UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Muon capture in ²⁸Si Moftah, Belal Ali 1996

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_1996-091333.pdf [ 7.16MB ]
Metadata
JSON: 831-1.0084982.json
JSON-LD: 831-1.0084982-ld.json
RDF/XML (Pretty): 831-1.0084982-rdf.xml
RDF/JSON: 831-1.0084982-rdf.json
Turtle: 831-1.0084982-turtle.txt
N-Triples: 831-1.0084982-rdf-ntriples.txt
Original Record: 831-1.0084982-source.json
Full Text
831-1.0084982-fulltext.txt
Citation
831-1.0084982.ris

Full Text

MUON CAPTURE IN  2 8  Si  By BELAL ALI MOFTAH B . S c , University of Winnipeg, 1988 M . S c , University of British Columbia, 1991  A T H E S I S 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 THE REQUIREMENTS F O R T H E D E G R E E OF DOCTOR OF PHILOSOPHY  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 w i t h a specific nuclear de-excitation 7-ray following exclusive m u o n capture on  2 8  Si,  i n order to determine the size of the induced-pseudoscalar coupling constant gp of the weak hadronic current. T h e correlation is observed v i a the Doppler-broadened line shape of the 1229 k e V A l 7-ray, w h i c h is measured using a pair of C o m p t o n 2 8  suppressed intrinsic G e detectors.  Significant background suppression is achieved  through the use of a coincidence technique i n w h i c h the 1229 k e V 7-ray of interest is 'tagged' by the subsequent 942 k e V 7-ray i n the cascade, w h i c h is detected i n an array of 24 N a l ( T l ) scintillators. T h r o u g h 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 . 3 6 0 ± 0 . 0 5 9 i n good agreement w i t h a recent measurement at D u b n a . 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 quenching of the induced-pseudoscalar coupling constant i n  2 8  S i i n comparison w i t h 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  intriguing results has yet to be assessed fully. A measurement of the correlation coefficient a of the 2171 k e V ( 1  +  —> 2 ) +  7-ray has solved the enigma of the u n p h y s i c a l result that was found by a previous experiment. In a d d i t i o n , the lifetime of the 2201 k e V A l level has been measured 2 8  ii  C o n t e n t s  Abstract  ii  List of Tables  vii  List of Figures  ix  Acknowledgements 1  Introduction  1  1.1  The M u o n  1  1.2  Mesic A t o m s  3  1.3  Weak Interactions  8  1.3.1  2  xii  T h e Weak-Interaction H a m i l t o n i a n  10  1.4  Constraints on the Weak C o u p l i n g Constants  13  1.5  Nuclear R e n o r m a l i z a t i o n of gp  18  1.6  Observables Sensitive to the Induced Pseudoscalar C o u p l i n g  19  1.6.1  M u o n C a p t u r e by H y d r o g e n  20  1.6.2  M u o n C a p t u r e by C o m p l e x Nuclei  22  T h e 7 — 1/ A n g u l a r C o r r e l a t i o n  27  2.1  Introduction  27  2.2  Fujii-PrimakofF A p p r o x i m a t i o n  29  2.3  B e y o n d the F P A  30  2.4  Full Calculation Models  34  2.4.1  34  M o d e l I : Ciechanowicz iv  3  4  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  T h e M e t h o d of Measurement  43  2.6  A n c i l l a r y Reactions  46  Description of the Experiment  51  3.1  B e a m P r o d u c t i o n and the M 9 B C h a n n e l  51  3.2  E x p e r i m e n t a l Arrangement  52  3.3  Detection System  55  3.4  Electronics and D a t a A c q u i s i t i o n  58  3.4.1  Telescope Logic  59  3.4.2  T h e Compton-suppressed G e r m a n i u m Logic  59  3.4.3  T h e N a l ( T l ) Logic  63  3.4.4  Strobe Logic  63  3.4.5  Data Acquisition Control  64  Technical problems  67  4.1  Basic Interactions i n 7-ray Detectors  67  4.2  N e u t r o n Effects i n Ge-detectors  70  4.3  Coincidence Technique  74  4.4  T h e Detector Response F u n c t i o n  79  4.5  S l o w i n g - D o w n Effects  85  4.6  F i n i t e S o l i d - A n g l e Effects  88  4.7  T h e Peak F i t t i n g Programs  '  5 Data Analysis  89  91  5.1  Introduction  91  5.2  Cuts  92  v  6  7  5.2.1  T i m e of the M u o n C u t  92  5.2.2  Compton-Suppression C u t  94  5.2.3  Rise T i m e C o r r e c t i o n  96  5.2.4  Time-Coincidence Cut  102  5.2.5  E n e r g y - G a t e d Coincidence  102  5.3  Acceptances of Detectors  106  5.4  Cascade Feeding  109  5.5  Background Subtraction  123  5.6  Parameter R e c a p i t u l a t i o n  126  5.7  A n a l y s i s of the Doppler-broadened Peaks  127  5.7.1  T h e 2171 k e V line  127  5.7.2  T h e 1229 k e V line  130  5.7.3  T h e 1229 k e V and the 2171 k e V Simultaneous F i t  132  Discussion of Results  134  6.1  The 7 — v Angular Correlation  134  6.2  T h e Induced Pseudoscalar C o u p l i n g  139  Conclusions  146  Bibliography  150  Appendix  160  A  160  T h e P e a k - F i t t i n g C o m p u t e r P r o g r a m Listing  vi  List of Tables 1.1  Properties of the m u o n  1.2  Theoretical a n d experimental weak coupling constants for the elementary process of m u o n capture  17  S u m m a r y of values of gp/gA m u o n capture i n hydrogen  22  1.3 1.4  2  as determined from measurements of  S u m m a r y of values of gp/gA determined by comparing experimental results w i t h theoretical predictions of radiative m u o n capture ( R M C ) i n complex nuclei  24  S u m m a r y of values of gp/gA as determined from measurements of m u o n capture ( O M C ) i n complex nuclei  25  2.1  N e u t r o n multiplicities following m u o n capture o n S i  50  3.1  Properties of the N a l ( T l ) counters  57  4.1  N e u t r o n induced 7-ray lines i n g e r m a n i u m isotopes  74  4.2  Chi-squared fit to the 1332 k e V C o peak for different response functions  82  1.5  6 0  5.1  Acceptances of the N a l ( T l ) coverage for the two G e detectors  106  5.2  M u o n i c X - r a y acceptance d a t a for the G e l detector  108  5.3  M u o n i c X - r a y acceptance d a t a for the Ge2 detector  109  5.4  Input parameters needed i n the least-squares fitting function for the two G e detectors  126  5.5  Results of the best fit to the 2171 k e V 7-ray lines i n b o t h G e detectors. 129  5.6  Results of the i n d i v i d u a l fits to the 1229 k e V 7-ray peak for each G e detector  130  Effect of the instrumental resolution of the detectors o n a  130  5.7  vii  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 k e V a n d the 1229 k e V 7-ray lines i n b o t h G e detectors  132  6.1  A comparative s u m m a r y of the the measurements of the 7 — 1/ angular correlation experiments 138  6.2  S u m m a r y of the extracted values of gp/gA from the 7 — ^ angular correlation experiments  vm  142  L i s t  o f  F i g u r e s  1.1  T y p i c a l muonic cascades  1.2  T h e single p i o n exchange diagram  16  1.3  gp/gA  determined from the rate of radiative m u o n capture measurements  25  T h e m u l t i p o l a r i t y factor F as a function of the m i x i n g ratio, S = E2/M1  33  Dependence of the amplitude ratio transition  36  2.1 2.2  a  6  s  x  on  gp/gA  f ° the 1229 k e V r  2.3  Dependence of a o n  2.4  Dependence of [3i on  gp/gA  f ° the 1229 k e V transition  38  2.5  Dependence of B on  gp/gA  for the 1229 k e V transition  39  2.6  K i n e m a t i c s of m u o n capture reaction o n S i  2.7  T h e shape of a Doppler-broadened 7-ray d i s t r i b u t i o n a n d the effect  2  gp/gA  f ° the 1229 k e V transition r  r  2 8  of the 7 — v angular correlation coefficient, a 2.8  28  2.9 2.10  37  44  45  Si(7r,7) Al c o n t i n u u m subtracted spectrum  47  2 8  S i ( d , H e ) A l excitation energy spectrum  48  2 8  S i ( p , n ) P excitation energy spectrum  48  3.1  T h e T R I U M F cyclotron a n d beamlines  53  3.2  Layout of the T R I U M F M 9 channel  54  3.3  E x p e r i m e n t a l setup  56  3.4  Complete electronic logic diagram  60  3.5  T i m i n g s a n d definitions of events for G e l a n d associated electronics.  61  3.6  A schematic flow of the data through the D a t a A c q u i s i t i o n System. .  65  28  2  2 8  2 8  ix  4.1  C o m p a r i s o n of the measured energy spectra for a H P G e detector a n d a N a l ( T l ) scintillator (source data)  4.2 4.3 4.4 4.5 4.6  71  G a m m a - r a y spectrum from muons stopping i n a silicon target. 2 8  . . .  73  S i gamma-ray energy spectrum, i n the vicinity of the 1229 k e V peak. 75  T h e p r o d u c t i o n of the 1229 k e V g a m m a ray i n dence technique  2 8  A 1 and the coinci77  S i gamma-ray energy spectra before and after the i m p o s i t i o n of the coincidence requirement  78  Various components of the response function fitted to the 1332 k e V C o peak  81  2 8  6 0  4.7  Energy dependence of the parameters of the response function for G e l . 84  4.8  Stopping-power curves for  4.9  Relationship between the stopping-power a n d the extracted lifetime from the 2171 k e V 7-ray line  87  5.1  T i m e of the m u o n spectrum for G e l w i t h a Si target  93  5.2  Ge2 C o m p t o n suppressed a n d unsuppressed spectrum t y p i c a l for m u o n capture on a Si target  94  5.3  D a t a removed by the N a l ( T l ) C o m p t o n suppressor  95  5.4  C o m p t o n suppressed and unsuppressed  95  5.5  G e l leading edge spectra corresponding to different discriminator thresholds for a t y p i c a l /j.Si r u n  97  Ge2 leading edge spectra corresponding to different d i s c r i m i n a t o r thresholds for t y p i c a l pSi r u n  98  5.6  2 8  A 1 ions i n ' S i m e d i u m n a  6 0  C o spectrum of G e l  86  5.7  P l o t of the corrected a n d uncorrected centroid channel of the 1173 k e V 7 rays as a function of their rise-time for the G e l detector. . . . 100  5.8  P l o t of the corrected a n d non-corrected centroid channel of the 1173 k e V 7 rays as a function of their rise-time for the Ge2 detector. . . . 100  5.9  T y p i c a l t i m i n g coincident spectra for one of the N a l ( T l ) detectors.  5.10 Time-coincidence spectrum for a N a l ( T l ) annulus segment 5.11 A dual-peak fit to the  6 0  C o 7-ray lines i n B A R 3  . 101 103 105  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 k e V 5.15 P a r t of the  2 8  2 8  A l level d i a g r a m  S i 7-ray energy spectrum for the G e l detector  5.16 P a r t s of the S i 7-ray energy spectrum for the G e l detector showing the 903 k e V a n d 3075 k e V peaks  112 113  2 8  5.17 5.18  115  S i 7-ray energy spectra obtained w i t h a N a l detector, w i t h a n d without coincidence requirement  117  S i 7-ray energy spectrum of N a l overlaid on a G e l detector singles spectrum  118  2 8  2 8  5.19 Singles S i gamma-ray energy spectra obtained w i t h the two G e detectors, i n the vicinity of the Doppler-broadened 2171 k e V peak. . 121 2 8  5.20  S i coincidence energy spectra obtained w i t h the two G e detectors, in the vicinity of the Doppler-broadened 2171 k e V peak  122  S i gamma-ray energy spectra of the G e l detector: (a)singles a n d (b)coincidence w i t h the 'side-band' background subtraction  124  S i gamma-ray energy spectra of the Ge2 detector: (a)singles and (b)coincidence w i t h the 'side-band' background subtraction  125  5.23 T h e interplay of the slowing-down, the angular correlation, a n d the instrumental resolution effects  128  5.24 T h e best simultaneous fit to a l l four spectra  133  5.21 5.22  6.1  2 8  2 8  2 8  T h e measured 7 — 1/ angular correlation coefficient a compared to the theoretical calculations  xi  141  Acknowledgements F i r s t a n d foremost I acknowledge the Help of T h e Creator A l l a h s.w. without W h o m none of this w o u l d be possible. I w o u l d like to extend m y sincere gratitude a n d appreciation to m y supervisor Professor D a v i d F . Measday for his guidance, advice a n d encouragement throughout this work. I a m greatly indebted to D r . D a v i d S. A r m s t r o n g who acted as the spokesman of the experiment as well as m y unofficial second supervisor. Special thanks are given to D r s . T . P . Gorringe and S. Stanislaus. T h e i r suggestions a n d discussions during the entire course of this work were well appreciated. I w o u l d like to thank m y fellow graduate students E r m i a s Gete a n d 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 . M a k o t o , R . J a c o t - G u i l l a r m o d a n d P . Weber for their assistance a n d contribution to the progress of the experiment. I w o u l d like also to thank Professors J . Deutsch, H . W . Fearing, M . D . Hasinoff, B . G . T u r r e l l a n d C . E . W a l t h a m for their advice and detailed reading of the manuscript. I a m grateful, for financial support d u r i n g m y work, to the Secretariat of Scientific Research of L i b y a a n d to the N a t u r a l Sciences a n d E n g i n e e r i n g Research C o u n c i l of C a n a d a . F i n a l l y , I would like to thank m y family, a n d i n p a r t i c u l a r m y wife G h a r s a , for their continuous support and encouragement throughout this work.  xn  Chapter 1 Introduction 1.1  The M u o n M u o n s were first discovered i n studies of cosmic rays w i t h cloud-chambers  and Geiger counters by Street a n d Stevenson (1937) [1] a n d A n d e r s o n a n d Neddermeyer (1938) [2,3]. T w o years earlier, particles of approximately the m u o n mass h a d been postulated by Y u k a w a [4] as quanta of the nuclear  field.  However, it  turned out that the observed cosmic-ray particles were not the Y u k a w a ones; for, although they have about the right mass, they do not interact strongly w i t h nuclei [5]. Shortly thereafter, the pion was discovered [6] i n photographic emulsions. Subsequent experiments demonstrated that it was the pion, not the m u o n w h i c h is the Y u k a w a particle. T h e m u o n , it turned out, was the decay product of the pion. T h e properties of the m u o n can be s u m m a r i z e d by describing it as a "heavy electron", for the only fundamental attribute that distinguishes a m u o n from elect r o n is its mass (about 207 times the electronic mass). W e now realize that there are three charged leptons, the electron, m u o n and tau, w h i c h are each members of different families (generations). T h e only obvious difference between t h e m is their masses. T h e muons are point-like leptons, w h i c h 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 m u o n (after [7]).  1 2  Spin Mass  m „ = 105.658389 ± 0.000034 M e V / c  Charge  = q  M e a n lifetime M a g n e t i c moment  2  e  r = (2.19703 ± 0.00004) x l O " fi = 1.001165923 ± 0.000000008  6  s eh/2m„  of the m u o n . M u o n s are usually produced ( as i n the present work ) by the decay of charged pions, i.e.  7T+  -> ^+ + ^  (1.1)  7T~  —* fl~ +  (1.2)  T h e m u o n has been an i m p o r t a n t test particle not only i n the various branches of physics, but i n several other science fields as well, see the review article of Scheck [8] a n d references therein. In the present work muons are used as probes of the weak interaction. Free muons n o r m a l l y decay into electrons a n d two neutrinos as follows  p  +  —y e  +  + u + e  p~ —• e~ +V  e  +  (1.3) (1.4)  T h i s three-particle decay scheme is consistent w i t h 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  m u o n , the m a x i m u m energy of the electron - when the two neutrinos escape i n the same direction, opposite to that of the electron - is given by E=  "  e  —  v  = 52.83 MeV  e  (1.5)  a n d hence is consistent w i t h the previous sentence, i.e. m „ + m „ e  ~ 0. Today, upper  limits on these masses are m „ < 4.5 e V [9] a n d m „ < 0.27 M e V [10] as obtained e  M  from the t r i t i u m experiments a n d the IT —• uv^ decay experiments respectively. Other m u o n decay modes, i n c l u d i n g forbidden lepton family number violating modes are listed, along w i t h their b r a n c h i n g ratios [7], below u~ — •  e  + v +  -  + 7  e  e" -rue + v» + e+ + e~ e  1.2  _  + i/ +F e  M  0.014 ± 0 . 0 0 4 (3.4 ± 0.4) x 1 0 < 0.012  (1.6) - 5  (1.7) (1.8)  e-+7  <4.9xlO  e - + e + + e-  < 1.0 x 1 0 "  1 2  (1.10)  e~+2  < 7.2 x 1 0 "  u  (1.11)  7  - 1 1  (1.9)  Mesic Atoms In a mesic - o r more generally e x o t i c - atom, a negatively charged particle  replaces one of the o r b i t a l electrons of an o r d i n a r y electronic counterpart. T o date five such atoms have been successfully observed. These are pionic, kaonic, muonic, hyperonic and antiprotonic atoms.  Mesic atoms have been k n o w n for more than  four decades. T h e first experimental i n d i c a t i o n 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 i n light elements. A t the same time Wheeler [11] a n d F e r m i and Teller [12] theoretically argued that mesic atoms should exist since the atomic cascading time ( ~ 1 0  - 1 3  s ) is short compared w i t h the lifetime of the particles involved.  T h e first demonstration of the existence of mesic atoms was made using muons from cosmic r a d i a t i o n [13]. Mesic a t o m physics has become an i m p o r t a n t tool for understanding nuclear properties. It can also provide information about the elementary particles themselves and their interactions. Indeed the field of mesic atoms involves molecular, atomic, nuclear a n d particle physics; see references [14,15,16] for applications of mesic atoms. T w o technological advances of recent years have reactivated the interest i n this field. These are the establishment of high-flux meson factories (e.g. T R I U M F , L A M P F , S I N ) and the development of high-resolution solid state detectors. T h e properties of the exotic atoms are not only similar to each other, but are rather closely related to those of the hydrogen atom. T h i s is due to the dominant role of the electromagnetic interaction. However, exotic atoms have two i m p o r t a n t characteristics considerably different from those of their electronic counterparts. These are consequences of the great difference i n mass between the particles involved a n d the electron. F o r example, the lightest of these particles, the m u o n , is 207 times as heavy as an electron. These characteristics follow from the fact that -for the same q u a n t u m n u m b e r s - t h e energy levels (radii) of the orbits are (inversely) p r o p o r t i o n a l to the mass of the o r b i t a l particles.  F o r 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 a n d hence are m u c h better suited for p r o b i n g nuclear properties t h a n the atomic electrons. A l t h o u g h the discussion below is specific for m u o n i c atoms, which is relevant for this work, most of the general processes - apart from strong interaction p h e n o m e n a - are essentially the same for a l l exotic atoms. M u o n s enter the target w i t h energies of the order of tens of M e V (~20 M e V i n our case), and possess velocities (u ) greater than those of the valence electrons M  4  (y  e  ~ a c , where a is the fine structure constant). T h e y lose most of their energy  through collision w i t h atomic electrons a n d then come to a stop w i t h i n ~ 1 0 s . _ 9  Once the m u o n comes to a stop, it w i l l be captured at a few tens of e V [17] by a target a t o m into a high o r b i t a l angular m o m e n t u m state, ejecting electrons and forming a m u o n i c atom. T h e atomic capture is roughly described by the socalled " Z - l a w " [12] i n w h i c h the capture rate is taken to be p r o p o r t i o n a l to the nuclear charge, Z . Following its capture, the m u o n ( w i t h i n ~ 1 0  - 1 4  s ) w i l l be inside  the K - s h e l l electron orbit at a p r i n c i p a l q u a n t u m number given a p p r o x i m a t e l y by n  ~ (m^/m )  1/2  M  e  ~ 14.  Since a l l of the low-lying "muonic" states are unoccupied, the m u o n cascades down ( w i t h i n ~ 1 0  - 1 3  ) from n ~ 1 4 to the Is q u a n t u m state. In this cascade, the  m u o n w i l l interact w i t h outer electrons a n d w i l l lose energy through A u g e r processes. However, as the t r a n s i t i o n energy increases r a p i d l y ( ~ 1/n ), this interaction is 3  no longer i m p o r t a n t a n d E l radiative transitions (muonic X - r a y s ) dominate; see F i g u r e 1.1. Effectively a l l of the negative muons captured i n the atomic orbits reach the Is orbit. T h i s is consistent w i t h the previously stated time-scales needed for the formation of muonic atoms. Once a negative m u o n reaches the lowest Is state, it either decays (equation 1.4) or gets captured by the nucleus v i a the elementary reaction,  \i~ + P -> n +  (1-12)  which i n 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, (designated by the asterisk) usually a giant resonance state. 5  A distinct feature of the rate of reaction (1.13) is its strong Z-dependence. T h i s was observed b y the experiment of Conversi et al. [5] a n d its form was later unravelled by Wheeler [18]. I n a simple model, he assumed that a l l protons, Z , i n the nucleus can interact independently w i t h the p a n d that the interaction is p r o p o r t i o n a l to the p r o b a b i l i t y of finding the p, i n the nucleus (|Vv(0)| )- N o w 2  since the p is observed from the Is state w h i c h has, for hydrogenic atom, the wave function [19] Vv(r) = y/zyiralexp  [-Zr/a,]  (1.14)  where a is the first muonic B o h r radius, the capture rate is M  K^Z where an effective charge (Z^j) A l t h o u g h this Z  4  (1.15)  A e } }  was used to account for the finite nucleus size.  l a w is a good a p p r o x i m a t i o n for light nuclei, theoretical m u o n  capture rates are usually obtained from the Primakoff the Goulard-Primakoff  formula  formula  [20] or its extension,  [21].  T h e rate of the capture ( A ) a n d decay ( A j ) are connected b y c  A = A + QA t  c  (1.16)  d  where A (=7-) is the total disappearance rate a n d Q is the Huff factor to take act  count of the fact that the m u o n is b o u n d i n the a t o m (see the Nuclear Muon  Capture  review article b y M u k h o p a d h y a y [22] for the differences i n the decay rates of the free and the b o u n d m u o n decays a n d their causes) . U t i l i z i n g the C P T theorem, i.e. Ad = -^p = - ^ J T , the measurement of the nuclear capture rate ( A ) amounts simply c  to the determination of the b o u n d m u o n lifetime ( T - ) i n the relevant material. m  T h i s h a d been demonstrated at T R I U M F b y S u z u k i et al. [23,24] who measured the lifetime of p~ i n 50 elements plus 8 isotopes and deduced the associated capture rates. A more recent experiment [25] i m p r o v e d the data for isotopes of u r a n i u m and 7  n e p t u n i u m . F o r 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 distinction - a t low energy- among the four k n o w n fundamental  interactions is established by b o t h 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 a n d the gravitational ones i n contrast to the other two, namely the strong a n d the weak interactions. T h e interaction strengths are usually expressed i n terms of dimensionless coup l i n g constants. R o u g h l y speaking, the weak interaction is 10 a n d 1 0 9  12  times weaker  t h a n the electromagnetic a n d the strong interactions respectively (the fourth interaction, gravity, is completely negligible at the level of particle physics since it is some t h i r t y orders of magnitude weaker t h a n any other interaction). T h e apparent weakness of the weak interaction is ascribed to the very short range associated w i t h its massive exchange particles,  a n d Z° (and their couplings to leptons a n d  quarks). A l t h o u g h these (three) processes appear to be different, their m a t h e m a t i c a l formulation is very similar. T h e y are a l l described by gauge theories; i.e. theories i n w h i c h fermions interact by the exchange of spin 1 gauge bosons. In fact, advances toward the ' u l t i m a t e ' unification of these interactions are underway. U n l i k e strong interactions, the weak interactions can involve b o t h hadrons as well as leptons. Furthermore, the weak interaction operates at approximately the same strength among these particles, a phenomenological observation k n o w n 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 Y u k a w a picture, the range of an interaction, R , is associated w i t h the reciprocal of the mass of its propagator.  Thus R k wea  ~ Mw  ~ 1 0 ~ m a n d R on 1 8  str  ~ rn'  1  g  ~ 1.4 x 1 0  _ 1 5  m while the  range of the electromagnetic interaction, w i t h M = 0 , is infinite. In contrast to the 7  electromagnetic a n d strong interactions, the weak interaction violates some of the s y m m e t r y principles a n d conservation rules such as the discrete operations P (parity), C (charge conjugation) a n d T (time reversal) . F o r further discussions of these principles a n d their experimental status, see references [27,28]. T h e weak interactions were observed about a century ago through the discovery of r a d i o a c t i v i t y by Becquerel. However, the establishment of the weak interaction as a separate and independent process was rather gradual. In 1934 F e r m i [29] proposed the first theory of the weak interaction based on a four-fermion point i n teraction to describe the beta decay of nuclei, e.g. n —> p + e~ + T7 . T h i s theory e  remains to this day, w i t h a few modifications. It took about 14 years from the origi n a l formulation of F e r m i on nuclear (3 decay before the p-e u  universality" sprang  forth through the work of Pontecorvo [30] a n d P u p p i [31] leading to the hypothesis of the Universal Fermi Interaction.  T h i s has now been extended to the r as well.  D u r i n g 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. T h i s interaction was later generalized ( G a m o w and Teller, 1936 [32]) to a linear c o m b i n a t i o n of the five bilinear quantities: vector ( V ) as well as scalar (S), pseudoscalar ( P ) , a x i a l vector ( A ) and tensor ( T ) . In contrast to the original V form w h i c h allows transitions between nuclear states of equal angular m o m e n t u m ( F e r m i transitions), this general interaction could couple nuclear states differing by one unit of angular m o m e n t u m ( G a m o w Teller transitions). T h e t u r n i n g point i n the construction of the weak interaction came about, as 9  a possible way out of the so-called (6 — r ) puzzle, w i t h the questioning of p a r i t y conservation i n the weak interaction by Lee a n d Y a n g [33] and later by the verification of its v i o l a t i o n 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 o d d a n d even p a r i t y amplitudes have roughly the same magnitudes and give what is called the principle of maximum  parity  violation.  The V - A theory is not the whole t r u t h , rather its success lies i n e x p l a i n i n g a large class of weak phenomena.  It is now generally regarded as a p a r t i c u l a r case  of the extremely successful Weinberg-Salam-Glashow electroweak theory [35,36,37] (also k n o w n as the S t a n d a r d M o d e l of the electroweak theory) i n the l i m i t of low energy.  T h e electroweak theory is a renormalizable gauge theory unifying weak  a n d electromagnetic interactions i n one m a t h e m a t i c a l framework. Its success was highlighted by the p r e d i c t i o n of the neutral weak current discovered at C E R N i n 1973 [38], and by the prediction of the W a n d Z massive gauge bosons discovered 10 years later, also at C E R N [39]. T h e S t a n d a r d M o d e l now also encompasses the description of the strong interactions by q u a n t u m chromodynamics ( Q C D ) .  1.3.1  The Weak-Interaction Hamiltonian A s noted above, weak processes, such as m u o n capture, at low m o m e n t u m  transfer q (q ~ m 2  2  <C Myy), are adequately described by the V - A theory and hence  w i l l be used i n this section. W i t h i n this picture, the weak interaction H a m i l t o n i a n , at a given space-time p o i n t ( x ) , has the form  H{x) = -^=j{(x) v2 where G — 1.16639(2) x 1 0 J  A  - 5  GeV  - 2  J\x)  + h.c.  (1.17)  [40] is the effective weak coupling constant a n d  is the weak four-current which can be decomposed into hadronic a n d leptonic  components, v i z . : 10  J (x) = J\ + J[  (1.18)  A  Accordingly, weak processes are usually classified as • pure leptonic, where only leptons are involved, • semileptonic, where b o t h leptons a n d hadrons are involved, a n d • hadronic, i n w h i c h only hadrons are involved. M u o n capture is one example of a semi-leptonic process a n d is the one considered i n this work.  