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Muon capture in ²⁸Si Moftah, Belal Ali 1996

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M U O N C A P T U R E I N 2 8 S i By B E L A L A L I M O F T A H B . S c , University of Winnipeg, 1988 M . S c , University of British Columbia, 1991 A THESIS S U B M I T T E D IN 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 OF D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Physics We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A February 1996 © Belal A l i Moftah, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) A b s t r a c t A measurement has been made of the angular correlation of the neutrino wi th a specific nuclear de-excitation 7-ray following exclusive muon capture on 2 8 S i , i n order to determine the size of the induced-pseudoscalar coupling constant gp of the weak hadronic current. The correlation is observed v ia the Doppler-broadened line shape of the 1229 keV 2 8 A l 7-ray, which is measured using a pair of Compton-suppressed intrinsic Ge detectors. Significant background suppression is achieved through the use of a coincidence technique in which the 1229 keV 7-ray of interest is 'tagged' by the subsequent 942 keV 7-ray in the cascade, which is detected in an array of 24 N a l ( T l ) scintillators. Through a detailed attention to the detectors' response functions as well as the use of background-subtracted spectra, the 7 — 1/ correlation coefficient a is found to be 0 .360±0 .059 i n good agreement wi th a recent measurement at Dubna . The result obtained yields the value of gp/gA = 0 .0±3 .2 when compared to the latest calculation of the 7 — u angular correlation, suggesting a massive quench-ing of the induced-pseudoscalar coupling constant in 2 8 S i i n comparison wi th the value expected for a free nucleon. However other available calculations give the values gp/gA — 5 .3±2 .0 and gp/gA = 4 . 2 ± 2 . 5 , but the model-dependence of these intr iguing results has yet to be assessed fully. A measurement of the correlation coefficient a of the 2171 keV ( 1 + —> 2 + ) 7-ray has solved the enigma of the unphysical result that was found by a previous experiment. In addit ion, the lifetime of the 2201 keV 2 8 A l level has been measured i i C o n t e n t s Abstract ii List of Tables vii List of Figures ix Acknowledgements xii 1 Introduction 1 1.1 The M u o n 1 1.2 Mesic Atoms 3 1.3 Weak Interactions 8 1.3.1 The Weak-Interaction Hami l ton ian 10 1.4 Constraints on the Weak Coupl ing Constants 13 1.5 Nuclear Renormalizat ion of gp 18 1.6 Observables Sensitive to the Induced Pseudoscalar Coupl ing 19 1.6.1 M u o n Capture by Hydrogen 20 1.6.2 M u o n Capture by Complex Nuclei 22 2 The 7 — 1/ Angular Correlation 27 2.1 Introduction 27 2.2 Fujii-PrimakofF Approximat ion 29 2.3 Beyond the F P A 30 2.4 F u l l Calcula t ion Models 34 2.4.1 M o d e l I : Ciechanowicz 34 iv 2.4.2 M o d e l II :Parthasarathy and Sridhar 35 2.4.3 M o d e l III : K u z ' m i n at al 42 2.5 The Me thod of Measurement 43 2.6 Anc i l l a ry Reactions 46 3 Description of the Experiment 51 3.1 Beam Product ion and the M 9 B Channel 51 3.2 Exper imental Arrangement 52 3.3 Detection System 55 3.4 Electronics and Da ta Acquis i t ion 58 3.4.1 Telescope Logic 59 3.4.2 The Compton-suppressed German ium Logic 59 3.4.3 The N a l ( T l ) Logic 63 3.4.4 Strobe Logic 63 3.4.5 Da ta Acquis i t ion Cont ro l 64 4 Technical problems 67 4.1 Basic Interactions in 7-ray Detectors 67 4.2 Neutron Effects i n Ge-detectors 70 4.3 Coincidence Technique 74 4.4 The Detector Response Funct ion 79 4.5 Slowing-Down Effects 85 4.6 F in i te Sol id-Angle Effects 88 4.7 The Peak F i t t i n g Programs ' 89 5 Data Analysis 91 5.1 Introduction 91 5.2 Cuts 92 v 5.2.1 T ime of the M u o n Cu t 92 5.2.2 Compton-Suppression Cut 94 5.2.3 Rise T i m e Correct ion 96 5.2.4 Time-Coinc idence Cut 102 5.2.5 Energy-Ga ted Coincidence 102 5.3 Acceptances of Detectors 106 5.4 Cascade Feeding 109 5.5 Background Subtract ion 123 5.6 Parameter Recapi tulat ion 126 5.7 Analysis of the Doppler-broadened Peaks 127 5.7.1 The 2171 keV line 127 5.7.2 The 1229 keV line 130 5.7.3 The 1229 keV and the 2171 keV Simultaneous F i t 132 6 Discussion of Results 134 6.1 The 7 — v Angula r Correlat ion 134 6.2 The Induced Pseudoscalar Coupl ing 139 7 Conclusions 146 Bibliography 150 Appendix 160 A T h e Peak-Fitt ing Computer P r o g r a m Listing 160 v i L i s t o f T a b l e s 1.1 Properties of the muon 2 1.2 Theoretical and experimental weak coupling constants for the ele-mentary process of muon capture 17 1.3 Summary of values of gp/gA as determined from measurements of muon capture i n hydrogen 22 1.4 Summary of values of gp/gA determined by comparing experimental results wi th theoretical predictions of radiative muon capture ( R M C ) in complex nuclei 24 1.5 Summary of values of gp/gA as determined from measurements of muon capture ( O M C ) i n complex nuclei 25 2.1 Neutron multiplici t ies following muon capture on Si 50 3.1 Properties of the N a l ( T l ) counters 57 4.1 Neutron induced 7-ray lines in germanium isotopes 74 4.2 Chi-squared fit to the 1332 keV 6 0 C o peak for different response func-tions 82 5.1 Acceptances of the N a l ( T l ) coverage for the two Ge detectors 106 5.2 Muon ic X - r a y acceptance data for the G e l detector 108 5.3 Muon ic X - r a y acceptance data for the Ge2 detector 109 5.4 Input parameters needed i n the least-squares fitting function for the two Ge detectors 126 5.5 Results of the best fit to the 2171 keV 7-ray lines i n both Ge detectors. 129 5.6 Results of the ind iv idua l fits to the 1229 keV 7-ray peak for each Ge detector 130 5.7 Effect of the instrumental resolution of the detectors on a 130 v i i 5.8 Effect of the different 'side-band -background subtractions on a. . . . 131 5.9 Results of the best fit to a l l four spectra: the 2171 keV and the 1229 keV 7-ray lines in both Ge detectors 132 6.1 A comparative summary of the the measurements of the 7 — 1/ angular correlation experiments 138 6.2 Summary of the extracted values of gp/gA from the 7 — ^ angular correlation experiments 142 v m L i s t o f F i g u r e s 1.1 Typ ica l muonic cascades 6 1.2 The single pion exchange diagram 16 1.3 gp/gA a s determined from the rate of radiative muon capture mea-surements 25 2.1 The mult ipolar i ty factor F as a function of the mix ing ratio, S = E2/M1 33 2.2 Dependence of the amplitude ratio x on gp/gA f ° r the 1229 keV transit ion 36 2.3 Dependence of a on gp/gA f ° r the 1229 keV transit ion 37 2.4 Dependence of [3i on gp/gA f ° r the 1229 keV transit ion 38 2.5 Dependence of B2 on gp/gA for the 1229 keV transit ion 39 2.6 Kinemat ics of muon capture reaction on 2 8 S i 44 2.7 The shape of a Doppler-broadened 7-ray dis t r ibut ion and the effect of the 7 — v angular correlation coefficient, a 45 2.8 2 8Si(7r,7) 2 8Al continuum subtracted spectrum 47 2.9 2 8 S i ( d , 2 H e ) 2 8 A l excitation energy spectrum 48 2.10 2 8 S i ( p , n ) 2 8 P excitation energy spectrum 48 3.1 The T R I U M F cyclotron and beamlines 53 3.2 Layout of the T R I U M F M 9 channel 54 3.3 Exper imental setup 56 3.4 Complete electronic logic diagram 60 3.5 Timings and definitions of events for G e l and associated electronics. 61 3.6 A schematic flow of the data through the D a t a Acquis i t ion System. . 65 ix 4.1 Comparison of the measured energy spectra for a H P G e detector and a N a l ( T l ) scintil lator (source data) 71 4.2 Gamma-ray spectrum from muons stopping in a silicon target. . . . 73 4.3 2 8 S i gamma-ray energy spectrum, in the vicini ty of the 1229 keV peak. 75 4.4 The product ion of the 1229 keV gamma ray i n 2 8 A 1 and the coinci-dence technique 77 4.5 2 8 S i gamma-ray energy spectra before and after the imposi t ion of the coincidence requirement 78 4.6 Various components of the response function fitted to the 1332 keV 6 0 C o peak 81 4.7 Energy dependence of the parameters of the response function for G e l . 84 4.8 Stopping-power curves for 2 8 A 1 ions in n a ' S i medium 86 4.9 Relationship between the stopping-power and the extracted lifetime from the 2171 keV 7-ray line 87 5.1 T i m e of the muon spectrum for G e l wi th a Si target 93 5.2 Ge2 Compton suppressed and unsuppressed spectrum typical for muon capture on a Si target 94 5.3 Da ta removed by the N a l ( T l ) Compton suppressor 95 5.4 Compton suppressed and unsuppressed 6 0 C o spectrum of G e l 95 5.5 G e l leading edge spectra corresponding to different discriminator thresholds for a typical /j.Si run 97 5.6 Ge2 leading edge spectra corresponding to different discriminator thresholds for typical pSi run 98 5.7 Plot of the corrected and uncorrected centroid channel of the 1173 keV 7 rays as a function of their rise-time for the G e l detector. . . . 100 5.8 Plot of the corrected and non-corrected centroid channel of the 1173 keV 7 rays as a function of their rise-time for the Ge2 detector. . . . 100 5.9 Typ ica l t iming coincident spectra for one of the N a l ( T l ) detectors. . 101 5.10 Time-coincidence spectrum for a N a l ( T l ) annulus segment 103 5.11 A dual-peak fit to the 6 0 C o 7-ray lines in B A R 3 105 5.12 Acceptance curve for the G e l detector 110 5.13 Acceptance curve for the Ge2 detector 110 5.14 Cascade feeding of the 2201 keV 2 8 A l level diagram 112 5.15 Part of the 2 8 S i 7-ray energy spectrum for the G e l detector 113 5.16 Parts of the 2 8 S i 7-ray energy spectrum for the G e l detector showing the 903 keV and 3075 keV peaks 115 5.17 2 8 S i 7-ray energy spectra obtained wi th a N a l detector, w i th and without coincidence requirement 117 5.18 2 8 S i 7-ray energy spectrum of N a l overlaid on a G e l detector singles spectrum 118 5.19 Singles 2 8 S i gamma-ray energy spectra obtained wi th the two Ge detectors, i n the vicini ty of the Doppler-broadened 2171 keV peak. . 121 5.20 2 8 S i coincidence energy spectra obtained wi th the two Ge detectors, in the vicini ty of the Doppler-broadened 2171 keV peak 122 5.21 2 8 S i gamma-ray energy spectra of the G e l detector: (a)singles and (b)coincidence wi th the 'side-band' background subtraction 124 5.22 2 8 S i gamma-ray energy spectra of the Ge2 detector: (a)singles and (b)coincidence wi th the 'side-band' background subtraction 125 5.23 The interplay of the slowing-down, the angular correlation, and the instrumental resolution effects 128 5.24 The best simultaneous fit to al l four spectra 133 6.1 The measured 7 — 1/ angular correlation coefficient a compared to the theoretical calculations 141 x i Acknowledgements Firs t and foremost I acknowledge the Help of The Creator A l l a h s.w. without W h o m none of this would be possible. I would like to extend my sincere gratitude and appreciation to my supervisor Professor D a v i d F . Measday for his guidance, advice and encouragement throughout this work. I am greatly indebted to D r . D a v i d S. Armst rong who acted as the spokesman of the experiment as well as my unofficial second supervisor. Special thanks are given to Drs . T . P . Gorringe and S. Stanislaus. Thei r suggestions and discussions during the entire course of this work were well appreciated. I would like to thank my fellow graduate students Ermias Gete and Trevor Stocki as well as J . Bauer, J . Evans, B . L . Johnson, S. K a l v o d a , R . Porter, B . Siebels, M . Makoto , R . Jaco t -Gui l la rmod and P. Weber for their assistance and contribution to the progress of the experiment. I would like also to thank Professors J . Deutsch, H . W . Fearing, M . D . Hasinoff, B . G . Turre l l and C . E . W a l t h a m for their advice and detailed reading of the manuscript. I am grateful, for financial support during my work, to the Secretariat of Scientific Research of L i b y a and to the Na tura l Sciences and Engineering Research Counc i l of Canada. Final ly , I would like to thank my family, and in part icular my wife Gharsa, for their continuous support and encouragement throughout this work. xn Chapter 1 Introduction 1.1 The Muon Muons were first discovered i n studies of cosmic rays wi th cloud-chambers and Geiger counters by Street and Stevenson (1937) [1] and Anderson and Nedder-meyer (1938) [2,3]. T w o years earlier, particles of approximately the muon mass had been postulated by Yukawa [4] as quanta of the nuclear field. However, it turned out that the observed cosmic-ray particles were not the Yukawa ones; for, although they have about the right mass, they do not interact strongly wi th nu-clei [5]. Shortly thereafter, the pion was discovered [6] in photographic emulsions. Subsequent experiments demonstrated that it was the pion, not the muon which is the Yukawa particle. The muon, it turned out, was the decay product of the pion. The properties of the muon can be summarized by describing it as a "heavy electron", for the only fundamental attribute that distinguishes a muon from elec-tron is its mass (about 207 times the electronic mass). We now realize that there are three charged leptons, the electron, muon and tau, which are each members of different families (generations). The only obvious difference between them is their masses. The muons are point-like leptons, which experience the electromagnetic and the weak interactions but not the strong interaction. Table 1.1 gives some properties 1 Table 1.1: Properties of the muon (after [7]). Spin 1 2 Mass m „ = 105.658389 ± 0.000034 M e V / c 2 Charge = q e M e a n lifetime r = (2.19703 ± 0.00004) x l O " 6 s Magnet ic moment fi = 1.001165923 ± 0.000000008 eh/2m„ of the muon. Muons are usually produced ( as i n the present work ) by the decay of charged pions, i.e. 7T+ -> ^+ + ^ (1.1) 7T~ —* fl~ + (1.2) The muon has been an important test particle not only in the various branches of physics, but i n several other science fields as well , see the review article of Scheck [8] and references therein. In the present work muons are used as probes of the weak interaction. Free muons normally decay into electrons and two neutrinos as follows p+ —y e+ + ue + (1.3) p~ —• e~ +Ve + (1.4) This three-particle decay scheme is consistent wi th the observational fact that the resulting electron spectrum is a continuum. Furthermore, the masses of the neutrinos must be small (when compared to their charged counterparts) as a direct consequence of momentum-energy conservation. In the rest frame of the decaying 2 muon, the max imum energy of the electron - when the two neutrinos escape in the same direction, opposite to that of the electron - is given by E= " e — v e = 52.83 MeV (1.5) and hence is consistent wi th the previous sentence, i.e. m „ e + m „ ~ 0. Today, upper l imits on these masses are m „ e < 4.5 eV [9] and m„ M < 0.27 M e V [10] as obtained from the t r i t ium experiments and the IT —• uv^ decay experiments respectively. Other muon decay modes, including forbidden lepton family number violat ing modes are listed, along wi th their branching ratios [7], below u~ — • e - + ve + + 7 0.014 ± 0 . 0 0 4 (1.6) e" -rue + v» + e+ + e~ (3.4 ± 0.4) x 1 0 - 5 (1.7) e _ + i / e + F M < 0.012 (1.8) e-+7 < 4 . 9 x l O - 1 1 (1.9) e - + e + + e- < 1.0 x 1 0 " 1 2 (1.10) e ~ + 2 7 < 7.2 x 1 0 " u (1.11) 1.2 Mesic Atoms In a mesic -o r more generally exot ic- atom, a negatively charged particle replaces one of the orbi ta l electrons of an ordinary electronic counterpart. To date five such atoms have been successfully observed. These are pionic, kaonic, muonic, hyperonic and antiprotonic atoms. Mesic atoms have been known for more than four decades. The first experimental indicat ion of such atoms can be traced back to the work of Conversi et al. i n 1947 [5] who measured the ratio of nuclear absorption of negative muons in light elements. A t the same time Wheeler [11] and Fermi and Teller [12] theoretically argued that mesic atoms should exist since the atomic cas-cading time (~ 1 0 - 1 3 s ) is short compared wi th the lifetime of the particles involved. The first demonstration of the existence of mesic atoms was made using muons from cosmic radiat ion [13]. Mesic atom physics has become an important tool for understanding nuclear properties. It can also provide information about the elementary particles them-selves and their interactions. Indeed the field of mesic atoms involves molecular, atomic, nuclear and particle physics; see references [14,15,16] for applications of mesic atoms. T w o technological advances of recent years have reactivated the interest in this field. These are the establishment of high-flux meson factories (e.g. T R I U M F , L A M P F , SIN) and the development of high-resolution solid state detectors. The properties of the exotic atoms are not only similar to each other, but are rather closely related to those of the hydrogen atom. Th i s is due to the dominant role of the electromagnetic interaction. However, exotic atoms have two important characteristics considerably different from those of their electronic counterparts. These are consequences of the great difference in mass between the particles involved and the electron. For example, the lightest of these particles, the muon, is 207 times as heavy as an electron. These characteristics follow from the fact that -for the same quantum numbers- the energy levels (radii) of the orbits are (inversely) proport ional to the mass of the orbital particles. For example, the diameter (energy) of the muonic hydrogen atom is 1/207th (207 times) that of the electronic counterpart. Therefore, these particles spend more time inside the nucleus and hence are much better suited for probing nuclear properties than the atomic electrons. Al though the discussion below is specific for muonic atoms, which is rele-vant for this work, most of the general processes - apart from strong interaction phenomena- are essentially the same for a l l exotic atoms. Muons enter the target wi th energies of the order of tens of M e V (~20 M e V i n our case), and possess velocities (uM) greater than those of the valence electrons 4 (ye ~ ac, where a is the fine structure constant). They lose most of their energy through collision wi th atomic electrons and then come to a stop wi th in ~ 1 0 _ 9 s . Once the muon comes to a stop, it w i l l be captured at a few tens of eV [17] by a target atom into a high orbi tal angular momentum state, ejecting electrons and forming a muonic atom. The atomic capture is roughly described by the so-called "Z- law" [12] i n which the capture rate is taken to be proport ional to the nuclear charge, Z. Following its capture, the muon (within ~ 1 0 - 1 4 s ) w i l l be inside the K-she l l electron orbit at a pr incipal quantum number given approximately by n M ~ (m^/me)1/2 ~ 14. Since al l of the low-lying "muonic" states are unoccupied, the muon cascades down (within ~ 1 0 - 1 3 ) from n~14 to the Is quantum state. In this cascade, the muon w i l l interact wi th outer electrons and w i l l lose energy through Auger processes. However, as the transit ion energy increases rapidly (~ 1/n 3), this interaction is no longer important and E l radiative transitions (muonic X-rays) dominate; see Figure 1.1. Effectively a l l of the negative muons captured i n the atomic orbits reach the Is orbit. This is consistent wi th the previously stated time-scales needed for the formation of muonic atoms. Once a negative muon reaches the lowest Is state, it either decays (equation 1.4) or gets captured by the nucleus v ia the elementary reaction, \i~ + P -> n + (1-12) which in the nuclear environment becomes pT + (A, Z) -> (A, Z - 1)* + u„ (1.13) The daughter nucleus (A,Z-1) , is often left i n one of its excited states, (desig-nated by the asterisk) usually a giant resonance state. 5 A distinct feature of the rate of reaction (1.13) is its strong Z-dependence. Th i s was observed by the experiment of Conversi et al. [5] and its form was later unravelled by Wheeler [18]. In a simple model, he assumed that al l protons, Z, i n the nucleus can interact independently wi th the p and that the interaction is proport ional to the probabil i ty of finding the p, i n the nucleus (|Vv(0)|2)- Now since the p is observed from the Is state which has, for hydrogenic atom, the wave function [19] Vv(r) = y/zyiralexp [-Zr/a,] (1.14) where a M is the first muonic Bohr radius, the capture rate is K ^ Z A e } } (1.15) where an effective charge (Z^j) was used to account for the finite nucleus size. Al though this Z4 law is a good approximation for light nuclei, theoretical muon capture rates are usually obtained from the Primakoff formula [20] or its extension, the Goulard-Primakoff formula [21]. The rate of the capture ( A c ) and decay (Aj) are connected by At = Ac + QAd (1.16) where At (=7-) is the total disappearance rate and Q is the Huff factor to take ac-count of the fact that the muon is bound i n the atom (see the Nuclear Muon Capture review article by Mukhopadhyay [22] for the differences in the decay rates of the free and the bound muon decays and their causes) . Ut i l i z ing the C P T theorem, i.e. Ad = -^p = - ^ J T , the measurement of the nuclear capture rate ( A c ) amounts simply to the determination of the bound muon lifetime (T m -) i n the relevant material . This had been demonstrated at T R I U M F by Suzuki et al. [23,24] who measured the lifetime of p~ in 50 elements plus 8 isotopes and deduced the associated capture rates. A more recent experiment [25] improved the data for isotopes of uranium and 7 neptunium. For example, the mean lifetimes of muons i n C , Si and P b are 2026.3, 756.0 and 72.3 ns respectively. 1.3 Weak Interactions B y now the dist inction -at low energy- among the four known fundamental interactions is established by both the large differences i n their relative strengths as well as by their distinct properties. T w o of these interactions whose effects are evident i n everyday life are the electromagnetic and the gravitat ional ones i n contrast to the other two, namely the strong and the weak interactions. The interaction strengths are usually expressed in terms of dimensionless cou-pl ing constants. Roughly speaking, the weak interaction is 10 9 and 10 1 2 times weaker than the electromagnetic and the strong interactions respectively (the fourth inter-action, gravity, is completely negligible at the level of particle physics since it is some thir ty orders of magnitude weaker than any other interaction). The appar-ent weakness of the weak interaction is ascribed to the very short range associated wi th its massive exchange particles, and Z° (and their couplings to leptons and quarks). Al though these (three) processes appear to be different, their mathematical formulation is very similar. They are a l l described by gauge theories; i.e. theories i n which fermions interact by the exchange of spin 1 gauge bosons. In fact, advances toward the 'ul t imate ' unification of these interactions are underway. Unl ike strong interactions, the weak interactions can involve both hadrons as well as leptons. Furthermore, the weak interaction operates at approximately the same strength among these particles, a phenomenological observation known as the universality of the weak interaction [22,26]. Other properties that distinguish the weak interaction include its range and 8 behaviour under symmetry principles. W i t h i n the Yukawa picture, the range of an interaction, R , is associated wi th the reciprocal of the mass of its propaga-tor. Thus Rweak ~ Mw ~ 1 0 ~ 1 8 m and Rstrong ~ rn'1 ~ 1.4 x 1 0 _ 1 5 m while the range of the electromagnetic interaction, wi th M 7 = 0 , is infinite. In contrast to the electromagnetic and strong interactions, the weak interaction violates some of the symmetry principles and conservation rules such as the discrete operations P (par-i ty) , C (charge conjugation) and T (time reversal) . For further discussions of these principles and their experimental status, see references [27,28]. The weak interactions were observed about a century ago through the discov-ery of radioact ivi ty by Becquerel. However, the establishment of the weak interac-t ion as a separate and independent process was rather gradual. In 1934 Fermi [29] proposed the first theory of the weak interaction based on a four-fermion point in -teraction to describe the beta decay of nuclei, e.g. n —> p + e~ + T7e. Th i s theory remains to this day, wi th a few modifications. It took about 14 years from the orig-ina l formulation of Fermi on nuclear (3 decay before the up-e universali ty" sprang forth through the work of Pontecorvo [30] and P u p p i [31] leading to the hypothesis of the Universal Fermi Interaction. Th i s has now been extended to the r as well . Dur ing the next two decades, theoretical effort was directed toward finding out the correct structure of the weak interactions. Fermi's first theory was based on direct four-fermion coupling of vector type only. This interaction was later gen-eralized (Gamow and Teller, 1936 [32]) to a linear combination of the five bilinear quantities: vector (V) as well as scalar (S), pseudoscalar (P) , axia l vector (A) and tensor (T) . In contrast to the original V form which allows transitions between nu-clear states of equal angular momentum (Fermi transitions), this general interaction could couple nuclear states differing by one unit of angular momentum (Gamow-Teller transitions). The turning point in the construction of the weak interaction came about, as 9 a possible way out of the so-called (6 — r ) puzzle, wi th the questioning of pari ty conservation i n the weak interaction by Lee and Yang [33] and later by the verifi-cation of its violat ion by W u et al. [34]. Subsequent experiments established the so-called universal V - A (vector-axial vector) character of the weak interactions. In this framework, the odd and even pari ty amplitudes have roughly the same magni-tudes and give what is called the principle of maximum parity violation. The V - A theory is not the whole t ruth, rather its success lies i n explaining a large class of weak phenomena. It is now generally regarded as a part icular case of the extremely successful Weinberg-Salam-Glashow electroweak theory [35,36,37] (also known as the Standard Mode l of the electroweak theory) in the l imi t of low energy. The electroweak theory is a renormalizable gauge theory unifying weak and electromagnetic interactions in one mathematical framework. Its success was highlighted by the prediction of the neutral weak current discovered at C E R N in 1973 [38], and by the prediction of the W and Z massive gauge bosons discovered 10 years later, also at C E R N [39]. The Standard M o d e l now also encompasses the description of the strong interactions by quantum chromodynamics ( Q C D ) . 