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Low energy elastic scattering and the pionic atom anomaly Hanna, Mark 1988

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LOW ENERGY ELASTIC SCATTERING AND THE PIONIC ATOM ANOMALY B y MARK HANNA B.Sc. (Hons ) , C o n c o r d i a Un i ve r s i t y , 1986 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S D e p a r t m e n t of P h y s i c s W e accept th i s thes i s as c o n f o r m i n g t o the r equ i r ed s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA Sep tember 1988 © M a r k H a n n a , 1988 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 of The University of British Columbia Vancouver, Canada Date QCTOJSeg DE-6 (2/88) A b s t r a c t D i f f e r e n t i a l cross sect ions fo r the e la s t i c s c a t t e r i n g o f l o w energy (20 M e V ) po s i t i v e a n d nega t i ve p i on s f r o m 1 2 C a n d 4 0 C a have been measu red . T h i s mea su rement was p e r f o r m e d at TRIUMF u s i n g t he QQD l ow energy p i o n spec t r omete r at 11 angles be tween 45° a n d 125°. T h e 7r+ 1 2 C d a t a are i n g o o d agreement w i t h [ O B G + 8 3 ] , as are the T r " 1 2 C d a t a w i t h [ W B B + 8 7 ] , i n d i c a t i n g t h a t the o v e r a l l n o r m a l i z a t i o n of the cross sect ions is good . Howeve r , t he IT* 4 0 C a d a t a do no t agree w e l l w i t h the p r e v i o u s l y p u b l i s h e d d a t a o f [ W M R + 8 8 ] . A n o p t i c a l p o t e n t i a l m o d e l , whose pa r amete r s were d e t e r m i n e d f r o m p i o n i c a t o m d a t a , was u sed t o p r ed i c t these d i f fe rent i a l cross sect ions. T w o sets of p a r a m e t e r s were u sed i n the m o d e l . O n e set was d e t e r m i n e d f r o m fits t o " n o r m a l " p i o n i c a t o m d a t a , w h i l e ano the r set was e x t r a c t e d f r o m fits t o " a n o m a l o u s " d a t a whe re l e ve l sh i f t s a n d w i d t h s do not c o m p a r e t o t h e o r e t i c a l va lues o b t a i n e d w h e n the n o r m a l set o f pa ramete r s is used. In a l l t he above cases, the e x p e r i m e n t a l d a t a best fits t he o p t i c a l m o d e l p r ed i c t i o n s of [ FG80,F r i 88b ] w h e n the n o r m a l set of p a r a m e t e r s is used. T h e q u a l i t a t i v e agreement o f the d a t a t o the " n o r m a l " o p t i c a l m o d e l p r e d i c t i o n s i nd i ca te s t h a t the p i o n i c a n o m a l y effects do no t e x t e n d t o po s i t i v e p i o n energy values. i i T a b l e o f C o n t e n t s Abstract ii List of Tables v List of Figures vi Acknowledgements vii I Introduction 1 L l The Pionic Atom 1 1.2 The Anomaly 3 1.3 Motivation for this Experiment 4 1.4 Experiment 373 5 II Theoretical Description 6 II. 1 Nuclear Scattering 6 11.2 The Optical Potential Model 7 11.3 Outline of the 7r-Nucleus Potential 8 11.4 Potential Model Calculations 10 III The Experiment 15 111.1 TRIUMF's M13 Low Energy IT - n Channel 15 111.2 Q Q D Spectrometer 18 111.3 Wire Chamber Design and Calibration 20 111.4 Momentum Calibrations 23 111.4.1 Target Traceback Coefficients 24 111.4.2 Magnet Transfer Coefficients 24 iii 111.5 Targets 26 111.6 Data Acquisition 27 IV Data Analysis 31 IV. 1 Off-Line Analysis Program 31 IV. 2 Event Definitions 31 IV.2.1 Spectrometer Events 32 IV.2.2 Beam Sample Events 34 IV.3 Cross Sections 34 IV.4 Cross Section Errors 40 IV.5 Results and Comparisons 41 IV.6 Optical Potential Model Comparisons 50 IV. 7 Conclusion 50 Bibliography 56 A Traceback Fitting 60 iv L i s t o f T a b l e s I Fermi three parameter fit values for nuclear density distributions in 1 2 C and 4 0 C a 11 II Optical potential parameters obtained from fits to normal and anomalous pionic atom data 13 III Optical potential parameters extrapolated to 20MeV 14 IV Specifications of targets used in Exp. 373 28 V Measured 7 r + differential cross sections for 1 2 C at 20 MeV. . . . . 