T h e leptonic current T h e leptonic current i n m u o n capture has the V - A structure, (1.19) w i t h 75 = 27o7i7273 a n d where ipj a n d j \ are the lepton fields a n d the D i r a c 7matrices respectively. T h e (1 — 75) i n J[ automatically selects left-handed leptons consistent w i t h the fact that weak interaction couples only to left-handed particles (and right-handed anti-particles). T h e V - A form of the leptonic current has been stringently tested using purely leptonic processes, such as m u o n decay.  T h e hadronic current T h e details of the hadronic weak current,  are not as clearly established as  that of the leptonic one. T h i s is due to the presence of the strong interaction w h i c h induces extra structure i n the hadronic weak currents. In fact, most m u o n capture experiments are concerned w i t h the unravelling of this induced structure.  11  In terms of the V - A structure and the G e l l - M a n n - C a b i b b o universality hypothesis [41,42], the hadronic current for the semi-leptonic weak process of m u o n capture can be w r i t t e n as j£ = cosO (Vx-A ) c  (1.20)  x  with  9M  . .gs  2M  m„  .  V\ = iip.  N  9P  l  SU7A75 H  a n d o\  v  = !(7A7"  respectively.  m,  75tfA +  where M / v a n d m  —  *9T  2M,N  (1.21)  (1.22)  are the nucleon a n d lepton masses  M  T h e angle 6 is the C a b i b b o angle, introduced to account for the C  m i x i n g between the quark generations, i.e.  to account for the different rates i n  the strangeness conserving a n d non-conserving weak decay processes. T h e g  a  coupling "constants" w h i c h are functions of q  2  are  (q\ — n\ — p\, where n\ and p\  are the neutron a n d the p r o t o n 4-momenta respectively). T h e terms gv and gA are the conventional vector a n d axial vector coupling constants. These two are the only ones that contribute i n the q =0 l i m i t , such as i n /3-decay. T h e other four 2  coupling constants gM, gs, gp and gj are the induced ones.  T h e y measure the  strength of the induced weak magnetic (o-\„), the scalar, the pseudoscalar (75) and the tensor(<7Ai,75) currents respectively. T h e consequences of the induced strong interaction 'dressing' are a change i n the strength of the conventional components of the weak current a n d the appearance of new components, the contributions of w h i c h are p r o p o r t i o n a l to the m o m e n t u m transfer (q) a n d consequently contribute only negligibly i n /?-decay a n d electron capture, but show up clearly i n m u o n capture where the 4 - m o m e n t u m transfer can be m u c h larger.  12  1.4  Constraints on the Weak Coupling Constants The determination of weak couplings is based u p o n several general theoretical  constraints. A brief description of these constraints along w i t h 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 a n d isospin invariance) , the induced currents i n V\ a n d A\ are expected to have a c o m m o n G - p a r i t y transformation. However, the gs a n d gx induced currents transform w i t h opposite G - p a r i t y to the other vector a n d axial vector terms respectively and hence must vanish, i.e., gs = 0,  (1.23)  9T = 0.  (1.24)  These two currents are classified by W e i n b e r g [43] as "second-class" currents as opposed to the other "first-class" currents. T h e "second-class" currents have been investigated [44,45], however, no convincing evidence of their existence has been found  2  [22]. Up-to-date limits on these currents are given by Grenacs [46].  A n o t h e r 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 F e y n m a n a n d 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 a n d hence the vanishing of its divergence, i.e., dxV = 0. x  (1.25)  The small time reversal noninvariance implied by C P violation in the neutral kaon system (at the level of 10~ ) is entirely negligible here [27]. lr  3  Since 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. 2  13  A p p l y i n g directly to equation (1.21) w o u l d give 9s = 0,  (1.26)  reinforcing the exclusion of "second-class" currents. Furthermore, C V C relates the weak vector a n d the isovector electromagnetic currents.  T h i s allows the weak magnetic coupling constants, gv a n d gM, to be  expressed i n terms of electromagnetic form factors w h i c h can be measured w i t h precision i n electron scattering experiments. A t low energy (q —> 0) the calculated 2  values are = 1.0  (1.27)  = 3.706.  (1.28)  gv  g  M  C V C is contained w i t h i n the S t a n d a r d M o d e l a n d 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]. Deutsch et al. [51] have used the p a r t i a l capture rate i n 1 6  1 6  F o r example,  0 to the 1~ state i n  N to test the C V C hypothesis, a n d found m u t u a l agreement w i t h i n 13%. Fur-  thermore, the b r a n c h i n g ratio of x  +  beta-decay, ( 1 . 0 2 5 ± 0 . 0 3 4 ) x l 0 , was found to - 8  agree well w i t h the calculated C V C prediction, 1 . 0 7 x l 0 ~ ; see [7] a n d references 8  therein. M o r e recently, the C h a l k R i v e r collaboration [52] have produced a precise measurement of the  1 0  C superallowed F e r m i /?-decay b r a n c h whose Ft-value, when  added to the previously obtained F t values from eight other nuclei, removed any discernible trends w i t h the nuclear charge Z . F i n a l l y , 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 r e m a i n i n g coupling constants, gA a n d gp. It states that the weak hadronic axial current (A\) is only conserved for massless pions and can be related to the p i o n field, (f) , as follows: v  (1.29) 14  T h e P C A C hypothesis, applied to the expression for A\, yields the relation 2M  - q^  = ^< f"  2  N9A  . )  G  (1  30  between the weak axial and weak pseudoscalar coupling constants, the measured pion decay constant (/„.) a n d the strong coupling constant for the pion-nucleon vertex  I  (G NN)V  n  the low energy l i m i t (q  —> 0), this relation reduces to the  2  Goldberger-Treiman [55] estimate for the axial vector coupling constant: g  A  where G  v N N  =  ^  f  N  = 1.32 ± 0.02  N  = 13.40 ± 0.8 [56] a n d f  (1.31)  = 130.7 ± 0.1 ± 0.36 M e V [7] have been  v  used. T h i s estimate is slightly different from the measured value of g  A  = 1.2573 ±  0.0028 [7]. T h e 5% difference between the two values is called the G o l d b e r g e r - T r e i m a n discrepancy a n d is addressed i n references [56,57]. T h e G o l d b e r g e r - T r e i m a n relation can be used to yield a prediction for the induced pseudoscalar coupling,i.e.  gp =  '  2M m ' N  q  2  +  )1  ra _ 2  gA  (1.32)  Clearly, g is a strong function of q and, furthermore, is d o m i n a t e d by the 2  p  pion pole, i n d i c a t i n g that the pseudoscalar c o n t r i b u t i o n is p r e d o m i n a n t l y due to the single p i o n exchange between the leptons and hadrons; see F i g u r e 1.2. For the m u o n capture process on the p r o t o n (equation 1.12), the two b o d y kinematics fix the four-momentum transfer at q = 0.88m , 2  2  (1.33)  w h i c h when substituted into equation (1.32) yields the G o l d b e r g e r - T r e i m a n value, viz., g  P  = 7.1g  A  15  = 8.9.  (1.34)  F i g u r e 1.2: T h e single p i o n exchange d i a g r a m (the source of the weak pseu doscalar coupling). T h e enlarged vertex represents the compos ite quark structure of the hadronic current.  16  Table 1.2: Theoretical a n d experimental weak coupling constants for the elementary process of m u o n capture.  Coupling constant  Theoretical prediction  Experimental measurement  9v  1.0 3.706 0 1.32 8.44 0  1.000 ± 0.0015 [61] 3.78 ± 0.22 [46] 0.0005 ± 0.0031 [62] 1.2573 ± 0.0028 [7] 8.7 ± 1.9 [60] 0.06 ± 0.49 [62]  9M  9s 9A 9P 9T  Technique used fj, decay /?-decay, ^-capture /?-decay neutron 8-decay /^-capture i n H /3-decay, A = 1 2  Invoking a weaker P C A C , Wolfenstein [58] extended the work of Goldberger and T r e i m a n a n d obtained g  P  = 8A  (1.35)  k n o w n as the Wolfenstein estimate a n d was recommended by M u k h o p a d h y a y [22] a an i n p u t for the pseudoscalar coupling constant. M o r e recently, a new calculation of gp [59] confirmed the work of Wolfenstein a n d gave the very precise prediction g  P  = 8.44 ± 0 . 2 3  (1.36)  where the error comes m a i n l y from the uncertainty i n G^NNW h i l e theoretical calculations of gp c l a i m accuracy of better t h a n 3%, the weighted average of a l l experimental measurements w i t h the free nucleon [60], namely gp/gA — 6.9 ± 1.5 - w h i l e i n accord w i t h the theoretical p r e d i c t i o n s - carries an uncertainty i n excess of 20%. Moreover, the most accurate single measurement contributing to this average has a 43% error. Table 1.2 gives the calculated a n d measured values of the six weak coupling constants along w i t h the techniques used. A s can be seen i n this table, a l l coupling constants, except gp, are well determined by experiment a n d i n rather good agreement w i t h the theory for the  17  elementary process of m u o n capture on the proton.  C l e a r l y 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 p o o r l y - k n o w n gp, is complicated a n d is discussed i n 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 a n d gM, r e m a i n constant when the nucleon is embedded i n the nucleus. In contrast, the weak axial hadronic vector current (A\)  couplings, gA a n d gp, may  change reflecting changes i n the nucleon structure i n the nuclear m e d i u m . 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. T h e a x i a l vector a n d pseudoscalar components of the weak current, i n the nucleus, can be effectively modified i n several ways [22,63,64]. T h e scattering of v i r t u a l pions by other nucleons can reduce their effective range a n d thus alter their propagators.  In a d d i t i o n , the induced p o l a r i z a t i o n of the nuclear m e d i u m around  the p i o n - e m i t t i n g nucleon can modify their interaction. These two effects can be accounted for by replacing the mass of the p i o n , m , n  tex,  GITNN-,  a n d the pion-nucleon ver-  of the p i o n field by effective mass a n d coupling respectively. C o m b i n i n g  these two effects, along w i t h other uncertain ones, E r i c s o n a n d collaborators [63] obtained the following expressions for the effective weak axial vector a n d pseudoscalar components i n terms of the bare ones, 9A  =  [1 + -  )9A  (1.37)  and 2M  N  m  A  q + ml 2  18  (1.38)  where an ~ —0.9 is interpreted as a measure of the nucleon polarizability a n d m  x  is the effective mass of the p i o n inside the nucleus. These effects would lead to dramatic renormalizations of the coupling constants i n the nucleus; i.e. g~ = 0.76g A  (1.39)  A  and g  P  = 0.35flf  (1.40)  P  at the m o m e n t u m transfer appropriate for m u o n capture. Some experimental deviations of g  A  i n nuclei from the p r i m i t i v e free-nucleon  value have been interpreted as evidence for modification of the weak axial vector coupling. E x a m i n i n g the Gamow-Teller m i r r o r /^-decays, W i l k i n s o n [65] deduced g~ = (0.899 ± 0 . 0 3 5 ) ^ .  (1.41)  A  Furthermore, the 'unobserved' Gamow-Teller strength i n the studies of the small angle (p,n) reactions [66], now being reinforced by analogous (n,p) measurements [67], has been interpreted i n terms of strong r e n o r m a l i z a t i o n of g  A  i n complex nuclei;  yielding g~ = 1.00 ± 0.002 [66,68]. A  T h e experimental situation of gp and its possible r e n o r m a l i z a t i o n w i l l be discussed next.  1.6  Observables Sensitive to the Induced Pseudoscalar Coupling A s stated earlier, the sensitivity of some observables i n m u o n capture to the i l l -  determined pseudoscalar coupling (gp), comes about due to the large characteristic m o m e n t u m transfer, q, i n m u o n capture as compared to, say, /?-decay or e-capture. T h i s can clearly be seen i n the axial vector m a t r i x element, equation (1.22), where the whole pseudoscalar coupling t e r m is p r o p o r t i o n a l to q. Furthermore Since the 19  dominant c o n t r i b u t i o n to the pseudoscalar coupling comes from the single p i o n exchange diagram, F i g u r e (1.2), a n d is given by the G o l d b e r g e r - T r e i m a n relation equation (1.32), gp is a strong function of q . 2  W h i l e several gp-sensitive observables exist [69], one is faced w i t h two difficulties i n measuring them. F i r s t l y , the outgoing particles (n,i/) i n the capture process are h a r d to detect. Secondly, these observables are, to some degree, dependent on the nuclear structure, i.e. the i n i t i a l a n d final wavefunctions. T h e only observables avoiding the second difficulty - a l t h o u g h adversely affected by the Z l a w - are the 4  ones from m u o n capture on hydrogen. In the context of m u o n capture, processes (1.12) a n d (1.13) are referred to as ordinary m u o n capture ( O M C ) as opposed to their m u c h less probable versions H~ + p^n  + v t + y,  (1-42)  t  and fi- + (A,Z)  ( , Z - 1)* + !/„ + 7  (1.43)  a  known as radiative m u o n capture ( R M C ) . A p a r t from its rarity (branching ratio ~ 1 0  - 4  for heavy nuclei), R M C has  several advantages over O M C i n determining gp. W h i l e the four-momentum transfer (q ) 2  is fixed at q  2  ~0.88 m  2  i n O M C , it could approach - m  2  for R M C at the  m a x i m u m photon energy. T h i s i n t u r n enhances the pseudoscalar c o n t r i b u t i o n i n 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 T h e advantage of the basic m u o n capture on hydrogen, i n contrast to that i n  complex nuclei, is the absence of the m a n y uncertainties arising from the treatment 20  of the nuclear response function. T h e observables measured i n O M C on hydrogen are the capture rates from the singlet ( A s ) 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 time, the radiative m u o n capture on hydrogen. T h e i r measured value of gp, determined from the photon energy spectrum, has just been reported as [74]  g = (10.0 ± 0 . 9 ± 0 . 3 ) f l u P  (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 transition rate i n muonic molecular hydrogen, \  o p  .  It should perhaps be remarked that our group [76] is proposing to measure this transition rate on w h i c h the above R M C value of gp depends, yet to a lesser degree than the O M C on hydrogen, and for w h i c h there has only been one measurement [77,78], A  = (4.1 ± 1.4) x 10 s , w h i c h is significantly different than the 4  o p  theoretical prediction [79], A  _ 1  = (7.1 ± 1.2) x 10 s . Depending on the value of X 4  o p  _ 1  OP  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 w i t h the other O M C measurements) or with the P C A C prediction; the later, i f taken at face value, w o u l d demand new physics of our understanding of semileptonic weak interactions [76]. T h e experimental results to date on the value of gp (as gp/gA) deduced from m u o n capture (both O M C and R M C ) i n hydrogen are summarized i n Table 1.3. T h i s table represents the only information avaliable to date on the induced pseudoscalar coupling constant for the free nucleon. 21  Table 1.3: S u m m a r y of values of gp/gA as determined from measurements of ordinary m u o n capture ( O M C ) i n hydrogen. T h e values quoted for O M C are from the analysis of B a r d i n et al. [77]. Experiment  9P/9A  O M C ( n e u t r o n ) on l i q u i d H [72] O M C ( n e u t r o n ) on l i q u i d H [73] O M C ( n e u t r o n ) on gaseous H [70] O M C ( n e u t r o n ) on gaseous H [71] O M C ( e l e c t r o n ) on l i q u i d H [78] O M C W o r l d Average R M C ( p h o t o n ) on l i q u i d H [74] 2  2  2  2  2  2  4.8 ± 6 . 3 8.7 ± 3 . 4 8.2 ± 3 . 1 6.3 ± 4 . 7 5.6 ± 2 . 4 6.9 ± 1.5 10.0 ± 0 . 9 ± 0 . 3  In a d d i t i o n to p r o v i d i n g the best values for gp, m u o n 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 D u e to the low capture p r o b a b i l i t y i n hydrogen ( ~ 1 0 ) , efforts - b o t h - 3  experimentally and theoretically- have concentrated on heavier nuclei . 3  In complex nuclei, the induced pseudoscalar interaction influence several experimentallyaccessible observables. In nuclear R M C , these include the often measured observable, i.e. the R M C / O M C branching ratio as well as the p h o t o n energy spectrum, the photon asymmetry w i t h respect to the m u o n spin direction, and the photon circular p o l a r i z a t i o n . T h e 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 limit on gp, and hence do not show up i n Table 1.4. T h e review article by G m i t r o and T r u o l [80] on R M C provides a beautiful s u m m a r y of the theoretical as well as the experimental aspects of these observables. A s discussed in section 1.5, measurements of gp in nuclei are also interesting in view of investigating its eventual renormalization, i.e. the influence of the nuclear medium on the axial weak current. 3  22  D u e to the inherent difficulty of these measurements, caused p r i m a r i l y by the low yield ( Z law), the only R M C data available , before the T R I U M F hydrogen 4  experiment result, are for nuclei w i t h A > 12. Table 1.4 provides a s u m m a r y of 9P/9A  as determined from R M C i n complex nuclei. A few comments on Table 1.4, are i n order. O n e is that the extracted values  of gp/gA  are not consistent a n d often do not agree w i t h the G o l d b e r g e r - T r e i m a n  expectation. M o r e i m p o r t a n t l y , these results suggest a substantial enhancement i n light nuclei and quenching i n heavy nuclei. T h i s can clearly be seen by p l o t t i n g gp as a function of nuclear mass, see F i g u r e 1.3.  T h i s trend for gp does indeed  show a tantalizing i n d i c a t i o n of a quenching of the pseudoscalar coupling for heavy nuclei. However, the interpretation of these measurements are fraught w i t h nuclear structure uncertainties . T h i s has been reinforced by the fact (Table 1.4) that the 4  same experiment, when compared to different theoretical models, can lead to very different extracted values oi gp. W e shall come back to this topic. Due to the low radiative branching ratio ( ~ 10~ ) combined w i t h the many po5  tential large background sources, efforts also have concentrated on O M C i n heavier nuclei. A l l i n a l l , aside from 7 — u angular correlation w h i c h is the subject of this thesis and w i l l be discussed later, five different O M C observables have produced 'quoteworthy' measured values of gp i n five l i g h t - t o - m e d i u m nuclei, as shown i n Table 1.5.  A m o n g these are the polarizations of the residual nucleus along the  direction of the m u o n spin (P ) av  and along the direction of its recoil ( P L ) , the rate  of m u o n capture to specific final states ( A , ) , a n d the hyperfine dependence of m u o n capture ( A / A _ ) . +  T h e apparent concentration of efforts on 4  1 2  C is due to the existence of several  Some doubts were cast on the validity of the theory of Ref. [94] by Fearing and Welsh [95].  23  Table 1.4: S u m m a r y of values of gp/gA determined b y comparing experimental results w i t h theoretical predictions of radiative m u o n capture ( R M C ) i n 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 A r m s t r o n g et al. [83]  Nucleus i2  C  16Q  n  Dobeli et al. [87] Frischknecht et al. [88] Average [89] A r m s t r o n g et al. [83]  H  11 11 4 0  Ca  11  11 11  A r m s t r o n g et al. [92] n  11  11  11 11  Frischknecht et al. [93] Dobeli et al [87] Average [89]  11 11  Theory [84] [85] [86] [86] [86] [90] [91] [85] [91] [85] [90] [86] [86] [90]  QP/QA  16.2 t ; ? 13.6 _[% 7.3 ± 0 . 9 8.4 ± 1.9 13.5 ± 1.5 9.0 ± 0 . 8 5.7 ± 0 . 8 4.6 ± 1.8 5.9 ± 0 . 8 5.0 ± 1.7 7.8 ± 0 . 9 4.6 ± 0 . 9 u  +  6-3 t\i 8.1 ± 0 . 3  Dobeli et al. [87]  natp  [94]  3.0 ± 1.3  Bergbusch [82]  nat^j.  [94]  -0-4  A r m s t r o n g et al. [92]  nat  M o  [94]  nat  o.o  Bergbusch [82]  A g  [94]  o  A r m s t r o n g et al. [92]  nat  S n  [94]  Ho  [94]  - 0 . 5 ± 1.4  [94] [94]  < 0.2  D o b e l i et al. [87] A r m s t r o n g et al. [92] D o b e l i et al. [87]  1 6 5  e  natpt) 209  24  Bi  i  o.o  tii +0.6  -0.7  t\i  0.2 ± 1.1  50  100  150  Nuclear F i g u r e 1.3:  200  250  Mass (A)  as determined from the rate of radiative m u o n capture measurements.  gp/gA  Table 1.5: S u m m a r y of values of gp/gA capture ( O M C ) i n complex nuclei. Reference Deutsch et al. [96] Possoz et al. [97,98]  Nucleus 1 2  as determined from measurements of m u o n  Observable  Theory  A+/Ap  [22]? [99] [100,101] [102] [103]  C  55  55  55  55  55  55  K u n o et al. [104] M i l l e r et al. [105] Roesch et al. [106,107]  55  A, Paw/Pi  55  55  55  55  M a s u d a et al. [110] Towner et al. [112] Gorringe et al. [114]  55  1 6 Q 2 3  Na  A /A_ +  25  55  [102] [108] [109] [103] [111] . [113] [115]  9P/9A  < 15 7.1 ± 2 . 7 13.6 ± 2 . 1 15 ± 4 10.3 t l i tli ±2.5 ± 1.7 ± 1.7 ± 1.8 ± 1.9 7-9  IO.I  8.5 9.4 7.2 9.1 8.5  7-6  ^  observables along w i t h other weak and electromagnetic processes involving the same or analogous nuclear states i n the A = 12 nuclei; and consequently leading to better understanding of the nuclear structure uncertainties therein. U n l i k e R M C , O M C data are consistent w i t h the G o l d b e r g e r - T r e i m a n estimate; for example, the most recent determination of the coupling constant, made by our group [114] , from the A / A _ hyperfine dependence of m u o n capture on 5  +  2 3  N a , is  consistent w i t h the free nucleon value and i n disagreement w i t h the i n t r i g u i n g renorm a l i z a t i o n suggested by the R M C data. M o r e recently, Delorme a n d E r i c s o n [117] argued that the massive quenching of gp i n nuclei they h a d predicted [118], due to a r e n o r m a l i z a t i o n of the p-wave pion nucleon interaction, is nearly totally counterbalanced by a large enhancement of the s-wave p i o n nucleon interaction. To conclude, the possible r e n o r m a l i z a t i o n of the induced pseudoscalar coupling constant i n nuclear m e d i u m is a topic that has continued to fuel interest i n the field of low-energy weak interactions i n general a n d m u o n capture i n particular. Yet, it is obviously too early to draw any definitive conclusion. C l e a r l y more precise experimental determinations - a n d / o r experiments w i t h different characteristic uncertainties a n d dependences on nuclear s t r u c t u r e - of gp are i n h i g h demand. T h a t is the m a i n m o t i v a t i o n 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.  A recent comparison [116] of the Na(/x ,v) and N a ( n , p ) reactions tested the nuclear model used and gave confidence in the extracted gp/gA from the data. 5  23  23  26  C h a p t e r  T h e  2.1  2  7 — 1/ A n g u l a r  C o r r e l a t i o n  Introduction The previous chapter outlined the demand for other reliable observables for  the determination of the pseudoscalar coupling (gp), that is, observables w h i c h offer b o t h sensitivity to gp a n d a lesser dependence u p o n nuclear structure uncertainties. One such observable is the angular correlation between the neutrino and a specific nuclear de-excitation gamma-ray following m u o n capture on certain nuclei. U n l i k e most m u o n capture observables, the 7 — v angular correlation has the advantage of being an exclusive process, that is, the final states are resolved a n d hence avoids some (but not all) nuclear structure uncertainties. T h e angular correlation i n the emitted de-excitation gamma-ray is generated by the difference i n the m u o n capture p o p u l a t i o n of the allowed magnetic substates of the daughter nucleus. In the case of an allowed m u o n capture sequence,  fi~ +  (A,Z)  (a,z  - ly + v*  + 7,  (2.1)  the angular correlation can be written, using the n o t a t i o n of Parthasarathy and  27  Sridhar [100,119], as follows:  = J(0) [1 + a P ( c o s c V ) +  • )(  2  7  • i>)P (costV) + & ( £ • ) (  7  2  7  7  • v)}. (2.2)  where P  2  is the Legendre p o l y n o m i a l , a , fli and /3 are the correlation coefficients, 2  /} is the m u o n spin direction, and 9  1U  is the angle between the unit vectors along  the photon and neutrino momenta, 7 and E q u a t i o n 2.2 are i n order. 7  respectively.  T w o remarks about  T h e first is that i n the case of unpolarized muons, or  - r a y detection at 90° to the m u o n polarization direction, the terms involving (/i - ) 7  drop out and the correlation is characterized by only one coefficient (a).  T h e other  is that the effects of the coefficients (5\ and B are suppressed by the small residual 2  polarization of the m u o n after the atomic capture and cascade (about 15 % i n Si). In general, the three correlation coefficients depend u p o n 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 enhanced, whilst the sensitivity to nuclear physics uncertainties is m i n i m i z e d . One such case is the allowed 0 —>1 —>0 m u o n capture sequence, +  \x~ +  2 8  Si(0 )->^ +  +  28  +  +  A1*(2201 k e V , 1 ) +  v„ +  28  A1**(972.6 k e V , 0+)  +7. (2.3)  (Details concerning this cascade w i l l be discussed i n sections 4.3 and Figure 4.4.) T h e theory of the v — 7 correlation following nuclear m u o n capture was originally studied by P o p o v and collaborators [120,121,122,123,124,125]  and later ex-  amined by Ciechanowicz [109] a n d P a r t h a s a r a t h y and S r i d h a r [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, B and B ) on the p o o r l y determined weak cour  2  pling gp i n these models, w i t h emphasis on the m u o n capture sequence of interest, 28  i.e. E q u a t i o n 2.3.  2.2  Fujii-Primakoff Approximation T h e angular correlation coupling constants depend u p o n the weak coupling  constants as well as on the nuclear m a t r i x elements.  O n l y i n the so-called Fujii-  Primakoff a p p r o x i m a t i o n (FPA)[127] do the m a t r i x elements cancel out and the angular correlation dependence on gp springs forth.  In this a p p r o x i m a t i o n , one  assumes only an allowed capture, i.e. the neutrino carries away no angular moment u m (s-wave neutrinos only), and one ignores the nucleon-momentum dependent terms (also k n o w n as recoil or relativistic Hamiltonian.  terms) i n the effective  Fujii-Primakoff  These terms come from the non-relativistic reduction of the weak  interaction L a g r a n g i a n w h i c h was carried out by Fujii and P r i m a k o f f [127], using a F o l d y - W o u t h u y s e n transformation [128]. T h e y developed an approximate effective H a m i l t o n i a n for m u o n capture suitable for use w i t h non-relativistic nuclear wavefunctions. In the F P A , and for a pure M l 7-ray such as E q u a t i o n 2.3, one obtains :  « = 3G\ J n rGpZ —r IGpGA < -\G  (2-4)  P  Gp 3G\ + Gp — 3G  A  —  (2.5)  IGPGA  Gp  3G\ + Gp —  (2.6)  IGPGA  where Gp and GA are the Fujii-Primakoff form factors, w h i c h are linear combinations of the standard semileptonic coupling constants, i.e.  G  A  =g  GP = (g  P  A  -(gv  - g  A  + g M ) ^  - g v -  29  9M)^J^  (2.7)  •  (2.8)  Hence, under the F P A , the coefficients a , 8\, a n d B are functions of only 2  GP/GAI  or equivalently gp/gA p r o v i d e d gv a n d #M  a  r  e  k n o w n . Furthermore, exam-  i n i n g the above Equations (2.4, 2.5 and 2.6), one obtains simple relations, namely,  1+  A  = 3 + B F  2  (2.9)  ± VTT^}\  (2.io)  L  and & = - [ V i ^  w h i c h indicate that, among the three correlation coefficients, only one is  linearly  independent, or, i n alternative language, that a l l three depend on only one variable. However, it is expected that the F P A is an oversimplification a n d is inadequate for reliable extraction of gp.  2.3  Beyond the F P A  Some weak nuclear-structure dependence w o u l d surface u p o n the i n c o r p o r a t i o n of the recoil terms a n d / o r the higher p a r t i a 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 p a r t i a l m u o n capture by spinless nuclei i n terms of the m u l t i p o l e expansion of the weak hadronic currents [120,121,122,123,124,125], a n d finds that i n the case of the 0 + A l + m u o n capture sequence, the v — 7 angular correlation, among other observables, is described by a complex ratio of two independent m u l t i p o l e amplitudes ( ^ ) a n d can be characterized by two parameters x a n d  as follows:  = 57  Hsl 30  <-> 2n  <-> 22i  w i t h relative phase <f> = 0 or TT. T h e amplitude ratio x describes the relative m u o n capture feeding of the components of the excited state (e.g. 2201 k e V 1  +  in Equa-  tion 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 w i t h the following relationships among the correlation coefficients:  a  =  ( ' >  F  2  1 + x + 2x cos < , 2+ x  13  2  Pi = F [  (2.14)  y  2  2 — x — 2Fx cos 6 2  =  A  2  +  .