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Radiative muon capture of oxygen, aluminum, silicon, titanium, zirconium, and silver Bergbusch, Paul C. 1995

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R A D I A T I V E M U O N C A P T U R E O N O X Y G E N , A L U M I N U M , SILICON, T I T A N I U M , Z I R C O N I U M , A N D SILVER B y Paul C . Bergbusch B.Sc . (Honors), Simon Fraser University, 1993 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A October 1995 © Paul C . Bergbusch, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t i sh Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of Br i t i sh Columbia 6224 Agricul tural Road Vancouver, Canada V 6 T 1W5 Date: Abstract The photon spectra from radiative muon capture ( R M C ) on oxygen, a luminum, silicon, t i tanium, zirconium, and silver have been measured for photon energies greater than 57 M e V using a cylindrical pair spectrometer at T R I U M F . Ratios of R M C to ordinary muon capture ( O M C ) and values of the weak induced pseudoscalar coupling constant (specifically, gp/gA) have been calculated using the observed photon spectra in conjunction wi th theoretical nuclear R M C models. Present discrepancies between different theoretical calculations mean that experimental values of gp cannot yet be used to make meaningful tests of the partially conserved axial current hypothesis ( P C A C ) . However, trends in R M C / O M C and gp/gA with atomic number indicate an extreme sensitivity of R M C , and nuclear weak vir tual pion currents i n particular, to Paul i blocking. A future R M C experiment at T R I U M F wi th the nickel isotopes 5 8 N i , 6 0 N i , and 6 2 N i wi l l help to clarify the present theoretical situation. Results from R M C on H and 3 H e are forthcoming shortly from T R I U M F , and wi l l provide direct tests of P C A C . 11 Table of Contents Abstract 1 1 Table of Contents iii List of Figures v List of Tables vii 1 Introduction 1 1.1 Weak interactions 2 1.2 Radiative muon capture 7 2 Theory 11 2.1 Elementary R M C 11 2.2 Nuclear R M C 13 2.2.1 Quenching of gp 13 2.2.2 Nuclear response 14 3 Experiment 17 3.1 T R I U M F muon and pion beams 17 3.2 Targets 18 3.3 R M C spectrometer 19 3.3.1 Drift chamber 21 3.3.2 Inner wire chamber 21 3.3.3 Beam, trigger, and cosmic ray counters 22 3.4 Data acquisition 23 in 4 A n a l y s i s 26 4.1 Backgrounds 26 4.2 Counting R M C photons — 7 rate analysis 28 4.2.1 Software cuts • 28 4.2.2 Cut efficiencies 31 4.2.3 Absolute photon acceptance 34 4.3 Counting stops — R rate analysis 37 4.3.1 S T O P scaler 37 4.3.2 Correction factors 37 4.4 Addi t ion of uncertainties 41 5 R e s u l t s 43 5.1 Calculation of R~, and gp/gA 43 5.2 0, A l , S i , and T i 46 5.3 Zr and A g 54 6 D i s c u s s i o n 59 6.1 59 6.2 gP 64 7 C o n c l u s i o n s 67 B i b l i o g r a p h y 69 A p p e n d i x A Quant i t i e s used in the ca lculat ions of a n d gp/gA 74 A p p e n d i x B D e r i v a t i o n of equat ions (1.8) a n d (1.9) 76 i v ( List of Figures 2.1 Feynman diagrams contributing to the elementary R M C process 12 3.1 Global view of the R M C spectrometer 20 3.2 Cross section of the R M C spectrometer 20 4.1 R M C photon spectrum at various levels of software cuts 30 4.2 R M C photon spectrum after all software cuts. 31 4.3 Time spectrum of hits in one A counter 32 4.4 The zclose distribution 36 4.5 Comparison of a Monte Carlo R P C photon spectrum to experiment 36 5.1 x2 vs- kmax for fits of Primakoff spectra to experiment. 45 5.2 Comparison of experimental R M C on Si and A l to closure spectral shapes. . 48 5.3 Comparison of experimental R M C on 0 and T i to closure spectral shapes. . 49 5.4 N 7 vs. Ry and gP/gA for R M C on 0 and T i 50 5.5 Ry vs. gp/gA for R M C on 0 and T i . 51 5.6 Comparison of experimental R M C on 0 to theoretical spectra 52 5.7 Comparison of experimental R M C on T i to theoretical spectra 53 5.8 Comparison of experimental R M C on Zr and Ag to closure spectral shapes. 55 5.9 N 7 vs. Ay and gp/gA for R M C on Zr and Ag. . 56 5.10 Comparison of experimental R M C on Zr and Ag to theoretical spectra. . . . 57 v 6.1 Hy vs. Z ' 62 6.2 Fly vs. a (neutron excess) 62 6.3 Experimental FLy vs. Z compared with theory 63 6.4 gp/gA as a function of atomic number 66 vi List of Tables 1.1 The four fundamental forces in nature 1 1.2 gp results from measurements of O M C on H2 8 1.3 gp/gA results from measurements of O M C on nuclei 9 3.1 Dimensions of R M C targets 19 5.1 O M C capture fractions and mean muon capture lifetimes 44 5.2 Ry, gp/gA, and kmax for 0, A l , S i , and T i 58 5.3 R^y, gp/gA, and kmax for Zr and A g 58 6.1 R^, kmax, and the x2 of fit as determined by the shape method 61 6.2 gp/gA and as determined by the integral method 65 A . l Quantities used in the calculations of Ry and gp/gA 75 v i i Chapter 1 Introduction Physicists classify four fundamental forces in nature: strong, electromagnetic, weak, and gravitational. The gravitational and electromagnetic forces are observed on a macroscopic scale, but the strong and weak forces are observed only at the subatomic level. The four forces are listed in order of decreasing strength in table 1.1. The electromagnetic and weak forces have been unified into a single description known as Weinberg-Salam-Glashow S U ( 2 ) x U ( l ) electroweak theory [2, 3, 4], or the Standard Model of electroweak interactions. This model has been very successful, most notably in its prediction and subsequent discovery of the W and Z intermediate vector bosons [5]. However, questions sti l l exist concerning the form of the weak interaction at low energies, particularly when strongly-interacting particles (hadrons) are involved. The present study tests the form of the weak interaction in the presence of the strong interaction by investigating the semi-leptonic weak process of radiative muon capture on nuclei. force coupling strength [1] strong electromagnetic weak gravitational as(Mz) = 0.116 a = 1/137 GF/(kc)3 = 1.16639 x 1 0 - 5 G e V - 2 GN = 6.7071 x 1 0 - 3 9 7 i c ( G e V / c 2 ) - 2 Table 1.1: The four fundamental forces in nature. 1 Chapter 1. Introduction 2 1.1 Weak interactions For weak processes at low energy, where the square of the momentum transfer is much less than the square of the mass of the mediating particle (i.e., q2 <C Mw for a charged weak process such as muon capture), the W propagator —> -Ar in the Feynman amplitude, and the interaction can be assumed to take place at a point. Furthermore, the strength of the weak interaction (see table 1.1) is such that it can be treated successfully by perturbation theory. The amplitude of a weak process is therefore calculated in perturbation theory from the "contact" interaction [6] HI(x) = ?Lj»(x)4(x) (i.l) where GF is the Fermi weak interaction coupling constant, and J01 is the weak 4-vector current. This contact interaction involving self-interacting weak currents was first proposed by Fermi in 1934 to describe the nuclear /3-decay process [7, 8]. It allows a weak process to be completely determined once the currents are specified. For a semi-leptonic process, the weak current J01 is decomposed into leptonic and hadronic parts: Ja(x) = J^(x) + J»{x). (1.2) The experimental data on a wide range of leptonic and semi-leptonic processes indicate that the lepton fields enter the interaction in a purely " V — A " (vector — axial vector) form [6], J«(x) = £ ? z ( * h « ( i -75)^ 0*) ( i .3) I 4LV) = £ft,(*h«(l " 75)lM*) (1-4) I where I labels the charged lepton fields e, p, and r ; v\ labels the corresponding neutrino fields; ifri(x) is a Dirac field linear in the absorption operators of the l~ leptons and the creation operators of the l+ leptons; and the ^ a are Dirac matrices wi th 75 = 270717273-The weak hadronic current is less well-known. However, retaining V — A structure, the weak hadronic current may be written as [9] = [Vi0)(x) - 4 O ) (*)]cos0 c + [V^(x) - AW(x)}SmOc (1.5) Chapter 1. Introduction 3 where Oc is the Cabibbo angle which represents weak flavour mixing of the quarks, and the superscripts on Va and Aa indicate strangeness-conserving (AS = 0) or strangeness-changing (| AS"! = 1) currents. For a process such as muon capture, which involves only u and d quarks, JH(x) = [va(x) - M*)] c o s ° c (1.6) where the superscript (0) has been dropped from Va and Aa. The Va and Aa terms may be parameterized in order to retain Lorentz covariance of the weak current. The available covariants are the 4-momentum transfer qa and the Dirac matrices -ya. For ordinary muon capture on the proton, u~ + p —• n + (1.7) the quantized fields available for the construction of the weak hadronic current are the proton and neutron fields i/)p(x) and ij)n(x). Therefore, the most general covariant expressions for Va and Aa are [10] (see appendix B) Va(x) = -i^v(x)[Fv(q2)la - FM(q2)aaXqX + iFs{q2)qa]iPn(x) (1.8) Aa(x) = i~$p(x)[FA(q2)lal5 + iFP(q2)-f5qa + FT(q2)lbaaXqX]^n(x) (1.9) where oa\ = | [ 7 a , 7 A ] , a n d the F{ are "form factors" which depend on q2 and account for the composite structure of nucleons. Fy and FA are the "bare" vector and axial vector form fac-tors; FM, FS, Fp, and Fj are the "induced" weak magnetic, scalar, pseudoscalar, and tensor form factors, whose associated currents are said to be induced by strong interactions inside the nucleon. The form factors are, in general, weak functions of q2, and so a corresponding set of coupling constants is usually defined: 9v{q2) = Fy(q2) (1.10) gM{q2) = 2MFM(q2) (1.11) gs(q2) = mlFs(q2) (1.12) gA(q2) = FA(q2) (1.13) Chapter 1. Introduction 4 gP(q2) = mtFpiq2) (1.14) gT{q2) = 2MFT(q2) (1.15) where M is the nucleon mass and m; is the lepton mass (for muon capture, / = / / " ) . Experimental and theoretical constraints on the weak form factors are as follows: • T ime reversal invariance - the assumption that the semileptonic weak interaction, E q . (1.1), is time reversal invariant means that all weak form factors are real. • G parity invariance - under G parity (a rotation in isospin space followed by charge conjugation: G = Ce1*12), the bare weak vector and axial vector currents transform as [11] GV[A]& = V[-A]. (1.16) The induced weak magnetism and pseudoscalar currents also transform in this manner, and are called first-class currents. The induced scalar and tensor currents transform in the opposite manner, GV[A]G* = -V[A] (1.17) and are called second-class currents. The strong vector and axial vector currents con-serve G parity, so the weak currents induced by the strong interaction are expected to transform under G parity in the same manner as the bare weak vector and axial vector currents. That is, G parity invariance of the strong interaction requires that second-class currents vanish: Fs(q2) = FT{q2) = 0. (1.18) • Conserved vector current hypothesis ( C V C ) - in electrodynamics, the equality of the observed (i.e. renormalized) charges of the positron (which is a lepton) and proton (a hadron) is attributed to the equality of the bare charges and the divergence-free nature of the electromagnetic (vector) current. Analogously, the observed near-equality of the muon (lepton) decay constant G^ and hadronic decay vector coupling constants is Chapter 1. Introduction 5 attributed to the (near) equality of the bare vector couplings and the divergence-free nature of Va(x) in E q . (1.8): daVa(x) = 0. (1.19) A suitable divergence-free vector current does exist, namely, the conserved isospin current of strong interactions, Ja{x). The C V C hypothesis consists of identifying Va(x) and V^(x) wi th the (1 + i2) and (1 — i2) components of Ja. Furthermore, the third (^ 3 = 0) component of Ja is part of the electromagetic current of hadrons according to the Gel l -Mann-Nishi j ima-Nakano [12, 13] relation Q = I3 + Y/2 (1.20) where Q is the electromagnetic charge, Y is the hypercharge, and 7 3 is the third com-ponent of isospin carried by a current. Therefore, there exist relationships between the semileptonic weak form factors and the (well-known from electron scattering experi-ments) hadronic electromagnetic form factors. In the q2 —* 0 l imi t [10], Fv(q2) = l (1.21) Fs{q2) = 0 (1.23) where M is the nucleon mass, and nP and //„ are the anomalous magnetic moments of the proton and neutron respectively. • f3 decay of the neutron - a value for F A ( ^ 2 ) at q2 = 0 has been measured using neutron j3-decay [1]: FA(0) = 1.2573 ± 0.0028 (1.24) Therefore, we are left with the induced pseudoscalar form factor Fp(q2) which is the focus of the present experimental study. The axial current given in E q . (1.9) is clearly not conserved; otherwise, TV~ — > fCV^ decay could not occur. However, the axial current is made almost divergenceless by making the Chapter 1. Introduction 6 partially conserved axial current hypothesis ( P C A C ) , which relates the divergence of the axial current to the pion field [14, 15], daAa(x) = Uml^{x) (1.25) where /„• is the pion decay constant. App ly ing P C A C to E q . (1.9) using E q . (1.18), and using the Dirac equation gives 2MFA(q2) - q2FP(q2) = yftffZG*™ . (1.26) q2 + ra£ where G^NN is the strong coupling constant for the pion-nucleon vertex, M is the nucleon mass, and is the pion mass. In the q2 —> 0 l imi t , and assuming that the pion-nucleon coupling varies slowly with q2 (i.e., G^NN^) = G^NN^2) = G^NN), E q . (1.26) reduces to the Goldberger-Treiman relation [16] ^ ( 0 ) = (1.27) Assuming FA(q2) = FA(0) and using E q . (1.27) in E q . (1.26) gives or, using Eqs. (1.13) and (1.14) for