42 VI Measured 7T~ differential cross sections for 1 2 C at 20 MeV 43 VII Measured TT+ differential cross sections for 4 0 C a at 20 MeV 44 VIII Measured ir~ differential cross sections for 4 0 C a at 20 MeV 45 IX Optical potential model fit summary 55 v L i s t o f F i g u r e s 1 T h e effect o f the s t r ong i n t e r a c t i o n o n l eve l sh i f t s a n d w i d t h s . . . . 3 2 S c h e m a t i c l a you t of the TRIUMF M13 p i o n c h a n n e l 16 3 S c h e m a t i c v i e w o f the QQD s pec t r omete r 19 4 P o s i t i o n i n f o r m a t i o n o b t a i n e d f r o m x a n d y T D C d i f ference d a t a . . 22 5 T h e ^-d i f ference ( D D I F ) cut 27 6 Ta r ge t h o l d e r u sed for the C a l c i u m target 28 7 T h e e x p e r i m e n t a l e lec t ron i c log ic 30 8 T y p i c a l t ime-o f - f l i gh t s p e c t r u m fo r M13 a n d QQD 33 9 D i f fe rence i n y p o s i t i o n i n W C 4 a n d W C 5 a n d b o x tex t 35 10 B e a m samp le s p e c t r u m for po s i t i ve a n d nega t i ve p o l a r i t y r un s . . . . 36 11 A c o m p a r i s o n be tween [OBG+83] a n d TRIUMF T T + 1 2 C d a t a . . . 46 12 A c o m p a r i s o n be tween [WBB+87 ] a n d TRIUMF TT~ 1 2 C d a t a . . . 47 13 A c o m p a r i s o n between [ W M R + 8 8 ] a n d TRIUMF 7r+ 4 0 C a d a t a . . 48 14 A c o m p a r i s o n be tween [ W M R + 8 8 ] a n d TRIUMF TT~ 4 0 C a d a t a . . 49 15 N o r m a l a n d anoma lou s p red i c t i on s c o m p a r e d t o 7r + 1 2 C d a t a . . . . 51 16 N o r m a l a n d anoma lou s p r ed i c t i o n s c o m p a r e d t o TT~ 1 2 C d a t a . . . . 52 17 N o r m a l a n d anoma lou s p r ed i c t i o n s c o m p a r e d t o 7 r + 4 0 C a d a t a . . . . 53 18 N o r m a l a n d anoma lou s p r ed i c t i o n s c o m p a r e d t o TT~ 4 0 C a d a t a . . . . 54 v i A c k n o w l e d g e m e n t s I would like to gratefully acknowledge Marty Rozon, who's help throughout the production of this thesis has been greatly appreciated. His guidance and patience during what seemed like an eternal period of analysis, and his knowledge of experimental physics has proven invaluable. I wish to thank my supervisor, Dick Johnson, first and foremost for his persistence in getting me to complete this project, also for his ability to keep his unique sense of humor, sending me to Lake Louise, and making incredible culinary delicacies (not to mention his fearless handling of sharks). More thanks go to Eli Friedman and Oded Meirav for there assistance with the theoretical aspect of this document. Special thanks go to the "semi-cool dude", Rinaldo Rui, for seeing the lighter side of a problem and keeping me laughing, and to "the baron", Rigo Olsweski, (sorry about the van) whose computer wizardry and beer drinking is second to none. Additional thanks go to Andrew (le hot racer!), John (la premiere etoile!), Chris (where's the beef?), Reena, Jean, Vesna, Grant (don worry, be appy), Peter, Martin, Angela, Niall, Dave and to those killer ping pong paddle pushers Gio, Hon, and Richard. I couldn't have gone through all those late nights without you, thanks a million for the