*>  •  < ' > 2  15  where the factor F depends on spin sequence and the m u l t i p o l a r i t y of the deexcitation 7-ray. In particular, F  =  1  for 1 + ^ 0 +  (2.16)  and  where 8 is the m u l t i p o l a r i t y m i x i n g parameter given by the E2/M1  ratio.  Several comments about the above Equations are i n order: • T h e F P A a p p r o x i m a t i o n , Equations 2.4, 2.5 and 2.6, falls out of Equations 2.13, 2.14 and 2.15 as s i m p l y being the prediction  • Unlike fi\ and f3 , a is independent of the phase </>. 2  • For the spin sequence 0 A 1+  0 , Equations 2.9 and 2.10 of the previous  +  +  section connecting a , / ? i and (3 can be deduced from Equations 2.13, 2.14 2  31  and 2.15. Oziewicz points out that a measurement of the sign of B w o u l d 1  constitute a measurement of m u o n neutrino helicity, or i n other words, a negative sign for B\ w o u l d i m p l y the emission of a right-handed neutrino. • A l s o , Equations 2.13, 2.14 and 2.15 along w i t h Equations 2.9 a n d 2.10 i m p l y the following b o u n d a r y conditions - 1 < a < 0.5  (2.19)  0 < 8 < 1.5  (2.20)  - 1.5 < & < 1-5,  (2.21)  X  violations of w h i c h w o u l d i m p l y time-reversal violation. • A consequence of the m u l t i p o l a r i t y m i x i n g factor F is a suppression of the correlations i n the 1 1  +  0  +  A 2  +  7-ray transitions as opposed to the pure M l  transitions.  +  • Furthermore, from E q u a t i o n 2.17, one has the following bounds (see F i g ure 2.1): - 0.4 < F < +1 , on the 1  +  2  +  (2.22)  7-ray transitions.  In summary, even when one goes beyond the F u j i i - P r i m a k o f f a p p r o x i m a t i o n , the dependence of the angular correlation on gp remains a n d a l l three correlation coefficients a, B\ and B are functions of only one (nuclear structure dependent) pa2  rameter x\ thus there is only one independent coefficient, m o d u l o the sign ambiguity from <j>.  32  2  4  6  =  6  E2/M1  8  10  ratio  F i g u r e 2.1: T h e m u l t i p o l a r i t y factor F as a function of the m i x i n g 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 i n  2 8  A l , that go beyond the Fujii-Primakoff a p p r o x i m a t i o n , to calculate  numerically the dependence of the angular correlation coefficients as a function of the pseudoscalar coupling constant.  These result i n what we herein-after refer  to as 'full' calculation models. T h e y 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. T h i s  model was an application of the m u l t i p o l e theory of P o p o v [120,121,122,123,124] a n d was basically an outgrowth of Oziewicz's work [125]. In this model, Ciechanowicz calculates two multipole amplitudes (A a n d M) for the 1229 k e V transition (equation 2.3) as well as the 1342 k e V transition, i.e.  u~ +  2 8  Si(0 ) +  + A1*(1373 k e V 1+) 28  i/„ + A1**(30.6 k e V 2+) +7, (2.23) 28  by i n c l u d i n g a l l recoil terms up to order 1 / M i n the effective H a m i l t o n i a n , i.e. second order terms i n M are omitted. T h e m u o n wave function used is a realistic one, t a k i n g into account the extended nuclear charge d i s t r i b u t i o n . T h e D i r a c E q u a t i o n was numerically solved for the nuclear charge d i s t r i b u t i o n . T h e  2 8  S i and  2 8  A l wave-  functions w h i c h were used were from W i l d e n t h a l , DeVoigt and M c G r o r y [130,131] w i t h configuration m i x i n g taken into account. For the 1229 k e V 7-ray, the two m u l t i p o l e amplitudes are given as: A = -19.58 g  A  M = -23.86 g  A  + 0.216 g  v  + 4.954 g  v  34  + 1.373 g  P  - 0.019 g , P  (2.24)  (2.25)  and hence their m u l t i p o l e ratio x, u t i l i z i n g gv = — 0 . 7 8 4 ^ [132], becomes: x (2201,1+) = 0.712 - - 9P/9A ^ l + 6 . 8 x 10-* g /g 1  q Q7Q  (  ;  P  y  2  2  G  ) J  A  Ciechanowicz also studies the effect of the shape of the m u o n wave function on gp by using a constant m u o n wave function over the nuclear volume, a r r i v i n g at what he calls the non-relativistic a p p r o x i m a t i o n ( N R A ) :  A = —19.80(7,4 + 0.940  M = -25.40 g  A  (2.27)  + 6.920 gv,  (2.28)  or equivalently, x ( 2 2 0 1 , 1 ) = 0.642 - 0.0305 g /g , +  P  A  (2.29)  which is not to be confused w i t h the F u j i i - P r i m a k o f f a p p r o x i m a t i o n (equation 2.18) as the former still includes the recoil terms i n the effective H a m i l t o n i a n , a n d differs slightly from relation 2.26 by a C o u l o m b correction i n the m u o n wavefunction. T h e mutipole ratio x for the 1229 k e V t r a n s i t i o n from these three calculations is shown i n F i g u r e 2.2 for comparison. T h e angular correlation coefficients  1  a, 8\, a n d 8  2  can then s i m p l y be calcu-  lated from relations 2.13, 2.14 and 2.15, a n d are presented i n Figures 2.3, 2.4 a n d 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  Sridhar  In this m o d e l [119,100,101], the 7 — v angular correlation is developed using the density m a t r i x formalism of Devanathan a n d S u b r a m a n i a n [133]. In the first paper [119], P a r t h a s a r a t h y a n d S r i d h a r consider the capture of unpolarized muons i n the  2 8  S i process of E q u a t i o n 2.3 a n d hence they are only  There 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. 1  35  9p/9a F i g u r e 2.2: Dependence of the multipole ratio on gp/gA for the 1229 k e V transition, as calculated by Ciechanowicz [129]. A l s o shown are the non-relativistic a p p r o x i m a t i o n ( N R A ) as well as the Fujii-Primakoff approximation ( F P A ) .  36  F i g u r e 2:3: Dependence of a on gp/gA for the 1229 k e V transition, as calculated by Ciechanowicz [129], Parthasarathy and Sridhar [100] as well as K u z ' m i n et al. [126]. A l s o shown is the FujiiPrimakoff a p p r o x i m a t i o n .  37  1.0  _j  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  1  1  i  i  i  i  i i _  Legcmd —e— - Model : Cie Model : P&S FPA  —*—  0.8  5 0.6  -  0.4  0.2  0.0  i  r  20  F i g u r e 2.4: Dependence of fi\ on gp/gA for the 1229 k e V transition, as calculated by Ciechanowicz [129] and P a r t h a s a r a t h y and Sridhar [100]. A l s o shown is the F u j i i - P r i m a k o f f a p p r o x i m a t i o n .  38  F i g u r e 2.5: Dependence of B on gp/gA for the 1229 k e V transition, as calculated by Ciechanowicz as well as Parthasarathy and Sridhar. A l s o shown is the Fujii-Primakoff a p p r o x i m a t i o n . 2  39  concerned w i t h the coefficient a.. In the second paper [100], they extend the same formalism to the capture of polarized muons i n the same process, a n d hence include  /?i and B . 2  Including b o t h higher p a r t i a l wave neutrinos as well as nucleon-momentum dependent terms i n the Fujii-Primakoff effective H a m i l t o n i a n , they derive expressions for a , 81 and f3 i n terms of reduced nuclear m a t r i x elements i n the particle-hole 2  model of D o n n e l l y a n d W a l k e r [134]. E x a m i n i n g these expressions under the FujiiPrimakoff a p p r o x i m a t i o n ( F P A ) , they obtain the same relations as the ones given i n Equations 2.4, 2.5 a n d 2.6 earlier. T h e y also derive simple relations l i n k i n g the correlation coefficients to the average and the l o n g i t u d i n a l polarizations of the recoiling nucleus; namely , 2  - a = 1 + ^PL,  (2.30)  l3 = 1 - ^P  (2.31)  1  av  and fa = - 1 + \p  av  - \P ,  (2.32)  L  as well as P  L  = -\{fa  + B ).  (2.33)  2  T h e y also note that these relations are rigorously true, a n d independent of nuclear structure; even when one includes nucleon-momentum dependent terms a n d higher p a r t i a l 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 a n d S r i d h a r then numerically evaluate the reduced nuclear m a t r i x elements appearing i n the expressions of a [119] a n d R\ a n d i3 [100] using the 2  the wave functions of D o n n e l l y a n d Walker [134]. T h e y give their numerical results i n a tabular form, c l a i m i n g that their results for the correlation coefficients These relations were derived first by Bernabeu in 1976 (see Ref. [22]), however, on different grounds. 2  40  are very different from those of Ciechanowicz [129]. However, as A r m s t r o n g [135] has pointed out, these claims are s i m p l y not true, as can be seen i n Figures 2.3, 2.4 a n d 2.5 where their results are plotted along w i t h those of Ciechanowicz. Hence, their calculation seems to agree fairly well w i t h 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. T h i s error is pointed out by the authors themselves i n Ref. [25] of the subsequent paper [100], a n d corrected values of a are given therein . 3  A s for the case of /?i a n d B  2:  the claimed disagreement is due to the use of the  wrong choice of (f> i n Ciechanowicz's theory. T h e choice (f) = ir (in Equations 2.13, 2.14 & 2.15) gives values for fli a n d B that are i n reasonable agreement w i t h those 2  of Parthasarathy and Sridhar, see Figures 2.4 & 2.5. T h i s unfortunate choice of <f>, seems to have caused several authors, amongst them are M u k h o p a d h y a y [22] as well as E r a m z h y a n [136], to arrive at the same misleading conclusion about the disagreement between the two models. It is interesting to further note that Parthasarathy a n d Sridhar apparently make no statement about relation (2.10), i n fact their results seem to satisfy this relation , w h i c h indicates that only one of the correlation coefficients is linearly 4  independent, i n agreement w i t h Oziewicz and Ciechanowicz. Parthasarathy a n d S r i d h a r also claim that among the three correlation coefficients,  B is nearly nuclear-model insensitive. However, this is only true i n the 2  region where B is nearly flat, i.e. 2  B is less gp dependent as well. 2  Furthermore,  since only one of the coefficients is independent, one expects a l l coefficients to have Y e t , 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.". Except for a small region around gp/gA ~ 17.5 where a violates the 'rigorous' limit (equation 2.4). If not absorbed in the uncertainty of their model, this may suggest some numerical problem with the model. 3  4  41  the same gp model-dependence. In the t h i r d 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 III  : K u z ' m i n at al.  Recently, K u z ' m i n , O v i c h i n n i k o v a and Tetereva [126,137], from the Joint Institute for Nuclear Research ( J I N R ) i n D u b n a , calculated the independent transition amplitudes (the effective m u o n capture m a t r i x elements) as well as the angular correlation coefficients for the 7-ray transitions of interest following m u o n capture on 2 8  S i i n the framework of the nuclear shell model. T h e y have developed a procedure  to calculate nuclear single particle m a t r i x elements applicable for any h a r m o n i c oscillator wave function. A l l possible linear velocity dependent terms are included i n the effective m u o n capture H a m i l t o n i a n . W i t h i n this unified approach, they c l a i m that their code can use any shellmodel nuclear wave functions. T h e nuclear wave functions used i n their most complete model were calculated w i t h the well tested [138] shell m o d e l code O X B A S H [115] In a d d i t i o n to the O X B A S H 'full' sd-shell m o d e l nuclear wave functions, they use several sets of wave functions; i n c l u d i n g those of W i l d e n t h a l et al. [130,131] and Donnelly and W a l k e r [134] w h i c h are used i n the previous two models. T h e y are able to reproduce the calculations of Ciechanowicz as well as of Parthasarathy and Sridhar. Consequently, they ascribe the discrepancy between their m o d e l 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 U n i v e r s i t y of K e n t u c k y have used a similar code [139] using a full Is — Od valence nucleon space and U S D residual interaction [140]. W e have recently used it to predict p a r t i a l rates and hyperfine dependences of m u o n capture on 42  2 3  N a [141]; and found that the  calculated capture rates to the six excited states i n  2 3  N e are i n general agreement  w i t h the measured ones [114]. T h i s same p r o g r a m , 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, F i g u r e 2.3 shows this dependence along w i t h the other two models. A s can be seen from F i g u r e 2.3, m o d e l III ( w i t h the more realistic m o d e r n wave functions a n d complete effective H a m i l t o n i a n ) 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 p r o m i s i n g 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 i m i t a t i o n , typical of a l l m u o n capture calculations, that can be serious. T h a t is the possible overestimation of the velocity dependent terms i n the effective H a m i l t o n i a n due to the use of harmonic-oscillator single particle wave functions w h i c h are used i n a l l three models.  T o test this  suspicion, they are i n the process of replacing the harmonic-oscillator potential by a finite one such as a W o o d s - S a x o n potential.  2.5  The Method of Measurement T h e experimental m e t h o d to measure the 7 — v angular correlation has been  suggested by Grenacs et al. [142]. T h e y pointed out that, since the daughter nucleus recoils against the neutrino after m u o n capture, the angular correlation between the neutrino a n d the emitted 7 ray is equivalent - w i t h i n a factor of 7T- to the angular correlation between the nuclear recoil a n d the 7 ray. T h e observed energy of the emitted 7 ray is related to the transition energy E i n the e m i t t i n g nucleus by the 0  43  F i g u r e 2.6: K i n e m a t i c s of m u o n capture reaction on  2 8  Si.  Doppler E q u a t i o n : £  7  = £(,(1 -(3 cos 6)  (2.34)  where 8 is the velocity of the recoiling nucleus at the moment of the 7 transition (Figure 2.6). T h e i n i t i a l recoil velocity for the 2201 k e V A l excited state (E* ) is 2 8  xt  given by : /  m  u  — iTT-e +  8 = »/2 — -  msi  — —  0  V  — m\\  "Ui  — E*  f  —  —  BE  + 1 - 1 = 0.0037484  (2.35)  where BE is the m u o n b i n d i n g energy of the atomic Is orbit i n Si (0.539 M e V ) . T h e correlation function ( E q u a t i o n 2.2) can be w r i t t e n as an energy distrib u t i o n of the emitted 7 rays. Consequently, the measurement of this d i s t r i b u t i o n 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 w h i c h (p. -7) = 0 for the reasons aforementioned, i.e. easy to measure, same nuclear m o d e l dependence and no need to measure residual muons p o l a r i z a t i o n . F i g u r e 2.7 shows the shape of a Doppler-broadened 7-ray d i s t r i b u t i o n w i t h the angular correlation coefficient, a , effects, see also F i g u r e 5.23. O f course, this m e t h o d is valid only for a transition from a short lived state, i n which the daughter nucleus decays in-flight, and hence there is no appreciable slowing-down effects. T h i s is fairly well satisfied i n the case of the ( 1 , 2.2 M e V ) level for w h i c h the lifetime is about 65 fs, see ref+  44  \ /  \ /  \ -  \  /  IL<5g<sinidl  a = +0.25 • - - - a = -0.25 E (1-/?o) 0  E  0  E (1+/? ) 0  0  Gamma —ray Energy (E ) F i g u r e 2.7: T h e shape of a Doppler-broadened 7-ray d i s t r i b u t i o n a n d the effect of the 7 — 1/ angular correlation coefficient, a (effects due to the resolution function of the detector are not i n c l u d e d here).  erence [143] and section (5.7). T h u s b o t h decays from this level, the 1229 k e V a n d 2171 k e V 7-rays are suitable candidates. Measurements of the Doppler-broadened lineshape of the 1229 k e V 7-ray was performed by M i l l e r et al. [144,145] at S R E L i n the late 1960's a n d just recently by B r u d a n i n et al. [146] at J I N R . A full account of these measurements, a n d the way they compare to this work, along w i t h their effect on the above theoretical models w i l l be given i n C h a p t e r 6.  45  2.6  Ancillary Reactions A s is usually the case, different nuclear reactions complement and supplement  one another. F o r example, the 7 - r a y spectra obtained following m u o n capture alone do not give us directly the energies of the excited states of the residual nuclei. T h e importance of such knowledge, for the purpose of this work, stems from the fact that we w o u l d like to know whether higher excited levels of the 2201 k e V  2 8  A l level of interest.  2 8  A l cascade-feed into  Such a possibility w o u l d spoil the goal of this  work. T h i s topic w i l l be discussed more thoroughly i n the following chapters. T h e 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 , H e ) w h i c h give us the 2  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], N i i z e k i et al. [147] a n d  E r a m z h y a n et al. [148]. These reactions are similar because they a l l tend to excite 1  +  transitions, often called G a m o w - T e l l e r ( G T ) by analogy to the  from beta-decay.  Unfortunately, the only d a t a to our knowledge on  are from B h a r u t h - R a m et al. [149] at E  n  E  n  nomenclature 2 8  Si(n,p) Al 2 8  = 21.6 M e V and from B r a d y et al. [150] at  = 60 M e V . B o t h h a d poor resolution ( A ~ 1 M e V ) and very low statistics, so are  not too helpful. T h e S i ( 7 r , 7 ) A l reaction was studied twenty years ago at P S I , but 2 8  2 8  the data were never published as far as we know. Nevertheless, a silicon spectrum was presented i n the 1976 annual report [151] a n d is reproduced i n F i g u r e 2.8. T h e 7 - r a y resolution is about 1 M e V a n d the statistics are sufficient. T h e difference is that for a stopped 7r~, the absorption tends to occur p r e d o m i n a n t l y from the atomic 2p state not the Is state [17,152]. The  2 8  S i ( d , H e ) A l was studied by Sakai a n d collaborators [147]. It poten2  2 S  tially h a d better energy resolution t h a n the ( ^ , 7 ) reaction. A n excitation-energy  46  E  Figure 2.8:  45  S i ( 7 r , 7 ) A l c o n t i n u u m s u b t r a c t e d spectrum taken from P . T r u o l et al.[151]. T h e levels i n S i excited by (e,e') at 180° are also 1 levels a n d are m a r k e d at the b o t t o m , see Schneider et al. [153]. T h e level of interest i n this experiment is at 2201 k e V e x c i t a t i o n energy i n A 1 .  28  28  2 8  +  2 8  47  300  F i g u r e 2.9:  2 8  S i ( d , H e ) A l excitation energy spectrum at 270 M e V and 0° 2  2 8  obtained by Niizeki et a?. [147].  >  28  16  Si(p,n) P 28  6 jo o  0  2  A  6  0  10  12  14  16  10  E x c i t a t i o n Energy (MeV)  F i g u r e 2.10:  S i ( p , n ) P excitation energy spectrum at 136 M e V and 0.2° from A n d e r s o n et al. [154]. We have received similar d a t a at 120 M e V from J . R a p a p o r t .  2 8  2 8  48  spectrum at  = 270 M e V and at 0° is extracted from their paper and is shown  i n F i g u r e 2.9. T h e S i ( p , n ) reaction has also been studied by A n d e r s o n et al [154] 28  at E  p  = 136 M e V and 0.2°. T h e i r spectrum is shown i n F i g u r e 2.10. (Note that  there w i l l be C o u l o m b differences i n the levels.) T h e comparison of the (p,n) and ( d , H e ) spectra is striking. A s i m i l a r i t y among these spectra is the excitation of the 2  1  +  levels, just as i n m u o n capture.  show a strong 1  +  However, while the (p,n) and ( d , H e ) spectra 2  peak a r o u n d 4.8 M e V , the (7r, ) spectrum has a strong level at 7  6.2 M e V (which is still below the neutron break-up threshold). Other reactions that are very useful i n determining the the  2 7  A l ( n , 7 ) A l reaction, see Schmidt et al. [155], and the 2 8  2 9  2 8  A 1 level scheme are  S i ( d , H e ) A l reaction, 3  2 8  see Vernotte et al. [156]. T h e y b o t h attain high resolution and can unravel the level scheme quite accurately.  However, the (n,7) reaction tends to populate different  levels (particularly those w i t h h i g h spin), whereas m u o n capture tends to populate mainly 1  +  as well as 0~, 1~, 2~ and 2  +  levels, while the ( d , H e ) is a different  reaction again as well as operating on a different target.  3  Some things are clear  however. F i r s t is that the ( / i , v) reaction w i l l excite several levels above 2.2 M e V w h i c h are b o u n d , and therefore could cascade into the 2201 k e V level of interest. T h e indications are that the feeding to these levels is 1.5 to 2 times that to the 2201 k e V level a n d / o r 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]. T h u s 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 m u o n capture w i t h  the precision that we need. T h i s topic w i l l be revisited i n section 5.4 i n connection w i t h the cascade-feeding p r o b l e m . M a c D o n a l d et al  [157] give the neutron m u l t i p l i c i t y - d i s t r i b u t i o n following  m u o n capture i n several targets. Table 2.1 is a s u m m a r y of the results (properly corrected for the detector efficiency) for S i , w h i c h is not incompatible w i t h Table II 49  Table 2.1: al. [157].  N e u t r o n multiplicities of m u o n capture on S i , from M a c D o n a l d  Neutron multiplicity  Yield  0 1 2  0.362±0.065 0.472±0.092 0.160±0.070  3  0.006±0.008  of M i l l e r et al. [144].  50  Resulting nuclei 2 8  27  26  2 5  Al, A1, A1,  2 7  Al,  2 4  2 6  2 5  Mg, Mg, Mg, Mg,  2 4  2 3  2 2  2 1  Na Na Na Na  et  Chapter 3 Description of the Experiment The  experiment described here has been performed at the superconduct-  ing channel ( M 9 B ) of the T r i - U n i v e r s i t y M e s o n F a c i l i t y ( T R I U M F ) i n Vancouver, C a n a d a . It was performed d u r i n g the course of a two-week cyclotron r u n i n November 1992.  T h e experimental setup was p r i m a r i l y designed for the second phase  of experiment 570 whose a i m was to measure the angular correlation between the neutrino a n d a D o p p l e r broadened de-excitation g a m m a ray i n  2 8  A l following m u o n  capture on S i [159] through a coincidence technique. T h e coincidence technique is 2 8  intended to suppress backgrounds underneath the g a m m a 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 M 9 B Channel The  T R I U M F accelerator is a six-sector focusing cyclotron w i t h a special  feature being the acceleration of negative H~ ions which, i n t u r n , permits a convenient extraction of the beam. T h i s is done by passing the beam through a t h i n carbon foil to strip the two electrons a n d hence the resulting beam of positive ions curves i n the opposite direction i n the magnetic field a n d exits the cyclotron.  51  A s shown i n F i g u r e 3.1, two stripping foils, located 180° apart, are used to feed the two m a i n beam lines B L 4 ( P ) a n d B L 1 ( P ) w i t h proton beams of current u p to 140 fiA a n d energy range of 183 to 520 M e V . T h e first beam line is dedicated to proton-induced reactions while the other is used for p i meson production. T h e secondary m u o n beam used i n this experiment was produced when the p r i m a r y proton beam i n the meson beamline impinges on the meson p r o d u c t i o n 1 A T 2 target, typically a water cooled strip of b e r y l l i u m 10 c m thick i n the beam direction a n d 5 m m x 15 m m i n cross sectional area.  T h e p r i m a r y proton beam is n o r m a l l y  delivered i n 3 nsec pulses every 43 nsec a n d w i t h 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 a n d other p r i m a r y a n d secondary beamlines. T h e m u o n beam used was obtained from the new M 9 B channel w h i c h incorporates a superconducting solenoid to collect l o w - m o m e n t u m polarized muons from the decay of pions.  T h i s p i o n contamination of the beam was less than 0.2 % ,  and the fraction of electrons i n the beam was about 20 % . N e w l y installed slits on the channel helped focus the beam further. A b o u t 1.2 x 10 fj,~/sec 5  at m e d i u m  m o m e n t u m ( ~ 60 M e V / c ) were required to stop i n our targets. T h i s m o m e n t u m is equivalent to 20 M e V muons w h i c h have a range of 3 g / c m  2  i n silicon. F i g u r e 3.2  shows the layout of M 9 channel, w i t h its two legs M 9 A a n d M 9 B .  3.2  Experimental Arrangement T h e experimental setup is shown i n F i g u r e 3.3. It consisted of a conventional  beam telescope of three scintillation counters, S i , S2 a n d S3 w i t h dimensions of 15.2 c m x 20 c m x 0.32 c m , 7.6 c m diameter x 0.16 c m a n d 10.2 c m x 10.2 c m x 0.32 c m respectively. S i a n d S2 w i t h S3 i n anti-coincidence defined a m u o n stop i n  52  BEAMLINES A N D EXPERIMENTAL FACILITIES. ISAC UNDER CONSTRUCTION.  TR 30 ISOTOPE • PRODUCTION CYCLOTRON  to  PROTON HALL EXTENSION  d  i  s SH  TISOLEXPERIMENTAL AREA  U  TRINAT'  t3  i—i P5  MESON HALL SERVICE ANNEX  u (H  FEB96  OPTICALLY PUMPED POLARIZED ION SOURCE (OPPIS)  •  POLARIZED ION SOURCE  M15(/z)  i—I  fa-.  co  t o RF S e p a r a t o r and IPC  Superconducting solenoid  M9B M9A  t  f-=  •  3—  I  •  Target 1AT2  Figure 3.2: Layout of the T R I U M F M9 channel. 54  the target. T h e t i m i n g of a m u o n stop was defined by S2, w h i c h was mounted close to the target and h a d 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 m a i n target used was n a t u r a l Si (92.2%  2 8  S i , 4.7%  2 9  S i a n d 3.1% S i ) i n 3 0  granular form. Its dimensions was 6.3 c m diameter by 1 c m thickness. It was set at 45° to the beam and to the Ge detectors axis i n order to m i n i m i z e the thickness traversed by g a m m a rays and m a x i m i z e the thickness traversed by muons.  This  orientation also reduces the bremsstrahlung from muon-decay electrons. A variety of targets were used to study backgrounds as well as to determine the acceptances of the G e detectors. T h e y were: C , Mg(granules), Al(6061), Cr(granules), Fe, N i , C u , Ge(powder), M o , S n , A u , P b as well as N a l ( p o w d e r ) and stainless steel(316).  3.3  Detection System The  7—ray detection system used i n this experiment (Figure 3.3) had  two p r i n c i p a l arms: a pair of high p u r i t y g e r m a n i u m detectors ( H P G e ) along w i t h several N a l ( T l ) counters. T h e two H P G e crystals were the p r i n c i p a l detectors i n this experiment and must measure the 1229 k e V 7-ray line emitted from the capture of muons on S i . These g e r m a n i u m detectors, G e l and Ge2, were of 44% and 30% relative efficiency ( w i t h respect to a 7.62 c m diameter by 7.62 c m long N a l ( T l ) ) w i t h system resolution at 1.3 M e V of about 1.9 k e V and 2.0 k e V full w i d t h at half m a x i m u m ( F W H M ) respectively. U n d e r experimental conditions^', e. i n beam, these resolutions became 2.2 k e V and 2.4 k e V F W H M respectively. T h e G e detectors were located at 90° w i t h 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  Polyethylene sleeve  Lead collimator  PM's  Nal Compton suppressors  Lead collimator  Figure 3.3: Experimental setup.  56  Table 3.1: Properties of the N a l ( T l ) counters.  N a l ( T l ) Counter Bl B2 B3 B4 NI N2 N4 N5 N7 N8 N9 NIO  Dimensions (in cm) 10.2 x 10.2 x 20.3 10.2 x 10.2 x 20.3 10.2 x 10.2 x 20.3 10.2 x 10.2 x 20.3 12.7 x 15.2 12.7 x 15.2 12.7 x 15.2 12.7 x 10.2 12.7 x 7.2 7.6 x 7.6 5.1 x 5.1 5.1 x 5.1  Resolution (%) F W H M 9.9 9.6 8.0 7.7 9.6 8.3 7.6 9.3 7.1 6.1 9.9 8.5  Furthermore, the two G e detectors were surrounded by Compton-suppression devices, S A a n d S B respectively. E a c h of the suppressors was constructed out of six segmented arrays of N a l ( T l ) crystals arranged i n coaxial geometry a n d each segment was viewed by a phototube. T h e basic a i m of the N a l ( T l ) annulus is to detect the g a m m a rays that are C o m p t o n scattered out of the G e detector -before depositing all their energy- a n d reject the related events by operating the annulus a n d the G e detector i n anticoincidence mode. T y p i c a l performance of the C o m p t o n suppression system w i l l be illustrated i n Figures 5.2, 5.3 and 5.4. T h e two Compton-suppressed Ge detectors were located 19.4 c m a n d 14.7 c m respectively from the target. The other a r m of the detection system consisted of an array of 12 N a l ( T l ) counters (not a l l are shown i n F i g u r e 3.3) arranged around the target i n order to increase the total g a m m a ray tagging efficiency. T h e purpose of these detectors is to select events i n which a coincident 7-ray of 942 k e V is emitted w i t h the 1229 k e V line of interest. T h e properties of these counters are given i n Table 3.1. The first four counters, B l t h r o u g h B 4 , are rectangular bars while the other eight are c y l i n d r i c a l i n shape. T h e cited resolutions are the "in-beam" F W H M at 57  the 1332 k e V C o g a m m a ray. 6 0  In a d d i t i o n to their basic a i m as C o m p t o n suppressors for the surrounded G e detectors, the N a l ( T l ) annuli can be operated i n coincidence mode w i t h the opposing Ge detectors a n d consequently increasing the tagging efficiency of the N a l ( T l ) a r m .  