the coupling constants, _ 2 A ^ W ( , 2 9 ) q2 + ml For ordinary muon capture on the proton, q2 — 0.88m 2 giving the Goldberger-Treiman value for gP — - 6.78 (1.30) 9A or, using Eqs. (1.13) and (1.24), #P = 8.52. (1.31) A n objective of the present study is to test this value of gp and hence the validity of P C A C by measuring the process of radiative muon capture on nuclei. Chapter 1. Introduction 7 1.2 Radiative muon capture In order to accurately measure the pseudoscalar contribution to the weak hadronic current, it is important to observe a weak process where this contribution is as large as possible. The form of E q . (1.29) suggests a process which involves a heavy lepton. Also, the fact that the induced couplings enter into the semileptonic weak interaction proportional to q2 (see Eqs. (1.1), (1-6), (1-8), and (1.9) ) suggests a process with large 4-momentum transfers. Final ly, the form of E q . (1.29) suggests a process wi th momentum transfers near the pion pole. These criteria heavily favour tauon or muon capture processes over f3 decay or electron capture processes. Muon capture processes are chosen for study due to the availability of very high quality muon beams. The weak process of ordinary muon capture (hereafter referred to as O M C ) on the proton, is one possible process for observing the induced pseudoscalar current. A low velocity muon becomes bound in an atomic orbital, and cascades in less than 1 ns to the Is muonic orbital. This is followed by decay of the muon, or capture on the proton to produce a final state neutron and muon neutrino. Measurements of most t/p-dependent observables (e.g. polar-izations, angular correlations) are l imited by the detectability of these final state particles. A simple observable such as the muon capture rate can be measured, but in light nuclei where specific final states can be isolated, the measurement is difficult because the capture probability is small . In fact, only 1 0 - 3 of the muons which stop in a hydrogen target wi l l undergo O M C ; the rest w i l l decay according to which is called "Miche l " decay. Measurements of gp as determined by measuring the O M C rate on hydrogen are shown in table 1.2. Whi l e all results agree with the Goldberger-Treiman value for gp, it should be noted that the uncertainties are large. Measurements of the O M C rate in nuclei are made easier by the fact that the rate for muon capture goes as Z^r t, where \i + p - » n + v, (1.32) H -> e + ue + (1.33) Chapter 1. Introduction 8 Reference Target 9P 9P/9A Bleser et al. 1962 [17] l iquid H2 6.0 ± 8 . 0 4.8 ± 6 . 4 Rothberg et al. 1963 [18] l iquid H2 11.0 ± 4 . 3 8.7 ± 3 . 4 Alber ig i Quaranta et al. 1969 [19] H 2 gas 10.3 ± 3 . 9 8.2 ± 3 . 1 Byst r i t i sk i i et al. 1974 [20] H 2 gas 7.9 ± 5 . 9 6.3 ± 4 . 7 Bard in et al. 1981 [21] l iquid H 2 7.1 ± 3 . 0 5.6 ± 2 . 4 Average 8.7 ± 1 . 9 6.9 ± 1.5 Table 1.2: Summary of gp results from measurements of O M C on H 2 , as presented by Bard in et al. [22]. Zeff is an effective nuclear charge. However, nuclear structure details dominate the capture rate in complex nuclei, thereby obscuring any dependence of the capture rate on gp. Other observables, such as the capture rate to a specific final nuclear state, are less senstive to nuclear structure and have been used wi th carbon, oxygen, and silicon to extract values of gp/gA- In the case of 1 2 C , values of gp/gA have been extracted by measuring the average polarization of the recoiling nucleus along the muon-spin direction (Pav), the longitudinal polarization of the recoiling nucleus (PL), and the capture rate to the 1 2 B ground state. For 1 6 0 , the O M C rate to the first excited state in 1 6 N , relative to the /3-decay rate back to the 1 6 0 ground state, has been measured. Final ly , for 2 8 S i , observation of the angular correlation between the O M C neutrino and nuclear de-excitation gamma rays has provided values of gp/gA- Unfortunately, the extracted values of gp/gA depend strongly on the theory used to describe the nuclear O M C process under investigation. Results are shown in table 1.3. A complementary semileptonic weak process for measurement of the induced pseudoscalar current is radiative muon capture (hereafter referred to as R M C ) on the proton, p~ +p-+n + + 7. (1.34) R M C has two distinct advantages over O M C . First , the photon in the final state is easy to detect so that an R M C branching ratio and an R M C photon energy spectrum can be measured. Bo th of these observables are directly related to gp. Second, the momentum transfer of R M C on the (stationary) proton is variable, ranging from q2 = 0.88m 2 at 0 M e V photon energy to — m2 at max imum photon energy. This is because there are three bodies Chapter 1. Introduction 9 Reference Nucleus Method Theory ' 9P/9A Possoz et al. [23, 24] 1 2 C P 1 av [25] [26, 27] [28] [29] 7.1 ± 2 . 7 13.6 ± 2 . 1 15 ± 4 l O ^ 2 . ; * Kuno et al. [30] 1 2 C P 1 av [29] 10.1+& Mil le r et al. [31] 1 2 C A , [28] 8.5 ± 2 . 5 Roesch et al. [32, 33] 1 2 C Pavl PL [34] [35] [29] 9.4 ± 1.7 7.2 ± 1.7 9.1 ± 1.8 Guichon et al. [36], Hamel et al. [37] 16Q [38] 11 - 12 Mi l l e r et al. [39] 2 8 S i v — 7 [26, 27] [40] 12.9 ± 3 . 9 - 1 . 9 ± 3 . 1 Table 1.3: Summary of gp/gA results from measurements of O M C on nuclei. P A V refers to measurement of the average polarization of the recoiling nucleus along the muon-spin direction; P L refers to measurement of the longitudinal polarization of the recoiling nucleus; A M and Ap refer to measurement of the O M C and /5-decay rates to specific nuclear states; and v — 7 refers to measurement of the angular correlation between the O M C neutrino and nuclear de-excitation gamma rays. in the final state, in contrast with O M C on the proton where the two-body final state fixes q2 at 0 .88m 2 . Experimentally, it is the high energy R M C photons that are observed (see section 4.1), so the observed momentum transfer in R M C is close to the pion pole, resulting in an enhancement of the pseudoscalar current by a factor of about 3 over O M C . However, R M C has the disadvantage of being a rare process. The R M C / O M C ratio for the high energy end of the R M C photon spectrum is on the order of 1 0 - 5 . Therefore, measurement of R M C requires careful assessment of backgrounds. Also, due to the rarity of the process, nuclear R M C measurements are typically inclusive measurements, which means that theoretical nuclear structure calculations, used in the extraction of experimental results, are important when interpreting these results. In the present experiment, R M C photons are observed v ia pair conversion in lead. Track reconstruction of the e + e _ pairs in a cylindrical drift chamber allows for measurement of the differential and integrated R M C photon energy spectra. In this study, R M C on six different nuclei is studied. Oxygen is chosen in order to compare wi th earlier oxygen results from PSI Chapter 1. Introduction 10 [41, 42], as well as a previous oxygen result from T R I U M F [43] which used a time projection chamber (as opposed to the drift chamber described in section 3.3.1) and a slightly different photon trigger (2D instead of > ID - see section 3.4). Aluminum and silicon are chosen in order to verify an "isotope effect" (see section 6) seen in previous aluminum and silicon results [44]. Titanium is chosen for comparison with previous calcium results [41, 43, 45, 46, 47] to see if an "isotope effect" might also be present here. Zirconium and silver are chosen in order to investigate the decreasing trend in the R M C / O M C ratio with increasing atomic number (see section 6). The next chapter will review the R M C theory which applies to the nuclei in the present study. In chapter 3 the experimental apparatus is discussed. Chapter 4 gives the detailed data analysis procedure, and all results are presented in chapter 5. Discussion and interpre-tation of results take place in chapter 6. Chapter 2 Theory Calculations of R M C on the free proton and in nuclei are discussed below. Whi le the elementary R M C process is well understood, the necessarily complex nuclear R M C process is more theory-dependent. A brief overview of elementary R M C theory is given is section 2.1. A more detailed dicussion of the nuclear R M C theory wi th which the present study is concerned is given i n section 2.2.2. A n extensive review of R M C theory has been made by Gmi t ro and Tru61 [48]. 2.1 E l e m e n t a r y R M C There have been three distinct approaches to the calculation of the amplitude for the ele-mentary R M C process, p~ + p — • n + i/p + 7. (2.1) A l l three yield consistent results, so the elementary R M C amplitude is believed to be well understood. The first of these approaches is the evaluation of Feynman diagrams (i.e. perturbation theory). The standard diagrams [49, 50, 51] are shown in figure 2.1, and include radiation from the charge of the muon, the charge and magnetic moment of the proton, the magnetic moment of the neutron, and the charge of the exchanged vir tual pion. The exchange diagram and the vertex-radiating diagram are needed to maintain gauge invariance of the amplitude. The relative strength of the pseudoscalar amplitude in the R M C process is largely determined 11 Chapter 2. Theory 12 P M p p /x P M P M Figure 2.1: Feynman diagrams contributing to the elementary R M C process. by the relative strength of the pion exchange diagram. The second approach, followed by Adler and Dothan [52] and later by Chr is t i l l in and Servadio [53], involves application of the soft photon theorem of Low [54] to the weak axial vector current. Bo th groups of authors show that the elementary R M C amplitude can be expressed in terms of the non-radiative amplitude ( O M C ) and the divergence of the axial vector current, and their results are in close agreement. Furthermore, the results of this second approach do not differ significantly [55] from the first diagrammatic approach discussed aboye. The final approach, undertaken by Hwang and Primakoff [56] and extended by Gmi t ro and Ovchinnikova [57], is the construction of the elementary R M C amplitude from current conservation laws (conservation of electromagnetic current ( C E C ) , C V C , and P C A C ) , the requirement of gauge invariance, and a "linearity hypothesis" which constrains the form of the weak form factors. Init ial results were at odds wi th the diagrammatic approach ( R M C rates were a factor of two lower for a given value of gp); however, this discrepancy was attributed to problems with the linearity hypothesis, which were subsequently resolved [57]. Chapter 2. Theory 13 2.2 Nuclear R M C Once the elementary R M C process has been calculated, it is necessary to "embed" it in the nucleus in order to calculate the nuclear R M C process p-{Z,A)-+{Z-l,A)ult<y. (2.2) The simplest way to do this is to sum the elementary amplitude incoherently over all protons in the nucleus, i.e., to treat the interaction in impulse approximation ( IA) . The success of the IA requires that interactions involving two nucleons (via meson-exchange currents) are negligible. A l l of the nuclear R M C models discussed in section 2.2.2 use this approximation in conjunction wi th an appropriate nuclear response, except for Gmi t ro et al. [58] who use ,a "modified impulse approximation". Before discussing calculation of the nuclear response, it is important to consider effects of the nuclear medium on the weak pseudoscalar current which may result in deviations of gp from the Goldberger-Treiman value. 2.2.1 Quenching of gp As stated in section 2.1, the pseudoscalar amplitude in R M C is sensitive to the pion exchange diagram in figure 2.1. Therefore, nuclear effects which disturb the pion current, such as nucleon polarizability, short-range nucleon-nucleon correlations, and Paul i blocking of final-state neutrons, are expected to modify the value of gp in a nucleus. Scattering of an emitted vir tual pion by other nucleons in the nucleus is taken into account by replacing m2 in the pion propagator by an effective pion mass [59] i -\- oc where a is the nucleon polarizability. Also, the pion-nucleon coupling is modified by writ ing GnNN = (1 + £oc)GvNN (2.4) where £ accounts for short-range nucleon-nucleon correlations and Paul i blocking. Combi-nation of the nuclear effects then results in a large "renormalization" of g& and gp: ~9A=9A{\+IOL\ (2.5) Chapter 2. Theory 14 (2.6) The renormalized gp/gA ratio does not depend on £, so with a = —0.75 for infinite nuclear matter [60], and q2 = 0.88m 2 for O M C , Surface effects for finite nuclei are expected to reduce this "quenching" of gp, but it is clear that the weak pseudoscalar current is significantly affected by the nuclear medium. 