1.3.1 The Weak-Interaction Hamiltonian As noted above, weak processes, such as muon capture, at low momentum transfer q (q2 ~ m 2 <C Myy), are adequately described by the V - A theory and hence wi l l be used in this section. W i t h i n this picture, the weak interaction Hamil tonian , at a given space-time point(x) , has the form H{x) = -^=j{(x) J\x) + h.c. (1.17) v 2 where G — 1.16639(2) x 1 0 - 5 G e V - 2 [40] is the effective weak coupling constant and J A is the weak four-current which can be decomposed into hadronic and leptonic components, v iz . : 10 JA(x) = J\ + J[ (1.18) Accordingly, weak processes are usually classified as • pure leptonic, where only leptons are involved, • semileptonic, where both leptons and hadrons are involved, and • hadronic, in which only hadrons are involved. M u o n capture is one example of a semi-leptonic process and is the one considered i n this work. T h e leptonic current The leptonic current i n muon capture has the V - A structure, wi th 75 = 27o7i7273 and where ipj and j \ are the lepton fields and the Dirac 7-matrices respectively. The (1 — 75) i n J[ automatically selects left-handed leptons consistent wi th the fact that weak interaction couples only to left-handed particles (and right-handed anti-particles). The V - A form of the leptonic current has been stringently tested using purely leptonic processes, such as muon decay. T h e hadronic current The details of the hadronic weak current, are not as clearly established as that of the leptonic one. This is due to the presence of the strong interaction which induces extra structure i n the hadronic weak currents. In fact, most muon capture experiments are concerned wi th the unravelling of this induced structure. (1.19) 11 In terms of the V - A structure and the Ge l l -Mann-Cab ibbo universality hy-pothesis [41,42], the hadronic current for the semi-leptonic weak process of muon capture can be writ ten as j£ = cosOc(Vx-Ax) (1.20) wi th V\ = iip. and o\v = !(7A7" — where M/v and m M are the nucleon and lepton masses respectively. The angle 6C is the Cabibbo angle, introduced to account for the mix ing between the quark generations, i.e. to account for the different rates in the strangeness conserving and non-conserving weak decay processes. The ga are coupling "constants" which are functions of q2 (q\ — n\ — p\, where n\ and p\ are the neutron and the proton 4-momenta respectively). The terms gv and gA are the conventional vector and axial vector coupling constants. These two are the only ones that contribute in the q2=0 l imi t , such as in /3-decay. The other four coupling constants gM, gs, gp and gj are the induced ones. They measure the strength of the induced weak magnetic (o-\„), the scalar, the pseudoscalar (75) and the tensor(<7Ai,75) currents respectively. The consequences of the induced strong interaction 'dressing' are a change in the strength of the conventional components of the weak current and the appearance of new components, the contributions of which are proport ional to the momentum transfer (q) and consequently contribute only negligibly in /?-decay and electron capture, but show up clearly i n muon capture where the 4-momentum transfer can be much larger. . 9M . .gs 2MN m „ (1.21) l9P S U 7 A 7 5 H 75tfA + *9T m, 2M, N (1.22) 12 1.4 Constraints on the Weak Coupling Constants The determination of weak couplings is based upon several general theoretical constraints. A brief description of these constraints along wi th some experimental evidence for their validity is given here. T i m e r e v e r s a l invariance of the weak interaction amplitude dictates that a l l coupling constants i n equations (1.21,1.22) are real 1 . Since the strong interaction is invariant under G - p a r i t y (combined charge conjugation and isospin invariance) , the induced currents i n V\ and A\ are expected to have a common G-pari ty transformation. However, the gs and gx induced cur-rents transform wi th opposite G-par i ty to the other vector and axial vector terms respectively and hence must vanish, i.e., gs = 0, (1.23) 9T = 0. (1.24) These two currents are classified by Weinberg [43] as "second-class" currents as opposed to the other "first-class" currents. The "second-class" currents have been investigated [44,45], however, no convincing evidence of their existence has been found 2 [22]. Up-to-date l imits on these currents are given by Grenacs [46]. Another useful constraint on the weak coupling constants is the c o n s e r v e d v e c t o r c u r r e n t h y p o t h e s i s ( C V C ) postulated by Feynman and G e l l - M a n n [47]. In direct analogy to the conservation of electric charge i n electromagnetic theory, it postulates that the vector part of the weak current is conserved and hence the vanishing of its divergence, i.e., dxVx = 0. (1.25) l r The small time reversal noninvariance implied by C P violation in the neutral kaon system (at the level of 10~3) is entirely negligible here [27]. 2Since gs — 0 is given by C V C also, a non-vanishing gr would be a test of the existence of "second-class" currents and consequently that of G-parity violation. 13 A p p l y i n g directly to equation (1.21) would give 9s = 0, (1.26) reinforcing the exclusion of "second-class" currents. Furthermore, C V C relates the weak vector and the isovector electromagnetic currents. Th i s allows the weak magnetic coupling constants, gv and gM, to be expressed i n terms of electromagnetic form factors which can be measured wi th precision i n electron scattering experiments. A t low energy (q2 —> 0) the calculated values are C V C is contained wi th in the Standard M o d e l and has been well supported by experimental evidence [48]. A number of tests have been applied, a l l of which together strongly confirm C V C at the level of 10% or so [49,50]. For example, Deutsch et al. [51] have used the par t ia l capture rate i n 1 6 0 to the 1~ state i n 1 6 N to test the C V C hypothesis, and found mutua l agreement wi th in 13%. Fur-thermore, the branching ratio of x+ beta-decay, ( 1 . 0 2 5 ± 0 . 0 3 4 ) x l 0 - 8 , was found to agree well w i th the calculated C V C prediction, 1 .07x l0~ 8 ; see [7] and references therein. More recently, the Chalk River collaboration [52] have produced a precise measurement of the 1 0 C superallowed Fermi /?-decay branch whose Ft-value, when added to the previously obtained F t values from eight other nuclei, removed any discernible trends wi th the nuclear charge Z. Final ly , the p a r t i a l l y c o n s e r v e d a x i a l v e c t o r c u r r e n t h y p o t h e s i s ( P C A C ) [53,54] places restrictions on the other remaining coupling constants, gA and gp. It states that the weak hadronic axial current (A\) is only conserved for massless pions gv = 1.0 (1.27) gM = 3.706. (1.28) and can be related to the pion field, (f)v, as follows: (1.29) 14 The P C A C hypothesis, applied to the expression for A\, yields the relation 2MN9A - q2^ = ^<Gf" (1.30) between the weak axial and weak pseudoscalar coupling constants, the measured pion decay constant (/„.) and the strong coupling constant for the pion-nucleon vertex (GVNN)- I n the low energy l imi t (q2 —> 0), this relation reduces to the Goldberger-Treiman [55] estimate for the axial vector coupling constant: gA = f ^ N N = 1.32 ± 0.02 (1.31) where G v N N = 13.40 ± 0.8 [56] and fv = 130.7 ± 0.1 ± 0.36 M e V [7] have been used. This estimate is slightly different from the measured value of gA = 1.2573 ± 0.0028 [7]. The 5% difference between the two values is called the Goldberger-Treiman discrepancy and is addressed in references [56,57]. The Goldberger-Treiman relation can be used to yield a prediction for the induced pseudoscalar coupling,i.e. ' 2MNm)1' gp = gA- (1.32) q2 + ra2_ Clearly, gp is a strong function of q2 and, furthermore, is dominated by the pion pole, indicat ing that the pseudoscalar contr ibut ion is predominantly due to the single pion exchange between the leptons and hadrons; see Figure 1.2. For the muon capture process on the proton (equation 1.12), the two body kinematics fix the four-momentum transfer at q2 = 0 .88m 2 , (1.33) which when substituted into equation (1.32) yields the Goldberger-Treiman value, viz . , gP = 7.1gA = 8.9. (1.34) 15 Figure 1.2: The single pion exchange diagram (the source of the weak pseu doscalar coupling). The enlarged vertex represents the compos ite quark structure of the hadronic current. 16 Table 1.2: Theoretical and experimental weak coupling constants for the elementary process of muon capture. Coupl ing Theoretical Exper imenta l Technique constant prediction measurement used 9v 1.0 1.000 ± 0.0015 [61] fj, decay 9M 3.706 3.78 ± 0.22 [46] /?-decay, ^-capture 9s 0 0.0005 ± 0.0031 [62] /?-decay 9A 1.32 1.2573 ± 0.0028 [7] neutron 8-decay 9P 8.44 8.7 ± 1.9 [60] /^-capture in H 9T 0 0.06 ± 0.49 [62] /3-decay, A = 1 2 Invoking a weaker P C A C , Wolfenstein [58] extended the work of Goldberger and Tre iman and obtained gP = 8A (1.35) known as the Wolfenstein estimate and was recommended by Mukhopadhyay [22] a an input for the pseudoscalar coupling constant. More recently, a new calculation of gp [59] confirmed the work of Wolfenstein and gave the very precise prediction gP = 8.44 ± 0 . 2 3 (1.36) where the error comes mainly from the uncertainty in G^NN-W h i l e theoretical calculations of gp c la im accuracy of better than 3%, the weighted average of a l l experimental measurements wi th the free nucleon [60], namely gp/gA — 6.9 ± 1.5 -whi le i n accord wi th the theoretical predict ions- carries an uncertainty in excess of 20%. Moreover, the most accurate single measurement contributing to this average has a 43% error. Table 1.2 gives the calculated and measured values of the six weak coupling constants along wi th the techniques used. A s can be seen in this table, al l coupling constants, except gp, are well de-termined by experiment and in rather good agreement wi th the theory for the 17 elementary process of muon capture on the proton. Clear ly a more precise mea-surement of gp is needed for the free nucleon; see section 1.6.1 for a discussion of a recent measurement. For protons i n complex nuclei, the present situation for these couplings, espe-cially for the poorly-known gp, is complicated and is discussed in the next sections. 1.5 Nuclear Renormalization of gp A s a consequence of the C V C hypothesis, the weak hadronic vector current (V\) couplings, gv and gM, remain constant when the nucleon is embedded i n the nucleus. In contrast, the weak axial hadronic vector current (A\) couplings, gA and gp, may change reflecting changes i n the nucleon structure i n the nuclear medium. In fact, the possible modifications of the gp/gA i n the nucleus have motivated a fair amount of theoretical as well as experimental efforts. The axial vector and pseudoscalar components of the weak current, i n the nucleus, can be effectively modified in several ways [22,63,64]. The scattering of v i r tua l pions by other nucleons can reduce their effective range and thus alter their propagators. In addit ion, the induced polar izat ion of the nuclear medium around the pion-emitt ing nucleon can modify their interaction. These two effects can be accounted for by replacing the mass of the pion, mn, and the pion-nucleon ver-tex, GITNN-, of the pion field by effective mass and coupling respectively. Combin ing these two effects, along wi th other uncertain ones, Er icson and collaborators [63] ob-tained the following expressions for the effective weak axial vector and pseudoscalar components i n terms of the bare ones, 9 A = [1 + - )9A (1.37) and 2MN m A q2 + ml (1.38) 18 where an ~ —0.9 is interpreted as a measure of the nucleon polarizabil i ty and m x is the effective mass of the pion inside the nucleus. These effects would lead to dramatic renormalizations of the coupling constants i n the nucleus; i.e. g~A = 0.76gA (1.39) and gP = 0.35flfP (1.40) at the momentum transfer appropriate for muon capture. Some experimental deviations of gA in nuclei from the pr imit ive free-nucleon value have been interpreted as evidence for modification of the weak axial vector coupling. Examin ing the Gamow-Teller mirror /^-decays, W i l k i n s o n [65] deduced g~A = (0.899 ± 0 . 0 3 5 ) ^ . (1.41) Furthermore, the 'unobserved' Gamow-Teller strength i n the studies of the small an-gle (p,n) reactions [66], now being reinforced by analogous (n,p) measurements [67], has been interpreted i n terms of strong renormalization of gA i n complex nuclei; yielding g~A = 1.00 ± 0.002 [66,68]. The experimental si tuation of gp and its possible renormalization w i l l be dis-cussed next. 1.6 Observables Sensitive to the Induced Pseu-doscalar Coupling A s stated earlier, the sensitivity of some observables i n muon capture to the i l l -determined pseudoscalar coupling (gp), comes about due to the large characteristic momentum transfer, q, i n muon capture as compared to, say, /?-decay or e-capture. This can clearly be seen i n the axial vector matr ix element, equation (1.22), where the whole pseudoscalar coupling term is proport ional to q. Furthermore Since the 19 dominant contribution to the pseudoscalar coupling comes from the single pion exchange diagram, Figure (1.2), and is given by the Goldberger-Treiman relation equation (1.32), gp is a strong function of q2. W h i l e several gp-sensitive observables exist [69], one is faced wi th two difficul-ties in measuring them. Firs t ly , the outgoing particles (n,i/) in the capture process are hard to detect. Secondly, these observables are, to some degree, dependent on the nuclear structure, i.e. the in i t i a l and final wavefunctions. The only observables avoiding the second difficulty -a l though adversely affected by the Z 4 l a w - are the ones from muon capture on hydrogen. In the context of muon capture, processes (1.12) and (1.13) are referred to as ordinary muon capture ( O M C ) as opposed to their much less probable versions H~ + p^n + vtt + y, (1-42) and fi- + (A,Z) ( a , Z - 1)* + !/„ + 7 (1.43) known as radiative muon capture ( R M C ) . Apar t from its rari ty (branching ratio ~ 1 0 - 4 for heavy nuclei), R M C has several advantages over O M C in determining gp. W h i l e the four-momentum transfer (q2) is fixed at q2 ~0.88 m 2 i n O M C , it could approach - m 2 for R M C at the max imum photon energy. This in turn enhances the pseudoscalar contribution in equation 1.32 by up to a factor of 3.5 over that of O M C . One other advantage of R M C is the relatively straightforward detection of the 7 ray involved as compared to n and v. 1.6.1 Muon Capture by Hydrogen The advantage of the basic muon capture on hydrogen, in contrast to that in complex nuclei, is the absence of the many uncertainties arising from the treatment 20 of the nuclear response function. The observables measured i n O M C on hydrogen are the capture rates from the singlet (As) and the triplet (AT) hyperfine states [70,71] and the orthomolecular capture rate (AOM) [72,73] through the direct detection and measurement of the emitted neutrons, see Table 1.3. Despite the extremely low radiative branching ratio (7.9 x 10~ 8 [74]), ex-periment E452 at T R I U M F [75] has successfully measured, for the first t ime, the radiative muon capture on hydrogen. Thei r measured value of gp, determined from the photon energy spectrum, has just been reported as [74] gP = (10.0 ± 0 . 9 ± 0 . 3 ) f l u (1.44) where the first error includes statistical and systematic errors, while the second error is due to the uncertainty i n the ortho-para transit ion rate i n muonic molecular hydrogen, \ o p . It should perhaps be remarked that our group [76] is proposing to measure this transit ion rate on which the above R M C value of gp depends, yet to a lesser degree than the O M C on hydrogen, and for which there has only been one mea-surement [77,78], A o p = (4.1 ± 1.4) x 10 4 s _ 1 , which is significantly different than the theoretical prediction [79], A o p = (7.1 ± 1.2) x 10 4 s _ 1 . Depending on the value of XOP adopted, the T R I U M F R M C measurement (equation 1.44) is i n conflict either with the most accurate O M C measurement (and i n poor agreement wi th the other O M C measurements) or with the P C A C prediction; the later, i f taken at face value, would demand new physics of our understanding of semileptonic weak interactions [76]. The experimental results to date on the value of gp (as gp/gA) deduced from muon capture (both O M C and R M C ) i n hydrogen are summarized i n Table 1.3. This table represents the only information avaliable to date on the induced pseu-doscalar coupling constant for the free nucleon. 21 Table 1.3: Summary of values of gp/gA as determined from measurements of ordi-nary muon capture ( O M C ) in hydrogen. The values quoted for O M C are from the analysis of B a r d i n et al. [77]. Experiment 9P/9A OMC(neu t ron ) on l iquid H 2 [72] OMC(neu t ron ) on l iquid H 2 [73] OMC(neu t ron ) on gaseous H 2 [70] OMC(neu t ron ) on gaseous H 2 [71] OMC(elec t ron) on l iqu id H 2 [78] 4.8 ± 6 . 3 8.7 ± 3 . 4 8.2 ± 3 . 1 6.3 ± 4 . 7 5.6 ± 2 . 4 O M C W o r l d Average 6.9 ± 1.5 R M C ( p h o t o n ) on l iqu id H 2 [74] 10.0 ± 0 . 9 ± 0 . 3 In addit ion to providing the best values for gp, muon capture experiments i n hydrogen have provided convincing tests for the fi-e universality and of the V - A form of the weak interaction [22]. 1.6.2 Muon Capture by Complex Nuclei Due to the low capture probabil i ty in hydrogen (~ 1 0 - 3 ) , efforts - b o t h experimentally and theoretically- have concentrated on heavier nuclei 3 . In complex nuclei, the induced pseudoscalar interaction influence several experimentally-accessible observables. In nuclear R M C , these include the often measured observ-able, i.e. the R M C / O M C branching ratio as well as the photon energy spectrum, the photon asymmetry wi th respect to the muon spin direction, and the photon cir-cular polarizat ion. The 7 asymmetry observable is rather insensitive to the specific details of the nuclear model [80], but a l l previous measurements [81] suffer from too low statistics to set a useful l imit on gp, and hence do not show up i n Table 1.4. The review article by Gmi t ro and Truol [80] on R M C provides a beautiful summary of the theoretical as well as the experimental aspects of these observables. 3 A s discussed in section 1.5, measurements of gp in nuclei are also interesting in view of inves-tigating its eventual renormalization, i.e. the influence of the nuclear medium on the axial weak current. 22 Due to the inherent difficulty of these measurements, caused pr imar i ly by the low yield ( Z 4 law), the only R M C data available , before the T R I U M F hydrogen experiment result, are for nuclei wi th A > 12. Table 1.4 provides a summary of 9P/9A as determined from R M C in complex nuclei. A few comments on Table 1.4, are i n order. One is that the extracted values of gp/gA are not consistent and often do not agree w i th the Goldberger-Treiman expectation. More importantly, these results suggest a substantial enhancement in light nuclei and quenching i n heavy nuclei. This can clearly be seen by plot t ing gp as a function of nuclear mass, see Figure 1.3. Th i s trend for gp does indeed show a tantalizing indicat ion of a quenching of the pseudoscalar coupling for heavy nuclei. However, the interpretation of these measurements are fraught wi th nuclear structure uncertainties 4 . This has been reinforced by the fact (Table 1.4) that the same experiment, when compared to different theoretical models, can lead to very different extracted values oi gp. We shall come back to this topic. Due to the low radiative branching ratio (~ 10~ 5) combined wi th the many po-tential large background sources, efforts also have concentrated on O M C in heavier nuclei. A l l in a l l , aside from 7 — u angular correlation which is the subject of this thesis and wi l l be discussed later, five different O M C observables have produced 'quoteworthy' measured values of gp i n five l ight- to-medium nuclei, as shown in Table 1.5. A m o n g these are the polarizations of the residual nucleus along the direction of the muon spin (Pav) and along the direction of its recoil (PL ) , the rate of muon capture to specific final states ( A , ) , and the hyperfine dependence of muon capture ( A + / A _ ) . The apparent concentration of efforts on 1 2 C is due to the existence of several 4Some doubts were cast on the validity of the theory of Ref. [94] by Fearing and Welsh [95]. 23 Table 1.4: Summary of values of gp/gA determined by comparing exper-imental results wi th theoretical predictions of radiative muon capture ( R M C ) in complex nuclei a . "The R M C / O M C ratio for A l , Si and T i have been measured recently; however, no nuclear R M C models have yet been attempted for these nuclei to extract gp/gA [82]. Reference Nucleus Theory QP/QA Armst rong et al. [83] i 2 C [84] 16.2 t u ;? 1 6 Q [85] 13.6 +_[% n [86] 7.3 ± 0 . 9 Dobel i et al. [87] H [86] 8.4 ± 1.9 Frischknecht et al. [88] 11 [86] 13.5 ± 1.5 Average [89] 11 [90] 9.0 ± 0 . 8 Armst rong et al. [83] 4 0 C a [91] 5.7 ± 0 . 8 11 11 [85] 4.6 ± 1.8 Armst rong et al. [92] 11 [91] 5.9 ± 0 . 8 n 11 [85] 5.0 ± 1.7 11 11 [90] 7.8 ± 0 . 9 Frischknecht et al. [93] 11 [86] 4.6 ± 0 . 9 Dobel i et al [87] 11 [86] 6-3 t\i Average [89] 11 [90] 8.1 ± 0 . 3 Dobel i et al. [87] n a t p e [94] 3.0 ± 1.3 Bergbusch [82] nat^j. [94] -0 -4 Armst rong et al. [92] n a t M o [94] o.o t i i Bergbusch [82] n a t A g [94] o i +0.6 -0.7 Armst rong et al. [92] n a t S n [94] o.o t\i Dobel i et al. [87] 1 6 5 H o [94] - 0 . 5 ± 1.4 Armst rong et al. [92] natpt) [94] < 0.2 Dobel i et al. [87] 209B i [94] 0.2 ± 1.1 24 5 0 1 0 0 1 5 0 2 0 0 Nuc lea r M a s s (A) 2 5 0 Figure 1.3: gp/gA as determined from the rate of radiative muon capture measurements. Table 1.5: Summary of values of gp/gA as determined from measurements of muon capture ( O M C ) in complex nuclei. Reference Nucleus Observable Theory 9P/9A Deutsch et al. [96] A + / A - [22]? < 15 Possoz et al. [97,98] 55 55 55 K u n o et al. [104] Mi l l e r et al. [105] Roesch et al. [106,107] 55 55 Masuda et al. [110] 1 2 C p 55 55 55 55 A , P a w / P i 55 55 55 [99] [100,101] [102] [103] 55 [102] [108] [109] [103] [111] 7.1 ± 2 . 7 13.6 ± 2 . 1 15 ± 4 10.3 t l i IO.I t l i 8.5 ± 2 . 5 9.4 ± 1.7 7.2 ± 1.7 9.1 ± 1.8 8.5 ± 1.9 Towner et al. [112] 1 6 Q . [113] 7-9 Gorringe et al. [114] 2 3 N a A + / A _ [115] 7-6 ^ 25 observables along wi th other weak and electromagnetic processes involving the same or analogous nuclear states in the A = 12 nuclei; and consequently leading to better understanding of the nuclear structure uncertainties therein. Unl ike R M C , O M C data are consistent wi th the Goldberger-Treiman estimate; for example, the most recent determination of the coupling constant, made by our group [114]5, from the A + / A _ hyperfine dependence of muon capture on 2 3 N a , is consistent wi th the free nucleon value and i n disagreement wi th the intr iguing renor-malizat ion suggested by the R M C data. More recently, Delorme and Er icson [117] argued that the massive quenching of gp in nuclei they had predicted [118], due to a renormalization of the p-wave pion nucleon interaction, is nearly totally counter-balanced by a large enhancement of the s-wave pion nucleon interaction. To conclude, the possible renormalization of the induced pseudoscalar cou-pling constant i n nuclear medium is a topic that has continued to fuel interest in the field of low-energy weak interactions in general and muon capture i n particular. Yet, it is obviously too early to draw any definitive conclusion. Clear ly more precise experimental determinations - a n d / o r experiments wi th different characteristic un-certainties and dependences on nuclear s tructure- of gp are i n high demand. That is the ma in motivation for the this work. In the next chapter, the theory directly relevant to the observable of this work, the 7 — 1/ angular correlation, w i l l be given. 5 A recent comparison [116] of the 2 3Na(/x ,v) and 2 3 Na(n,p) reactions tested the nuclear model used and gave confidence in the extracted gp/gA from the data. 26 C h a p t e r 2 T h e 7 — 1/ A n g u l a r C o r r e l a t i o n 2.1 Introduction The previous chapter outlined the demand for other reliable observables for the determination of the pseudoscalar coupling (gp), that is, observables which offer both sensitivity to gp and a lesser dependence upon nuclear structure uncertainties. One such observable is the angular correlation between the neutrino and a specific nuclear de-excitation gamma-ray following muon capture on certain nuclei. Unl ike most muon capture observables, the 7 — v angular correlation has the advantage of being an exclusive process, that is, the final states are resolved and hence avoids some (but not all) nuclear structure uncertainties. The angular correlation i n the emitted de-excitation gamma-ray is generated by the difference in the muon capture populat ion of the allowed magnetic substates of the daughter nucleus. In the case of an allowed muon capture sequence, the angular correlation can be writ ten, using the notation of Parthasarathy and fi~ + (A,Z) (a,z - ly + v* + 7 , (2.1) 27 Sridhar [100,119], as follows: = J(0) [1 + aP 2 (coscV) + • 7 ) ( 7 • i>)P2(costV) + & ( £ • 7 ) ( 7 • v)}. (2.2) where P 2 is the Legendre polynomial , a , fli and /32 are the correlation coefficients, /} is the muon spin direction, and 91U is the angle between the unit vectors along the photon and neutrino momenta, 7 and respectively. T w o remarks about Equat ion 2.2 are i n order. The first is that in the case of unpolarized muons, or 7 - r a y detection at 90° to the muon polarization direction, the terms involving (/i - 7 ) drop out and the correlation is characterized by only one coefficient (a). The other is that the effects of the coefficients (5\ and B2 are suppressed by the small residual polarizat ion of the muon after the atomic capture and cascade (about 15 % in Si) . In general, the three correlation coefficients depend upon the weak coupling constants, the spin-parity sequence, the kinematics and the nuclear wavefunctions. In certain cases, however, the dependence of the angular correlation on gp is en-hanced, whilst the sensitivity to nuclear physics uncertainties is minimized. One such case is the allowed 0+—>1+—>0+ muon capture sequence, \x~ + 2 8 S i ( 0 + ) - > ^ + 2 8A1*(2201 keV, 1 + ) v„ + 2 8A1**(972.6 keV, 0+) +7. (2.3) (Details concerning this cascade wi l l be discussed in sections 4.3 and Figure 4.4.) The theory of the v — 7 correlation following nuclear muon capture was orig-inal ly studied by Popov and collaborators [120,121,122,123,124,125] and later ex-amined by Ciechanowicz [109] and Parthasarathy and Sridhar [100,119] and most recently by a D u b n a group [126]. In the discussion that follows, we provide a review of these theoretical developments and i n particular the dependence of the angular correlation coupling constants (a, Br and B2) on the poorly determined weak cou-pling gp i n these models, w i th emphasis on the muon capture sequence of interest, 28 i.e. Equat ion 2.3. 