3.4  Electronics and Data Acquisition The data acquisition system can be divided into two functional subsystems: the  trigger logic a n d the data acquisition control. T h e former m a y further be d i v i d e d into four distinct blocks: the telescope logic, the Compton-suppressed G e logic, the N a l ( T l ) logic a n d the strobe logic. T h e latter, the data acquisition control system, consisted of three components: the m a i n 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 a n d process data from a l l of the detection components as well as to m o n i t o r their performance. It was designed to examine stopped muons and detected g a m m a ray events a n d to determine w h i c h of those should be analyzed. E a c h time the trigger logic received an event detected i n either g e r m a n i u m detectors, the corresponding N a l ( T l ) suppressor a n d the scintillation telescope were examined. If the event was not vetoed by the suppressor and it satisfied the stop definition, a valid strobe was generated by w h i c h relevant information from the detection system was digitized and recorded on magnetic tape a n d part of the data was analyzed immediately for on-line m o n i t o r i n g of the experiment. A complete electronic diagram is shown i n F i g u r e 3.4. T h e overall t i m i n g of the logic signals associated w i t h one of the G e detectors is shown i n F i g u r e 3.5.  58  3.4.1  Telescope Logic A schematic of the telescope logic is i n c l u d e d i n F i g u r e 3.4. In this d i a g r a m  the triangles represent logical fan-in/fan-out units, w h i c h are essentially O R gates w i t h many outputs. T h e analog signals from scintillators S i , S2 a n d S3, were input to quad discriminators w h i c h i n t u r n 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 T i m e - t o - D i g i t a l Converters ( T D C s ) . A n o t h e r signal was delayed 5 microseconds (/jsec) a n d provided the input signal to a U B C R o u t e r box [162]. T h i s router, also called pulse separator, accepts a t r a i n of up to four logic pulses w i t h i n a time w i n d o w , splits t h e m a n d routes the i n d i v i d u a l pulses into four separate outputs: A l , A 2 , A 3 a n d A 4 . Hence it, along w i t h a T D C permits recording of m u l t i p l e time spectra; so that it w i l l give times of up to four muons w i t h i n a 2.5 /jsec gate, i.e. i f there is only 1 m u o n i n the 2.5 /jsec gate, its time relative to the T D C start, 7, w i l l be i n channel A = l , i f there are 2 muons, their times w i l l be i n A = l and A = 2 etc. T h e last m u o n stop entering the router (i.e. the one i n the highest channel) was the m u o n most likely to have caused the g a m m a ray event (or the strobe). T h e t h i r d signal from the "S1-S2-53" coincidence unit was used to initiate an u p d a t i n g 2.5 fisec pile-up gate for the two m a i n coincidence units, T R I G l and T R I G 2 . T h i s 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 g e r m a n i u m detectors was sent to a charge sensitive  preamplifier i n order to m a x i m i z e its signal-to-noise ratio a n d m a t c h impedance.  59  TDC atop  Passive Splitter  Nal=  - TDC stop  CFD  1  -| AMP  y.  ADC  to STARBURST—LAM  Busy signal  MIM-OUT  "INHIBIT^  G.G  G.G  G.G (2-5MS)  . A D C gates  UBC R  A=1. - , ' O  O o A=2 (fi u A=3 «-t' o T 3  Q  A=4  R  V.S. CS.  HLT1  > „  —-m-> o H m <=  D  I  5»  w  J  TDC stop  = F,  tu c n C <p O CO ->  ADC  V.S.  cs.  F i g u r e 3.4: Complete electronic logic d i a g r a m . T h e symbols used are as follows: V S = v i s u a l scaler, C S = c a m a c scaler, G G = g a t e generator, B i t = b i t register; others are explained i n the text. 60  to  /ii  y  /i2  jti3  /i4  PU  -2.5/is-  (extended gate) Gel SA SA Veto Gel-SA  TRIG1  u  INHIBIT  S1-S2-S3 (delayed  5iis  dead time  /ii  /i2  IT  IT  /z3  M  ROUTER gate  A=1 time A=2 time A=3 time A=4 time F i g u r e 3.5: T i m i n g s a n d definitions of events for G e l a n d associated electronics.  61  E a c h preamplifier provided two distinct output signals: energy a n d time signals. One energy signal was shaped, amplified a n d pole-zero adjusted by a spectroscopy a m p l i f i e r ( O R T E C 572 and Tennelec T C 2 4 1 for G e l a n d 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 A n a l o g - t o - D i g i t a l C o n v e r t e r ( A D C ) , where its energy dependent a m p l i t u d e was digitized and recorded. T h i s amplification a n d digitization was done i n the experimental area to reduce the effect of noise on the signal cables. T h e other preamplifier energy signal was passed through a high-threshold d i s c r i m i n a t o r ( O V L ) i n order to reject overload pulses i n the G e detector.  These are likely M i c h e l electrons from m u o n  decay a n d / o r their bremsstrahlung radiations going into the G e detector.  These  pulses w o u l d otherwise be overflow events i n the G e A D C a n d w o u l d have increased the computer(processing) dead time unnecessarily. T h e t i m i n g signal, obtained from the other preamplifier outputs, was fed to a t i m i n g filter amplifier ( O R T E C model 474 T F A ) where it was shaped and amplified 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 a n d subsequently passed to the corresponding coincidence units, TRIG1/TRIG2.  Constant fraction d i s c r i m i n a t o r s ( C F D ) are used to produce fast  logical signals w i t h a baseline crossover nearly independent of the input signal amplitude. T h e use of two C F D s -one w i t h low-threshold defining the time a n d the other w i t h high-threshold establishing energy t h r e s h o l d - was found to further reduce the dependence of the pickoff time on the amplitude a n d rise time of the input pulses. T h e other T F A output was split into four identical pulses, each of which was sent to a leading-edge discriminator w i t h a different threshold. T h e 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 i m i n g signal from the g e r m a n i u m . T h e reason for this circuit was to measure the rise time of the pulses i n order to improve the detector resolutions, 62  b o t h energy a n d t i m i n g . T h i s w i l l be discussed later i n the thesis, section 5.2.3. T h e energy a n d time of each of the N a l ( T l ) suppressors' segments were digitized a n d recorded b y A D C s a n d T D C s for the offline analysis respectively. In addition, their logic signals were fanned-in together to provide the suppressor time w h i c h was used as a veto to their respective g e r m a n i u m detectors, i.e. G e l - S ' A a n d Ge2SB  coincidence units.  3.4.3  The Nal(Tl)  Logic  T h e analog signals from each of the 12 N a l ( T l ) counters was passively split-up into two signals. One was amplified a n d passed to a peak-sensing A D C . T h e other was sent to a constant-fraction discriminator a n d then to the input of a T D C to provide the coincidence time relative to the strobe(Ge events).  3.4.4  Strobe  Logic  E v e r y time a Compton-free g a m m a event ( G e l - S ' A or Ge2-SB) either G e detector following a m u o n stop i n the target or T R I G 2 ) was generated.  ("/i"),  was detected b y  a good event ( T R I G l  T h e events from the two triggers, T R I G l or T R I G 2 ,  were t i m e d together a n d input into a quad logic fan-in/fan-out u n i t ( L e C r o y 429) denning the event " S T R O B E " , see F i g u r e 3.4. Several strobe output signals were used from this u n i t . One output was used to generate front-panel interrupts  2  to the Starburst a n d subsequently started the data  acquisitions. A least-delayed strobe output was sent to a fan-in unit k n o w n as the " I N H I B I T " unit; where a computer busy signal(generated b y O R T E C N D 0 2 7 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. The Starburst handler (TWOTRAN) responds to the Front-panel interrupts in half the time (~120 /xsec) it responds to a C A M A C Graded Look-at me,LAM (200-250 psec). 2  63  T h e inhibit signal was used as a veto into the T R I G l a n d T R I G 2 coincidences to stop further events while one was being processed by the d a t a acquisition. Once the C A M A C modules were read by the computer, the N I M driver released the inhibit and opened the electronics for the next event. O t h e r strobe signals were stretched by a d u a l gate generators(LeCroy 222) and used to gate three 11-bit A D C s together w i t h the two A D 4 1 3 A D C slots. Others were used to start five 8-channel T D C s ( L e C r o y 2228). A n o t h e r strobe output was sent to a gate generator from which a 2.5 fisec N I M output a n d a delayed signal were respectively used as m u o n gate a n d reset signal into the U B C R o u t e r box where the input was p r o v i d e d by the m u o n stop coincidence " / J " .  3.4.5  Data Acquisition Control T h e 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 a n d a P D P - 1 1 front-end-processor ( C E S 2180 Starburst, herein-after a n d -before, s i m p l y called Starburst). F i g u r e 3.6 represents a schematic flow of the data t h r o u g h the V D A C S system. T h e V A X computer allowed the user to control aspects of the experiment such as data logging and online m o n i t o r i n g together w i t h downloading the user-defined T W O T R A N p r o g r a m to the Starburst w h i c h i n t u r n 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 a n d the T W O T R A N p r o g r a m 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 a n d w r i t t e n to 8 m m tape: • Energy Signals of : the two G e detectors, the 24 N a l ( T l ) detectors, a n d the three scintillation counters.  64  CAMAC Hardware  'electronics' r  CAMAC Commands  Experimental Data  PDP-11 (TWOTRAN)  FEP  Buffered Data  Control Commands  Data Logging  Online Monitoring  ¥ /  N  Figure 3.6: A schematic flow of the d a t a t h r o u g h the D a t a A c q u i s i t i o n System.  65  •  Timing  Signals of : the 24 N a l ( T l ) detectors, the three scintillation counters,  the four router signals and the 8 G e 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 a d d i t i o n to the above information, the following C A M A C scaler values were recorded at the end of each r u n : S I , S2, S3, S1-S2, S1-S2-53, > " , Gel-SA, T R I G l , TRIG2, "STROBE", Nal SUM, C L O C K ,  66  CLOCK-INHIBIT.  Ge2~SB,  Chapter 4 Technical problems T h i s chapter describes some of the technical problems which were encountered i n the data analysis of the experiment. In a d d i t i o n , some relevant basic concepts w i l l be highlighted. B o t h of these w i l l help i n the understanding of the two subsequent chapters. E a c h subsection is likely to be rather brief, a n d sometimes slightly isolated, since a complete description of these topics is provided by excellent works, some of w h i c h are cited i n the bibliography.  4.1  1  Basic Interactions in 7-ray Detectors G a m m a - r a y s interact w i t h matter i n various ways. However, only three p r i m a r y  processes, namely photoelectric  absorption,  Compton scattering and pair  production,  have any real significance for gamma-ray detection. In a l l these processes, g a m m a rays transfer a l l or part of their energy to the detection m e d i u m by generating free electrons. It is these secondary electrons that cause the ionization w h i c h is actually measured i n the photon detectors. Before discussing this detection, we shall present a brief s u m m a r y of these basic processes p r o d u c i n g the secondary electrons. In the photoelectric  absorption  process, a gamma-ray ejects a b o u n d electron  a n d imparts to it a l l of its energy . T h e ejected photoelectron has a kinetic energy 1  A  1  minute amount of energy is imparted to the associated atom to conserve energy and  67  given by the incident gamma-ray energy, E , y  minus its b i n d i n g energy. A s a result  the electron w i l l create a vacancy i n an atomic shell causing the a t o m to de-excite, liberating the b i n d i n g energy i n the form of characteristic X - r a y s or A u g e r electrons. B o t h the photoelectron energy as well as its b i n d i n g 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 m e d i u m . T h i s fact makes the photoelectric absorption an ideal and preferred mode of interaction for the measurement of the original 7-ray energy. T h e p r o b a b i l i t y for photoelectric absorption is approximately p r o p o r t i o n a l to - ^ 3 - , where Z is the atomic number of the material. In the Compton free atomic electron  scattering  process, the incident gamma-ray scatters from a  transferring part of its energy ( £ " ) to the electron (E )  2  7  departing w i t h the rest, E'.  and  e  These energies depend on the angle, 8, of the scattered  g a m m a ray as follows:  EP  E = E-, — Es,i = E~ — & 1 + - ^ ( 1 - c o s 8) 7  where m c 0  2  7  7  (4.1) y  '  is the energy corresponding to the rest mass of the electron. Hence,  the amount of energy deposited i n the surrounding material extends from zero, corresponding to a grazing angle of 8 — 0°, to a m a x i m u m energy corresponding to a head-on collision angle of 8 = 180° ( C o m p t o n edge).  T h e p r o b a b i l i t y for  C o m p t o n 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 C o m p t o n cross section is approximately p r o p o r t i o n a l to J k For large g e r m a n i u m detectors, as m a n y as half the events i n the full energy peak are actually a C o m p t o n scattering followed by a separate photoelectron event. In the pair production  process, a g a m m a ray w i t h energy i n excess of 1.022  momentum. A good approximation for a loosely-bound outer atomic electron with typical orbital energy usually small compared to the incident gamma-ray energy 2  68  M e V disappears, creating an electron-positron pair sharing the excess energy as kinetic energy. T h e positron almost always reacts w i t h an electron i n the m e d i u m and annihilates w i t h the simultaneous emission of two 0.511 M e V photons.  The  amount of energy deposited i n the m e d i u m i n this process depends o n the subsequent absorption of b o t h , one or neither of the 0.511 M e V a n n i h i l a t i o n photons. Consequently, a discrete amount of energy E^, E~, — m c  2  0  or E^ — 2moc  2  correspond-  ing to photopeak, single-escape peak or double-escape peak, gets absorbed i n the m e d i u m respectively. T h e atomic cross section for pair p r o d u c t i o n increases w i t h increasing Z a n d E-y.  For a gamma-rays of 4 or 5 M e V i n a s m a l l detector these  three peaks are quite clear. In our detector the escape peaks are n o r m a l l y rejected because of the C o m p t o n suppressors. T h e pair p r o d u c t i o n process becomes dominant only for h i g h energy gammarays while the photoelectric absorption process is the dominant mode of interaction for 7-rays of relatively low energies. T h e C o m p t o n 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 m e d i u m i n w h i c h incident g a m m a rays have a reasonable p r o b a b i l i t y 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 w i t h its density a n d volume are i m p o r t a n t .  Since the cross section for the preferred mode of interaction, the  photoelectric absorption, varies approximately as Z elements w i t h h i g h atomic numbers.  4 - 5  , the trend is to incorporate  over Ge since Zj = 53 compared to Za  T h i s consideration favors N a l ( T l ) (or C s l ) e  = 32.  T h e 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 p a t h of the "secondary" electrons i n G e detectors a n d photoelectrons released by light emitted i n excited molecular states i n N a l ( T l ) detectors. T h e number of information carriers should be linearly p r o p o r t i o n a l to (and as large as possible for) a given incident r a d i a t i o n to m i n i m i z e the statistical c o n t r i b u t i o n to the energy resolution. T h e average energy required to produce an information carrier (photoelectron) i n N a l ( T l ) is at least two orders of magnitude greater t h a n that required to produce one (electron-hole) pair i n G e detectors . 3  D u e to their excellent energy resolution, G e detectors are now being used i n v i r t u a l l y a l l gamma-ray spectroscopy that involves complex energy spectra. T h e i r superior resolution is i l l u s t r a t e d i n F i g u r e 4.1 i n contrast to N a l ( T l ) scintillators. T h e G e energy spectra are complicated by the presence of prominent continua due to p a r t i a l escape of i n i t i a l gamma-ray energy from the active volume of the detector.  T o reduce such continua, C o m p t o n 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 a n d detector characteristics, the reader is referred to the two excellent textbooks by K n o l l [164] and D e b e r t i n a n d 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  G e detectors ought to be addressed. These are fast neutron damage a n d neutroninduced peaks, b o t h of w h i c h can affect the detectors' performance. It has been found that G e detectors are susceptible to r a d i a t i o n damage (see reference [164]).  Fast neutrons can knock-out g e r m a n i u m atoms p r o d u c i n g pre-  dominantly isolated defects w i t h i n the active volume of the hyperpure G e detector. 3  3 eV per electron-hole pair; 300-1000 eV per photoelectron.  70  Figure 4.1: C o m p a r i s o n 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 ) scintillators (upper curve). T h e prominent peaks at 662 k e V a n d 1173 k e V and 1332 k e V are from C s a n d C o sources respectively. T h e bumps at 460, 950 and 1100 k e V are the C o m p t o n edges w h i c h become bumps because of the suppressors. 1 3 7  71  6 0  These defects act as "hole" traps, thus increasing the v a r i a t i o n of the amount  and  time of charge collected per pulse. T h e former variation, incomplete charge collection, 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) a n d hence worsens the t i m i n g 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 i m p r o v e d damage hardness over the conventional ptype ones. T h i s finding was to some extent expected from geometry considerations. M o s t of the interactions occur i n the outer p o r t i o n (periphery) of the detector because of the geometry of the closed-end coaxial detectors. T h u s 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.  T h u s i n 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. T h e other effect on G e detectors i n 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 g e r m a n i u m isotopes i n the detectors, by inelastic neutron scattering, followed by emission of conversion electrons a n d X - r a y s w h i c h are totally absorbed w i t h i n the Ge crystals. T y p i c a l neutron-induced 7-rays i n G e at 596 k e V a n d 693 k e V are evident for the in-beam energy spectrum of F i g u r e 4.2. T h e skewness towards higher energies of these peaks happens because some of the nuclear recoil energy is converted into electron-hole pairs [168] that a d d to  72  ~~i I i i  4 0 0  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 600  8 0 0  Energy  1000  1200  (keV)  Figure 4.2: G a m m a - r a y spectrum from muons stopping i n a silicon target. T h e neutron induced 7-rays i n g e r m a n i u m are evident.  73  Table 4.1: N e u t r o n induced 7-ray lines i n g e r m a n i u m isotopes. Those i n brackets were not observed i n this experiment.  Ge isotope 70 72 73 74 76  N a t u r a l A b u n d a n c e (%) 20.5 27.4 7.8 36.5 7.8  (n,n') induced 7-rays (keV) 1039.6, 1215.6, (1708.2) 629.9, 691.5, 834.1, 1464.1 868.0 595.9, 608.4, 1204.3, (1482.6) 562.9, 1108.4, 1410.1, (1539.1)  the associated transition energy. ( U p to 30 or 40 k e V 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. T h e y have been extensively investigated earlier by C h a s m a n et a l . [169] a n d others [168,170]. A complete list of these neutron-induced lines is given i n Table 4.1 under their respective isotope. T h e presence of such spurious peaks can interfere w i t h the analysis of other nearby 7-rays of interest . In fact, it was p a r t l y these induced peaks, i n p a r t i c u l a r 4  the lines at 1204 k e V a n d 1216 k e V , w h i c h spoiled the first phase of this experiment and p r o m p t e d the modification of the experimental technique, discussed i n the next section.  4.3  Coincidence Technique A s mentioned earlier, the a i m of this experiment is to measure the angular  correlation between the neutrino a n d the 1229 k e V Doppler-broadened g a m m a ray in  2 8  A l following m u o n capture on  2 8  S i . T h i s a i m was pursued i n the first phase of  the experiment (December 1989) by the measurement of the line shape of the 1229 k e V peak i n the "singles" energy spectrum. U n l i k e pion capture where the responsible neutrons originate directly from the prompt absorption of the pion, one cannot use time-of-flight techniques to discriminate against the (n,n'7) peaks 4  74  J  10'  I  L.  _J  I  l_  1  1  1  _1  I  I  1  1  l_  So  O  Lr  10  1  1  1  1150  1  1  1200  F  1250  Energy F i g u r e 4.3:  1  1  1300  1  r  1350  (keV)  S i gamma-ray energy spectrum (binned by 4) obtained w i t h G e l detector, i n the vicinity of the Doppler-broadened 1229 k e V peak.  2 8  75  F i g u r e 4.3 shows a gamma-ray energy spectrum obtained w i t h a silicon target for the G e l detector, i n the v i c i n i t y of the 1229 k e V peak. A plateau-like structure below a n d possibly extending underneath the line of interest is evident. T h e overlap of this plateau w i t h the peak of interest made the e x t r a c t i o n of the angular correlat i o n from the line shape very problematic. T h i s plateau was clearly present i n our i n i t i a l r u n [160] as well as i n other previous experiments from other laboratories. One can t r y to fit the singles spectra, using various assumptions for the behaviour of the background underneath the 1229 k e V peak, to extract the angular correlation coefficient(s). However such an approach is very prone to systematic errors. Nevertheless, it m a y be warranted to do so as a complementary approach. T h i s concern was the p r i m a r y m o t i v a t i o n for the adoption of a coincidence technique. In a d d i t i o n to suppressing the c o n t i n u u m background underneath the 1229 k e V peak of interest, the coincidence technique gives i n f o r m a t i o n about potential problems due to cascade-feeding of the t r a n s i t i o n of interest, see section 5.4. F i g u r e 4.4 illustrates the coincidence technique. It is based o n the following: each of the 1229.1 k e V gamma-rays from the (2201.5 k e V 1+ —-> 972.4 k e V 0+) t r a n s i t i o n is followed by a 942.1 k e V gamma-ray from the (972.4 k e V 0  +  —> 30.6  k e V 2 ) transition. B y detecting these 942 k e V gammas i n the N a l crystals sur+  r o u n d i n g the silicon target , the 1229 k e V line observed i n the two H P G e detectors 5  can be "tagged". T h e effect of this m e t h o d is to preferentially select the 1229 k e V g a m m a rays that are i n coincidence w i t h 942 k e V gammas, i.e. those that actually arise from the reaction of interest. T w o relevant facts s u p p o r t i n g the above coincidence scenario s h o u l d be highlighted.  B o t h stem from the nature of the intermediary level (972.4 k e V , 0 , 43 +  ps). in muon capture 7-ray spectra. Typical 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. 5  76  0  +  0  F i g u r e 4.4: T h e p r o d u c t i o n of the 1229 k e V g a m m a ray i n coincidence technique.  • F i r s t , its spin-parity of 0  +  2 8  A 1 and the  leads to an isotopic d i s t r i b u t i o n of the 942 k e V  line w i t h respect to the 1229 k e V g a m m a ray. T h u s 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 k e V line shape. T h i s allows a relatively tight coincidence windows around the 942 k e V tagging-line i n the N a l energy spectra.  F i g u r e 4.5 shows t y p i c a l S i gamma-ray energy spectrum before and after the 2 8  imposition of the energy-gated coincidence requirement w i t h the N a l crystals. conveys the crux of the coincidence method.  It  L i t t l e more needs to be said other  than emphasizing the complete suppression of the 7-rays that are not associated w i t h genuine cascades, such as the 1294 k e V air activation 7-ray line from the decay  77  20000 o)  Singles  1  1  3  2  keV  Mg  15000  c o o  1294 k e V  AtFe^P-OS)  Ar  10000  5000  1050  1100  1150  1200  1300  1250 Energy  1350  (keV)  1000  800  b) Coincidence  600-  1250 Energy ( k e V )  Figure 4.5:  S i gamma-ray energy spectra, i n the v i c i n i t y of the 1229 k e V peak, before a n d after the i m p o s i t i o n of the energy-gated coincidence requirement w i t h the N a l crystals.  2 8  78  1350  of  4 1  A r , and muonic X - r a y lines. O n the other h a n d , real coincidences from 7-7 6  cascades following m u o n capture on  2 8  S i (or even background) were expected and  are seen i n the figure, e.g. the 1229 k e V  2 8  A l a n d the 1132 k e V  2 6  M g lines. (The  particle h i t t i n g the N a l can be a neutron as well.) The reduction i n 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 i n statistical error) was more t h a n 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 r u n i n a "singles" mode w i t h the coincidence requirement applied later i n the software rather t h a n at the hardware stage. T h e advantage of this was the possibility of analysing the singles spectrum w i t h its h i g h statistics and thus p r o v i d i n g a supplementary measurement to that of the coincidence spectrum. T h i s topic w i l l be revisited i n chapter 5.  Other related topics deferred to  that chapter include: the remaining flat background i n the coincidence spectra, the photopeak efficiency of the N a l a r m , along w i t h 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. A s discussed i n section 4.1, the preferred mode of interaction, the photoelectric absorption, is not the only process c o n t r i b u t i n g to the pulse-height spectra i n G e detectors, nor does it always yield pulses corresponding to full-energy peaks. 6  E.g.  40  A r ( l % in air) + n - *  4 1  Ar(1.8 hrs)  79  /T +  41  K*(1294 keV).  (4.2)  Furthermore, photon interactions i n the detector surroundings may a d d to the pulseheight spectra; see section 5.2 for their suppression. T h e net effect of the different interaction processes i n a n d around the G e detectors is a complex response function for a mono-energetic 7-ray as well as a complex pulse-height spectrum. T h e full energy photopeaks, C o m p t o n edges along w i t h flat c o n t i n u u m as well as single- and double-escape peaks m a y contribute to such spectrum, see for example F i g u r e 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 i m p l y called the peak, a n d the "background" i n its vicinity. F i g u r e 4.6 shows the 1332 k e V C o peak, extracted from F i g u r e 4.1, together 6 0  w i t h various components of one of the complex a n a l y t i c a l line shapes [171] to illustrate the correspondent contributions to the peak. T h e labelled components i n the figure are as follows: 1. Flat continuum w h i c h is due to the C o m p t o n scattering, bremsstrahlung losses and escaping photoelectrons from the sensitive volume of the detector for higher energy gamma-rays. 2. Step-like component produced m a i n l y by the escape of p r i m a r y 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, w h i c h is due to t r a p p i n g effects i n a d d i t i o n to incomplete charge collection. 