2.2.2 Nuclear response In the I A , calculation of nuclear R M C requires only the elementary R M C amplitude and some method to account for in i t ia l and final state nuclear wavefunctions. The quantities of experimental interest that are calculated in a given theoretical model are the R M C / O M C ratio as a function of gp, and the shape of the R M C photon spectrum. The R M C / O M C ratio is observed experimentally for photons wi th k > 57 M e V , and this partial R M C / O M C ratio is defined as i i , . Ry is calculated (instead of pure R M C ) in order to remove some dependency on nuclear effects which are common to both R M C and O M C . Also, the inclusive R M C branching ratio is calculated (rather than R M C to an exclusive final state) due to the experimental difficulty in measuring exclusive R M C branching ratios. The nuclear models discussed below are relevant to the nuclei in the present study. • Closure model: this was the first model used to calculate nuclear R M C . The final state nucleus is assigned an average excitation energy Eav, and the nuclear R M C matrix ele-ment is evaluated at this transition energy. Fi ts to experimental data are accomplished by treating Eav and gp as free parameters. Various authors [50, 61, 62, 63, 64, 65] have calculated R M C branching ratios and spectra in the closure approximation, but the extracted values of gp are extremely sensitive to Eav. Therefore, this model is not used in the present study to determine values of gp. However, a closure R M C spectral shape, E q . (5.1), is used here as a representative R M C photon spectrum in order to calculate (2.7) Chapter 2. Theory 15 values of FLy. This polynomial shape was first derived in the closure approximation by Primakoff [66] by considering only the muon-radiating diagram in figure 2.1. • Fermi gas model: this model avoids the closure approxmation by using nuclear response functions calculated in the Fermi gas model. Chr is t i l l in et al. [55] take nucleon-nucleon correlations into account by introducing an effective nucleon mass M* as a free parameter. The value of M* ( M * = 0 .5M) is fixed by fitting model-predicted O M C rates to experiment; the model then reproduces O M C rates to better than 10% for a large number of nuclei with Z > 42. This model is used in the present study to determine values of FLy and gp for the heavy nuclei zirconium and silver. • Semi-phenomenological "realistic" nuclear excitation model: this model avoids the closure approximation by using giant dipole ( G D R ) and giant quadrupole ( G Q R ) res-onances for the final state nuclei. Values of FLy for a given value of gp are ~ 30% lower than those predicted by the closure model, bringing theoretical results into closer agree-ment with experiment (assuming the Goldberger-Treiman value for gp). Chris t i l l in 's [67] calculations of R M C on calcium are used to determine values of FLy and gp for t i tanium. Chr is t i l l in and Gmitro 's [68] calculations of R M C on oxygen are used to determine the same quantities for the oxygen target. • Modified impulse approximation ( M I A ) : Gmi t ro et al. [58] venture beyond the im-pulse approximation by considering nucleon-nucleon interactions v ia meson-exchange currents ( M E C ' s ) at the electromagnetic vertex in figure 2.1. The M E C ' s are accounted for using constraints which follow from continuity of the electromagnetic current. A microscopic model (shell model) is used for the in i t ia l and final state nuclei, and theo-retical predictions are similar to those of the phenomenological model described above. The M I A was attempted because a purely microscopic model [58] for oxygen and cal-c ium predicted values of FLy (for a given value of gp) which were roughly double those predicted by the phenomenological model. Gmi t ro et al [58] calculate R M C on oxygen and calcium, so the M I A is used here to determine values of FLy and gp for oxygen and Chapter 2. Theory 16 t i tanium. • Sum rule techniques: Roig and Navarro [69] use SU(4) symmetry, a Hartree-Foch scheme for the target ground state, and sum rule techniques (which are particularly adapted to the analysis of inclusive processes) to calculate R M C on carbon, oxygen, and calcium. Their values of Ry are the lowest for a given value of gp, and are the closest to the experimental values (assuming the Goldberger-Treiman value for gp). This model is used to determine values of gp for oxygen and t i tanium. To date, nuclear responses to R M C have not been calculated in a rigorous manner for alu-minum, silicon, or t i tanium. However, as mentioned above, the nuclear response of calcium has been calculated, and is used in this study as an approximation to the response of t i -tanium. Calculations for a luminum and silicon have been made in the Fermi gas model of Fearing and Welsh [70], but this Fermi gas model is only expected to be useful for Z > 20 nu-clei. Fearing and Welsh utilize a relativistic mean field theory approach, where a relativistic Fermi gas model is used to describe medium-heavy nuclei, and a local density approxima-tion along wi th realistic nuclear density distributions are used to relate the R M C process in infinite nuclear matter to finite nuclei. However, the a im of this model is to assess the reliability wi th which gp can be extracted from experimental data, rather than make explicit predictions. In fact, implementing the realistic nuclear density distributions results in O M C rates that are significantly higher than the experimental values. Moreover, a number of theoretical R M C calculations in nuclei have been done, and are used in the present study to extract experimental values of the partial R M C / O M C ratio, Ry, and the induced weak pseudoscalar coupling constant, gP. Ry and gp are sensitive to the pion exchange current in figure 2.1, so nucleon polarizability, short-range nucleon-nucleon correlations, and Paul i blocking in nuclei wi l l affect the values of Ry and gp as extracted from nuclear R M C . Chapter 3 Experiment The experiment was carried out in the meson hall at the T R I U M F laboratory in Vancouver, Br i t i sh Columbia , Canada. Hardware requirements were: producing an intense, collimated u~ beam; stopping this beam in targets of physical interest; detecting and high resolution tracking of R M C photons using pair conversion in lead; detecting beam, decay, and cosmic ray particles in scintillation counters; and writ ing data to tape for analysis. 3.1 T R I U M F m u o n a n d p i o n b e a m s Production of the primary proton beam at T R I U M F begins with injection of unpolarized H ~ ions in 5-ns-long pulses every 43.3 ns into the T R I U M F cyclotron. These ions are accelerated to 500 M e V at the cyclotron's outer edge, where thin carbon foils strip off the electrons to create H + (protons). The positively charged protons have opposite curvature (to H~) in the cyclotron's magnetic field, which results in their extraction from the cyclotron into proton beam lines wi th very high efficiency. The primary 500 M e V proton beam has a macroscopic duty factor of 100%, and a typical current of between 100 and 140 fiA. Pions and their decay products (muons and electrons) are produced when the primary proton beam impinges upon the meson production target 1AT2, typically a beryl l ium strip extending 10 cm in the proton beam direction. The M 9 A beam channel views 1AT2 at 135° to the proton beam direction and has an acceptance solid angle of 25 msr. The R M C spec-trometer (described in section 3.3) is mounted at the end of beam channel M 9 A . Addi t ional information on the T R I U M F cyclotron, beam lines, and beam channels can be found in the 17 Chapter 3. Experiment 18 T R I U M F users handbook [71]. The muon beam used in this experiment is produced from pions decaying near the pro-duction target (called cloud muons). After transport by a series of dipole and quadrupole magnets to the R M C spectrometer, the cloud muon beam has a raw particle composition of 7r//x ~ 1 and e//x ~ 15. However, for R M C studies, an R F separator [72] utilizes the differences in particle time-of-flights to suppress the pion and electron contents of the beam to Tr/fi ~ 1 0 - 3 and e/p ~ 5 x 1 0 - 2 . For radiative pion capture (hereafter referred to as R P C ) studies, the separated 7r~ beam composition at the spectrometer is typically 96.3% 7r-, 2.9% e~, and 0.8% fi~. The muon beam has a momentum of 65 M e V / c , Sp/p = 8%, a spot size of 5 x 5 c m 2 at the beam counters (see section 3.3.3), and a typical stopping rate in the targets of 5.0 x 10 5 s _ 1 . The pion beam has a momentum of 81 M e V / c and a typical stopping rate in the targets of 5.6 x 10 5 s _ 1 . 3.2 Targets Six nuclear targets are used in the present R M C study: oxygen, a luminum, silicon, t i tanium, zirconium, and silver. The targets are composed of the naturally occurring elements, and their dimensions are listed in table 3.1. The oxygen target consists of l iquid D 2 0 in a polyethylene bag, held in the beam pipe by a lucite ring surrounded by a styrofoam holder. L iqu id D 2 0 is chosen for its sufficient muon stopping power, and D 2 0 is used instead of H 2 0 because the R P C photon background arising from the small ir~ content of the p~ beam (see section 3.1) and the charge exchange reaction, E q . (4.1), is three orders of magnitude smaller for the deuteron than it is for the proton [73]. Furthermore, R M C on the deuteron is totally insignificant compared to that on oxygen. The silicon target consists of granular silicon in a polypropylene container, held in the beam pipe by a styrofoam holder. The aluminum, zirconium, and silver targets consist of several metallic plates, and the t i tanium target of metallic shavings, all held in the beam pipe Chapter 3. Experiment 19 Target Shape Dimensions No. of foils Spacing of foils 0 disk 14.4 cm dia. x 3.0 cm thick 1 -A l disk 8.5 cm dia. x 0.15 cm thick 10 0.87 cm Si disk 8.6 cm dia. x 2.0 cm thick 1 -T i disk 15.5 cm dia. x 20.0 cm thick 1 -Zr square 10.0 cm x 10.0 cm x 0.025 cm 15 0.69 cm A g square 10.0 cm x 10.0 cm x 0.037 cm 7 0.96 cm Table 3.1: Dimensions of R M C targets. A l l targets are mounted perpendicular to the beam axis. by styrofoam holders. The motivation for using grains, plates, and shavings (as opposed to solid targets) is to reduce photon conversion in the target (increasingly significant for heavier targets). A carbon target is used for calibration of the spectrometer (see section 4.2.3). It is composed of a single disk of graphite, 15.4 cm in diameter and 1.9 cm thick. 3.3 R M C spectrometer The various components of the R M C spectrometer are shown in figures 3.1 and 3.2. The spectrometer has cylindrical symmetry, and an acceptance solid angle of 37r. Starting in the target and extending radially outwards, a photon passes undetected through an A counter, an A' counter, and a B counter. The photon then converts into an e+e~ pair in the lead converter which is sandwiched between the B and C scintillators. The e + e~ pair leaves a signal in one or more C counters, and leaves two distinct circular patterns of cell hits in the drift chamber. One or more D counters may be hit, depending on the topology and energy of the e+e~ pair. Conversely, a charged particle, such as a Miche l electron, is detected in one of more of the A, A', and B counters (called "veto" counters); is detected in a C counter; leaves a single circular track in the drift chamber; and may hit a D counter. The components of the spectrometer are discussed in more detail below. Chapter 3. Experiment 20 Figure 3.1: Global view of the R M C spectrometer, as configured for a hydrogen target. For the nuclear targets of the present study, the l iquid prot ium refrigerator is replaced by a downstream veto scintillation counter. Figure 3.2: Cross section of the R M C spectrometer, showing a two-photon event. Chapter 3. Experiment 21 3.3.1 Drift chamber The drift chamber is used for track reconstruction of e +e~ pairs arising from photon conver-sion in a 1.0 m m thick cylindrical lead converter. The thickness of the lead converter is chosen to provide efficient photon conversion and minimal momentum degradation of the e + e~ pair. The drift chamber, shown in cross section in figure 3.2, is composed of 4 cylindrical superlay-ers of 56, 64, 72, and 80 cells in a uniform magnetic field of 0.235T ± 0.5%. The cells range from 1.947 to 2.081 cm across at their radial midpoints, and contain 6 sense wires, each of which is instrumented wi th a discriminator and a LeCroy 1879 Fastbus pipelined T D C for mult ihi t information in 2 ns bins. The sense wires are alternately staggered right and left of the cell midplane in order to resolve the left/right drift ambiguity. The wires in superlayers 1, 2, and 4 are strung axially, and provide x and y track coordinates. The wires of superlayer 3 are strung axially, followed by rotation of the end plates in opposite directions by 3.5°, so that the wires lie at a stereo angle of 7° with respect to the drift chamber axis. Superlayer 3 provides z track coordinates. The drift chamber uses a gas mixture of 49.9% argon, 49.9% ethane, and 0.2% ethanol in order to achieve a drift velocity of 50 ^im/ns at electric fields near 2 k V / c m . The hits in a cell are fit to a straight line, and the line segments in one axial layer are linked to segments in the other two axial layers to form circular x-y tracks. The radius of a track and the magnetic field in the drift chamber give the x-y component of the particle's momentum according to p(eV/c) = \q\ • Ti(Tesla) • r(cm) • 3 x 10 6 (3.1) where \q\ is the magnitude of the particle's charge in units of the proton charge e. More information on the design and performance of the drift chamber, as well as the track fitting technique, can be found elsewhere [74, 75]. 