2.2 Fujii-Primakoff Approximation The angular correlation coupling constants depend upon the weak coupling constants as well as on the nuclear mat r ix elements. Only i n the so-called Fuj i i -Primakoff approximation (FPA)[127] do the matr ix elements cancel out and the angular correlation dependence on gp springs forth. In this approximation, one assumes only an allowed capture, i.e. the neutrino carries away no angular momen-tum (s-wave neutrinos only), and one ignores the nucleon-momentum dependent terms (also known as recoil or relativistic terms) in the effective Fuj i i -Pr imakoff Hamil tonian . These terms come from the non-relativistic reduction of the weak interaction Lagrangian which was carried out by Fuji i and Primakoff [127], using a Foldy-Wouthuysen transformation [128]. They developed an approximate effec-tive Hami l ton ian for muon capture suitable for use wi th non-relativistic nuclear wavefunctions. In the F P A , and for a pure M l 7-ray such as Equat ion 2.3, one obtains : « = J G P n r Z r < (2-4) (2.5) (2.6) 3G\ -\- Gp — IGpGA Gp 3G\ + Gp — IGPGA 3GA — Gp 3G\ + Gp — IGPGA where Gp and GA are the Fuj i i -Primakoff form factors, which are linear combina-tions of the standard semileptonic coupling constants, i.e. G A = g A - ( g v + g M ) ^ (2.7) GP = (gP - g A - g v - 9M)^J^ • (2.8) 29 Hence, under the F P A , the coefficients a , 8\, and B2 are functions of only GP/GAI or equivalently gp/gA provided gv and #M a r e known. Furthermore, exam-ining the above Equations (2.4, 2.5 and 2.6), one obtains simple relations, namely, 1 + A = F3L + B2 (2.9) and & = - [ V i ^ ± VTT^}\ (2.io) which indicate that, among the three correlation coefficients, only one is linearly independent, or, in alternative language, that a l l three depend on only one variable. However, it is expected that the F P A is an oversimplification and is inadequate for reliable extraction of gp. 2.3 Beyond the F P A Some weak nuclear-structure dependence would surface upon the incorporation of the recoil terms and/or the higher par t ia l waves for the neutrino. Despite that, the dependence of the angular correlation coefficients on the induced pseudoscalar coupling survives. Several authors have carried out such an calculation. In Ref.[125] Oziewicz examines extensively the theory of par t ia l muon cap-ture by spinless nuclei in terms of the mult ipole expansion of the weak hadronic currents [120,121,122,123,124,125], and finds that in the case of the 0 + A l + muon capture sequence, the v — 7 angular correlation, among other observables, is de-scribed by a complex ratio of two independent mult ipole amplitudes ( ^ ) and can be characterized by two parameters x and as follows: = 57 <2-n> Hsl <2-i2> 30 with relative phase <f> = 0 or TT. The amplitude ratio x describes the relative muon capture feeding of the components of the excited state (e.g. 2201 keV 1 + in Equa-t ion 2.3), and hence would be the same for any other de-excitation 7-ray starting from the same nuclear state. Oziewicz then goes on to deduce the angular correlation formula (equation 2.2) along wi th the following relationships among the correlation coefficients: a = F ( 2' 1 3> 1 + x2 + 2x cos < 2 + x 2 Pi = F [ , y (2.14) 2 — x2 — 2Fx cos 6 A = 2 + .*> • < 2 ' 1 5 > where the factor F depends on spin sequence and the mult ipolar i ty of the de-excitation 7-ray. In particular, F = 1 for 1 + ^ 0 + (2.16) and where 8 is the mul t ipolar i ty mixing parameter given by the E2/M1 ratio. Several comments about the above Equations are i n order: • The F P A approximation, Equations 2.4, 2.5 and 2.6, falls out of Equations 2.13, 2.14 and 2.15 as s imply being the prediction • Unlike fi\ and f32, a is independent of the phase </>. • For the spin sequence 0 + A 1+ 0 + , Equations 2.9 and 2.10 of the previous section connecting a, /?i and (32 can be deduced from Equations 2.13, 2.14 31 and 2.15. Oziewicz points out that a measurement of the sign of B1 would constitute a measurement of muon neutrino helicity, or i n other words, a negative sign for B\ would imply the emission of a right-handed neutrino. • Also , Equations 2.13, 2.14 and 2.15 along wi th Equations 2.9 and 2.10 imply the following boundary conditions - 1 < a < 0.5 (2.19) 0 < 8X < 1.5 (2.20) - 1.5 < & < 1-5, (2.21) violations of which would imply time-reversal violat ion. • A consequence of the mult ipolar i ty mixing factor F is a suppression of the correlations in the 1 + A 2 + 7-ray transitions as opposed to the pure M l 1 + 0 + transitions. • Furthermore, from Equat ion 2.17, one has the following bounds (see F i g -ure 2.1): - 0.4 < F < +1 , (2.22) on the 1 + 2 + 7-ray transitions. In summary, even when one goes beyond the Fuj i i -Pr imakoff approximation, the dependence of the angular correlation on gp remains and al l three correlation coefficients a, B\ and B2 are functions of only one (nuclear structure dependent) pa-rameter x\ thus there is only one independent coefficient, modulo the sign ambiguity from <j>. 32 2 4 6 8 6 = E2/M1 ratio 10 Figure 2.1: The mult ipolar i ty factor F as a function of the mix ing ratio, 6 = E2/M1, as calculated by Oziewicz [125]. 33 2.4 Full Calculation Models There appear to be three 'comprehensive' analyses, for the transition(s) of interest in 2 8 A l , that go beyond the Fuj i i -Primakoff approximation, to calculate numerically the dependence of the angular correlation coefficients as a function of the pseudoscalar coupling constant. These result in what we herein-after refer to as ' fu l l ' calculation models. They are from different groups and use different approaches as well as different nuclear wave functions. 2.4.1 Model I : Ciechanowicz The first full treatment was performed by Ciechanowicz [129] i n 1976. This model was an application of the multipole theory of Popov [120,121,122,123,124] and was basically an outgrowth of Oziewicz's work [125]. In this model, Ciechanowicz calculates two multipole amplitudes (A and M) for the 1229 keV transit ion (equa-tion 2.3) as well as the 1342 keV transit ion, i.e. u~ + 2 8 S i ( 0 + ) + 2 8A1*(1373 keV 1+) i/„ + 2 8A1**(30.6 keV 2+) +7, (2.23) by including a l l recoil terms up to order 1 / M in the effective Hamil tonian , i.e. sec-ond order terms in M are omitted. The muon wave function used is a realistic one, taking into account the extended nuclear charge distr ibution. The Di rac Equat ion was numerically solved for the nuclear charge dis tr ibut ion. The 2 8 S i and 2 8 A l wave-functions which were used were from Wi lden tha l , DeVoigt and M c G r o r y [130,131] wi th configuration mix ing taken into account. For the 1229 keV 7-ray, the two multipole amplitudes are given as: A = -19 .58 gA + 0.216 gv + 1.373 gP (2.24) M = -23 .86 gA + 4.954 gv - 0.019 gP, (2.25) 34 and hence their mult ipole ratio x, u t i l iz ing gv = —0.784^ [132], becomes: x (2201,1+) = 0.712 1 -q-Q7Q9P/9A ( 2 2 G ) ^ ; l + 6 . 8 x 10-* gP/gA y J Ciechanowicz also studies the effect of the shape of the muon wave function on gp by using a constant muon wave function over the nuclear volume, arr iving at what he calls the non-relativistic approximation ( N R A ) : A = —19.80(7,4 + 0.940 (2.27) M = -25 .40 gA + 6.920 gv, (2.28) or equivalently, x (2201 ,1 + ) = 0.642 - 0.0305 gP/gA, (2.29) which is not to be confused wi th the Fuj i i -Primakoff approximation (equation 2.18) as the former s t i l l includes the recoil terms in the effective Hamil tonian , and differs slightly from relation 2.26 by a Coulomb correction i n the muon wavefunction. The mutipole ratio x for the 1229 keV transit ion from these three calculations is shown in Figure 2.2 for comparison. The angular correlation coefficients 1 a, 8\, and 82 can then simply be calcu-lated from relations 2.13, 2.14 and 2.15, and are presented i n Figures 2.3, 2.4 and 2.5 respectively. 2.4.2 M o d e l II : P a r t h a s a r a t h y a n d S r i d h a r In this model [119,100,101], the 7 — v angular correlation is developed using the density matr ix formalism of Devanathan and Subramanian [133]. In the first paper [119], Parthasarathy and Sridhar consider the capture of unpolarized muons i n the 2 8 S i process of Equat ion 2.3 and hence they are only 1There is a typo in [129] in the line after Equation (16), where the values of F for the 1229 keV and 1342 keV transitions have been mixed up. 35 9p/9a Figure 2.2: Dependence of the multipole ratio on gp/gA for the 1229 keV transition, as calculated by Ciechanowicz [129]. Also shown are the non-relativistic approximation ( N R A ) as well as the Fuj i i -Primakoff approximation ( F P A ) . 36 Figure 2:3: Dependence of a on gp/gA for the 1229 keV transit ion, as calcu-lated by Ciechanowicz [129], Parthasarathy and Sridhar [100] as well as K u z ' m i n et al. [126]. Also shown is the Fuj i i -Primakoff approximation. 37 5 1.0 0 . 8 0 . 6 -0 . 4 0 . 2 0 . 0 _ j i i i i i i i i i i i i i i i i i i i i i i _ Legcmd —e— - Model : Cie —*— Model : P&S FPA i 1 1 r 2 0 Figure 2.4: Dependence of fi\ on gp/gA for the 1229 keV transit ion, as cal-culated by Ciechanowicz [129] and Parthasarathy and Sridhar [100]. Also shown is the Fuj i i -Pr imakoff approximation. 38 Figure 2.5: Dependence of B2 on gp/gA for the 1229 keV transit ion, as cal-culated by Ciechanowicz as well as Parthasarathy and Sridhar. Also shown is the Fuj i i -Primakoff approximation. 39 concerned wi th the coefficient a.. In the second paper [100], they extend the same formalism to the capture of polarized muons in the same process, and hence include /?i and B2. Including both higher par t ia l wave neutrinos as well as nucleon-momentum de-pendent terms in the Fuj i i -Primakoff effective Hami l ton ian , they derive expressions for a , 81 and f32 in terms of reduced nuclear mat r ix elements in the particle-hole model of Donnelly and Walker [134]. Examin ing these expressions under the Fuj i i -Primakoff approximation ( F P A ) , they obtain the same relations as the ones given in Equations 2.4, 2.5 and 2.6 earlier. They also derive simple relations l inking the cor-relation coefficients to the average and the longi tudinal polarizations of the recoiling nucleus; namely 2 , - a = 1 + ^PL, (2.30) l31 = 1 - ^Pav (2.31) and fa = - 1 + \pav - \PL, (2.32) as well as PL = -\{fa + B2). (2.33) They also note that these relations are rigorously true, and independent of nuclear structure; even when one includes nucleon-momentum dependent terms and higher part ial wave neutrinos. Furthermore, these relations yield back relation 2.9 as well as the boundary conditions found by Oziewicz (Equations 2.13, 2.14, 2.15). Parthasarathy and Sridhar then numerically evaluate the reduced nuclear ma-tr ix elements appearing i n the expressions of a [119] and R\ and i32 [100] using the the wave functions of Donnel ly and Walker [134]. They give their numerical re-sults in a tabular form, claiming that their results for the correlation coefficients 2These relations were derived first by Bernabeu in 1976 (see Ref. [22]), however, on different grounds. 40 are very different from those of Ciechanowicz [129]. However, as Armst rong [135] has pointed out, these claims are s imply not true, as can be seen in Figures 2.3, 2.4 and 2.5 where their results are plotted along wi th those of Ciechanowicz. Hence, their calculation seems to agree fairly well wi th that of Ciechanowicz, especially near the P C A C value for gp. In the case of a , the discrepancy appears to be due to a numerical error i n the first paper [119], which caused the results for a to be significantly wrong. This error is pointed out by the authors themselves in Ref. [25] of the subsequent paper [100], and corrected values of a are given therein 3 . As for the case of /?i and B2: the claimed disagreement is due to the use of the wrong choice of (f> in Ciechanowicz's theory. The choice (f) = ir ( in Equations 2.13, 2.14 & 2.15) gives values for fli and B2 that are in reasonable agreement wi th those of Parthasarathy and Sridhar, see Figures 2.4 & 2.5. Th i s unfortunate choice of <f>, seems to have caused several authors, amongst them are Mukhopadhyay [22] as well as Eramzhyan [136], to arrive at the same misleading conclusion about the disagreement between the two models. It is interesting to further note that Parthasarathy and Sridhar apparently make no statement about relation (2.10), i n fact their results seem to satisfy this relation 4 , which indicates that only one of the correlation coefficients is linearly independent, i n agreement wi th Oziewicz and Ciechanowicz. Parthasarathy and Sridhar also claim that among the three correlation coef-ficients, B2 is nearly nuclear-model insensitive. However, this is only true in the region where B2 is nearly flat, i.e. B2 is less gp dependent as well . Furthermore, since only one of the coefficients is independent, one expects a l l coefficients to have 3 Yet , the authors fail to alter the conclusions in [119], the first of which (conclusion i) contains the claimed disagreement with Ciechanowicz [129], in fact they state the contrary: " The conclusions in I [119] remain unaltered.". 4 Except for a small region around gp/gA ~ 17.5 where a violates the 'rigorous' limit (equa-tion 2.4). If not absorbed in the uncertainty of their model, this may suggest some numerical problem with the model. 41 the same gp model-dependence. In the th i rd paper [101], Parthasarathy and Sridhar examine the effect of meson exchange current ( M E C ) corrections on the correlation coefficients, and find them to be generally rather small . 2 . 4 . 3 M o d e l I I I : K u z ' m i n at al. Recently, K u z ' m i n , Ovichinnikova and Tetereva [126,137], from the Joint Insti-tute for Nuclear Research ( J I N R ) i n Dubna , calculated the independent transit ion amplitudes (the effective muon capture matr ix elements) as well as the angular cor-relation coefficients for the 7-ray transitions of interest following muon capture on 2 8 S i in the framework of the nuclear shell model. They have developed a procedure to calculate nuclear single particle matr ix elements applicable for any harmonic os-cillator wave function. A l l possible linear velocity dependent terms are included i n the effective muon capture Hamil tonian . W i t h i n this unified approach, they cla im that their code can use any shell-model nuclear wave functions. The nuclear wave functions used i n their most com-plete model were calculated wi th the well tested [138] shell model code O X B A S H [115] In addit ion to the O X B A S H ' fu l l ' sd-shell model nuclear wave functions, they use several sets of wave functions; including those of Wi lden tha l et al. [130,131] and Donnelly and Walker [134] which are used i n the previous two models. They are able to reproduce the calculations of Ciechanowicz as well as of Parthasarathy and Sridhar. Consequently, they ascribe the discrepancy between their model and those of the previous models to the use of different nuclear structure, i.e. wave functions. It should perhaps be remarked that our collaborators at the Universi ty of Kentucky have used a similar code [139] using a full Is — Od valence nucleon space and U S D residual interaction [140]. We have recently used it to predict par t ia l rates and hyperfine dependences of muon capture on 2 3 N a [141]; and found that the 42 calculated capture rates to the six excited states i n 2 3 N e are i n general agreement wi th the measured ones [114]. Th i s same program, when applied to the j — u angular correlation, yields the same dependence of the correlation coefficients on gp/gA as that obtained by the D u b n a group, Figure 2.3 shows this dependence along wi th the other two models. As can be seen from Figure 2.3, model III (with the more realistic modern wave functions and complete effective Hamil tonian) gives a dependence of a on 9P/<JA that is weaker near the P C A C value for gp; this makes a measurement of a less promising as a means of measuring gp. Moreover, i n this model, the calculated values of the correlation coefficients are larger for a constant gp/gA as compared to the other two models. K u z ' m i n et al. [137] pointed out a l imi ta t ion, typical of al l muon capture calculations, that can be serious. That is the possible overestimation of the velocity dependent terms in the effective Hamil tonian due to the use of harmonic-oscillator single particle wave functions which are used in a l l three models. To test this suspicion, they are in the process of replacing the harmonic-oscillator potential by a finite one such as a Woods-Saxon potential . 2.5 The Method of Measurement The experimental method to measure the 7 — v angular correlation has been suggested by Grenacs et al. [142]. They pointed out that, since the daughter nucleus recoils against the neutrino after muon capture, the angular correlation between the neutrino and the emitted 7 ray is equivalent - w i t h i n a factor of 7T - to the angular correlation between the nuclear recoil and the 7 ray. The observed energy of the emitted 7 ray is related to the transit ion energy E 0 i n the emitt ing nucleus by the 43 Figure 2.6: Kinemat ics of muon capture reaction on 2 8 S i . Doppler Equat ion: £ 7 = £(,(1 -(3 cos 6) (2.34) where 8 is the velocity of the recoiling nucleus at the moment of the 7 t ransit ion (Figure 2.6). The in i t ia l recoil velocity for the 2201 keV 2 8 A l excited state (E*xt) is given by : / m u — iTT-e + msi — m\\ — E*f — BE 80 = »/2 —- — — — + 1 - 1 = 0.0037484 (2.35) V " U i where BE is the muon binding energy of the atomic Is orbit i n Si (0.539 M e V ) . The correlation function (Equat ion 2.2) can be wri t ten as an energy distr i-but ion of the emitted 7 rays. Consequently, the measurement of this dis tr ibut ion constitutes a measurement of the 7 — 1/ angular correlation. In the present work, we choose to measure a, as opposed to the other correlation coefficients, for which (p. -7) = 0 for the reasons aforementioned, i.e. easy to measure, same nuclear model dependence and no need to measure residual muons polarizat ion. Figure 2.7 shows the shape of a Doppler-broadened 7-ray dis t r ibut ion wi th the angular correlation coefficient, a , effects, see also Figure 5.23. O f course, this method is val id only for a transit ion from a short l ived state, i n which the daughter nucleus decays in-flight, and hence there is no appreciable slowing-down effects. Th i s is fairly well satisfied in the case of the ( 1 + , 2.2 M e V ) level for which the lifetime is about 65 fs, see ref-44 / / - / \ \ \ \ IL<5g<sinidl a = +0.25 • - - - a = -0.25 E0(1-/?o) E 0 E 0(1+/? 0) Gamma —ray Energy (E ) Figure 2.7: The shape of a Doppler-broadened 7-ray dis t r ibut ion and the effect of the 7 — 1/ angular correlation coefficient, a (effects due to the resolution function of the detector are not included here). erence [143] and section (5.7). Thus both decays from this level, the 1229 keV and 2171 keV 7-rays are suitable candidates. Measurements of the Doppler-broadened lineshape of the 1229 keV 7-ray was performed by Mi l l e r et al. [144,145] at S R E L in the late 1960's and just recently by Brudan in et al. [146] at J I N R . A full account of these measurements, and the way they compare to this work, along wi th their effect on the above theoretical models w i l l be given i n Chapter 6. 45 2.6 Ancillary Reactions A s is usually the case, different nuclear reactions complement and supplement one another. For example, the 7-ray spectra obtained following muon capture alone do not give us directly the energies of the excited states of the residual nuclei. The importance of such knowledge, for the purpose of this work, stems from the fact that we would like to know whether higher excited levels of 2 8 A l cascade-feed into the 2201 keV 2 8 A l level of interest. Such a possibility would spoil the goal of this work. This topic w i l l be discussed more thoroughly in the following chapters. The nuclear level corresponding to a certain state of the residual nucleus can be distinguished by other reactions such as (n,p), (TT,J) and (d , 2 He) which give us the excited states directly, because the role of the v is played by a detectable particle. Comparisons of these reactions show that they are similar though not identical to the (u,u) reaction; see for example Siebels et al. [116], Ni izek i et al. [147] and Eramzhyan et al. [148]. These reactions are similar because they al l tend to excite 1 + transitions, often called Gamow-Teller ( G T ) by analogy to the nomenclature from beta-decay. Unfortunately, the only data to our knowledge on 2 8 S i ( n , p ) 2 8 A l are from B h a r u t h - R a m et al. [149] at En = 21.6 M e V and from B r a d y et al. [150] at En = 60 M e V . B o t h had poor resolution ( A ~ 1 M e V ) and very low statistics, so are not too helpful. The 2 8 S i ( 7 r , 7 ) 2 8 A l reaction was studied twenty years ago at PSI , but the data were never published as far as we know. Nevertheless, a sil icon spectrum was presented i n the 1976 annual report [151] and is reproduced in Figure 2.8. The 7-ray resolution is about 1 M e V and the statistics are sufficient. The difference is that for a stopped 7r~, the absorption tends to occur predominantly from the atomic 2p state not the Is state [17,152]. The 2 8 S i ( d , 2 H e ) 2 S A l was studied by Sakai and collaborators [147]. It poten-t ial ly had better energy resolution than the ( ^ , 7 ) reaction. A n excitation-energy 46 E 45 Figure 2.8: 2 8 Si (7r,7 ) 2 8 Al cont inuum subtracted spectrum taken from P. Truo l et al.[151]. The levels i n 2 8 S i excited by (e,e') at 180° are also 1 + levels and are marked at the bot tom, see Schneider et al. [153]. The level of interest i n this experiment is at 2201 keV excitat ion energy in 2 8 A 1 . 47 3 0 0 Figure 2.9: 2 8 S i ( d , 2 H e ) 2 8 A l excitation energy spectrum at 270 M e V and 0° obtained by Ni izeki et a?. [147]. > 16 6 jo o 28 Si(p,n) 2 8P 0 2 A 6 0 10 12 14 16 10 E x c i t a t i o n Energy (MeV) Figure 2.10: 2 8 S i ( p , n ) 2 8 P excitation energy spectrum at 136 M e V and 0.2° from Anderson et al. [154]. We have received similar data at 120 M e V from J . Rapaport . 48 spectrum at = 270 M e V and at 0° is extracted from their paper and is shown i n Figure 2.9. The 2 8 Si (p ,n ) reaction has also been studied by Anderson et al [154] at E p = 136 M e V and 0.2°. The i r spectrum is shown i n Figure 2.10. (Note that there wi l l be Coulomb differences i n the levels.) The comparison of the (p,n) and (d, 2 He) spectra is str iking. A similari ty among these spectra is the excitation of the 1 + levels, just as in muon capture. However, while the (p,n) and (d , 2 He) spectra show a strong 1 + peak around 4.8 M e V , the (7r,7) spectrum has a strong level at 6.2 M e V (which is st i l l below the neutron break-up threshold). Other reactions that are very useful in determining the 2 8 A 1 level scheme are the 2 7 A l ( n,7 ) 2 8 A l reaction, see Schmidt et al. [155], and the 2 9 S i ( d , 3 H e ) 2 8 A l reaction, see Vernotte et al. [156]. They both attain high resolution and can unravel the level scheme quite accurately. However, the (n,7) reaction tends to populate different levels (particularly those wi th high spin), whereas muon capture tends to populate mainly 1 + as well as 0~, 1~, 2~ and 2 + levels, while the (d , 3 He) is a different reaction again as well as operating on a different target. Some things are clear however. Fi rs t is that the ( / i , v) reaction wi l l excite several levels above 2.2 M e V which are bound, and therefore could cascade into the 2201 keV level of interest. The indications are that the feeding to these levels is 1.5 to 2 times that to the 2201 keV level and/or below, see M a c D o n a l d et al. [157] and M i l l e r et al [145,158]. Thus the (TT~ ,7) reaction gives an accurate impression of the relative feeding even though it may be wrong i n the details. Secondly the information presently available for 2 8 A l is not sufficient to be able to identify the levels fed by muon capture wi th the precision that we need. This topic w i l l be revisited i n section 5.4 i n connection wi th the cascade-feeding problem. M a c D o n a l d et al [157] give the neutron mul t ip l ic i ty-dis t r ibut ion following muon capture in several targets. Table 2.1 is a summary of the results (properly corrected for the detector efficiency) for S i , which is not incompatible wi th Table II 49 Table 2.1: Neutron multiplici t ies of muon capture on S i , from M a c D o n a l d et al. [157]. Neutron mul t ip l ic i ty Y i e l d Result ing nuclei 0 0 .362±0 .065 2 8 A l , 2 7 M g , 2 4 N a 1 0 .472±0 .092 2 7 A 1 , 2 6 M g , 2 3 N a 2 0 .160±0 .070 2 6 A 1 , 2 5 M g , 2 2 N a 3 0 .006±0 .008 2 5 A l , 2 4 M g , 2 1 N a of M i l l e r et al. [144]. 50 Chapter 3 Description of the Experiment The experiment described here has been performed at the superconduct-ing channel ( M 9 B ) of the Tr i -Univers i ty Meson Faci l i ty ( T R I U M F ) i n Vancouver, Canada. It was performed during the course of a two-week cyclotron run i n Novem-ber 1992. The experimental setup was pr imar i ly designed for the second phase of experiment 570 whose a im was to measure the angular correlation between the neutrino and a Doppler broadened de-excitation gamma ray i n 2 8 A l following muon capture on 2 8 S i [159] through a coincidence technique. The coincidence technique is intended to suppress backgrounds underneath the gamma ray of interest discovered in the first phase of the experiment(December 1989) [160]. Such backgrounds make the extraction of an angular correlation from the line shape very problematic. 3.1 Beam Production and the M9B Channel The T R I U M F accelerator is a six-sector focusing cyclotron wi th a special feature being the acceleration of negative H~ ions which, i n turn, permits a con-venient extraction of the beam. This is done by passing the beam through a th in carbon foil to strip the two electrons and hence the resulting beam of positive ions curves in the opposite direction in the magnetic field and exits the cyclotron. 51 A s shown in Figure 3.1, two str ipping foils, located 180° apart, are used to feed the two main beam lines B L 4 ( P ) and B L 1 ( P ) wi th proton beams of current up to 140 fiA and energy range of 183 to 520 M e V . The first beam line is dedicated to proton-induced reactions while the other is used for pi meson production. The secondary muon beam used i n this experiment was produced when the pr imary proton beam in the meson beamline impinges on the meson production 1AT2 target, typically a water cooled strip of bery l l ium 10 cm thick in the beam direction and 5 m m x 15 m m i n cross sectional area. The pr imary proton beam is normally delivered in 3 nsec pulses every 43 nsec and wi th a 99% duty factor. See T R I U M F users handbook [161] for more information on the T R I U M F cyclotron and other pr imary and secondary beamlines. The muon beam used was obtained from the new M 9 B channel which incor-porates a superconducting solenoid to collect low-momentum polarized muons from the decay of pions. This pion contamination of the beam was less than 0.2 %, and the fraction of electrons in the beam was about 20 %. Newly installed slits on the channel helped focus the beam further. Abou t 1.2 x 105fj,~/sec at medium momentum (~ 60 M e V / c ) were required to stop i n our targets. Th i s momentum is equivalent to 20 M e V muons which have a range of 3 g / c m 2 i n sil icon. Figure 3.2 shows the layout of M 9 channel, wi th its two legs M 9 A and M 9 B . 3.2 Experimental Arrangement The experimental setup is shown i n Figure 3.3. It consisted of a conventional beam telescope of three scinti l lat ion counters, S i , S2 and S3 wi th dimensions of 15.2 cm x 20 cm x 0.32 cm, 7.6 cm diameter x 0.16 cm and 10.2 cm x 10.2 cm x 0.32 cm respectively. S i and S2 wi th S3 in anti-coincidence defined a muon stop i n 52 BEAMLINES A N D EXPERIMENTAL FACILITIES. ISAC UNDER CONSTRUCTION. PROTON HALL EXTENSION TISOL-EXPERIMENTAL AREA TRINAT' FEB96 OPTICALLY PUMPED POLARIZED ION SOURCE (OPPIS) TR 30 ISOTOPE • PRODUCTION CYCLOTRON MESON HALL SERVICE ANNEX POLARIZED ION SOURCE M15(/z) to d i s SH U t 3 i — i P5 u ( H • i—I fa-. co Superconducting solenoid M9B to RF Separator and IPC M9A t f-= • 3 — I • Target 1AT2 Figure 3.2: Layout of the T R I U M F M9 channel. 