5. High-energy tail, w h i c h is usually due to pile-up effects a n d 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  T—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—r~i—r ^—i—r -  1 5 6 0  1 5 8 0  1 6 0 0  1 6 2 0  1 6 4 0  I  I  I  I  I  I  I  I  I  I  1 6 6 0  C o peak.  81  I  L  1 6 8 0 1 7 0 0  F i g u r e 4.6: V a r i o u s components of the response function, R F 3 , fitted to 6 0  I  i—i—|—i—i—i—i—|—i—i—i—i—|—i—r  Channel  the 1332 k e V  I  Table 4.2: Q u a l i t y of fit to the 1332 k e V C o peak for different response functions. 6 0  peak-shape function RFl:Gaussian R F 2 : G a u s s i a n w i t h joined tails [152] R F 3 : G a u s s i a n w i t h tails [171]  # of parameters 3 5 9  x 817.7 313.7 197.6 2  X 5.33 2.03 1.32 2  R  In addition, an i m p r o p e r l y adjusted pole-zero i n the m a i n amplifier a n d / o r i n sufficient base-line restoration can contribute to the low- a n d / o r high-energy t a i l components. T h e first two components are usually grouped together under the name "background" while the last three components are labeled as the "peak". A variety of response functions have been t r i e d out. It was found that the best representation of the background components consists of a constant representing the flat c o n t i n u u m a n d a gaussian-convoluted step-function given by the complementary error function [171]: | e r f c [ ( x - X )/y/2a]  (4.3)  0  w i t h amplitude S, gaussian w i d t h a, channel number x, and peak center XQ. T h e peak-shape function representing the peak components fall into two categories, those w h i c h retain the m a i n gaussian t e r m w i t h smoothly added tails on b o t h sides, a n d those w h i c h are a s u m of several functional terms. T w o peak functions, along w i t h the gaussian function, corresponding to the above two categories are given i n Table 4.2 showing the quality-of-fit to the 1332 k e V C o peak. Note 6 0  that for such peak w i t h large number of counts,  1 . 5 x l 0 , a n d large fitting range 5  of channels, one needs to a d d i n more tailing terms to b r i n g the reduced chi-square value, x p,i closer to 1.0. 2  T h e simple gaussian representation is not a sufficiently accurate representation. T h e complexity of the t h i r d function a n d 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. T h e 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 function of the response function of our detectors. It is represented by P [P +2(x-X )}/2a  e  L  L  2  0  -(x-X f/2a  e  2  0  x - X  R  R  <  - P L < X - X  P [P -2(x-X )]/2a  e  0  0  2  X - X  0  >  -P 0  >  L  PR  PR  where PL a n d P R are the j o i n i n g points of the left a n d right tails measured from the center of the peak(gaussian) respectively. T h e parameterization of the response function was carried out as follows: first, the step function amplitude, 5 , was estimated w i t h the help of c a l i b r a t i o n sources (  1 3 7  C s and  6 0  C o ) . T h e n fixing S at its estimated values, the other parameters, 7  cr, PL a n d P R were determined by the use of in-beam strong peaks through-out the energy spectra.  F o r these results, analytical expressions giving these parameters  as a function of the 7-ray energy were determined. A s an illustration, the p a r a m eterization of these parameters for G e l is shown i n F i g u r e 4.7. A consistent a n d reasonable set of values is obtained throughout the energy range of interest for b o t h 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 F i g u r e 4.7 for G e l a n d a similar set of curves for Ge2, see section 5.6. 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%. 7  83  F i g u r e 4.7: Energy dependence of the parameters of the response function, R F 2 for the G e l detector. T o convert to k e V , these parameters should be m u l t i p l i e d by 0.36 k e V / c h a n n e l .  84  4.5  Slowing-Down Effects T h e Doppler-broadened 7-rays of interest are emitted from nuclei recoiling  i n matter, and hence their energies are shifted depending on the  instantaneous  velocities of the nuclei at the time of emission. T h e nuclei are slowed down by the m e d i u m , and so the D o p p l e r 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 k n o w n . In the case of the nuclear level of interest, i.e.  28  A1(2201 k e V , 1 ) , there have +  been two previous attempts to measure its lifetime. B o t h measurements utilized the Doppler-shift attenuation m e t h o d ( D S A M ) as applied to the  2 7  A l ( d , p ) reaction, by  measuring the D o p p l e r shift of the 7-rays emitted i n coincidence w i t h the protons. T h e reported values are (35 ± 10)fs [172] and (120 ± 70)fs [173], w h i c h is confusing though not totally inconsistent. A major contribution to the uncertainty i n 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.  T h e total stopping powers may be d i v i d e d into two parts:  the  electronic stopping power and the "nuclear" stopping power. T h e y give the energy loss due to the interaction of ions w i t h the target electrons and w i t h the target nuclei respectively . T h e former dominates at h i g h i o n energies (above 100 k e V ) , 8  the latter at low energies.  F i g u r e 4.8 shows the two parts as calculated by the  Biersack and Ziegler code(1989) [175] for our reaction i n a Si nat  2 8  S i ( ^ , ^ ) A 1 , i.e 2 8  2 8  A l ions  m e d i u m . T h e i n i t i a l recoil energy for the 2201 k e V level is 187 k e V and  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]. s  85  J  3  I  I  L  J  I  l_  I  J  I  I  J  l _  I  |_  V  CP  2  >  CD  x Lcgcmd TRIM ZB total ZB nuclear ZB electronic  0  T  0  50  100  150  Energy  (keV)  F i g u r e 4.8: Stopping-power curves for A l ions i n Si m e d i u m . T h e arrow corresponds to the i n i t i a l recoil velocity, /?o=0.0037, ?.e. 187 k e V . 2 8  nat  hence one must take b o t h parts into account. See reference [176] for a comprehensive treatment of this topic. T h e slowing-down effects, along w i t h the lack of a reasonable measurement of the level lifetime, complicate the extraction of the angular correlation coefBcient(s) from the observed 1229 k e V 7-ray. T h 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. O u r analysis indicates that the slowingdown effect is of central importance a n d cannot be neglected.  86  T h e energy-loss  200  _i  120  100  i  i  i  i  i  i  1  1  1  1  1  1  i  i  1  r  i  i  i  i  i  i  i  i_  -  80  ^  60  4 0 -  20  1  1  1.0  1  1  1  1.5  1  1  1  1  1  1  1  1  2.0  2.5  3.5  3.0  4.0  dE/clX ( M e V / m g / c m ) F i g u r e 4.9: Relationship between the stopping-power and the extracted lifetime from the 2171 k e V 7-ray line. R o u g h l y 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 M o n t e C a r l o code T R I M ( T R a n s p o r t of Ions i n M a t t e r ) [175], as can be seen i n F i g u r e 4.8. stopping-power,  Different shapes of the  were used to fit this function, but the details do not matter.  T h i s topic w i l l be further discussed i n section 5.6 i n conjunction w i t h the lifetime measurement of the 2201 k e V  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 i o n does not slow down very m u c h . A l s o , the absolute value of the  directly affects the value of the lifetime w h i c h is extracted(see F i g u r e 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 a n d / o r  the solid angle of the G e detector on the measured angular correlation coefficient, a, i.e. to investigate the sensitivity of the angular correlation to the other correlation coefficients, f3\ a n d f3\. A s noted earlier, i n this experiment, the 7 rays are detected at 90° to the fi p o l a r i z a t i o n direction and hence f3\ a n d (3 terms, involving 2  (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 w h i c h (p. • 7) ^ 0 due to solid-angle effects. To check such effects, we used a M o n t e C a r l o p r o g r a m to generate the Dopplerbroadened spectrum of the 1229 k e V 7-ray of interest for various values of a , f3\, 02 and the angle 9^ (i.e.  the angle that ideally is 90°). F o r 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, a n d that they a l 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), a n d the angle varies by ± 2 5 ° over the detector. T h i s is for G e l , and the effect is smaller for Ge2. T h e resulting spectra were then fit using M I N U I T , w i t h /3\, f3 a n d cos# 2  71/  a l l fixed i n the  fit to 0, i.e. the ideal (not the actual) values. C o m p a r i n g the fitted values of " a " to the actual value used i n generating the spectrum showed the following: i n the above extreme case (and w i t h (3\ a n d L3 b o t h at their m a x i m u m allowed values), 2  the extracted a is reduced by only 10% from its actual value. F o r 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\ a n d B on the 2  88  spectrum, i.e. the chi-squared of the fit gets a bit worse w i t h systematic deviations between the fit a n d the data, but the extracted value of the correlation coefficient of interest, a , is almost completely unaffected. T h i s result allowed the neglect of the other correlation coefficients i n our  fitting  function. Furthermore, it eliminated any complications i n the analysis due to the residual m u o n spin p o l a r i z a t i o n 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 p r o g r a m w i t h user-written subroutines reads the V D A C S - f o r m a t t e d tapes a n d sets up various energy a n d time histograms. These histograms were further examined, analyzed a n d 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 a n d P H Y S I C A were very useful for m a n i p u l a t i o n , a n d p l o t t i n g of data.  fitting  E x p e r i m e n t a l data of the Doppler-broadened gamma-rays  of interest were fit to a functional form representing the (double) convolutions of the response f u n c t i o n ( R F ) , angular correlation f u n c t i o n ( A C ) a n d the slowing-down function(SD) given i n sections 4.4, 2.5 a n d 4.5 respectively. The convolutions h a d to be carried out numerically. T w o quadrature methods were tried: Simpson's composite rule a n d the I M S L D T W O D Q routine. T h e y gave consistent results. T h e latter, an adaptive quadrature scheme, was adopted  9  The best values of the u n k n o w n parameters along w i t h their uncertainties a n d correlations are obtained by m i n i m i z i n g the difference(chi-square) between the (theoretical) functional form a n d the experimental data. T h i s was carried out w i t h Simpson'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. 9  89  the use of the C E R N M I N U I T package/programme. T h e various input parameters needed i n the least-squares fitting p r o g r a m w i l l be summarized i n section 5.6.  90  Chapter 5 Data Analysis 5.1  Introduction T h e principle goal of the data analysis i n this work is the extraction of  the angular correlation coefficient ( A C C ) from the 1229 k e V Doppler-broadened gamma-ray i n  2 8  A l following m u o n capture on  2 8  Si.  T h i s task is accomplished i n  two m a i n stages: the o p t i m i z a t i o n of the energy and time gamma-ray spectra and the analysis of the principle Doppler-broadened gamma-ray lines of interest. T h e former aims at producing "clean" G e energy spectra appropriate for the line shape analysis. T h i s is achieved by i m p o s i n g conditions (or cuts) on the d a t a (5.2) as well as doing side-band background subtraction (5.5). T h e second stage, the line shape analysis, includes the estimation of the lifetime of the c o m m o n level (2201 k e V , 1+,  2 8  A l ) from the 2171 k e V line (5.7.1) and the extraction of the angular  correlation coefficient from the 1229 k e V line (5.7.2). O t h e r topics include cascade feeding (section 5.4) and 7-ray efficiency (section 5.3). T h e off-line analysis of the data was carried out using the T R I U M F D a t a A n a l ysis Centre V A X cluster of computers, m a i n l y two V A X 4000/100's named E R I C H and R E G along w i t h several Vaxstations. Various computer programs were used for different aspects of the d a t a analysis. In the first step i n the d a t a analysis, the sorting analysis p r o g r a m version of D I S P L A Y [177] w i t h user-written subroutines  91  read the V D A C S - f o r m a t t e d tapes a n d set up various energy a n d time histograms. These histograms were further examined, analyzed a n d 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 m a n i p u l a t i o n , fitting and p l o t t i n g of data.  5.2  Cuts T h i s section outlines the different conditions imposed on the data to select  events. T h e choice of what cuts to use, a n d how restrictive to make them, was to some extent a matter of judgement since, i n 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 G e detectors can be divided into three categories depending  u p o n their time relative to the m u o n stopping i n the target, i.e. stop pulse (see /3.4.1).  the last m u o n  T h e first category contains the G e signals w h i c h are i n  prompt coincidence w i t h a stopped m u o n and constitute essentially only X - r a y s from the muonic cascade. T h e corresponding energy spectra are called the "prompt" spectra. "Delayed" spectra, recorded w i t h i n the m u o n lifetime i n the relevant target preferentially contain delayed 7 events, resulting from nuclear m u o n capture i n the target. T h e t h i r d category constitutes the "background" spectra. It contains the Ge signals which are not related i n time to a m u o n stop. A typical plot of M U S T O P t i m i n g d i s t r i b u t i o n is shown i n F i g u r e 5.1.  It is  characterized by the above three categories: a large spike due to prompt muonic X rays, a long decaying exponential t a i l w i t h a slope characteristic of the m u o n disappearance lifetime i n the Si target (756 ns) and a flat r a n d o m background. B y choosing different time windows on the M U S T O P spectrum, energy spectra were reconstructed offline to reduce b a c k g r o u n d and increase the signal to noise ratio as 92  400  600  800  1000  Channel  1200  1400  number  F i g u r e 5.1: T i m e of the m u o n spectrum for G e l w i t h a Si target. T h e dispersion is about 2.67 n s / c h (time goes i n the "backwards" direction). T h e periodic ripple is due to the differential nonlinearity of the 20 M H z T D C used while the dip (kink) around the p r o m p t peak is due to the suppressor vetoing some events.  93  I I I  I  I  I  500  I  I  I  I  1  I  1  I  I  I  I  I  I  I  I  I  I  I  I  I  I  1000 1500 Gamma Ray Energy (keV)  I  I  I  I  I  I  1—L  2000  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 i n 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 hardwired 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  _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 _  10  > CD  C Z5  o o 10 -  icr  I j I ! I 1I I I I I | I I I I I I I I 1| I I 500 1000 1500 Gamma Ray Energy (keV)  2000  F i g u r e 5.3: D a t a removed by the N a l ( T l ) C o m p t o n suppressor from the previous figure.  400  800 1200 1600 Gamma Ray Energy (keV)  2000  Figure 5.4: C o m p t o n suppressed and unsuppressed C o spectrum of G e l . Note that the C o m p t o n edges become b u m p s i n the suppressed spectrum. (There is a discriminator cut off below 500 k e V . ) 6 0  95  demonstrated i n F i g u r e 5.3, w h i c h shows the data that was actually removed from the unsuppressed spectrum b y the N a l ( T l ) suppressor. Note however that the 511 k e V peak due to fi a n n i h i l a t i o n is prominent because it comes i n pairs. A n o t h e r +  effect is that the C o m p t o n edges become broad peaks i n the suppressed spectra; this c a n be seen more clearly i n F i g u r e 5.4. T h i s is due to gamma-rays scattered back out of the G e entrance holes i n the suppressors, a n d hence not detected i n the respective N a l annulus ( C o m p t o n edge events correspond to incident 7 rays being backscattered toward their direction of origin, i.e. head-on C o m p t o n scattering w i t h 6 = 7r). T h e C o m p t o n suppression factor is usually defined as the ratio of unsuppressed to suppressed data i n the c o n t i n u u m . T h e average factor obtained for a t y p i c a l inbeam S i spectrum (e.g. F i g u r e 5.2), i n the vicinity of the 1229 k e V 7-ray, is around 5.7 a n d 8.4 i n G e l a n d Ge2 respectively.  5.2.3  Rise Time Correction It is well k n o w n that the amount a n d time of the charge collected i n a G e  detector are not constant b u t depend o n the gamma-ray interaction position i n the detector, presence of defects i n the G e lattice, a n d non-uniformities i n the electric field i n the crystal as well as the detector geometry [164]. These variations produce variable height a n d shape i n the pulse for monoenergetic 7-rays i n G e detectors a n d hence limit their energy a n d time resolutions respectively. T h i s means that the time for these pulses to rise to any given voltage w o u l d be different. In this experiment, besides the use of two constant-fraction discriminators-one w i t h low-threshold defining the time a n d the other w i t h high-threshold establishing energy cutoff- to reduce the dependence of the pickup time o n the amplitude a n d rise time of the input pulses, we have used a leading edge (rise-time) method to 96  Chonnel number  Channel number  F i g u r e 5.5: G e l leading edge spectra corresponding to different d i s c r i m i nator thresholds for a t y p i c a l / / S i r u n .  measure the rise time of the pulses, i n order to compensate for these variations. T h i s is based on the following: 1. T h e G e pulses are split into four identical pulses using a linear fan-out, and sent to 4 leading-edge discriminators w i t h different threshold levels. T h e output pulses are then sent to stop T D C clocks, the start for these T D C s being the constant fraction t i m i n g signal from the germanium; see the block diagram of the electronic arrangement i n F i g u r e 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  a) Ge2 LE1  20-.  C D15 H CJ ~c 10-i oo 35 oc _ Ui  1  100  100  200 Channel  200 Channel  300 number  400  500  300 number  100  200 Channel  300 number  500  200 Channel  300 number  500  Figure 5.6: Ge2 leading edge spectra corresponding to different d i s c r i m i nator thresholds for typical yuSi r u n .  98  2. T h e L E spectra were subtracted from each other to find the most appropriate rise-time spectra. These were L E 3 - 1 a n d L E 4 - 1 for G e l a n d Ge2 respectively. ( G e l L E 4 threshold was set too h i g h for our peak of interest). 3. T h e Ge energy and N a l - G e t i m i n g coincidence spectra were then sorted by setting 10 windows (bins) on the rise-time spectra L E 3 - 1 a n d L E 4 - 1 obtained above. T o find the correlation between the energy spectra and the rise-time spectra, the centroids of several 7 - r a y lines were determined a n d 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. T h i s factor was then applied to the energy spectra for each pulse w h i c h h a d a non-standard rise time. T h i s was done for each Ge detector. F i g u r e 5.7 a n d F i g u r e 5.8 show this correlation for the 1173 k e V  6 0  C o 7 - r a y ( / i beam on) along w i t h the rise-time corrected d a t a for the  G e l and Ge2 detectors respectively. E n e r g y resolution improvements of up to 100 e V F W H M were obtained for the region of interest. A n even more useful effect was the time resolution improvement obtained v i a the correlation between the centroids of the t i m i n g coincidence spectra a n d the rise-time spectra. T h i s correlation was found i n the same way as above to create a "corrected" t i m i n g spectra. T h i s correlation relation was then a p p l i e d to the time spectrum of every N a l for a specific Ge detector, since the correlation is a function only of the G e signal. T h e significance of this correction m e t h o d can be seen i n F i g u r e 5.9, w h i c h shows t y p i c a l t i m i n g spectra w i t h a n d w i t h o u t the rise-time correction. T h e corrected spectra are more s y m m e t r i c a n d have better resolutions as well as better signal-to-background ratios. T h e low t a i l on the Ge2 uncorrected spectrum is m a i n l y due to low-energy g a m m a rays, such as the 400 k e V fiK  a  99  i n S i , w h i c h h a d pulses  1604 -_ "D  0  1  o  1602  > cu  - 1600 Legend  1598  o  Raw  data  •  LEC  data  -  Fit t o Fit  100  200  raw d a t a  to L E C d a t a n  300  1  r~  400  500  Ge2 L E 4 - 1 C e n t r o i d s  F i g u r e 5.7: P l o t of the corrected a n d uncorrected centroid channel of the 1173 k e V 7 rays as a function of their rise-time for the G e l detector.  1604 cn  S 1602 > cu  -i  1600 -_  1598 -  o  Row  data  •  LEC  data  -  Fit to Fit  -j  100  200  !  raw d a t a  to L E C data  j _  T  300  400  500  Ge2 L E 4 - 1 C e n t r o i d s  F i g u r e 5.8: P l o t of the corrected and non-corrected centroid channel of the 1173 k e V 7 rays as a function of their rise-time for the Ge2 detector. 100  Channel number  Channel number  F i g u r e 5.9: T y p i c a l t i m i n g coincident spectra for one of the N a l ( T l ) detectors, B A R 4 . c) and d) are the rise-time corrected spectra for a) a n d b) for G e l and Ge2 respectively.  too small to fire the h i g h Ge2 L E 4 discriminator and consequently they w o u l d not be i n the "corrected" spectra. T h e apparent loss of counts is not a p r o b l e m , since it only effects low-energy events i n the G e detectors, but not the 1229 k e V signal a n d its vicinity. Average improvements i n time resolutions obtained were 1.5 ns a n d 2.8 ns at F W H M (or 10% a n d 20% ) for G e l a n d 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. T h e effect of this correction is to allow for relatively tighter coincidence w i n dows around the G e - N a l time coincidence peak a n d hence it reduces the r a n d o m 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 p r o m p t peaks to select G e events w h i c h were i n coincidence w i t h N a l ( T l ) crystals, see F i g u r e 5.9. Since the two G e detectors, defining the strobe, were out-of-time w i t h each other, a window on each N a l ( T l ) h a d to be set i n d i v i d u a l l y depending on w h i c h G e fired. T h e coincidence windows were around 3 a F W H M wide. In a d d i t i o n to the 12 N a l ( T l ) crystals used i n coincidence w i t h the two Ge detectors, we made use of the two N a l ( T l ) a n n u l i as secondary coincidence detectors. F i g u r e 5.10 shows a typical time-coincidence spectrum for one of the N a l ( T l ) annulus segments, S B 3 , i l l u s t r a t i n g the double use of the N a l ( T l ) annuli. T h e suppressed a n d otherwise huge peak ( t S B 3 . G e 2 ) is composed of the C o m p t o n events i n coincidence w i t h the surrounded Ge-detector operated i n the anti-coincidence mode while the central peak ( t S B 3 . G e l ) corresponds to events when the other (opposing) G e set the trigger. Hence to increase the tagging coincidence-efficiency of the N a l ( T l ) a r m , 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 a d d i t i o n to the requirement that the opposing G e detector is the trigger a n d not the other. G a m m a - r a y lines w h i c h are not associated w i t h real coincidences, such as the air activation line from the decay of  4 1  A r at 1294 k e V , are suppressed after the N a l  time-coincidence requirement is applied.  5.2.5  Energy—Gated Coincidence  T h e coincidence technique was described i n section 4.3. T h e 942 k e V gammarays are selected by p u t t i n g 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  tSB3-Ge2  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  1  i i i i i i i i ! | i" i" "i""r1 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 1  0  100  200  Channel  300  400  number  Figure 5.10: Time-coincidence spectrum for N a l ( T l ) annulus segment, S 3 B . Note the two peaks resulting from operating the segment i n coincidence and anti-coincidence mode w i t h the opposing and surrounded G e detector respectively.  103  500  each of the 24 N a l crystals whose A D C spectra h a d to be calibrated beforehand. D u e 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 ( F i g u r e 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 fitting the peak positions of the two  6 0  C o 7-rays i n the in-beam source runs to a dual-gaussian  function. A s an illustrative example, this dual fit to these lines of one of the N a l detectors, B A R 3 , is shown on F i g u r e 5.11. T h i s c a l i b r a t i o n gave rough estimates of the peak positions of the 'tagging' 7 ray of interest, along w i t h estimates of their widths i n the N a l spectra.  T h e reason for not using the better resolved spectra  from the source-only c a l i b r a t i o n runs was due to the rate dependence effects; see for example, F i g u r e 5.11. In the second method, the coincidence technique was used to 'tag' the 942 k e V instead of the 1229 k e V g a m m a rays (as is done i n section 5.4), i.e. by setting gates around the 1229 k e V 7-ray line i n the two G e detectors a n d consequently observing the 942 k e V 7-ray line i n the N a l detectors (see F i g u r e 5.17b). T h e two methods gave consistent results for most of the runs and most of the N a l crystals, w i t h the latter technique being the one w h i c h was u l t i m a t e l y used. T h e energy coincidence cut yields a signal-to-noise ratio at the 1229 k e V peak of about 4.0 (4.8) , w h i c h is a factor of 17 (20) better for G e l (Ge2) t h a n 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 k e V centroids. T h e use of the corresponding G e 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 n u m b e r  F i g u r e 5.11: A dual-peak fit to the 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 a n d the gain. 6 0  105  Table 5.1: Acceptances of the N a l ( T l ) coverage for the two Ge detectors.  Line 1173 k e V 1332 k e V  singles 108630±713 104609±720  Gel coincidence  eAft (%)  16749±296 15009±287  15.4±0.4 14.3±0.4  singles 86736±659 77384±625  Ge2 coincidence 11644±241 9434±212  e A O (%) 13.4±0.4 12.2±0.4  on the scintillators' T D C s a n d A D C s were utilized at some stages of the analysis, p r i n c i p a l l y at the singles stage. However, their overall effect was m i n o r 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 g a m m a rays from a  6 0  C o source  were used. T h i s was done by measuring the ratio of counts i n the 1173 k e V a n d 1332 k e V peaks i n the Ge coincidence a n d G e singles 7-ray spectra, i.e. w i t h a n d without an energy window imposed on the 1332 k e V a n d 1173 k e V full-energy peaks i n the N a l spectra respectively. T h e results are tabulated i n Table 5.1. Hence we adopt ( 1 4 . 9 ± 0 . 4 ) % a n d ( 1 2 . 8 ± 0 . 