3.3.2 Inner wire chamber The inner wire chamber ( IWC) is a dual-coordinate multiwire proportional chamber located inside the inner radius of the drift chamber. Its primary purpose is to provide a second Chapter 3. Experiment 22 z coordinate for charged particle tracks in the drift chamber, so that the z component of a particle's momentum can be determined. The I W C consists of 768 axial anode wires arranged cylindrically and spaced by 2.21 m m , bounded on the inner and outer radii by cathode layers, each consisting of 384 spiral strips of a luminum supported by mylar. On the inner and outer cathode layers the aluminum strips are spaced by 3.06 m m and 3.19 m m , and lie at angles of —45° and +45° to the anode wires, respectively. The chamber uses a gas mixture of 77.6% argon, 22.2% isobutane, and 0.2% freon bubbled through methylal , with the cathode strips at ground and the anode wires at +3.4 k V . The z coordinate of a track point is found from the intersection of the hit anode wire and the centroid of the induced pulses on a local strip cluster from at least one of the cathode planes. More information on the design and performance of the I W C can be found elsewhere [74, 76]. 3.3.3 B e a m , t r i g g e r , a n d c o s m i c r a y c o u n t e r s Four identical scintillation counters, of total thickness 0.635 cm, are located approximately 17 cm upstream of the target center. Coincident hits between these four counters are used to count beam particles, and the size of the signal in any one beam counter is used to discriminate between electrons, muons, and pions. A scintillation counter is also located downstream of the target, and serves as a veto in the definition of a particle stop. F ive cylindrical layers of azimuthally segmented scintillation counters surround the target. The first three layers, the A, A', and B counters, are located inside the lead converter; therefore, they act as photon vetos because a photon event should not leave a signal in any one of these counters. The A and A' rings are each segmented into 4 arc-shaped counters, and the inner A r ing is rotated wi th respect to the outer A' ring by 45° so that the joints do not overlap. The B ring is segmented into 12 counters. The C ring is segmented into 12 counters, and is located just outside the lead converter; therefore, it is used in the identification of successful photon conversions. The D ring is segmented into 16 counters, and is located outside the drift chamber. The C and D counters are used together to identify hit patterns characteristic of an e + e~ pair. Chapter 3. Experiment 23 The top face of the spectrometer, and the upper parts of al l four sides, are covered by a total of six scintillator-drift chamber units, each unit consisting of a rectangular scintillation counter and two rectangular drift chambers. This array of scintillators and drift chambers is used to reject photon events which arise due to cosmic ray showers. More information on the beam, trigger, and cosmic ray counters can be found elsewhere [74]. 3.4 Data acquisition T D C and A D C data from the component detectors of the R M C spectrometer is broken up into several Y B O S data "banks" and writ ten to tape when at least one of three trigger conditions is met: • 7 trigger - this trigger selects e +e~ pairs arising from photons entering the lead converter. It is designed to maximize the acceptance of R M C and R P C photons, and minimize the acceptance of non-photon events, so that a CPU-manageable trigger rate is achieved. It also provides some rejection of single track events, such as Miche l electrons that are missed by the A, A', or B veto counters and cosmic rays that are missed by the cosmic ray counters, through the definition TRIGy = STROBE^ • EVENT • AEC (3.2) where STROBE^ = £ ( A + A') • • £ C - > ID EVENT = valid C - D pattern coincidence for an e+e~ pair AHC = "analog hit counter"; valid pattern of drift chamber cell hits for an e+e~ pair. The > ID condition, which requires that at least one the particles in an e+e~ pair reach the D scintillators, rejects low energy photons such as X-rays resulting from cascade of a bound muon to the Is muonic orbital . • Q rate trigger - this trigger selects a fraction of the charged particles leaving signals in the A, A', or B veto counters. It is designed to accept some of the charged particles Chapter 3. Experiment 24 arising from muon decay in the target and cosmic rays, and is used for background and normalization studies of the R M C spectrometer. It is defined by TRIGQ = Q1 + Q2 (3.3) where QI = £ B • £ C • £ D • Q Q2 = > l ( ^ - f - A ' ) - £ C - £ £ > - Q and Q is a variable width gate, usually set so that the Q trigger rate is several percent of the 7 trigger rate. • R r a t e t r i g g e r - this trigger selects a random sample of beam particles. If no charged particle trigger occurs during the Q gate, then another gate, called the R gate, is generated. If at least two out of four beam counters fire in coincidence with this gate, an R rate trigger occurs. The width of the R gate is usually adjusted so that the R trigger rate is about 10% of the 7 trigger rate. R-rate data is used to monitor the pion, muon, and electron contents of the beam, beam counter efficiency, and drift chamber behaviour. It is also used directly in the calculation of R M C rates, since it provides information on the accuracy of the muon S T O P scaler, defined by S T O P M - = 1 • 2 • 3 • 4 • RF,j, • V + A + A' • MV (3.4) 4 = 4-fold beam counter coincidence = correct R F separator t iming for a muon = downstream veto counter = Master Veto - due to equipment faults, computer busy, or low beam rate. More information on R M C data acquisition and monitoring can be found elsewhere [74]. R M C data for the oxygen, aluminum, and silicon targets was taken in January-February 1992. Data for the t i tanium, zirconium, and silver targets was taken in May-June 1994. The 1992 data had a faulty discriminator on the 1 • 2 • 3 • 4 • RF^ Fastbus T D C , but T D C where 1 - 2 - 3 RF» V MV Chapter 3. Experiment 25 data was sti l l available due to the acquisition of both C A M A C and Fastbus T D C data for al l spectrometer components. Data on tape is unpacked, decoded, and analyzed using the R M C O F I A (Radiative Muon Capture OFfline Interactive Analysis) analysis package developed at T R I U M F . The raw data on tape is first passed through a "skim" analysis in order to reject 7 events that are unworthy of full analysis due to insufficient or poor drift chamber data. The remaining 7 events, and the R and Q rate events, are written to three separate tapes for further analysis. The 7 and R rate analyses are pertinent to the calculation of nuclear R M C , and are discussed in sections 4.2 and 4.3. Chapter 4 Analysis 4.1 Backgrounds The goal of the R M C data analysis is to extract, for a given nucleus, an R M C photon spectrum, and to use this spectrum to determine Ry (the R M C / O M C ratio for k > 57 M e V ) which, when compared to theoretical predictions, gives a value for gp. In order to extract the R M C photon spectrum, a number of backgrounds must be completely removed by hardware and software cuts, while maintaining good R M C photon acceptance. These backgrounds are o radiative pion capture ( R P C ) on the nuclear target, as well as other materials in the spectrometer. P ion capture on the proton at rest has two major branches: TT~+P—>ir° + n (60.4%) T T 0 — > 7 + 7 (98.8%) (4.1) T T " + p —> 7 + n (39.6%) (4.2) R P C on nuclei has a branching ratio ~ 2 x 1 0 - 2 per captured pion, whereas R M C has a branching ratio ~ 10~ 5 per captured muon. The R P C photon spectrum peaks around 115 M e V , but extends down into the observable R M C photon spectrum, so it must be removed without compromising the number of R M C photons. This is achieved by minimizing the R P C background by suppressing the pion content of the beam as outlined in section 3.1, and rejecting photon events that occur in prompt coincidence 26 Chapter 4. Analysis 27 wi th a beam particle, because R P C is a strong process whereas R M C is weak. A typical nuclear R P C photon spectrum is shown later in this chapter in figure 4.5. • bremsstrahlung of muon-decay (Michel) electrons, and radiative muon decay ( R M D ) : —> e _ F e ^ 7 . For free muons, both of these processes have a kinematic endpoint of 53 M e V , so this unavoidable background is removed by rejecting photons with energies less than 57 M e V (to allow for spectrometer energy resolution). The observable R M C photon spectrum is then l imited to energies greater than 57 M e V . However, muon binding effects and any high-energy tail in the spectrometer response may produce muon decay photons wi th energies greater than 57 M e V . B y simulating the complete muon decay photon spectrum (i.e. radiative muon decay and bremsstrahlung for bound and free muons) in the spectrometer Monte Carlo (see section 4.2.3), the number of muon decay photons wi th energies greater than 57 M e V is found to be ~0.8% of the observed R M C spectrum for the oxygen target, and <0.2% of the observed R M C spectra for al l other targets considered here. This is consistent with the number of (non-RMC) photons observed above 100 M e V after al l cuts and extrapolated back to 57 M e V for each of the targets. Therefore, the Monte Carlo accurately simulates this background, which is significant only for the oxygen target. However, in the case of the oxygen and silicon targets, this background is removed along wi th R M C photons from non-target materials by running a p~ beam on the "empty" targets (see next i tem and section 5.2). Furthermore, by running a p+ beam on a l iquid hydrogen target so that all capture processes are "turned off" leaving only muon-decay processes, the contribution to the energy spectrum above 57 M e V due to the high-energy tail in the spectrometer response has been found to be negligible [44]. Moreover, the background from muon decay photons in the observable R M C energy region (>57 M e V ) is only significant for the oxygen target, and in this case it is successfully removed. • R M C on non-target materials, most importantly the target container if the nuclear material under investigation is i n l iquid or granular form. This background is estimated Chapter 4. Analysis 28 by running the fi beam on the empty target container, and observing the number of R M C photons per incident muon. • cosmic rays. The vast majority of all photons arising due to electromagnetic showers from cosmic ray particles are rejected by software cuts (see section 4.2.1). The re-maining cosmic ray background after these cuts has been measured to be 2.3 ± 0.4 photons/day in the energy range of interest (57-100 M e V ) [44]. Based on the livetimes of the R M C data collections for the targets considered here, this background is 0.23% of the R M C signal for 0 , and < 0.10% for al l other targets. Therefore, the after-cuts cosmic ray background is negligible. Furthermore, in the case of the oxygen and sil-icon targets, any cosmic ray background is removed along wi th R M C photons from non-target materials by running a \T beam on the "empty" targets. 4.2 C o u n t i n g R M C photons — 7 rate analys is In order to determine R^, the absolute number of R M C photons and the absolute number of muon stops must be known. The absolute number of R M C photons is found by applying software cuts to the 7 rate data; by determining the efficiencies of these software cuts (also using 7 rate data); by finding the absolute photon acceptance of the spectrometer (using Monte Carlo data), and by calibrating this acceptance to a well-known radiative capture process, namely, R P C on carbon. 4.2.1 Software cuts The software cuts are described in the order in which they are applied to the data: 1. veto cut — reject photons wi th hits i n A, A ' , or B, in a 60 ns t ime window around STROBE-,. This cut is a repeat of the hardware veto, E q . (3.2), and is intended to remove a large number of bremsstrahlung photons arising from Miche l electrons. Chapter 4. Analysis 29 2. cstrobe cut — reject photons that do not fire a C counter. This is necessary in order to remove a large fraction of non-photon events. The photon spectrum from a fi~ beam incident upon the silicon target, after the veto and cstrobe cuts, is shown in figure 4.1. 3. tracking cut — reject photons based on e + and e~ track fitting parameters. These parameters include the number of points used in the fit, the x2 of the fit, the proximity of the track to I W C hits, and the distance from the center of the spectrometer to the apex of the track. This cut improves the photon energy resolution of the spectrometer, and removes many bremsstrahlung and cosmic ray photons. 4. photon cut — reject photons based on physical parameters of the observed photon. These parameters include the relative geometries of the P b converter and the tracks of a e +e~ pair; the z component (i.e. the component along the beam axis) of the closest distance between the path of the extrapolated photon and the center of the spectrometer; and the difference in the z components of the e + and e~ momenta. This cut removes photons which have geometrical sources other than the nuclear target. The photon spectrum from a fi~ beam incident upon the silicon target, after al l cuts up to and including the photon cut, is shown in figure 4.1. 