54 the target. The t iming of a muon stop was defined by S2, which was mounted close to the target and had its discriminator threshold set just below muons. A polyethylene-sleeved lead collimator was used to collimate the beam as well as to shield the detectors from neutrons and background radiations. The main target used was natural Si (92.2% 2 8 S i , 4.7% 2 9 S i and 3.1% 3 0 S i ) in granular form. Its dimensions was 6.3 cm diameter by 1 cm thickness. It was set at 45° to the beam and to the Ge detectors axis i n order to minimize the thickness traversed by gamma rays and maximize the thickness traversed by muons. Th i s orientation also reduces the bremsstrahlung from muon-decay electrons. A variety of targets were used to study backgrounds as well as to deter-mine the acceptances of the Ge detectors. They were: C , Mg(granules), Al(6061), Cr(granules), Fe, N i , C u , Ge(powder), M o , Sn, A u , P b as well as Nal(powder) and stainless steel(316). 3.3 Detection System The 7—ray detection system used i n this experiment (Figure 3.3) had two pr incipal arms: a pair of high puri ty germanium detectors ( H P G e ) along wi th several N a l ( T l ) counters. The two H P G e crystals were the pr incipal detectors in this experiment and must measure the 1229 keV 7-ray line emitted from the capture of muons on S i . These germanium detectors, G e l and Ge2, were of 44% and 30% relative efficiency (with respect to a 7.62 cm diameter by 7.62 cm long N a l ( T l ) ) w i th system resolution at 1.3 M e V of about 1.9 keV and 2.0 keV full w id th at half max imum ( F W H M ) respectively. Under experimental conditions^', e. i n beam, these resolutions became 2.2 keV and 2.4 keV F W H M respectively. The Ge detectors were located at 90° wi th respect to the beam direction 1 . ^ h e reason for this specific angle was to drop out the terms involving (/t -7) in equation 2.2 and so that the 7 — v correlation would essentially be characterized by only one coefficient, a. 55 P o l y e t h y l e n e sleeve L e a d c o l l i m a t o r L e a d c o l l i m a t o r PM's N a l C o m p t o n s u p p r e s s o r s Figure 3.3: Experimental setup. 56 Table 3.1: Properties of the N a l ( T l ) counters. N a l ( T l ) Counter Dimensions (in cm) Resolution (%) F W H M B l 10.2 x 10.2 x 20.3 9.9 B 2 10.2 x 10.2 x 20.3 9.6 B3 10.2 x 10.2 x 20.3 8.0 B4 10.2 x 10.2 x 20.3 7.7 N I 12.7 x 15.2 9.6 N2 12.7 x 15.2 8.3 N4 12.7 x 15.2 7.6 N5 12.7 x 10.2 9.3 N7 12.7 x 7.2 7.1 N8 7.6 x 7.6 6.1 N9 5.1 x 5.1 9.9 NIO 5.1 x 5.1 8.5 Furthermore, the two Ge detectors were surrounded by Compton-suppression devices, S A and S B respectively. Each of the suppressors was constructed out of six segmented arrays of N a l ( T l ) crystals arranged i n coaxial geometry and each segment was viewed by a phototube. The basic a im of the N a l ( T l ) annulus is to detect the gamma rays that are Compton scattered out of the Ge detector -before depositing al l their energy- and reject the related events by operating the annulus and the Ge detector in anticoincidence mode. T y p i c a l performance of the Compton suppression system w i l l be i l lustrated i n Figures 5.2, 5.3 and 5.4. The two Compton-suppressed Ge detectors were located 19.4 cm and 14.7 cm respectively from the target. The other arm of the detection system consisted of an array of 12 N a l ( T l ) counters (not al l are shown i n Figure 3.3) arranged around the target i n order to increase the total gamma ray tagging efficiency. The purpose of these detectors is to select events in which a coincident 7-ray of 942 keV is emitted wi th the 1229 keV line of interest. The properties of these counters are given i n Table 3.1. The first four counters, B l through B4 , are rectangular bars while the other eight are cyl indr ical in shape. The cited resolutions are the "in-beam" F W H M at 57 the 1332 keV 6 0 C o gamma ray. In addit ion to their basic a im as Compton suppressors for the surrounded Ge detectors, the N a l ( T l ) annuli can be operated i n coincidence mode wi th the opposing Ge detectors and consequently increasing the tagging efficiency of the N a l ( T l ) a rm. 3.4 Electronics and Data Acquisition The data acquisition system can be divided into two functional subsystems: the trigger logic and the data acquisition control. The former may further be divided into four distinct blocks: the telescope logic, the Compton-suppressed Ge logic, the N a l ( T l ) logic and the strobe logic. The latter, the data acquisition control system, consisted of three components: the main V D A C S system, the front-end processor and the C A M A C modules. The overall function of the data acquisition system was to collect and process data from all of the detection components as well as to monitor their performance. It was designed to examine stopped muons and detected gamma ray events and to determine which of those should be analyzed. Each time the trigger logic received an event detected i n either germanium detectors, the corresponding N a l ( T l ) suppressor and the scinti l lat ion telescope were examined. If the event was not vetoed by the suppressor and it satisfied the stop definition, a val id strobe was generated by which relevant information from the detection system was digitized and recorded on magnetic tape and part of the data was analyzed immediately for on-line monitor ing of the experiment. A complete electronic diagram is shown i n Figure 3.4. The overall t iming of the logic signals associated wi th one of the Ge detectors is shown i n Figure 3.5. 58 3.4.1 Telescope Logic A schematic of the telescope logic is included i n Figure 3.4. In this diagram the triangles represent logical fan-in/fan-out units, which are essentially O R gates wi th many outputs. The analog signals from scintillators S i , S2 and S3, were input to quad discriminators which i n turn produced an output logic pulse for every input signal that crossed a fixed threshold level. Incident particles were determined by a (S1-S2) coincidence while stopped muons i n the target were defined by a (S1-S2-53) coincidence, where S3 means an anticoincidence of S3. Three "S1-S2-53" output signals were generated. One was used to stop the Time- to-Dig i ta l Converters ( T D C s ) . Another signal was delayed 5 microseconds (/jsec) and provided the input signal to a U B C Router box [162]. This router, also called pulse separator, accepts a t ra in of up to four logic pulses wi th in a time window, splits them and routes the indiv idual pulses into four separate outputs: A l , A 2 , A 3 and A 4 . Hence it , along wi th a T D C permits recording of mult iple time spectra; so that it w i l l give times of up to four muons wi th in a 2.5 /jsec gate, i.e. i f there is only 1 muon i n the 2.5 /jsec gate, its time relative to the T D C start, 7, w i l l be in channel A = l , i f there are 2 muons, their times w i l l be i n A = l and A = 2 etc. The last muon stop entering the router (i.e. the one i n the highest channel) was the muon most likely to have caused the gamma ray event (or the strobe). The th i rd signal from the "S1-S2-53" coincidence unit was used to init iate an updating 2.5 fisec pile-up gate for the two main coincidence units, T R I G l and T R I G 2 . This gate was extended for 2.5 /jsec each time a signal was received. 3.4.2 The Compton-suppressed Germanium Logic The signal from each germanium detectors was sent to a charge sensitive preamplifier i n order to maximize its signal-to-noise ratio and match impedance. 59 TDC atop Nal= Passive Splitter CFD 1 - TDC stop -| AMP y. ADC to STARBURST—LAM G.G Busy signal "INHIBIT^  MIM-OUT G.G . A D C gates G.G (2-5MS) U B C A=1. -, R ' O o A=2 O u A=3 3 (fi «-t-T ' o A=4 to Q R HLT1 > „ —-m-> o H D I 5» w m <= tu cn C <p O CO = F, J V.S. CS. - > ADC V.S. cs. TDC stop Figure 3.4: Complete electronic logic diagram. The symbols used are as follows: V S = v i s u a l scaler, CS=camac scaler, GG=ga te generator, B i t=b i t reg-ister; others are explained i n the text. 60 / i i /i2 jti3 /i4 y PU (extended gate) Gel SA SA Veto Gel-SA S1-S2-S3 (delayed 5iis ROUTER gate A=1 time A=2 time A=3 time A=4 time -2.5/is-TRIG1 u INHIBIT dead time /ii /i2 /z3 M IT IT Figure 3.5: T imings and definitions of events for G e l and associated elec-tronics. 61 Each preamplifier provided two distinct output signals: energy and time signals. One energy signal was shaped, amplified and pole-zero adjusted by a spec-troscopy a m p l i f i e r ( O R T E C 572 and Tennelec TC241 for G e l and Ge2 respectively) whose output was fed into high-speed, high-resolution O R T E C A D 4 1 3 A C A M A C Analog- to-Digi ta l Conve r t e r (ADC) , where its energy dependent amplitude was digi-tized and recorded. Th i s amplification and digi t izat ion was done i n the experimental area to reduce the effect of noise on the signal cables. The other preamplifier energy signal was passed through a high-threshold d i sc r imina to r (OVL) i n order to reject overload pulses in the Ge detector. These are likely Miche l electrons from muon decay and/or their bremsstrahlung radiations going into the Ge detector. These pulses would otherwise be overflow events i n the Ge A D C and would have increased the computer(processing) dead time unnecessarily. The t iming signal, obtained from the other preamplifier outputs, was fed to a t iming filter amplifier ( O R T E C model 474 T F A ) where it was shaped and am-plified before being split into two identical signals. One of these T F A signals was applied to constant fraction discriminators ( T E N N E L E C T C 4 5 5 / O R T E C 935) for fast-time pickoff and subsequently passed to the corresponding coincidence units, T R I G 1 / T R I G 2 . Constant fraction d i sc r imina tors (CFD) are used to produce fast logical signals wi th a baseline crossover nearly independent of the input signal am-plitude. The use of two C F D s -one wi th low-threshold defining the time and the other wi th high-threshold establishing energy threshold- was found to further re-duce the dependence of the pickoff time on the amplitude and rise time of the input pulses. The other T F A output was split into four identical pulses, each of which was sent to a leading-edge discriminator wi th a different threshold. The output was then sent to the input of a T D C , the start for this T D C being the " S T R O B E " , i.e. the constant fraction t iming signal from the germanium. The reason for this circuit was to measure the rise time of the pulses i n order to improve the detector resolutions, 62 both energy and t iming. This w i l l be discussed later i n the thesis, section 5.2.3. The energy and time of each of the N a l ( T l ) suppressors' segments were dig-i t ized and recorded by A D C s and T D C s for the offline analysis respectively. In addit ion, their logic signals were fanned-in together to provide the suppressor time which was used as a veto to their respective germanium detectors, i.e. G e l - S ' A and Ge2SB coincidence units. 3.4.3 T h e N a l ( T l ) L o g i c The analog signals from each of the 12 N a l ( T l ) counters was passively split-up into two signals. One was amplified and passed to a peak-sensing A D C . The other was sent to a constant-fraction discriminator and then to the input of a T D C to provide the coincidence time relative to the strobe(Ge events). 3 .4.4 S t r o b e L o g i c Every time a Compton-free gamma event ( G e l - S ' A or Ge2-SB) was detected by either Ge detector following a muon stop in the target ( " / i " ) , a good event ( T R I G l or T R I G 2 ) was generated. The events from the two triggers, T R I G l or T R I G 2 , were t imed together and input into a quad logic fan-in/fan-out un i t (LeCroy 429) denning the event " S T R O B E " , see Figure 3.4. Several strobe output signals were used from this unit . One output was used to generate front-panel interrupts 2 to the Starburst and subsequently started the data acquisitions. A least-delayed strobe output was sent to a fan-in unit known as the " I N H I B I T " unit; where a computer busy signal(generated by O R T E C ND027 N I M Driver) was also fed to form system-busy inhibits . Since it took several microseconds for the data acquisition system to produce the busy signal, a 120 //sec protection gate was created at the end of the strobe signal to cover this gap. 2The Starburst handler (TWOTRAN) responds to the Front-panel interrupts in half the time (~120 /xsec) it responds to a CAMAC Graded Look-at me,LAM (200-250 psec). 63 The inhibi t signal was used as a veto into the T R I G l and T R I G 2 coincidences to stop further events while one was being processed by the data acquisition. Once the C A M A C modules were read by the computer, the N I M driver released the inhibi t and opened the electronics for the next event. Other strobe signals were stretched by a dual gate generators(LeCroy 222) and used to gate three 11-bit A D C s together wi th the two A D 4 1 3 A D C slots. Others were used to start five 8-channel T D C s (LeCroy 2228). Another strobe output was sent to a gate generator from which a 2.5 fisec N I M output and a delayed signal were respectively used as muon gate and reset signal into the U B C Router box where the input was provided by the muon stop coincidence " /J" . 3.4.5 Data Acquisition Control The data acquisition was regulated by the T R I U M F V D A C S system which consisted of a V A X station 3200 and a P D P - 1 1 front-end-processor ( C E S 2180 Starburst, herein-after and -before, s imply called Starburst). Figure 3.6 represents a schematic flow of the data through the V D A C S system. The V A X computer allowed the user to control aspects of the experiment such as data logging and online moni tor ing together wi th downloading the user-defined T W O T R A N program to the Starburst which i n turn was responsible for the actual real-time data acquisition from the C A M A C modules. "Strobe" events caused the Starburst to be activated and the T W O T R A N program then collected the relevant data from the C A M A C modules. For each strobe event, the following information (total of 69 data words) were digitized and writ ten to 8 m m tape: • Energy Signals of : the two Ge detectors, the 24 N a l ( T l ) detectors, and the three scinti l lat ion counters. 64 CAMAC Hardware 'electronics' Experimental Data r CAMAC Commands PDP-11 ( T W O T R A N ) FEP Buffered Data Control Commands D a t a L o g g i n g O n l i n e M o n i t o r i n g ¥ / N Figure 3.6: A schematic flow of the data through the Da ta Acquis i t ion System. 65 • Timing Signals of : the 24 N a l ( T l ) detectors, the three scinti l lat ion counters, the four router signals and the 8 Ge leading edge thresholds. • Event Bit patterns for: S i , S2, S3, S1-S2-53, T R I G l , T R I G 2 , 0 V L 1 , 0 V L 2 , P U 1 , and P U 2 . In addit ion to the above information, the following C A M A C scaler values were recorded at the end of each run: S I , S2, S3, S1-S2, S1-S2-53, > " , Gel-SA, Ge2~SB, T R I G l , T R I G 2 , " S T R O B E " , N a l S U M , C L O C K , CLOCK-INHIBIT. 66 Chapter 4 Technical problems This chapter describes some of the technical problems which were encountered i n the data analysis of the experiment. In addit ion, some relevant basic concepts w i l l be highlighted. B o t h of these w i l l help in the understanding of the two subsequent chapters. Each subsection is l ikely to be rather brief, and sometimes slightly isolated, since a complete description of these topics is provided by excellent works, some of which are cited i n the bibliography. 1 4.1 Basic Interactions in 7-ray Detectors Gamma-rays interact wi th matter i n various ways. However, only three pr imary processes, namely photoelectric absorption, Compton scattering and pair production, have any real significance for gamma-ray detection. In a l l these processes, gamma rays transfer a l l or part of their energy to the detection medium by generating free electrons. It is these secondary electrons that cause the ionization which is actually measured in the photon detectors. Before discussing this detection, we shall present a brief summary of these basic processes producing the secondary electrons. In the photoelectric absorption process, a gamma-ray ejects a bound electron and imparts to it a l l of its energy 1 . The ejected photoelectron has a kinetic energy 1A minute amount of energy is imparted to the associated atom to conserve energy and 67 given by the incident gamma-ray energy, Ey, minus its b inding energy. A s a result the electron w i l l create a vacancy in an atomic shell causing the atom to de-excite, l iberat ing the binding energy in the form of characteristic X-rays or Auger electrons. B o t h the photoelectron energy as well as its b inding energy are generally fully absorbed i n the surrounding material so that a l l of the original 7-ray energy is transferred to the medium. Th i s fact makes the photoelectric absorption an ideal and preferred mode of interaction for the measurement of the original 7-ray energy. The probabil i ty for photoelectric absorption is approximately proport ional to - ^ 3 - , where Z is the atomic number of the material . In the Compton scattering process, the incident gamma-ray scatters from a free atomic electron 2 transferring part of its energy (£" 7 ) to the electron (Ee) and departing wi th the rest, E'. These energies depend on the angle, 8, of the scattered gamma ray as follows: E EP- = E-, — Es,i = E~ — & (4.1) 7 7 7 1 + - ^ ( 1 - c o s 8) y ' where m 0 c 2 is the energy corresponding to the rest mass of the electron. Hence, the amount of energy deposited in the surrounding material extends from zero, corresponding to a grazing angle of 8 — 0°, to a max imum energy corresponding to a head-on collision angle of 8 = 180° (Compton edge). The probabil i ty for Compton scattering at an angle is given by the famous K l e i n - N i s h i n a formula [163]. For a gamma-ray energy i n excess of 0.5 M e V , the total Compton cross section is approximately proport ional to J k For large germanium detectors, as many as half the events in the full energy peak are actually a Compton scattering followed by a separate photoelectron event. In the pair production process, a gamma ray wi th energy i n excess of 1.022 momentum. 2 A good approximation for a loosely-bound outer atomic electron with typical orbital energy usually small compared to the incident gamma-ray energy 68 M e V disappears, creating an electron-positron pair sharing the excess energy as kinetic energy. The positron almost always reacts wi th an electron in the medium and annihilates wi th the simultaneous emission of two 0.511 M e V photons. The amount of energy deposited i n the medium i n this process depends on the subse-quent absorption of both, one or neither of the 0.511 M e V annihi lat ion photons. Consequently, a discrete amount of energy E^, E~, — m0c2 or E^ — 2moc2 correspond-ing to photopeak, single-escape peak or double-escape peak, gets absorbed i n the medium respectively. The atomic cross section for pair product ion increases wi th increasing Z and E-y. For a gamma-rays of 4 or 5 M e V i n a small detector these three peaks are quite clear. In our detector the escape peaks are normal ly rejected because of the Compton suppressors. The pair production process becomes dominant only for high energy gamma-rays while the photoelectric absorption process is the dominant mode of interaction for 7-rays of relatively low energies. The Compton scattering dominates over the range of energy between the two processes. In order for a detector to serve as a good gamma-ray spectrometer, it must carry out two distinct functional steps. F i r s t , it must act as a conversion medium i n which incident gamma rays have a reasonable probabil i ty of interacting to yield "secondary" electrons(positrons) v i a one or more of the aforementioned processes. Second, it must function as a conventional detector for these electrons. In the first step, the atomic number of the detector material along wi th its density and volume are important . Since the cross section for the preferred mode of interaction, the photoelectric absorption, varies approximately as Z 4 - 5 , the trend is to incorporate elements wi th high atomic numbers. Th i s consideration favors N a l ( T l ) (or Cs l ) over Ge since Zj = 53 compared to Zae = 32. The second step, the detection of the "secondary" electrons, consists of the production and the subsequent collection of information carriers; electron-hole pairs 69 created along the path of the "secondary" electrons i n Ge detectors and photoelec-trons released by light emitted in excited molecular states i n N a l ( T l ) detectors. The number of information carriers should be linearly proport ional to (and as large as possible for) a given incident radiat ion to minimize the statistical contribution to the energy resolution. The average energy required to produce an information carrier (photoelectron) i n N a l ( T l ) is at least two orders of magnitude greater than that required to produce one (electron-hole) pair i n Ge detectors 3 . Due to their excellent energy resolution, Ge detectors are now being used i n vir tual ly a l l gamma-ray spectroscopy that involves complex energy spectra. Thei r superior resolution is i l lustrated i n Figure 4.1 i n contrast to N a l ( T l ) scintillators. The Ge energy spectra are complicated by the presence of prominent continua due to part ial escape of in i t i a l gamma-ray energy from the active volume of the detector. To reduce such continua, Compton suppression devices made up from N a l crystals were used i n this experiment. For more extensive discussion of the basic interactions i n gamma-ray detectors and detector characteristics, the reader is referred to the two excellent textbooks by K n o l l [164] and Debert in and Helmer [165]. 4.2 Neutron Effects in Ge-detectors In the neutron environment of the T R I U M F accelerator, two distinct effects on Ge detectors ought to be addressed. These are fast neutron damage and neutron-induced peaks, both of which can affect the detectors' performance. It has been found that Ge detectors are susceptible to radiat ion damage (see reference [164]). Fast neutrons can knock-out germanium atoms producing pre-dominantly isolated defects wi th in the active volume of the hyperpure Ge detector. 3 3 eV per electron-hole pair; 300-1000 eV per photoelectron. 70 Figure 4.1: Compar ison of the measured energy spectra for one of the H P G e detectors (lower curve) and for one of the N a l ( T l ) scin-tillators (upper curve). The prominent peaks at 662 keV and 1173 keV and 1332 keV are from 1 3 7 C s and 6 0 C o sources respec-tively. The bumps at 460, 950 and 1100 keV are the Compton edges which become bumps because of the suppressors. 71 These defects act as "hole" traps, thus increasing the variat ion of the amount and time of charge collected per pulse. The former variation, incomplete charge col-lection, causes deterioration i n the highly-prized energy resolution, while the latter variation i n collection time gives rise to different pulse rise times (if detrapping takes place) and hence worsens the t iming resolution of the detector, see Section 5.2.3 on the rise time correction. It has been experimentally proven [166,167] that n-type(reverse electrode) H P G e detectors offer greatly improved damage hardness over the conventional p-type ones. Th i s finding was to some extent expected from geometry considerations. Most of the interactions occur i n the outer port ion (periphery) of the detector because of the geometry of the closed-end coaxial detectors. Thus a p-type detector having the conventional outer peripheral contact as n+ (positive electrode) forces the holes to originate and travel a larger average distance to the negative p+ contact. Thus in the p-type configuration, the hole collection process dominates the signal, whereas i n the n-type configuration, the electron collection process dominates the signal. Following the above reasoning, the two H P G e detectors used i n the present experiment were of n-type; even though this type is more expensive. The other effect on Ge detectors in the presence of neutrons is the appearance of a number of spurious peaks i n the 7-ray spectra. These neutron-induced peaks appear as a consequence of excitation of various excited states of the nuclei of the germanium isotopes i n the detectors, by inelastic neutron scattering, followed by emission of conversion electrons and X-rays which are totally absorbed wi th in the Ge crystals. Typ ica l neutron-induced 7-rays i n Ge at 596 keV and 693 keV are evident for the in-beam energy spectrum of Figure 4.2. The skewness towards higher energies of these peaks happens because some of the nuclear recoil energy is converted into electron-hole pairs [168] that add to 72 ~~i I i i i — i — i — i — I — i — I — I — i — i — i — i — i — i — i — i — i — I — i — i — i — i — i — i — i — i — r 4 0 0 6 0 0 8 0 0 1 0 0 0 Energy (keV) 1 2 0 0 Figure 4.2: Gamma-ray spectrum from muons stopping in a silicon target. The neutron induced 7-rays in germanium are evident. 73 Table 4.1: Neutron induced 7-ray lines i n germanium isotopes. Those i n brackets were not observed i n this experiment. Ge isotope Natura l Abundance (%) (n,n') induced 7-rays (keV) 70 20.5 1039.6, 1215.6, (1708.2) 72 27.4 629.9, 691.5, 834.1, 1464.1 73 7.8 868.0 74 36.5 595.9, 608.4, 1204.3, (1482.6) 76 7.8 562.9, 1108.4, 1410.1, (1539.1) the associated transit ion energy. (Up to 30 or 40 keV electron equivalent can be added.) These background peaks are seen i n most in-beam gamma-ray measure-ments that follow reactions involving neutrons as outgoing particles. They have been extensively investigated earlier by Chasman et al . [169] and others [168,170]. A complete list of these neutron-induced lines is given i n Table 4.1 under their respective isotope. The presence of such spurious peaks can interfere wi th the analysis of other nearby 7-rays of interest 4 . In fact, it was part ly these induced peaks, in part icular the lines at 1204 keV and 1216 keV, which spoiled the first phase of this experiment and prompted the modification of the experimental technique, discussed in the next section. 4.3 Coincidence Technique A s mentioned earlier, the a im of this experiment is to measure the angular correlation between the neutrino and the 1229 keV Doppler-broadened gamma ray i n 2 8 A l following muon capture on 2 8 S i . This a im was pursued i n the first phase of the experiment (December 1989) by the measurement of the line shape of the 1229 keV peak i n the "singles" energy spectrum. 