4 ) % as the N a l ( T l ) acceptances for the G e l a n d Ge2 detectors respectively. In a d d i t i o n , the N a l ( T l ) acceptances were calculated using the 942 k e V  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, i n good agreement w i t h the adopted values. T h e 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 G e detectors (energy dependent), several muonic X - r a y spectra from various targets were used. T h e acceptance for a 7-ray  106  is simply defined by the formula :  where i V is the number of X - r a y s detected, JV is the number of associated stopped 7  M  muons and Yy is the yield of the X - r a y per stopped m u o n . T h e yields, Yy, were obtained from Vogel [178], H a r t m a n n et al. [179] and v o n E g i d y et al. [180]. T h e number of X - r a y s detected was determined, from the area (Area) of the X - r a y energy peak, as follows: N^ = Areaf f,f . a  The quantity f  a  (5.2)  v  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 H u b b l e [181], viz. f  = e , where x was taken to be 1/2 the +tlx  a  target thickness. T h e second factor (//) is the lost-strobe correction factor, a n d takes account of the fact that although the strobe scaler begins counting immediately at the start of each r u n , the events are not w r i t t e n to tape u n t i l the I / O channel is set-up. Consequently some events are lost. T h i s factor is obtained by c o m p a r i n g the number of events on tape to the number of strobes. T h e t h i r d 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 m u o n i c X - r a y , 7-ray, neutron or electron firing the associated N a l C o m p t o n suppressor, thus rejecting the event. T h i s effect is measured by counting coincidences between opposing G e - N a l suppressor pairs. E q u a l l y well, to arrive at N^, the total number of m u o n stops (S1-S2-S3), had to be corrected for computer processing dead-time (fd)-  T h i s correction was  determined by the ratio of the livetime clock scaler (CLOCK-INHIBIT)  a n d the  free r u n n i n g clock ( C L O C K ) . Furthermore, d u r i n g the course of the experiment (run 144), it was discovered that the veto scintillator 5 3 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  952 1094 1255 1425 1510 1522 1780 2178 2547 3450  Pb(4f->3d)  Y,  Area  fa  U  f/  1-2-3 (10 ) 5.169 1.973 10.01 3.107 0.722 10.01 3.184 3.184 5.169 0.709  U  fs3  8  Cr(2p^ls) Fe(2p^ls) Ni(2p^ls) Cu(2p^ls) Fe(3p^ls) Ge(2p-*ls) Ge(3p—>ls)  Pb(3d^2p) Sn(2p-+ls)  0.684 0.715 0.716 0.740 0.783 0.082 0.853 0.066 0.802 0.921  56557 1.10 35443 1.20 129669 1.10 52498 1.11 10530 1.04 15523 1.11 47239 1.10 3015 1.10 40004 1.05 5211 1.03  1.13 1.05 1.05 1.04 1.07 1.02 1.08 1.05 1.14 1.09  1.09 1.03 1.03 1.07 1.15 1.03 1.06 1.06 1.09 1.03  (IO- ) 4  0.80 0.8 2.71±0.19 0.76 1.0 3.26±0.25 0.73 0.8 2.69±0.12 0.68 1.0 2.82±0.25 0.84 1.0 2.38±0.30 0.73 0.8 2.77±0.23 0.79 1.0 2.19±0.15 0.79 1.0 1.76±0.35 0.80 0.8 1.57±0.12 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 a n d after the 5 3 replacement for the same target and same m u o n flux. Hence, i n c o r p o r a t i n g the above correction factors, the acceptance is given by AO e A i l  A  r  e  a  f f> f»  (*<i\  a  = (si.52.33)/</»y • 7  ( 5  '  3 )  Tables 5.2 and 5.3 show the acceptances, along w i t h the aforementioned corrections, calculated from the various m u o n i c X - r a y s for the G e l a n d Ge2 detectors respectively. Figures 5.12 a n d 5.13 show the corresponding acceptance plots as a function of energy (E),  w i t h fits of the form :  where a a n d b are constants determined i n the fits to the data; and are given i n the figures 5.12 a n d 5.13. W e attribute the inconsistencies to the uncertainties i n the definition of a m u o n stop a n d the variation of the sizes a n d shapes of the targets as well as the different energy thresholds of the N a l detectors. These unfortunately 108  Table 5.3: M u o n i c X - r a y acceptance data for the G e 2 detector. E (keV) 347 413 436 447 453 952 1094 1255 1425 1510 1522 1780 2178 2547 3450  /iX-ray  Area  Y,  fa  f„  f/  1-2-3 (10 ) 11.04 11.04 11.04 11.04 11.04 5.169 1.973 10.01 3.107 0.722 10.01 3.184 3.184 5.169 0.709  U  fs3  8  Al(2p-»ls) Al(3p->ls) Al(4p-»ls) Al(5p-»ls) Al(6p->ls) Pb(4f-+3d) Cr(2p—>ls) Fe(2p^ls) Ni(2p->ls) Cu(2p^ls) Fe(3p^ls) Ge(2p^ls) Ge(3p—>ls) Pb(3d^2p) Sn(2p—>ls)  0.796 0.079 0.049 0.041 0.026 0.684  469051 42790 25521 21131 11573 41598 39912 91501 52498 7392 9916 34839 2270 19902 2543  0.715 0.716 0.740 0.783 0.082 0.853 0.066 0.802 0.921  1.14 1.14 1.13 1.13 1.13 1.10 1.20 1.10 1.11 1.04 1.11 1.10 1.10 1.05 1.03  1.04 1.04 1.04 1.04 1.04 1.44 1.17 1.16 1.16 1.23 1.09 1.29 1.22 1.43 1.27  1.09 1.09 1.09 1.09 1.09 1.09 1.03 1.03 1.07 1.15 1.03 1.06 1.06 1.09 1.03  eAtt (lO" ) 6.89±0.12 6.34±0.30 6.04±0.42 4  0.68 0.68 0.68 0.68 0.68 0.80 0.76 0.73 0.68 0.84 0.73 0.79 0.79 0.80 0.85  0.8 0.8 0.8 0.8 0.8 0.8 1.0 0.8 1.0 1.0 0.8 1.0 1.0 0.8 1.0  5.98±0.45 5.16±0.68 2.54±0.22 4.09±0.31 2.10±0.16 3.14±0.29 1.92±0.25 1.89±0.22 1.93±0.17 1.54±0.35 0.98±0.16 0.52±0.35  were not considered priorities d u r i n g the r u n . W e have checked w i t h an efficiency curve i n the E G & G O R T E C m a n u a l (page 474) and has a s i m i l a r energy dependence to ours.  5.4  Cascade Feeding One mechanism that could distort the 1229 7-ray, a n d thereby spoil the inter-  pretation of the Doppler-broadened lineshape measurement i n terms of gp, is the indirect population of the 2201 k e V A 1 level of interest from higher-lying excited 2 8  states. ( T h e neutron threshold for A 1 is quite high at 7725.18 k e V [155].) M a n y 2 8  levels above 2201 k e V have been reported to be populated i n t h e r m a l neutron capture, i.e.  2 7  A l ( n , 7 ) A r , see S c h m i d t et al. [155] and the c o m p e n d i u m of E n d t [143]. 2 8  Since the ground state of the  2 7  A 1 has J = 5 / 2 , the levels of ? r  +  2 8  A l excited by the  capture of thermal neutrons w i l l tend to have a higher spin t h a n those produced i n  109  8 eAQ  Ge1  0.563  =  823.1+E  6 I  < i  ^  4  2  0  "1  0  1000  I  I  2 0 0 0  Energy  I  I  I  I  I  3 0 0 0  4 0 0 0  (keV)  F i g u r e 5.12: Acceptance curve for the G e l detector. i  '  i  i  I  1000  i  i  i  i  I  i  i  i  i  2 0 0 0  Energy  3 0 0 0  (keV)  F i g u r e 5.13: Acceptance curve for the Ge2 detector. 110  I  i  i  i  i_  4 0 0 0  m u o n capture ( m a i n l y 1  +  as well as 0~, 1~, 2~ a n d 2  al. [156] studied the p r o t o n pickup reaction, i.e. several other 1  +  2 9  +  levels). Recently, Vernotte et  S i ( d , H e ) A l , and have reported 3  2 8  levels omitted by Schmidt et a/., for example levels at 3542 k e V ,  4115 k e V , a n d 4846 k e V . T h e y also assigned the 3015 k e V level J  7 r  =l  +  (which h a d  also been done by Lawergren and Beyea [182], but forgotten). F i g u r e 5.14 shows some of the potential  2 8  A l levels populated i n various re-  actions (and potentially populated i n m u o n capture), w h i c h are k n o w n to decay to the 2201 k e V level of interest.  There is no evidence of such p o p u l a t i o n i n m u o n  capture ; but the effect could easily have been overlooked or missed, so it is of 1  central importance to rule out such a possibility. To investigate the possible cascade-feeding of the 2201 k e V level from upper states, a search for peaks at 4218, 3791, 3659 and 904 k e V (corresponding to the 6420, 5992, 5861 and 3105 k e V levels) i n the Ge singles spectrum was performed. F i g u r e 5.15 shows part of a singles spectrum obtained after a d d i n g the low-gain (high-energy) m u o n Si runs. F o r the 6420 k e V level, there is no significant peak i n this spectrum at 4218 k e V ( B R = 1 2 % ) , nor at 4281 k e V a decay to the 2139 k e V level ( B R = 6 1 % ) . F o r the 5992 k e V level, there seems to be a peak a r o u n d 3791 k e V , however, there is no peak at 4372 k e V w h i c h occurs w i t h a similar b r a n c h i n g ratio as the 3791 k e V transition. Furthermore, the 5992 k e V state w h i c h has T = 2 is not populated i n either the (n,7) nor the ( d , H e ) reactions. (There is a strength there 3  i n the ( T T ^ ) a n d ( d , H e ) reactions however, but one w o u l d not expect a T = 2 state 2  to be excited). F o r the 5861 k e V level, there is no peak at 3659 k e V ( B R = 2 5 % ) nor at 5861 k e V , the direct transition to the ground state w h i c h has a b r a n c h i n g ratio of 60%. T h e most probable candidate is the 3105 k e V 1  +  state because it is fed by  Singles measurements [144,146] used indirect arguments to indicate that the feeding is small and consequently neglected such an effect. 1  Ill  E  J  (keV)  12%  (1.2)  +  6419.8  43% 0  5992.4  +  (2.3)  25%  +  > 00  5860.8  > 1)  cn O  r o  CD  in  csl  CO  75%  25%  3105  > m r o  o 79%  \l  >  2201.46  >  CO  CD  00 CD  o  V  rO  30.64  F i g u r e 5.14: Cascade feeding of the 2201 k e V A 1 level (for a complete energy level diagram, see P . M . E n d t [143] a n d Vernotte et al [156].) 2 8  112  I I I I I I I I I  3500  I  I I I I I I I I I  4000  I  I I I I I I I I I  4500  I  I I I I I I I I I  5000  I  I I I I I I I I I  5500  I  I I I I I I I I I  6000  6500  Energy (keV) F i g u r e 5.15: P a r t of the S i 7-ray energy spectrum for the G e l detector. U p p e r arrows refer to 7-ray transitions potentially feeding the 2201 k e V level of interest, as seen i n F i g u r e 5.14, while the lower arrows are for other associated transitions of the same levels. 2 8  113  similar reactions such as (^,7) and (p,n), see section 2.6.  In a d d i t i o n , two 7-  ray peaks at 3075 k e V and around 903 k e V are seen, see F i g u r e 5.16. However, their relative strengths do not m a t c h the branching ratios given i n F i g u r e 5.14; i.e. (^903 ) ( J A Q ^ ) ~ (6)(2-3) ~ 14 compared to a ratio of only 3 from F i g u r e 5.14 . In 3  5  2  Ge  addition, their shapes are quite different; while the 3075 k e V peak has a Dopplerbroadening of 24 k e V F W H M , the line at 903 k e V is not broadened at a l l (a broadened peak underneath the 903 k e V w o u l d have a very small contribution) nor does it get enhanced by an energy gate around the 2171 k e V 7-ray i n the N a l energy spectra. (The lifetime of the 3105 k e V has not been measured). Note that Schmidt et al. [155] do not include the 3105 k e V state nor list the 903 k e V 7-ray. However, they list a 7-ray at 3075.6 k e V w i t h a footnote that it is not in the level scheme . 3  We conclude from the singles spectrum that the unbroadened 903 k e V 7-ray that we observe has another, but u n k n o w n , origin. Because the broadened 903 k e V line is not observed, we also question the origin of the 3075 k e V line. A direct search for cascade feeding can be made through the coincidence technique by reversing the roles of the detectors, i.e.  by imposing an energy window  on the 1229 k e V 7-ray line i n the two G e detectors and examining the coincident N a l spectra, see F i g u r e 5.17b. O n l y our expected 942 k e V line is apparent. (The dotted lines show a t y p i c a l 942 k e V energy coincidence gate used to tag the 1229 k e V 7-rays as described i n section 4.3.) A n even more sensitive check can be made by gating on the more prolific 2171 k e V 7-ray w h i c h has an 79% b r a n c h i n g ratio compared to 16% for the 1229 k e V 7-ray. T h i s is shown i n F i g u r e 5.17c. Furthermore, F i g u r e 5.17d shows this spectrum after subtracting out the raw spectrum (suitably normalized); and again there is no evidence (peaks or dips) for the potenT h e 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. 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. 2  3  114  600  700  800  900  Energy i .  i  i  i i i  i  1000  1100  1200  (keV) i  i  ,  i  i  ,  i  i  i  i  i  i  3075  iu l 2800  2900  3000  3100  Energy  3200  3300  I 3400  (keV)  F i g u r e 5.16: P a r t s of the S i 7-ray energy spectrum for the G e l detector showing the 903 k e V a n d 3075 k e V peaks. 2 8  115  t i a l transitions seen i n F i g u r e 5.14, nor for any others. W e postulate that the large number of events at the low energy are related to the 400 k e V //-mesic X - r a y , but the events are cut off by the hard-wired N a l d i s c r i m i n a t o r . Note that the bumps at 0.85, 1.0, 1.7 and 2.1 M e V i n F i g u r e 5.17c correspond to the strong peaks at 843 keV from from 2 8  2 6  2 7  A 1 , 1014 k e V from  2 7  A 1 , combined 1779 k e V from  M g , and the cluster of peaks at 2108 k e V  A l a n d 2211 k e V  2 7  2 8  2 8  S i a n d 1808 k e V  A l , 2138 k e V  2 8  A l , 2171 k e V  A l respectively. These bumps can be identified from the sin-  gles G e spectrum w h i c h closely parallels the N a l spectrum, see F i g u r e 5.18. These peaks are relatively strong i n the singles spectra a n d are expected to show up i n the coincidence spectra due to accidentals a n d / o r real coincidences. These bumps disappear i n F i g u r e 5.17d, i n d i c a t i n g that most or a l l of the events are accidental. To put limits on cascade-feeding of the 2201 k e V level, a n d hence limits on the feeding of the 1229 k e V 7-ray of interest, the 2171 k e V g a m m a rays were used. T h i s was done by measuring the ratio of counts i n the 2171 k e V peak i n the G e coincidence  (NQ ) 171  a n d G e singles (Ng ) 171  spectra. Hence, the coincident 7-rays  per 1229 k e V 7-ray is given by :  iV  2171  1  (5.5)  NI  171  where eAQ, is the N a l acceptance as determined i n the preceding section. L i m i t s on feeding from the 3105 k e V state a n d nearby levels were obtained by i m p o s i n g an energy window around the 903 k e V i n the N a l energy spectra and taking N Q  1 7 1  to be the r e m a i n i n g 2171 k e V 7-rays i n the G e detectors. E q u a t i o n 5.5  then gives 702 ± 225  1  220700 ± 1200  L0.149 ± 0.004  = 0.021 ± 0.007  (5.6)  and 229 ± 80 131400 ± 950 J L0.128 ± 0.004 116  = 0.014 ± 0 . 0 0 5  (5.7)  F i g u r e 5.17:  S i 7-ray energy spectra obtained w i t h one of the N a l detectors, B A R 1 : (a) without energy gate on the G e detectors; (b) & (c) after gating on the 1229 k e V a n d 2171 k e V transitions i n the G e detectors respectively; a n d (d) spectrum (a) n o r m a l ized & subtracted from spectrum (c). Spectra for the other N a l crystals are similar.  2 8  117  J  I  I  I  I  500  I  I  I  I  I  I  I  I  I  I  I  I  1000 Gamma  F i g u r e 5.18:  I  I  I  I  I  I  I  I  I  1500 Ray Energy  I  I  I  I  I  I  I  I  1 I  I  I  2000  I  2500  (keV)  S i 7-ray energy spectrum of F i g u r e 5.17a overlaid on a G e l detector singles spectrum (from r u n 136).  2 8  as the l i m i t s on cascade-feeding from levels i n the vicinity of the 3105 k e V for the G e l and Ge2 detectors respectively. T h e p o p u l a t i o n of the 2201 k e V level by a few "strong" h i g h energy 7-rays is ruled out (by F i g u r e 5.17d at the few percent level). Since the threshold for is 7.7 M e V , such possible 7-rays p o p u l a t i n g the 2.2 M e V from  bound  2 8  Al  states w o u l d  have a m a x i m u m energy of 5.5 M e V and hence w o u l d be seen i n F i g u r e 5.17. A remaining possibility is the p o p u l a t i o n of the 2201 k e V level by a large number of weak cascades. To put a l i m i t on such a possibility, t i m i n g coincidence between the G e - N a l spectra, along w i t h a software energy threshold on N a l spect r a of ~ 0.6 M e V , were required. T h e energy threshold ensured a u n i f o r m 7-ray efficiency and also reduced contamination from the 400 k e V /i Si(2p—>ls) X - r a y s . _  Taking N Q  1 7 1  to be the remaining 2171 k e V 7-rays i n the G e detectors after these  requirements (Figure 5.20), the l i m i t s on coincident events per 1229 k e V 7-ray for  118  each detector are: 5500 ± 160 L220700 ± 1200  0.149 ± 0 . 0 0 4 1  0.167 ± 0 . 0 0 7  (5.8)  and 2570 ± 150 131400 ± 9 5 0  .0.128 ± 0 . 0 0 4 J  = 0.15 ± 0 . 0 1 .  (5.9)  However, before taking these results as l i m i t s on cascade-feeding of the level of interest from higher levels, an estimate of non-cascade feeding coincidences h a d to be determined and subtracted off. There were two types of non-cascade feeding coincidences: ' r a n d o m ' and target/beam-related coincidences. T h e former measured the coincident room-backgrounds and was determined from the 1294 k e V  4 1  A r air  activation 7-ray peak to be : [0.0063 ± 0.0010]  1 L0.149 ± 0.004  = 0.042 ± 0 . 0 0 7  (5.10)  0.045 ± 0 . 0 1 0  (5.11)  and [0.0057 ± 0.0013]  L0.128 ± 0.004J  for the G e l a n d Ge2 detectors respectively. S u b t r a c t i n g these ' r a n d o m ' coincidences events from the values obtained i n equations 5.8 a n d 5.9, we obtained the following limits : (0.167 ± 0.007) - (0.042 ± 0.007) = 0.12 ± 0.01  (5.12)  (0.15 ± 0.01) - (0.045 ± 0.010) = 0.10 ± 0.02  (5.13)  and  respectively, w h i c h i f taken at face value would indicate some feeding of the level of interest. T h e other type of non-cascade feeding coincidences is more complicated to estimate; a n d is due to, but not restricted to, pile-up effects.  A 400 k e V muonic  X - r a y precedes each m u o n capture 7-ray. N o w i f a smaller energy event (7, e, or n) 119  comes along accidentally, it w i l l trigger the N a l threshold a n d register a prompt signal. T h u s the accidentals are enhanced. Such coincidences are difficult to estimate, as this effect was not discovered u n t i l the data analysis was underway. However, redoing the above calculation using the muonic X - r a y s (//Fe(2p—>ls)), instead of the 1294 k e V  4 1  A r 7-ray, as the measure of non-cascade feeding coincidences, gave  the following results: [0.0242 ± 0.0027]  1 0.149 ± 0 . 0 0 4  0.162 ± 0 . 0 1 9  (5.14)  and [0.0192 ± 0.0030]  L0.128 ± 0.004J  0.150 ± 0 . 0 2 4  (5.15)  for the G e l a n d Ge2 detectors respectively, a n d consequently the corresponding limits on cascade-feeding w o u l d be : (0.167 ± 0.007) - (0.162 ± 0.019) = 0.005 ± 0.026  (5.16)  (0.153 ± 0.010) - (0.150 ± 0.024) = 0.003 ± 0.034 ,  (5.17)  and  consistent w i t h no cascade-feeding. Since this muonic X - r a y is i n real coincidence w i t h only very low-energy X - r a y s , it constitutes a better i n d i c a t i o n of the noncascade feeding coincidences t h a n does the 1294 k e V 7-ray. However, this estimate may be an underestimation of the feeding as the lifetime of muons i n Fe is shorter t h a n that i n Si a n d hence the feeding is somewhere between the 0 and 11% ( E q u a tions 5.16, 5.17 a n d 5.12, 5.13). It should also be remarked that the shape of the 2171 k e V 7-ray peak i n the coincidence spectra is very similar to that i n the singles spectra. F i g u r e 5.19 shows the singles gamma-ray energy spectra, i n the v i c i n i t y of the 2171 k e V peak, for the two Ge detectors; while F i g u r e 5.20 shows the corresponding spectra after the imposition of the aforementioned G e - N a l t i m i n g coincidence. There is no evidence 120  8-  (a)  2171 keV f  N  A > CD 1  >CD <D  26  A1  Ge1  CO  4  o  1  C\2  J  V 1  f  A 1  CO  /  \  [  J V  /  f  N  c o o 0 2050  2100  2150  /  i  2200  2250  2300  2250  2300  G a m m a Ray Energy (keV)  (b)  Ge2 All  CM  " \  ^  ^ V \  \  >  2 1 1 ke  4-  / 1  CM IO  CM  A ™  rr  o c_>  0 2050  2100  2150  2200  Gamma Ray Energy (keV)  F i g u r e 5.19: Singles S i gamma-ray energy spectra obtained w i t h the two Ge detectors, i n the vicinity of the Doppler-broadened 2171 keV peak. 2 8  for a sharp peak i n the centre, for example, which w o u l d happen if there was feeding from a long-lived state. A c o n t r i b u t i o n of only 0.5% was obtained on such feeding, see section 5.7.2.  Note that some lines i n  2 7  A 1 are enhanced i n the coincidence  spectrum because they are part of cascades a n d 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 r a n d o m . T h e 2171 k e V line is still there though relatively weaker. In concluding this section, a stringent l i m i t at the level of 2% on cascadefeeding from the most probable level (3105 k e V ) was set. belief that the overall cascade-feeding limit is less t h a n 5%. 121  Furthermore, it is our  2050  2100  2150  2200  Gamma  2050  2100  Energy  Ray  Energy  2150  2250  2300  2250  2300  (keV)  2200  Gamma  F i g u r e 5.20 :  Ray  (keV)  S i coincidence energy spectra obtained w i t h the two G e detectors, i n the vicinity of the Doppler-broadened 2171 k e V peak.  2 8  122  5.5  Background Subtraction T h e best coincidence spectra were obtained after o p t i m i z i n g the various cuts,  a n d i n p a r t i c u l a r the energy-gated cut. A l t h o u g h the coincidence m e t h o d greatly improved on the singles spectra; there remained some background under a n d near the peak of interest, i n c l u d i n g a bit of the infamous 'plateau', see F i g u r e 4.5. In a d d i t i o n to r a n d o m coincidences, the flat component of the r e m a i n i n g background i n the coincidence spectrum is due mostly to n-7 a n d n - n coincidences. These events are due to the neutrons following m u o n capture, w i t h associated nuclear 7rays. T h i s is supported by the fact that the N a l - G e coincidence t i m i n g spectra show a 10% or so background underneath the coincidence peak (see F i g u r e 5.9) as well as the enhancement of the characteristic Ge(n,n') lines at, e.g. 596 k e V and 692 k e V . T h e remaining structure under or near the peak of interest is due to the C o m p t o n tails of higher-lying peaks i n the N a l spectrum. B o t h types of b a c k g r o u n d can be subtracted off by using a 'side-band' backg r o u n d subtraction. T h i s 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 k e V 'tagging' 7-ray line. T h e reason for this c o m b i n a t i o n is to m i m i c closely the background underneath the peaks of interest, albeit losing a few genuine 1229 k e V 7 rays (in Ge) w h i c h are i n coincidence w i t h 942 k e V C o m p t o n events picked i n the "below-942" N a l w i n d o w . E a c h w i n d o w has the same w i d t h (in k e V ) as the central 942 k e V w i n d o w , i.e. four standard deviation wide. T h e corresponding - "below942" a n d "above-942"-  G e spectra are added, n o r m a l i z e d , a n d then subtracted from  the m a i n 942 keV-gated coincidence spectra to produce the background-free  G e en-  ergy spectra (herein-after referred to as the A B - s p e c t r a ) , Figures 5.21 a n d 5.22, ready for the lineshape analysis. It should be noted, however, that the details of this subtraction are not critical; see section 5.7.2.  123  800  I  I  900  1000  I  I  I  1100  1200  1300  1400  Gamma Ray Energy (keV)  (b)  1  i  4  800  900  1000  1100  1200  ;  L  1300  1400  Gamma Ray Energy (keV)  Figure 5.21:  S i gamma-ray energy spectra of the G e l detector: (a)singles and (b)coincidence w i t h the 'side-band' background subtraction. E v i d e n t is the flat background around the 1229 k e V peak of interest.  2 8  124  o o r  - 2 700  800  900  1000  1100  1200  1300  Gamma Ray Energy (keV)  F i g u r e 5.22:  S i gamma-ray energy spectra of the Ge2 detector: (a)singles and (b)coincidence w i t h the 'side-band' background subtraction. E v i d e n t is the fiat background a r o u n d the 1229 k e V peak of interest.  2 8  125 (  Table 5.4: Input parameters needed i n the least-squares fitting function for the two Ge detectors, cr, PL a n d PR are i n channels.  Gel 7-ray Line 1229 k e V 2171 k e V  Po 0.0037484c 0.0037484c  dE dX  ( MeV \ V mg/cm > 2  2.2 2.2  cr  PL  PR  S  2.46 3.44  2.76 3.49  3.35 4.30  0.0011 0.0011  a  PL  PR  5  3.15 4.24  4.19 4.54  5.23 5.47  0.0014 0.0017  Ge2 7-ray L i n e 1229 k e V 2171 k e V  Bo  dE dX  1 MeV V mg/cm  0.0037484c 0.0037484c  2  \ '  2.2 2.2  These spectra have the following merits: • the remaining "plateau" is gone, • the other remaining single-line backgrounds nearby are gone, a n d • the 1229 k e V line of interest seems to sit on a perfectly flat background, a n d hence one could really believe a lineshape based o n such a spectrum.  5.6  Parameter Recapitulation T h i s section summarizes the various input parameters needed i n the least-  squares fitting program to fit the experimental line shapes of the two transitions of interest: the 2171 k e V a n d 1229 k e V . O f the fifteen (15) parameters needed i n the least-squares three (/?i,  fitting  program  8 , a n d cos# ,) were set to 0, see section 4.6. T h e response function 2  7i  parameters (a, PL, PR, a n d S) of b o t h G e detectors were fixed at their estimated values following their parameterization, see section 4.4. A n o t h e r two, B a n d j^-, 0  were fixed to their values as determined i n sections 2.5 a n d 4.5 respectively. These values are summarized i n Table 5.4 for the two lines of interest, a n d for the two G e detectors respectively. 126  1  In a d d i t i o n , 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 k e V and the 1229 k e V transitions, it was found that the fitting parameters a and r were highly correlated and tended to offset each other d u r i n g fitting, especially for the 2171 k e V 7-ray line, for which a is negative. T h e interplay of the three effects: the slowing-down, the angular correlation, and the response function is illustrated i n F i g u r e 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 k e V lines i n the subtracted coincidence spectra, the Doppler-broadened lines of interest were fit i n two approaches.  In the first approach, the 2171 k e V peaks i n b o t h  Ge detectors were fit simultaneously to extract the value of the lifetime, r , of the c o m m o n nuclear level (2201 k e V ) . T h e n , w i t h r fixed at this value, the 'side-band' background subtracted 1229 k e V 7-rays of b o t h G e detectors were fit j o i n t l y to extract the angular correlation coefficient, a.  In the second approach, the  fitting  parameters (in p a r t i c u l a r a and r ) were obtained from a simultaneous fit to a l l four spectra, i.e. b o t h 7-ray lines (2171 k e V and 1229 k e V ) of b o t h G e detectors.  5.7.1  The 2171 keV line  T h e 2171 k e V 7-ray, coming from the same level as the 1229 k e V 7-ray, turned out to be a critical indicator of whether the recoiling nucleus, A 1 , slows d o w n and 2 8  hence is crucial for o b t a i n i n g the lifetime of the level. It has a five times higher yield than the 1229 k e V 7-ray. Moreover, it is apparently background free, i.e. has a flat background, so the singles spectra w i t h m u c h higher statistics could be used; 127  -I  2 0 0  I  l_  I  I  _1  _l  l_  I  I  J  l_  I  I  I  l_  I  1 5 0 -  ™  100H  Pi  o °  50H  0  •50  ~~i—'— — — —i— — — — —i— — — — —i— — —  1220  1  1  1  1 2 2 5  1  1  1  1  1 2 3 0  1  1  1  1  1 2 3 5  1  1  r  1240  Gamma—ray Energy (keV) 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 k e V 7-ray lines i n b o t h G e detectors. r(fs)  «2171  61.0±3.7  -0.058±0.025  x  2  207.78  GLOBAL  XR  1.065  0.967  0.969  see F i g u r e 5.19. A s illustrated i n F i g u r e 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 w i t h the high statistics i n the 2171 k e V lines, allowed the determination of b o t h effects w i t h sufficient confidence. V a r y i n g r and  0:2171,  i n addition to the two amplitudes ( N ) , i n the  of the 2171 k e V 7-ray spectra of the two G e detectors allowed their extraction.  simultaneous  Table 5.5 shows the results of the best joint fit to the 2171 k e V 7-  ray peaks. T h e a ^ m value cited for the 2171 k e V is the product of the correlation coefficient (a) and the E 2 / M 1 m i x i n g factor (F). are the x  2  fitting  a  n  d the reduced \ \ p.d.f.  angular  C i t e d i n Table 5.5  reflecting the quality of fit. A l s o cited i n  the table are the M I N U I T correlation coefficients, p(r,a)  and p  ^  measuring  GLOBAL  the correlations between the variable parameters. T h e fact that these correlation coefficients are large demonstrates the high correlation (as discussed above) among the fitting parameters, i n particular between r and a.  However, such values of  the correlation coefficients are still less t h a n one i n absolute value , and hence 4  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 . 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. 4  129  Table 5.6: Results of the i n d i v i d u a l fits to the 1229 k e V 7-ray peak for each G e detector. r(fs) Gel 61.0 Ge2 61.0 weig ited a  a  x 169.90 186.47  0.385±0.065 0.318±0.107  pGLOBAL  XR  2  0.862 0.898  0.253 0.260  0.367±0.055  Table 5.7: Effect of the i n s t r u m e n t a l resolution of the detectors on the angular correlation coefficient, a  a  W-  a  1.0(7  W + 1.0<7  5.7.2  x 373.49 372.22 371.31 2  0.35.8±0.052 0.361±0.052 0.364±0.053  XR  0.881 0.878 0.876  The 1229 keV line  H o l d i n g the lifetime ( r ) of the 2201 k e V level fixed, a fit was performed on the 1229 k e V gamma-ray peak i n the 'side-band' background-subtracted spectrum of each G e detector to extract values of the angular correlation coefficient from w h i c h a weighted mean of a was determined (see Table 5.6). T h e same result, w i t h i n ~ 0 . 3 <7, was obtained i n a simultaneous fit to the 1229 k e V lines i n b o t h G e detectors. T h e same results, w i t h i n ~ 0 . 0 5 cr, were obtained whether E 0 a n d U B were fixed or allowed to vary. Furthermore, varying the i n s t r u m e n t a l resolutions of the detectors w i t h i n their uncertainties changed the value of a by <  0.1a.  