5. random cut — reject photons constructed from mismatched e + and e~ tracks. The drift chamber has a "memory" time of 250 ns; hence it is possible for a false photon to be constructed from the e + track of an asymmetric photon conversion in the P b converter and the e~ track of a Michel decay or photon conversion in the target, where the e~ track passes through the drift chamber outside the veto cut t ime window, but inside a window up to 250 ns before and up to 250 ns after STROBE-,. Other possibilities for false photon construction involve the e + track of a photon conversion in the target, and the e~ track from a Miche l decay or photon conversion in the P b converter. Unl ike the veto cut, the identities of the e + and e~ particles are constrained here by the A, A', B hit pattern. Chapter 4. Analysis 30 Photon Energy (MeV) Photon Energy (MeV) Figure 4.1: Photon spectrum from a [i~ beam incident upon the silicon target, at various levels of software cuts: (a) after al l cuts up to and including the cstrobe (unshaded) and photon cuts (shaded); (b) after al l cuts up to and including the cosmic (unshaded) and prompt cuts (shaded). There are no photons below ~30 M e V because the spectrometer photon acceptance goes to zero here. This is a result of the 7 trigger condition that at least one of the tracks of a conversion e + e~ pair reach the D scinillators (see E q . (3.2) ). 6. cosmic cut — reject photons induced by cosmic ray particles. These events are pri-mari ly identified by hit patterns in the cosmic ray drift chambers, and further identified by hit patterns in the C and D counters (a high energy cosmic ray particle w i l l leave a straight track through the spectrometer, so that opposite C and D counters are likely to fire). The photon spectrum from a u~ beam incident upon the silicon target, after al l cuts up to and including the cosmic cut, is shown in figure 4.1. 7. prompt cut — reject photons from R P C on the target. R P C is a strong process (as opposed to R M C which is weak), so that R P C photons are identified by prompt 7-beam particle coincidences. The photon spectrum from a fi~ beam incident upon the silicon target, after all cuts up to and including the prompt cut, is shown in figure 4.1. 8. energy cut — after al l previous cuts, there is st i l l a large peak centered at 53 M e V in the photon energy spectrum, which is due to bremsstrahlung of Miche l electrons, Chapter 4. Analysis 31 3028 150 200 P h o t o n E n e r g y ( M e V ) Figure 4.2: Photon spectrum from a u~ beam incident upon the silicon target, after all software cuts. The bremsstrahlung and R P C photon contributions have been eliminated, as seen by comparison with figure 4.1. and radiative muon decay photons. Therefore, al l photons wi th energy less than 57 M e V are rejected (to allow for finite energy resolution), and only the partial R M C photon spectrum above 57 M e V is retained for calculation of results and comparison with theory. The resulting photon spectrum, shown in figure 4.2, is clearly a pure R M C photon spectrum, although there are a few n o n - R M C photons at high energy (>100 M e V ) . These photons are u~ decay photons and, as stated in section 4.1, occur in a number which is significant only for the oxygen target in which case they are successfully removed. 4.2.2 Cut efficiencies The software cuts described in section 4.2.1 are designed to completely remove n o n - R M C photons. In so doing however, they also remove some R M C photons. These R M C photons are accounted for by calculating the following cut efficiencies. Chapter 4. Analysis 32 t i m e ( n s ) Figure 4.3: T ime spectrum of hits in one A counter for events passing all cuts up to the random cut in the R M C on Si data, where 0 ns is the time of STROBE-,. Off-time regions S and T are used in the combined A, A1, and B t ime spectrum of events passing all cuts up to and including the prompt cut to calculate the efficiency of the veto cut, ev. ev: corrects for R M C photons cut by the hardware and software veto cuts, by looking in two off-time windows on opposite sides of STROBE^ (regions S and T in figure 4.3), each with the same width as the veto cut (60 ns), to see if there is a hit in A , A', or B (due to Michel electrons, beam particles, or noise) for events passing all cuts up to and including the prompt cut. £ " = 1 - 2 • (# +events) ( 4 " 3 ) where S,T = number of events wi th a hit in A, A', or B in the S,T off-time window. ecs: corrects for good photons cut by the cstrobe cut, because a C counter was inefficient. ecs = C counter efficiency = 0.976 ± 0.005. (4.4) e r: corrects for R M C photons cut by the random cut by finding the number of prompt photons (which are ult imately cut by the prompt cut) in the energy range 50 - 80 M e V Chapter 4. Analysis 33 that fail the random cut. The prompt photons in this energy range are representative of R M C photons because they have energies, or topologies, similar to reconstructed R M C photons. / # prompt 7, 50 - 80 M e V , that are cut € r ~ ~ V # prompt 7, 50 - 80 M e V e c: corrects for R M C photons cut by the cosmic cut by finding the number of prompt photons (which are ultimately cut by the prompt cut) in the energy range 100 - 140 M e V that fail the cosmic cut. The prompt photons in this energy range consist solely of photons from a single-photon reaction; R P C photons in the 50 - 80 M e V range may come from a two-photon reaction, E q . (4.1), so even if only one photon is reconstructed in this range, it is likely that opposite D counters wi l l have fired, to which the cosmic cut is sensitive. Therefore, only the prompt photons in the energy range 100 - 140 M e V are representative of how many R M C photons are cut by the cosmic cut. _ / # prompt 7, 100 - 140 M e V , that are cut t c ~ ~ V # prompt 7, 100 - 140 M e V e p: corrects for R M C photons cut by the prompt cut. This correction factor is found by fitting the spectrum of photons cut by the prompt cut to the function a • RMC + b • RPC, where RMC is the final R M C spectrum (i.e., after the energy cut), RPC is the pure R P C spectrum for the target under investigation, and a and b are variable parameters. ep = (l + a) \ (4.7) If the R M C : R P C and RMC:bremsstrahlung photon ratios i n the prompt cut photon spectrum are very small (true for nuclei wi th Z < 6), ep is found by doing a Poisson calculation, given a beam rate and the bremsstrahlung photon production rate (i.e., the free muon decay rate). This is how ep is calculated when analyzing R M C runs on the "empty" targets (i.e., the styrofoam and plastic containers). 1-e,- is the fraction of R M C photons cut by the ith cut. Furthermore, (after a i r cuts) = N*xp is a number of R M C photons (all n o n - R M C photons have been removed). Therefore, Chapter 4. Analysis 34 application of only one correction factor (e.g., e r) to the experimental number of R M C photons gives N e x p = N t r u e _ N t r u e ^ _ ^ = . ^ ^ g ) SO jyexp N t r u e = ^ After application of e r, we are st i l l left with a pure number of R M C photons. Appl icat ion of all the correction factors then yields N t r u e = 7 _ ( 4 1 0 ) €-V ' ^CS ' ^-T * " Note: when analyzing R P C data (e.g., for the purposes outlined in section 4.2.3), the random, cosmic, prompt, and energy cuts are not used, because the number of random, cosmic, and bremsstrahlung photons in the R P C spectrum is negligible (this is what allows the use of R P C photons in the calculation of e r and ec above). Furthermore, R P C photons are, by definition, prompt. Therefore, in the R P C case, jyexp KRPC = -f^2-- (4 .H) *v ^cs 4.2.3 Absolute photon acceptance The intrinsic photon acceptance of the R M C spectrometer is found by means of a Monte Carlo simulation, which uses the framework of the C E R N software package G E A N T [77]. The Monte Carlo simulation is broken down into three stages. Firs t , a specially tailored Monte Carlo program developed at T R I U M F for dealing wi th beam optics, called R E V M O C [78], is used to simulate the muon beam entering the spectrometer. A second Monte Carlo simu-lation, the target Monte Carlo, uses this beam to determine the muon stopping distribution in the target. The accuracy of R E V M O C and the target Monte Carlo is verified by com-paring Monte Carlo predicted stopping fractions to experiment. Final ly , in the spectrometer Monte Carlo, a theoretical R M C photon spectrum is generated isotropically from the muon stopping distribution in the target. A l l important aspects of the spectrometer and target, Chapter 4. Analysis 35 including geometry, chemical and physical properties of materials, and trigger conditions, are input into the Monte Carlo code. The Monte Carlo generated data is written to disk in Y B O S format, and analyzed with R M C O F I A using the identical tracking and photon cuts as were used in the analysis of the experimental R M C data (the Monte Carlo generated data contains only R M C photons, and does not take detector inefficiency into account, so the veto, cstrobe, random, cosmic, and prompt cuts are redundant). The number of Monte Carlo photons after analysis, compared to the original number of photons generated in the target, gives the absolute photon acceptance of the spectrometer, for a given theoretical input R M C photon spectrum. The calibration of the spectrometer (i.e., the normalization of the absolute photon accep-tance, as determined by Monte Carlo, to a well-known radiative process) is achieved using the radiative pion capture photon spectrum from carbon [79]. A t the beginning and/or end of each experimental running period for a target, the beam is switched from u~ to 7 T ~ , and R P C on C data is taken. This R P C data is analyzed using exactly the same cuts as were used in the corresponding R M C analysis. However, due to the differing geometries of the carbon and R M C targets, the photon cut parameter zclose (the z component of the closest approach of the path of the extrapolated photon to the center of the spectrometer) differs between the two analyses. A typical zclose distribution, that for R M C (and R P C , due to the small n~ content of the fi~ beam) on Si , is shown i n figure 4.4. Also shown in figure 4.4 is the zclose distribution wi th the prompt photons and the "wraparound" photons (one of the tracks of a e + e _ pair did not reach the radius of the D scintillators) removed. In both the R M C analysis for a given target, and its corresponding R P C on C analysis, the peak in the zclose distribution corresponding respectively to R M C or R P C photons originating in the target is fit to a gaussian function, and the acceptable zclose window is typically set to the mean ±3cr of this gaussian (in the case of the t i tanium target and its corresponding R P C on C analysis, this window is ±2cr) . After analysis, the agreement between the shapes of the Monte Carlo and experimental R P C on C photon spectra is excellent. This agreement is shown in figure 4.5 for the R P C on C data taken during the R M C on Si data acquisition Chapter 4. Analysis 36 Figure 4.4: The zclose distribution for R M C (and R P C , due to the small TT~ content of the u~ beam) on silicon, zclose is the z component of the distance of closest approach of the path of the extrapolated photon to the center of the spectrometer. The shaded area excludes prompt photons and "wraparound" photons (one of the tracks of a e +e~ pair did not reach the radius of the D scintillators). Photon Energy (MeV) Figure 4.5: Comparison of the known R P C on C photon spectrum [79] after convolution with the spectrometer Monte Carlo and analysis by R M C O F I A (error bars), to the experimental R P C on C photon spectrum (stars). The spectra have been normalized so that each has the same integral number of counts, and scaled so that the size of the error bars (Poisson statistics) reflects the accuracy of the spectrometer Monte Carlo in reproducing experimental data. Chapter 4. Analysis 37 period. The experimental R P C on C branching ratios, calculated for each nuclear R M C target, are in reasonable agreement wi th the weighted average of three mutually consistent measurements [79, 80]: (1.83 ± 0.06)%. The ratios (listed as 11F" in table A . l ) of the experi-mental R P C on C branching ratio values to this given value are applied to the experimental R M C branching ratios in order to calibrate the spectrometer, i.e., in order to account for the small inaccuracy in the absolute photon acceptance of the spectrometer as determined by Monte Carlo. The absolute number of muon stops is found from the S T O P scaler and associated corrections as determined from R rate data and Monte Carlo simulations of muon and pion beams. The muon S T O P scaler is constructed from a coincidence between the four beam counters 1 • 2 • 3 • 4 and the R F separator t ime window for a muon RF^, in turn put into coincidence with MV and V + A + A', where V is the downstream veto counter. Therefore, in order to determine the absolute number of muon stops, it is imperative to know the muon tagging accuracy of RF^ when presented with the raw pion beam (/j,~:ir~:e~ ~ 1:1:10); the muon detection accuracy of 1 -2-3-4 when presented wi th the muon beam (f/,~:ir~:e~ ~ 1000:1:50); and the detection efficiencies of V, A, and A'. The determination and application of the relevant correction factors are discussed in section 4.3.2. 