4 Unlike pion capture where the responsible neutrons originate directly from the prompt absorp-tion of the pion, one cannot use time-of-flight techniques to discriminate against the (n,n'7) peaks 74 10' J I L . _J I l_ _1 I I l_ So O 10 Lr 1 1 1 1 1 F 1150 1 2 0 0 1 2 5 0 1 3 0 0 Energy (keV) 1 1 1 1 1 1 1 1 r 1 3 5 0 Figure 4.3: 2 8 S i gamma-ray energy spectrum (binned by 4) obtained wi th G e l detector, in the vicini ty of the Doppler-broadened 1229 keV peak. 75 Figure 4.3 shows a gamma-ray energy spectrum obtained wi th a silicon target for the G e l detector, in the vic ini ty of the 1229 keV peak. A plateau-like structure below and possibly extending underneath the line of interest is evident. The overlap of this plateau wi th the peak of interest made the extraction of the angular correla-t ion from the line shape very problematic. Th i s plateau was clearly present i n our in i t i a l run [160] as well as i n other previous experiments from other laboratories. One can try to fit the singles spectra, using various assumptions for the be-haviour of the background underneath the 1229 keV peak, to extract the angular correlation coefficient(s). However such an approach is very prone to systematic er-rors. Nevertheless, it may be warranted to do so as a complementary approach. This concern was the pr imary motivat ion for the adoption of a coincidence technique. In addit ion to suppressing the continuum background underneath the 1229 keV peak of interest, the coincidence technique gives information about potential problems due to cascade-feeding of the transit ion of interest, see section 5.4. Figure 4.4 illustrates the coincidence technique. It is based on the following: each of the 1229.1 keV gamma-rays from the (2201.5 keV 1+ —-> 972.4 keV 0+) transit ion is followed by a 942.1 keV gamma-ray from the (972.4 keV 0 + —> 30.6 keV 2 + ) transition. B y detecting these 942 keV gammas i n the N a l crystals sur-rounding the silicon target 5 , the 1229 keV line observed i n the two H P G e detectors can be "tagged". The effect of this method is to preferentially select the 1229 keV gamma rays that are in coincidence wi th 942 keV gammas, i.e. those that actually arise from the reaction of interest. T w o relevant facts supporting the above coincidence scenario should be high-lighted. B o t h stem from the nature of the intermediary level (972.4 keV, 0 + , 43 ps). in muon capture 7-ray spectra. 5Typical Nal energy-spectra in the 942 keV region along with a typical gate used to tag the 1229 keV 7-rays are shown in Figure 5.17. 76 0+ 0 Figure 4.4: The production of the 1229 keV gamma ray i n 2 8 A 1 and the coincidence technique. • F i rs t , its spin-parity of 0 + leads to an isotopic dis tr ibut ion of the 942 keV line wi th respect to the 1229 keV gamma ray. Thus allowing the placement of large number of N a l crystals anywhere around the target. • Second, its relatively long (43 ps) lifetime does not cause Doppler-broadening of the 942 keV line shape. Th i s allows a relatively tight coincidence windows around the 942 keV tagging-line i n the N a l energy spectra. Figure 4.5 shows typical 2 8 S i gamma-ray energy spectrum before and after the imposi t ion of the energy-gated coincidence requirement wi th the N a l crystals. It conveys the crux of the coincidence method. Li t t l e more needs to be said other than emphasizing the complete suppression of the 7-rays that are not associated wi th genuine cascades, such as the 1294 keV air activation 7-ray line from the decay 77 2 0 0 0 0 15000 c 10000 o o 5 0 0 0 o) S ing les 1 1 3 2 keV Mg 1050 1100 1150 AtFe^P-OS) 1200 1294 keV A r 1250 Energy (keV) 1300 1350 1000 8 0 0 6 0 0 -b) Coincidence 1250 Energy (keV) 1350 Figure 4.5: 2 8 S i gamma-ray energy spectra, i n the vicini ty of the 1229 keV peak, before and after the imposi t ion of the energy-gated co-incidence requirement wi th the N a l crystals. 78 of 4 1 A r 6 , and muonic X - r a y lines. O n the other hand, real coincidences from 7-7 cascades following muon capture on 2 8 S i (or even background) were expected and are seen i n the figure, e.g. the 1229 keV 2 8 A l and the 1132 keV 2 6 M g lines. (The particle hi t t ing the N a l can be a neutron as well.) The reduction in the number of counts of the peak of interest i n the coincidence spectra due to the finite N a l coincidence-efficiency (and the consequent increase in statistical error) was more than offset by the gains due to the suppression of the backgrounds beneath the peak. However, as discussed earlier, the data acquisition circuitry was designed to run in a "singles" mode wi th the coincidence requirement applied later in the software rather than at the hardware stage. The advantage of this was the possibil i ty of analysing the singles spectrum wi th its high statistics and thus providing a supplementary measurement to that of the coincidence spectrum. This topic w i l l be revisited i n chapter 5. Other related topics deferred to that chapter include: the remaining flat background in the coincidence spectra, the photopeak efficiency of the N a l arm, along wi th total coincidence efficiency, as well as the signal to noise improvement obtained. 4.4 The Detector Response Function In order to determine the angular correlation coefficients from the shape of the line of interest, an accurate representation of the detector response function is of central importance. As discussed in section 4.1, the preferred mode of interaction, the photoelec-tric absorption, is not the only process contributing to the pulse-height spectra i n Ge detectors, nor does it always yield pulses corresponding to full-energy peaks. 6 E.g. 4 0 A r ( l % in air) + n - * 4 1 Ar(1.8 hrs) / T + 41K*(1294 keV). (4.2) 79 Furthermore, photon interactions i n the detector surroundings may add to the pulse-height spectra; see section 5.2 for their suppression. The net effect of the different interaction processes i n and around the Ge detectors is a complex response function for a mono-energetic 7-ray as well as a complex pulse-height spectrum. The full energy photopeaks, Compton edges along wi th flat continuum as well as single- and double-escape peaks may contribute to such spectrum, see for example Figure 4.1. To determine the line shape, it is useful to understand the origin of the various contributions to the full-energy "peak", herein-after s imply called the peak, and the "background" i n its vicinity. Figure 4.6 shows the 1332 keV 6 0 C o peak, extracted from Figure 4.1, together wi th various components of one of the complex analytical line shapes [171] to il lus-trate the correspondent contributions to the peak. The labelled components i n the figure are as follows: 1. Flat continuum which is due to the Compton scattering, bremsstrahlung losses and escaping photoelectrons from the sensitive volume of the detector for higher energy gamma-rays. 2. Step-like component produced mainly by the escape of pr imary or secondary electrons (energy) from the sensitive volume of the detector. 3. Gaussian component representing the full-energy peak produced by "ideal" photoelectric absorptions. 4. Low-energy tail, which is due to trapping effects in addit ion to incomplete charge collection. 5. High-energy tail, which is usually due to pile-up effects and high counting rates. 80 J I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I L T — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — r ~ i — r - ^ — i — r i — i — | — i — i — i — i — | — i — i — i — i — | — i — r 1 5 6 0 1 5 8 0 1 6 0 0 1 6 2 0 1 6 4 0 1 6 6 0 1 6 8 0 1 7 0 0 Channel Figure 4.6: Various components of the response function, R F 3 , fitted to the 1332 keV 6 0 C o peak. 81 Table 4.2: Qual i ty of fit to the 1332 keV 6 0 C o peak for different response functions. peak-shape function # of parameters x2 X2R R F l : G a u s s i a n 3 817.7 5.33 RF2:Gauss ian wi th joined tails [152] 5 313.7 2.03 RF3:Gauss ian wi th tails [171] 9 197.6 1.32 In addit ion, an improperly adjusted pole-zero i n the main amplifier and/or in-sufficient base-line restoration can contribute to the low- and/or high-energy ta i l components. The first two components are usually grouped together under the name "back-ground" while the last three components are labeled as the "peak". A variety of response functions have been tr ied out. It was found that the best representation of the background components consists of a constant representing the flat continuum and a gaussian-convoluted step-function given by the complementary error function [171]: | e r fc [ (x - X0)/y/2a] (4.3) wi th amplitude S, gaussian wid th a, channel number x, and peak center XQ. The peak-shape function representing the peak components fall into two cat-egories, those which retain the main gaussian term wi th smoothly added tails on both sides, and those which are a sum of several functional terms. T w o peak func-tions, along wi th the gaussian function, corresponding to the above two categories are given in Table 4.2 showing the quality-of-fit to the 1332 keV 6 0 C o peak. Note that for such peak wi th large number of counts, 1 . 5 x l 0 5 , and large fitting range of channels, one needs to add i n more tai l ing terms to br ing the reduced chi-square value, x2p,i closer to 1.0. The simple gaussian representation is not a sufficiently accurate representa-tion. The complexity of the th i rd function and the absence of good (i.e. strong and 82 clean) peaks throughout the energy spectra made the energy dependence of this function uncertain for the in-beam spectra. The second peak-shape function, R F 2 , was found to form as good a representation of the detector response as was needed. Furthermore, it was easier to parameterize, i.e. to determine its parameters as a function of the 7-ray energy. Consequently, it was adopted as the peak-shape func-t ion of the response function of our detectors. It is represented by where PL and PR are the joining points of the left and right tails measured from the center of the peak(gaussian) respectively. The parameterization of the response function was carried out as follows: first, the step function amplitude, 5 , was estimated wi th the help of calibration sources ( 1 3 7 C s and 6 0 C o ) 7 . Then fixing S at its estimated values, the other parameters, cr, PL and PR were determined by the use of in-beam strong peaks through-out the energy spectra. For these results, analytical expressions giving these parameters as a function of the 7-ray energy were determined. A s an i l lustrat ion, the param-eterization of these parameters for G e l is shown in Figure 4.7. A consistent and reasonable set of values is obtained throughout the energy range of interest for both Ge detectors, see Table 5.4. In the subsequent analysis of the Doppler-broadened lines of interest, these parameters were fixed at the estimated values taken from Figure 4.7 for G e l and a similar set of curves for Ge2, see section 5.6. 7 W e find that S is around 0.1% of the peak height which is smaller than that observed by others who have found typically 0.5%. We ascribe this to our large detector, viz. ~ 40%. ePL[PL+2(x-X0)}/2a2 e-(x-X0f/2a2 ePR[PR-2(x-X0)]/2a2 x - X 0 < - P L - P L < X - X 0 > PR X - X 0 > PR 83 Figure 4.7: Energy dependence of the parameters of the response function, R F 2 for the G e l detector. To convert to keV, these parameters should be mul t ip l ied by 0.36 keV/channel . 84 4.5 Slowing-Down Effects The Doppler-broadened 7-rays of interest are emitted from nuclei recoiling in matter, and hence their energies are shifted depending on the instantaneous velocities of the nuclei at the time of emission. The nuclei are slowed down by the medium, and so the Doppler broadening of the emitted 7-rays depends on the velocity at the time of emission and thus on the lifetime of the particular level. Hence information about the lifetime may be obtained from the shape of the 7-ray line, provided the slowing-down time is known. In the case of the nuclear level of interest, i.e. 2 8 A1(2201 keV, 1 + ) , there have been two previous attempts to measure its lifetime. B o t h measurements uti l ized the Doppler-shift attenuation method ( D S A M ) as applied to the 2 7 A l ( d , p ) reaction, by measuring the Doppler shift of the 7-rays emitted in coincidence wi th the protons. The reported values are (35 ± 10)fs [172] and (120 ± 70)fs [173], which is confusing though not totally inconsistent. A major contribution to the uncertainty in these measurements- and i n a l l D S A M lifetime measurements- has been the inexact knowledge of the slowing-down effects (or stopping powers). These effects have been studied since the beginning of the century. The total stopping powers may be divided into two parts: the electronic stopping power and the "nuclear" stopping power. They give the energy loss due to the interaction of ions wi th the target electrons and wi th the target nuclei respectively 8 . The former dominates at high ion energies (above 100 keV) , the latter at low energies. Figure 4.8 shows the two parts as calculated by the Biersack and Ziegler code(1989) [175] for our reaction 2 8 S i ( ^ , ^ ) 2 8 A 1 , i.e 2 8 A l ions i n a natSi medium. The in i t ia l recoil energy for the 2201 keV level is 187 keV and s T h e nuclear stopping component can be separated because the heavy recoiling target nuclei can be considered to be unconnected to their lattice during the passage of the ions, and the interaction can be treated as the kinetic scattering of two screened particles [174]. 85 3 J I I L J I I l_ J I I l _ J I |_ CP 2 > CD x V 0 0 5 0 Lcgcmd TRIM ZB total ZB nuclear ZB electronic T 100 150 Energy (keV) 2 0 0 Figure 4.8: Stopping-power curves for 2 8 A l ions i n natSi medium. The ar-row corresponds to the in i t i a l recoil velocity, /?o=0.0037, ?.e. 187 keV. hence one must take both parts into account. See reference [176] for a comprehensive treatment of this topic. The slowing-down effects, along wi th the lack of a reasonable measurement of the level lifetime, complicate the extraction of the angular correlation coefBcient(s) from the observed 1229 keV 7-ray. Th i s comes about due to the fact that there is some coupling between these effects on the line shape of the Doppler-broadened gamma-ray of interest, see also figure 5.23. Our analysis indicates that the slowing-down effect is of central importance and cannot be neglected. The energy-loss 86 1 2 0 1 0 0 -^ 8 0 6 0 4 0 -2 0 _i i i i i i i i i i i i i i i i i_ 1.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 1.5 2 . 0 2 . 5 3 . 0 3 . 5 dE/c lX ( M e V / m g / c m ) 4 . 0 Figure 4.9: Relationship between the stopping-power and the extracted lifetime from the 2171 keV 7-ray line. Roughly r d E / d X = 140 f s . M e V . c m ^ m g " 1 . history of the 2 8 A l ions as a function of their velocity (energy) was simulated using electronic and nuclear stopping powers by the Monte Car lo code T R I M ( T R a n s p o r t of Ions in Mat ter) [175], as can be seen i n Figure 4.8. Different shapes of the stopping-power, were used to fit this function, but the details do not matter. This topic w i l l be further discussed i n section 5.6 i n conjunction wi th the lifetime measurement of the 2201 keV 2 8 A l level. Here, it suffices to stress the fact that the choice of j^- as a function of velocity does not influence the lifetime value significantly because the ion does not slow down very much. Al so , the absolute value of the directly affects the value of the lifetime which is extracted(see Figure 4.9), but it does N O T affect the angular correlation coefficients. 87 4.6 Finite Solid-Angle Effects In this section, we wish to consider the effects of finite target size and/or the solid angle of the Ge detector on the measured angular correlation coefficient, a, i.e. to investigate the sensitivity of the angular correlation to the other corre-lat ion coefficients, f3\ and f3\. A s noted earlier, in this experiment, the 7 rays are detected at 90° to the fi polarizat ion direction and hence f3\ and (32 terms, involving (fj, • 7), drop out i n the correlation function, equation (2.2). However, i n a finite target/detector system, there exist parts of the detector for which (p. • 7) ^ 0 due to solid-angle effects. To check such effects, we used a Monte Car lo program to generate the Doppler-broadened spectrum of the 1229 keV 7-ray of interest for various values of a , f3\, 02 and the angle 9^ (i.e. the angle that ideally is 90°). For each spectrum, the gammas were generated over a range of these angles. F r o m the geometrical factors, the largest possible angle that the detector could be off from 90° is about 15° (which is very conservative, as it assumes that the muons a l l stopped i n the back of the target, and that they al l stopped on the extreme edge of the target, and that the target was not on the detector center-line, but off by 1 cm), and the angle varies by ± 2 5 ° over the detector. This is for G e l , and the effect is smaller for Ge2. The resulting spectra were then fit using M I N U I T , wi th /3\, f32 and cos# 7 1 / al l fixed i n the fit to 0, i.e. the ideal (not the actual) values. Compar ing the fitted values of " a " to the actual value used in generating the spectrum showed the following: i n the above extreme case (and wi th (3\ and L32 both at their m a x i m u m allowed values), the extracted a is reduced by only 10% from its actual value. For a more reasonable estimate of the probable angle, i.e. off by 5°, the effect on a is on the order of 2%, i.e. completely negligible. Note that for the extreme values, one can see the effect of f3\ and B2 on the 88 spectrum, i.e. the chi-squared of the fit gets a bit worse wi th systematic deviations between the fit and the data, but the extracted value of the correlation coefficient of interest, a , is almost completely unaffected. This result allowed the neglect of the other correlation coefficients i n our fitting function. Furthermore, it eliminated any complications in the analysis due to the residual muon spin polarizat ion at the time of capture. 4.7 The Peak Fitting Programs Various computer programs were used for different aspects of the data analysis. In the first step i n the data analysis, the U K I E D I S P L A Y [177] sorting analysis program wi th user-written subroutines reads the V D A C S - f o r m a t t e d tapes and sets up various energy and time histograms. These histograms were further examined, analyzed and fit using D I S P L A Y . Several T R I U M F general-purpose routines such as P L O T D A T A , E D G R and P H Y S I C A were very useful for manipulat ion, fitting and plot t ing of data. Exper imenta l data of the Doppler-broadened gamma-rays of interest were fit to a functional form representing the (double) convolutions of the response funct ion(RF) , angular correlation funct ion(AC) and the slowing-down function(SD) given in sections 4.4, 2.5 and 4.5 respectively. The convolutions had to be carried out numerically. T w o quadrature methods were tried: Simpson's composite rule and the I M S L D T W O D Q routine. They gave consistent results. The latter, an adaptive quadrature scheme, was adopted 9 The best values of the unknown parameters along wi th their uncertainties and correlations are obtained by min imiz ing the difference(chi-square) between the (theoretical) functional form and the experimental data. This was carried out wi th 9Simpson's rule uses equally spaced nodes and hence is not the most efficient method (in accuracy and speed) for gamma-ray peaks, where the interval of integration contains regions with both large and small functional variations. On the contrary, the adaptive quadrature methods use smaller stepsize for large variation regions and vice versa. 89 the use of the C E R N M I N U I T package/programme. The various input parameters needed i n the least-squares fitting program wi l l be summarized i n section 5.6. 90 Chapter 5 Data Analysis 5.1 Introduction The principle goal of the data analysis i n this work is the extraction of the angular correlation coefficient ( A C C ) from the 1229 keV Doppler-broadened gamma-ray i n 2 8 A l following muon capture on 2 8 S i . Th i s task is accomplished i n two main stages: the opt imizat ion of the energy and time gamma-ray spectra and the analysis of the principle Doppler-broadened gamma-ray lines of interest. The former aims at producing "clean" Ge energy spectra appropriate for the line shape analysis. This is achieved by imposing conditions (or cuts) on the data (5.2) as well as doing side-band background subtraction (5.5). The second stage, the line shape analysis, includes the estimation of the lifetime of the common level (2201 keV, 1+, 2 8 A l ) from the 2171 keV line (5.7.1) and the extraction of the angular correlation coefficient from the 1229 keV line (5.7.2). Other topics include cascade feeding (section 5.4) and 7-ray efficiency (section 5.3). The off-line analysis of the data was carried out using the T R I U M F Da ta A n a l -ysis Centre V A X cluster of computers, mainly two V A X 4000/100's named E R I C H and R E G along wi th several Vaxstations. Various computer programs were used for different aspects of the data analysis. In the first step i n the data analysis, the sorting analysis program version of D I S P L A Y [177] wi th user-written subroutines 91 read the V D A C S - f o r m a t t e d tapes and set up various energy and time histograms. These histograms were further examined, analyzed and fit using D I S P L A Y . Besides D I S P L A Y , several T R I U M F general-purpose routines such as P L O T D A T A , E D G R , and P H Y S I C A were very useful for manipulat ion, fitting and plot t ing of data. 5.2 Cuts This section outlines the different conditions imposed on the data to select events. The choice of what cuts to use, and how restrictive to make them, was to some extent a matter of judgement since, in general, each cut w i l l reject a certain number of good events. 5.2.1 Time of the Muon Cut Signals from the Ge detectors can be divided into three categories depending upon their time relative to the muon stopping i n the target, i.e. the last muon stop pulse (see /3.4.1). The first category contains the Ge signals which are i n prompt coincidence wi th a stopped muon and constitute essentially only X-rays from the muonic cascade. The corresponding energy spectra are called the "prompt" spectra. "Delayed" spectra, recorded wi th in the muon lifetime i n the relevant target preferentially contain delayed 7 events, resulting from nuclear muon capture i n the target. The th i rd category constitutes the "background" spectra. It contains the Ge signals which are not related in time to a muon stop. A typical plot of M U S T O P t iming distr ibut ion is shown i n Figure 5.1. It is characterized by the above three categories: a large spike due to prompt muonic X rays, a long decaying exponential ta i l w i th a slope characteristic of the muon disappearance lifetime in the Si target (756 ns) and a flat random background. B y choosing different time windows on the M U S T O P spectrum, energy spectra were reconstructed offline to reduce background and increase the signal to noise ratio as 92 4 0 0 6 0 0 8 0 0 1000 1200 Channe l number 1400 Figure 5.1: T ime of the muon spectrum for G e l wi th a Si target. The dispersion is about 2.67 ns /ch (time goes in the "backwards" direction). The periodic ripple is due to the differential non-linearity of the 20 M H z T D C used while the dip (kink) around the prompt peak is due to the suppressor vetoing some events. 93 I I I I I I I I I I 1 I 1 I I I I I I I I I I I I I I I I I I I I 1—L 500 1000 1500 2000 Gamma Ray Energy (keV) Figure 5.2: Ge2 Compton suppressed (lower) and unsuppressed (upper) spectrum typical for muon capture on a Si target. well as to help in identifying the gamma ray lines. 5.2.2 Compton-Suppression Cut The data recorded on the tape during this experiment included the time in-formation for each event detected in each segment of the two Nal(Tl) suppressors. Generally the suppressor's cut was hardwired loosely so as not to veto too many Ge events. It was tightened and optimized later in the software during the offline analysis. In order to see the effect of this cut, the suppressor vetos were not hard-wired for some runs. Instead, all the data were written to tape and then different suppressor cuts could be established offline to reject the Compton-scattered events. Figure 5.2 shows a part of a Si spectrum with and without the suppressor cut. One notes that the suppressor cut preferentially rejects events in the continuum, without affecting the full-energy events, except for accidental coincidences. This is 94 10 > CD C Z5 o o 10 -ic r _l I I I u _1 I I I I I I I I I 1 I I 1 I 1 I I I I I I I I I I L _ I j I ! I 1 I I I I I | I I I I I I I I 1 | I I 5 0 0 1000 1500 Gamma Ray Energy (keV) 2 0 0 0 Figure 5.3: Da ta removed by the N a l ( T l ) Compton suppressor from the previous figure. 4 0 0 8 0 0 1200 1600 Gamma Ray Energy (keV) 2 0 0 0 Figure 5.4: Compton suppressed and unsuppressed 6 0 C o spectrum of G e l . Note that the Compton edges become bumps i n the suppressed spectrum. (There is a discriminator cut off below 500 keV.) 95 demonstrated in Figure 5.3, which shows the data that was actually removed from the unsuppressed spectrum by the N a l ( T l ) suppressor. Note however that the 511 keV peak due to fi+ annihi lat ion is prominent because it comes i n pairs. Another effect is that the Compton edges become broad peaks i n the suppressed spectra; this can be seen more clearly i n Figure 5.4. This is due to gamma-rays scattered back out of the Ge entrance holes in the suppressors, and hence not detected in the respective N a l annulus (Compton edge events correspond to incident 7 rays being backscattered toward their direction of origin, i.e. head-on Compton scattering wi th 6 = 7r). The Compton suppression factor is usually defined as the ratio of unsuppressed to suppressed data in the continuum. The average factor obtained for a typical in-beam Si spectrum (e.g. Figure 5.2), in the vicini ty of the 1229 keV 7-ray, is around 5.7 and 8.4 i n G e l and Ge2 respectively. 5.2.3 Rise Time Correction It is well known that the amount and time of the charge collected in a Ge detector are not constant but depend on the gamma-ray interaction posit ion i n the detector, presence of defects i n the Ge lattice, and non-uniformities in the electric field in the crystal as well as the detector geometry [164]. These variations produce variable height and shape i n the pulse for mono-energetic 7-rays i n Ge detectors and hence l imit their energy and time resolutions respectively. This means that the time for these pulses to rise to any given voltage would be different. In this experiment, besides the use of two constant-fraction discriminators-one wi th low-threshold defining the time and the other wi th high-threshold establishing energy cutoff- to reduce the dependence of the pickup time on the amplitude and rise time of the input pulses, we have used a leading edge (rise-time) method to 96 Chonnel number Channel number Figure 5.5: G e l leading edge spectra corresponding to different discr imi-nator thresholds for a typical / /S i run. measure the rise time of the pulses, in order to compensate for these variations. This is based on the following: 1. The Ge pulses are split into four identical pulses using a linear fan-out, and sent to 4 leading-edge discriminators wi th different threshold levels. The out-put pulses are then sent to stop T D C clocks, the start for these T D C s being the constant fraction t iming signal from the germanium; see the block dia-gram of the electronic arrangement i n Figure 3.4. Figures 5.5 and 5.6 show leading edge spectra ( L E i = i ; ^ ) for the / /S i runs for G e l and Ge2 detectors respectively. 