T h e resolution (in channels) at the 1229 k e V line for G e l is 2.46(5) a n d for Ge2 is 3.15(6), which correspond to 2.09 a n d 2.38 k e V F W H M . Table 5.7 shows the dependence of a on the i n s t r u m e n t a l resolution of the detectors, i n the simultaneous fit to the 1229 k e V peaks. E v e n i f the g e r m a n i u m resolutions were a few a off, the effect on the final result is m u c h smaller t h a n the fitting error. T o 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 i n the first column.  Spectrum A-spectrum B-spectrum AB-spectrum  1229 k e V signal 9,334 8,325 8,603  a 0.320±0.052 0.386±0.058 0.361±0.052  x 365.03 395.76 372.22 2  Xn 0.865 0.938 0.878 2  following procedure was done. In a d d i t i o n to the master spectra (the A B - s p e c t r a ) obtained i n section 5.5, the "above-942" a n d the "below-942" G e spectra were individually normalized a n d subtracted from the m a i n 942 keV-gated coincidence spect r a to produce another two sets of the 'side-band' background-subtracted spectra, herein-after referred to as the A - s p e c t r a a n d B-spectra respectively. Likewise, these two spectra were fit i n the same way as was done w i t h the A B - s p e c t r a , w i t h the results are shown i n Table 5.8. A s can be seen i n this Table, this subtraction affects the final value of a , but the three results agree w i t h each other w i t h i n associated uncertainties. T h e s m a l l variation i n the number of the 7-rays under the 1229 k e V peaks is due to the variation of the 942 k e V C o m p t o n events picked i n the associated N a l energy windows. F i n a l l y we check the effect of cascade feeding of m a x i m a l intensity on a. A n y cascade feeding w o u l d be either from short-lived or long-lived states. W h i l e the former create c o n t r i b u t i o n w i t h similar Doppler-broadening, the latter w o u l d create a sharp (gaussian) peak i n the center of the 2171 k e V (and 1229 k e V ) , the presence of which was very sensitive to a .  T o test the effect of feeding from short-lived  states, the 1229 k e V d a t a was refitted after subtracting a 5% Doppler-broadened contribution, w i t h a shape corresponding to cv=0.  T h i s is the overall l i m i t on  cascade-feeding as found earlier. T h e effect on a for the 1229 k e V was about 5%, which is m u c h less t h a n other errors. T o put a l i m i t on feeding from long-lived states, the 2171 k e V lines were refitted, i n c l u d i n g a gaussian peak to the overall 131  Table 5.9: Results of the best fit to a l l four spectra: the 2171 k e V and the 1229 k e V 7-ray lines i n b o t h G e detectors ( A B subtraction).  r(fs) 60.8±3.4  fitting  a 0.360±0.059  F -0.165±0.080  x 581.82  XR  p(r,a)  0.934  0.461  2  P(T,F)  0.929  pGLOBAL 0.965  function a n d a l i m i t of ( 0 . 5 8 ± 0 . 2 7 ) % was obtained. T h i s w o u l d affect a by  about 5% w h i c h is again less t h a n the errors quoted on a.  5.7.3  The 1229 keV and the 2171 keV Simultaneous Fit T h e previous fitting procedure was repeated also on a l l four spectra simulta-  neously. T h e fitting function therefore contained the following free parameters: the four amplitudes ( N ) , the four background levels ( U B ) , the four peak positions ( E 0 ) , the 2201 k e V lifetime ( r ) , the angular correlation coefficient (a) a n d the E 2 / M 1 m u l t i p o l a r i t y m i x i n g factor (F) of the 2171 k e V 7-ray. T h i s approach h a d the advantage of lower correlations among the various parameters.  Furthermore the  x  2  surface ( m i n i m u m ) becomes deeper and easier to locate. T h e final spectra a n d fits are shown i n F i g u r e 5.24 while the results are given i n Table 5.9. T h e same results, w i t h i n O.lcr, were obtained whether E 0 a n d U B were fixed or allowed to vary. Furthermore, varying the i n s t r u m e n t a l resolutions of the detectors w i t h their uncertainties changed the values of r a n d a by < 0.3o\ T h e values of a , T, F determined i n the above sections were i n good agreement. Due to the aforementioned reasons, the final results are taken to be those of Table 5.9, v i z . : a = 0.360 ± 0.059 ,  (5.18)  r = 60.8 ± 3.4 fs ,  (5.19)  F = —0.165 ± 0.080 .  (5.20)  132  cn  o f\j o ov o  a  ro  m  Counts/keV ro 01 OJ  o cn o o o o o o o o i » • • • 1 1 • • • * * » • • 11 •  o cn i o o ( o o I • 11 • « 1 1 1 1 • 11  -r* Ol o  O o  Q  too  K) O O O  OJ  O O O  ^ o o 0  Counts/keV CJI O)  1  o o o  •I•  • I  00 o o o  o o o  o o o  I•••  ro o °5o  O  3 3 o IO O  a coo  IO CD' o  ii  11  I  i ii iI i i Counts/keV Cn o _i  i  I  o  , i  i  o  I_I  L_i  ai  o  o  > i > t  I Ol  o  i  ro  i  oo  Counts/keV ro ro Ol o . 1..  o o  Ol o  Ol o o .. i . . . cr  .  ro ro  o  o  CD  O  o  (D  3 3 a m a  ro  or o  re < ro o  ro or  o  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 , i n 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 i n C h a p t e r 2. In this Chapter, we shall take the next step w h i c h 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 w i t h those obtained i n previous experiments.  6.1  The 7 — v Angular Correlation  O n l y two other measurements exist for the 7 — 1/ angular correlation following m u o n capture on  2 8  Si.  B o t h were singles measurements.  T h e first was the pioneering  measurement of M i l l e r et al. [144,145], using the N . A . S . A . Space R a d i a t i o n Effects L a b o r a t o r y ( S R E L ) synchrocyclotron i n the late 1960's. T h e y used a n a t u r a l S i target as well as an isotopically pure Si02 target. T h e 1229 k e V g a m m a rays were 28  detected i n a 10% G e ( L i ) detector at 90° w i t h respect to the m u o n spin direction, so that the p r i m a r y sensitivity was to the coefficient a (which M i l l e r et al  refer  to as a ° ) . W h i l e their detector was located at 90° to the b e a m axis, they retained some sensitivity to Q\ and Q due to the large dimensions of the targets used (which 2  134  were set at 45° to the beam a n d to the Ge detector, v i z . Table II a n d F i g u r e 20 i n Ref. [145]). T h e i r quoted value of a was  a  =  0.15 ± 0.25 a n d  a  = 0.29 ± 0.30 ,  (6.1)  for the n a t u r a l Si and Si02 targets respectively. 28  T h e y also considered correlations for the 0  +  —> 1 —> 2 +  +  (e.g.  the 2171 k e V )  transitions. T h e y concluded that the extracted correlations were inconsistent a n d could not be reconciled w i t h theory, regardless of the 8 m i x i n g ratio. M i l l e r [145]: "the correlation  coefficients  are experimentally  and large, while predicted to be small but positive. developed by Popov et a l . , this observation to these 0  +  —> 1  +  —> 2  +  transitions  T o quote  found to be negative  Within the context of the theory  is inexplicable.  thus remain  The situation  a mystery."  1  .  with regard  W e now realize  that this situation came about because they d i d not include the lifetime of the 2201 k e V level i n the analysis. Nevertheless, they h a d observed, for the first time, the Doppler broadened 7-ray transitions, w h i c h are suitable for analysis i n terms of the 7 — v angular correlation. In a d d i t i o n , they were able to put a useful limits on the p o p u l a t i o n of the 2201 k e V state of interest from the  29  S i ( / t , ^ n ) reaction through  the use of an isotopically enriched SiG*2 target. Such p o p u l a t i o n w o u l d confuse 29  the 7 — 1/ angular correlation analysis. ( N a t u r a l Si contains 4.67% S i as compared 2 9  to 92.2% S i ) . M i l l e r et al. [145] observed no captures i n 2 8  2 9  S i which resulted i n a  transition through the 2201 k e V level, a n d gave a contamination l i m i t of < 0.015 due to the presence of S i i n n a t u r a l silicon target. 2 9  The  only other measurement of the 7 — v angular correlation was carried  out recently by B r u d a n i n et al. [146] at the Joint Institute for Nuclear Research ( J I N R ) i n D u b n a . T h e y used two H P G e detectors to detect the 1229 k e V 7-rays at two angles, 60° a n d 120°, w i t h respect to the m u o n p o l a r i z a t i o n axis, w h i c h 1  However, the error bars are quite significant and the inconsistency is not so serious.  135  provided t h e m w i t h sensitivity to the correlation coefficients B and B as well. T h e x  2  residual m u o n p o l a r i z a t i o n was measured to be ( 1 0 . 2 5 ± 0 . 2 5 ) % for their beam, for which the i n i t i a l p o l a r i z a t i o n was given as ~ 7 0 % .  T h e m u o n p o l a r i z a t i o n has  been measured i n n a t u r a l Si to be ( 1 5 ± 2 ) % ( A s t b u r y et al. [185]) a n d i n  2 8  S i to  be ( 1 6 ± 4 ) % (Weissenberg [186]) of the i n i t i a l p o l a r i z a t i o n respectively. U s i n g a •pure gaussian for the detector response function, the corresponding 7-ray spectra obtained from the two G e detectors were corrected for energy c a l i b r a t i o n shifts. T h e i r 7-ray spectra are similar to ours i n that they show the plateau-like structure next to the 7-ray peak of interest.  T h e y proposed a complex structure for the  background underneath the 1229 k e V 7-ray peak, composed of three components: 1204 k e V , 1222 k e V a n d 1238 k e V 7 rays from the  7 4  G e ( n , n ' ) , A 1 and 2 7  7  56  Fe(n,n')  reactions respectively. A n o t h e r source believed to contribute to this region, yet not included i n their background structure, is the 1216 k e V 7-ray due to the Ge(n,n'7) 70  reaction . B r u d a n i n et al. then performed a simultaneous fit to the 1229 k e V and 2  the 2171 k e V 7-ray peaks i n the energy spectra of the two G e detectors i n the two angular positions. Due to their low statistics, it was impossible to fit these 7-ray peaks separately . T h e i r result is quoted for a m u l t i p o l e parameter to be 3  x  B  = 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 w h i c h was reported i n the 1995 W E I N Conference [187], but they obtained a similar result, 7-ray lines at 1216 keV and 1224 keV seen in muon capture on stainless steel may also contribute 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. T h e 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). 2  3  136  viz. XB = 0.225 ± 0.026. T h e 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 G e detectors were mechanically moved between the two angles, while i n the second experiment this was done by rotating the m u o n spin i n an external magnetic Note that i n their notation the correlation coefficients a, f3\ and B are given  field.  2  as a , 62 and (a + |ci) respectively. Moreover, the expressions for the correlation 2  coefficients as a function of the m u l t i p o l e parameter ( B r u d a n i n et al. Equations 14, 15, 16 & 1 7 ) are that of E r a m z h y a n et al. [148]; a n d are different from Equations 2.13,  2.14 & 2.15 of C h a p t e r 2, w h i c h are used by Oziewicz [125] as well as by  Ciechanowicz [129]. T h i s is due to the use of different definitions of the m u l t i p o l e parameter. In fact the two are connected by 1 + 2x  2 B  1 + \x\  +  2^/2x  B  V2x  (6.4)  B  Table 6.1 lists some relevant parameters of these two experiments and shows how they compare w i t h 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 extracted results.  T h e S / N entry is the signal-to-noise ratio a n d is calculated as  the background-subtracted area (N^) of the peak d i v i d e d by the area of the background underneath the peak. T h e F W H M of the G e detector(s) is the full w i d t h at half m a x i m u m of the corresponding 7-ray peak. T h e angular correlation coefficient a for the 2171 k e V is the product of a for the 1229 k e V a n d the E2/M1  mixing  factor F as given by E q u a t i o n 2.13. A s can be seen i n Table 6.1, our experiment is superior i n m a n y respects to b o t h experiments; a factor of 4.8 increase i n data, a factor of 4 (or 2) improvement i n S / N ratio a n d an experimental resolution i m provement of 30%. Moreover, an extra factor of 5 improvement is gained i n the S / N ratio after a p p l y i n g the coincidence requirement; albeit losing some signal. Note the relatively poor resolutions of the detectors i n the experiment of B r u d a n i n et 137  Table 6.1: A comparative summary of the three measurements of the 7 — v angular correlation following muon capture on S i , M i l l e r et al. [144], B r u d a n i n et al. [146] a n d the present work. T h e entries i n the Table are discussed i n some detail i n the text. 2 8  Experiment  Mode  (keV)  7V (10 )  S/N  FWHM (keV)  a  7  3  2171  Miller Brudanin present work  singles singles singles  73 73 355  0.83 1.8 3.5  4.0 3.7/4.0 2.9/3.2  - 0.37±0.10 - 0.123±0.062 - 0.058±0.025  1229  Miller Brudanin present work  singles singles coincidence  13 26 9.6  0.24 0.5 4.5  2.6 3.0/3.0 2.1/2.4  ±0.15±0.25 +0.307±0.041 +0.360±0.059  r (fs) 38.2±2.8 60.8±3.4±9.1  al. [146], an important parameter for the lineshape analysis. One also notes that the 2171 k e V peaks i n their Figure 4 are much wider than what they should be; i.e. ~ 22 k e V instead of ~ 2(3 E 0  = 2(0.00375)(2171) = 16 k e V . However, because we  0  focus on our coincidence result, our statistical error is larger than that of B r u d a n i n et ai.  (We also have no sensitivity to Q\ and B .) 2  A s was discussed earlier, the 2171 k e V 7-ray became crucial for obtaining the lifetime of the 2201 k e V level. T h e 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 i n the energy loss ( f § ) of the  2 8  A l ions i n S i , as obtained from the T R I M program [175]. It is important to  note that i n the fitting routine this does not need to be included; just the statistical error on the lifetime for the 2171 k e V line, which is transferred to an error i n the coupling constants for the 1229 k e V line. A recent comparison [188] of measured stopping powers of S i ions i n A l w i t h the T R I M prediction indicates some system2 9  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 m u c h as 20%, 138  { i  and hence the lifetime value may have to be raised by 10% or so, which can be done using F i g u r e 4.9. O u r lifetime measurement is to be compared w i t h previous values of ( 3 5 ± 1 0 ) fs and ( 1 2 0 ± 7 0 ) fs as measured i n References [172] a n d [173]. B r u d a n i n et al. quote a lifetime of ( 3 8 . 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 ( 4 8 . 5 ± 2 . 8 ) fs.  6.2  The Induced Pseudoscalar Coupling W e t u r n now to the extracted values of a. F o r the the 2171 k e V 7-ray peak,  the corresponding F factors are -2.5, -0.40 a n d -0.17 for the M i l l e r et al, B r u d a n i n et al. and the present experiments respectively. In fact one can o b t a i n a value of the m i x i n g ratio 8 by using the F value a n d F i g u r e 2.1. W e 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. T h i s value is to be compared w i t h 8 = 0.74 ± 0.29 a n d 8 = 0.14 ± 0.33 ,  (6.7)  as obtained i n the two different experiments of the D u b n a group respectively . W e 4  note that the measured value of F i n our experiment cuts the curve of F i g u r e 2.1 at a second region, v i z . 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 k e V 7-ray peak, a l l three measurements are seen to be i n reasonable agreement. T h e large errors associated w i t h the B r u d a n i n 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. 4  139  value of M i l l e r et al. is a reflection of the poor conditions of their b e a m as well as the low signal-to-noise ratio a n d the moderate instrumental resolution. It should perhaps be noted that they performed their least-squares fit o n background-subtracted peaks. There was no raw spectrum of the 1229 k e V 7-ray peak given, nor was there any mention of how such background subtraction was carried out. Nevertheless, they h a d set the stage for the future 7 — v angular correlation experiments. T h e error on a obtained by B r u d a n i n et al. is slightly smaller t h a n our error. T h e i r use of different angles to the m u o n p o l a r i z a t i o n axis (i.e. the forwardbackward spectra that are sensitive to the B\ a n d f3 parameters) was an advantage 2  and m a y have given them some confidence i n their result. However, their quoted errors do not seem to include some systematic contributions due to, but not restricted to, the effects described above, e.g. background subtraction, detector response function a n d energy-losses. In contrast, we believe that our rather detailed attention to the response function a n d the slowing-down effects as well as o u r use of backgroundfree spectra render these errors m u c h smaller. It is therefore our contention that the difference i n the quality of the data of the two experiments is not adequately reflected i n the relative values of the quoted errors. We t u r n now to a discussion of the pseudoscalar weak coupling gp/gA as extracted from our data a n d from the other experiments, a n d how they compare w i t h each other a n d w i t h the P C A C prediction. T o o b t a i n gp/g , one needs to compare A  w i t h the calculations of section 2.4. F o r convenience, the various theoretical calculations (i.e. F i g u r e 2.3) are overlaid i n F i g u r e 6.1 against the measured correlation coefficient obtained i n this experiment to deduce the pseudoscalar coupling constant. Furthermore, to put the present results into perspective, Table 6.2 presents a s u m m a r y of the extracted values of gp/gA from our data, along w i t h those extracted from the other experiments.  140  .  ,  ,  i  i  i  i  i  i  i  i  i  L—J  I  I  1  I  I  I  I  l_  •5-  .4  1  .3  •2H Legend  .1  —*—  .0  FPA Model I: Cie Model I!: P&S Model III: Kuzmin i  1  1  r  20  *9 /g. P  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 S i for the 1229 keV 7-ray. 2 8  Experiment  Miller et al. [144]  Brudanin et al. [146]  present work  Correlation Coefficient a  +0.15±0.25  +0.307±0.041  +0.360±0.059  Theoretical model !)• Fujii-Primakoff Approximation  Deduced values of gp/gA +7.7±1.2  Ciechanowicz [109]  +4.5±8.0 +0.4±6.2  +3.4±1.0  +9.5±2.4 +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 a n d how they compare w i t h each other and w i t h the P C A C prediction, let us address a difficulty i n interpreting the measurement of M i 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. M i l l e r et al. themselves extracted gp/gA  from  their data using their value of cv a n d using (essentially) the Fujii-Primakoff app r o x i m a t i o n . T h e i r weighted average value of a , from the isotopically pure  28  SiG"2  a n d the n a t u r a l Si data is 0 . 2 1 ± 0 . 1 9 , and comparing this to the Fujii-Primakoff a p p r o x i m a t i o n gives the value gp/g  A  = 5 ± 8 that they quote i n their paper [144].  Note that the three more complete calculations (Ciechanowicz, Parthasarathy a n d Sridhar a n d K u z ' m i n et al.) h a d 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 w i t h these calcula-  tions, then one gets a m u c h lower value of g /gA,  v i z . Table 6.2. B u t then, one  P  asks, where do the numbers of  - 4.9 < g /g P  A  < 1-2  and  g /g P  A  = 13.5±|;| ,  (6.8)  that are given i n the papers of Ciechanowicz [109] a n d P a r t h a s a r a t h y a n d S r i d har [100], respectively, come from? T h e answer is that Ciechanowicz compares w i t h M i l l e r ' s a, 3^ a n d 3 simultaneously, a n d i n fact his number m a i n l y comes from 8\ 2  and 3 , not a; however it is not far from the 0 ± 6 that one gets from a alone, w i t h 2  his calculation. P a r t h a s a r a t h y a n d Sridhar, however, use only 3 i n their compari2  son w i t h M i l l e r et at, a n d ignore the fact that this value of gp/gA gives values for a a n d B that are very different t h a n what M i l l e r et al. measured, a n d ignore the fact 1  that there is a second solution for gp/gA that gives the same f3 a n d agrees better 2  w i t h a and B . T h i s can clearly be seen i n Table I ( " M o d e l II") i n their paper [100]. 1  A m u c h better agreement w i t h a, 3 and 3 is achieved for gp/gA around -2.5 t h a n X  2  for gp/gA around +12.5 :  142  M i l l e r et al. a 0.21 ± 0 . 1 9 pi 0.02 ± 0 . 0 3 P 1.12 ± 0 . 1 0  gp/gA  2  = —2.5 0.21 0.04 1.17  Hence the preferred solution is  gp/gA  gp/gA  = +12.5 0.49 0.32 1.16  around -2.5, i.e. consistent w i t h that of  Ciechanowicz. T h u s , we believe that the value i n E q u a t i o n 6.8 from P a r t h a s a r a t h y and Sridhar is taken from the wrong solution. However, the error bars of M i l l e r et al. for Pi a n d p have to be treated w i t h scepticism since : 2  • T h e effects of b o t h Pi a n d p on the gamma-ray spectrum are washed out by 2  the small residual m u o n p o l a r i z a t i o n , v i z . 15%. • Pi a n d p have very similar effects on the gamma-ray line shape, a n d therefore 2  the fitted values are likely to be highly correlated; also one has to a d d i n the error i n the measurement of this p o l a r i z a t i o n . • O u r M o n t e C a r l o / f i t t i n g studies show that one cannot extract such small errors o n Pi a n d p w i t h M i l l e r ' s statistics (even b y fitting to his data). 2  T h i s w o u l d mean that the value extracted b y Ciechanowicz should have larger errors, a n d should p r o b a b l y be about the value of 0 ± 6 from a alone. So we t h i n k that it is safest to use only M i l l e r ' s value of a, a n d one then obtains the values tabulated i n Table 6.2 above for the various theoretical models. Brudanin  et al,  i n their final result, quoted only the  value as obtained  gp/gA  from the Ciechanowicz calculation; claiming that the P a r t h a s a r a t h y a n d Sridhar calculation is inconsistent a n d less reliable "as correspond,  as they should,  to the same values  their  correlation  of the parameter  coefficients XB"  do  not  [146]. However,  a careful comparison between the values of the correlation coefficients (a, Pi a n d P ) corresponding to the experimental range of XB a n d curves corresponding to 2  the Parthasarathy a n d S r i d h a r calculation gives consistent values of  143  gp/gA  within  the uncertainties, contrary to the above c l a i m . T h u s the conclusion d r a w n by 5  B r u d a n i n et al. that "no physical solution exists for the combination  in the x-region  (a-\-^ci(=  B )) 2  of interest" is surprising, as noted b y P a r t h a s a r a t h y [189], i n his  comments o n the B r u d a n i n et al. measurement.  Note that the K u z ' m i n et al.  calculation was underway when B r u d a n i n et al. published their results, a n d so no comparison w i t h this calculation was i n c l u d e d . T h e gp/gA values as extracted from all theoretical calculations using the B r u d a n i n et al. experiment are also i n c l u d e d i n 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 various theoretical models, using a l l three measurements are consistent w i t h each other w i t h i n the (rather large) quoted uncertainties. Furthermore i f one adopts the FujiiPrimakoff A p p r o x i m a t i o n , a l l three experiments would agree w i t h the GoldbergerT r e i m a n expectation ( P C A C - p r e d i c t i o n ) of gp/gA  ~ 7 (see section 1.5). T h e ex-  tracted values of gp/gA from a l l three "complete" calculations using the B r u d a n i n et al. results is i n a strong disagreement w i t h the G o l d b e r g e r - T r e i m a n estimate and i n d i c a t i n g a strong downward r e n o r m a l i z a t i o n 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 , is i n conflict w i t h the G o l d b e r g e r - T r e i m a n 6  value only when compared to the K u z ' m i n et al. model. A l l i n a l l , the extracted values of gp/gA are on the lower side of the canonical value of Goldberger a n d T r e i m a n , suggesting a sizable quenching of gp/gA i n 2 8  S i . T h u s , the overall situation is somewhat confusing. It is apparent that more  theoretical efforts are required to assess the model-dependence of these i n t r i g u i n g 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. T h e comments made previously comparing the quality of the quoted error bars in the two experiments applies here as well and need not be repeated. 5  6  144  results.  Several theorists at the recent W E I N conference were very uneasy w i t h  this situation and felt that it was necessary to look at the calculations m u c h more critically. T h u s although further experiments might be worthwhile, the m a i n focus should be on o b t a i n i n g a better understanding of the theoretical uncertainties.  145  Chapter 7 Conclusions T h e angular correlation of the neutrino w i t h the 1229 k e V 7-ray from the deexcitation of the 2201 k e V 1  +  level i n A l , following exclusive m u o n capture on 2 8  2 8  Si  has been measured, i n order to determine the magnitude of the induced-pseudoscalar coupling constant gp of the weak hadronic current. T h e correlation was observed by measuring the energy d i s t r i b u t i o n of the 1229 k e V 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 i n w h i c h the 1229 k e V 7-ray is 'tagged' by the subsequent 942 keV  7-ray (100% branching ratio) i n the cascade. A pair of Compton-suppressed  high p u r i t y g e r m a n i u m detectors were used to detect the Doppler-broadened 7rays, while an array of 24 N a l ( T l ) scintillators detected the 942 k e V 7-rays. A s a result, very clean spectra were obtained; and a dependable measurement of the angular correlation was made. In a d d i t i o n , a signal-to-background ratio of 4.5 was obtained, which is to be compared w i t h ratios of 0.5 and 0.24 observed i n the other two experiments [144,146]. T h e correlation coefficient a was found to be 0 . 3 6 0 ± 0 . 0 5 9 , i n good agreement w i t h the recent measurement of B r u d a n i n 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 results are consistent w i t h the P C A C prediction of gp/gA — 7, but suggest a possible quenching of the induced-pseudoscalar coupling constant i n  2 8  S i . However, a com-  parison w i t h 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. Furthermore, when the same result is compared w i t h the Fujii-Primakoff approximation, a value of gp/gA = 9 . 5 ± 2 . 4 is found to be quite consistent w i t h the P C A C value (and w i t h an unrenormalized value of gp). W e suppose that the more recent calculations are more dependable, but the model-dependence of these i n t r i g u i n g results remains to be assessed. Clearly more work is required on the theoretical side. In addition, the more prolific 2171 k e V 7-ray de-exciting from the 2201 k e V c o m m o n level proved to be quite useful; its Doppler-broadened line shape was used to measure the lifetime of this level, v i z . , 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 ( M i l l e r et al. [144,145]). T h e coincidence technique has provided a thorough investigation of possible cascade feeding of the c o m m o n level of interest from h i g h - l y i n g excited states. Such feeding w o u l d act as another source for the 1229 k e V 7-rays, besides the direct p r o d u c t i o n from the c/p-sensitive spin sequence of S i ( 0 ) A A1*(2201 k e V 1 : A 2 8  +  28  +  973 k e V 0 ) ; and could distort the 7 — 1/ angular correlation, a n d thereby spoil +  the interpretation of the Doppler-broadened lineshape measurement i n terms of gp. Stringent l i m i t s on such feeding are obtained for the first time, thus reducing systematic errors of the angular correlation. T h e p r i m a r y l i m i t a t i o n of the present experiment was a c o m b i n a t i o n of low statistics and highly correlated variables. T h e former was due to the loss of the 1229 k e V 7-ray signal after the i m p o s i t i o n of the coincidence requirement. Higher 147  statistics m a y be gained either by r u n n i n g longer a n d / o r o p t i m i z i n g the detection system, e.g. by increasing the efficiency of the N a l ( T l ) a r m . T h e later l i m i t a t i o n was due to the strong dependence of the correlation coefficients on r a n d 8; a n d can be eliminated by independent measurements of either or b o t h of these parameters. Overcoming these limitations i n future coincidence experiments w o u l d enable a more accurate measurement of the correlation coefficient(s) and u l t i m a t e l y improve the precision on the extracted pseudoscalar coupling constant. In addition, a periodic use of a series of k n o w n 7-ray sources throughout a wide range of energies would help to determine the response function as well as the acceptances of the G e detectors more accurately. ( D u r i n g the experiment we d i d not realise how important the 2171 k e V line w o u l d be.) It w o u l d also be useful to o b t a i n good statistics from m u o n capture on an isotopically enriched S i target to 2 9  measure the yield (and i f needed the shape) of the 1229 k e V 7-rays emitted i n the Si(n,vn)  reaction; a n d to confirm the results of M i l l e r et al. [144,145]. F i n a l l y , this  29  technique may be extended to other nuclei i n w h i c h the transitions are sensitive to g  P  (e.g.  2 0  F).  