4.3.2 Correction factors Cbm- corrects for missed stops due to inefficiencies of the beam counters. A term could also be added here to account for the efficiency of the downstream veto counter, but because this efficiency has been measured to be ~ 97% and the vast majority of muons stop in the target, any correction to the number of stops due to the efficiency of the 4.3 Counting \i stops R rate analysis 4.3.1 STOP scaler Chapter 4. Analysis 38 downstream veto counter may be neglected. n4 + n3 n4 + n\ (4.12) where ra4 $ of 4-fold beam counter coincidences nZ = # of 3-fold beam counter coincidences ral = # of instances where a 1-fold beam counter coincidence occurs in the dead time window previous to a 3-fold coincidence. When this happens, the STOP scaler can still see the 4-fold coincidence, (if it occurs), but the 1 • 2 • 3 • 4 • RF^ Fastbus T D C , used in R rate analysis, cannot. CREVMOC'' corrects for u~ stops which occur in the target container or beam counter 4, and not in the target material itself, as determined by a Monte Carlo simulation of the \i~ beam incident upon the entire target (container and all). where N c o n t — number of stops (per incident beam particle) in the target container Ntarg = number of stops (per incident beam particle) in the target material. C0'- corrects for over-counting of a~ stops, due to e~ in the beam. e~ are identified by dE/dx in the beam counters, and counted per Fastbus 1 • 2 • 3 • 4 • RF^ T D C entry. This correction involves the relative probabilities for e~ and u~, of the same momentum, to make a valid STOP. However, these probabilities are equal in first order because, for u~ running, the vast majority of p~ stop in the target, thereby making a valid STOP, and the vast majority of e~ pass through the target and downstream veto counter, leaving a below-threshold signal in the veto counter, making a valid STOP (Note: for 7T~ running, the similar masses of the %~ and u~ mean that they also have roughly the (4.13) NbmA = number of stops (per incident beam particle) in beam counter 4 Chapter 4. Analysis 39 same probability for making a valid S T O P ) . ft = ( l + f £ ) - \ (4.14) Cu: corrects for under-counting of p~ stops, due to u~ missing an RF tag. These RF-muons are identified by dE/dx in the beam counters, and counted per Fastbus 1 • 2 • 3 • 4 • RF^ T D C entry. c. = (!-*"' j»J*y**y\ (4.15) C M : corrects for multiple particles per beam burst, i.e., one or more muons are missed due to the presence of another muon. B y counting the number of additional Fastbus 1 • 2 • 3 • 4 • RF^ entries in 14 subsequent beam buckets, rumult, per original Fastbus 1 • 2 • 3 • 4 • RFp entry (assuming that al l of the additional hits are muons - the beam counter A D C ' s are read only once per event, so beam particle identity is unknown for additional beam particles in an R-rate event), the number of additional muons in a beam bucket in which a muon has already been detected is rumult j'14. / # additional unseen p~ \ ~x Cm=(l — j . (4.16) Equivalently, with r — beam rate, and At = 43 ns (beam bucket width - see section 3.1), C™ = 1^7^- . (4-17) Cmd' corrects the number of p~ stops, due to a p~ stop being vetoed by a Miche l e~ hi t t ing the downstream veto counter (V), or a p~ stop being added by a Miche l e~ returning to fire al l four beam counters in coincidence wi th RF^. This correction is found by doing a Monte Carlo simulation of Miche l decays from the p~ stopping distribution in the target (also determined by Monte Carlo). Cw = (l + ( # 6 " ^  b n t 3 ' 2 Z " 7 ( # e " ^  V ) • (0.16). (1 - e,))" (4.18) V TT ^ enter A / where V is the downstream veto counter, and l - e r is the fraction of R M C photons that have a Michel e~ track pointing to the C counter that fired as a result of the R M C Chapter 4. Analysis 40 photon. The Miche l e~ is detected if there is a hit in any one of three possible A and A' counters (mapped back from the C counter that fired) in a 500 ns window around STROBE^. The factor of 0.16 in E q . (4.18) arises from the fact that there are a total of 8 A and A' counters, and the fact that 1 • 2 • 3 • 4 • RF^ and V + A + A' must fall wi thin 30 ns of each other in order to form a S T O P (§ • jffc = 0.16). Unlike the experimental number of R M C photons after cuts, N^xp, the experimental number of muon stops contains "bad" stops (i.e., non-muon stops which must be removed first). so But so NZPPS = N„- + Ne- = / V > ( l + ^ ) (4.19) iyexp N»- = 1 + = NZPVS • C0. (4.20) K- = Ngood + N d e c a y s = Ngood(l + ^py±) (4.21) V "good / N -Ngood = " = JV> • Cm*. (4.22) ^ i -LV decays / IV good Then, because Cbm, Cm, and Cu are defined such that 1-C 6"^, 1 - C ~ : , and l - C " 1 represent fractions of good muon stops lost, Nllops,alI = Ng00d • Cbm ' Cm • Cu. (4.23) But so K:;s>all = Ntarg + Ncont + Nbm4 = Ntarg ( l + N™*+ (4.24) \. "targ / ftjtrue M — \jtrue _ stops,all f.jtrue s~\ (A o r \ "targ - "stops ~ i , / » r T l u w i _ " stops,all " CREVMOC- (4.2o) 1 T {"cont T "bmijl"targ Combining Eqs. (4.20), (4.22), (4.23), and (4.25), Nltops = Ntteps ' Cbm • CREVMOC • C0 • Cu • Cm • Cmd- (4.26) Chapter 4. Analysis 41 Note: in the analysis of RPC data, where the number of pion stops is used, the correction factors to the pion STOP scaler data are analogous to those above for the muon STOP scaler data, except that RF^ is replaced by RF„, and Cmd is replaced by Cpd-Cpd'. corrects the number of TT" stops, where the ir~ decays before stopping in the target, but the STOP definition is still satisfied (r x - = 26 ns, whereas r^- = 2.2 / L i s , so this correction is insignificant for p~ beam). This correction factor is determined by means of a Monte Carlo simulation of the TT~ beam incident upon the target. n ft . (# ^~ stopping in target or b m 4 ) w - d e c a 2 / \ - 1 O p d = l l + — — — : : — - . (4.2 () V # 7r stopping in target / The derivation of the true number of 7r~ stops in RPC data is similar to the above for p~ stops in R M C data, and is given by N^-stops — ^T-stopS' Cbm • CREVMOC • C 0 • C u • C m • Cpd- (4.28) 4.4 Addition of uncertainties The e,- efficiencies are all statistically independent; therefore, their uncertainties are added in quadrature when calculating R^. The e,- involve quantities of the form X = n-i/N, where ni is a subset of N, with n\ + n 2 = N. The uncertainty in X, dX, is easily calculated to be 1 V / 2 / 1 \ 1 / 2 dX CREVMOC and Cpd are calculated by independent Monte Carlo simulations, and are statis-tically independent of all other correction factors. Cmd is found by doing a Monte Carlo simulation and by using the value, of er; hence the uncertainty to be added in quadrature is (d[er • Cmd})2 = (Cmd • der + er • dCmdf. (4.30) dCpd and dCmd are found by varying the z location of the target in their respective Monte Carlo simulations. dCREVMOC is found by comparing Monte Carlo data to experimental data for runs involving empty target containers, and determining the accuracy with which Chapter 4. Analysis 42 the Monte Carlo predicts the detection efficiency of the downstream veto counter (which is independent of target). C0, Cu, and Cm are not statistically independent of one another (unless, as in the case of the 0, A l , and Si data, C0 and Cu are calculated using C A M A C T D C data, but Cm is st i l l calculated using Fastbus T D C data, due to a faulty discriminator on the 1 • 2 • 3 • 4 • RF^ Fastbus T D C - see section 3.4. In this case, Cm is independent of all other correction factors, but C0 and Cu are not independent of each other). The uncertainty that is added in quadrature to the other uncertainties involved in calculating Ry is The uncertainty in the absolute photon acceptance depends on the form of the R M C photon spectrum input in the Monte Carlo. If the spectrum is taken bin by bin from theory, then the uncertainty is found by varying the Monte Carlo determined muon stopping distribution in the target. The two extremes of a point stopping distribution in the middle of the target, and a uniform stopping distribution throughout the target, are used to generate photons in the spectrometer Monte Carlo. This was done explicity for the A l and Si targets (and C for R P C ) , and the resulting uncertainty due to the accuracy of the Monte Carlo determination of the muon stopping distribtution was found to be < 2%. A n uncertainty of 2% was then applied to the absolute photon acceptance of the other targets. Because these two extremes in the muon stopping distribution are unrealistic, the uncertainty in the absolute photon acceptance is over- rather than underestimated. If the R M C photon spectrum is in analytic form with a variable parameter (specifically, kmax in the Primakoff polynomial , given in section 5.1), then the uncertainty in the absolute photon acceptance has a second contribution which is estimated by determining the values of the absolute photon acceptance when the high and low l imits on kmax are used for the R M C spectrum input in the Monte Carlo. The determination of the l imits on kmax is given in section 5.1. (4.31) Cu involves a quantity of the form Y = n\jni, wi th ri\ + n-i = N. In this case, (4.32) Chapter 5 Results 5.1 Calculation of R7 and gp/gA Two methods are used for the extraction of FLy and gp/gA from the experimental R M C data: the integral method and the shape method. The integral method is used when the nuclear R M C response has been calculated for the nucleus under investigation, and gives experimental values for both FLy and gp/gA- This is the case for 0, T i , Zr, and Ag. The shape method is used in all cases, and only gives values for FLy. The integral method involves passing the theoretical R M C spectrum through the Monte Carlo, normalizing the number of generated photons (i.e., the number of photons input into the Monte Carlo) to the experimental number of captured muons (i.e., the corrected number of muon stops multiplied by the O M C capture fraction - see table 5.1), analyzing the resulting spectrum with R M C O F I A , and obtaining a predicted number of observed R M C photons above 57 MeV as a function of both the theoretically calculated FLy and gp/gA-These functions are fit to simple polynomials. The intersection of the polynomial functions with the experimental number of R M C photons above 57 MeV gives experimental values for FLy and gP/gA-The shape method involves passing a representative theoretical R M C photon spectral shape through the Monte Carlo in order to determine the photon acceptance of the spec-trometer and thus a value for FLy. The spectral shape used is the closure or PrimakofF [66] 43 Chapter 5. Results 44 /capture T (ns) c 0.0777 ± 0.0007 2026.3 ± 1.5 0 0.1844 ± 0.0009 1795.4 ± 2.0 A l 0.6095 ± 0.0005 864.0 ± 1.0 Si 0.6587 ± 0.0005 756.0 ± 1.0 T i 0.8530 ± 0.0006 329.3 ± 1.3 Zr 0.9529 ± 0.0004 110.0 ± 1.0 A g 0.9634 ± 0.0006 87.0 ± 1.5 Table 5.1: O M C capture fractions and mean muon capture lifetimes in nuclear targets [81]. polynomial (see section 2.2.2), given by *M „ (i _ 2x + 2x2)x{l - x2) (5.1) ak where d G ^ is the differential R M C photon spectrum; and x — k/kmax, where k is the R M C photon energy and kmax is the maximum R M C photon energy. kmax is directly related to Eav, discussed in section 2.2.2. The value of kmax chosen for a given nucleus is that for which the Primakoff spectrum, after Monte Carlo convolution and R M C O F I A analysis, gives the best fit to the experimental R M C spectrum. The uncertainty in kmax, which contributes to the uncertainty in the absolute photon acceptance (see section 4.4), is found by normalizing each Monte Carlo spectrum to the experimental spectrum (i.e., mul t ip lying each Monte Carlo spectrum by a factor such that the Monte Carlo and experimental spectra have the same integrated number of photons); fitting each normalized Monte Carlo spectrum to the experimental spectrum and obtaining a value for the x2 goodness-of-fit statistic; plotting X2 vs. kmax and fitting this to a simple polynomial; and extracting the uncertainty in kmax using the fact that for a single free parameter, the 68.3% confidence interval is in the region Ax2 = 1 [82]. The x2 v s- kmax curve for the oxygen target is shown in figure 5.1. After R M C O F I A analysis, the number of R M C photons above 57 M e V , compared to the original number of Primakoff photons above 57 M e V generated in the Monte Carlo, gives the absolute R M C photon acceptance of the spectrometer. Wri t ing the photon acceptance as Acc, we Chapter 5. Results 45 86 87 88 89 90 km o x (MeV) Figure 5.1: x2 v s - kmax for fits of Primakoff spectra generated by Monte Carlo to the experimental R M C on 0 spectrum. The solid curve is a quadratic fit to the data points. have Acc- £ Gj= £ (5.2) j>57MeV i>57MeV j where Gj is the physical R M C photon energy spectrum in units of photons/MeV/capture , and A(i, j) is the absolute probability of a photon of energy Ej being reconstructed at an energy E ; . The experimentally observed "true" photon energy spectrum from R M C , N^™e, is related to the physical photon energy spectrum, Gj, by NT = N Z s • feature - £ ^ , j ) - Gj (5.3) j where J"capture is the fraction of muons that undergo O M C in the target (see table 5.1). Summing both sides of E q . (5.3) over i > 57 M e V , and using E q . (5.2), gives A r = £ Gj= 2cc7- Ntrue ( 5 - 4 ) j>57MeV J capture - T L ^ l y stops where R^ is the k > 57 M e V R M C / O M C ratio. In terms of experimentally observed numbers of photons and stops, this is pjexp "> = 7 Z-N- ( 5 ' 5 ) J capture ST-IA- -f>tries Chapter 5. Results 46 where, using Eqs. (4.10) and (4.26), Ntries — ^stops ' ' ^cs ' ^r ' €c ' eP • Cbm • CREVMOC • C 0 • C u - C m • Cmd- (5.6) Ntries is defined such that i V i r , e s • fcapture is the number of Monte Carlo photon "tries" that reduces to the observed experimental number of photons after Monte Carlo convolution and R M C O F I A analysis. The calibration of the spectrometer is then taken into account by multiplying Ry by F (see section 4.2.3 and appendix A) . Note : in R P C analysis, the R P C spectrum is observed at all