97 20-. CD  o _c 15 H CJ Ui ~c 3 10-i o o 5 1 100 a) Ge2 LE1 200 300 Channel number 400 500 200 300 Channel number 500 100 200 300 Channel number 100 200 300 Channel number 500 Figure 5.6: Ge2 leading edge spectra corresponding to different discr imi-nator thresholds for typical yuSi run. 98 2. The L E spectra were subtracted from each other to find the most appropriate rise-time spectra. These were L E 3 - 1 and L E 4 - 1 for G e l and Ge2 respectively. ( G e l L E 4 threshold was set too high for our peak of interest). 3. The Ge energy and N a l - G e t iming coincidence spectra were then sorted by setting 10 windows (bins) on the rise-time spectra LE3-1 and L E 4 - 1 obtained above. To find the correlation between the energy spectra and the rise-time spectra, the centroids of several 7 - ray lines were determined and plotted against the centers of the corresponding rise-time bins. A least-squares fit to a quadratic function was made for each curve to determine a correction factor. This factor was then applied to the energy spectra for each pulse which had a non-standard rise time. This was done for each Ge detector. Figure 5.7 and Figure 5.8 show this correlation for the 1173 keV 6 0 C o 7 - ray ( / i beam on) along wi th the rise-time corrected data for the G e l and Ge2 detectors respectively. Energy resolution improvements of up to 100 eV F W H M were obtained for the region of interest. A n even more useful effect was the time resolution improvement obtained v ia the correlation between the centroids of the t iming coincidence spectra and the rise-time spectra. Th i s correlation was found i n the same way as above to create a "corrected" t iming spectra. This correlation relation was then applied to the time spectrum of every N a l for a specific Ge detector, since the correlation is a function only of the Ge signal. The significance of this correction method can be seen in Figure 5.9, which shows typical t iming spectra wi th and without the rise-time correction. The cor-rected spectra are more symmetric and have better resolutions as well as better signal-to-background ratios. The low ta i l on the Ge2 uncorrected spectrum is mainly due to low-energy gamma rays, such as the 400 keV fiKa i n S i , which had pulses 99 1604 -_ " D 0 1 1602 o > cu - 1600 1598 100 200 Legend o Raw d a t a • L E C d a t a - Fit to raw data Fit to L E C d a t a n 1 r~ 300 Ge2 L E 4 - 1 Cent ro ids 400 500 Figure 5.7: Plot of the corrected and uncorrected centroid channel of the 1173 keV 7 rays as a function of their rise-time for the G e l detector. 1604 cn S 1602 > cu 1600 -_ 1598 -100 - i o Row d a t a • L E C data - Fit to raw data Fit to L E C data - j ! j _ T 200 300 400 Ge2 L E 4 - 1 Cent ro ids 500 Figure 5.8: Plot of the corrected and non-corrected centroid channel of the 1173 keV 7 rays as a function of their rise-time for the Ge2 detector. 100 Channel number Channel number Figure 5.9: Typ ica l t iming coincident spectra for one of the N a l ( T l ) detec-tors, B A R 4 . c) and d) are the rise-time corrected spectra for a) and b) for G e l and Ge2 respectively. too small to fire the high Ge2 L E 4 discriminator and consequently they would not be in the "corrected" spectra. The apparent loss of counts is not a problem, since it only effects low-energy events i n the Ge detectors, but not the 1229 keV signal and its vicinity. Average improvements i n time resolutions obtained were 1.5 ns and 2.8 ns at F W H M (or 10% and 20% ) for G e l and Ge2 respectively, thus giving time resolutions ranging from 7 ns to 15 ns F W H M for the 36 N a l - G e pairs. The effect of this correction is to allow for relatively tighter coincidence win-dows around the G e - N a l time coincidence peak and hence it reduces the random backgrounds i n the coincidence energy spectrum. 101 5.2.4 Time—Coincidence Cut Once the time-coincidence spectra were rise-time corrected, windows were placed on the prompt peaks to select Ge events which were in coincidence wi th N a l ( T l ) crystals, see Figure 5.9. Since the two Ge detectors, defining the strobe, were out-of-time wi th each other, a window on each N a l ( T l ) had to be set ind i -vidual ly depending on which Ge fired. The coincidence windows were around 3 a F W H M wide. In addit ion to the 12 N a l ( T l ) crystals used i n coincidence wi th the two Ge detectors, we made use of the two N a l ( T l ) annuli as secondary coincidence detec-tors. Figure 5.10 shows a typical time-coincidence spectrum for one of the N a l ( T l ) annulus segments, S B 3 , i l lustrat ing the double use of the N a l ( T l ) annuli . The sup-pressed and otherwise huge peak ( tSB3.Ge2) is composed of the Compton events in coincidence wi th the surrounded Ge-detector operated in the anti-coincidence mode while the central peak ( t S B 3 . G e l ) corresponds to events when the other (opposing) Ge set the trigger. Hence to increase the tagging coincidence-efficiency of the N a l ( T l ) arm, a cut is set to select the coincidence events i n the central peak of every N a l ( T l ) annulus segment i n addit ion to the requirement that the opposing Ge detector is the trigger and not the other. Gamma-ray lines which are not associated wi th real coincidences, such as the air activation line from the decay of 4 1 A r at 1294 keV, are suppressed after the N a l time-coincidence requirement is applied. 5.2.5 Energy—Gated Coincidence The coincidence technique was described in section 4.3. The 942 keV gamma-rays are selected by put t ing a window around its peak i n the energy spectrum of 102 i - J I I I L _ J L _ l I I I I I I I I I I I I I I I I l l I L_l l l I I L _ l I l I I I I I I I l I I L I tSB3-Ge2 1 i i i i i i i i ! | i" i" "i""r11 r~i i i | i i i i i i i i i | i i i i i i i i i j i i i i i i i i i 0 100 2 0 0 3 0 0 4 0 0 5 0 0 Channel number Figure 5.10: Time-coincidence spectrum for N a l ( T l ) annulus segment, S3B. Note the two peaks resulting from operating the segment in coincidence and anti-coincidence mode wi th the opposing and surrounded Ge detector respectively. 103 each of the 24 N a l crystals whose A D C spectra had to be calibrated beforehand. Due to the relatively poor resolution of N a l detectors (the energy resolution was 9-10% F W H M ) , the rate dependence of the gain (Figure 5.11), as well as the lack of well-determined 7-ray lines i n their spectra, two complementary methods were used for their energy calibrations. F i r s t , the N a l spectra were calibrated by fit t ing the peak positions of the two 6 0 C o 7-rays in the in-beam source runs to a dual-gaussian function. As an il lustrative example, this dual fit to these lines of one of the N a l detectors, B A R 3 , is shown on Figure 5.11. This cal ibrat ion gave rough estimates of the peak positions of the ' tagging' 7 ray of interest, along wi th estimates of their widths i n the N a l spectra. The reason for not using the better resolved spectra from the source-only calibrat ion runs was due to the rate dependence effects; see for example, Figure 5.11. In the second method, the coincidence technique was used to ' tag' the 942 keV instead of the 1229 keV gamma rays (as is done i n section 5.4), i.e. by setting gates around the 1229 keV 7-ray line i n the two Ge detectors and consequently observing the 942 keV 7-ray line in the N a l detectors (see Figure 5.17b). The two methods gave consistent results for most of the runs and most of the N a l crystals, w i th the latter technique being the one which was ul t imately used. The energy coincidence cut yields a signal-to-noise ratio at the 1229 keV peak of about 4.0 (4.8) , which is a factor of 17 (20) better for G e l (Ge2) than the previous measurement of M i l l e r et al. [145]. Three different windows, 4 standard deviations wide, were placed on each of the N a l detectors (and segments). These windows were placed around (-6cr,-2a), (-2<7,+2CJ) and (+2cr,-r-6fj) relative to the 942 keV centroids. The use of the corresponding Ge energy spectra w i l l be discussed later. Other cuts such as pile up ( P U ) rejection, overload cut, as well as cuts based 104 Channel number Figure 5.11: A dual-peak fit to the 6 0 C o 7-ray lines of B A R 3 spectra from the in-beam and no-beam source runs. Note the rate-dependent effects on the resolution and the gain. 105 Table 5.1: Acceptances of the N a l ( T l ) coverage for the two Ge detectors. G e l Ge2 Line singles coincidence eAft (%) singles coincidence e A O (%) 1173 keV 1332 keV 108630±713 104609±720 16749±296 15009±287 15 .4±0 .4 14 .3±0 .4 8 6 7 3 6 ± 6 5 9 7 7 3 8 4 ± 6 2 5 11644±241 9 4 3 4 ± 2 1 2 13 .4±0 .4 12 .2±0 .4 on the scintillators ' T D C s and A D C s were ut i l ized at some stages of the analysis, pr incipal ly at the singles stage. However, their overall effect was minor relative to the coincidence cut(s). 5.3 Acceptances of Detectors To determine the acceptance (eAQ, detection efficiency including geometrical effects) of the N a l ( T l ) detectors, the coincident gamma rays from a 6 0 C o source were used. This was done by measuring the ratio of counts in the 1173 keV and 1332 keV peaks i n the Ge coincidence and Ge singles 7-ray spectra, i.e. wi th and without an energy window imposed on the 1332 keV and 1173 keV full-energy peaks in the N a l spectra respectively. The results are tabulated in Table 5.1. Hence we adopt ( 1 4 . 9 ± 0 . 4 ) % and ( 1 2 . 8 ± 0 . 4 ) % as the N a l ( T l ) acceptances for the G e l and Ge2 detectors respectively. In addit ion, the N a l ( T l ) acceptances were calculated using the 942 keV 2 8 A l in-beam 7-ray to be ( 1 5 . 9 ± 0 . 5 ) % and ( 1 3 . 0 ± 0 . 5 ) % for the G e l and Ge2 detectors respectively, in good agreement wi th the adopted values. The N a l ( T l ) acceptance is expected to be fairly uniform for the region of interest, i.e. 0.5-3.0 M e V . To determine the acceptance of the Ge detectors (energy dependent), several muonic X- ray spectra from various targets were used. The acceptance for a 7-ray 106 is s imply defined by the formula : where i V 7 is the number of X-rays detected, JVM is the number of associated stopped muons and Yy is the yield of the X- ray per stopped muon. The yields, Yy, were obtained from Vogel [178], Har tmann et al. [179] and von Egidy et al. [180]. The number of X-rays detected was determined, from the area (Area) of the X - r a y energy peak, as follows: N^ = Areafaf,fv. (5.2) The quantity fa is a self-absorption correction to account for the different photon absorptions of the target materials. It was calculated from the mass attenuation coefficients (p/p) of Hubble [181], viz. fa = e+tlx, where x was taken to be 1/2 the target thickness. The second factor (//) is the lost-strobe correction factor, and takes account of the fact that although the strobe scaler begins counting immediately at the start of each run, the events are not wri t ten to tape unt i l the I / O channel is set-up. Consequently some events are lost. This factor is obtained by comparing the number of events on tape to the number of strobes. The th i rd correction made to the area of the muonic X - r a y peaks is the self-veto factor, / „ . It takes account of the fact that an interesting muonic X - r a y event might be vetoed by another muonic X-ray , 7-ray, neutron or electron firing the associated N a l Compton suppressor, thus rejecting the event. This effect is measured by counting coincidences between opposing G e - N a l suppressor pairs. Equal ly well , to arrive at N^, the total number of muon stops (S1-S2-S3), had to be corrected for computer processing dead-time (fd)- Th i s correction was determined by the ratio of the livetime clock scaler (CLOCK-INHIBIT) and the free running clock ( C L O C K ) . Furthermore, during the course of the experiment (run 144), it was discovered that the veto scintillator 53 was missing some muons 107 Table 5.2: M u o n i c X - r a y acceptance data for the G e l detector. E (keV) fi X - r a y Y, Area fa U f/ 1-2-3 (108) U fs3 ( IO- 4 ) 952 Pb(4f->3d) 0.684 56557 1.10 1.13 1.09 5.169 0.80 0.8 2.71±0.19 1094 C r ( 2 p ^ l s ) 0.715 35443 1.20 1.05 1.03 1.973 0.76 1.0 3.26±0.25 1255 Fe(2p^ l s ) 0.716 129669 1.10 1.05 1.03 10.01 0.73 0.8 2.69±0.12 1425 N i ( 2 p ^ l s ) 0.740 52498 1.11 1.04 1.07 3.107 0.68 1.0 2.82±0.25 1510 C u ( 2 p ^ l s ) 0.783 10530 1.04 1.07 1.15 0.722 0.84 1.0 2.38±0.30 1522 Fe(3p^ l s ) 0.082 15523 1.11 1.02 1.03 10.01 0.73 0.8 2.77±0.23 1780 Ge(2p-*ls) 0.853 47239 1.10 1.08 1.06 3.184 0.79 1.0 2.19±0.15 2178 Ge(3p—>ls) 0.066 3015 1.10 1.05 1.06 3.184 0.79 1.0 1.76±0.35 2547 Pb(3d^2p) 0.802 40004 1.05 1.14 1.09 5.169 0.80 0.8 1.57±0.12 3450 Sn(2p-+ls) 0.921 5211 1.03 1.09 1.03 0.709 0.85 1.0 0.92±0.35 and it was replaced thereafter by a bigger one. Consequently, another correction (/s3) to was needed for the earlier runs. It was estimated from the ratio of Sl-S2-5"3 and S1-S2 scalers before and after the 53 replacement for the same target and same muon flux. Hence, incorporat ing the above correction factors, the acceptance is given by A O A r e a fa f> f» (*<i\ e A i l = (si.52.33)/</»y 7 • ( 5 ' 3 ) Tables 5.2 and 5.3 show the acceptances, along wi th the aforementioned corrections, calculated from the various muonic X-rays for the G e l and Ge2 detectors respec-tively. Figures 5.12 and 5.13 show the corresponding acceptance plots as a function of energy (E), w i th fits of the form : where a and b are constants determined in the fits to the data; and are given in the figures 5.12 and 5.13. We attribute the inconsistencies to the uncertainties in the definition of a muon stop and the variation of the sizes and shapes of the targets as well as the different energy thresholds of the N a l detectors. These unfortunately 108 Table 5.3: Muonic X - r a y acceptance data for the Ge2 detector. E (keV) /iX-ray Y, Area fa f„ f/ 1-2-3 (108) U fs3 eAtt ( l O " 4 ) 347 Al (2p-» l s ) 0.796 469051 1.14 1.04 1.09 11.04 0.68 0.8 6.89±0.12 413 Al(3p->ls) 0.079 42790 1.14 1.04 1.09 11.04 0.68 0.8 6.34±0.30 436 Al (4p-» l s ) 0.049 25521 1.13 1.04 1.09 11.04 0.68 0.8 6.04±0.42 447 Al (5p-» l s ) 0.041 21131 1.13 1.04 1.09 11.04 0.68 0.8 5.98±0.45 453 Al(6p->ls) 0.026 11573 1.13 1.04 1.09 11.04 0.68 0.8 5.16±0.68 952 Pb(4f-+3d) 0.684 41598 1.10 1.44 1.09 5.169 0.80 0.8 2.54±0.22 1094 Cr(2p—>ls) 0.715 39912 1.20 1.17 1.03 1.973 0.76 1.0 4.09±0.31 1255 F e ( 2 p ^ l s ) 0.716 91501 1.10 1.16 1.03 10.01 0.73 0.8 2.10±0.16 1425 Ni(2p->ls) 0.740 52498 1.11 1.16 1.07 3.107 0.68 1.0 3.14±0.29 1510 C u ( 2 p ^ l s ) 0.783 7392 1.04 1.23 1.15 0.722 0.84 1.0 1.92±0.25 1522 F e ( 3 p ^ l s ) 0.082 9916 1.11 1.09 1.03 10.01 0.73 0.8 1.89±0.22 1780 G e ( 2 p ^ l s ) 0.853 34839 1.10 1.29 1.06 3.184 0.79 1.0 1.93±0.17 2178 Ge(3p—>ls) 0.066 2270 1.10 1.22 1.06 3.184 0.79 1.0 1.54±0.35 2547 Pb(3d^2p) 0.802 19902 1.05 1.43 1.09 5.169 0.80 0.8 0.98±0.16 3450 Sn(2p—>ls) 0.921 2543 1.03 1.27 1.03 0.709 0.85 1.0 0.52±0.35 were not considered priorities during the run. We have checked wi th an efficiency curve in the E G & G O R T E C manual (page 474) and has a s imilar energy depen-dence to ours. 5.4 Cascade Feeding One mechanism that could distort the 1229 7-ray, and thereby spoil the inter-pretation of the Doppler-broadened lineshape measurement in terms of gp, is the indirect population of the 2201 keV 2 8 A 1 level of interest from higher-lying excited states. (The neutron threshold for 2 8 A 1 is quite high at 7725.18 keV [155].) M a n y levels above 2201 keV have been reported to be populated in thermal neutron cap-ture, i.e. 2 7 A l ( n , 7 ) 2 8 A r , see Schmidt et al. [155] and the compendium of End t [143]. Since the ground state of the 2 7 A 1 has J ? r = 5 / 2 + , the levels of 2 8 A l excited by the capture of thermal neutrons wi l l tend to have a higher spin than those produced in 109 I 8 6 < 2 0 0.563 eAQ G e 1 = 823.1+E i 4 ^ "1 I I I I I I I 0 1 0 0 0 2 0 0 0 3 0 0 0 Energy (keV) 4 0 0 0 Figure 5.12: Acceptance curve for the G e l detector. i ' i i I i i i i I i i i i I i i i i_ 1 0 0 0 2 0 0 0 3 0 0 0 Energy (keV) Figure 5.13: Acceptance curve for the Ge2 detector. 110 4 0 0 0 muon capture (mainly 1 + as well as 0~, 1~, 2~ and 2 + levels). Recently, Vernotte et al. [156] studied the proton pickup reaction, i.e. 2 9 S i ( d , 3 H e ) 2 8 A l , and have reported several other 1 + levels omitted by Schmidt et a/., for example levels at 3542 keV, 4115 keV, and 4846 keV. They also assigned the 3015 keV level J 7 r = l + (which had also been done by Lawergren and Beyea [182], but forgotten). Figure 5.14 shows some of the potential 2 8 A l levels populated in various re-actions (and potentially populated in muon capture), which are known to decay to the 2201 keV level of interest. There is no evidence of such populat ion in muon capture 1 ; but the effect could easily have been overlooked or missed, so it is of central importance to rule out such a possibility. To investigate the possible cascade-feeding of the 2201 keV level from upper states, a search for peaks at 4218, 3791, 3659 and 904 keV (corresponding to the 6420, 5992, 5861 and 3105 keV levels) i n the Ge singles spectrum was performed. Figure 5.15 shows part of a singles spectrum obtained after adding the low-gain (high-energy) muon Si runs. For the 6420 keV level, there is no significant peak in this spectrum at 4218 keV (BR=12%) , nor at 4281 keV a decay to the 2139 keV level (BR=61%) . For the 5992 keV level, there seems to be a peak around 3791 keV, however, there is no peak at 4372 keV which occurs wi th a similar branching ratio as the 3791 keV transit ion. Furthermore, the 5992 keV state which has T = 2 is not populated i n either the (n,7) nor the (d, 3 He) reactions. (There is a strength there in the ( T T ^ ) and (d , 2 He) reactions however, but one would not expect a T = 2 state to be excited). For the 5861 keV level, there is no peak at 3659 keV (BR=25%) nor at 5861 keV, the direct transit ion to the ground state which has a branching ratio of 60%. The most probable candidate is the 3105 keV 1 + state because it is fed by 1 Singles measurements [144,146] used indirect arguments to indicate that the feeding is small and consequently neglected such an effect. I l l J (1.2) + 0 + (2.3) + 12% 4 3 % > 00 csl \l 25% cn O CD 75% > CO o r O > 1) r o in CO 25% > m r o o E (keV) 6419.8 5992 .4 5 8 6 0 . 8 3105 79% > CD 00 CD 2201.46 V 30 .64 Figure 5.14: Cascade feeding of the 2201 keV 2 8 A 1 level (for a complete energy level diagram, see P . M . Endt [143] and Vernotte et al [156].) 112 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 3 5 0 0 4 0 0 0 4 5 0 0 5 0 0 0 5 5 0 0 6 0 0 0 6 5 0 0 E n e r g y (keV) Figure 5.15: Part of the 2 8 S i 7-ray energy spectrum for the G e l detector. Upper arrows refer to 7-ray transitions potentially feeding the 2201 keV level of interest, as seen in Figure 5.14, while the lower arrows are for other associated transitions of the same levels. 113 similar reactions such as (^ ,7) and (p,n), see section 2.6. In addi t ion, two 7-ray peaks at 3075 keV and around 903 keV are seen, see Figure 5.16. However, their relative strengths do not match the branching ratios given i n Figure 5.14; i.e. (^903 ) ( J A Q 3 ^ 5 ) ~ (6)(2-3) ~ 14 compared to a ratio of only 3 from Figure 5.14 2 . In Ge addit ion, their shapes are quite different; while the 3075 keV peak has a Doppler-broadening of 24 keV F W H M , the line at 903 keV is not broadened at a l l (a broad-ened peak underneath the 903 keV would have a very small contribution) nor does it get enhanced by an energy gate around the 2171 keV 7-ray i n the N a l energy spectra. (The lifetime of the 3105 keV has not been measured). Note that Schmidt et al. [155] do not include the 3105 keV state nor list the 903 keV 7-ray. However, they list a 7-ray at 3075.6 keV wi th a footnote that it is not in the level scheme 3 . We conclude from the singles spectrum that the unbroadened 903 keV 7-ray that we observe has another, but unknown, origin. Because the broadened 903 keV line is not observed, we also question the origin of the 3075 keV line. A direct search for cascade feeding can be made through the coincidence tech-nique by reversing the roles of the detectors, i.e. by imposing an energy window on the 1229 keV 7-ray line in the two Ge detectors and examining the coincident N a l spectra, see Figure 5.17b. Only our expected 942 keV line is apparent. (The dotted lines show a typical 942 keV energy coincidence gate used to tag the 1229 keV 7-rays as described in section 4.3.) A n even more sensitive check can be made by gating on the more prolific 2171 keV 7-ray which has an 79% branching ratio compared to 16% for the 1229 keV 7-ray. This is shown i n Figure 5.17c. Further-more, Figure 5.17d shows this spectrum after subtracting out the raw spectrum (suitably normalized); and again there is no evidence (peaks or dips) for the poten-2 The branching ratios for the 3105 keV level were obtained in one experiment only, by Lawergren and Beyea [182]. The earlier compendium by Endt and Van der Leun [183] has a typographical error in Table 28.9. For the 3105 keV level the footnotes should be d,f. 3 I f one takes 3075.65 keV to be the transition energy into the 30.6 keV level, then the level energy is 3106.2 keV and therefore 3106.2 keV 2201.5 keV = 904.7 keV; yet another problem. 114 6 0 0 7 0 0 8 0 0 9 0 0 1000 1100 1200 Energy (keV) i . i i i i i i i i , i i , i i i i i i 3 0 7 5 iu l I 2 8 0 0 2 9 0 0 3 0 0 0 3100 3 2 0 0 3 3 0 0 3 4 0 0 Energy (keV) Figure 5.16: Parts of the 2 8 S i 7-ray energy spectrum for the G e l detector showing the 903 keV and 3075 keV peaks. 115 t ia l transitions seen in Figure 5.14, nor for any others. We postulate that the large number of events at the low energy are related to the 400 keV //-mesic X-ray , but the events are cut off by the hard-wired N a l discriminator. Note that the bumps at 0.85, 1.0, 1.7 and 2.1 M e V i n Figure 5.17c correspond to the strong peaks at 843 keV from 2 7 A 1 , 1014 keV from 2 7 A 1 , combined 1779 keV from 2 8 S i and 1808 keV from 2 6 M g , and the cluster of peaks at 2108 keV 2 8 A l , 2138 keV 2 8 A l , 2171 keV 2 8 A l and 2211 keV 2 7 A l respectively. These bumps can be identified from the sin-gles Ge spectrum which closely parallels the N a l spectrum, see Figure 5.18. These peaks are relatively strong i n the singles spectra and are expected to show up i n the coincidence spectra due to accidentals and/or real coincidences. These bumps disappear in Figure 5.17d, indicat ing that most or a l l of the events are accidental. To put l imits on cascade-feeding of the 2201 keV level, and hence l imits on the feeding of the 1229 keV 7-ray of interest, the 2171 keV gamma rays were used. This was done by measuring the ratio of counts i n the 2171 keV peak i n the Ge coincidence (NQ171) and Ge singles (Ng171) spectra. Hence, the coincident 7-rays per 1229 keV 7-ray is given by : 1 iV 2 1 7 1 NI171 (5.5) where eAQ, is the N a l acceptance as determined i n the preceding section. L imi t s on feeding from the 3105 keV state and nearby levels were obtained by imposing an energy window around the 903 keV i n the N a l energy spectra and taking N Q 1 7 1 to be the remaining 2171 keV 7-rays i n the Ge detectors. Equat ion 5.5 then gives and 702 ± 225 220700 ± 1200 229 ± 80 1 131400 ± 950 J L0.149 ± 0.004 L0.128 ± 0.004 = 0.021 ± 0.007 = 0.014 ± 0 . 0 0 5 (5.6) (5.7) 116 Figure 5.17: 2 8 S i 7-ray energy spectra obtained wi th one of the N a l detec-tors, B A R 1 : (a) without energy gate on the Ge detectors; (b) & (c) after gating on the 1229 keV and 2171 keV transitions i n the Ge detectors respectively; and (d) spectrum (a) normal-ized & subtracted from spectrum (c). Spectra for the other N a l crystals are similar. 117 J I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 I I I I 5 0 0 1000 1500 2 0 0 0 2 5 0 0 G a m m a R a y E n e r g y ( k e V ) Figure 5.18: 2 8 S i 7-ray energy spectrum of Figure 5.17a overlaid on a G e l detector singles spectrum (from run 136). as the l imits on cascade-feeding from levels in the vicini ty of the 3105 keV for the G e l and Ge2 detectors respectively. The populat ion of the 2201 keV level by a few "strong" high energy 7-rays is ruled out (by Figure 5.17d at the few percent level). Since the threshold for 2 8 A l is 7.7 M e V , such possible 7-rays populat ing the 2.2 M e V from bound states would have a m a x i m u m energy of 5.5 M e V and hence would be seen i n Figure 5.17. A remaining possibility is the populat ion of the 2201 keV level by a large number of weak cascades. To put a l imi t on such a possibility, t iming coincidence between the G e - N a l spectra, along wi th a software energy threshold on N a l spec-t ra of ~ 0.6 M e V , were required. The energy threshold ensured a uniform 7-ray efficiency and also reduced contamination from the 400 keV /i _Si(2p—>ls) X-rays . Taking N Q 1 7 1 to be the remaining 2171 keV 7-rays i n the Ge detectors after these requirements (Figure 5.20), the l imits on coincident events per 1229 keV 7-ray for 118 each detector are: 5500 ± 160 and L220700 ± 1200 2570 ± 150 0.149 ± 0 . 0 0 4 1 0.167 ± 0 . 0 0 7 131400 ± 9 5 0 = 0.15 ± 0 . 0 1 . (5.8) (5.9) .0.128 ± 0 . 0 0 4 J However, before taking these results as l imits on cascade-feeding of the level of interest from higher levels, an estimate of non-cascade feeding coincidences had to be determined and subtracted off. There were two types of non-cascade feeding co-incidences: ' random' and target/beam-related coincidences. The former measured the coincident room-backgrounds and was determined from the 1294 keV 4 1 A r air activation 7-ray peak to be : 1 and [0.0063 ± 0.0010] [0.0057 ± 0.0013] L0.149 ± 0.004 = 0.042 ± 0 . 0 0 7 0.045 ± 0 . 0 1 0 (5.10) (5.11) L0.128 ± 0.004J for the G e l and Ge2 detectors respectively. Subtracting these ' random' coincidences events from the values obtained in equations 5.8 and 5.9, we obtained the following l imits : (0.167 ± 0.007) - (0.042 ± 0.007) = 0.12 ± 0.01 (5.12) and (0.15 ± 0.01) - (0.045 ± 0.010) = 0.10 ± 0.02 (5.13) respectively, which i f taken at face value would indicate some feeding of the level of interest. The other type of non-cascade feeding coincidences is more complicated to estimate; and is due to, but not restricted to, pile-up effects. A 400 keV muonic X - r a y precedes each muon capture 7-ray. Now if a smaller energy event (7, e, or n) 119 comes along accidentally, it w i l l trigger the N a l threshold and register a prompt sig-nal . Thus the accidentals are enhanced. Such coincidences are difficult to estimate, as this effect was not discovered unt i l the data analysis was underway. However, redoing the above calculation using the muonic X-rays (//Fe(2p—>ls)), instead of the 1294 keV 4 1 A r 7-ray, as the measure of non-cascade feeding coincidences, gave the following results: 1 [0.0242 ± 0.0027] and [0.0192 ± 0.0030] 0.149 ± 0 . 0 0 4 0.162 ± 0 . 0 1 9 (5.14) 0.150 ± 0 . 