fun and laughs. Marcello (le golfeur extraordinaire) also deserves special mention for indirectly turning me into an menacing biker. Finally, I would like to sincerely thank my family for their patience and support throughout my academic years. I couldn't have done it without you. I would also like to express my deep appreciation to Christy, who's encouragement (and scolding for staying up late and eating junk food) kept me going during the final stages of this project. vii C h a p t e r I I n t r o d u c t i o n Man's curiosity for understanding the basic forces in nature has always been an insatiable one. Only in the last century has man begun to unravel the mysteries that lie deep in the heart of matter. In 1935 Yukawa [Yuk35] proposed that the nuclear force, which tightly binds the atomic nucleus, is mediated by some massive particle (massf«150 MeV), similar to the way that the electromagnetic interaction is mediated by massless photons. The pi-meson was first seen by Lattes et al. [LMP047] in photographic emulsions exposed to cosmic rays on top of a mountain. These pions, identified with the Yukawa pion, were postulated to exist in three charge states. The pion had to exist in positive, negative, and neutral charge states to account for the known strong interactions. These states were later confirmed experimentally. 1.1 The Pionic Atom Almost immediately after the discovery of the pion, the existence of mesic atoms were predicted by both Wheeler [Whe47] and Fermi and Teller [FT47]. A pionic atom is formed when a negatively charged pion is stopped in some material, then captured by an atom in a high atomic orbit. The pion, because of its large mass, orbits at levels much closer to the nucleus than that of a corresponding electron orbit. It then cascades down to lower energy orbits, first by Auger electron processes, and then by the emission of X-rays. Once the pion's wavefunction 1 over laps w i t h t he nuc lea r wave func t i on , the cascade s top s 1 . T h e p i o n t h e n i n te rac t s s t rong ly , a n d is ab so rbed b y the nuc leus . M o s t o f the p i o n ' s a t o m i c l i fe is spent at levels l ower t h a n t he B o h r o r b i t s of t he e l ec t ron . D u e t o the p i o n ' s la rge mass r e l a t i v e t o t he e l ec t r on , i t ' s o r b i t i n g levels a r o u n d the nuc leus are w e l l w i t h i n the lowest e l e c t r on she l l a n d t hu s c an be t r e a t e d l i ke a h y d r o g e n a t o m . T h e h i ghe r energy levels c a n be c a l c u l a t e d d i r e c t l y f r o m e l e c t r omagne t i c theory. However , w h e n the p i o n gets close e n o u g h t o t he nuc leus , i t s ta r t s t o feel t he effects o f the s t r ong force, a n d t h a t re su l t s i n a sh i f t a n d b r o a d e n i n g of the energy levels. T h e lower t he energy l e ve l , t he s t ronger the nuc l ea r i n t e r a c t i o n , a n d consequent l y o n l y the lowest energy l e ve l is s i gn i f i c an t l y a f fected before nuc lea r a b s o r p t i o n occur s . It is therefore the last ob se rvab le X - r a y t h a t is s i gn i f i c an t l y s h i f t ed a n d b r o a d e n e d i n energy f r o m t h a t of p u r e l y e l e c t r omagne t i c cons iderat ions . A s we l l as th i s sh i f t i n energy, t he lowest energy l e ve l is su scept ib le t o " a b s o r p t i v e b r o a d e n i n g " [Br i84]. T h i s ar ises due t o the s ho r t ened l i f e t i m e of t he p i o n i n a n o r b i t whe re the s t r ong force is encounte red . T h e