (−1.7 ± 1.2) × 10 −5 (+5.4 ± 7.5) × 10 −72364 127.3 0.45 0.47 0.95 1242 1 7 (+1.2 ± 1.2) × 10 −5 (+5.9 ± 7.5) × 10 −72385 126.4 0.45 0.47 0.95 1241 1 8 (−1.6 ± 1.2) × 10 −5 (+5.2 ± 7.5) × 10 −72386 126.9 0.45 0.47 0.95 1246 1 3 (+0.4 ± 1.2) × 10 −5 (+5.3 ± 7.5) × 10 −7Table C.14:

273Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2387 129.0 0.45 0.47 0.95 1246 1 3 (−1.8 ± 1.2) × 10 −5 (+4.6 ± 7.5) × 10 −72388 128.0 0.45 0.47 0.95 1246 1 3 (+1.1 ± 1.1) × 10 −5 (+5.0 ± 7.4) × 10 −72389 127.7 0.45 0.47 0.95 1222 2 5 (−0.5 ± 1.2) × 10 −5 (+4.8 ± 7.4) × 10 −72390 127.7 0.45 0.47 0.95 1245 1 4 (−0.2 ± 1.1) × 10 −5 (+4.7 ± 7.4) × 10 −72391 127.8 0.45 0.47 0.95 1249 1 0 (+1.3 ± 1.1) × 10 −5 (+5.2 ± 7.4) × 10 −72392 127.9 0.45 0.47 0.95 1248 1 1 (+1.2 ± 1.2) × 10 −5 (+5.7 ± 7.4) × 10 −72393 127.9 0.45 0.47 0.95 1242 1 7 (+0.8 ± 1.1) × 10 −5 (+6.0 ± 7.4) × 10 −72394 127.8 0.45 0.47 0.95 1249 1 0 (−1.4 ± 1.2) × 10 −5 (+5.4 ± 7.4) × 10 −72396 115.0 0.45 0.47 0.95 1029 2 216 (−0.8 ± 1.4) × 10 −5 (+5.1 ± 7.3) × 10 −72397 126.2 0.45 0.47 0.95 1200 1 49 (+0.4 ± 1.2) × 10 −5 (+5.2 ± 7.3) × 10 −72398 126.2 0.45 0.47 0.95 1223 0 27 (−3.5 ± 1.2) × 10 −5 (+3.8 ± 7.3) × 10 −72399 127.3 0.45 0.47 0.95 1210 1 30 (−0.8 ± 1.1) × 10 −5 (+3.5 ± 7.3) × 10 −72400 127.0 0.45 0.47 0.95 1237 1 12 (−0.7 ± 1.2) × 10 −5 (+3.2 ± 7.3) × 10 −72401 127.7 0.45 0.47 0.95 1210 1 39 (−0.1 ± 1.2) × 10 −5 (+3.2 ± 7.3) × 10 −72402 127.5 0.45 0.47 0.95 1221 1 28 (−0.6 ± 1.2) × 10 −5 (+2.9 ± 7.3) × 10 −72403 124.6 0.45 0.47 0.95 1140 1 65 (−0.9 ± 1.2) × 10 −5 (+2.6 ± 7.2) × 10 −72404 127.6 0.45 0.47 0.95 1235 0 15 (−0.5 ± 1.2) × 10 −5 (+2.4 ± 7.2) × 10 −72405 127.7 0.45 0.47 0.95 1229 2 20 (+0.7 ± 1.2) × 10 −5 (+2.6 ± 7.2) × 10 −72406 125.7 0.45 0.47 0.95 1221 1 28 (+1.2 ± 1.2) × 10 −5 (+3.1 ± 7.2) × 10 −72407 126.8 0.45 0.47 0.95 1240 1 9 (+1.0 ± 1.2) × 10 −5 (+3.4 ± 7.2) × 10 −72409 127.2 0.45 0.47 0.95 1247 0 3 (+1.4 ± 1.1) × 10 −5 (+4.0 ± 7.2) × 10 −72410 127.5 0.45 0.47 0.95 1246 0 4 (+1.7 ± 1.1) × 10 −5 (+4.6 ± 7.2) × 10 −7Table C.15:

274Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2411 127.6 0.45 0.47 0.95 1240 0 10 (+0.3 ± 1.1) × 10 −5 (+4.7 ± 7.2) × 10 −72412 124.8 0.45 0.47 0.95 680 0 570 (−0.1 ± 1.7) × 10 −5 (+4.7 ± 7.1) × 10 −72413 128.3 0.45 0.47 0.95 839 1 4 (−1.1 ± 1.4) × 10 −5 (+4.4 ± 7.1) × 10 −72415 129.7 0.45 0.47 0.95 1236 1 13 (−0.8 ± 1.2) × 10 −5 (+4.1 ± 7.1) × 10 −72416 129.3 0.45 0.47 0.95 1244 1 5 (−0.3 ± 1.2) × 10 −5 (+4.0 ± 7.1) × 10 −72417 128.9 0.45 0.47 0.95 1245 1 4 (+0.4 ± 1.2) × 10 −5 (+4.1 ± 7.1) × 10 −72420 127.1 0.45 0.47 0.95 1240 1 9 (−1.5 ± 1.2) × 10 −5 (+3.5 ± 7.1) × 10 −72421 127.6 0.45 0.47 0.95 1227 1 22 (+1.3 ± 1.1) × 10 −5 (+4.0 ± 7.1) × 10 −72422 127.3 0.45 0.47 0.95 1239 0 10 (−0.1 ± 1.2) × 10 −5 (+3.9 ± 7.1) × 10 −72423 125.1 0.45 0.47 0.95 1207 0 18 (−2.1 ± 1.2) × 10 −5 (+3.2 ± 7.0) × 10 −72427 128.4 0.45 0.47 0.95 1248 1 1 (+0.3 ± 1.1) × 10 −5 (+3.3 ± 7.0) × 10 −72428 128.8 0.45 0.47 0.95 1244 1 5 (−0.2 ± 1.2) × 10 −5 (+3.2 ± 7.0) × 10 −72429 128.8 0.45 0.47 0.95 1246 1 3 (+2.1 ± 1.1) × 10 −5 (+4.0 ± 7.0) × 10 −72430 129.0 0.45 0.47 0.95 1248 1 1 (−0.6 ± 1.2) × 10 −5 (+3.7 ± 7.0) × 10 −72431 129.7 0.45 0.47 0.95 1247 1 2 (+0.0 ± 1.1) × 10 −5 (+3.7 ± 7.0) × 10 −72432 129.2 0.45 0.47 0.95 1248 1 1 (+2.7 ± 1.2) × 10 −5 (+4.7 ± 7.0) × 10 −72433 129.5 0.45 0.47 0.95 1248 1 1 (+0.2 ± 1.1) × 10 −5 (+4.8 ± 7.0) × 10 −72434 129.3 0.45 0.47 0.95 1246 1 3 (+0.8 ± 1.2) × 10 −5 (+5.1 ± 6.9) × 10 −72435 129.7 0.45 0.47 0.95 1248 1 1 (+0.3 ± 1.1) × 10 −5 (+5.1 ± 6.9) × 10 −72436 129.4 0.45 0.47 0.95 1246 1 3 (+1.2 ± 1.1) × 10 −5 (+5.5 ± 6.9) × 10 −72437 129.5 0.45 0.47 0.95 1249 1 0 (+1.6 ± 1.1) × 10 −5 (+6.1 ± 6.9) × 10 −72438 129.6 0.45 0.47 0.95 1248 1 1 (−0.1 ± 1.1) × 10 −5 (+6.1 ± 6.9) × 10 −7Table C.16:

275Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2439 129.1 0.45 0.47 0.95 1246 1 3 (−1.6 ± 1.2) × 10 −5 (+5.5 ± 6.9) × 10 −72440 129.7 0.45 0.47 0.95 1247 1 2 (+1.0 ± 1.2) × 10 −5 (+5.8 ± 6.9) × 10 −72441 129.9 0.45 0.47 0.95 1249 1 0 (+0.0 ± 1.2) × 10 −5 (+5.8 ± 6.9) × 10 −72442 129.9 0.45 0.47 0.95 1244 3 3 (−1.4 ± 1.1) × 10 −5 (+5.3 ± 6.8) × 10 −72443 129.8 0.45 0.47 0.95 1249 1 0 (−0.2 ± 1.2) × 10 −5 (+5.2 ± 6.8) × 10 −72444 129.9 0.45 0.47 0.95 1246 2 2 (−1.1 ± 1.1) × 10 −5 (+4.8 ± 6.8) × 10 −72447 130.0 0.45 0.47 0.95 1245 1 4 (+1.2 ± 1.2) × 10 −5 (+5.2 ± 6.8) × 10 −72448 129.7 0.45 0.47 0.95 1247 1 2 (−2.2 ± 1.2) × 10 −5 (+4.4 ± 6.8) × 10 −72449 124.2 0.45 0.47 0.95 1236 2 5 (−0.0 ± 1.2) × 10 −5 (+4.4 ± 6.8) × 10 −72450 128.6 0.45 0.47 0.95 1237 1 12 (+1.3 ± 1.2) × 10 −5 (+4.8 ± 6.8) × 10 −72451 129.7 0.45 0.47 0.95 1245 1 4 (−2.2 ± 1.2) × 10 −5 (+4.1 ± 6.8) × 10 −72452 128.5 0.45 0.47 0.95 1241 2 7 (−1.0 ± 1.1) × 10 −5 (+3.7 ± 6.8) × 10 −72453 129.0 0.45 0.47 0.95 1245 2 3 (−1.9 ± 1.2) × 10 −5 (+3.1 ± 6.7) × 10 −72454 128.9 0.45 0.47 0.95 1244 2 4 (−0.6 ± 1.2) × 10 −5 (+2.9 ± 6.7) × 10 −72455 129.1 0.45 0.47 0.95 1246 3 1 (+0.2 ± 1.2) × 10 −5 (+2.9 ± 6.7) × 10 −72456 129.4 0.45 0.47 0.95 124 1 0 (−0.7 ± 3.8) × 10 −5 (+2.9 ± 6.7) × 10 −72457 130.1 0.45 0.47 0.95 1249 1 0 (−1.3 ± 1.1) × 10 −5 (+2.4 ± 6.7) × 10 −72458 130.2 0.45 0.47 0.95 1248 1 1 (−0.7 ± 1.1) × 10 −5 (+2.2 ± 6.7) × 10 −72459 130.1 0.45 0.47 0.95 1247 1 2 (−0.1 ± 1.1) × 10 −5 (+2.1 ± 6.7) × 10 −72462 130.4 0.45 0.47 0.95 1249 1 0 (−0.6 ± 1.2) × 10 −5 (+1.9 ± 6.7) × 10 −72463 130.3 0.45 0.47 0.95 1248 1 1 (+1.4 ± 1.1) × 10 −5 (+2.4 ± 6.7) × 10 −72464 130.5 0.45 0.47 0.95 1248 1 1 (+1.2 ± 1.1) × 10 −5 (+2.8 ± 6.6) × 10 −7Table C.17:

276Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2465 131.2 0.45 0.47 0.95 1247 1 2 (−0.7 ± 1.1) × 10 −5 (+2.6 ± 6.6) × 10 −72466 129.6 0.45 0.47 0.95 1244 3 3 (+0.6 ± 1.1) × 10 −5 (+2.8 ± 6.6) × 10 −72467 129.6 0.45 0.47 0.95 1246 1 3 (−0.5 ± 1.1) × 10 −5 (+2.6 ± 6.6) × 10 −72468 129.5 0.45 0.47 0.95 1247 2 1 (−0.3 ± 1.2) × 10 −5 (+2.5 ± 6.6) × 10 −72469 129.9 0.45 0.47 0.95 1249 1 0 (+1.4 ± 1.2) × 10 −5 (+2.9 ± 6.6) × 10 −72470 129.6 0.45 0.47 0.95 1249 1 0 (−1.3 ± 1.2) × 10 −5 (+2.5 ± 6.6) × 10 −72471 129.9 0.45 0.47 0.95 1248 1 1 (−1.0 ± 1.1) × 10 −5 (+2.1 ± 6.6) × 10 −72472 129.7 0.45 0.47 0.95 1247 1 2 (−0.2 ± 1.2) × 10 −5 (+2.1 ± 6.6) × 10 −72473 129.2 0.45 0.47 0.95 1249 1 0 (+1.9 ± 1.1) × 10 −5 (+2.7 ± 6.5) × 10 −72474 129.6 0.45 0.47 0.95 683 1 2 (+0.2 ± 1.6) × 10 −5 (+2.7 ± 6.5) × 10 −72478 129.6 0.45 0.47 0.95 900 1 349 (+0.5 ± 1.4) × 10 −5 (+2.8 ± 6.5) × 10 −72479 129.7 0.45 0.47 0.95 1232 1 17 (−0.4 ± 1.1) × 10 −5 (+2.7 ± 6.5) × 10 −72480 129.5 0.45 0.47 0.95 1248 1 1 (+1.0 ± 1.1) × 10 −5 (+3.0 ± 6.5) × 10 −72481 129.5 0.45 0.47 0.95 1248 1 1 (−1.2 ± 1.1) × 10 −5 (+2.6 ± 6.5) × 10 −72482 129.3 0.45 0.47 0.95 1249 1 0 (−1.2 ± 1.2) × 10 −5 (+2.2 ± 6.5) × 10 −72483 129.3 0.45 0.47 0.95 1247 1 2 (+2.0 ± 1.2) × 10 −5 (+2.8 ± 6.5) × 10 −72484 129.3 0.45 0.47 0.95 1248 1 1 (+2.5 ± 1.2) × 10 −5 (+3.6 ± 6.5) × 10 −72485 129.5 0.45 0.47 0.95 1247 1 2 (+1.5 ± 1.2) × 10 −5 (+4.1 ± 6.5) × 10 −72486 129.4 0.45 0.47 0.95 1248 1 1 (+1.4 ± 1.1) × 10 −5 (+4.5 ± 6.5) × 10 −72487 129.0 0.45 0.47 0.95 1246 1 3 (+0.5 ± 1.2) × 10 −5 (+4.6 ± 6.4) × 10 −72488 128.8 0.45 0.47 0.95 1248 1 1 (−2.2 ± 1.1) × 10 −5 (+3.9 ± 6.4) × 10 −72489 129.0 0.45 0.47 0.95 1247 1 2 (−0.9 ± 1.2) × 10 −5 (+3.6 ± 6.4) × 10 −7Table C.18:

277Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2490 128.6 0.45 0.47 0.95 1240 1 9 (+0.2 ± 1.2) × 10 −5 (+3.7 ± 6.4) × 10 −72493 128.6 0.45 0.47 0.95 1247 1 2 (−0.2 ± 1.2) × 10 −5 (+3.6 ± 6.4) × 10 −72494 128.4 0.45 0.47 0.95 1247 1 2 (+0.5 ± 1.2) × 10 −5 (+3.8 ± 6.4) × 10 −72495 128.5 0.45 0.47 0.95 1245 1 4 (−1.7 ± 1.1) × 10 −5 (+3.2 ± 6.4) × 10 −72496 128.5 0.45 0.47 0.95 1246 1 3 (−0.4 ± 1.2) × 10 −5 (+3.1 ± 6.4) × 10 −72497 128.3 0.45 0.47 0.95 1248 2 0 (−1.0 ± 1.2) × 10 −5 (+2.8 ± 6.4) × 10 −72498 128.1 0.45 0.47 0.95 1243 1 6 (+0.2 ± 1.2) × 10 −5 (+2.8 ± 6.4) × 10 −72499 128.1 0.45 0.47 0.95 1243 1 6 (+0.4 ± 1.2) × 10 −5 (+2.9 ± 6.3) × 10 −72500 127.9 0.45 0.47 0.95 1245 2 3 (+0.9 ± 1.2) × 10 −5 (+3.2 ± 6.3) × 10 −72501 127.8 0.45 0.47 0.95 1244 1 5 (−0.5 ± 1.2) × 10 −5 (+3.0 ± 6.3) × 10 −72502 127.9 0.45 0.47 0.95 1246 1 3 (−1.3 ± 1.2) × 10 −5 (+2.6 ± 6.3) × 10 −72503 128.2 0.45 0.47 0.95 1246 1 3 (−0.7 ± 1.2) × 10 −5 (+2.4 ± 6.3) × 10 −72504 128.1 0.45 0.47 0.95 1242 1 7 (+1.7 ± 1.2) × 10 −5 (+2.9 ± 6.3) × 10 −72507 127.4 0.45 0.47 0.95 1240 1 9 (+1.3 ± 1.2) × 10 −5 (+3.3 ± 6.3) × 10 −72508 127.8 0.45 0.47 0.95 1245 1 4 (+0.2 ± 1.2) × 10 −5 (+3.4 ± 6.3) × 10 −72509 127.5 0.45 0.47 0.95 1240 1 9 (−2.6 ± 1.2) × 10 −5 (+2.6 ± 6.3) × 10 −72510 127.3 0.45 0.47 0.95 1238 1 11 (+0.1 ± 1.2) × 10 −5 (+2.6 ± 6.3) × 10 −72511 126.7 0.45 0.47 0.95 1236 1 13 (−0.1 ± 1.1) × 10 −5 (+2.6 ± 6.3) × 10 −72512 127.5 0.45 0.47 0.95 1239 1 10 (+0.6 ± 1.2) × 10 −5 (+2.7 ± 6.2) × 10 −72513 128.7 0.45 0.47 0.95 1244 1 5 (+0.7 ± 1.2) × 10 −5 (+2.9 ± 6.2) × 10 −72514 128.5 0.45 0.47 0.95 1246 1 3 (−1.2 ± 1.1) × 10 −5 (+2.5 ± 6.2) × 10 −72515 128.3 0.45 0.47 0.95 1247 1 2 (+0.6 ± 1.2) × 10 −5 (+2.7 ± 6.2) × 10 −7Table C.19:

278Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2516 128.3 0.45 0.47 0.95 1248 1 1 (−0.2 ± 1.2) × 10 −5 (+2.6 ± 6.2) × 10 −72517 127.0 0.45 0.47 0.95 1240 1 9 (−1.9 ± 1.2) × 10 −5 (+2.1 ± 6.2) × 10 −72518 128.0 0.45 0.47 0.95 1239 1 10 (+1.2 ± 1.2) × 10 −5 (+2.4 ± 6.2) × 10 −72521 121.4 0.45 0.47 0.95 1243 1 6 (+0.3 ± 1.1) × 10 −5 (+2.5 ± 6.2) × 10 −72522 122.0 0.45 0.47 0.95 1243 1 6 (−1.1 ± 1.2) × 10 −5 (+2.2 ± 6.2) × 10 −72523 123.5 0.45 0.47 0.95 1247 1 2 (−0.9 ± 1.2) × 10 −5 (+2.0 ± 6.2) × 10 −72524 128.0 0.45 0.47 0.95 1240 1 9 (−1.9 ± 1.2) × 10 −5 (+1.4 ± 6.2) × 10 −72525 128.2 0.45 0.47 0.95 1245 1 4 (−0.2 ± 1.2) × 10 −5 (+1.4 ± 6.2) × 10 −72526 128.2 0.45 0.47 0.95 1245 1 4 (−0.4 ± 1.2) × 10 −5 (+1.2 ± 6.1) × 10 −72527 128.2 0.45 0.47 0.95 831 1 1 (−1.2 ± 1.5) × 10 −5 (+1.0 ± 6.1) × 10 −72540 128.7 0.45 0.47 0.95 1250 0 0 (+1.1 ± 1.2) × 10 −5 (+1.3 ± 6.1) × 10 −72541 128.2 0.45 0.47 0.95 1248 0 2 (+0.3 ± 1.1) × 10 −5 (+1.4 ± 6.1) × 10 −72542 127.9 0.45 0.47 0.95 1245 1 4 (−0.0 ± 1.1) × 10 −5 (+1.4 ± 6.1) × 10 −72543 127.9 0.45 0.47 0.95 1248 0 2 (−0.2 ± 1.2) × 10 −5 (+1.4 ± 6.1) × 10 −72544 127.9 0.45 0.47 0.95 1246 0 4 (−1.4 ± 1.2) × 10 −5 (+1.0 ± 6.1) × 10 −72545 128.2 0.45 0.47 0.95 1245 1 4 (+0.4 ± 1.2) × 10 −5 (+1.1 ± 6.1) × 10 −72546 128.6 0.45 0.47 0.95 1246 0 4 (−1.2 ± 1.2) × 10 −5 (+0.7 ± 6.1) × 10 −72547 129.4 0.45 0.47 0.95 1247 1 2 (−0.6 ± 1.2) × 10 −5 (+0.6 ± 6.1) × 10 −72548 129.2 0.45 0.47 0.95 1250 0 0 (+0.1 ± 1.1) × 10 −5 (+0.6 ± 6.1) × 10 −72549 129.3 0.45 0.47 0.95 1248 0 2 (+0.1 ± 1.2) × 10 −5 (+0.6 ± 6.1) × 10 −72550 129.5 0.45 0.47 0.95 1249 1 0 (−1.8 ± 1.2) × 10 −5 (+0.1 ± 6.0) × 10 −72551 129.2 0.45 0.47 0.95 1250 0 0 (−1.1 ± 1.2) × 10 −5 (−0.2 ± 6.0) × 10 −7Table C.20:

279Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2554 129.5 0.45 0.47 0.95 125 0 0 (+5.0 ± 4.0) × 10 −5 (−0.0 ± 6.0) × 10 −72555 129.6 0.45 0.47 0.95 1244 0 6 (+2.2 ± 1.2) × 10 −5 (+0.6 ± 6.0) × 10 −72556 126.6 0.45 0.47 0.95 1243 0 7 (−0.3 ± 1.2) × 10 −5 (+0.5 ± 6.0) × 10 −72557 127.8 0.45 0.47 0.95 1247 0 3 (−0.6 ± 1.1) × 10 −5 (+0.5 ± 6.0) × 10 −72558 127.5 0.45 0.47 0.95 1250 0 0 (+2.8 ± 1.2) × 10 −5 (+1.2 ± 6.0) × 10 −72560 127.8 0.45 0.47 0.95 1248 1 1 (+0.6 ± 1.2) × 10 −5 (+1.4 ± 6.0) × 10 −72561 127.8 0.45 0.47 0.95 1248 0 1 (+1.0 ± 1.2) × 10 −5 (+1.7 ± 6.0) × 10 −72562 127.7 0.45 0.47 0.95 1248 1 1 (+1.6 ± 1.2) × 10 −5 (+2.1 ± 6.0) × 10 −72563 128.5 0.45 0.47 0.95 1246 1 3 (+2.1 ± 1.1) × 10 −5 (+2.6 ± 6.0) × 10 −72564 130.0 0.45 0.47 0.95 1244 2 4 (+0.1 ± 1.2) × 10 −5 (+2.7 ± 6.0) × 10 −72567 129.4 0.45 0.47 0.95 1248 2 0 (−1.6 ± 1.1) × 10 −5 (+2.2 ± 6.0) × 10 −72568 129.0 0.45 0.47 0.95 1246 0 3 (+1.2 ± 1.1) × 10 −5 (+2.5 ± 6.0) × 10 −72569 116.6 0.45 0.47 0.95 1102 2 6 (−1.8 ± 1.3) × 10 −5 (+2.1 ± 5.9) × 10 −72570 123.0 0.45 0.47 0.95 656 1 1 (+2.3 ± 1.6) × 10 −5 (+2.4 ± 5.9) × 10 −72571 124.2 0.45 0.47 0.95 936 1 4 (+2.4 ± 1.4) × 10 −5 (+2.9 ± 5.9) × 10 −72572 128.4 0.45 0.47 0.95 1248 0 2 (+0.9 ± 1.2) × 10 −5 (+3.1 ± 5.9) × 10 −72573 127.3 0.45 0.47 0.95 1246 2 2 (−1.2 ± 1.1) × 10 −5 (+2.8 ± 5.9) × 10 −72574 129.5 0.45 0.47 0.95 1247 1 2 (−1.4 ± 1.2) × 10 −5 (+2.4 ± 5.9) × 10 −72575 129.3 0.45 0.47 0.95 1249 1 0 (+0.1 ± 1.2) × 10 −5 (+2.5 ± 5.9) × 10 −72576 129.8 0.45 0.47 0.95 1248 1 1 (+0.8 ± 1.2) × 10 −5 (+2.7 ± 5.9) × 10 −72577 129.6 0.45 0.47 0.95 1077 1 1 (+2.4 ± 1.3) × 10 −5 (+3.2 ± 5.9) × 10 −72580 129.5 0.45 0.47 0.95 1236 2 2 (+1.0 ± 1.2) × 10 −5 (+3.4 ± 5.9) × 10 −7Table C.21:

280Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2581 129.8 0.45 0.47 0.95 1247 1 2 (−1.1 ± 1.1) × 10 −5 (+3.1 ± 5.9) × 10 −72582 123.8 0.45 0.47 0.95 1183 1 7 (+1.8 ± 1.2) × 10 −5 (+3.6 ± 5.9) × 10 −72583 128.8 0.45 0.47 0.95 1244 1 5 (+0.2 ± 1.2) × 10 −5 (+3.6 ± 5.9) × 10 −72584 128.6 0.45 0.47 0.95 1242 1 7 (+1.9 ± 1.2) × 10 −5 (+4.1 ± 5.9) × 10 −72585 127.7 0.45 0.47 0.95 1242 1 7 (+0.2 ± 1.2) × 10 −5 (+4.1 ± 5.8) × 10 −72586 127.4 0.45 0.47 0.95 1237 1 12 (+0.6 ± 1.2) × 10 −5 (+4.3 ± 5.8) × 10 −72587 127.2 0.45 0.47 0.95 1242 1 7 (+2.1 ± 1.2) × 10 −5 (+4.8 ± 5.8) × 10 −72588 127.5 0.45 0.47 0.95 915 1 4 (+0.5 ± 1.4) × 10 −5 (+4.9 ± 5.8) × 10 −72683 129.2 0.45 0.47 0.95 1073 2 175 (−0.7 ± 1.3) × 10 −5 (+4.7 ± 5.8) × 10 −72684 128.9 0.45 0.47 0.95 1244 3 3 (−0.8 ± 1.2) × 10 −5 (+4.5 ± 5.8) × 10 −72688 127.6 0.45 0.47 0.95 1242 1 7 (−1.1 ± 1.1) × 10 −5 (+4.2 ± 5.8) × 10 −72689 127.9 0.45 0.47 0.95 1247 1 2 (+0.7 ± 1.2) × 10 −5 (+4.3 ± 5.8) × 10 −72690 129.0 0.45 0.47 0.95 1246 1 3 (−1.5 ± 1.1) × 10 −5 (+3.9 ± 5.8) × 10 −72691 129.1 0.45 0.47 0.95 1247 1 2 (−1.5 ± 1.2) × 10 −5 (+3.6 ± 5.8) × 10 −72692 129.3 0.45 0.47 0.95 1236 1 13 (+0.3 ± 1.2) × 10 −5 (+3.6 ± 5.8) × 10 −72694 129.4 0.45 0.47 0.95 1246 2 2 (−0.7 ± 1.1) × 10 −5 (+3.4 ± 5.8) × 10 −72695 129.1 0.45 0.47 0.95 826 1 423 (−1.1 ± 1.5) × 10 −5 (+3.3 ± 5.8) × 10 −72696 129.2 0.45 0.47 0.95 1247 1 2 (−0.8 ± 1.1) × 10 −5 (+3.0 ± 5.8) × 10 −72697 129.2 0.45 0.47 0.95 1248 1 1 (−0.2 ± 1.2) × 10 −5 (+3.0 ± 5.8) × 10 −72698 129.1 0.45 0.47 0.95 1246 2 2 (−0.1 ± 1.2) × 10 −5 (+3.0 ± 5.7) × 10 −72699 128.4 0.45 0.47 0.95 1249 1 0 (+0.3 ± 1.1) × 10 −5 (+3.0 ± 5.7) × 10 −7Table C.22:

281Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2700 128.6 0.45 0.47 0.95 1249 1 0 (−1.0 ± 1.1) × 10 −5 (+2.8 ± 5.7) × 10 −72701 129.0 0.45 0.47 0.95 1249 1 0 (−0.2 ± 1.2) × 10 −5 (+2.7 ± 5.7) × 10 −72702 129.1 0.45 0.47 0.95 1248 1 1 (−2.1 ± 1.1) × 10 −5 (+2.2 ± 5.7) × 10 −72703 129.1 0.45 0.47 0.95 1249 1 0 (+0.4 ± 1.1) × 10 −5 (+2.3 ± 5.7) × 10 −72704 129.3 0.45 0.47 0.95 1247 1 2 (−0.2 ± 1.1) × 10 −5 (+2.2 ± 5.7) × 10 −72705 129.1 0.45 0.47 0.95 1249 1 0 (+0.6 ± 1.1) × 10 −5 (+2.4 ± 5.7) × 10 −72706 128.9 0.45 0.47 0.95 1245 1 4 (+0.7 ± 1.2) × 10 −5 (+2.5 ± 5.7) × 10 −72708 129.0 0.45 0.47 0.95 1190 1 59 (−0.6 ± 1.1) × 10 −5 (+2.4 ± 5.7) × 10 −72709 128.8 0.45 0.47 0.95 1246 1 3 (−0.0 ± 1.2) × 10 −5 (+2.4 ± 5.7) × 10 −72710 128.5 0.45 0.47 0.95 1247 2 1 (+0.6 ± 1.1) × 10 −5 (+2.5 ± 5.7) × 10 −72711 128.5 0.45 0.47 0.95 1245 1 4 (−0.5 ± 1.1) × 10 −5 (+2.4 ± 5.7) × 10 −72712 128.5 0.45 0.47 0.95 1248 1 1 (−1.4 ± 1.2) × 10 −5 (+2.0 ± 5.7) × 10 −72713 128.4 0.45 0.47 0.95 1247 1 2 (+0.9 ± 1.2) × 10 −5 (+2.2 ± 5.6) × 10 −72714 128.4 0.45 0.47 0.95 1246 1 3 (+0.5 ± 1.2) × 10 −5 (+2.4 ± 5.6) × 10 −72731 126.8 0.45 0.47 0.95 1244 1 5 (+0.2 ± 1.2) × 10 −5 (+2.4 ± 5.6) × 10 −72732 126.4 0.45 0.47 0.95 1242 1 7 (−0.6 ± 1.2) × 10 −5 (+2.2 ± 5.6) × 10 −72733 127.4 0.45 0.47 0.95 1245 1 4 (−0.6 ± 1.1) × 10 −5 (+2.1 ± 5.6) × 10 −72734 126.9 0.45 0.47 0.95 1243 1 6 (+0.2 ± 1.2) × 10 −5 (+2.1 ± 5.6) × 10 −72735 127.3 0.45 0.47 0.95 1247 1 2 (+1.2 ± 1.2) × 10 −5 (+2.4 ± 5.6) × 10 −72736 126.7 0.45 0.47 0.95 1248 1 1 (−1.0 ± 1.2) × 10 −5 (+2.2 ± 5.6) × 10 −72737 129.5 0.45 0.47 0.95 1243 1 6 (+0.9 ± 1.1) × 10 −5 (+2.4 ± 5.6) × 10 −7Table C.23:

282Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2738 129.0 0.45 0.47 0.95 1193 3 10 (−1.1 ± 1.2) × 10 −5 (+2.1 ± 5.6) × 10 −72739 128.6 0.45 0.47 0.95 1248 1 1 (+0.9 ± 1.2) × 10 −5 (+2.3 ± 5.6) × 10 −72740 128.4 0.45 0.47 0.95 1248 1 1 (+0.7 ± 1.2) × 10 −5 (+2.5 ± 5.6) × 10 −72741 128.1 0.45 0.47 0.95 1248 1 1 (+0.4 ± 1.1) × 10 −5 (+2.6 ± 5.6) × 10 −72743 128.4 0.45 0.47 0.95 1247 1 2 (−0.4 ± 1.2) × 10 −5 (+2.5 ± 5.6) × 10 −72744 128.5 0.45 0.47 0.95 1245 2 3 (−1.1 ± 1.1) × 10 −5 (+2.2 ± 5.6) × 10 −72745 128.5 0.45 0.47 0.95 1243 1 6 (+3.0 ± 1.2) × 10 −5 (+2.9 ± 5.6) × 10 −72746 128.4 0.45 0.47 0.95 1246 1 3 (−1.5 ± 1.1) × 10 −5 (+2.5 ± 5.5) × 10 −72747 128.3 0.45 0.47 0.95 1246 1 3 (+0.0 ± 1.2) × 10 −5 (+2.5 ± 5.5) × 10 −72748 127.7 0.45 0.47 0.95 1247 1 2 (+0.1 ± 1.1) × 10 −5 (+2.5 ± 5.5) × 10 −72749 127.9 0.45 0.47 0.95 1247 1 2 (−0.5 ± 1.2) × 10 −5 (+2.4 ± 5.5) × 10 −72750 127.6 0.45 0.47 0.95 1242 1 7 (−1.1 ± 1.2) × 10 −5 (+2.2 ± 5.5) × 10 −72751 127.6 0.45 0.47 0.95 1249 1 0 (+1.0 ± 1.1) × 10 −5 (+2.4 ± 5.5) × 10 −72752 127.4 0.45 0.47 0.95 1246 1 3 (−0.6 ± 1.1) × 10 −5 (+2.2 ± 5.5) × 10 −72753 127.1 0.45 0.47 0.95 1239 1 10 (−0.2 ± 1.2) × 10 −5 (+2.2 ± 5.5) × 10 −72755 126.9 0.45 0.47 0.95 1179 1 70 (−1.5 ± 1.2) × 10 −5 (+1.9 ± 5.5) × 10 −72756 127.0 0.45 0.47 0.95 1246 1 3 (−0.1 ± 1.2) × 10 −5 (+1.9 ± 5.5) × 10 −72757 127.2 0.45 0.47 0.95 1237 0 12 (−1.1 ± 1.2) × 10 −5 (+1.6 ± 5.5) × 10 −72758 127.1 0.45 0.47 0.95 1244 1 5 (−1.6 ± 1.2) × 10 −5 (+1.3 ± 5.5) × 10 −72759 127.3 0.45 0.47 0.95 1246 2 2 (+0.3 ± 1.2) × 10 −5 (+1.3 ± 5.5) × 10 −72760 127.3 0.45 0.47 0.95 1247 1 2 (−0.1 ± 1.1) × 10 −5 (+1.3 ± 5.5) × 10 −7Table C.24:

283Thin Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2761 126.8 0.45 0.47 0.95 1245 1 4 (+0.0 ± 1.2) × 10 −5 (+1.3 ± 5.5) × 10 −72762 126.5 0.45 0.47 0.95 1244 1 5 (+1.0 ± 1.2) × 10 −5 (+1.5 ± 5.5) × 10 −72763 126.5 0.45 0.47 0.95 1244 1 5 (+0.2 ± 1.2) × 10 −5 (+1.6 ± 5.4) × 10 −72764 126.1 0.45 0.47 0.95 1243 1 6 (+0.3 ± 1.1) × 10 −5 (+1.7 ± 5.4) × 10 −72765 125.7 0.45 0.47 0.95 1239 1 10 (+1.5 ± 1.2) × 10 −5 (+2.0 ± 5.4) × 10 −72768 128.1 0.45 0.47 0.95 1239 1 10 (+0.6 ± 1.2) × 10 −5 (+2.1 ± 5.4) × 10 −72769 128.6 0.45 0.47 0.95 1247 1 2 (+1.0 ± 1.2) × 10 −5 (+2.3 ± 5.4) × 10 −72770 127.9 0.45 0.47 0.95 1243 1 6 (−1.2 ± 1.2) × 10 −5 (+2.1 ± 5.4) × 10 −72771 128.0 0.45 0.47 0.95 1245 1 4 (−1.2 ± 1.2) × 10 −5 (+1.8 ± 5.4) × 10 −72772 127.6 0.45 0.47 0.95 1231 1 18 (−0.3 ± 1.2) × 10 −5 (+1.7 ± 5.4) × 10 −72773 127.7 0.45 0.47 0.95 1240 1 9 (+1.4 ± 1.1) × 10 −5 (+2.0 ± 5.4) × 10 −72774 127.1 0.45 0.47 0.95 1244 1 5 (+0.6 ± 1.2) × 10 −5 (+2.2 ± 5.4) × 10 −72775 127.5 0.45 0.47 0.95 1242 1 7 (−1.6 ± 1.2) × 10 −5 (+1.8 ± 5.4) × 10 −72776 127.6 0.45 0.47 0.95 1247 1 2 (−1.9 ± 1.2) × 10 −5 (+1.4 ± 5.4) × 10 −72777 127.0 0.45 0.47 0.95 1248 1 1 (−0.6 ± 1.2) × 10 −5 (+1.2 ± 5.4) × 10 −72778 126.7 0.45 0.47 0.95 762 1 487 (+0.6 ± 1.5) × 10 −5 (+1.3 ± 5.4) × 10 −72779 127.0 0.45 0.47 0.95 1249 1 0 (+1.5 ± 1.2) × 10 −5 (+1.6 ± 5.4) × 10 −72780 126.7 0.45 0.47 0.95 1248 1 1 (+2.0 ± 1.2) × 10 −5 (+2.1 ± 5.4) × 10 −72781 126.7 0.45 0.47 0.95 929 1 3 (+0.8 ± 1.4) × 10 −5 (+2.2 ± 5.4) × 10 −72782 127.7 0.45 0.47 0.95 882 2 3 (+0.2 ± 1.4) × 10 −5 (+2.2 ± 5.4) × 10 −72785 126.2 0.45 0.47 0.95 1244 1 5 (−0.4 ± 1.2) × 10 −5 (+2.1 ± 5.4) × 10 −72786 128.5 0.45 0.47 0.95 1249 1 0 (−1.6 ± 1.1) × 10 −5 (+1.8 ± 5.3) × 10 −72787 128.2 0.45 0.47 0.95 1246 2 2 (−0.1 ± 1.2) × 10 −5 (+1.7 ± 5.3) × 10 −7Table C.25:

C.2 Thick Aluminum TargetThick Target SummaryTotal Number of good runs : 555Total Number of sequences from all runs : 682302Number of good sequences : 664484Number of lost sequences from bad spin state : 611 ( 0.1% loss)Number of lost sequences from beam cuts : 17207 ( 2.5% loss)Total Number of lost sequences : 17818 ( 2.6% loss)Average Run Asymmetry Error : 1.0 × 10 −5Average Run Asymmetry Error RMS : 4.2 × 10 −6Average 3He Polarization : 0.42Average 3He Polarization RMS : 5.1 × 10 −3Average Energy Average Beam Polarization : 0.61Average Energy Average Beam Polarization RMS : 4.0 × 10 −3Average Beam Current : 121 µAAverage Beam Current RMS : 2.6 µATable C.26:284

285Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut3865 112.1 0.43 0.46 0.95 1239 1 10 (−3.7 ± 9.9) × 10 −6 (−3.7 ± 9.9) × 10 −63866 111.9 0.43 0.46 0.95 1246 1 3 (+8.9 ± 9.4) × 10 −6 (+2.6 ± 6.8) × 10 −63867 112.9 0.43 0.46 0.95 1246 1 3 (−10.9 ± 9.8) × 10 −6 (−1.9 ± 5.6) × 10 −63868 112.6 0.43 0.46 0.95 1243 2 5 (+1.3 ± 9.8) × 10 −6 (−1.1 ± 4.9) × 10 −63869 112.5 0.43 0.46 0.95 1247 1 2 (−10.5 ± 9.9) × 10 −6 (−3.0 ± 4.4) × 10 −63870 112.1 0.43 0.46 0.95 1244 1 5 (−22.0 ± 9.4) × 10 −6 (−6.2 ± 4.0) × 10 −63871 111.8 0.43 0.46 0.95 1242 1 7 (−6.0 ± 9.9) × 10 −6 (−6.1 ± 3.7) × 10 −63872 111.7 0.43 0.46 0.95 1242 1 7 (−0.3 ± 9.8) × 10 −6 (−5.4 ± 3.4) × 10 −63873 111.8 0.43 0.46 0.95 1245 1 4 (−0.5 ± 1.0) × 10 −5 (−5.4 ± 3.3) × 10 −63874 112.4 0.43 0.46 0.95 1248 2 0 (+20.6 ± 9.9) × 10 −6 (−2.8 ± 3.1) × 10 −63875 112.0 0.43 0.46 0.95 1244 1 5 (+3.2 ± 9.6) × 10 −6 (−2.2 ± 2.9) × 10 −63876 112.0 0.43 0.46 0.95 1240 1 9 (−1.1 ± 1.0) × 10 −5 (−3.0 ± 2.8) × 10 −63879 112.3 0.43 0.46 0.95 1228 2 20 (+0.4 ± 1.0) × 10 −5 (−2.5 ± 2.7) × 10 −63880 112.0 0.43 0.46 0.95 1236 1 13 (−5.1 ± 9.6) × 10 −6 (−2.7 ± 2.6) × 10 −63881 111.6 0.43 0.46 0.95 1233 1 16 (−9.3 ± 9.9) × 10 −6 (−3.1 ± 2.5) × 10 −63882 111.1 0.43 0.46 0.95 1237 1 12 (+8.7 ± 9.6) × 10 −6 (−2.4 ± 2.5) × 10 −63883 110.9 0.43 0.46 0.95 1230 1 19 (+3.0 ± 9.9) × 10 −6 (−2.0 ± 2.4) × 10 −63892 111.9 0.43 0.46 0.95 1248 0 2 (−17.0 ± 9.7) × 10 −6 (−2.9 ± 2.3) × 10 −63893 111.8 0.43 0.46 0.95 1248 0 2 (−11.5 ± 9.2) × 10 −6 (−3.3 ± 2.2) × 10 −63894 112.8 0.43 0.46 0.95 1248 0 2 (−2.2 ± 9.6) × 10 −6 (−3.3 ± 2.2) × 10 −63895 112.8 0.43 0.46 0.95 1246 1 3 (+8.2 ± 9.9) × 10 −6 (−2.7 ± 2.1) × 10 −6Table C.27:

286Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut3896 113.0 0.43 0.46 0.95 1247 1 2 (+0.9 ± 9.7) × 10 −6 (−2.6 ± 2.1) × 10 −63897 112.2 0.43 0.46 0.95 1245 0 5 (−7.9 ± 9.8) × 10 −6 (−2.8 ± 2.0) × 10 −63898 112.7 0.43 0.46 0.95 1244 0 6 (−1.9 ± 9.6) × 10 −6 (−2.8 ± 2.0) × 10 −63899 112.5 0.43 0.46 0.95 1246 0 4 (−4.3 ± 9.6) × 10 −6 (−2.8 ± 2.0) × 10 −63900 113.1 0.43 0.46 0.95 1247 1 2 (+0.8 ± 9.9) × 10 −6 (−2.7 ± 1.9) × 10 −63917 113.7 0.44 0.46 0.95 122 1 2 (+2.2 ± 3.2) × 10 −5 (−2.6 ± 1.9) × 10 −63918 112.9 0.44 0.46 0.95 1243 1 6 (−6.0 ± 9.7) × 10 −6 (−2.7 ± 1.9) × 10 −63919 113.2 0.44 0.46 0.95 1246 1 3 (+4.8 ± 9.2) × 10 −6 (−2.4 ± 1.8) × 10 −63920 113.8 0.44 0.46 0.95 1245 1 4 (+9.3 ± 9.4) × 10 −6 (−2.0 ± 1.8) × 10 −63921 113.3 0.44 0.46 0.95 1246 1 3 (−4.9 ± 9.2) × 10 −6 (−2.1 ± 1.8) × 10 −63922 113.1 0.44 0.46 0.95 1245 1 4 (+5.1 ± 9.5) × 10 −6 (−1.9 ± 1.7) × 10 −63923 113.9 0.44 0.46 0.95 1244 1 5 (−1.7 ± 9.3) × 10 −6 (−1.9 ± 1.7) × 10 −63924 113.8 0.44 0.46 0.95 1245 2 4 (−24.1 ± 9.3) × 10 −6 (−2.6 ± 1.7) × 10 −63925 113.8 0.45 0.47 0.95 1236 1 13 (+8.9 ± 9.2) × 10 −6 (−2.2 ± 1.7) × 10 −63959 111.8 0.44 0.47 0.95 1243 1 6 (−0.0 ± 9.5) × 10 −6 (−2.2 ± 1.6) × 10 −63960 111.7 0.44 0.47 0.95 1228 1 21 (−1.2 ± 1.0) × 10 −5 (−2.4 ± 1.6) × 10 −63968 111.8 0.44 0.47 0.95 1235 2 13 (−6.3 ± 9.6) × 10 −6 (−2.5 ± 1.6) × 10 −63969 111.8 0.44 0.47 0.95 1212 1 37 (+10.4 ± 9.6) × 10 −6 (−2.2 ± 1.6) × 10 −63974 109.4 0.44 0.47 0.95 1185 1 64 (+5.8 ± 9.7) × 10 −6 (−2.0 ± 1.5) × 10 −63975 109.2 0.44 0.47 0.95 1181 0 68 (+15.1 ± 9.8) × 10 −6 (−1.6 ± 1.5) × 10 −63976 110.9 0.44 0.47 0.95 1214 1 35 (+7.8 ± 9.5) × 10 −6 (−1.4 ± 1.5) × 10 −63977 113.5 0.44 0.47 0.95 1242 1 7 (+3.6 ± 9.4) × 10 −6 (−1.3 ± 1.5) × 10 −6Table C.28:

287Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut3978 112.2 0.44 0.47 0.95 1229 1 20 (+4.0 ± 9.4) × 10 −6 (−1.1 ± 1.5) × 10 −63979 112.5 0.44 0.47 0.95 1174 2 11 (+0.8 ± 1.0) × 10 −5 (−1.0 ± 1.5) × 10 −63980 107.8 0.44 0.47 0.95 1236 1 13 (−6.0 ± 9.6) × 10 −6 (−1.1 ± 1.4) × 10 −63981 110.4 0.44 0.47 0.95 1232 1 17 (+3.5 ± 9.6) × 10 −6 (−1.0 ± 1.4) × 10 −63982 112.7 0.44 0.47 0.95 1230 1 19 (+20.8 ± 9.3) × 10 −6 (−0.5 ± 1.4) × 10 −63983 113.8 0.44 0.46 0.95 1240 1 9 (+6.5 ± 9.5) × 10 −6 (−0.4 ± 1.4) × 10 −64279 124.2 0.42 0.45 0.95 1191 2 57 (−7.3 ± 9.7) × 10 −6 (−0.5 ± 1.4) × 10 −64280 124.1 0.42 0.45 0.95 1162 1 87 (−7.2 ± 9.7) × 10 −6 (−0.6 ± 1.4) × 10 −64281 124.1 0.42 0.45 0.95 1164 2 84 (+11.0 ± 9.6) × 10 −6 (−0.4 ± 1.4) × 10 −64282 124.3 0.42 0.45 0.95 1190 1 59 (+0.9 ± 9.8) × 10 −6 (−0.4 ± 1.3) × 10 −64283 124.6 0.42 0.45 0.95 1200 1 49 (+8.9 ± 9.5) × 10 −6 (−0.2 ± 1.3) × 10 −64284 124.5 0.42 0.45 0.95 1196 1 53 (+0.2 ± 9.5) × 10 −6 (−0.2 ± 1.3) × 10 −64285 124.4 0.42 0.45 0.95 1184 2 64 (+4.8 ± 9.5) × 10 −6 (−0.1 ± 1.3) × 10 −64286 124.2 0.42 0.45 0.95 1226 1 23 (−6.1 ± 9.7) × 10 −6 (−0.2 ± 1.3) × 10 −64287 124.0 0.42 0.45 0.95 1212 1 37 (+5.8 ± 9.4) × 10 −6 (−0.1 ± 1.3) × 10 −64288 123.9 0.42 0.45 0.95 1177 1 72 (+4.8 ± 9.5) × 10 −6 (−0.0 ± 1.3) × 10 −64289 123.9 0.42 0.45 0.95 1183 1 66 (−0.4 ± 9.5) × 10 −6 (−0.0 ± 1.3) × 10 −64290 123.8 0.42 0.45 0.95 1202 0 47 (+0.4 ± 9.9) × 10 −6 (−0.0 ± 1.2) × 10 −64291 123.8 0.42 0.45 0.95 1184 1 65 (+22.7 ± 9.8) × 10 −6 (+0.3 ± 1.2) × 10 −64294 123.9 0.42 0.45 0.95 1137 3 69 (+0.1 ± 10.0) × 10 −6 (+0.3 ± 1.2) × 10 −64295 123.8 0.42 0.45 0.95 1173 1 76 (+1.3 ± 9.8) × 10 −6 (+0.3 ± 1.2) × 10 −64296 123.9 0.42 0.45 0.95 1176 2 72 (+4.2 ± 9.4) × 10 −6 (+0.4 ± 1.2) × 10 −6Table C.29:

288Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4297 123.8 0.42 0.45 0.95 1181 1 68 (+7.0 ± 9.2) × 10 −6 (+0.5 ± 1.2) × 10 −64298 122.9 0.42 0.45 0.95 1200 1 49 (+16.7 ± 9.4) × 10 −6 (+0.7 ± 1.2) × 10 −64299 124.0 0.42 0.45 0.95 1209 1 40 (+2.3 ± 9.6) × 10 −6 (+0.7 ± 1.2) × 10 −64300 123.9 0.42 0.45 0.95 1213 1 36 (−20.7 ± 9.3) × 10 −6 (+0.4 ± 1.2) × 10 −64301 123.7 0.42 0.45 0.95 1206 0 43 (−4.7 ± 9.8) × 10 −6 (+0.4 ± 1.2) × 10 −64302 123.7 0.42 0.45 0.95 1219 1 30 (+15.9 ± 9.3) × 10 −6 (+0.6 ± 1.2) × 10 −64303 123.4 0.42 0.45 0.95 1051 0 197 (+1.7 ± 1.3) × 10 −5 (+0.8 ± 1.1) × 10 −64304 123.7 0.42 0.45 0.95 1199 1 50 (−6.7 ± 10.0) × 10 −6 (+0.7 ± 1.1) × 10 −64306 123.8 0.42 0.45 0.95 1200 1 49 (−8.3 ± 9.8) × 10 −6 (+0.6 ± 1.1) × 10 −64309 123.8 0.42 0.45 0.95 1205 1 44 (+8.2 ± 10.0) × 10 −6 (+0.7 ± 1.1) × 10 −64310 123.6 0.42 0.45 0.95 1209 2 39 (+15.2 ± 9.9) × 10 −6 (+0.9 ± 1.1) × 10 −64311 123.4 0.42 0.45 0.95 1185 1 64 (−3.2 ± 9.6) × 10 −6 (+0.8 ± 1.1) × 10 −64312 123.6 0.42 0.45 0.95 1175 1 74 (−0.4 ± 9.6) × 10 −6 (+0.8 ± 1.1) × 10 −64313 123.3 0.42 0.45 0.95 1163 0 86 (−1.4 ± 1.0) × 10 −5 (+0.6 ± 1.1) × 10 −64314 123.5 0.42 0.45 0.95 1157 1 92 (−0.1 ± 9.7) × 10 −6 (+0.6 ± 1.1) × 10 −64315 122.7 0.42 0.45 0.95 1199 1 50 (+3.8 ± 9.9) × 10 −6 (+0.6 ± 1.1) × 10 −64316 122.3 0.42 0.45 0.95 1220 1 29 (−15.8 ± 9.6) × 10 −6 (+0.4 ± 1.1) × 10 −64317 122.4 0.42 0.45 0.95 1212 1 37 (−0.6 ± 9.8) × 10 −6 (+0.4 ± 1.1) × 10 −64318 122.2 0.42 0.45 0.95 1202 1 47 (+2.3 ± 9.6) × 10 −6 (+0.4 ± 1.1) × 10 −64319 122.2 0.42 0.45 0.95 1216 1 33 (−3.6 ± 9.8) × 10 −6 (+0.4 ± 1.1) × 10 −64320 122.0 0.42 0.45 0.95 1193 1 56 (−9.5 ± 9.7) × 10 −6 (+0.3 ± 1.1) × 10 −64321 122.4 0.42 0.45 0.95 1199 1 50 (+10.9 ± 9.7) × 10 −6 (+0.4 ± 1.0) × 10 −6Table C.30:

289Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4322 122.4 0.42 0.45 0.95 1206 1 43 (−0.6 ± 1.0) × 10 −5 (+0.3 ± 1.0) × 10 −64325 122.5 0.42 0.45 0.95 122 1 2 (+0.8 ± 3.8) × 10 −5 (+0.3 ± 1.0) × 10 −64326 122.2 0.42 0.45 0.95 1192 1 57 (−8.2 ± 9.5) × 10 −6 (+0.2 ± 1.0) × 10 −64327 122.3 0.42 0.45 0.95 1206 1 43 (−1.7 ± 9.6) × 10 −6 (+0.2 ± 1.0) × 10 −64329 121.8 0.42 0.45 0.95 1182 2 66 (+0.7 ± 1.0) × 10 −5 (+0.3 ± 1.0) × 10 −64330 122.6 0.42 0.45 0.95 1189 1 60 (+9.2 ± 9.2) × 10 −6 (+0.4 ± 1.0) × 10 −64331 122.6 0.42 0.45 0.95 1175 1 74 (−8.3 ± 9.7) × 10 −6 (+0.3 ± 1.0) × 10 −64332 122.6 0.42 0.45 0.95 1188 1 61 (−5.8 ± 9.5) × 10 −6 (+0.2 ± 1.0) × 10 −64333 122.5 0.42 0.45 0.95 1184 2 64 (−8.4 ± 9.8) × 10 −6 (+0.1 ± 1.0) × 10 −64334 122.6 0.42 0.45 0.95 1173 1 76 (+11.5 ± 9.7) × 10 −6 (+2.7 ± 10.0) × 10 −74335 122.5 0.42 0.45 0.95 1181 1 68 (−7.0 ± 9.7) × 10 −6 (+1.9 ± 9.9) × 10 −74336 122.5 0.42 0.45 0.95 1186 0 63 (+1.0 ± 1.0) × 10 −5 (+2.9 ± 9.9) × 10 −74337 122.4 0.42 0.45 0.95 1207 2 41 (+2.0 ± 9.4) × 10 −6 (+3.1 ± 9.8) × 10 −74338 122.9 0.42 0.45 0.95 1203 2 46 (−3.4 ± 9.8) × 10 −6 (+2.7 ± 9.8) × 10 −74339 122.8 0.42 0.45 0.95 1205 2 43 (−2.0 ± 9.7) × 10 −6 (+2.5 ± 9.7) × 10 −74340 122.7 0.42 0.45 0.95 1177 1 72 (−15.9 ± 9.8) × 10 −6 (+1.0 ± 9.7) × 10 −74341 122.5 0.42 0.45 0.95 1192 1 57 (−22.7 ± 9.8) × 10 −6 (−1.2 ± 9.6) × 10 −74342 122.5 0.42 0.45 0.95 1190 1 59 (+6.5 ± 9.6) × 10 −6 (−0.6 ± 9.6) × 10 −74343 122.6 0.42 0.45 0.95 1181 1 68 (+6.2 ± 9.9) × 10 −6 (−0.0 ± 9.5) × 10 −74344 122.5 0.42 0.45 0.95 1178 1 71 (−0.3 ± 9.5) × 10 −6 (−0.1 ± 9.5) × 10 −74345 122.5 0.42 0.45 0.95 1191 1 58 (−6.7 ± 9.7) × 10 −6 (−0.7 ± 9.4) × 10 −74349 122.5 0.42 0.45 0.95 1177 1 72 (−3.4 ± 9.7) × 10 −6 (−1.0 ± 9.4) × 10 −7Table C.31:

290Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4350 121.8 0.42 0.45 0.95 1171 1 78 (−9.1 ± 9.9) × 10 −6 (−1.8 ± 9.3) × 10 −74351 120.1 0.42 0.45 0.95 1195 2 53 (−3.2 ± 9.5) × 10 −6 (−2.1 ± 9.3) × 10 −74352 122.7 0.42 0.45 0.95 1178 2 71 (−12.7 ± 9.6) × 10 −6 (−3.2 ± 9.3) × 10 −74358 124.8 0.42 0.45 0.95 1231 0 18 (−5.6 ± 9.3) × 10 −6 (−3.7 ± 9.2) × 10 −74359 124.6 0.42 0.45 0.95 1231 2 17 (−0.6 ± 9.5) × 10 −6 (−3.7 ± 9.2) × 10 −74360 124.2 0.42 0.45 0.95 1228 1 21 (−11.7 ± 9.3) × 10 −6 (−4.7 ± 9.1) × 10 −74361 124.2 0.42 0.45 0.95 1232 1 17 (+10.9 ± 9.9) × 10 −6 (−3.7 ± 9.1) × 10 −74364 124.1 0.42 0.45 0.95 1243 1 6 (+12.3 ± 9.5) × 10 −6 (−2.5 ± 9.0) × 10 −74365 123.8 0.42 0.45 0.95 1230 1 19 (+0.6 ± 1.0) × 10 −5 (−2.0 ± 9.0) × 10 −74366 123.6 0.42 0.45 0.95 1221 1 28 (−12.2 ± 9.4) × 10 −6 (−3.0 ± 9.0) × 10 −74367 123.6 0.42 0.45 0.95 1238 1 11 (−3.1 ± 9.6) × 10 −6 (−3.3 ± 8.9) × 10 −74368 123.7 0.42 0.45 0.95 1231 1 18 (−9.1 ± 9.3) × 10 −6 (−4.0 ± 8.9) × 10 −74369 123.7 0.42 0.45 0.95 1232 1 17 (−1.2 ± 9.6) × 10 −6 (−4.1 ± 8.8) × 10 −74370 123.7 0.42 0.45 0.95 1237 1 12 (+15.7 ± 9.3) × 10 −6 (−2.7 ± 8.8) × 10 −74371 123.7 0.42 0.45 0.95 1162 2 86 (+3.7 ± 9.8) × 10 −6 (−2.4 ± 8.8) × 10 −74372 123.8 0.42 0.45 0.95 1099 1 150 (+1.6 ± 9.9) × 10 −6 (−2.3 ± 8.7) × 10 −74373 123.8 0.42 0.45 0.95 1105 1 144 (−0.0 ± 1.0) × 10 −5 (−2.3 ± 8.7) × 10 −74374 123.7 0.42 0.45 0.95 1081 2 155 (−0.2 ± 1.0) × 10 −5 (−2.3 ± 8.7) × 10 −74375 123.8 0.42 0.45 0.95 889 1 360 (+0.4 ± 1.3) × 10 −5 (−2.1 ± 8.7) × 10 −74376 123.6 0.42 0.45 0.95 1045 1 204 (−0.5 ± 1.1) × 10 −5 (−2.4 ± 8.6) × 10 −74385 123.6 0.42 0.45 0.95 1114 2 85 (+0.4 ± 1.0) × 10 −5 (−2.1 ± 8.6) × 10 −74386 121.7 0.42 0.45 0.95 1075 2 174 (+1.6 ± 1.0) × 10 −5 (−1.0 ± 8.6) × 10 −7Table C.32:

291Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4387 121.2 0.42 0.45 0.95 1119 1 130 (+4.3 ± 9.9) × 10 −6 (−0.7 ± 8.5) × 10 −74388 121.5 0.42 0.45 0.95 1207 1 42 (+7.0 ± 9.6) × 10 −6 (−0.1 ± 8.5) × 10 −74389 120.3 0.42 0.45 0.95 1182 1 67 (+7.0 ± 10.0) × 10 −6 (+0.4 ± 8.5) × 10 −74390 120.1 0.42 0.45 0.95 1194 2 55 (−4.4 ± 9.9) × 10 −6 (+0.1 ± 8.4) × 10 −74391 119.4 0.42 0.45 0.95 1181 1 68 (+0.6 ± 1.0) × 10 −5 (+0.5 ± 8.4) × 10 −74394 119.0 0.42 0.45 0.95 1051 2 45 (+0.2 ± 1.1) × 10 −5 (+0.6 ± 8.4) × 10 −74395 119.2 0.42 0.45 0.95 919 2 35 (−0.0 ± 1.2) × 10 −5 (+0.6 ± 8.4) × 10 −74396 122.1 0.42 0.45 0.95 1028 1 40 (+1.3 ± 1.0) × 10 −5 (+1.4 ± 8.3) × 10 −74397 124.3 0.42 0.45 0.95 1203 1 46 (−11.9 ± 9.6) × 10 −6 (+0.5 ± 8.3) × 10 −74398 124.4 0.42 0.45 0.95 1214 1 35 (−0.1 ± 9.6) × 10 −6 (+0.5 ± 8.3) × 10 −74399 124.4 0.42 0.45 0.95 1212 1 37 (+0.6 ± 9.5) × 10 −6 (+0.5 ± 8.2) × 10 −74400 124.2 0.42 0.45 0.95 1202 1 47 (−1.9 ± 9.3) × 10 −6 (+0.4 ± 8.2) × 10 −74401 124.7 0.42 0.45 0.95 1231 1 18 (−4.7 ± 9.3) × 10 −6 (+0.1 ± 8.2) × 10 −74402 124.9 0.42 0.45 0.95 1234 1 15 (+3.1 ± 9.2) × 10 −6 (+0.3 ± 8.2) × 10 −74403 124.5 0.42 0.45 0.95 1228 1 21 (+9.7 ± 9.4) × 10 −6 (+1.0 ± 8.1) × 10 −74404 124.3 0.42 0.45 0.95 1235 1 14 (+9.0 ± 9.5) × 10 −6 (+1.6 ± 8.1) × 10 −74405 124.3 0.42 0.45 0.95 1234 1 15 (−8.6 ± 9.5) × 10 −6 (+1.0 ± 8.1) × 10 −74408 124.1 0.42 0.45 0.95 1234 1 15 (+19.5 ± 9.6) × 10 −6 (+2.3 ± 8.0) × 10 −74409 124.0 0.42 0.45 0.95 1238 1 11 (−5.0 ± 9.3) × 10 −6 (+2.0 ± 8.0) × 10 −74410 124.1 0.42 0.45 0.95 1231 1 18 (−4.3 ± 9.8) × 10 −6 (+1.7 ± 8.0) × 10 −74411 124.3 0.42 0.45 0.95 1240 1 9 (+0.6 ± 9.6) × 10 −6 (+1.7 ± 7.9) × 10 −74412 123.9 0.42 0.45 0.95 1238 1 11 (+7.1 ± 9.1) × 10 −6 (+2.2 ± 7.9) × 10 −7Table C.33:

292Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4413 123.7 0.42 0.45 0.95 1242 1 7 (−0.2 ± 9.9) × 10 −6 (+2.1 ± 7.9) × 10 −74414 123.7 0.42 0.45 0.95 1230 1 19 (−9.0 ± 9.5) × 10 −6 (+1.5 ± 7.9) × 10 −74415 124.1 0.42 0.45 0.95 1220 1 29 (−11.5 ± 9.7) × 10 −6 (+0.8 ± 7.8) × 10 −74416 123.8 0.42 0.45 0.95 1237 1 12 (+3.4 ± 9.4) × 10 −6 (+1.0 ± 7.8) × 10 −74417 123.6 0.42 0.45 0.95 1224 1 25 (+8.7 ± 9.7) × 10 −6 (+1.5 ± 7.8) × 10 −74418 123.3 0.42 0.45 0.95 1234 1 15 (−8.1 ± 9.5) × 10 −6 (+1.0 ± 7.8) × 10 −74419 123.7 0.42 0.45 0.95 1231 1 18 (+5.0 ± 9.6) × 10 −6 (+1.3 ± 7.7) × 10 −74420 123.4 0.42 0.45 0.95 1231 1 18 (−12.8 ± 9.4) × 10 −6 (+0.5 ± 7.7) × 10 −74431 122.5 0.42 0.45 0.95 1230 1 19 (−11.1 ± 9.8) × 10 −6 (−0.2 ± 7.7) × 10 −74432 122.0 0.42 0.45 0.95 1199 1 50 (+0.7 ± 9.7) × 10 −6 (−0.2 ± 7.7) × 10 −74433 121.7 0.42 0.45 0.95 1208 2 40 (−3.6 ± 9.9) × 10 −6 (−0.4 ± 7.6) × 10 −74434 122.0 0.42 0.45 0.95 1218 1 31 (+21.3 ± 9.8) × 10 −6 (+0.9 ± 7.6) × 10 −74435 122.4 0.42 0.45 0.95 1206 2 42 (−11.3 ± 10.0) × 10 −6 (+0.2 ± 7.6) × 10 −74436 121.4 0.42 0.45 0.95 1210 1 39 (−16.5 ± 9.5) × 10 −6 (−0.8 ± 7.6) × 10 −74437 122.6 0.42 0.45 0.95 1211 1 38 (+1.5 ± 9.5) × 10 −6 (−0.7 ± 7.5) × 10 −74438 122.6 0.42 0.45 0.95 1221 1 28 (+0.2 ± 9.7) × 10 −6 (−0.7 ± 7.5) × 10 −74439 122.5 0.42 0.45 0.95 1218 1 31 (−12.9 ± 9.5) × 10 −6 (−1.4 ± 7.5) × 10 −74440 122.5 0.42 0.45 0.95 1207 1 42 (+4.9 ± 9.5) × 10 −6 (−1.1 ± 7.5) × 10 −74441 122.1 0.42 0.45 0.95 1211 1 38 (−0.7 ± 9.5) × 10 −6 (−1.2 ± 7.5) × 10 −74442 122.6 0.42 0.45 0.95 1217 1 32 (−19.6 ± 9.7) × 10 −6 (−2.3 ± 7.4) × 10 −74443 122.6 0.42 0.45 0.95 1213 1 36 (+8.8 ± 9.4) × 10 −6 (−1.8 ± 7.4) × 10 −74444 122.8 0.42 0.45 0.95 1222 1 27 (−5.8 ± 9.7) × 10 −6 (−2.1 ± 7.4) × 10 −7Table C.34:

293Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4447 122.7 0.42 0.45 0.95 1232 1 17 (−7.1 ± 9.8) × 10 −6 (−2.5 ± 7.4) × 10 −74448 122.6 0.42 0.45 0.95 1215 1 34 (+1.2 ± 9.7) × 10 −6 (−2.4 ± 7.3) × 10 −74449 122.3 0.42 0.45 0.95 1208 1 41 (−5.8 ± 9.5) × 10 −6 (−2.8 ± 7.3) × 10 −74450 122.5 0.42 0.45 0.95 1214 1 35 (−4.7 ± 9.6) × 10 −6 (−3.0 ± 7.3) × 10 −74451 122.5 0.42 0.45 0.95 1219 1 30 (−12.6 ± 9.7) × 10 −6 (−3.7 ± 7.3) × 10 −74452 123.1 0.42 0.45 0.95 1223 1 26 (−4.7 ± 9.5) × 10 −6 (−4.0 ± 7.3) × 10 −74453 123.1 0.42 0.45 0.95 1228 1 21 (+17.7 ± 9.3) × 10 −6 (−2.9 ± 7.2) × 10 −74454 123.3 0.42 0.45 0.95 1225 1 24 (+2.1 ± 9.5) × 10 −6 (−2.8 ± 7.2) × 10 −74455 123.2 0.42 0.45 0.95 1231 1 18 (−11.5 ± 9.6) × 10 −6 (−3.4 ± 7.2) × 10 −74456 123.4 0.42 0.45 0.95 1227 1 22 (+5.1 ± 9.6) × 10 −6 (−3.1 ± 7.2) × 10 −74457 123.3 0.42 0.45 0.95 1227 3 20 (+15.2 ± 9.4) × 10 −6 (−2.3 ± 7.2) × 10 −74458 122.0 0.42 0.45 0.95 1178 0 71 (−1.0 ± 1.3) × 10 −5 (−2.8 ± 7.2) × 10 −74461 121.5 0.42 0.45 0.95 1223 0 27 (−6.2 ± 10.0) × 10 −6 (−3.1 ± 7.1) × 10 −74462 121.5 0.42 0.45 0.95 1230 0 20 (+17.2 ± 9.8) × 10 −6 (−2.2 ± 7.1) × 10 −74463 121.1 0.42 0.45 0.95 672 0 24 (−0.0 ± 1.4) × 10 −5 (−2.2 ± 7.1) × 10 −74464 122.0 0.42 0.45 0.95 1211 1 38 (+0.9 ± 9.6) × 10 −6 (−2.1 ± 7.1) × 10 −74465 122.5 0.42 0.45 0.95 1230 1 19 (−20.0 ± 9.7) × 10 −6 (−3.2 ± 7.1) × 10 −74466 122.2 0.42 0.45 0.95 1217 2 31 (+1.6 ± 9.3) × 10 −6 (−3.1 ± 7.0) × 10 −74467 122.2 0.42 0.45 0.95 1233 1 16 (−4.6 ± 9.6) × 10 −6 (−3.3 ± 7.0) × 10 −74468 122.1 0.42 0.45 0.95 1232 1 17 (+29.8 ± 9.6) × 10 −6 (−1.7 ± 7.0) × 10 −74469 121.9 0.42 0.45 0.95 1230 1 19 (+9.0 ± 9.6) × 10 −6 (−1.2 ± 7.0) × 10 −74470 121.7 0.42 0.45 0.95 1069 1 54 (−0.4 ± 1.0) × 10 −5 (−1.4 ± 7.0) × 10 −7Table C.35:

294Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4471 122.5 0.42 0.45 0.95 1213 1 36 (−3.4 ± 9.7) × 10 −6 (−1.6 ± 7.0) × 10 −74472 122.2 0.42 0.45 0.95 1196 1 53 (+6.4 ± 9.6) × 10 −6 (−1.3 ± 6.9) × 10 −74475 123.1 0.42 0.45 0.95 1234 1 15 (−2.3 ± 9.8) × 10 −6 (−1.4 ± 6.9) × 10 −74476 123.1 0.42 0.45 0.95 1210 1 39 (−11.6 ± 9.9) × 10 −6 (−1.9 ± 6.9) × 10 −74480 123.6 0.42 0.45 0.95 1221 2 27 (−0.4 ± 1.0) × 10 −5 (−2.1 ± 6.9) × 10 −74481 121.2 0.42 0.45 0.95 1236 1 13 (−5.7 ± 9.1) × 10 −6 (−2.4 ± 6.9) × 10 −74482 122.5 0.42 0.45 0.95 1236 1 13 (−0.3 ± 9.7) × 10 −6 (−2.4 ± 6.9) × 10 −74483 122.9 0.42 0.45 0.95 1238 1 11 (−9.0 ± 9.7) × 10 −6 (−2.9 ± 6.8) × 10 −74484 122.9 0.42 0.45 0.95 1231 1 18 (−25.0 ± 9.6) × 10 −6 (−4.1 ± 6.8) × 10 −74485 123.0 0.42 0.45 0.95 1236 1 13 (+0.0 ± 9.6) × 10 −6 (−4.1 ± 6.8) × 10 −74486 122.9 0.42 0.45 0.95 1236 1 13 (−9.4 ± 9.3) × 10 −6 (−4.5 ± 6.8) × 10 −74487 122.8 0.42 0.45 0.95 1237 1 12 (−0.1 ± 9.4) × 10 −6 (−4.5 ± 6.8) × 10 −74488 122.9 0.42 0.45 0.95 1241 1 8 (+5.2 ± 9.5) × 10 −6 (−4.2 ± 6.7) × 10 −74489 122.7 0.42 0.45 0.95 1234 2 14 (−6.9 ± 9.4) × 10 −6 (−4.5 ± 6.7) × 10 −74490 122.5 0.42 0.45 0.95 1236 1 13 (−23.8 ± 9.8) × 10 −6 (−5.7 ± 6.7) × 10 −74491 122.6 0.42 0.45 0.95 1238 1 11 (−6.8 ± 9.7) × 10 −6 (−6.0 ± 6.7) × 10 −74492 122.6 0.42 0.45 0.95 1243 1 6 (−0.2 ± 9.6) × 10 −6 (−6.0 ± 6.7) × 10 −74493 122.6 0.42 0.45 0.95 1231 1 18 (+8.8 ± 9.6) × 10 −6 (−5.5 ± 6.7) × 10 −74494 122.7 0.42 0.45 0.95 1234 1 15 (+3.8 ± 9.7) × 10 −6 (−5.3 ± 6.7) × 10 −74495 122.8 0.42 0.45 0.95 1227 1 22 (+9.9 ± 9.5) × 10 −6 (−4.8 ± 6.6) × 10 −74501 123.0 0.42 0.45 0.95 1244 1 5 (+6.5 ± 9.5) × 10 −6 (−4.5 ± 6.6) × 10 −74502 123.2 0.42 0.45 0.95 1233 1 16 (+2.1 ± 9.7) × 10 −6 (−4.4 ± 6.6) × 10 −7Table C.36:

295Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4503 123.2 0.42 0.45 0.95 1233 1 16 (−0.4 ± 9.4) × 10 −6 (−4.4 ± 6.6) × 10 −74504 123.4 0.42 0.45 0.95 1241 1 8 (−5.6 ± 9.8) × 10 −6 (−4.6 ± 6.6) × 10 −74505 123.4 0.42 0.45 0.95 1232 1 17 (+3.2 ± 9.3) × 10 −6 (−4.4 ± 6.6) × 10 −74506 123.5 0.42 0.45 0.95 1236 1 13 (+0.3 ± 9.9) × 10 −6 (−4.4 ± 6.5) × 10 −74507 123.6 0.42 0.45 0.95 1236 1 13 (−10.3 ± 9.4) × 10 −6 (−4.9 ± 6.5) × 10 −74508 123.6 0.42 0.45 0.95 1241 1 8 (−5.8 ± 9.6) × 10 −6 (−5.1 ± 6.5) × 10 −74509 123.5 0.42 0.45 0.95 1239 1 10 (+6.4 ± 9.9) × 10 −6 (−4.8 ± 6.5) × 10 −74510 123.3 0.42 0.45 0.95 1245 1 4 (+0.9 ± 9.5) × 10 −6 (−4.7 ± 6.5) × 10 −74511 123.2 0.42 0.45 0.95 1237 1 12 (+7.9 ± 9.5) × 10 −6 (−4.4 ± 6.5) × 10 −74512 123.3 0.42 0.45 0.95 1187 2 62 (+8.8 ± 9.9) × 10 −6 (−4.0 ± 6.5) × 10 −74513 123.2 0.42 0.45 0.95 1151 1 98 (−1.1 ± 9.9) × 10 −6 (−4.0 ± 6.4) × 10 −74514 123.0 0.42 0.45 0.95 1183 1 66 (+6.9 ± 9.9) × 10 −6 (−3.7 ± 6.4) × 10 −74515 123.4 0.42 0.45 0.95 1162 1 87 (+1.1 ± 1.0) × 10 −5 (−3.2 ± 6.4) × 10 −74516 123.4 0.42 0.45 0.95 1161 0 88 (−14.0 ± 9.6) × 10 −6 (−3.8 ± 6.4) × 10 −74517 123.3 0.42 0.45 0.95 1201 1 48 (−15.2 ± 9.4) × 10 −6 (−4.4 ± 6.4) × 10 −74520 123.4 0.42 0.45 0.95 1181 1 68 (−0.5 ± 1.1) × 10 −5 (−4.6 ± 6.4) × 10 −74521 123.1 0.42 0.45 0.95 1201 1 48 (−0.1 ± 9.8) × 10 −6 (−4.6 ± 6.4) × 10 −74522 122.8 0.42 0.45 0.95 1223 1 26 (+1.0 ± 9.7) × 10 −6 (−4.5 ± 6.3) × 10 −74523 122.8 0.42 0.45 0.95 1222 3 25 (−2.1 ± 9.3) × 10 −6 (−4.6 ± 6.3) × 10 −74524 122.6 0.42 0.45 0.95 1210 1 39 (−1.3 ± 9.5) × 10 −6 (−4.6 ± 6.3) × 10 −74525 122.5 0.42 0.45 0.95 1221 1 28 (+10.0 ± 9.8) × 10 −6 (−4.2 ± 6.3) × 10 −74526 122.1 0.42 0.45 0.95 1239 1 10 (−3.5 ± 9.6) × 10 −6 (−4.3 ± 6.3) × 10 −7Table C.37:

296Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4527 122.4 0.42 0.45 0.95 1235 1 14 (−8.3 ± 9.4) × 10 −6 (−4.6 ± 6.3) × 10 −74528 122.2 0.42 0.45 0.95 1226 1 23 (+22.1 ± 9.4) × 10 −6 (−3.7 ± 6.3) × 10 −74529 122.3 0.42 0.45 0.95 1216 1 33 (+14.3 ± 9.8) × 10 −6 (−3.1 ± 6.2) × 10 −74530 122.3 0.42 0.45 0.95 1219 1 30 (+10.6 ± 9.9) × 10 −6 (−2.6 ± 6.2) × 10 −74531 122.2 0.42 0.45 0.95 1229 1 20 (+10.3 ± 9.9) × 10 −6 (−2.2 ± 6.2) × 10 −74532 122.5 0.42 0.45 0.95 1230 1 19 (−0.2 ± 9.4) × 10 −6 (−2.2 ± 6.2) × 10 −74533 122.4 0.42 0.45 0.95 1239 1 10 (−5.1 ± 9.4) × 10 −6 (−2.4 ± 6.2) × 10 −74537 122.2 0.42 0.45 0.95 1234 1 15 (+2.6 ± 9.6) × 10 −6 (−2.3 ± 6.2) × 10 −74538 122.0 0.42 0.45 0.95 1227 1 22 (−2.8 ± 9.6) × 10 −6 (−2.4 ± 6.2) × 10 −74539 122.0 0.42 0.45 0.95 1235 1 14 (−6.8 ± 9.9) × 10 −6 (−2.7 ± 6.2) × 10 −74540 121.9 0.42 0.45 0.95 1233 1 16 (−9.0 ± 9.6) × 10 −6 (−3.0 ± 6.1) × 10 −74541 120.6 0.42 0.45 0.95 1234 1 15 (−8.5 ± 9.7) × 10 −6 (−3.3 ± 6.1) × 10 −74542 122.3 0.42 0.45 0.95 1236 2 12 (−0.1 ± 1.0) × 10 −5 (−3.4 ± 6.1) × 10 −74543 122.1 0.42 0.45 0.95 1232 1 17 (−1.7 ± 9.8) × 10 −6 (−3.4 ± 6.1) × 10 −74546 122.2 0.42 0.45 0.95 1235 1 14 (−8.0 ± 9.5) × 10 −6 (−3.7 ± 6.1) × 10 −74547 122.2 0.42 0.45 0.95 1231 2 17 (+22.4 ± 9.7) × 10 −6 (−2.8 ± 6.1) × 10 −74548 122.2 0.42 0.45 0.95 1228 1 21 (−14.8 ± 9.1) × 10 −6 (−3.4 ± 6.1) × 10 −74549 122.3 0.42 0.45 0.95 1237 1 12 (−3.0 ± 9.7) × 10 −6 (−3.5 ± 6.1) × 10 −74550 122.3 0.42 0.45 0.95 1233 1 16 (−0.5 ± 9.4) × 10 −6 (−3.5 ± 6.0) × 10 −74551 122.2 0.42 0.45 0.95 411 3 8 (+1.7 ± 1.6) × 10 −5 (−3.3 ± 6.0) × 10 −74554 121.6 0.42 0.45 0.95 12 1 0 (+2.0 ± 9.8) × 10 −5 (−3.3 ± 6.0) × 10 −74555 121.8 0.42 0.45 0.95 1233 1 16 (−9.3 ± 9.7) × 10 −6 (−3.6 ± 6.0) × 10 −7Table C.38:

297Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4556 121.4 0.42 0.45 0.95 1239 1 10 (+1.6 ± 9.2) × 10 −6 (−3.5 ± 6.0) × 10 −74557 121.1 0.42 0.45 0.95 1240 1 9 (−0.3 ± 9.4) × 10 −6 (−3.5 ± 6.0) × 10 −74558 121.1 0.42 0.45 0.95 1233 1 16 (−19.6 ± 9.7) × 10 −6 (−4.3 ± 6.0) × 10 −74559 120.5 0.42 0.45 0.95 1228 1 21 (−4.8 ± 9.5) × 10 −6 (−4.4 ± 6.0) × 10 −74560 121.1 0.42 0.45 0.95 1237 1 12 (+7.5 ± 9.6) × 10 −6 (−4.1 ± 6.0) × 10 −74561 121.0 0.42 0.45 0.95 1242 1 7 (+2.2 ± 9.5) × 10 −6 (−4.0 ± 6.0) × 10 −74562 121.3 0.42 0.45 0.95 1235 1 14 (+7.5 ± 9.9) × 10 −6 (−3.7 ± 6.0) × 10 −74563 121.3 0.42 0.45 0.95 1239 1 10 (−2.6 ± 9.8) × 10 −6 (−3.8 ± 5.9) × 10 −74564 121.3 0.42 0.45 0.95 1234 1 15 (−28.9 ± 9.9) × 10 −6 (−4.9 ± 5.9) × 10 −74565 121.1 0.42 0.45 0.95 1231 1 18 (+10.8 ± 9.7) × 10 −6 (−4.5 ± 5.9) × 10 −74566 121.2 0.42 0.45 0.95 1234 1 15 (+9.5 ± 9.5) × 10 −6 (−4.1 ± 5.9) × 10 −74567 121.1 0.42 0.45 0.95 1237 1 12 (+7.4 ± 9.6) × 10 −6 (−3.8 ± 5.9) × 10 −74568 120.9 0.42 0.45 0.95 273 2 4 (+0.4 ± 2.0) × 10 −5 (−3.8 ± 5.9) × 10 −74571 121.3 0.42 0.45 0.95 1236 0 14 (+6.7 ± 9.9) × 10 −6 (−3.5 ± 5.9) × 10 −74572 120.7 0.42 0.45 0.95 1228 1 21 (+1.9 ± 1.0) × 10 −5 (−2.8 ± 5.9) × 10 −74573 121.3 0.42 0.45 0.95 1235 1 14 (−20.6 ± 9.6) × 10 −6 (−3.5 ± 5.9) × 10 −74574 121.6 0.42 0.45 0.95 1239 1 10 (+0.7 ± 9.4) × 10 −6 (−3.5 ± 5.9) × 10 −74575 121.9 0.42 0.45 0.95 1234 1 15 (+13.9 ± 9.3) × 10 −6 (−3.0 ± 5.8) × 10 −74576 121.8 0.42 0.45 0.95 1240 1 9 (−1.7 ± 9.5) × 10 −6 (−3.0 ± 5.8) × 10 −74577 121.8 0.42 0.45 0.95 1238 1 11 (+3.9 ± 9.4) × 10 −6 (−2.9 ± 5.8) × 10 −74578 121.8 0.42 0.45 0.95 1237 1 12 (−4.1 ± 9.7) × 10 −6 (−3.0 ± 5.8) × 10 −74581 121.9 0.42 0.45 0.95 1237 1 12 (−5.5 ± 10.0) × 10 −6 (−3.2 ± 5.8) × 10 −7Table C.39:

298Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4582 121.9 0.42 0.45 0.95 1244 1 5 (−9.9 ± 9.7) × 10 −6 (−3.5 ± 5.8) × 10 −74583 122.1 0.42 0.45 0.95 1236 1 13 (−18.0 ± 9.6) × 10 −6 (−4.2 ± 5.8) × 10 −74584 122.1 0.42 0.45 0.95 1240 1 9 (−11.5 ± 9.6) × 10 −6 (−4.6 ± 5.8) × 10 −74585 122.2 0.42 0.45 0.95 1237 2 12 (+3.0 ± 9.6) × 10 −6 (−4.5 ± 5.8) × 10 −74586 122.1 0.42 0.45 0.95 1236 1 13 (+11.1 ± 9.7) × 10 −6 (−4.0 ± 5.7) × 10 −74587 122.2 0.42 0.45 0.95 1245 1 4 (−6.7 ± 9.4) × 10 −6 (−4.3 ± 5.7) × 10 −74588 122.1 0.42 0.45 0.95 1237 1 12 (−14.3 ± 9.7) × 10 −6 (−4.8 ± 5.7) × 10 −74589 122.2 0.42 0.45 0.95 1237 1 12 (−0.6 ± 1.0) × 10 −5 (−4.9 ± 5.7) × 10 −74590 121.9 0.42 0.45 0.95 1240 1 9 (+0.8 ± 9.3) × 10 −6 (−4.9 ± 5.7) × 10 −74591 121.9 0.42 0.45 0.95 1244 1 5 (+8.3 ± 9.6) × 10 −6 (−4.6 ± 5.7) × 10 −74592 122.0 0.42 0.45 0.95 1242 1 7 (−2.2 ± 9.8) × 10 −6 (−4.7 ± 5.7) × 10 −74593 122.1 0.42 0.45 0.95 1246 1 3 (−11.5 ± 9.6) × 10 −6 (−5.0 ± 5.7) × 10 −74594 122.1 0.42 0.45 0.95 1183 2 7 (+4.8 ± 9.6) × 10 −6 (−4.9 ± 5.7) × 10 −74595 122.5 0.42 0.45 0.95 158 2 1 (−3.2 ± 2.5) × 10 −5 (−5.0 ± 5.7) × 10 −74610 122.1 0.42 0.45 0.95 1246 1 3 (+8.7 ± 9.4) × 10 −6 (−4.7 ± 5.7) × 10 −74611 122.0 0.42 0.45 0.95 1239 2 9 (+4.0 ± 9.4) × 10 −6 (−4.5 ± 5.6) × 10 −74612 122.1 0.42 0.45 0.95 954 1 8 (+1.2 ± 1.0) × 10 −5 (−4.2 ± 5.6) × 10 −74613 122.4 0.42 0.45 0.95 1135 1 15 (+11.0 ± 9.7) × 10 −6 (−3.8 ± 5.6) × 10 −74614 122.7 0.42 0.45 0.95 1247 1 2 (+21.3 ± 9.8) × 10 −6 (−3.1 ± 5.6) × 10 −74615 122.2 0.42 0.45 0.95 1239 1 10 (+8.9 ± 9.5) × 10 −6 (−2.8 ± 5.6) × 10 −74616 120.1 0.42 0.45 0.95 1240 1 9 (+4.1 ± 9.7) × 10 −6 (−2.6 ± 5.6) × 10 −74618 122.3 0.42 0.45 0.95 1245 1 4 (+4.0 ± 9.3) × 10 −6 (−2.5 ± 5.6) × 10 −7Table C.40:

299Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4619 122.4 0.42 0.45 0.95 1243 1 6 (+10.0 ± 9.3) × 10 −6 (−2.1 ± 5.6) × 10 −74620 122.3 0.42 0.45 0.95 1244 1 5 (+6.3 ± 9.4) × 10 −6 (−1.9 ± 5.6) × 10 −74621 122.3 0.42 0.45 0.95 1242 1 7 (−15.0 ± 9.8) × 10 −6 (−2.4 ± 5.6) × 10 −74622 122.4 0.42 0.45 0.95 1239 0 10 (−2.5 ± 9.4) × 10 −6 (−2.5 ± 5.5) × 10 −74626 122.9 0.42 0.45 0.95 1225 2 23 (−4.2 ± 9.3) × 10 −6 (−2.6 ± 5.5) × 10 −74627 123.2 0.42 0.45 0.95 1228 1 21 (+1.9 ± 9.6) × 10 −6 (−2.6 ± 5.5) × 10 −74628 123.9 0.42 0.45 0.95 1231 1 18 (+16.0 ± 9.5) × 10 −6 (−2.0 ± 5.5) × 10 −74629 123.0 0.42 0.45 0.95 1231 1 18 (−11.5 ± 9.2) × 10 −6 (−2.4 ± 5.5) × 10 −74630 122.8 0.42 0.45 0.95 1236 1 13 (+22.8 ± 9.1) × 10 −6 (−1.6 ± 5.5) × 10 −74631 122.5 0.42 0.45 0.95 1241 1 8 (+0.7 ± 9.1) × 10 −6 (−1.6 ± 5.5) × 10 −74632 121.9 0.42 0.45 0.95 1245 1 4 (−1.3 ± 9.1) × 10 −6 (−1.7 ± 5.5) × 10 −74633 122.8 0.42 0.45 0.95 1239 1 10 (+1.3 ± 9.4) × 10 −6 (−1.6 ± 5.5) × 10 −74634 122.9 0.42 0.45 0.95 1236 2 13 (+1.0 ± 9.3) × 10 −6 (−1.6 ± 5.5) × 10 −74635 122.8 0.42 0.45 0.95 1193 1 56 (+1.7 ± 1.3) × 10 −5 (−1.0 ± 5.5) × 10 −74636 123.2 0.42 0.45 0.95 1246 1 3 (−19.6 ± 9.5) × 10 −6 (−1.7 ± 5.5) × 10 −74653 122.3 0.42 0.45 0.95 1238 1 11 (+10.5 ± 9.6) × 10 −6 (−1.3 ± 5.4) × 10 −74654 122.0 0.42 0.45 0.95 1229 1 20 (−13.0 ± 9.4) × 10 −6 (−1.7 ± 5.4) × 10 −74655 122.0 0.42 0.45 0.95 1231 1 18 (−7.5 ± 9.5) × 10 −6 (−1.9 ± 5.4) × 10 −74656 121.8 0.42 0.45 0.95 1217 1 32 (+7.3 ± 9.6) × 10 −6 (−1.7 ± 5.4) × 10 −74657 121.9 0.42 0.45 0.95 1160 1 89 (−0.8 ± 9.8) × 10 −6 (−1.7 ± 5.4) × 10 −74658 121.8 0.42 0.45 0.95 1170 1 79 (−11.8 ± 9.6) × 10 −6 (−2.1 ± 5.4) × 10 −74659 121.4 0.42 0.45 0.95 1210 1 39 (−9.5 ± 9.8) × 10 −6 (−2.4 ± 5.4) × 10 −7Table C.41:

300Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4660 121.4 0.42 0.45 0.95 1199 1 50 (−4.8 ± 9.4) × 10 −6 (−2.5 ± 5.4) × 10 −74661 121.7 0.42 0.45 0.95 1207 1 42 (−0.7 ± 9.3) × 10 −6 (−2.5 ± 5.4) × 10 −74662 122.4 0.42 0.45 0.95 1232 1 17 (−1.7 ± 9.6) × 10 −6 (−2.6 ± 5.4) × 10 −74663 122.7 0.42 0.45 0.95 1229 3 18 (−20.9 ± 9.1) × 10 −6 (−3.2 ± 5.4) × 10 −74664 122.4 0.42 0.45 0.95 1227 1 22 (+9.0 ± 8.8) × 10 −6 (−2.9 ± 5.3) × 10 −74665 122.7 0.42 0.45 0.95 1232 1 17 (+4.1 ± 9.6) × 10 −6 (−2.8 ± 5.3) × 10 −74668 122.9 0.42 0.45 0.95 1235 1 14 (+15.4 ± 9.7) × 10 −6 (−2.3 ± 5.3) × 10 −74669 122.7 0.42 0.45 0.95 1242 1 7 (+5.7 ± 9.3) × 10 −6 (−2.1 ± 5.3) × 10 −74670 122.6 0.42 0.45 0.95 1228 1 21 (−13.9 ± 9.2) × 10 −6 (−2.5 ± 5.3) × 10 −74671 122.8 0.42 0.45 0.95 1234 1 15 (−12.0 ± 9.3) × 10 −6 (−2.9 ± 5.3) × 10 −74672 122.4 0.42 0.45 0.95 1242 1 7 (−11.3 ± 9.7) × 10 −6 (−3.2 ± 5.3) × 10 −74673 122.6 0.42 0.45 0.95 1241 1 8 (+7.6 ± 9.5) × 10 −6 (−3.0 ± 5.3) × 10 −74674 122.9 0.42 0.45 0.95 1236 1 13 (+0.4 ± 9.4) × 10 −6 (−3.0 ± 5.3) × 10 −74675 122.7 0.42 0.45 0.95 1239 1 10 (−4.2 ± 9.3) × 10 −6 (−3.1 ± 5.3) × 10 −74676 122.5 0.42 0.45 0.95 1242 1 7 (+8.0 ± 9.4) × 10 −6 (−2.8 ± 5.3) × 10 −74677 122.2 0.42 0.45 0.95 1228 1 21 (−2.0 ± 9.4) × 10 −6 (−2.9 ± 5.3) × 10 −74678 121.9 0.42 0.45 0.95 1221 1 28 (−6.5 ± 9.6) × 10 −6 (−3.1 ± 5.2) × 10 −74679 121.6 0.42 0.45 0.95 1220 1 29 (−9.9 ± 9.8) × 10 −6 (−3.4 ± 5.2) × 10 −74680 122.2 0.42 0.45 0.95 1230 1 19 (+7.0 ± 9.1) × 10 −6 (−3.1 ± 5.2) × 10 −74683 122.2 0.42 0.45 0.95 1235 1 14 (−10.2 ± 9.8) × 10 −6 (−3.4 ± 5.2) × 10 −74684 122.6 0.42 0.45 0.95 1233 1 16 (−0.9 ± 9.3) × 10 −6 (−3.4 ± 5.2) × 10 −74685 122.2 0.42 0.45 0.95 1235 1 14 (+6.1 ± 9.8) × 10 −6 (−3.3 ± 5.2) × 10 −7Table C.42:

301Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4686 121.8 0.42 0.45 0.95 1225 1 24 (−10.0 ± 9.4) × 10 −6 (−3.5 ± 5.2) × 10 −74687 122.3 0.42 0.45 0.95 1238 1 11 (−13.7 ± 9.8) × 10 −6 (−3.9 ± 5.2) × 10 −74688 121.1 0.42 0.45 0.95 1236 1 13 (+5.4 ± 9.3) × 10 −6 (−3.8 ± 5.2) × 10 −74689 122.0 0.42 0.45 0.95 1239 1 10 (+8.4 ± 9.4) × 10 −6 (−3.5 ± 5.2) × 10 −74690 122.3 0.42 0.45 0.95 1234 1 15 (−2.9 ± 9.1) × 10 −6 (−3.6 ± 5.2) × 10 −74691 122.2 0.42 0.45 0.95 1237 1 12 (+5.4 ± 9.1) × 10 −6 (−3.4 ± 5.2) × 10 −74692 121.9 0.42 0.45 0.95 1235 1 14 (−18.5 ± 9.5) × 10 −6 (−3.9 ± 5.2) × 10 −74693 121.5 0.42 0.45 0.95 1230 1 19 (−3.5 ± 9.4) × 10 −6 (−4.0 ± 5.1) × 10 −74694 121.8 0.42 0.45 0.95 1223 1 26 (−4.5 ± 9.4) × 10 −6 (−4.1 ± 5.1) × 10 −74695 121.8 0.42 0.45 0.95 1232 1 17 (+13.7 ± 9.6) × 10 −6 (−3.7 ± 5.1) × 10 −74696 119.9 0.42 0.45 0.95 1224 1 25 (−15.5 ± 9.7) × 10 −6 (−4.2 ± 5.1) × 10 −74699 121.4 0.42 0.45 0.95 1228 1 21 (−3.0 ± 9.6) × 10 −6 (−4.2 ± 5.1) × 10 −74700 121.6 0.42 0.45 0.95 1227 0 22 (+2.6 ± 9.3) × 10 −6 (−4.1 ± 5.1) × 10 −74701 121.2 0.42 0.45 0.95 1208 1 41 (−3.3 ± 9.8) × 10 −6 (−4.2 ± 5.1) × 10 −74702 121.7 0.42 0.45 0.95 1232 1 17 (−1.4 ± 9.2) × 10 −6 (−4.3 ± 5.1) × 10 −74703 122.4 0.42 0.45 0.95 1236 2 12 (−4.7 ± 8.9) × 10 −6 (−4.4 ± 5.1) × 10 −74704 122.8 0.42 0.45 0.95 1186 1 63 (+0.7 ± 1.3) × 10 −5 (−4.2 ± 5.1) × 10 −74705 122.6 0.42 0.45 0.95 1230 1 19 (+3.4 ± 9.2) × 10 −6 (−4.1 ± 5.1) × 10 −74706 122.3 0.42 0.45 0.95 1237 1 12 (+3.7 ± 9.2) × 10 −6 (−3.9 ± 5.1) × 10 −74707 122.9 0.42 0.45 0.95 1073 1 11 (−4.2 ± 9.9) × 10 −6 (−4.0 ± 5.1) × 10 −74708 121.9 0.42 0.45 0.95 1236 1 13 (−5.1 ± 9.4) × 10 −6 (−4.2 ± 5.1) × 10 −74709 122.8 0.42 0.45 0.95 1239 1 10 (−8.8 ± 9.4) × 10 −6 (−4.4 ± 5.0) × 10 −7Table C.43:

302Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4713 121.8 0.42 0.45 0.95 1239 1 10 (−4.6 ± 9.5) × 10 −6 (−4.5 ± 5.0) × 10 −74714 121.4 0.42 0.45 0.95 1237 2 12 (+8.0 ± 9.4) × 10 −6 (−4.3 ± 5.0) × 10 −74715 121.4 0.42 0.45 0.95 1237 1 12 (−1.6 ± 9.7) × 10 −6 (−4.3 ± 5.0) × 10 −74716 121.0 0.42 0.45 0.95 1221 1 28 (−6.0 ± 9.6) × 10 −6 (−4.5 ± 5.0) × 10 −74717 121.0 0.42 0.45 0.95 1236 1 13 (−3.7 ± 9.6) × 10 −6 (−4.6 ± 5.0) × 10 −74718 121.2 0.42 0.45 0.95 1236 1 13 (+12.6 ± 9.3) × 10 −6 (−4.2 ± 5.0) × 10 −74719 121.8 0.42 0.45 0.95 1242 1 7 (−0.6 ± 9.4) × 10 −6 (−4.2 ± 5.0) × 10 −74720 122.0 0.42 0.45 0.95 435 2 5 (−1.8 ± 1.5) × 10 −5 (−4.4 ± 5.0) × 10 −74785 120.0 0.42 0.45 0.95 1206 1 43 (−1.1 ± 1.0) × 10 −5 (−4.6 ± 5.0) × 10 −74786 120.9 0.42 0.45 0.95 1215 1 34 (−6.8 ± 9.7) × 10 −6 (−4.8 ± 5.0) × 10 −74787 121.4 0.42 0.45 0.95 1202 1 47 (−4.3 ± 9.7) × 10 −6 (−4.9 ± 5.0) × 10 −74788 120.5 0.42 0.45 0.95 1171 1 78 (−15.6 ± 9.5) × 10 −6 (−5.3 ± 5.0) × 10 −74789 119.6 0.42 0.45 0.95 1160 3 87 (−17.2 ± 9.9) × 10 −6 (−5.7 ± 5.0) × 10 −74790 119.1 0.42 0.45 0.95 1166 1 83 (−1.2 ± 1.0) × 10 −5 (−6.0 ± 5.0) × 10 −74791 120.0 0.42 0.45 0.95 1185 1 64 (−0.2 ± 1.0) × 10 −5 (−6.0 ± 5.0) × 10 −74792 122.2 0.42 0.45 0.95 1240 1 9 (+10.9 ± 9.4) × 10 −6 (−5.7 ± 4.9) × 10 −74793 122.1 0.42 0.45 0.95 1231 1 18 (−3.2 ± 9.6) × 10 −6 (−5.8 ± 4.9) × 10 −74794 121.9 0.42 0.45 0.95 1235 1 14 (−13.4 ± 9.4) × 10 −6 (−6.1 ± 4.9) × 10 −74795 121.6 0.42 0.45 0.95 1240 1 9 (−4.4 ± 9.4) × 10 −6 (−6.2 ± 4.9) × 10 −74796 121.3 0.42 0.45 0.95 1222 1 27 (+14.9 ± 9.4) × 10 −6 (−5.8 ± 4.9) × 10 −74797 121.3 0.42 0.45 0.95 1228 1 21 (+10.2 ± 9.4) × 10 −6 (−5.5 ± 4.9) × 10 −74798 121.5 0.42 0.45 0.95 1234 1 15 (+4.6 ± 9.5) × 10 −6 (−5.4 ± 4.9) × 10 −7Table C.44:

303Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4799 121.1 0.42 0.45 0.95 1226 1 23 (+0.1 ± 1.0) × 10 −5 (−5.4 ± 4.9) × 10 −74802 120.7 0.42 0.45 0.95 1223 1 26 (+0.4 ± 1.0) × 10 −5 (−5.3 ± 4.9) × 10 −74803 120.1 0.42 0.45 0.95 1233 1 16 (−4.2 ± 9.7) × 10 −6 (−5.3 ± 4.9) × 10 −74804 119.8 0.42 0.45 0.95 1196 1 53 (−2.3 ± 9.8) × 10 −6 (−5.4 ± 4.9) × 10 −74805 120.0 0.42 0.45 0.95 1138 0 111 (−0.3 ± 1.0) × 10 −5 (−5.4 ± 4.9) × 10 −74806 119.8 0.42 0.45 0.95 1138 0 111 (−13.0 ± 9.8) × 10 −6 (−5.7 ± 4.9) × 10 −74807 120.8 0.42 0.45 0.95 1196 1 53 (−5.8 ± 9.7) × 10 −6 (−5.9 ± 4.9) × 10 −74808 121.6 0.42 0.45 0.95 1226 1 23 (+10.5 ± 9.3) × 10 −6 (−5.6 ± 4.9) × 10 −74809 121.4 0.42 0.45 0.95 1238 2 10 (+4.1 ± 9.6) × 10 −6 (−5.5 ± 4.9) × 10 −74810 121.6 0.42 0.45 0.95 1233 2 15 (−0.9 ± 9.5) × 10 −6 (−5.5 ± 4.8) × 10 −74811 121.6 0.42 0.45 0.95 1234 1 15 (−3.9 ± 9.6) × 10 −6 (−5.6 ± 4.8) × 10 −74812 121.4 0.42 0.45 0.95 1238 1 11 (+11.6 ± 9.3) × 10 −6 (−5.3 ± 4.8) × 10 −74813 121.1 0.42 0.45 0.95 1239 1 10 (−7.1 ± 9.4) × 10 −6 (−5.4 ± 4.8) × 10 −74814 120.8 0.42 0.45 0.95 1235 1 14 (+10.8 ± 9.3) × 10 −6 (−5.1 ± 4.8) × 10 −74817 121.3 0.42 0.45 0.95 1239 1 10 (−0.7 ± 1.0) × 10 −5 (−5.3 ± 4.8) × 10 −74818 120.7 0.42 0.45 0.95 1233 1 16 (−9.4 ± 9.8) × 10 −6 (−5.5 ± 4.8) × 10 −74819 121.3 0.42 0.45 0.95 1232 1 17 (+16.2 ± 9.5) × 10 −6 (−5.1 ± 4.8) × 10 −74820 121.3 0.42 0.45 0.95 1228 1 21 (+13.4 ± 9.4) × 10 −6 (−4.8 ± 4.8) × 10 −74821 121.0 0.42 0.45 0.95 1230 1 19 (−16.4 ± 9.6) × 10 −6 (−5.1 ± 4.8) × 10 −74822 121.1 0.42 0.45 0.95 1234 1 15 (−2.3 ± 9.7) × 10 −6 (−5.2 ± 4.8) × 10 −74823 121.4 0.42 0.45 0.95 1241 3 6 (−15.3 ± 9.5) × 10 −6 (−5.6 ± 4.8) × 10 −74824 121.5 0.42 0.45 0.95 1244 1 5 (−13.7 ± 9.4) × 10 −6 (−5.9 ± 4.8) × 10 −7Table C.45:

304Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4825 121.8 0.42 0.45 0.95 1237 1 12 (+10.2 ± 9.6) × 10 −6 (−5.6 ± 4.8) × 10 −74826 121.7 0.42 0.45 0.95 1243 1 6 (−12.0 ± 9.5) × 10 −6 (−5.9 ± 4.8) × 10 −74827 121.8 0.42 0.45 0.95 1239 1 10 (−16.3 ± 9.4) × 10 −6 (−6.3 ± 4.8) × 10 −74828 121.6 0.42 0.45 0.95 1240 1 9 (−0.0 ± 9.4) × 10 −6 (−6.3 ± 4.7) × 10 −74829 121.7 0.42 0.45 0.95 1239 1 10 (−7.8 ± 9.5) × 10 −6 (−6.5 ± 4.7) × 10 −74833 120.3 0.42 0.45 0.95 1238 1 11 (+15.3 ± 9.4) × 10 −6 (−6.1 ± 4.7) × 10 −74834 121.4 0.42 0.45 0.95 1238 1 11 (−2.5 ± 9.5) × 10 −6 (−6.1 ± 4.7) × 10 −74835 122.4 0.42 0.45 0.95 1246 1 3 (−8.8 ± 9.6) × 10 −6 (−6.3 ± 4.7) × 10 −74836 121.2 0.42 0.45 0.95 1239 2 9 (+12.0 ± 9.5) × 10 −6 (−6.0 ± 4.7) × 10 −74837 121.1 0.42 0.45 0.95 1243 1 6 (+11.3 ± 9.5) × 10 −6 (−5.7 ± 4.7) × 10 −74838 121.0 0.42 0.45 0.95 1239 1 10 (−0.0 ± 9.6) × 10 −6 (−5.7 ± 4.7) × 10 −74839 121.3 0.42 0.45 0.95 1241 2 7 (−1.3 ± 9.4) × 10 −6 (−5.7 ± 4.7) × 10 −74840 121.3 0.42 0.45 0.95 1239 2 9 (−3.7 ± 9.8) × 10 −6 (−5.8 ± 4.7) × 10 −74841 121.4 0.42 0.45 0.95 1132 2 2 (+0.1 ± 1.0) × 10 −5 (−5.8 ± 4.7) × 10 −74842 121.1 0.42 0.45 0.95 1235 1 14 (−6.7 ± 9.8) × 10 −6 (−5.9 ± 4.7) × 10 −74843 121.3 0.42 0.45 0.95 1232 2 16 (−0.7 ± 9.9) × 10 −6 (−5.9 ± 4.7) × 10 −74846 121.2 0.42 0.45 0.95 1235 1 14 (−5.9 ± 9.6) × 10 −6 (−6.0 ± 4.7) × 10 −74847 120.9 0.42 0.45 0.95 1231 1 18 (+3.6 ± 9.4) × 10 −6 (−5.9 ± 4.7) × 10 −74848 120.8 0.42 0.45 0.95 1235 1 14 (+12.1 ± 9.5) × 10 −6 (−5.6 ± 4.7) × 10 −74849 120.9 0.42 0.45 0.95 1239 1 10 (+2.6 ± 9.6) × 10 −6 (−5.6 ± 4.7) × 10 −74850 120.4 0.42 0.45 0.95 1235 1 14 (−0.9 ± 9.6) × 10 −6 (−5.6 ± 4.7) × 10 −74851 120.4 0.42 0.45 0.95 1194 1 55 (+0.2 ± 1.3) × 10 −5 (−5.5 ± 4.6) × 10 −7Table C.46:

305Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4852 120.4 0.42 0.45 0.95 1224 1 25 (+7.6 ± 9.5) × 10 −6 (−5.3 ± 4.6) × 10 −74853 120.7 0.42 0.45 0.95 1234 1 15 (+18.1 ± 9.5) × 10 −6 (−4.9 ± 4.6) × 10 −74854 120.1 0.42 0.45 0.95 785 2 22 (+1.2 ± 1.2) × 10 −5 (−4.7 ± 4.6) × 10 −74855 119.3 0.42 0.45 0.95 1222 0 28 (−2.2 ± 9.9) × 10 −6 (−4.7 ± 4.6) × 10 −74856 120.7 0.42 0.45 0.95 1210 1 38 (+2.2 ± 9.5) × 10 −6 (−4.7 ± 4.6) × 10 −74857 121.7 0.42 0.45 0.95 1178 1 61 (+8.1 ± 9.8) × 10 −6 (−4.5 ± 4.6) × 10 −74858 121.4 0.42 0.45 0.95 1205 1 44 (−2.3 ± 9.9) × 10 −6 (−4.5 ± 4.6) × 10 −74859 121.8 0.42 0.45 0.95 1210 1 39 (+11.5 ± 9.6) × 10 −6 (−4.3 ± 4.6) × 10 −74862 123.1 0.42 0.45 0.95 1236 1 13 (+4.2 ± 9.9) × 10 −6 (−4.2 ± 4.6) × 10 −74863 124.0 0.42 0.45 0.95 1246 1 3 (+27.1 ± 9.6) × 10 −6 (−3.5 ± 4.6) × 10 −74864 124.1 0.42 0.45 0.95 1241 1 8 (−12.2 ± 9.6) × 10 −6 (−3.8 ± 4.6) × 10 −74865 124.2 0.42 0.45 0.95 1242 3 5 (+11.5 ± 9.2) × 10 −6 (−3.5 ± 4.6) × 10 −74866 124.1 0.42 0.45 0.95 1244 1 5 (−0.1 ± 9.6) × 10 −6 (−3.5 ± 4.6) × 10 −74867 124.0 0.42 0.45 0.95 1243 1 6 (−8.9 ± 9.6) × 10 −6 (−3.7 ± 4.6) × 10 −74868 124.1 0.42 0.45 0.95 1247 1 2 (+4.2 ± 9.3) × 10 −6 (−3.6 ± 4.6) × 10 −74869 124.0 0.42 0.45 0.95 1247 1 2 (−4.1 ± 9.5) × 10 −6 (−3.7 ± 4.6) × 10 −74870 122.6 0.42 0.45 0.95 1241 1 8 (−3.1 ± 9.4) × 10 −6 (−3.8 ± 4.6) × 10 −74871 123.2 0.42 0.45 0.95 1212 1 37 (+1.6 ± 9.7) × 10 −6 (−3.7 ± 4.6) × 10 −74872 125.6 0.42 0.45 0.95 1245 1 4 (−6.8 ± 9.6) × 10 −6 (−3.9 ± 4.6) × 10 −74873 126.3 0.42 0.45 0.95 1247 1 2 (+7.3 ± 9.4) × 10 −6 (−3.7 ± 4.5) × 10 −74874 126.3 0.42 0.45 0.95 1249 1 0 (+14.5 ± 9.3) × 10 −6 (−3.3 ± 4.5) × 10 −74875 126.7 0.42 0.45 0.95 1240 1 9 (−17.4 ± 9.4) × 10 −6 (−3.7 ± 4.5) × 10 −7Table C.47:

306Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4876 126.6 0.42 0.45 0.95 1248 1 1 (−5.6 ± 9.1) × 10 −6 (−3.8 ± 4.5) × 10 −74879 124.9 0.42 0.45 0.95 1244 2 4 (−16.3 ± 9.8) × 10 −6 (−4.2 ± 4.5) × 10 −74880 124.2 0.42 0.45 0.95 1245 1 4 (+3.1 ± 9.1) × 10 −6 (−4.1 ± 4.5) × 10 −74881 123.5 0.42 0.45 0.95 1240 1 9 (−12.4 ± 9.2) × 10 −6 (−4.4 ± 4.5) × 10 −74882 123.2 0.42 0.45 0.95 1242 2 6 (−6.4 ± 9.6) × 10 −6 (−4.5 ± 4.5) × 10 −74883 123.4 0.42 0.45 0.95 1243 3 4 (+3.1 ± 9.4) × 10 −6 (−4.4 ± 4.5) × 10 −74884 123.4 0.42 0.45 0.95 1247 1 2 (+7.0 ± 9.0) × 10 −6 (−4.3 ± 4.5) × 10 −74885 123.2 0.42 0.45 0.95 1247 2 1 (+7.2 ± 9.3) × 10 −6 (−4.1 ± 4.5) × 10 −74886 123.1 0.42 0.45 0.95 1247 1 2 (+14.9 ± 9.2) × 10 −6 (−3.8 ± 4.5) × 10 −74945 121.4 0.42 0.45 0.95 1222 1 27 (−5.7 ± 9.9) × 10 −6 (−3.9 ± 4.5) × 10 −74946 119.5 0.42 0.45 0.95 1212 1 37 (+1.7 ± 9.4) × 10 −6 (−3.8 ± 4.5) × 10 −74947 119.3 0.42 0.45 0.95 1229 1 20 (−0.5 ± 9.5) × 10 −6 (−3.8 ± 4.5) × 10 −74948 119.3 0.42 0.45 0.95 1233 1 16 (−12.4 ± 9.5) × 10 −6 (−4.1 ± 4.5) × 10 −74949 119.2 0.42 0.45 0.95 1233 1 16 (−3.2 ± 9.6) × 10 −6 (−4.2 ± 4.5) × 10 −74950 119.1 0.42 0.45 0.95 1229 1 20 (−0.3 ± 9.4) × 10 −6 (−4.2 ± 4.5) × 10 −74951 118.7 0.42 0.45 0.95 1229 1 20 (−1.3 ± 9.3) × 10 −6 (−4.2 ± 4.5) × 10 −74952 118.1 0.42 0.45 0.95 1167 1 42 (+2.5 ± 9.8) × 10 −6 (−4.1 ± 4.4) × 10 −74953 114.6 0.42 0.45 0.95 1200 1 49 (+0.1 ± 9.6) × 10 −6 (−4.1 ± 4.4) × 10 −74954 117.4 0.42 0.45 0.95 1186 1 63 (+5.3 ± 9.6) × 10 −6 (−4.0 ± 4.4) × 10 −74955 117.3 0.42 0.45 0.95 1159 1 90 (+1.1 ± 1.0) × 10 −5 (−3.8 ± 4.4) × 10 −74956 117.0 0.42 0.45 0.95 1084 1 165 (+12.2 ± 9.9) × 10 −6 (−3.5 ± 4.4) × 10 −74957 116.7 0.42 0.45 0.95 1118 1 131 (+0.4 ± 1.0) × 10 −5 (−3.4 ± 4.4) × 10 −7Table C.48:

307Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4958 116.4 0.42 0.45 0.95 1094 1 155 (+0.1 ± 1.0) × 10 −5 (−3.4 ± 4.4) × 10 −74961 118.8 0.42 0.45 0.95 1224 1 25 (+8.1 ± 9.8) × 10 −6 (−3.3 ± 4.4) × 10 −74962 118.1 0.42 0.45 0.95 1216 3 31 (+12.8 ± 9.8) × 10 −6 (−3.0 ± 4.4) × 10 −74963 117.9 0.42 0.45 0.95 1204 1 45 (+5.3 ± 9.6) × 10 −6 (−2.9 ± 4.4) × 10 −74964 117.8 0.42 0.45 0.95 1217 1 32 (+3.5 ± 9.6) × 10 −6 (−2.8 ± 4.4) × 10 −74965 118.3 0.42 0.45 0.95 1212 1 37 (−2.4 ± 9.9) × 10 −6 (−2.8 ± 4.4) × 10 −74966 120.3 0.42 0.45 0.95 1234 1 15 (+6.6 ± 9.5) × 10 −6 (−2.7 ± 4.4) × 10 −74967 121.2 0.42 0.45 0.95 1247 1 2 (−9.1 ± 9.6) × 10 −6 (−2.9 ± 4.4) × 10 −74968 121.6 0.42 0.45 0.95 1246 1 3 (−3.1 ± 9.4) × 10 −6 (−2.9 ± 4.4) × 10 −74969 121.9 0.42 0.45 0.95 1246 2 2 (−5.3 ± 9.3) × 10 −6 (−3.0 ± 4.4) × 10 −74970 121.7 0.42 0.45 0.95 1243 1 6 (−3.7 ± 8.9) × 10 −6 (−3.1 ± 4.4) × 10 −74971 122.1 0.42 0.45 0.95 1247 1 2 (−8.7 ± 9.5) × 10 −6 (−3.3 ± 4.4) × 10 −74972 122.3 0.42 0.45 0.95 1248 1 1 (+10.0 ± 9.6) × 10 −6 (−3.1 ± 4.4) × 10 −74973 122.4 0.42 0.45 0.95 1247 1 2 (+1.7 ± 9.0) × 10 −6 (−3.0 ± 4.4) × 10 −74977 111.0 0.42 0.45 0.95 1249 1 0 (+7.6 ± 9.7) × 10 −6 (−2.9 ± 4.4) × 10 −74978 121.0 0.42 0.45 0.95 1246 1 3 (+12.4 ± 9.8) × 10 −6 (−2.6 ± 4.4) × 10 −74979 121.6 0.42 0.45 0.95 1248 1 1 (−8.7 ± 9.4) × 10 −6 (−2.8 ± 4.3) × 10 −74980 122.2 0.42 0.45 0.95 1248 1 1 (−11.3 ± 9.4) × 10 −6 (−3.0 ± 4.3) × 10 −74981 122.6 0.42 0.45 0.95 1248 1 1 (+7.6 ± 9.5) × 10 −6 (−2.8 ± 4.3) × 10 −74982 122.5 0.42 0.45 0.95 1248 1 1 (−15.4 ± 9.6) × 10 −6 (−3.1 ± 4.3) × 10 −74983 122.5 0.42 0.45 0.95 1247 1 2 (+8.9 ± 9.1) × 10 −6 (−3.0 ± 4.3) × 10 −74984 122.8 0.42 0.45 0.95 1249 1 0 (−1.4 ± 9.2) × 10 −6 (−3.0 ± 4.3) × 10 −7Table C.49:

308Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut4985 122.8 0.42 0.45 0.95 1249 1 0 (+3.9 ± 9.4) × 10 −6 (−2.9 ± 4.3) × 10 −74986 122.9 0.42 0.45 0.95 1248 1 1 (−16.6 ± 9.3) × 10 −6 (−3.2 ± 4.3) × 10 −74987 123.1 0.42 0.45 0.95 1249 1 0 (−11.0 ± 9.5) × 10 −6 (−3.4 ± 4.3) × 10 −74988 122.8 0.42 0.45 0.95 1248 1 1 (−8.6 ± 9.6) × 10 −6 (−3.6 ± 4.3) × 10 −74989 122.8 0.42 0.45 0.95 1248 1 1 (−5.6 ± 9.3) × 10 −6 (−3.7 ± 4.3) × 10 −74993 122.0 0.42 0.45 0.95 1245 1 4 (−8.0 ± 10.0) × 10 −6 (−3.9 ± 4.3) × 10 −74994 122.3 0.42 0.45 0.95 1245 1 4 (+19.2 ± 9.3) × 10 −6 (−3.5 ± 4.3) × 10 −74995 122.5 0.42 0.45 0.95 1248 1 1 (+17.8 ± 9.2) × 10 −6 (−3.1 ± 4.3) × 10 −74996 122.6 0.42 0.45 0.95 1247 1 2 (+3.7 ± 9.6) × 10 −6 (−3.0 ± 4.3) × 10 −74997 122.8 0.42 0.45 0.95 1247 1 2 (+6.9 ± 9.5) × 10 −6 (−2.9 ± 4.3) × 10 −74998 121.5 0.42 0.45 0.95 1241 1 8 (+30.4 ± 9.8) × 10 −6 (−2.3 ± 4.3) × 10 −74999 120.5 0.42 0.45 0.95 1242 1 7 (+0.3 ± 1.1) × 10 −5 (−2.2 ± 4.3) × 10 −75000 120.2 0.42 0.45 0.95 751 1 492 (+2.3 ± 1.5) × 10 −5 (−1.9 ± 4.3) × 10 −75002 120.2 0.42 0.45 0.95 786 2 16 (+2.9 ± 1.3) × 10 −5 (−1.6 ± 4.3) × 10 −75003 119.5 0.42 0.45 0.95 1237 1 12 (+0.6 ± 9.8) × 10 −6 (−1.5 ± 4.3) × 10 −75007 120.2 0.42 0.45 0.95 1237 1 12 (−3.2 ± 9.5) × 10 −6 (−1.6 ± 4.3) × 10 −75008 120.2 0.42 0.45 0.95 1240 1 9 (+8.2 ± 9.8) × 10 −6 (−1.4 ± 4.3) × 10 −75009 120.0 0.42 0.45 0.95 1238 1 11 (+13.1 ± 9.4) × 10 −6 (−1.2 ± 4.3) × 10 −75010 120.1 0.42 0.45 0.95 1239 1 10 (−11.3 ± 9.3) × 10 −6 (−1.4 ± 4.2) × 10 −75011 120.2 0.42 0.45 0.95 1243 1 6 (+0.9 ± 9.2) × 10 −6 (−1.4 ± 4.2) × 10 −75012 120.4 0.42 0.45 0.95 1241 1 8 (+4.9 ± 9.5) × 10 −6 (−1.3 ± 4.2) × 10 −75013 120.6 0.42 0.45 0.95 1242 1 7 (−9.8 ± 9.5) × 10 −6 (−1.5 ± 4.2) × 10 −7Table C.50:

309Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut5014 120.6 0.42 0.45 0.95 1247 1 2 (+1.0 ± 9.7) × 10 −6 (−1.4 ± 4.2) × 10 −75015 120.5 0.42 0.45 0.95 1243 1 6 (−5.2 ± 9.6) × 10 −6 (−1.5 ± 4.2) × 10 −75016 120.5 0.42 0.45 0.95 1181 1 68 (−12.8 ± 9.9) × 10 −6 (−1.8 ± 4.2) × 10 −75017 117.0 0.42 0.45 0.95 790 1 459 (−0.7 ± 1.2) × 10 −5 (−1.9 ± 4.2) × 10 −75018 118.5 0.42 0.45 0.95 747 2 501 (−0.8 ± 1.2) × 10 −5 (−2.0 ± 4.2) × 10 −75019 120.4 0.42 0.45 0.95 971 1 278 (+0.2 ± 1.1) × 10 −5 (−1.9 ± 4.2) × 10 −75020 121.4 0.42 0.45 0.95 1153 1 96 (−6.6 ± 9.6) × 10 −6 (−2.0 ± 4.2) × 10 −75021 121.6 0.42 0.45 0.95 1115 2 133 (−11.2 ± 9.8) × 10 −6 (−2.2 ± 4.2) × 10 −75024 121.9 0.42 0.45 0.95 1176 1 73 (−4.4 ± 9.9) × 10 −6 (−2.3 ± 4.2) × 10 −75025 121.8 0.42 0.45 0.95 1177 1 72 (+0.5 ± 1.0) × 10 −5 (−2.2 ± 4.2) × 10 −75026 122.4 0.42 0.45 0.95 1186 1 63 (−2.6 ± 9.6) × 10 −6 (−2.3 ± 4.2) × 10 −75027 122.2 0.42 0.45 0.95 1185 1 64 (−0.4 ± 9.6) × 10 −6 (−2.3 ± 4.2) × 10 −75028 122.6 0.42 0.45 0.95 1205 0 44 (−2.9 ± 9.6) × 10 −6 (−2.3 ± 4.2) × 10 −75029 122.8 0.42 0.45 0.95 1212 2 36 (−15.9 ± 9.6) × 10 −6 (−2.6 ± 4.2) × 10 −75030 122.9 0.42 0.45 0.95 1216 1 33 (+6.0 ± 9.6) × 10 −6 (−2.5 ± 4.2) × 10 −75031 123.0 0.42 0.45 0.95 127 1 2 (−0.7 ± 2.9) × 10 −5 (−2.5 ± 4.2) × 10 −75035 123.6 0.42 0.45 0.95 1217 1 32 (−0.5 ± 1.0) × 10 −5 (−2.6 ± 4.2) × 10 −75036 124.0 0.42 0.45 0.95 1226 2 22 (+3.2 ± 9.5) × 10 −6 (−2.5 ± 4.2) × 10 −75037 124.5 0.42 0.45 0.95 1240 1 9 (−1.1 ± 9.2) × 10 −6 (−2.5 ± 4.2) × 10 −75038 124.8 0.42 0.45 0.95 1241 1 8 (+20.7 ± 9.3) × 10 −6 (−2.1 ± 4.2) × 10 −75039 124.8 0.42 0.45 0.95 1222 1 27 (+2.0 ± 9.4) × 10 −6 (−2.1 ± 4.2) × 10 −75040 123.7 0.42 0.45 0.95 1222 1 27 (+5.4 ± 9.6) × 10 −6 (−2.0 ± 4.2) × 10 −7Table C.51:

Thick Aluminum Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut5041 123.8 0.42 0.45 0.95 1216 1 33 (+0.5 ± 9.9) × 10 −6 (−2.0 ± 4.2) × 10 −75042 124.0 0.42 0.45 0.95 1212 2 37 (−14.8 ± 9.4) × 10 −6 (−2.2 ± 4.1) × 10 −75043 124.1 0.42 0.45 0.95 1210 1 39 (−9.7 ± 9.4) × 10 −6 (−2.4 ± 4.1) × 10 −75044 124.0 0.42 0.45 0.95 1172 0 77 (+14.4 ± 9.8) × 10 −6 (−2.2 ± 4.1) × 10 −75045 124.4 0.42 0.45 0.95 1175 1 74 (−12.9 ± 9.7) × 10 −6 (−2.4 ± 4.1) × 10 −75046 124.7 0.42 0.45 0.95 1160 1 89 (−15.7 ± 9.4) × 10 −6 (−2.7 ± 4.1) × 10 −7Table C.52:310

C.3 CCl 4 TargetCCl 4 SummaryTotal Number of good runs : 27Total Number of sequences from all runs : 33547Number of good sequences : 32441Number of lost sequences from bad spin state : 33 ( 0.1% loss)Number of lost sequences from beam cuts : 1073 ( 3.2% loss)Total Number of lost sequences : 1106 ( 3.3% loss)Average Run Asymmetry Error : 1.0 × 10 −5Average Run Asymmetry Error RMS : 8.0 × 10 −7Average 3He Polarization : 0.44Average 3He Polarization RMS : 1.1 × 10 −3Average Energy Average Beam Polarization : 0.62Average Energy Average Beam Polarization RMS : 1.0 × 10 −3Average Beam Current : 100 µAAverage Beam Current RMS : 2.3 µATable C.53:311

312CCl 4 Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut1747 99.8 0.44 0.62 0.95 1213 2 35 (+0.1 ± 1.0) × 10 −5 (+0.1 ± 1.0) × 10 −51748 99.6 0.44 0.62 0.95 1235 0 15 (+1.9 ± 1.1) × 10 −5 (+10.0 ± 7.5) × 10 −61749 96.1 0.44 0.62 0.95 913 0 337 (+1.7 ± 1.2) × 10 −5 (+11.9 ± 6.4) × 10 −61750 95.8 0.44 0.62 0.95 841 2 387 (+2.7 ± 1.3) × 10 −5 (+15.0 ± 5.8) × 10 −61751 98.4 0.44 0.62 0.95 1204 1 45 (+2.6 ± 1.1) × 10 −5 (+17.5 ± 5.0) × 10 −61752 99.0 0.44 0.62 0.95 1244 3 3 (+1.1 ± 1.0) × 10 −5 (+16.2 ± 4.5) × 10 −61753 98.6 0.44 0.62 0.95 1243 1 6 (−0.1 ± 1.0) × 10 −5 (+13.4 ± 4.1) × 10 −61754 97.6 0.44 0.62 0.95 1243 1 6 (+2.6 ± 1.0) × 10 −5 (+15.1 ± 3.8) × 10 −61755 99.8 0.44 0.62 0.95 1246 1 3 (+2.9 ± 1.0) × 10 −5 (+16.8 ± 3.6) × 10 −61756 100.4 0.44 0.62 0.95 1210 1 39 (+1.6 ± 1.0) × 10 −5 (+16.8 ± 3.4) × 10 −61757 99.9 0.44 0.62 0.95 1241 1 8 (+2.1 ± 1.0) × 10 −5 (+17.2 ± 3.2) × 10 −61758 100.0 0.44 0.62 0.95 1247 1 2 (+15.9 ± 9.9) × 10 −6 (+17.0 ± 3.1) × 10 −61759 100.0 0.44 0.62 0.95 1249 1 0 (+3.6 ± 1.0) × 10 −5 (+18.6 ± 2.9) × 10 −61760 100.0 0.44 0.62 0.95 1246 1 3 (+2.4 ± 1.0) × 10 −5 (+19.0 ± 2.8) × 10 −61761 100.0 0.44 0.62 0.95 1247 1 2 (+35.2 ± 9.8) × 10 −6 (+20.1 ± 2.7) × 10 −61762 99.9 0.44 0.62 0.95 1247 1 2 (+15.5 ± 9.9) × 10 −6 (+19.8 ± 2.6) × 10 −61763 99.9 0.44 0.62 0.95 1243 3 4 (+3.6 ± 1.0) × 10 −5 (+20.8 ± 2.5) × 10 −61764 99.8 0.44 0.62 0.95 1248 1 1 (+2.7 ± 1.0) × 10 −5 (+21.2 ± 2.5) × 10 −61765 99.8 0.44 0.62 0.95 1248 1 1 (+1.8 ± 1.0) × 10 −5 (+21.0 ± 2.4) × 10 −61766 100.1 0.44 0.62 0.95 1249 1 0 (+2.0 ± 1.0) × 10 −5 (+20.9 ± 2.3) × 10 −61767 99.9 0.44 0.62 0.95 1249 1 0 (+1.1 ± 1.0) × 10 −5 (+20.5 ± 2.3) × 10 −6Table C.54:

CCl 4 Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut1768 99.9 0.44 0.62 0.95 1245 2 3 (+8.6 ± 10.0) × 10 −6 (+19.9 ± 2.2) × 10 −61769 97.0 0.44 0.62 0.95 898 2 167 (+1.9 ± 1.2) × 10 −5 (+19.9 ± 2.2) × 10 −61770 101.4 0.44 0.62 0.95 1248 1 1 (+0.2 ± 1.0) × 10 −5 (+19.1 ± 2.1) × 10 −61771 101.7 0.44 0.62 0.95 1247 1 2 (+1.9 ± 1.0) × 10 −5 (+19.1 ± 2.1) × 10 −61772 101.5 0.44 0.62 0.95 1249 1 0 (+2.6 ± 1.0) × 10 −5 (+19.4 ± 2.0) × 10 −61773 101.3 0.44 0.62 0.95 1248 1 1 (+1.6 ± 1.0) × 10 −5 (+19.2 ± 2.0) × 10 −6Table C.55:313

C.4 Cu TargetCu SummaryTotal Number of good runs : 17Total Number of sequences from all runs : 20135Number of good sequences : 20087Number of lost sequences from bad spin state : 19 ( 0.1% loss)Number of lost sequences from beam cuts : 29 ( 0.1% loss)Total Number of lost sequences : 48 ( 0.2% loss)Average Run Asymmetry Error : 1.1 × 10 −5Average Run Asymmetry Error RMS : 4.8 × 10 −6Average 3He Polarization : 0.44Average 3He Polarization RMS : 2.1 × 10 −4Average Energy Average Beam Polarization : 0.63Average Energy Average Beam Polarization RMS : 3.0 × 10 −4Average Beam Current : 130 µAAverage Beam Current RMS : 0.5 µATable C.56:314

315Cu Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2828 128.9 0.44 0.63 0.95 1247 1 2 (+0.2 ± 1.0) × 10 −5 (+0.2 ± 1.0) × 10 −52829 129.0 0.44 0.63 0.95 1247 1 2 (+1.1 ± 1.1) × 10 −5 (+6.5 ± 7.4) × 10 −62830 129.1 0.44 0.63 0.95 1246 1 3 (+1.6 ± 1.0) × 10 −5 (+9.6 ± 6.0) × 10 −62831 129.2 0.44 0.63 0.95 1247 1 2 (−1.2 ± 1.0) × 10 −5 (+4.3 ± 5.2) × 10 −62832 129.4 0.44 0.63 0.95 1247 1 2 (+15.1 ± 9.7) × 10 −6 (+6.5 ± 4.6) × 10 −62833 129.3 0.44 0.63 0.95 1246 1 3 (+0.4 ± 1.0) × 10 −5 (+6.0 ± 4.2) × 10 −62834 129.5 0.44 0.63 0.95 1249 1 0 (−1.0 ± 1.0) × 10 −5 (+3.8 ± 3.9) × 10 −62835 129.5 0.44 0.63 0.95 1249 1 0 (−6.4 ± 9.8) × 10 −6 (+2.5 ± 3.6) × 10 −62836 130.1 0.44 0.63 0.95 1247 2 1 (+3.6 ± 10.0) × 10 −6 (+2.6 ± 3.4) × 10 −62837 130.1 0.44 0.63 0.95 1247 1 2 (−13.6 ± 9.9) × 10 −6 (+1.0 ± 3.2) × 10 −62840 130.1 0.44 0.63 0.95 133 1 1 (+1.0 ± 3.1) × 10 −5 (+1.1 ± 3.2) × 10 −62851 130.6 0.44 0.63 0.95 1246 1 3 (+0.9 ± 1.0) × 10 −5 (+1.8 ± 3.0) × 10 −62852 130.5 0.44 0.63 0.95 1247 1 2 (−0.3 ± 1.0) × 10 −5 (+1.4 ± 2.9) × 10 −62853 130.5 0.44 0.63 0.95 1247 1 2 (−5.6 ± 9.9) × 10 −6 (+0.9 ± 2.8) × 10 −62854 130.6 0.44 0.63 0.95 1248 1 1 (−0.0 ± 1.0) × 10 −5 (+0.8 ± 2.7) × 10 −62855 130.3 0.44 0.63 0.95 1247 2 1 (+0.0 ± 1.0) × 10 −5 (+0.7 ± 2.6) × 10 −62856 130.2 0.44 0.63 0.95 1247 1 2 (+0.8 ± 1.0) × 10 −5 (+1.2 ± 2.5) × 10 −6Table C.57:

C.5 In TargetIn SummaryTotal Number of good runs : 7Total Number of sequences from all runs : 8587Number of good sequences : 8551Number of lost sequences from bad spin state : 7 ( 0.1% loss)Number of lost sequences from beam cuts : 29 ( 0.3% loss)Total Number of lost sequences : 36 ( 0.4% loss)Average Run Asymmetry Error : 6.0 × 10 −6Average Run Asymmetry Error RMS : 4.8 × 10 −7Average 3He Polarization : 0.45Average 3He Polarization RMS : 2.0 × 10 −4Average Energy Average Beam Polarization : 0.63Average Energy Average Beam Polarization RMS : 2.0 × 10 −4Average Beam Current : 126 µAAverage Beam Current RMS : 2.0 µATable C.58:316

Cu Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2863 126.9 0.45 0.47 0.95 1082 1 4 (−3.8 ± 6.9) × 10 −6 (−3.8 ± 6.9) × 10 −62873 125.3 0.45 0.47 0.95 1241 1 8 (−2.4 ± 6.4) × 10 −6 (−3.1 ± 4.7) × 10 −62874 127.4 0.45 0.47 0.95 1247 1 2 (−3.2 ± 6.0) × 10 −6 (−3.1 ± 3.7) × 10 −62875 126.6 0.45 0.47 0.95 1243 1 6 (−2.7 ± 6.0) × 10 −6 (−3.0 ± 3.1) × 10 −62876 126.8 0.45 0.47 0.95 1247 1 2 (−3.9 ± 5.5) × 10 −6 (−3.2 ± 2.7) × 10 −62878 125.3 0.45 0.47 0.95 1247 1 2 (−2.3 ± 5.7) × 10 −6 (−3.0 ± 2.5) × 10 −62879 125.8 0.45 0.47 0.95 1244 1 5 (−4.4 ± 5.4) × 10 −6 (−3.2 ± 2.3) × 10 −6Table C.59:317

C.6 B 4 C TargetIn SummaryTotal Number of good runs : 11Total Number of sequences from all runs : 12542Number of good sequences : 12491Number of lost sequences from bad spin state : 12 ( 0.1% loss)Number of lost sequences from beam cuts : 39 ( 0.3% loss)Total Number of lost sequences : 51 ( 0.4% loss)Average Run Asymmetry Error : 9.0 × 10 −6Average Run Asymmetry Error RMS : 4.8 × 10 −7Average 3He Polarization : 0.46Average 3He Polarization RMS : 4.6 × 10 −3Average Energy Average Beam Polarization : 0.64Average Energy Average Beam Polarization RMS : 3.4 × 10 −3Average Beam Current : 126 µAAverage Beam Current RMS : 2.0 µATable C.60:318

319Cu Run-By-Run Data SummaryRun Curr 3 He Beam RFSF Good Spin st. Beam Run Asymmetry Running Asymmetry(µA) Pol. Pol. Eff. Seq. Cut Cut2882 124.7 0.45 0.47 0.95 120 1 4 (−0.4 ± 2.4) × 10 −5 (−0.4 ± 2.4) × 10 −52883 124.6 0.45 0.47 0.95 1244 1 5 (+8.8 ± 7.3) × 10 −6 (+7.7 ± 7.0) × 10 −62884 123.9 0.45 0.47 0.95 1239 1 10 (−1.3 ± 7.3) × 10 −6 (+3.4 ± 5.0) × 10 −62885 124.9 0.45 0.47 0.95 1241 1 8 (+5.2 ± 7.0) × 10 −6 (+4.0 ± 4.1) × 10 −62886 125.2 0.45 0.47 0.95 1163 1 3 (+4.5 ± 7.5) × 10 −6 (+4.1 ± 3.6) × 10 −62951 126.5 0.46 0.48 0.95 1247 1 2 (−11.0 ± 7.5) × 10 −6 (+1.1 ± 3.2) × 10 −62952 128.0 0.46 0.48 0.95 1248 1 1 (−6.0 ± 8.0) × 10 −6 (−0.1 ± 3.0) × 10 −62953 127.9 0.46 0.48 0.95 1248 1 1 (+0.0 ± 7.3) × 10 −6 (−0.1 ± 2.8) × 10 −62954 128.1 0.46 0.48 0.95 1247 2 1 (+7.3 ± 7.3) × 10 −6 (+0.8 ± 2.6) × 10 −62955 126.5 0.46 0.48 0.95 1247 1 2 (−9.5 ± 7.4) × 10 −6 (−0.3 ± 2.5) × 10 −62956 125.2 0.46 0.48 0.95 1247 1 2 (+15.7 ± 7.5) × 10 −6 (+1.3 ± 2.3) × 10 −6Table C.61:

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