energies. Therefore, the R P C branching ratio is given by R P C ~ AccN ~ ^ > 1 Y -K~ tries where, using Eqs. (4.11) and (4.28), N-x-tries = ^T-stops ' t v ' t c s ' Cbm ' CREVMOC • C0 • C u • C m ' Cpd- (5-8) 5.2 O , A l , Si , and T i No theoretical R M C photon spectra, with associated values of R-y and gp/gA, are currently available for aluminum or silicon nuclei (the theory of Fearing and Welsh [70] is only realistic for heavy nuclei). Therefore, the shape method is used to determine R^, and values for 9P/9A are not obtained for A l and Si. However, three calculations of R M C on 0 [58, 68, 69] are available for direct comparison with experiment. Similarly, three calculations of R M C on Ca [58, 67, 69] are available for comparison with R M C on T i data (assuming that the nuclear response of T i is similar to that of Ca). Therefore, the integral method is used to obtain values of gp/gA, and both the integral and shape methods are used to obtain values of 7^, for 0 and T i . In the special case of Roig and Navarro [69], R M C photon spectra are not available, so a modified integral method is used to determine values of gp: theoretical values of Ry are plotted vs. gp and fit to a simple polynomial; an experimental value for gp is found from the intersection of this polynomial function with the experimental value of Ry as determined by the shape method. The nuclear R M C models used for 0 and Ca are discussed in section 2.2.2. Chapter 5. Results 47 A l l quantities involved in the calculation of FLy and gp/gA for 0, A l , S i , and T i are shown in appendix A , table A . l . As stated in section 3.2, the Si target is granular, and held in place by a polypropylene, (CaH^n, and mylar, (C5rl402)n, container. The oxygen target is composed of l iquid D2O, held in place by a polyethylene bag, ( C H 2 ) „ , and a lucite ring, (CsHsC^n- Because the muon capture lifetimes in C , 0, and Si are similar (see table 5.1), R M C data was taken using the empty D2O and Si targets, to determine the fraction of R M C photons coming from the container in each case. The number of R M C photons from the empty target is normalized to the number of 1 • 2 • 3 • 4 • RF^ • MV hits for the full and empty targets, and corrected (via Monte Carlo) for the number of muons which stop in the container when the container is full as opposed to empty. The resulting fraction of R M C photons from the target container is shown in table A . l as empfrac. The a luminum target is composed of a luminum plates, and the t i tanium target of metallic T i shavings, both held in place by styrofoam target holders. Because the styrofoam is of very low density, its R M C photon contribution is negligible. The photon acceptance, given as Acc in table A . l , is that found from the shape method. Comparisons of the PrimakofF R M C spectra (after Monte Carlo convolution and R M C O F I A analysis) to the experimental R M C spectra for Si , A l , 0, and T i are shown in figures 5.2 and 5.3. Values of FL, and gp/gA-, obtained for 0 and T i using the integral method, are read directly off figure 5.4. Values of gp for 0 and T i , obtained using FLy as determined by the shape method, are read directly off figure 5.5. Comparisons of the theoretical R M C on 0 and C a spectra (after Monte Carlo convolution and R M C O F I A analysis) to the experimental R M C on 0 and T i photon spectra are shown in figures 5.6 and 5.7. F ina l numerical results are shown in table 5.2. Chapter 5. Results 48 15 250 r a Si l icon vmax 85 MeV 88 MeV 95 MeV 60 70 80 90 Photon Energy (MeN 00 ^ 250 \ 200 c 150 S 100 50 0 -1 I L Aluminum nax a 87 MeV b 88 MeV c 89 MeV _ l I I— L_ 60 70 80 90 Photon Energy (MeN 100 Figure 5.2: Comparison of the experimental R M C on Si and A l photon spectra with the closure spectral shapes given by Eq. (5.1) after these shapes have been convoluted by the spectrometer Monte Carlo, and analyzed by R M C O F I A . The solid line in each figure is the spectral shape for the best fit value of kmax. Chapter 5. Results 49 > CD CO 250 200 150 ° 100 50 0 Oxygen k max 86 MeV 87 MeV 90 MeV 60 70 80 90 Photon Energy (MeN 00 100 Photon Energy (MeV) Figure 5.3: Comparison of the experimental R M C on 0 and T i photon spectra with the closure spectral shapes given by Eq. (5.1) after these shapes have been convoluted by the spectrometer Monte Carlo, and analyzed by R M C O F I A . The solid line in each figure is the spectral shape for the best fit value of kmax. Chapter 5. Results 50 1900 0.12 0.14 0.16 x 10 2600 2400 2200 r 2000 1800 1 600 1400 1200 Titanium G O T C -3 -2 -1 9 P / 9 A Figure 5.4: Number of R M C photons above 57 M e V , N 7 , vs. R^ and gp/gA for the R M C on oxygen theories of Gmi t ro et al. ( G O T ) [58] and Chr is t i l l in and Gmi t ro (CG) [68], and the R M C on C a theories of Gmi t ro et al. ( G O T ) [58] and Chr is t i l l in (C) [67]. The hatched regions are the experimental number of R M C photons above 57 M e V for oxygen and T i . Chapter 5. Results 51 g P / g , Figure 5.5: R-, vs. gp/gA for the R M C on 0 and C a theory of Roig and Navarro (RN) [69]. The hatched regions are the experimental values of Ry for 0 and T i as determined by the shape method. Chapter 5. Results 52 > CD 250 200 CO 150 J 100 50 0 Oxygen _ ! 1 L_ _1 I I L 60 70 80 Photon Energy GOT gp/gA a 4.5 b 7.5 C 12.0 d 1 6.0 90 1 1 00 > CD 250 r 200 ^ 150 F J 100 50 0 a H \ \ Oxygen TTT M a b CG gp/gA 6.8 13.6 _ l 1 L_ 60 70 80 90 Photon Energy (MeN 00 Figure 5.6: Comparison of the experimental R M C on O photon spectrum wi th the theories of Gmi t ro et al. ( G O T ) [58] and Chr is t i l l in and Gmi t ro (CG) [68]. The theoretical spectra have been convoluted by the spectrometer Monte Carlo, analyzed by R M C O F I A , and / normalized to reflect the results in figure 5.4. Chapter 5. Results 53 > CD 200 CO 150 c Z3 o 100 o 50 0 a i M C ' V ' \ T + + 4 Titanium GOT gp/gA V V+ a b c 4.5 7.5 12.0 ++++ _ i i i _ 60 70 80 Photon Energy 90 100 eV) cu 200 CO 150 c o 100 o 50 I c 0 Titanium 60 70 80 Photon Energy (Me C gp/gA a -6.8 b 0 c 6.8 d 13.6 90 1 00 Figure 5.7: Comparison of the experimental R M C on T i photon spectrum with the R M C on C a theories of Gmi t ro et al. ( G O T ) [58] and Chr is t i l l in (C) [67]. The theoretical spectra have been convoluted by the spectrometer Monte Carlo, analyzed by R M C O F I A , and normalized to reflect the results in figure 5.4. Chapter 5. Results 54 5.3 Zr and Ag Calculations of R M C on M o and Sn [55] and calculations of R M C on Zr and A g [70] are available for comparison wi th experimental R M C on Zr and A g data (assuming that the nuclear responses of M o and Sn [55] are similar to those of Zr and Ag) . Either theory may be used to determine experimental values of R^ and gp /g& for Zr and A g using the integral method, but only the M o and Sn theory of Chr is t i l l in tt al. [55] is used, for the reasons discussed in section 2.2.2 and in order to compare with previous results [44]. A l l quantities involved in the calculation of R^ and gp/gA for Zr and A g are shown in appendix A , table A . l . The photon acceptance, given as Acc in table A . l , is that found from the shape method. Comparisons of the Primakoff R M C spectra (after Monte Carlo convolution and R M C O F I A analysis) to the experimental R M C spectra for Zr and A g are shown in figure 5.8. Values of R^ and gp/gA, obtained for Zr and A g using the integral method, are read directly off figure 5.9. Comparisons of the theoretical R M C on M o and Sn spectra (after Monte Carlo convolution and R M C O F I A analysis) to the experimental R M C on Zr and A g photon spectra are shown in figure 5.10. F ina l numerical results are shown in table 5.3. Chapter 5. Results 55 > CD CO c Z5 O O 100 80 60 40 20 0 a b c _ l I I I I L Zirconium k a 86 MeV b 87 MeV c 89 MeV 60 70 80 90 Photon Energy (Me' 00 > CD CO C 13 O o 80 60 40 20 0 b iO i c \. _! I l _ Silver k max a 86 MeV b 87 MeV c 89 MeV I I I I I I I - J b=l= l_ 60 70 80 90 Photon Energy (MeN 100 Figure 5.8: Comparison of the experimental R M C on Zr and A g photon spectra with the closure spectral shapes given by E q . (5.1) after these shapes have been convoluted by the spectrometer Monte Carlo, and analyzed by R M C O F I A . The solid line in each figure is the spectral shape for the best fit value of kmax. Chapter 5. Results 56 Figure 5.9: Number of R M C photons above 57 M e V , N-y, vs. Ry and gp/gA for the R M C on M o and Sn theory of Chr is t i l l in et al. (CRS) [55]. The hatched regions are the experimental number of R M C photons above 57 M e V for Zr and A g . Chapter 5. Results 57 > CD CO 125 100 75 c Z5 ° 50 25 0 c b a T \ H , . _ l I I I L 60 Zirconium Photon Energy (Me CRS gP/gA a -6.8 b 0 c 6.8 d 13.6 - J I L_ 100 > CD CO C Z 5 O o Silver 60 70 80 Photon Energy CRS gp/gA a -6.8 b 0 c 6.8 d U-13.6 90 1 100 F i g u r e 5.10: C o m p a r i s o n of the e x p e r i m e n t a l R M C o n Z r a n d A g p h o t o n spec t ra w i t h the R M C on M o a n d S n theory of C h r i s t i l l i n et al. ( C R S ) [55]. T h e theore t i ca l spec t r a have been convo lu t ed b y the spec t romete r M o n t e C a r l o , a n a l y z e d by R M C O F I A , and n o r m a l i z e d to reflect the results i n figure 5.9. Chapter 5. Results 58 Z Ry ( l O " 5 ) 9P/9A kmax ( M e V ) Theory 0 8 1.47 ±0.05 0.1-1.2 — Gmit ro et al. [58] 1.37 ±0.06 4.1 ±0.2 — Chris t i l l in and Gmi t ro [68] 1.61 ±0.16 — 87.1 ± 1.9 Primakoff [66] 1.61 ±0.16 6.otl:? — Roig and Navarro [69] A l 13 1.36 ±0.12 — 88.5 ±1.2 Primakoff [66] Si 14 1.96 ±0.20 — 87.8 ± 1.7 Primakoff [66] T i 22 1.42 ±0.04 < 0 — Gmit ro et al. [58] 1.19 ±0.07 < 0 — Chris t i l l in [67] 1.28 ±0.10 — 88.1 ±2.0 Primakoff [66] 1.28 ±0.10 2.9183 — Roig and Navarro [69] Table 5.2: Experimental values of Ry, gp/gA, and kmax for O, A l , S i , and T i . Z Ry ( 1 0 ~ 5 ) 9P/9A kmax ( M e V ) Theory Zr 40 1.28 ± 0 . 0 7 1.26 ± 0 . 1 1 87.1 ± 2 . 0 Chris t i l l in et al. [55] Primakoff [66] A g 4 7 1.21 ± 0 . 0 6 1.16 ± 0 . 1 0 o i +0.6 z - i - 0 . 7 87 .3 ± 1.9 Chris t i l l in et al. [55] Primakoff [66] Table 5.3: Experimental values of Ry, gp/gA, and kmax for Zr and A g . Chapter 6 Discussion 6.1 R7 Due to the model dependency of nuclear R M C (see table 5.2), the final FLy results of the present study are those found using the closure or Primakoff polynomial (i.e., the shape method described in section 5.1). These results, along wi th previous results also obtained in the closure model, are presented in table 6.1. The widely different values of FLy for Si and A l , as observed by Armstrong et al. [44], are confirmed by the present results. Al though Armstrong et al. and the present experiment used the same experimental apparatus, this agreement is a validation of the experimental and analytical procedures of each, because Armstrong et al. used a 2D photon trigger (see E q . (3.2) ), whereas the present experiment used a > ID trigger. Furthermore, Armstrong et al. used an analytic form for their detector response and acceptance, whereas in the present experiment, the detector acceptance was modelled independently for each nuclear target in the Monte Carlo. The ~ 3<r difference between the values of FLy for A l and Si is qualitatively explained by Paul i blocking of the final state neutron, which reduces the available phase space for both R M C and O M C . R M C , wi th its 3-body final state, is suppressed more than O M C ; therefore, a decrease in FLy (= R M C / O M C ) is expected for nuclei such as A l which have neutron excesses. This "isotope effect" was first noted by Primakoff [66]. T i also has a neutron excess (whereas C a does not), so the ~ 4cr difference between the present FLy result for T i , measured here for the first t ime, and the previous result for C a (see table 6.1) is a further example of this isotope effect. 