u n c e r t a i n t y p r i n c i p l e states t ha t AE • At > h, a n d hence t he l i ne w i d t h is b r oadened . E x p e r i m e n t a l l y , these effects are v i e w e d as a L o r e n t z i a n b r o a d e n e d X - r a y l i ne , whose c en t r o i d is sh i f ted f r o m a p o s i t i o n e x p e c t e d f r o m a p u r e l y e l e c t r omagne t i c i n t e r a c t i o n . T h i s is s hown i n f i gu re 1. T h e t h e o r e t i c a l de s c r i p t i on o f the l eve l sh i f t s a n d w i d t h s i n p i o n i c a t oms was i n i t i a l l y d educed f r o m low energy i r ~ — nuc l eon s c a t t e r i n g d a t a u s i n g first o r d e r p e r t u r b a t i o n theory. T h i s p r o ved t o be i n a c cu r a t e s ince the m o d i f i c a t i o n o f the p i o n w a v e f u n c t i o n due t o the s t r ong shor t range p ion -nuc leus i n t e r a c t i o n ( l ead i ng t o ab s o rp t i on ) , was no t a ccoun ted for. T h i s l ed t o a s em i - phenomeno l og i c a l 1 ' Stops ' is a rather strong word. If the negative pion reaches the Is level (light atoms) obviously the cascade stops since it 's at the ground state. Above 2 0 N a the cascade doesn't exactly 'stop' when the wavefunctions overlap, rather nuclear absorption competes and rapidly dominates. In actuality, absorption may occur from different levels in the same atom. This is indicated experimentally by the small intensity of low transition X-rays in larger atoms. 2 UNPERTURBED GAUSSIAN LINESHAPE LORENTZIAN BROADENED LINESHAPE SHIFT Figure 1: T h e effect of the strong interaction on level shifts and widths. approach using mult iple scattering theory. The result was a nonlocal pion-nuclear potential derived by Er icson and Er icson [EE66]. T h e predictions of the potential model agreed satisfactorily w i t h experimental level shift and w i d t h data i n pionic atoms for elements throughout the periodic table. T h e optical potential model w i l l be discussed i n following chapter. 1.2 T h e A n o m a l y T h e negative pion-nucleus optical potential has been very successful i n reproducing experimental results on strong interaction level shifts and widths i n pionic atoms [Fri88b]. In the past few years, improved experimental techniques have made it possible to study levels that were previously too broad and too weak to measure w i t h any precision 2 . These levels, whose p ion wavefunction overlap 2The last observable X-ray (i.e. 2p —• Is) in medium to heavy nuclei (Na or Mg) is naturally weak since the cross section for absorption is already large at the 2p level. It is this transition that m o r e w i t h t he nuc l ea r w a v e f u n c t i o n t h a n those o f p rev i ou s l y s t u d i e d levels, have been mea su r ed u s i n g C o m p t o n suppres s ion spec t romete r s [Br i84]. The se t ypes of spec t romete r s gene ra l l y use a S o d i u m I od ide ( N a l ) o r B i s m u t h G e r m a n a t e ( B i 4 G e 3 0 i 2 ) de tec to r t o i den t i f y C o m p t o n s ca t te red e lect rons f r o m the m a i n s o l i d s tate detecto r . T h e s i gna l f r o m th i s detec to r is t h e n u sed as a v e t o fo r the C o m p t o n s ca t te red event, t hu s c r ea t i n g a s p e c t r u m i n w h i c h the t o t a l a b s o r p t i o n peak fo r the X - r a y s is m u c h m o r e p r onounced , d i s t i n c t X - r a y peak. O w i n g t o th i s e x p e r i m e n t a l advance, i t was observed t h a t a s u b s t a n t i a l f r a c t i o n of levels i n t he Is, 2p, a n d 3d range, have w i d t h s na r rower t h a n e s t i m a t e d by the o p t i c a l m o d e l . The se w i d t h s were seen t o s a t u r a t e 3 w i t h i n c rea s i n g Z. A l s o , t he sh i f t o f m a n y of these levels were l a rge r t h a n p r e d i c t e d i n d i c a t i n g a repu l s i ve p o t e n t i a l . N e w d a t a m a y shed some l i gh t t owa rd s e x p l a i n i n g these p h e n o m e n a i f t h e y r e a l l y ex i s t . 