In summary, a dependable measurement of gp/gA seems still to elude us because of systematic errors a n d theoretical uncertainties. A similar situation exists w i t h radiative m u o n capture on nuclei for w h i c h the measurements are reasonably consistent [92,82], yet the theoretical uncertainties are clearly considerable a n d have been discussed recently by Fearing a n d Welsh [95]. T o a d d to the confusion the recent experimental result for radiative m u o n capture on hydrogen gives a high value for gp/gA,  v i z . 10.0 ± 0.9 ± 0.3 [74]. T h e only measurements for w h i c h there is  less controversy are the o l d ones measuring other observables for  1 2  C and  1 6  0 which  give values of gp/gA slightly larger t h a n the G o l d b e r g e r - T r e i m a n number.  Thus  although further experiments might be worthwhile, especially capture on new nuclei or measuring new observables, the m a i n focus should be on o b t a i n i n g a better 148  understanding of why the calculations of b o t h R M C and nuclear m u o n capture give inconsistent and unlikely values of gp/gA-  149  Bibliography [1] J . C . Street a n d E . C . Stevenson. Phys.  Rev., 52:1003, (1937).  [2] C D . A n d e r s o n a n d S . H . Neddermeyer. Phys.  Rev., 51:884, (1937).  [3] C D . A n d e r s o n a n d S . H . Neddermeyer. Phys.  Rev., 54:88, (1938).  [4] H . Yukawa. Physico-Mathematical [5] M . Conversi et al. Phys.  Society of Japan,  Rev., 71:209, (1947).  [6] C . M . G . Lattes et al. Nature(London), [7] Particle D a t a G r o u p . Phys. [8] F . Scheck. Phys.  159:694, (1947).  Rev. D, 50:1173, (1994).  Rep., 30C:187, (1978).  [9] E . W . O t t e n et al. Nucl.  [10] B . Jeckelman et al. Phys. [11] J . Wheeler. Phys.  17:48, (1935).  Phys.  Lett,  News,  Vol. 5, 1:11,  (1995).  B335:326, (1994).  Rev., 71:320, (1947).  [12] E . F e r m i and E . Teller. Phys. [13] W . Y . C h a n g . Phys.  Rev., 72:399, (1947).  Rev., 75:1315, (1949).  [14] H . K o c h . M u o n i c a n d hadronic atoms. In J . B . W a r r e n , editor, Nuclear Particle  Physics  at Intermediate  Energies.  [15] S. Devons a n d I. D u e r d o t h . Adv. Nucl.  P l e n u m Press, (1976).  Phys.,  2:295, (1969).  [16] J . Hufner et al. M u o n i c atoms. In V . Hughes a n d C S . W u , editors, Physics Vol. I. A c a d e m i c Press, N Y , (1975). [17] F . J . H a r t m a n n . Phys., Atomic [18] J . A . Wheeler. Rev. Mod. Phys., [19] L . I . Schiff. Quantum  Mechanics.  [20] H . Primakoff. Rev. Mod. Phys.,  Nuclei,  58:1729, (1995).  21:133, (1949). M c G r a w - H i l l , (1955). 31:802, (1959). 150  and  Muonic  [21] B . G o u l a r d a n d H . Primakoff. Phys. [22] N . C . M u k h o p a d h y a y . Phys. [23] T . S u z u k i .  A Systematic  Rev. C, 83:2034, (1974).  Rep., 3 0 C : 1 , (1977).  Study  of Muon  Capture.  P h D thesis, 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 , (1980). unpublished. [24] T . S u z u k i , D . F . Measday, a n d J . P. Roalsvig. Phys. [25] H . Hanscheid et al. Z. Phys., [26] L . A . Schaller et al. Nucl.  Rev. C, 35:2212, (1987).  A 3 3 5 : l , (1990).  Phys.,  A300:225, (1978).  [27] R . E . M a r s h a k , R i a z u d d i n , and C P . R y a n . Theory of Weak Interactions Particle Physics. J o h n W i l e y a n d Sons, Inc., (1969). [28] D . B a i l i n . Weak Interactions. [29] E . F e r m i . Z Phys.,  A d a m Hilger L t d , B r i s t o l , (1982).  88:161, (1934).  [30] B . Pontecorvo. Phys. [31] G . P u p p i . Nuovo  Rev., 72:246, (1947).  Cimento,  5:587, (1948).  [32] G . G a m o w a n d E . Teller. Phys.  Rev., 49:895, (1936).  [33] T . D . Lee and C . N . Y a n g . Phys.  Rev., 104:254, (1956).  [34] C S . W u et al. Phys. [35] S. Glashow. Nucl.  Rev., 105:1413, (1957).  Phys.,  [36] S. Weinberg. Phys. [37] A . S a l a m .  in  22:579, (1961).  Rev. Lett., 19:1264, (1967).  Elementary  Particle  Physics  (Nobel  Symp.  No.  8),  page  A l m q v i s t a n d W i l s e l l , Stockholm, (1968). [38] F . J . Hasert et al. Phys.  Lett,  46B:121, (1973).  [39] G . A r n i s o n et al. ( U A 1 collaboration). Phys. [40] B . N . Taylor. Phys.  Lett,  [41] M . G e l l - M a n n . Phys.  Lett,  B 2 3 9 : l l l , (1990).  Rev., 1257:1067, (1962).  [42] N . C a b i b b o . Phys.  Rev. Lett,  [43] S. Weinberg. Phys.  Rev., 112:1375, (1958).  [44] P . Langacker. Phys. [45] H . Stremnitzer. Phys.  10:531, (1963).  Rev. D, 14:2340, (1976). Rev. D, 10:1327, (1974).  122B:103, (1983).  367.  [46] L . Grenacs. Ann. Rev. Nucl. Part. Sci., 35:455, (1985). [47] R . P . F e y n m a n a n d M . G e l l - M a n n . Phys. Rev., 109:193, (1953). [48] W . K . M c F a r l a n e et al. Phys. Rev. D, 32:547, (1985). [49] D . H . W i l k i n s o n . Weak coupling constants i n nuclei. In H . O h t s u b o M . M o r i t a , H . E j i r i a n d T . Sato, editors, YAMADA CONFERENCE XXIII: Nuclear Weak Process and Nuclear Structure, pages 1-45. W o r l d Scientific, (1989). [50] J . C . H a r d y et al. Nucl. Phys., B509:429, (1990). [51] J . P . Deutsch a n d L . Grenacs. In IV International Conference Physics and Nuclear Structure, Dubna, USSR, (1971).  on High  Energy  [52] G . Savard et al. Phys. Rev. Lett., 74:1521, (1995). [53] M . G e l l - M a n n a n d M . Levy. Nuovo Cimento,  16:705, (1960).  [54] Y . N a m b u . Phys. Rev. Lett., 4:380, (1960). [55] M . L . Goldberger a n d S . B . T r e i m a n . Phys. Rev., 110:1178, (1958). [56] S . A . C o o n a n d M . D . Scaldron. Phys. Rev. D, 42:2256, (1990). [57] C A . D o m i n g u e z . Phys. Rev. D, 25:1937, (1982). [58] L . Wolfenstein. High Energy Physics and Nuclear Structure, P l e n u m Press, N e w Y o r k , (1970), page 661.  ed. S.  Devons.  [59] N . K a i s e r V . B e r n a r d a n d U l f - G . Meissner. Phys. Rev. D, 50:6899, (1994). [60] G . B a r d i n et al. Nucl. Phys., A352:365, (1981). a n d references therein. [61] J - M G a i l l a r d a n d G . Saurage. Ann. Rev. Nucl. Part. Sci,  34:351, (1984).  [62] M . O k a and K . K u b o d e r a . Phys. Lett., 90B:45, (1980). [63] J . Delorme et al. Ann.  of Phys, 102:273, (1976).  [64] M . E r i c s o n . Progress in Nuclear and Particle Oxford, (1978).  Physics.  [65] D . H . W i l k i n s o n . Nucl. Phys., A225:365, (1974). [66] C D . G o o d m a n . Nucl. Phys., A374:241c, (1982). [67] K . P . Jackson et al. Phys. Lett., 201B:25, (1988). [68] M . R h o . Ann. Rev. Nucl. Part. Sci., 34:531, (1984).  152  , page 67. P e r g a m o n ,  [69] V . Devanathan. Observables i n nuclear m u o n capture. I n H . O h t s u b o M . M o r i t a , H . E j i r i a n d T . Sato, editors, YAMADA CONFERENCE XXIII: Nuclear Weak Process and Nuclear Structure, pages 83-92. W o r l d Scientific, (1989). [70] A . A l b e r i g i Q u a r a n t a et al. Phys. Rev., 177:2118, (1969). [71] V . N . B y s t r i t s k i i et al. Sov. J. Phys. JETP, 39:19, (1974). [72] E . J . Bleser et al. Phys. Rev. Lett., 8:288, (1962). [73] J . E . R o t h b e r g et al. Phys. Rev., 132:2664, (1963). [74] G . J o n k m a n s et al., (1995). p u b l i c a t i o n i n preparation. [75] M . Hasinoff (spokesman), (1986). T R I U M F proposal 452. [76] D . S . A r m s t r o n g a n d T . P . Gorringe (spokespersons), (1995). E766: T R I U M F proposal submitted to the T R I U M F E E C . [77] G . B a r d i n et al. Phys. Lett., B l O 4 : 3 2 0 , (1981). [78] G . B a r d i n et al. Nucl. Phys., A 3 5 2 : 3 6 5 , (1981). [79] D . D . Bakalov et al. Nucl. Phys., A 3 8 4 : 3 0 2 , (1982). [80] M . G m i t r o a n d P . T r u o l . Adv. Nucl. Phys., 18:241, (1987). [81] A . J . Larabee et al. In P . D e p o m m i e r , editor, In Proceedings of the International Conference on Weak and Electromagnetic Interactions in Nuclei, page 641. E d i t i o n Frontieres, (1989). [82] P . C . Bergbusch. Radiative Muon Capture on 0, Al, Si, Ti, Zr, and Ag, and the Induced Weak Pseudoscalar Current. M a s t e r ' s thesis, 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 , (1995). unpublished. [83] D . S . A r m s t r o n g S. A h m a d , R . A . B u r n h a m , T . P . Gorringe, M . D . Hasinoff, A . J . Larabee, C . E . W a l t h a m , G . Azuelos, J . A . M a c d o n a l d , T . N u m a o , J M . Poutissou, M . Blecher, D . H . W r i g h t , E . T . H . Clifford, J . S u m m h a m m e r , P. D e p o m m i e r , R . Poutissou, H . M e s , a n d B . C . R o b e r t s o n . Phys. Rev. C, 43:1425, (1991). [84] M . G m i t r o , S. S. K a m a l o v , F . Simkovic, a n d A . A . Ovchinnikova. Nucl. Phys., A 5 O 7 : 7 0 7 , (1990). [85] M . G m i t r o , A . A . Ovchinnikova, a n d T . V . Tetereva. Nucl. Phys., A 4 5 3 : 6 8 5 , (1986). [86] P . C h r i s t i l l i n a n d M . G m i t r o . Phys. Lett., 1 5 O B : 5 0 , (1985). 153  [87] M . D 6 b e l i et al. Phys. Rev. C, 37:1633, (1988). [88] A . Frischknecht et al. Phys. Rev. C, 38:1996, (1988). [89] M . D . Hasinoff. R a d i a t i v e m u o n capture a n d the induced pseudoscalar coup l i n g constant. In C M Class a n d L . C o h e n , editors, Variation on Nuclear Themes, pages 165-186. W o r l d Scientific, (1991). [90] F . R o i g and J . Navarro. Phys. Lett, B236:393, (1990). [91] P. C h r i s t i l l i n . Nucl. Phys., A362:391, (1981). [92] D . S . A r m s t r o n g , A . Serna-Angel, S. A h m a d , G . Azuelos, W . B e r t l , M . Blecher C . Q . C h e n , P . D e p o m m i e r , T . v o n Egidy, T . P . Gorringe, M . D . Hasinoff, R . S . Henderson, A . J . Larabee, J . A . M a c d o n a l d , S . C . M c D o n a l d , J M . Poutissou, R . Poutissou, B . C . R o b e r t s o n , D . G . Sample, G . N . Taylor, D . H . W r i g h t , and N . S . Z h a n g . Phys. Rev. C, 46:1094, (1991). [93] A . Frischknecht et al. Phys. Rev. C, 32:1506, (1988). [94] P. C h r i s t i l l i n , M . R o s a - C l o t , and S. Servadio. Nucl. Phys., A345:331, (1980). [95] H . W . Fearing a n d M . S . Welsh. Phys. Rev. C, 46:2077, (1992). [96] J . P . Deutsch, L . Grenacs, J . L e h m a n n , P . L i p n i k , a n d P. C . M a c q . Lett, 28B:178, (1968).  Phys.  [97] A . Possoz et al. Phys. Lett, 50B:438, (1974). [98] A . Possoz et al. Phys. Lett, 70B:265, (1977). [99] L . Foldy a n d J . D . Walecka. Phys. Rev. B, 133:1339, (1965). [100] R . Parthasarathy a n d V . N . Sridhar. Phys. Rev. C, 23:861, (1981). [101] R . Parthasarathy a n d V . N . Sridhar. Phys. Lett,  106B:363, (1981).  [102] B . Holstein. Phys. Rev. D, 13:2499, (1976). [103] M . F u k u i et al. Prog. Theor. Phys., 70:827, (1983). [104] Y . K u n o et al. Phys. Lett, 29:270, (1984). [105] G . H . M i l l e r , M . Eckhause, F . R . K a n e , P . M a r t i n , a n d R . E . W e l s h . Lett, 41B:50, (1972). [106] L . P h . Roesch et al. Phys. Rev. Lett, 46:1507, (1981). [107] L . P h . Roesch et al. Phys. Lett, 107B:31, (1981).  154  Phys.  [108] M . K o b a y a s h i , N . Ohtsuka, H . Ohtsubo, A 3 1 2 : 3 7 7 , (1978).  Ann.  Phys.,  XIII.  World  Rev. Lett., 43:1083, (1974).  [ I l l ] M . M o r i t a et al. In A . P a s c o l i n i , editor, Particles Scientific, (1993). [112] I.S. Towner. therein.  Nucl.  A 3 7 2 : 4 4 4 5 , (1981).  [109] S. Ciechanowicz. Nucl. Phys., [110] Y . M a s u d a et al. Phys.  and M . M o r i t a .  Rev.  Nucl.  Part.  Sci.,  and Nuclei  36:115, (1986).  [113] W . C . H a x t o n and C . Johnson. Phys. Rev. Lett,  and references  65:1325, (1990).  [114] T . P . Gorringe, B . L . Johnson, D . S . A r m s t r o n g , J . Bauer, M . A . K o v a s h , M . D . Hasinoff, D . F . Measday, B . A . M o f t a h , R . Porter, and D . H . W r i g h t . Phys. Rev. Lett, 72:3472, (1994). [115] A . Echegoyan, W . M . D . M c R a e , and B . A . B r o w n . Technical report, M S U N S C L Report N o . 524, Oxford-Buenos Aires shell model code O X B A S H 8 2 , (1984). (unpublished). [116] B . Siebels, T . P . Gorringe, W . P . A l f o r d , J . Bauer, J . Evens, S. E l - K a t e b , K . P . Jackson, A . T r u d e l , and S.Yen. Phys. Rev. C, 5 2 : 1 , (1994). [117] J . Delorme and M . E r i c s o n . Phys. Rev. C, 4 9 : R 1 7 6 3 , (1994). [118] M . E r i c s o n et al. Phys.  Lett,  B 4 5 : 1 9 , (1973).  [119] R . P a r t h a s a r a t h y and V . N . Sridhar. Phys. Rev. C, 18:1796, (1978). [120] N . P . P o p o v . Sov. Phys.  JETP,  17:1130, (1963).  [121] Z . Oziewicz and N . P . P o p o v . Phys.  Lett,  15:903, (1965).  [122] G . M . Bukhvostov and N . P . P o p o v . Sov. J. Nucl. Phys., 6:903, (1968). [123] G . M . B u k h a t and N . P . P o p o v . Sov. Phys.  JETP,  [124] A . P . Bukhvostov and N . P . P o p o v . Nucl. Phys.,  19:1200, (1964). A 1 4 7 : 3 8 5 , (1970).  [125] Z . Oziewicz. M u o n capture phenomenology (spin zero targets). report, J I N R report E4-8350, D u b n a , (1974). (unpublished).  Technical  [126] K . Junker, V . A . K u z ' m i n , A . A . O v i c h i n n i k o v a , and T . V . Tetereva. In H . E j i r i , T . K i s h i m o t o , and T . Sato, editors, Weak and Electromagnetic Interactions in Nuclei (WEIN 95). W o r l d Scientific, (1995). C a l c u l a t i o n of the G a m m a N e u t r i n o A n g u l a r C o r r e l a t i o n Coefficients of the O r d i n a r y M u o n Capture on Si. 2 8  155  [127] A . Fujii a n d H . Primakoff. Nuovo Cimento, 12:327, (1959). [128] L . L . F o l d y a n d S. A . Wouthuysen. Phys. Rev., 78:29, (1950). [129] S. Ciechanowicz. Nucl. Phys., A267:472, (1976). [130] B . H . W i l d e n t h a l a n d J . B . M c G r o r y . Phys. Rev., C7:714, (1973). [131] M . J . A . DeVoigt a n d B . H . W i l d e n t h a l . Nucl. Phys., A2O6:305, (1973). [132] M . M o r i t a . Beta Decay and Muon Capture. W . A . B e n j a m i n , Inc., (1973). [133] V . Devanathan a n d P . R . S u b r a m a n i a n . Ann. of Phys (N.Y.), 92:25, (1975). [134] T . W . D o n n e l l y a n d G . E . W a l k e r . Ann. of Phys (N.Y.), 60:209, (1970). [135] D . S . A r m s t r o n g , (1992). G a m m a - N e u t r i n o A n g u l a r C o r r e l a t i o n i n M u o n C a p ture o n S i , E x p t . 570 Notes o n Theory, unpublished. 2 8  [136] R . A . E r a m z h y a n . M u o n capture: present status a n d perspectives. I n Weak and Electromagnetic Interactions in Nuclei (WEIN 92). W o r l d Scientific, (1992). [137] V . A . K u z ' m i n a n d A . A . Ovchinnikova a n d T . V . Tetereva 1995, private communication. [138] B . A . B r o w n , a n d B . H . W i l d e n t h a l . Ann. Rev. Nucl. Part. Sci., 38:29, (1988). [139] T . P . Gorringe, (1993). M U C A P T U R E program: calculates m u o n capture rates a n d associated observables (Private communication). [140] B . W i l d e n t h a l . At. Data & Nucl. Data Tables, 33:347, (1985). [141] T . P . Gorringe, B . L . Johnson, D . S . A r m s t r o n g , J . Bauer, P . G u m p l i n g e r , M . D . Hasinoff, M . A . K o v a s h , D . F . Measday, B . A . M o f t a h , R . Porter, a n d D . H . W r i g h t . Phys. Lett, B309:241, (1993). [142] L . Grenacs, J . P . Deutsch, P . L i p n i k , a n d P. C . M a c q . Nucl. Instrum. Methods, 58:164, (1968). [143] P . M E n d t . Nucl. Phys., A 5 2 1 : l , (1990). [144] G . H . M i l l e r , M . Eckhause, F . R . K a n e , P . M a r t i n , a n d R . E . W e l s h . Phys. Rev. Lett, 29:1194, (1972). [145] G . H . M i l l e r . Gamma Rays Following Negative Muon Capture In Medium Z Nuclei. P h D thesis, College of W i l l i a m a n d M a r y , (1972). u n p u b l i s h e d .  156  [146] V . B r u d a n i n , V . Egorov, T . F i l i p o v a , A . K a c h a l k i n , V . Kovaleno, A . Salam a t i n , Y u . Shitov, I. Stekl, , S. Vassiliev, V . V o r o b e l , T s . V y l o v , I. Y u t l a n d o v , Sh. Zaparov, J . P . Deutsch, R . Prieels, L . Grenacs, J . R a k , a n d C h . Briangon. Nucl. Phys., A587:577, (1995). T . N i i z e k i et al. Nucl. Phys., A577:37c, (1994). R . A . E r a m z h y a n , M . G m i t r o , R . A . Sakaev, a n d L . A . Tosunjan. Nucl. A290:294, (1977).  Phys.,  K . B h a r u t h - R a m et al. Nucl. Phys., A278:285, (1977). F . P. B r a d y et al. Phys. Rev. Lett, 48:860, (1982). P. T r u o l et al, S I N A n n u a l report 1976 page E 4 5 . A r i e T a a l . Deeply Bound Orbits in Pionic Atoms and the Optical Potential. P h D thesis, Technische Universiteit van Delft, (1989). unpublished. ;  R . Schneider, A . Richter, A . Schwierczinski, E . Spamer, O . T i t z e , " and W . Knupfer. Nucl Phys., A323:13, (1979). ;  B . D . A n d e r s o n , N . T a m i m i , A . R . B a l d w i n , M . Elaasar, R . Madey, D . M . M a n ley, M . M o s t a j a b o d d a ' v a t i , J . W . W a t s o n , and W . M . Zhang. Phys. Rev. C, 43:50, (1991). H . H . Schmidt et al. Phys. Rev. C, 25:2888, (1982). J . Vernotte et al. Phys. Rev. C, 49:1559, (1994). B . M a c D o n a l d , J . A . D i a z , S . N . K a p l a n , a n d R . V . P y l e . Phys. Rev. B, 5:1253, (1965). G . H . M i l l e r , M . Eckhause, P . M a r t i n , a n d R . E . W e l s h . Phys. Rev. C, 6:487, (1972). D . S . A r m s t r o n g (spokesperson), (1989). T R I U M F proposal 570. B . A . M o f t a h . A Study of X and Gamma Rays Following Muon Capture in Si. Master's thesis, T h e University of B r i t i s h C o l u m b i a , (1991). unpublished. 28  T R I U M F Users Executive C o m m i t t e e . T R I U M F Users H a n d b o o k . Technical report, T R I U M F , (1987). O r i g i n a l l y designed by J . - P . M a r t i n , University of M o n t r e a l ; M o d i f i e d by D . M a a s , University of B r i t i s h C o l u m b i a . C M . Davisson. Interaction of 7-radiation w i t h matter. In K . Siegbahn, editor, Beta- and Gamma-Ray Spectroscopy, page ch. II. Interscience, New Y o r k , (1955). 157  [164] G . F . K n o l l . Radiation Detection and Measurement. J o h n W i l e y & Sons, second edition, (1989). [165] K . D e b e r t i n a n d R . G . Helmer. Gamma-and-X-Ray Spectrometry with Semiconductor Detectors. N o r t h - H o l l a n d , (1988). [166] R . H . P e h l et al. IEEE Trans. Nucl. Sci. NS-26, 1:321, (1979). [167] H . W K r a n e r . IEEE Trans. Nucl. Sci. NS-27, 1:218, (1980). [168] R . L . B u n t i n g a n d J . J K r a u s h a a r . Nucl. Instrum. Methods, 118:565, (1974). [169] C . C h a s m a n et al. Nucl. Instrum. Methods, 37:1, (1965). [170] J . L . R o d d a et al. Nucl. Instrum. Methods, 74:224, (1969). [171] G . W . P h i l l i p s a n d K . W . M a r l o w . Nucl. Instrum. Methods, 137:525, (1976). [172] J . V . M a h e r et al. Phys. Rev. C, 5:1313,1322, (1972). [173] F . A . E l - A k a d et al. Nucl. Phys., A 2 8 3 : 1 2 , (1977). [174] J . F . Ziegler. Stopping Cross-sections for Energetic Ions in all Elements. Pergamon Press, (1980). [175] T R a n s p o r t of Ions i n M a t t e r code ( T R I M ) , 1989. Available from J . F . Ziegler, IBM-Research,28-0, Y o r k t o w n , N Y 10598 U S A . [176] J . P . Biersack J . F . Ziegler a n d U . L i t t m a r k . The Stopping and Range of Ions in Solids. P e r g a m o n Press, (1985). [177] B u x t o n L . Johnson. A spectra analysis program: D I S P L A Y . Initially w r i t t e n by B u x t o n L . Johnson (1988) a n d subsequently revised b y B e l a l A . M o f t a h (1993). [178] P . Vogel. Phys. Rev. A, 22:1600, (1980). [179] F . J . H a r t m a n n , T . v o n Egidy, R . B e r g m a n n , M . K l e b e r , H . - J . Pfeiffer, K . Springerr, a n d H . D a n i e l . Phys. Rev. Lett., 37:331, (1976). [180] T . v o n Egidy, W . Denk, R . B e r g m a n n , H . D a n i e l F . J . H a r t m a n n , J . J . Reidy, and W . W i l h e l m . Phys. Rev. A, 23:427, (1981). [181] J . H . H u b b l e . Int. J. Appl. Radiat. & Iso., 33:1269, (1982). [182] B . Lawergren a n d J . Beyea. Phys. Rev. C, 6:2082, (1972). [183] P . M E n d t a n d V a n der L e u n . Nucl. Phys., A 3 1 0 : l , (1978). [184] M I N U I T F u n c t i o n M i n i m i z a t i o n a n d E r r o r A n a l y s i s , C E R N Reference M a n ual, Version 92.1 ( M a r c h 1992). 158  [185] A . A s t b u r y et al. Proc. Phys. Soc, [186] A . O . Weissenberg. Muons.  78:1144, (1961).  N o r t h - H o l l a n d , (1967).  [187] V . B r u d a n i n , V . E g o r o v , T . F i l i p o v a , T . M a m e d o v , A . S a l a m a t i n , Y u . S h i tov, I. Stekl, V . V o r o b e l , T s . V y l o v , I. Y u t l a n d o v , S h . Zaparov, J . Deutsch, R . Prieels, L . Grenacs, a n d C h . B r i a n c o n . In H . E j i r i , H . K i s h i m o t o , a n d T . Sato, editors, Weak and Electromagnetic Interactions in Nuclei (WEIN 95). W o r l d Scientific, (1995). Investigation of spin-neutrino angular correlations i n the capture of polarized muons by silicon nuclei. [188] K . A r s t i l a , J . K e i n o n e n , and P . T i k k a n e n . 101:321, (1995).  Nucl.  Instrum.  [189] R . Parthasarathy, Comments on the Recent Measurement Capture by Si, 1995 (unpublished). 28  159  Methods,  of gp/gA  in  B  Muon  Appendix A The Peak-Fitting Computer Program Listing T h e p r o g r a m used to fit the 1229 k e V a n d 2171 k e V simultaneously is given here. C**  C**  *  C**  Program DOPFIT minimizes the c h i squared f o r a set of 1-D data *  C**  using a s t r a i g h t l i n e .  C**  found by MINUIT routines from CERN l i b r a r y CERNlib.  The c o e f f i c i e n t s and t h e i r e r r o r s are * *  C**  *  C**  Modified to use the IMSL DTWODQ routine and include Slowing-Down*  C**  e f f e c t s ( c o n s t a n t ) and the RF2 peak-shape response f u n c t i o n .  *  C** PROGRAM DOPFITc.IMSL 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)  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 u s i n g d o u b l e  precision.  *  C* CALL  MINUIT(FCN.O)  C* C*  Exit  program DOPFITc_IMSL.  *  C* CALL EXIT END C**  C**  *  C**  Subroutine  C**  minimized  C**  requires.  FCN c a l c u l a t e s t h e v a l u e o f t h e f u n c t i o n t o be or studied.  T h i s i s t h e FCN s u b r o u t i n e MINUIT  * * *  C**  *  C** SUBROUTINE FCN (NPAR,G,F,X,IFLAG) C* C*  D e c l a r e r e a l v a r i a b l e s as double  precision.  *  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  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  IFLAG  —  A MARKER WHOSE MEANING IS DESCRIBED BELOW:  !  1 = Initializing  PARAMETERS  entry.  Read i n a l l n e c e s s a r y  !  d a t a t o FCN, c a l c u l a t e c o n s t a n t s ,  !  input  !  VALUES  print  gradient.  Calculate the  !  d e r i v a t i v e s i n v e c t o r G and t h e f u n c t i o n  !  i n F at the point 3 = Terminating  summaries, o u t p u t  !  plot.  Write  f u n c t i o n value  out any s p e c i a l  graphs,  t a b l e s , e t c . f o r t h e minimum  4 = Normal e n t r y w i t h o u t  !  value  X.  entry.  ! !  and g r a p h  i f desired, etc.  2 = Normal e n t r y w i t h  !  special  gradient.  F at point  C a l c u l a t e only the  X.  !IMSL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! C* C*  Declare  Common B l o c k  PARAMETER  C* DIMENSION  XX(34)  COMMON /PARAMETER/  XX,GAIN,OFFSET,XDT  !IMSL!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! C* C*  Declare  local  subroutine  variables.  (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 c o m p u t a t i o n (added Y v a l u e s even f o r a s u b t r a c t e d treat  ERR(n)  —  errors properly,  spectra,  B. M o f t a h May 16,1994)  ERROR IN DATA POINT where n i s t h e d a t a p o i n t  number  C* C*  Initialize  pi.  *  C* PI=3.1415927  O C*  IFLAG = 1.  Initializing  entry.  Read i n a l l n e c e s s a r y  C*  d a t a t o FCN, c a l c u l a t e c o n s t a n t s , p r i n t  C*  desired,  and g r a p h i n p u t  etc.  special * i f  * *  C* IF  (IFLAG .EQ. 1) THEN  C* C*  Read i n d a t a f r o m i n p u t f i l e .  C* READ (5,*) NDATA1,GAIN1,0FFSET1 DO 100 I=1,NDATA1 READ (5,*) X V A L U E l ( I ) , Y V A L U E 1 ( I ) , Y E R R 1 ( 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 103  (5,*) XVALUE4(I),YVALUE4(I),YERR4(I)  CONTINUE  C* C*  Compute e r r o r o f each d a t a 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 o f 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 t h e c h i - s q u a r e d  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  f i t to the polynomial.  *  XX(I)=X(I) 115  CONTINUE BETAO=0.0037484  C  2171keV G e l 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) F I = F I +((YVALUE1(K)-YFIT1)/ERR1(K))**2. 120 C  CONTINUE 1229keV G e l 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 C  CONTINUE 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 C  CONTINUE 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  C*  summaries, o u t p u t  entry.  tables,  W r i t e o u t any s p e c i a l g r a p h s ,  e t c . f o r t h e minimum p l o t .  * *  C* IF  (IFLAG  .EQ. 3) THEN  WRITE (*,*) 'DOPPLER PEAK FITTTING COMPLETE' ENDIF C* C*  Exit  s u b r o u t i n e FCN.  *  C* RETURN END C**  C**  *  C**  F u n c t i o n F F computes t h e v a l u e o f t h e 2-D f u n c t i o n t o be  C**  i n t e g r a t e d by t h e IMSL code DTWODQ  C**  * * *  C**  166  C===========================2171 KeV Gel============================== DOUBLE PRECISION FUNCTION FF1  (X,Y)  C* C*  Declare r e a l  variables  as d o u b l e p r e c i s i o n .  *  C* IMPLICIT DOUBLE PRECISION DIMENSION  (A-H,0-Z)  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. X X ( 1 2 ) ) ) 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= ( X X ( 1 ) / ( X X ( 3 ) * 1 . 0 D - 1 5 ) ) * D E X P ( - Y / ( X X ( 3 ) * 1 . 0 D - 1 5 ) ) 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  variables  as d o u b l e p r e c i s i o n .  C* IMPLICIT DOUBLE PRECISION DIMENSION  (A-H,0-Z)  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. X X ( 1 9 ) ) ) 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= 1  (XX(13)/(XX(3)*1.OD-15))*DEXP(-Y/(XX(3)*1.OD-15)) *(A*CC+B*(X-XX(14))**2.*CC**3.)  *(PDT+STEP)  C* RETURN END C==============================2171 keV Ge2 ========================== DOUBLE PRECISION FUNCTION F F 3 (X,Y) C* C*  Declare r e a l variables  as d o u b l e 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 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(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= 1  (XX(28)/(XX(3)*1.OD-15))*DEXP(-Y/(XX(3)*1.OD-15)) *(A*CC+B*(X-XX(29))**2.*CC**3.)  *(PDT+STEP)  C* RETURN END C**  c** C**  * F u n c t i o n GG t o e v a l u a t e t h e l o w e r l i m i t s  of t h e i n n e r  C**  integral* *  C** DOUBLE PRECISION FUNCTION GG  (X)  C* C*  Declare r e a l  variables  as d o u b l e 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**  * F u n c t i o n HH t o e v a l u a t e t h e upper l i m i t s  C**  of the i n n e r  integral* *  C** DOUBLE PRECISION FUNCTION HH1  (X)  C* C*  Declare r e a l  v a r i a b l e s as d o u b l e 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 G e l ===================================== DOUBLE PRECISION FUNCTION HH2 (X) C* C*  Declare r e a l  variables  as d o u b l e 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  real variables  as d o u b l e p r e c i s i o n .  C*  171  *  IMPLICIT DOUBLE PRECISION  (A-H,0-Z)  DIMENSION XX(34) COMMON /PARAMETER/ XX,GAIN,OFFSET,XDT c* 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 variables  as d o u b l e p r e c i s i o n .  *  C* IMPLICIT DOUBLE PRECISION  (A-H.O-Z)  DIMENSION XX(34) C*  COMMON /PARAMETER/ XX,GAIN,OFFSET,XDT 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** C**  * F u n c t i o n ERFc c a l c u l a t e s  the value  function.  o f t h e COMPLEMENTARY e r r o r  * *  C**  *  172  c** FUNCTION ERFC(X) C* C*  Declare  real variables  as d o u b l e p r e c i s i o n .  *  C* IMPLICIT DOUBLE PRECISION  (A-H.O-Z)  C* C*  Calculate  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)*T1  0.3565638)*T+0.3193815) IF  100 110  (X) 100,110,110  PP=1.0-PP ERFC=1.-(PP*2-1.0)  C* C*  Exit function  ERF.  *  C* RETURN END  173  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0084982/manifest

Comment

Related Items