59 Chapter 6. Discussion 60 Phase space arguments also account for the decreasing trend of FLy wi th Z, shown in figure 6.1. As Z increases, the neutron Fermi momentum becomes significantly larger than that of the proton, resulting in less available phase space for R M C . The present FLy results for Zr and A g , measured here for the first time, are consistent with previous results for other high Z nuclei on the FLy vs Z plot. The present FLy result for 0 is ~ 25% lower than the previous results of Armstrong et al. [43] (see tables 6.1 and 6.2) and Dobeli et al. [41] (see table 6.2); and is ~ 60% lower than the previous result of Frischknecht et al. [42] (see table 6.2). Al though Armstrong etal. also did their work at T R I U M F , their C , O, and C a experiment uti l ized a t ime projection chamber ( T P C ) in place of the present drift chamber (section 3.3.1), and they found a beam rate-dependency in their photon acceptance (an exponential decrease in acceptance wi th particle flux in the T P C ) . They were only able to observe the functionality of this rate dependence for R P C on C and R M C on Ca , because the small R M C rates for C and 0 make observation of the beam rate-dependence of the photon acceptance impractical . However, they were able to simulate the observed rate-dependence of the R P C on C and R M C on C a acceptances in their Monte Carlo. Therefore, their Monte Carlo was used to correct the photon acceptance in the C and O cases, and their FLy result for O agrees wi th the previous result of Dobeli et al. [41]. Nevertheless, it may be possible that they overestimated the effect of the beam rate on the photon acceptance in the C and 0 cases, arriving at acceptance values that are too low and corresponding FLy values that are too high. Experimental values of FLy are plotted vs. neutron excess, a = (N - Z ) / A , in figure 6.2. The decrease in FLy wi th a is explained above by phase space arguments which result from Paul i blocking of the final state neutron. The low value of FLy for 0 compared to that of Si or C a (i.e., the other Z = N (a = 0) nuclei) is qualitatively explained by the larger energy gap between the filled proton shell and the empty neutron shell for 0 compared to that of Si or Ca . This energy gap reduces the available phase space for muon capture. Theoretical predictions of FLy vs. Z are compared to experimental results in figure 6.3. In figure 6.3(a), the theory of Chr is t i l l in et al. [55] agrees well wi th the experimental FLy vs. Chapter 6. Discussion 61 Z plot for Z > 40 and gp/gA = 0. The theories of Chr is t i l l in [67] and Chr is t i l l in and Gmi t ro [68] are included for comparison wi th the low Z results, and there is an intriguing agreement between experiment and theory for both the gp/gA — 0 and gp/gA — 6.8 curves, even though these two theoretical curves have distinctly different shapes at low Z. In figure 6.3(b), the theory of Fearing and Welsh [70] reproduces the shape of the experimental Ry vs. Z plot for Z > 40, but only after the theoretical values of Ry are scaled by 0.38. Comparison between theory and experiment for Z < 20 is included in the figure, but the model of Fearing and Welsh is not expected to be applicable for these nuclei [70]. Z a Ry ( l O " 5 ) kmax ( M e V ) X 2/d-o.f. Reference 0 8 0 1.61 ± 0 . 1 6 87.1 ± 1.9 1.46 present work A l 13 0.0364 1.36 ± 0 . 1 2 88.5 ± 1.2 1.58 present work 1.43 ± 0 . 1 3 90 ± 2 1.12 Armstrong et al. [44] Si 14 0.00304 1.96 ± 0 . 2 0 87.8 ± 1.7 1.9.1 present work 1.93 ±0.1 ,8 92 ± 2 1.73 Armstrong et al. [44] C a 20 0.00195 2.09 ± 0 . 1 9 93 ± 2 1.56 Armstrong et al. [44] T i 22 0.0810 1.28 ± 0 . 1 0 88.1 ± 2 . 0 1.45 present work Zr 40 0.123 1.26 ± 0 . 1 1 87.1 ± 2 . 0 1.31 present work M o 42 0.124 1.11 ± 0 . 1 1 90 ± 2 0.82 Armstrong et al. [44] A g 47 0.129 1.16 ± 0 . 1 0 87.3 ± 1 . 9 1.01 present work Sn 50 0.158 0.98 ± 0 . 0 9 87 ± 2 1.14 Armstrong et al. [44] P b 82 0.208 0.60 ± 0.07 84 ± 3 .. 0.85 Armstrong et al. [44] Table 6.1: Values of Ry, kmax, and the corresponding x2 °f fit as determined by the shape method (closure model), a — (N - Z ) / A is the neutron excess, where A is the atomic mass of the natural element [1], and N = A - Z . Chapter 6. Discussion 62 Figure 6.1: FL, vs. Z. Plot ted values are from table 6.1, where the solid circles are the present closure model results, and the solid squares are previous closure model results [44]. a Figure 6.2: Fly vs. a (neutron excess). Plot ted values are from table 6.1, where the solid circles are the present closure model results, and the solid squares are previous closure model results [44]. Chapter 6. Discussion 63 2.5 1 cr 5) 1.5 1 0.5 0 0 A * a • g P / g A = -6.8 o g P / g A = 0 A g P / g A = 6.8 o g P / g A = 1 3.6 '20' 40 Z (a) 8 60 80 100 2.5 r (10-5) 1 .5 0 •(b) - i i , 0 0 : 0 ? i r 0 8 0 0 0 :_ i 0 0 g P / g A = 6.8, scaled by 0.38 1 , 1 , 1 , , , 0 20 40 60 Z 80 100 Figure 6.3: Experimental Fly vs. Z compared wi th (a) the theories of Chr is t i l l in et al. [55] (Z > 40), Chr is t i l l in [67] (Z = 20), and Chr is t i l l in and Gmi t ro [68] (Z = 8); and (b) the theory of Fearing and Welsh [70], scaled by 0.38. Theoretical values of FLy for Z < 40 are shown in (b) even though the model of Fearing and Welsh is not expected to be applicable for such nuclei. Experimental values of FLy are shown by the solid circles (present closure model results) and solid squares (previous closure model results [44]), and are taken from table 6.1. Chapter 6. Discussion 64 6.2 gP The gp/gA results of this and other R M C experiments are shown in table 6.2 and plotted vs. atomic number in figure 6.4. The model dependence of nuclear R M C is clear in the cases of 0 and Ca , and indicates that nuclear R M C must have a better theoretical understanding before experiment can make any tests of P C A C . The present T i , Zr, and A g results indicate that gp may be quenched to a large degree in nuclei, although the gp/gA values for T i cannot be taken too seriously, as nuclear responses for C a were adopted in this case, and the assumption that the nuclear responses of C a and T i are similar, at least with respect to R M C , appears to be invalid when one considers the large difference in their Ry values (see section 6.1). The present gp/gA values for 0 are marginally lower than the Goldberger-Treiman value, and only display a slight model dependence. Previous gp/gA values from R M C on 0 are significantly higher, and, in the case of Armstrong et al. [43], are strongly model-dependent. The discrepancy between the present and previous [43] results as extracted from R M C on 0 is discussed in section 6.1. In figure 6.4, the decrease in gp/gA wi th Z for Z > 40 is model-dependent, and merely reflects the Z-dependent quenching of gp present in the Fermi gas model of Chr is t i l l in et al. [55] (this model was used to extract values of gp/gA for all of the Z > 40 results in figure 6.4). However, in the model of Chr is t i l l in et al, the values of gp/gA that are required to reproduce the experimental values of R-y (i.e., the values of gp/gA in table 6.2 for Z > 40) do not agree wi th the values of gp/gA required to reproduce the well-known O M C rates [81]. Also, although both Chr is t i l l in et al. and Fearing and Welsh [70] are able to reproduce the shape of the Ry vs. Z plot for Z > 40 (see figure 6.3), Fearing and Welsh are able to do so without the requirement of a Z-dependent quenching of gp (note, however, that Fearing and Welsh's absolute values of Ry are roughly 2.5 times larger than experimental values, assuming the Goldberger-Treiman value of gp). Fearing and Welsh also find that Ry is quite sensitive to various inputs in the Fermi gas model, which suggests that it is difficult to extract values of gp. Clearly then, the present theoretical situation for nuclear R M C is not such that experiment can make conclusive statements about gP and P C A C . Chapter 6. Discussion 65 Z 9P/9A Ry ( l O " 5 ) Theory Reference c 6 1 6 . 2 1 ^ 2 .33 ± 0 . 1 7 G O K S Armstrong et al. [43] 0 8 13 .5 ± 1.5 3 .8 ± 0 . 4 C G Frischknecht et al. [42] 8.4 ± 1 . 9 2 .44 ± 0 .47 C G Dobeli et al. [41] 7.3 ± 0 . 9 2 .22 ± 0 . 2 3 C G Armstrong et al. [43] 2.18 ± 0 . 2 1 G O T Armstrong et al. [43] 4.1 ± 0 . 2 1.37 ± 0 . 0 6 C G present work c 1+0.9 ° - i - 1 . 2 1.47 ± 0 . 0 5 G O T present work 6.oil:? 1.61 ± 0 . 1 6 R N present work C a 20 4 .6 ± 0 . 9 1.96 ± 0 . 2 0 C Frischknecht et al. [46] 6.5 ± 1 . 5 6-0115 9 occ+0.32 Z . O O _ 0 . 3 0 9 i c+0.27 C G O T Vi r tue et al. [47] Vir tue et al. [47] 6.3115 2.30 ± 0 . 2 1 C Dobeli et al. [41] 5.7 ± 0 . 8 2 .18 ± 0 . 1 6 C Armstrong et al. [43] 5.9 ± 0 . 8 4.611:1 2.21 ± 0 . 1 5 2 .04 ± 0 . 1 4 C G O T Armstrong et al. [44] Armstrong et al. [43] 5.0 ± 1 . 7 2 .07 ± 0 . 1 4 G O T Armstrong et al. [44] 7.8 ± 0 . 9 2 .09 ± 0 . 1 9 R N Armstrong et al. [44] T i 22 < 0 1.19 ± 0 . 0 7 C present work < 0 1.42 ± 0 . 0 4 G O T present work 2-9181 1.28 ± 0 . 1 0 R N present work Zr 40 -0.4l?3 1.28 ± 0 . 0 7 C R S present work M o 42 o.oij-? 1.26 ± 0 . 1 0 C R S Armstrong et al. [44] A g 4 7 o 1+0.6 z - x - 0 . 7 1.21 ± 0 . 0 6 C R S present work Sn 50 o.i^5 1.03 ± 0 . 0 8 C R S Armstrong et al. [44] P b 82 < 0.2 0 .60 ± 0 . 0 5 C R S Armstrong et al. [44] Table 6 .2: Values of gp/gA and Ry as determined by the integral method. " G O K S " refers to Gmi t ro et al. [83]; " C G " refers to Chr is t i l l in and Gmi t ro [68]; " G O T " referes to Gmi t ro et al. [58]; " R N " referes to Roig and Navarro [69]; " C " refers to Chr is t i l l in [67]; and " C R S " refers to Chr is t i l l in et al. [55]. Chapter 6. Discussion 66 QP/QA 15 10 5 0 6 0 20 A GOKS o CG • GOT • RN • C • CRS PCAC _i i i_ A 40 60 Z 80 100 Figure 6.4: gp/gA as a function of atomic number. Plot ted values are from table 6.2 (integral method results). The acronyms in the symbol legend correspond to the different theoretical models used to extract gp/gA- They are given in full in the caption to table 6.2. Chapter 7 Conclusions A test of the P C A C prediction of the induced weak pseudoscalar coupling constant, gp, has been made by measuring the photon spectrum from R M C in nuclei. Values of gp and the gp-dependent FLy (the R M C / O M C ratio for k > 57 M e V ) are extracted by comparing experimental results wi th theoretical predictions. Due to inconsistencies between different theoretical calculations, a meaningful test of P C A C using nuclear R M C is not yet possible. FLy has been measured in six nuclei: oxygen, aluminum, silicon, t i tanium, zirconium, and silver. The present results for a luminum and silicon agree with previous results [44], and indicate a sensitivity of FLy to Paul i blocking. The present FLy result for T i , when compared to previous results for Ca , further shows a sensitivity of FLy to Paul i blocking. The present FLy result for 0 is ~ 25% lower than previous results, but one previous result [43] may have an overestimated beam rate-dependent correction. The present results for zirconium and silver are consistent wi th previous high Z results, and indicate either a Z dependent quenching of gp to zero in heavy nuclei (Chris t i l l in et al. [55]), or Z independent effects which do not alter the form of the Goldberger-Treiman relation given i n E q . (1.29) (Fearing and Welsh [70]). 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Kamalov, F . Simkovic, and A . A . Ovchinnikova. Nucl Phys. A507, 707 (1990). Appendix A Quantities used in the calculations i ? 7 and gp/gA 74 Appendix A. Quantities used in the calculations of FL, and gp/gA 75 0 A l Si Nexp i V 7 > 5 7 2365 ± 49 2787 ± 53 3028 ± 55 jyexp ^ vstops 1.29881 x IO1 1 5.41003 x I O 1 0 4.01114 x I O 1 0 0.9613 ± 0.0006 0.9705 ± 0.0010 0.9730 ± 0 . 0 0 1 0 €cs 0.976 ± 0.005 0.976 ± 0.005 0.976 ± 0.005 er 0.943 ± 0.004 0.968 ± 0.005 0.968 ± 0 . 0 0 5 ec 0.957 ± 0.002 0.948 ± 0.004 0.940 ± 0.006 Cp 0.940 ± 0.013 0.927 ± 0.009 0.910 ± 0 . 0 0 7 Cbm 1.022 ± 0 . 0 0 4 1.019 ± 0 . 0 0 3 1.019 ± 0 . 0 0 2 CREVMOC 0.99558 ± 0.00004 0.99788 ± 0.00002 0.9771 ± 0.0002 C0 0.987 ± 0 . 0 0 3 0.991 ± 0 . 0 0 2 0.9901 ± 0 . 0 0 1 6 Cu 1.007 ± 0 . 0 0 2 1.0064 ± 0 . 0 0 1 9 1.0053 ± 0 . 0 0 1 3 Cm 1.026 ± 0 . 0 0 5 1.022 ± 0 . 0 0 4 1.021 ± 0 . 0 0 3 Cmd 1.0025 ± 0 . 0 0 0 3 1.00115 ± 0 . 0 0 0 1 9 1.0015 ± 0 . 0 0 0 3 empfrac 0.024 ± 0.008 — 0.050 ± 0 . 0 1 4 Acc 0.0068 ± 0.0006 0.0070 ± 0.0005 0.0066 ± 0.0006 F 0.92 ± 0.03 0.95 ± 0.03 0.95 ± 0 . 0 3 T i Zr A g Mexp •<V7>57 1644 ± 41 876 ± 30 743 ± 27 Ajexp ^ * stops 2.68948 x I O 1 0 1.49912 x I O1 0 1.56968 x I O 1 0 Zv 0.908 ± 0.003 0.912 ± 0 . 0 0 4 0.913 ± 0 . 0 0 5 ^cs 0.976 ± 0.005 0.976 ± 0.005 0.976 ± 0.005 er 0.983 ± 0.005 0.983 ± 0 . 0 0 2 0.985 ± 0.004 ec 0.951 ± 0 . 0 0 7 , 0.933 ± 0.003 0.929 ± 0.007 Cp 0.869 ± 0 . 0 1 1 0.78 ± 0.03 0.758 ± 0 . 0 1 7 Cbm 1.022 ± 0 . 0 0 6 1.0208 ± 0 . 0 0 1 4 1.15 ± 0 . 0 5 CREVMOC 0.98131 ± 0 . 0 0 0 1 8 0.9597 ± 0.0004 0.9682 ± 0.0003 C0 0.988 ± 0.002 0.9887 ± 0 . 0 0 1 0 0.983 ± 0 . 0 1 1 Cu 1.017 ± 0 . 0 0 5 1.0117 ± 0 . 0 0 1 1 1.008 ± 0 . 0 0 9 Cm 1.030 ± 0 . 0 1 0 1.0300 ± 0 . 0 0 1 7 1.017 ± 0 . 0 1 3 Cmd 1.0003 ± 0 . 0 0 0 2 1.0004 ± 0 . 0 0 0 2 1.0003 ± 0 . 0 0 0 2 empf rac — — — Acc 0.0071 ± 0.0003 0.0070 ± 0.0004 0.0063 ± 0.0003 F 0.95 ± 0.06 0.93 ± 0.04 1.03 ± 0 . 0 4 Table A . l : Quantities used in the calculations of FL, and gp/gA for 0 , A l , S i , T i , Zr, and A g . N^57 is the observed number of photons (> 57 M e V ) after al l software cuts, and N^gps is the raw (uncorrected) number of beam particle stops in the target. See sections 4.2.2, 4.2.3, 4.3.2, 4.4, and 5.2 for details on all quantities. Appendix B Derivation of equations (1.8) and (1.9) Comparing E q . (2.213a) of Marshak, Riazuddin, and Ryan ( M R R ) [10] wi th E q . (1.3) in the present thesis, and using E q . (2.228) of M R R makes it clear that the Jx(x) in E q . (2.229) of M R R is Jx\x) in the notation of the present thesis. Therefore, V\(0) and A\{0) in Eqs. (2.231a) and (2.231b) of M R R are actually VA +(0) and -A\(0) in the notation of the present thesis (compare E q . (2.229) in M R R with E q . (1.6) of this thesis). Hence, from Eqs. (2.231a) and (2.231b) of M R R with an obvious change in notation for the form factors, we have V*{x) = i$n(x)[Fv(q2)7a - FM(q2)aaXqX - iFs(q2)qa}^p(x) ( B . l ) Al(x) = -i^n(x)[FA(q2)7a75 + iFP(q2)l5qa - FT(q2)75aaXqX]rPp(x). (B.2) From M a n d l and Shaw [6] we have fl = 7o7*7o, 7s = 7s, il = 7s = h ( B -3) {7« ,7s} = 0, craX =-[ya,1x\, (B.4) ? = ^ 7 o , (B.5) and using the reality of the form factors (see section 1.1) we derive Eqs. (1.8) and (1.9): Va(x) = - ? ^ ) [ ^ ( ? 2 ) 7 l - ^ M ( ? 2 ) ^ A + JF5(92)9a]7oVn(x) (B.6) = -iV'J(a:)[Fy( (72)7o7a7o - FM{q2hoo-a\1oqX +iFs(q2)loloqaho^n(x) (B.7) = -$p(x)[Fv{q2)ia - F M ( q 2 ) a a X q x + iFs(q2)qo]M*) (B.8) 76 Derivation of equations (1.8) and (1.9) 77 Aa(x) = l^l(x)[FA(q2)47l-iFP(q2)4qa-FT(q2^ (B.9) = iV 'J(^)[-^4(9 2)757o7«7o ~ iFP(q2)-y51o-yoqa -FT(q2)^QO-axiQi5qX]lo^n{x) (B.10) = z ? p ( x ) [ F A ( « 3 2 ) 7 a 7 5 + i J Fp ( g 2 )75«?« + i ; T (9 2 )75^A9 A ]^n (a; ) ( B . l l ) 

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