1.3 Motivation for this Experiment O p t i c a l p o t e n t i a l mode l s have success fu l ly r e p r o d u c e d numerou s p i o n i c a t o m d a t a t h r o u g h o u t t he p e r i o d i c tab le . However , t hey cont inue t o p r e d i c t sh i f t s a n d w i d t h s a f a c t o r of two o r greater t h a t e x p e r i m e n t a l l y ob se rved fo r the t r a n s i t i o n t o the last obse rvab le l e ve l i n a p a r t i c u l a r a t o m . T h e anoma lou s sh i f t s appea r t o be e x p l a i n a b l e fo r va r i ou s reasons [BFG83 ,Sek82 ] , w h i c h ar i se f r o m a c a n c e l l a t i o n of effects f r o m di f ferent pa r t s of the o p t i c a l p o t e n t i a l [MS85]. The se pa r t s i n c l u d e r epu l s i ve 3-wave a n d the a t t r a c t i v e p-wave te rms . T h e anoma l ou s w i d t h s however, have r e m a i n e d w i t h o u t any reasonab le e x p l a n a t i o n . T h e u n e x p l a i n e d e x p e r i m e n t a l ex i s tence o f these w i d t h s i nd i ca te s t h a t there m a y be some i m p o r t a n t p h y s i c a l f a c t o r w h i c h was ove r l ooked d u r i n g the c o n s t r u c t i o n o f t he p o t e n t i a l . If t h i s is t r ue , we have a very i n c o m p l e t e u n d e r s t a n d i n g of l ow energy p i o n phys ic s . T o feels the largest strong interaction effects and so is a most sensitive test of theoretical models. 3 A s Z increases, the observed widths of the X-rays remains constant at a factor f» | of that predicted by theory. To date there is no explanation for this phenomenon. 4 determine whether or not the anomaly indeed exists at small positive pion energies, a low energy elastic scattering experiment was proposed to study elastic scattering from a nucleus that does not exhibit the anomaly and one that does. 1.4 Experiment 373 The pionic atom optical potentials can be successfully modified to model 7r-nucleus scattering at positive energies with some minor corrections for the energy increase. Therefore, we may predict differential cross sections for an angular distribution of a low energy elastic scattering experiment. Experimentally there has been no recognition of anomalous effects for elastic 7r-scattering data in the range of 30 - 50 MeV [SMC79,SMOY80,SM83]. This indicates, assuming the anomaly exists, that either the energies are too high to show the effect unambiguously, or discrepancies in the analyses are too large. Thus the experiment should be performed at energies < 30 MeV. An experiment employing the QQD spectrometer on TRIUMF's M13 beam line was executed. To determine whether or not the effects of the anomaly are observable at small positive energies, positive and negative pions were scattered from targets made of 4 0 C a and 1 2 C at 20 MeV. At this energy only a few partial waves contribute appreciably, namely the s and p waves. Calcium was chosen as an ideal target for two reasons; one being that due to its nuclear size, it should be large enough to exhibit the anomaly in the Is state, the other being that since it is an N=Z nucleus, no complications due to isospin arise. Thus the anomalous effect is expected to appear more distinctly than 7r-scattering from heavier nuclei. Carbon is not expected to exhibit the anomaly but is useful for calibration and normalization purposes. 