0 2 4 (5.15) L0.128 ± 0.004J for the G e l and Ge2 detectors respectively, and consequently the corresponding l imits on cascade-feeding would be : (0.167 ± 0.007) - (0.162 ± 0.019) = 0.005 ± 0.026 (5.16) and (0.153 ± 0.010) - (0.150 ± 0.024) = 0.003 ± 0.034 , (5.17) consistent wi th no cascade-feeding. Since this muonic X - r a y is in real coincidence wi th only very low-energy X-rays , it constitutes a better indicat ion of the non-cascade feeding coincidences than does the 1294 keV 7-ray. However, this estimate may be an underestimation of the feeding as the lifetime of muons i n Fe is shorter than that in Si and hence the feeding is somewhere between the 0 and 11% (Equa-tions 5.16, 5.17 and 5.12, 5.13). It should also be remarked that the shape of the 2171 keV 7-ray peak i n the coincidence spectra is very similar to that i n the singles spectra. Figure 5.19 shows the singles gamma-ray energy spectra, i n the vic ini ty of the 2171 keV peak, for the two Ge detectors; while Figure 5.20 shows the corresponding spectra after the imposi t ion of the aforementioned G e - N a l t iming coincidence. There is no evidence 120 8-<D 4 c o o 0 2 0 5 0 2171 keV 26A1 (a) Ge1 N f V > A > J 1 CD f CD 1 CO A / o 1 1 CO [ J V i C\2 / \ N f / 2100 2150 2 2 0 0 2 2 5 0 Gamma Ray Energy (keV) 2 3 0 0 4 -o c_> 0 2 0 5 0 (b) Ge2 CM All \ > " \ ^ / 1 ^ CM V \ IO A ™ 211 ke CM rr 2100 2150 2 2 0 0 2 2 5 0 Gamma Ray Energy (keV) 2 3 0 0 Figure 5.19: Singles 2 8 S i gamma-ray energy spectra obtained wi th the two Ge detectors, i n the vicini ty of the Doppler-broadened 2171 keV peak. for a sharp peak in the centre, for example, which would happen if there was feeding from a long-lived state. A contribution of only 0.5% was obtained on such feeding, see section 5.7.2. Note that some lines in 2 7 A 1 are enhanced in the coincidence spectrum because they are part of cascades and also have neutrons i n coincidence. The neutron-proton capture 7-ray at 2.2 M e V almost disappears because it is a true random. The 2171 keV line is s t i l l there though relatively weaker. In concluding this section, a stringent l imi t at the level of 2% on cascade-feeding from the most probable level (3105 keV) was set. Furthermore, it is our belief that the overall cascade-feeding l imit is less than 5%. 121 2 0 5 0 2100 2150 2 2 0 0 G a m m a Ray E n e r g y ( keV) 2 2 5 0 2 3 0 0 2 0 5 0 2100 2150 2 2 0 0 G a m m a Ray E n e r g y ( keV) 2 2 5 0 2 3 0 0 Figure 5.20 : 2 8 S i coincidence energy spectra obtained wi th the two Ge detec-tors, in the vicini ty of the Doppler-broadened 2171 keV peak. 122 5.5 Background Subtraction The best coincidence spectra were obtained after opt imizing the various cuts, and in part icular the energy-gated cut. A l though the coincidence method greatly improved on the singles spectra; there remained some background under and near the peak of interest, including a bit of the infamous 'plateau' , see Figure 4.5. In addit ion to random coincidences, the flat component of the remaining back-ground in the coincidence spectrum is due mostly to n-7 and n-n coincidences. These events are due to the neutrons following muon capture, wi th associated nuclear 7-rays. This is supported by the fact that the N a l - G e coincidence t iming spectra show a 10% or so background underneath the coincidence peak (see Figure 5.9) as well as the enhancement of the characteristic Ge(n,n') lines at, e.g. 596 keV and 692 keV. The remaining structure under or near the peak of interest is due to the Compton tails of higher-lying peaks in the N a l spectrum. B o t h types of background can be subtracted off by using a 'side-band' back-ground subtraction. This is accomplished by setting two N a l energy windows on each of the N a l detectors (and segments), one above and the other below the 942 keV 'tagging' 7-ray line. The reason for this combination is to mimic closely the background underneath the peaks of interest, albeit losing a few genuine 1229 keV 7 rays (in Ge) which are in coincidence wi th 942 keV Compton events picked i n the "below-942" N a l window. Each window has the same wid th (in keV) as the central 942 keV window, i.e. four standard deviation wide. The corresponding - "below-942" and "above-942"- Ge spectra are added, normalized, and then subtracted from the main 942 keV-gated coincidence spectra to produce the background-free Ge en-ergy spectra (herein-after referred to as the AB-spect ra) , Figures 5.21 and 5.22, ready for the lineshape analysis. It should be noted, however, that the details of this subtraction are not cri t ical; see section 5.7.2. 123 I I I I I 8 0 0 9 0 0 1000 1100 1200 1300 1400 Gamma Ray Energy (keV) i (b) ; 1 L 4 8 0 0 9 0 0 1000 1100 1200 1300 1400 Gamma Ray Energy (keV) Figure 5.21: 2 8 S i gamma-ray energy spectra of the G e l detector: (a)singles and (b)coincidence wi th the 'side-band' background subtrac-tion. Evident is the flat background around the 1229 keV peak of interest. 124 o o - 2 - r 7 0 0 8 0 0 9 0 0 1000 1100 1200 1300 Gamma Ray Energy (keV) Figure 5.22: 2 8 S i gamma-ray energy spectra of the Ge2 detector: (a)singles and (b)coincidence wi th the 'side-band' background subtrac-t ion. Evident is the fiat background around the 1229 keV peak of interest. 125 ( Table 5.4: Input parameters needed i n the least-squares fitting function for the two Ge detectors, cr, PL and PR are i n channels. G e l 7-ray Line Po dE ( MeV \ dX V mg/cm2 > cr PL PR S 1229 keV 2171 keV 0.0037484c 0.0037484c 2.2 2.2 2.46 3.44 2.76 3.49 3.35 4.30 0.0011 0.0011 Ge2 7-ray Line Bo dE 1 MeV \ dX V mg/cm2 ' a PL PR 5 1229 keV 2171 keV 0.0037484c 0.0037484c 2.2 2.2 3.15 4.24 4.19 4.54 5.23 5.47 0.0014 0.0017 These spectra have the following merits: • the remaining "plateau" is gone, • the other remaining single-line backgrounds nearby are gone, and • the 1229 keV line of interest seems to sit on a perfectly flat background, and hence one could really believe a lineshape based on such a spectrum. 5.6 Parameter Recapitulation This section summarizes the various input parameters needed i n the least-squares fit t ing program to fit the experimental line shapes of the two transitions of interest: the 2171 keV and 1229 keV. O f the fifteen (15) parameters needed i n the least-squares fitting program three (/?i, 82, and cos#7i,) were set to 0, see section 4.6. The response function parameters (a, PL, PR, and S) of both Ge detectors were fixed at their estimated values following their parameterization, see section 4.4. Another two, B0 and j^-, were fixed to their values as determined i n sections 2.5 and 4.5 respectively. These values are summarized i n Table 5.4 for the two lines of interest, and for the two Ge detectors respectively. 126 1 In addit ion, the response function parameters, the peak position (EO) and the background level ( U B ) , and | j| were sometimes varied and or fixed to see their effect on the quality of the fit. 5.7 Analysis of the Doppler-broadened Peaks W h i l e fitting the Doppler-broadened peaks of the 2171 keV and the 1229 keV tran-sitions, it was found that the fitting parameters a and r were highly correlated and tended to offset each other during fitting, especially for the 2171 keV 7-ray line, for which a is negative. The interplay of the three effects: the slowing-down, the angular correlation, and the response function is i l lustrated in Figure 5.23. It is clear that there is a strong correlation between a , r and a. Because of these strong correlations and the low statistics on the 1229 keV lines in the subtracted coincidence spectra, the Doppler-broadened lines of interest were fit i n two approaches. In the first approach, the 2171 keV peaks in both Ge detectors were fit simultaneously to extract the value of the lifetime, r , of the common nuclear level (2201 keV) . Then , wi th r fixed at this value, the 'side-band' background subtracted 1229 keV 7-rays of both Ge detectors were fit jo int ly to extract the angular correlation coefficient, a. In the second approach, the fitting parameters (in part icular a and r ) were obtained from a simultaneous fit to a l l four spectra, i.e. both 7-ray lines (2171 keV and 1229 keV) of both Ge detectors. 5.7.1 The 2171 keV line The 2171 keV 7-ray, coming from the same level as the 1229 keV 7-ray, turned out to be a cri t ical indicator of whether the recoiling nucleus, 2 8 A 1 , slows down and hence is crucial for obtaining the lifetime of the level. It has a five times higher yield than the 1229 keV 7-ray. Moreover, it is apparently background free, i.e. has a flat background, so the singles spectra wi th much higher statistics could be used; 127 2 0 0 1 5 0 -™ 1 0 0 H Pi o ° 5 0 H 0 -I I I l_ _1 I I l_ _l I I l_ J I I I l_ • 5 0 ~ ~ i — ' — 1 — 1 — 1 — i — 1 — 1 — 1 — 1 — i — 1 — 1 — 1 — 1 — i — 1 — 1 — r 1 2 2 0 1 2 2 5 1 2 3 0 1 2 3 5 Gamma—ray Energy (keV) 1 2 4 0 Figure 5.23: The interplay of the slowing-down, the angular correlation, and the instrumental resolution effects on a box spectrum for a Doppler-broadened 7-ray line. 128 Table 5.5: Results of the best fit to the 2171 keV 7-ray lines in both Ge detectors. r(fs) « 2 1 7 1 x 2 XR GLOBAL 61.0±3.7 -0 .058±0 .025 207.78 1.065 0.967 0.969 see Figure 5.19. A s i l lustrated i n Figure 5.23, the slowing-down effect is not dissimilar to the angular correlation coefficient effect, nevertheless, fixing the 7-ray response function from other 7-ray lines, coupled wi th the high statistics i n the 2171 keV lines, allowed the determination of both effects wi th sufficient confidence. Vary ing r and 0 : 2 1 7 1 , in addit ion to the two amplitudes (N) , i n the fitting of the 2171 keV 7-ray spectra of the two Ge detectors allowed their simultaneous extraction. Table 5.5 shows the results of the best joint fit to the 2171 keV 7-ray peaks. The a ^ m value cited for the 2171 keV is the product of the angular correlation coefficient (a) and the E 2 / M 1 mix ing factor (F). C i t ed i n Table 5.5 are the x2 a n d the reduced \ \ p.d.f. reflecting the quality of fit. Also cited in the table are the M I N U I T correlation coefficients, p(r,a) and pGLOBAL^ measuring the correlations between the variable parameters. The fact that these correlation coefficients are large demonstrates the high correlation (as discussed above) among the fitting parameters, i n part icular between r and a. However, such values of the correlation coefficients are st i l l less than one i n absolute value 4 , and hence allowed the simultaneous determination of r and a. O n the other hand, the strong correlation does mean that the a can be determined more precisely if r is measured independently. However, the present measurement is the best available for r . 4 M I N U I T Reference Manual [184] considers correlation coefficients of greater than 0.99 to be very close to one and indicate an illposed problem, i.e. an exceptionally difficult one, with more free parameters than can be determined by the model and the data. 129 Table 5.6: Results of the ind iv idua l fits to the 1229 keV 7-ray peak for each Ge detector. r ( f s ) a x2 XR pGLOBAL G e l 61.0 0 .385±0 .065 169.90 0.862 0.253 Ge2 61.0 0 .318±0 .107 186.47 0.898 0.260 weig i ted a 0 .367±0 .055 Table 5.7: Effect of the instrumental resolution of the detectors on the angular correlation coefficient, a a a x2 XR W- 1.0(7 W + 1.0<7 0.35.8±0.052 0 .361±0 .052 0 .364±0 .053 373.49 372.22 371.31 0.881 0.878 0.876 5.7.2 The 1229 keV line Holding the lifetime ( r ) of the 2201 keV level fixed, a fit was performed on the 1229 keV gamma-ray peak in the 'side-band' background-subtracted spectrum of each Ge detector to extract values of the angular correlation coefficient from which a weighted mean of a was determined (see Table 5.6). The same result, wi th in ~0.3 <7, was obtained in a simultaneous fit to the 1229 keV lines in both Ge detectors. The same results, wi th in ~0.05 cr, were obtained whether E0 and U B were fixed or allowed to vary. Furthermore, varying the instrumental resolutions of the detectors wi th in their uncertainties changed the value of a by < 0.1a. The resolution (in channels) at the 1229 keV line for G e l is 2.46(5) and for Ge2 is 3.15(6), which correspond to 2.09 and 2.38 keV F W H M . Table 5.7 shows the dependence of a on the instrumental resolution of the detectors, i n the simultaneous fit to the 1229 keV peaks. Even i f the germanium resolutions were a few a off, the effect on the final result is much smaller than the fitting error. To check the effect of different 'side-band -background subtractions on a , the 130 Table 5.8: Effect of the different 'side-band -background subtractions on a. See text for the definition of the three spectra in the first column. Spectrum 1229 keV signal a x2 X2n A-spect rum 9,334 0 .320±0 .052 365.03 0.865 B-spectrum 8,325 0 .386±0 .058 395.76 0.938 AB-spec t rum 8,603 0 .361±0 .052 372.22 0.878 following procedure was done. In addit ion to the master spectra (the AB-spectra) obtained in section 5.5, the "above-942" and the "below-942" Ge spectra were indi-vidually normalized and subtracted from the main 942 keV-gated coincidence spec-t ra to produce another two sets of the 'side-band' background-subtracted spectra, herein-after referred to as the A-spectra and B-spectra respectively. Likewise, these two spectra were fit in the same way as was done wi th the AB-spec t ra , w i th the results are shown i n Table 5.8. As can be seen in this Table, this subtraction affects the final value of a , but the three results agree wi th each other wi th in associated uncertainties. The small variation i n the number of the 7-rays under the 1229 keV peaks is due to the variation of the 942 keV Compton events picked i n the associated N a l energy windows. F ina l ly we check the effect of cascade feeding of max imal intensity on a. A n y cascade feeding would be either from short-lived or long-lived states. W h i l e the former create contr ibut ion wi th similar Doppler-broadening, the latter would create a sharp (gaussian) peak in the center of the 2171 keV (and 1229 keV) , the presence of which was very sensitive to a . To test the effect of feeding from short-lived states, the 1229 keV data was refitted after subtracting a 5% Doppler-broadened contribution, wi th a shape corresponding to cv=0. This is the overall l imi t on cascade-feeding as found earlier. The effect on a for the 1229 keV was about 5%, which is much less than other errors. To put a l imi t on feeding from long-lived states, the 2171 keV lines were refitted, including a gaussian peak to the overall 131 Table 5.9: Results of the best fit to a l l four spectra: the 2171 keV and the 1229 keV 7-ray lines in both Ge detectors ( A B subtraction). r(fs) a F x2 XR p(r,a) P(T,F) pGLOBAL 6 0 . 8 ± 3 . 4 0 .360±0 .059 -0 .165±0 .080 581.82 0.934 0.461 0.929 0.965 fitting function and a l imi t of ( 0 . 5 8 ± 0 . 2 7 ) % was obtained. This would affect a by about 5% which is again less than the errors quoted on a. 5.7.3 The 1229 keV and the 2171 keV Simultaneous Fit The previous fitting procedure was repeated also on a l l four spectra simulta-neously. The fitting function therefore contained the following free parameters: the four amplitudes (N) , the four background levels ( U B ) , the four peak positions (E0), the 2201 keV lifetime ( r ) , the angular correlation coefficient (a) and the E 2 / M 1 mult ipolar i ty mix ing factor (F) of the 2171 keV 7-ray. Th i s approach had the ad-vantage of lower correlations among the various parameters. Furthermore the x2 surface (minimum) becomes deeper and easier to locate. The final spectra and fits are shown in Figure 5.24 while the results are given i n Table 5.9. The same results, wi th in O.lcr, were obtained whether E0 and U B were fixed or allowed to vary. Furthermore, varying the instrumental resolutions of the detectors wi th their uncertainties changed the values of r and a by < 0.3o\ The values of a , T, F determined i n the above sections were i n good agree-ment. Due to the aforementioned reasons, the final results are taken to be those of Table 5.9, v iz . : a = 0.360 ± 0.059 , (5.18) r = 60.8 ± 3.4 fs , (5.19) F = —0.165 ± 0.080 . (5.20) 132 cn a m o o o f\j o o o o v o ro o °5-o O 3 3 o IO O a co-o IO CD' o Counts/keV ro ro 01 OJ o cn o cn i o o o o ( o o o o I i » • • • 1 1 • • • * * » • • 11 • • 11 • « 1 1 1 1 • 11 -r* Ol o Q i i 1 1 I i i i i I i i Counts/keV Cn o ai o o , o o o _i i I i i I_I L_i > i > t i i O o t o o K) OJ O O O O O O Counts/keV ^ CJI O) o o o o o o 0 o o 1 • I • • I I • • • o o o 00 o o o I Ol Counts/keV ro o -o o o o 3 3 a m a ro o r o re < ro o ro o r o ro ro Ol Ol o Ol o o . 1 . . o o o . . i . . . c r . ro ro CD O (D Figure 5.24: The best simultaneous fit to all four spectra: the 2171 keV (a) and the 1229 keV (b) 7-ray peaks from G e l and the 2171 keV (c) and the 1229 keV 7-ray peaks from Ge2. Notice the flat background levels underneath these peaks. 133 Chapter 6 Discussion of Results In the preceding chapter the measured 7 — v angular correlation coefficient, a , in the capture of polarized muons by 2 8 S i was determined from the experimental data. A description of the various theoretical models of the angular correlation was given in Chapter 2. In this Chapter, we shall take the next step which is to extract a value of the induced pseudoscalar coupling constant, gp/gA from this measurement. However before doing so, we shall first compare our results wi th those obtained i n previous experiments. 6.1 The 7 — v Angular Correlation Only two other measurements exist for the 7 — 1/ angular correlation following muon capture on 2 8 S i . B o t h were singles measurements. The first was the pioneering measurement of Mi l l e r et al. [144,145], using the N . A . S . A . Space Radia t ion Effects Laboratory ( S R E L ) synchrocyclotron i n the late 1960's. They used a natural Si target as well as an isotopically pure 28Si02 target. The 1229 keV gamma rays were detected in a 10% Ge(Li ) detector at 90° wi th respect to the muon spin direction, so that the pr imary sensitivity was to the coefficient a (which M i l l e r et al refer to as a° ) . Whi l e their detector was located at 90° to the beam axis, they retained some sensitivity to Q\ and Q2 due to the large dimensions of the targets used (which 134 were set at 45° to the beam and to the Ge detector, v iz . Table II and Figure 20 in Ref. [145]). Thei r quoted value of a was a = 0.15 ± 0.25 and a = 0.29 ± 0.30 , (6.1) for the natural Si and 28Si02 targets respectively. They also considered correlations for the 0 + —> 1 + —> 2 + (e.g. the 2171 keV) transitions. They concluded that the extracted correlations were inconsistent and could not be reconciled wi th theory, regardless of the 8 mix ing ratio. To quote M i l l e r [145]: "the correlation coefficients are experimentally found to be negative and large, while predicted to be small but positive. Within the context of the theory developed by Popov et a l . , this observation is inexplicable. The situation with regard to these 0 + —> 1 + —> 2 + transitions thus remain a mystery." 1 . We now realize that this situation came about because they d id not include the lifetime of the 2201 keV level in the analysis. Nevertheless, they had observed, for the first t ime, the Doppler broadened 7-ray transitions, which are suitable for analysis i n terms of the 7 — v angular correlation. In addit ion, they were able to put a useful l imits on the populat ion of the 2201 keV state of interest from the 2 9 Si ( / t , ^n) reaction through the use of an isotopically enriched 29SiG*2 target. Such populat ion would confuse the 7 — 1/ angular correlation analysis. (Natura l Si contains 4.67% 2 9 S i as compared to 92.2% 2 8 S i ) . M i l l e r et al. [145] observed no captures in 2 9 S i which resulted i n a transit ion through the 2201 keV level, and gave a contamination l imi t of < 0.015 due to the presence of 2 9 S i i n natural silicon target. The only other measurement of the 7 — v angular correlation was carried out recently by Brudan in et al. [146] at the Joint Institute for Nuclear Research ( J I N R ) i n Dubna . They used two H P G e detectors to detect the 1229 keV 7-rays at two angles, 60° and 120°, wi th respect to the muon polarizat ion axis, which 1 However, the error bars are quite significant and the inconsistency is not so serious. 135 provided them wi th sensitivity to the correlation coefficients Bx and B2 as well . The residual muon polarizat ion was measured to be ( 1 0 . 2 5 ± 0 . 2 5 ) % for their beam, for which the in i t i a l polarization was given as ~ 7 0 % . The muon polarizat ion has been measured i n natural Si to be ( 1 5 ± 2 ) % (Astbury et al. [185]) and i n 2 8 S i to be ( 1 6 ± 4 ) % (Weissenberg [186]) of the in i t i a l polarizat ion respectively. Using a •pure gaussian for the detector response function, the corresponding 7-ray spectra obtained from the two Ge detectors were corrected for energy calibrat ion shifts. Thei r 7-ray spectra are similar to ours in that they show the plateau-like structure next to the 7-ray peak of interest. They proposed a complex structure for the background underneath the 1229 keV 7-ray peak, composed of three components: 1204 keV, 1222 keV and 1238 keV 7 rays from the 7 4 G e ( n , n ' 7 ) , 2 7 A 1 and 5 6 Fe(n,n ' ) reactions respectively. Another source believed to contribute to this region, yet not included in their background structure, is the 1216 keV 7-ray due to the 7 0Ge(n,n'7) reaction 2 . B rudan in et al. then performed a simultaneous fit to the 1229 keV and the 2171 keV 7-ray peaks i n the energy spectra of the two Ge detectors in the two angular positions. Due to their low statistics, it was impossible to fit these 7-ray peaks separately 3 . Thei r result is quoted for a mult ipole parameter to be xB = 0.254 ± 0.034 , (6.2) but converting to our convention, a = 0.307 ± 0 . 0 4 1 . (6.3) It should be noted that the D u b n a group has done a second experiment which was reported in the 1995 W E I N Conference [187], but they obtained a similar result, 27-ray lines at 1216 keV and 1224 keV seen in muon capture on stainless steel may also con-tribute to this structure. A more careful identification of these peaks and their contribution to the background around the 1229 keV will be done when we analyze our singles spectra. 3 The main reason for this 'impossibility' is the high correlations among the fitted parameters as discussed in section 5.7, however our higher statistics in the singles spectra allowed us to fit the 2171 keV peaks separately to extract a and r simultaneously (section 5.7.1). 136 viz . XB = 0.225 ± 0.026. The difference between the two experiments was i n the definition of the forward-backward set up (i.e. j± • 7 angle). In the first experiment the two Ge detectors were mechanically moved between the two angles, while in the second experiment this was done by rotating the muon spin i n an external magnetic field. Note that in their notation the correlation coefficients a, f3\ and B2 are given as a 2 , 62 and (a + |ci) respectively. Moreover, the expressions for the correlation coefficients as a function of the multipole parameter (Brudan in et al. Equations 14, 15, 16 &17) are that of Eramzhyan et al. [148]; and are different from Equations 2.13, 2.14 & 2.15 of Chapter 2, which are used by Oziewicz [125] as well as by Ciechanowicz [129]. This is due to the use of different definitions of the mult ipole parameter. In fact the two are connected by 1 + 2x2B - 2^/2xB (6.4) 1 + \x\ + V2xB Table 6.1 lists some relevant parameters of these two experiments and shows how they compare wi th the present work for the 7-ray peaks of interest. A few comments on this table w i l l be made before the detailed discussion of the ex-tracted results. The S / N entry is the signal-to-noise ratio and is calculated as the background-subtracted area (N^) of the peak divided by the area of the back-ground underneath the peak. The F W H M of the Ge detector(s) is the full w id th at half max imum of the corresponding 7-ray peak. The angular correlation coefficient a for the 2171 keV is the product of a for the 1229 keV and the E2/M1 mix ing factor F as given by Equat ion 2.13. A s can be seen i n Table 6.1, our experiment is superior i n many respects to both experiments; a factor of 4.8 increase i n data, a factor of 4 (or 2) improvement in S / N ratio and an experimental resolution i m -provement of 30%. Moreover, an extra factor of 5 improvement is gained in the S / N ratio after applying the coincidence requirement; albeit losing some signal. Note the relatively poor resolutions of the detectors i n the experiment of Brudan in et 137 Table 6.1: A comparative summary of the three measurements of the 7 — v angular correlation following muon capture on 2 8 S i , Mi l le r et al. [144], Brudan in et al. [146] and the present work. The entries in the Table are discussed in some detail i n the { text. i Experiment Mode 7V7 S / N F W H M a r (keV) (10 3) (keV) (fs) M i l l e r singles 73 0.83 4.0 - 0 .37±0 .10 2171 Brudan in singles 73 1.8 3.7/4.0 - 0 .123±0 .062 3 8 . 2 ± 2 . 8 present work singles 355 3.5 2.9/3.2 - 0 .058±0 .025 6 0 . 8 ± 3 . 4 ± 9 . 1 M i l l e r singles 13 0.24 2.6 ± 0 . 1 5 ± 0 . 2 5 1229 Brudan in singles 26 0.5 3.0/3.0 + 0 . 3 0 7 ± 0 . 0 4 1 present work coincidence 9.6 4.5 2.1/2.4 + 0 . 3 6 0 ± 0 . 0 5 9 al. [146], an important parameter for the lineshape analysis. One also notes that the 2171 keV peaks in their Figure 4 are much wider than what they should be; i.e. ~ 22 keV instead of ~ 2(30E0 = 2(0.00375)(2171) = 16 keV. However, because we focus on our coincidence result, our statistical error is larger than that of Brudan in et ai. (We also have no sensitivity to Q\ and B2.) As was discussed earlier, the 2171 keV 7-ray became crucial for obtaining the lifetime of the 2201 keV level. The extracted value of the lifetime was T = 60.8 ± 3 . 4 ± 9 . 1 f s , (6.5) where the second error of 15% is due to the uncertainty in the energy loss ( f § ) of the 2 8 A l ions in S i , as obtained from the T R I M program [175]. It is important to note that in the fitting routine this does not need to be included; just the statistical error on the lifetime for the 2171 keV line, which is transferred to an error in the coupling constants for the 1229 keV line. A recent comparison [188] of measured stopping powers of 2 9 S i ions in A l wi th the T R I M prediction indicates some system-atic deviations between the two, for example, Figure 4.8 of Ref. [188] indicates a reduction i n the electronic stopping power (~ 1/2 of the total) of as much as 20%, 138 and hence the lifetime value may have to be raised by 10% or so, which can be done using Figure 4.9. Our lifetime measurement is to be compared wi th previous values of ( 3 5 ± 1 0 ) fs and ( 1 2 0 ± 7 0 ) fs as measured i n References [172] and [173]. Brudan in et al. quote a lifetime of (38 .2±2 .8 ) fs, but do N O T indicate what value of the the energy loss was used, nor do they give a systematic uncertainty for this value. Moreover, i n the second experiment they reported a lifetime of (48 .5±2 .8 ) fs. 6.2 The Induced Pseudoscalar Coupling We turn now to the extracted values of a. For the the 2171 keV 7-ray peak, the corresponding F factors are -2.5, -0.40 and -0.17 for the M i l l e r et al, B rudan in et al. and the present experiments respectively. In fact one can obtain a value of the mix ing ratio 8 by using the F value and Figure 2.1. We get 8 = 0.22 ± 0 . 0 9 ± 0 . 0 3 , (6.6) where the second error is the 15% error due to the energy loss. This value is to be compared wi th 8 = 0.74 ± 0.29 and 8 = 0.14 ± 0.33 , (6.7) as obtained in the two different experiments of the D u b n a group respectively 4 . We note that the measured value of F i n our experiment cuts the curve of Figure 2.1 at a second region, viz. 8 = 1.8±0.4. (The second region may be ruled out i f one takes the D u b n a results at face value, but it is not clear how they eliminated the second solution for their second experiment.) For the a values extracted from the 1229 keV 7-ray peak, al l three measure-ments are seen to be in reasonable agreement. The large errors associated wi th the 4 Brudanin et al. uses 8 as the varying parameter instead of F in their least-squares fit, which may be dangerous since 8 is double valued. 139 value of M i l l e r et al. is a reflection of the poor conditions of their beam as well as the low signal-to-noise ratio and the moderate instrumental resolution. It should per-haps be noted that they performed their least-squares fit on background-subtracted peaks. There was no raw spectrum of the 1229 keV 7-ray peak given, nor was there any mention of how such background subtraction was carried out. Nevertheless, they had set the stage for the future 7 — v angular correlation experiments. The error on a obtained by Brudan in et al. is slightly smaller than our er-ror. Thei r use of different angles to the muon polarizat ion axis (i.e. the forward-backward spectra that are sensitive to the B\ and f32 parameters) was an advantage and may have given them some confidence i n their result. However, their quoted er-rors do not seem to include some systematic contributions due to, but not restricted to, the effects described above, e.g. background subtraction, detector response func-t ion and energy-losses. In contrast, we believe that our rather detailed attention to the response function and the slowing-down effects as well as our use of background-free spectra render these errors much smaller. It is therefore our contention that the difference i n the quality of the data of the two experiments is not adequately reflected in the relative values of the quoted errors. We turn now to a discussion of the pseudoscalar weak coupling gp/gA as ex-tracted from our data and from the other experiments, and how they compare w i th each other and wi th the P C A C prediction. To obtain gp/gA, one needs to compare wi th the calculations of section 2.4. For convenience, the various theoretical calcu-lations (i.e. Figure 2.3) are overlaid i n Figure 6.1 against the measured correlation coefficient obtained in this experiment to deduce the pseudoscalar coupling con-stant. Furthermore, to put the present results into perspective, Table 6.2 presents a summary of the extracted values of gp/gA from our data, along wi th those extracted from the other experiments. 140 • 5 -.4 1 .3 • 2 H .1 .0 . , i , i i i i i i i i L—J I I 1 I I I I l_ Legend FPA Model I: Cie — * — Model I!: P&S Model III: Kuzmin i 1 1 r *9P/g. 2 0 Figure 6.1: The measured 7 — 1 / angular correlation coefficient a compared to the theories of Ciechanowicz [129], Parthasarathy & Sridhar [100] and Kuz'min et al. [126]. Also shown is the Fujii-Primakoff approximation. Table 6.2: Summary of the extracted values of gp/gA from the available theoretical calculations using the measured results of the 7 — 1/ angular correlation experiments of muon capture in 2 8 S i for the 1229 keV 7-ray. Experiment Miller et al. [144] Brudanin et al. [146] present work Correlation Coefficient a + 0 . 1 5 ± 0 . 2 5 + 0 . 3 0 7 ± 0 . 0 4 1 + 0 . 3 6 0 ± 0 . 0 5 9 Theoretical model !)• Deduced values of gp/gA Fujii-Primakoff Approximation + 4 . 5 ± 8 . 0 + 7 . 7 ± 1 . 2 + 9 . 5 ± 2 . 4 Ciechanowicz [109] + 0 . 4 ± 6 . 2 + 3 . 4 ± 1 . 0 + 5 . 3 ± 2 . 0 Parthasarathy and Sridhar [100] - 2 . 5 ± 8 . 2 + 2 . 0 ± 1 . 6 + 4 . 2 ± 2 . 5 Kuz'min et al. [126] - 7 . 6 ± 9 . 6 - 3 . 0 ± 2 . 0 0 . 0 ± 3 . 2 141 Before commenting on these results and how they compare wi th each other and wi th the P C A C prediction, let us address a difficulty in interpreting the mea-surement of Mi l l e r et al. There are three values of gp/gA given i n the literature from the M i l l e r et al. experiment. Mi l l e r et al. themselves extracted gp/gA from their data using their value of cv and using (essentially) the Fuji i -Primakoff ap-proximation. Thei r weighted average value of a , from the isotopically pure 2 8 SiG "2 and the natural Si data is 0 .21±0 .19 , and comparing this to the Fuji i -Primakoff approximation gives the value gp/gA = 5 ± 8 that they quote i n their paper [144]. Note that the three more complete calculations (Ciechanowicz, Parthasarathy and Sridhar and K u z ' m i n et al.) had not been done when M i l l e r et al. published their measurement. If the result of M i l l e r et al. for a is compared wi th these calcula-tions, then one gets a much lower value of gP/gA, v iz . Table 6.2. Bu t then, one asks, where do the numbers of - 4.9 < gP/gA < 1-2 and gP/gA = 13.5±|; | , (6.8) that are given in the papers of Ciechanowicz [109] and Parthasarathy and Sr id-har [100], respectively, come from? The answer is that Ciechanowicz compares wi th Mi l le r ' s a, 3^ and 32 simultaneously, and in fact his number mainly comes from 8\ and 32, not a; however it is not far from the 0 ± 6 that one gets from a alone, wi th his calculation. Parthasarathy and Sridhar, however, use only 32 i n their compari-son wi th M i l l e r et at, and ignore the fact that this value of gp/gA gives values for a and B1 that are very different than what M i l l e r et al. measured, and ignore the fact that there is a second solution for gp/gA that gives the same f32 and agrees better wi th a and B1. This can clearly be seen i n Table I ( "Mode l II") i n their paper [100]. A much better agreement wi th a, 3X and 32 is achieved for gp/gA around -2.5 than for gp/gA around +12.5 : 142 Mil l e r et al. gp/gA = —2.5 gp/gA = +12.5 a 0.21 ± 0 . 1 9 0.21 0.49 pi 0.02 ± 0 . 0 3 0.04 0.32 P2 1.12 ± 0 . 1 0 1.17 1.16 Hence the preferred solution is gp/gA around -2.5, i.e. consistent wi th that of Ciechanowicz. Thus , we believe that the value in Equat ion 6.8 from Parthasarathy and Sridhar is taken from the wrong solution. However, the error bars of M i l l e r et al. for Pi and p2 have to be treated wi th scepticism since : • The effects of both Pi and p2 on the gamma-ray spectrum are washed out by the small residual muon polarizat ion, v iz . 15%. • Pi and p2 have very similar effects on the gamma-ray line shape, and therefore the fitted values are likely to be highly correlated; also one has to add i n the error in the measurement of this polarization. • Our Monte Car lo / f i t t ing studies show that one cannot extract such small errors on Pi and p2 w i th Mi l l e r ' s statistics (even by fit t ing to his data). Th i s would mean that the value extracted by Ciechanowicz should have larger errors, and should probably be about the value of 0 ± 6 from a alone. So we think that it is safest to use only Mi l l e r ' s value of a, and one then obtains the values tabulated i n Table 6.2 above for the various theoretical models. Brudan in et al, i n their final result, quoted only the gp/gA value as obtained from the Ciechanowicz calculation; claiming that the Parthasarathy and Sridhar calculation is inconsistent and less reliable "as their correlation coefficients do not correspond, as they should, to the same values of the parameter XB" [146]. However, a careful comparison between the values of the correlation coefficients (a, Pi and P2) corresponding to the experimental range of XB and curves corresponding to the Parthasarathy and Sridhar calculation gives consistent values of gp/gA w i th in 143 the uncertainties, contrary to the above cla im 5 . Thus the conclusion drawn by Brudan in et al. that "no physical solution exists for the combination (a-\-^ci(= B2)) in the x-region of interest" is surprising, as noted by Parthasarathy [189], in his comments on the Brudan in et al. measurement. Note that the K u z ' m i n et al. calculation was underway when Brudan in et al. published their results, and so no comparison wi th this calculation was included. The gp/gA values as extracted from all theoretical calculations using the Brudan in et al. experiment are also included in Table 6.2. If the extracted values of gp/gA i n Table 6.2 are taken at face value, a few comments are i n order. One is that the values of gp/gA, as extracted from the vari-ous theoretical models, using a l l three measurements are consistent wi th each other wi th in the (rather large) quoted uncertainties. Furthermore if one adopts the Fuj i i -Primakoff Approximat ion , a l l three experiments would agree wi th the Goldberger-Treiman expectation (PCAC-pred i c t i on ) of gp/gA ~ 7 (see section 1.5). The ex-tracted values of gp/gA from al l three "complete" calculations using the Brudan in et al. results is in a strong disagreement wi th the Goldberger-Treiman estimate and indicat ing a strong downward renormalization of gp/gA (or a failure of P C A C ) . In contrast, the extracted values of gp/gA using our result, due to a higher central gp/'9A value as well as larger error bars 6 , is in conflict w i th the Goldberger-Treiman value only when compared to the K u z ' m i n et al. model. A l l in a l l , the extracted values of gp/gA are on the lower side of the canon-ical value of Goldberger and Treiman, suggesting a sizable quenching of gp/gA i n 2 8 S i . Thus, the overall situation is somewhat confusing. It is apparent that more theoretical efforts are required to assess the model-dependence of these intr iguing 5 T h i s is also consistent with the arguments outlined in Chapter 2, i.e. the Parthasarathy and Sridhar calculation is consistent with one independent correlation coefficient. 6 The comments made previously comparing the quality of the quoted error bars in the two experiments applies here as well and need not be repeated. 144 results. Several theorists at the recent W E I N conference were very uneasy wi th this situation and felt that it was necessary to look at the calculations much more critically. Thus although further experiments might be worthwhile, the ma in focus should be on obtaining a better understanding of the theoretical uncertainties. 145 Chapter 7 Conclusions The angular correlation of the neutrino wi th the 1229 keV 7-ray from the de-excitation of the 2201 keV 1 + level in 2 8 A l , following exclusive muon capture on 2 8 S i has been measured, in order to determine the magnitude of the induced-pseudoscalar coupling constant gp of the weak hadronic current. The correlation was observed by measuring the energy distr ibution of the 1229 keV Doppler-broadened 7-ray, as originally suggested by Grenacs et al. [142]. A potentially serious background near, and probably underneath the 7-ray of interest made the extraction of the angular correlation from the line shape problematic. Consequently, we adopted a coincidence technique in which the 1229 keV 7-ray is 'tagged' by the subsequent 942 keV 7-ray (100% branching ratio) in the cascade. A pair of Compton-suppressed high puri ty germanium detectors were used to detect the Doppler-broadened 7-rays, while an array of 24 N a l ( T l ) scintillators detected the 942 keV 7-rays. A s a result, very clean spectra were obtained; and a dependable measurement of the angular correlation was made. In addit ion, a signal-to-background ratio of 4.5 was obtained, which is to be compared wi th ratios of 0.5 and 0.24 observed in the other two experiments [144,146]. The correlation coefficient a was found to be 0 .360±0 .059 , in good agreement wi th the recent measurement of Brudan in et al. [146]. W h e n compared to the theo-146 retical models of Ciechanowicz [129] and Parthasarathy and Sridhar [100], our result gives the values for gp/gA of 5 .3±2 .0 and 4 . 2 ± 2 . 5 respectively. B o t h of these re-sults are consistent wi th the P C A C prediction of gp/gA — 7, but suggest a possible quenching of the induced-pseudoscalar coupling constant in 2 8 S i . However, a com-parison wi th the most recent calculation of K u z ' m i n et al. [126] yields the value of gp/gA — 0 . 0±3 .2 , suggesting a massive (downward) renormalization of gp. Further-more, when the same result is compared wi th the Fuj i i -Pr imakoff approximation, a value of gp/gA = 9 .5±2 .4 is found to be quite consistent wi th the P C A C value (and wi th an unrenormalized value of gp). We suppose that the more recent calculations are more dependable, but the model-dependence of these intr iguing results remains to be assessed. Clearly more work is required on the theoretical side. In addit ion, the more prolific 2171 keV 7-ray de-exciting from the 2201 keV common level proved to be quite useful; its Doppler-broadened line shape was used to measure the lifetime of this level, v iz . , r = 6 0 . 8 ± 3 . 4 ± 9 . 1 fs, as well as to solve the enigma of the unphysical result of the correlation coefficient a that was found by a previous experiment (Mi l le r et al. [144,145]). The coincidence technique has provided a thorough investigation of possible cascade feeding of the common level of interest from high-lying excited states. Such feeding would act as another source for the 1229 keV 7-rays, besides the direct production from the c/p-sensitive spin sequence of 2 8 S i ( 0 + ) A 2 8A1*(2201 keV 1 + : A 973 keV 0 + ) ; and could distort the 7 — 1/ angular correlation, and thereby spoil the interpretation of the Doppler-broadened lineshape measurement i n terms of gp. Stringent l imits on such feeding are obtained for the first t ime, thus reducing systematic errors of the angular correlation. The pr imary l imi ta t ion of the present experiment was a combination of low statistics and highly correlated variables. The former was due to the loss of the 1229 keV 7-ray signal after the imposi t ion of the coincidence requirement. Higher 147 statistics may be gained either by running longer and/or opt imizing the detection system, e.g. by increasing the efficiency of the N a l ( T l ) arm. The later l imi ta t ion was due to the strong dependence of the correlation coefficients on r and 8; and can be eliminated by independent measurements of either or both of these parameters. Overcoming these l imitations i n future coincidence experiments would enable a more accurate measurement of the correlation coefficient(s) and ult imately improve the precision on the extracted pseudoscalar coupling constant. In addit ion, a periodic use of a series of known 7-ray sources throughout a wide range of energies would help to determine the response function as well as the acceptances of the Ge detectors more accurately. (Dur ing the experiment we did not realise how important the 2171 keV line would be.) It would also be useful to obtain good statistics from muon capture on an isotopically enriched 2 9 S i target to measure the yield (and i f needed the shape) of the 1229 keV 7-rays emitted in the 29Si(n,vn) reaction; and to confirm the results of M i l l e r et al. [144,145]. Final ly , this technique may be extended to other nuclei i n which the transitions are sensitive to gP (e.g. 2 0 F ) . In summary, a dependable measurement of gp/gA seems st i l l to elude us be-cause of systematic errors and theoretical uncertainties. A similar si tuation exists wi th radiative muon capture on nuclei for which the measurements are reasonably consistent [92,82], yet the theoretical uncertainties are clearly considerable and have been discussed recently by Fearing and Welsh [95]. To add to the confusion the re-cent experimental result for radiative muon capture on hydrogen gives a high value for gp/gA, viz . 10.0 ± 0.9 ± 0.3 [74]. 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The coefficients and their errors are * C** found by MINUIT routines from CERN libr a r y CERNlib. * C** * C** Modified to use the IMSL DTWODQ routine and include Slowing-Down* C** effects(constant) and the RF2 peak-shape response function. * C** PROGRAM DOPFITc.IMSL C* C* Declare real variables as double precision. * C* IMPLICIT DOUBLE PRECISION (A-H,0-Z) EXTERNAL FCN C* C* Redefine the I/O stream, (default 5,6,7) * 160 c* CCC CALL MINTI0(1,2,2) C* C* C a l l MINUIT using double p r e c i s i o n . * C* CALL MINUIT(FCN.O) C* C* E x i t program DOPFITc_IMSL. * C* CALL EXIT END C** C** * C** Subroutine FCN c a l c u l a t e s the value of the f u n c t i o n to be * C** minimized or studied. This i s the FCN subroutine MINUIT * C** r e q u i r e s . * C** * C** SUBROUTINE FCN (NPAR,G,F,X,IFLAG) C* C* Declare r e a l v a r i a b l e s as double p r e c i s i o n . * C* IMPLICIT DOUBLE PRECISION (A-H.O-Z) C* C* Declare passed subroutine v a r i a b l e s . * C* DIMENSION G(34),X(34) 161 NPAR — THE NUMBER OF VARIABLE PARAMETERS G(15) — A VECTOR INTO WHICH THE DERIVATIVES ARE TO BE PUT F — THE FUNCTION VALUE CALCULATED IN FCN X(15) — A VECTOR CONTAINING THE EXTERNAL PARAMETER VALUES IFLAG — A MARKER WHOSE MEANING IS DESCRIBED BELOW: ! 1 = I n i t i a l i z i n g entry. Read i n a l l necessary s p e c i a l ! data to FCN, c a l c u l a t e constants, p r i n t and graph ! input i f d e s i r e d , e t c . ! 2 = Normal entry with gradient. C a l c u l a t e the ! d e r i v a t i v e s i n vector G and the f u n c t i o n value ! i n F at the point X. ! 3 = Terminating entry. Write out any s p e c i a l graphs, ! summaries, output t a b l e s , e t c . f o r the minimum ! p l o t . ! 4 = Normal entry without gradient. C a l c u l a t e only the ! f u n c t i o n value F at point X. !IMSL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! C* C* Declare Common Block PARAMETER C* DIMENSION XX(34) COMMON /PARAMETER/ XX,GAIN,OFFSET,XDT !IMSL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! C* C* Declare l o c a l subroutine v a r i a b l e s . (data v a r i a b l e s ) * C* DIMENSION XVALUE1(1024),YVALUE1(1024),ERR1(1024),YERR1(1024), 1 XVALUE2(1024),YVALUE2(1024),ERR2(1024),YERR2(1024), 2 XVALUE3(1024),YVALUE3(1024),ERR3(1024),YERR3(1024), 3 XVALUE4(1024),YVALUE4(1024),ERR4(1024),YERR4(1024) !IMSL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! EXTERNAL FF1,GG,HH1,DTWODQ,FF2,HH2,FF3,HH3,FF4,HH4 !IMSL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! NDATA — THE NUMBER OF DATA POINTS 162 XVALUE(n) — X DATA POINT VALUE YVALUE(n) — Y DATA POINT VALUE YERR(n) — Y DATA POINT VALUE f o r e r r o r computation (added Y values even f o r a subtracted spectra, t r e a t e r r o r s p r o p e r l y , B. Moftah May 16,1994) ERR(n) — ERROR IN DATA POINT where n i s the data point number C* C* I n i t i a l i z e p i . * C* PI=3.1415927 O C* IFLAG = 1. I n i t i a l i z i n g entry. Read i n a l l necessary s p e c i a l * C* data to FCN, c a l c u l a t e constants, p r i n t and graph input i f * C* d e s i r e d , e t c . * C* IF (IFLAG .EQ. 1) THEN C* C* Read i n data from input f i l e . * C* READ (5,*) NDATA1,GAIN1,0FFSET1 DO 100 I=1,NDATA1 READ (5,*) XVALUEl(I),YVALUE1(I),YERR1(I) 100 CONTINUE READ (5,*) NDATA2,GAIN2,0FFSET2 DO 101 I=1,NDATA2 READ (5,*) XVALUE2(I),YVALUE2(I),YERR2(I) 101 CONTINUE READ (5,*) NDATA3,GAIN3,0FFSET3 DO 102 I=1,NDATA3 READ (5,*) XVALUE3(I),YVALUE3(I),YERR3(I) 102 CONTINUE READ (5,*) NDATA4,GAIN4,0FFSET4 163 DO 103 I=1,NDATA4 READ (5,*) XVALUE4(I),YVALUE4(I),YERR4(I) 103 CONTINUE C* C* Compute e r r o r of each data p o i n t . * C* DO 110 I=1,NDATA1 ERR1(I)=DSQRT(YERR1(I)) 110 .CONTINUE DO 111 I=1,NDATA2 ERR2(I)=DSQRT(YERR2(I)) 111 CONTINUE DO 112 I=1,NDATA3 ERR3(I)=DSQRT(YERR3(I)) 112 CONTINUE DO 113 I=1,NDATA4 ERR4(I)=DSQRT(YERR4(I)) 113 CONTINUE C* C* End of i n i t i a l i z i n g entry. * C* ENDIF C* C* Now c a l c u l a t e the chi-squared f i t to the polynomial. * C* F = 0.0D0 F1=0.0D0 F2=0.0D0 F3=0.0D0 F4=0.0D0 ERRABS=0.1 ERRREL=0.0 IRULE=2 DO 115 1=1,34 164 XX(I)=X(I) 115 CONTINUE BETAO=0.0037484 C 2171keV Gel GAIN=GAIN1 0FFSET=0FFSET1 E0=GAIN*X(7)+0FFSET AA= X(7)-BETA0*E0/GAIN BB= X(7)+BETA0*E0/GAIN DO 120 K=l,NDATA1 XDT = XVALUEl(K) CALL DTWODQ (FF1,AA,BB,GG,HH1,ERRABS,ERRREL,IRULE,RESULT,ERREST) YFIT1=RESULT+X(10) FI = FI +((YVALUE1(K)-YFIT1)/ERR1(K))**2. 120 CONTINUE C 1229keV Gel GAIN=GAIN2 0FFSET=0FFSET2 E0=GAIN*X(14)+0FFSET AA= X(14)-BETA0*E0/GAIN BB= X(14)+BETA0*E0/GAIN DO 121 K=l,NDATA2 XDT = XVALUE2(K) CALL DTWODQ (FF2,AA,BB,GG,HH2,ERRABS,ERRREL,IRULE,RESULT,ERREST) YFIT2=RESULT+X(16) F2 = F2 +((YVALUE2(K)-YFIT2)/ERR2(K))**2. 121 CONTINUE C 2171keV Ge2 GAIN=GAIN3 0FFSET=0FFSET3 E0=GAIN*X(22)+0FFSET AA= X(22)-BETA0*E0/GAIN BB= X(22)+BETA0*E0/GAIN DO 122 K=l,NDATA3 XDT = XVALUE3(K) CALL DTWODQ (FF3,AA,BB,GG,HH3,ERRABS,ERRREL,IRULE,RESULT,ERREST) YFIT3=RESULT+X(25) F3 = F3 +((YVALUE3(K)-YFIT3)/ERR3(K))**2. 122 CONTINUE C 1229keV Ge2 165 GAIN=GAIN4 0FFSET=0FFSET4 E0=GAIN*X(29)+0FFSET AA= X(29)-BETA0*E0/GAIN BB= X(29)+BETA0*E0/GAIN DO 123 K=l,NDATA4 XDT = XVALUE4(K) CALL DTWODQ (FF4,AA,BB,GG,HH4,ERRABS,ERRREL,IRULE,RESULT,ERREST) YFIT4=RESULT+X(31) F4 = F4 +((YVALUE4(K)-YFIT4)/ERR4(K))**2. 123 CONTINUE F=F+F1+F2+F3+F4 !IMSL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! C* C* IFLAG = 3. Terminating entry. Write out any s p e c i a l graphs, * C* summaries, output t a b l e s , e t c . f o r the minimum p l o t . * C* IF (IFLAG .EQ. 3) THEN WRITE (*,*) 'DOPPLER PEAK FITTTING COMPLETE' ENDIF C* C* E x i t subroutine FCN. * C* RETURN END C** C** * C** Function FF computes the value of the 2-D f u n c t i o n to be * C** i n t e g r a t e d by the IMSL code DTWODQ * C** * C** 166 C===========================2171 KeV Gel============================== DOUBLE PRECISION FUNCTION FF1 (X,Y) C* C* Declare r e a l v a r i a b l e s as double p r e c i s i o n . * C* IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION XX(34) COMMON /PARAMETER/ XX,GAIN,OFFSET,XDT C* E0=GAIN*XX(7)+0FFSET BETA0=0.0037484 A= (l.-XX(20)*XX(2)/2.) B = (3.*XX(20)*XX(2)/2.)*(GAIN/(E0*BETA0))**2. Q=28*931.5*BETA0/(3.0D+08*XX(4)) CC=1/(1.-Y/Q) DEL=XDT-X IF (DEL .LT. - X X ( l l ) ) THEN PDT=DEXP(0.5*XX(11)*(XX(11)+2.*DEL)/XX(8)**2 . ) ELSEIF ((DEL .GE. - X X ( l l ) ) .AND. (DEL .LE. XX(12))) THEN PDT=DEXP(-0.5*(DEL/XX(8))**2.) ELSEIF (DEL .GT. XX(12)) THEN PDT=DEXP(0.5*XX(12)*(XX(12)-2.*DEL)/XX(8)**2.) ENDIF STEP=.5*XX(9)*ERFC(DEL/(SORT(2.)*XX(8))) FF1= (XX(1)/(XX(3)*1.0D-15))*DEXP(-Y/(XX(3)*1.0D-15)) 1 *(A*CC+B*(X-XX(7))**2.*CC**3.) *(PDT+STEP) C* RETURN END C===========================1229 KeV Gel============================== DOUBLE PRECISION FUNCTION FF2 (X,Y) C* C* Declare r e a l v a r i a b l e s as double p r e c i s i o n . * C* IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION XX(34) 167 COMMON /PARAMETER/ XX,GAIN,OFFSET,XDT C* EO=GAIN*XX(14)+OFFSET BETAO=0.0037484 A= (l.-XX(2)/2.) B = (3.*XX(2)/2.)*(GAIN/(E0*BETA0))**2. Q=28*931.5*BETA0/(3.0D+08*XX(4)) CC=1/(1.-Y/Q) DEL=XDT-X IF (DEL .LT. -XX(18)) THEN PDT=DEXP(0.5*XX(18)*(XX(18)+2.*DEL)/XX(17)**2.) ELSEIF ((DEL .GE. -XX(18)) .AND. (DEL .LE. XX(19))) THEN PDT=DEXP(-0.5*(DEL/XX(17))**2.) ELSEIF (DEL .GT. XX(19)) THEN PDT=DEXP(0.5*XX(19)*(XX(19)-2.*DEL)/XX(17)**2.) ENDIF STEP=.5*XX(15)*ERFC(DEL/(SQRT(2.)*XX(17))) FF2= (XX(13)/(XX(3)*1.OD-15))*DEXP(-Y/(XX(3)*1.OD-15)) 1 *(A*CC+B*(X-XX(14))**2.*CC**3.) *(PDT+STEP) C* RETURN END C==============================2171 keV Ge2 ========================== DOUBLE PRECISION FUNCTION FF3 (X,Y) C* C* Declare r e a l v a r i a b l e s as double p r e c i s i o n . * C* IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION XX(34) COMMON /PARAMETER/ XX,GAIN,OFFSET,XDT C* EO=GAIN*XX(22)+OFFSET BETAO=0.0037484 A= (l.-XX(20)*XX(2)/2.) B = (3.*XX(20)*XX(2)/2.)*(GAIN/(E0*BETA0))**2. Q=28*931.5*BETA0/(3.0D+08*XX(4)) CC=1/(1.-Y/Q) 168 DEL=XDT-X IF (DEL .LT. -XX(26)) THEN PDT=DEXP(0.5*XX(26)*(XX(26)+2.*DEL)/XX(23)**2.) ELSEIF ((DEL .GE. -XX(26)) .AND. (DEL .LE. XX(27))) THEN PDT=DEXP(-0.5*(DEL/XX(23))**2.) ELSEIF (DEL .GT. XX(27)) THEN PDT=DEXP(0.5*XX(27)*(XX(27)-2.*DEL)/XX(23)**2.) ENDIF STEP=.5*XX(24)*ERFC(DEL/(SQRT(2.)*XX(23))) FF3= (XX(21)/(XX(3)*1.0D-15))*DEXP(-Y/(XX(3)*1.0D-15)) 1 *(A*CC+B*(X-XX(22))**2.*CC**3.) *(PDT+STEP) C* RETURN END C======================i229keV Ge2 ================================ DOUBLE PRECISION FUNCTION FF4 (X,Y) C* C* Declare real variables as double precision. * C* IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION XX(34) COMMON /PARAMETER/ XX,GAIN,OFFSET,XDT C* E0=GAIN*XX(29)+0FFSET BETAO=0.0037484 A= (l.-XX(2)/2.) B = (3.*XX(2)/2.)*(GAIN/(E0*BETA0))**2. Q=28*931.5*BETA0/(3.0D+08*XX(4)) CC=1/(1.-Y/Q) DEL=XDT-X IF (DEL .LT. -XX(33)) THEN PDT=DEXP(0.5*XX(33)*(XX(33)+2.*DEL)/XX(32)**2.) ELSEIF ((DEL .GE. -XX(33)) .AND. (DEL .LE. XX(34))) THEN PDT=DEXP(-0.5*(DEL/XX(32))**2.) ELSEIF (DEL .GT. XX(34)) THEN PDT=DEXP(0.5*XX(34)*(XX(34)-2.*DEL)/XX(32)**2.) ENDIF STEP=.5*XX(30)*ERFC(DEL/(SQRT(2.)*XX(32))) 169 FF4= (XX(28)/(XX(3)*1.OD-15))*DEXP(-Y/(XX(3)*1.OD-15)) 1 *(A*CC+B*(X-XX(29))**2.*CC**3.) *(PDT+STEP) C* RETURN END C** c** * C** Function GG to evaluate the lower l i m i t s of the inner i n t e g r a l * C** * C** DOUBLE PRECISION FUNCTION GG (X) C* C* Declare r e a l v a r i a b l e s as double p r e c i s i o n . * C* IMPLICIT DOUBLE PRECISION (A-H.O-Z) C* GG = 0.0 RETURN END C** c** * C** Function HH to evaluate the upper l i m i t s of the inner i n t e g r a l * C** * C** DOUBLE PRECISION FUNCTION HH1 (X) C* C* Declare r e a l v a r i a b l e s as double p r e c i s i o n . * 170 c* IMPLICIT DOUBLE PRECISION (A-H.O-Z) DIMENSION XX(34) COMMON /PARAMETER/ XX,GAIN.OFFSET,XDT C* EO=GAIN*XX(7)+OFFSET BETAO=0.0037484 Q=28*931.5*BETA0/(3.0D+08*XX(4)) DUMMY=0.99999 HH1=DUMMY*Q*(1.-GAIN*ABS(X-XX(7))/(EO*BETAO)) C* RETURN END C================= 1229 KeV Gel ===================================== DOUBLE PRECISION FUNCTION HH2 (X) C* C* Declare r e a l v a r i a b l e s as double p r e c i s i o n . * C* IMPLICIT DOUBLE PRECISION (A-H.O-Z) DIMENSION XX(34) COMMON /PARAMETER/ XX,GAIN,OFFSET,XDT C* E0=GAIN*XX(14)+0FFSET BETAO=0.0037484 Q=28*931.5*BETA0/(3.0D+08*XX(4)) DUMMY=0.99999 HH2=DUMMY*Q*(1.-GAIN*ABS(X-XX(14))/(EO*BETAO)) C* RETURN END C================= 2171 KeV Ge2 ===================================== DOUBLE PRECISION FUNCTION HH3 (X) C* C* Declare r e a l v a r i a b l e s as double p r e c i s i o n . * C* 171 c* IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION XX(34) COMMON /PARAMETER/ XX,GAIN,OFFSET,XDT E0=GAIN*XX(22)+0FFSET BETAO=0.0037484 Q=28*931.5*BETA0/(3.0D+08*XX(4)) DUMMY=0.99999 HH3=DUMMY*Q*(1.-GAIN*ABS(X-XX(22))/(E0*BETA0)) C* RETURN END C================= 1229 KeV Ge2 ====================================== DOUBLE PRECISION FUNCTION HH4 (X) C* C* Declare r e a l v a r i a b l e s as double p r e c i s i o n . * C* IMPLICIT DOUBLE PRECISION (A-H.O-Z) DIMENSION XX(34) COMMON /PARAMETER/ XX,GAIN,OFFSET,XDT C* E0=GAIN*XX(29)+0FFSET BETA0=0.0037484 Q=28*931.5*BETA0/(3.0D+08*XX(4)) DUMMY=0.99999 HH4=DUMMY*Q*(1.-GAIN*ABS(X-XX(29))/(E0*BETA0)) C* RETURN END !IMSL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! C** C** * C** Function ERFc c a l c u l a t e s the value of the COMPLEMENTARY e r r o r * C** f u n c t i o n . * C** * 172 c** FUNCTION ERFC(X) C* C* Declare r e a l v a r i a b l e s as double p r e c i s i o n . * C* IMPLICIT DOUBLE PRECISION (A-H.O-Z) C* C* C a l c u l a t e value of e r r o r f u n c t i o n . * C* ZX=X*1.4142 AX=ABS(ZX) T=l.0/(1.0+0.2316419*AX) DD=0.3989423*EXP((-1.0)*AX**2/2.0) PP=1.0-DD*T*((((1.330274*T-1.821256)*T+1.781478)*T-1 0.3565638)*T+0.3193815) IF (X) 100,110,110 100 PP=1.0-PP 110 ERFC=1.-(PP*2-1.0) C* C* E x i t f u n c t i o n ERF. * C* RETURN END 173 

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