5 C h a p t e r I I T h e o r e t i c a l D e s c r i p t i o n E x p e r i m e n t a l mea su rement s o f e l a s t i ca l l y s ca t te red p i on s f r o m nuc l e i y i e l d d a t a essent ia l t o t he u n d e r s t a n d i n g of t he s t r ong i n t e r a c t i o n . T h i s d a t a takes t he f o r m of d i f f e ren t i a l cross sect ions w h i c h are t h e n u sed t o con s t r u c t t h e o r e t i c a l mode l s to e x p l a i n e x p e r i m e n t a l measurements . A b r i e f d e s c r i p t i o n of the va r i ou s processes i n v o l v e d i n d e t e r m i n i n g th i s l i n k are d i scussed i n the f o l l o w i n g sect ions. II. 1 Nuclear Scattering T h e i n t e r a c t i o n be tween t he p i o n a n d the ta rget nuc leus is represented by a p o t e n t i a l w h i c h depends o n t he r e l a t i v e c oo rd i n a t e r r e l a t i n g the p o s i t i o n o f the p i o n t o the centre-of -mass of the nuc leus. T h i s p o t e n t i a l is t h e n i n se r t ed i n t o the K l e i n - G o r d o n equa t i on . A m a t h e m a t i c a l de s c r i p t i on o f the s c a t t e r i n g o f a r e l a t i v i s t i c s p i n zero p a r t i c l e ( the p i on ) f r o m a nuc leus , c a n thus be expres sed as [Fri83] n2c\v2 + fc2)v = [2E(vn + ve) - v?]*b ( i ) where hk is t he c m . m o m e n t u m , E is t he c m . t o t a l energy o f the p i o n , Vc is t he C o u l o m b p o t e n t i a l due to the finite charge d i s t r i b u t i o n of t he nuc leus , a n d Vn is the nuc lea r p o t e n t i a l . Q u a d r a t i c t e rms i n v o l v i n g Vn are u s u a l l y o m i t t e d s ince t he i r effects are s m a l l i n c o m p a r i s o n to the l i nea r te rms . T h e s o l u t i o n of th i s equa t i on , ib, m u s t be regu l a r at the o r i g i n a n d have the a s y m p t o t i c f o r m o f a n i n c o m i n g p l a n e wave a n d a n o u t g o i n g s ca t te red sphe r i ca l wave [Jac70], i.e. $ exp(ikz) + /(fl) e x p(* f c r) (2) r 6 where 9 is the s c a t t e r i n g ang le i n the c m . s y s t em a n d f(9) is t he s c a t t e r i n g a m p l i t u d e . T h e a s y m p t o t i c f o r m of vb c an a l so be w r i t t e n i n s p h e r i c a l coo rd ina te s as [BJ77] 1> — • \ J2 i l + 1 ( 2 1 + l)Pi(cosO)[Mkr) ~ nm(kr)} (3) 1 i whe re (j>i(kr) a n d <fi(kr) represent i n c o m i n g a n d o u t g o i n g s p h e r i c a l waves, re spect i ve ly , a n d n/ represents the re f l ec t i on coe f f i c i en t 1 , u s u a l l y expres sed i n t e r m s o f a c o m p l e x phase sh i f t , i.e. n, = exp(2 i£j) (4) If | ni |= 1, t h e n t he i n t en s i t y of the i n c o m i n g a n d o u t g o i n g waves a re i d e n t i c a l , thus t he s c a t t e r i n g is p u r e l y e last ic . However , i f i ne l a s t i c a n d a b s o r p t i o n processes t ake p l ace t h e n the i n t en s i t y o f the o u t g o i n g wave w i l l be less t h a n t h a t of the i n c o m i n g wave. C o m p a r i n g equat ions 2, 3 a n d 4, we o b t a i n fo r t he s c a t t e r i n g a m p l i t u d e / ( * ) = ^ p 2 / + ! ) ( e x P ( 2 ^ ' ) " * ) (5) T h e d i f f e ren t i a l cross sec t i on for e las t i c s c a t t e r i n g c an t h e n be g i ven b y [Jac70] 3g=i /wr (•) II.2 The Optical Potential Model A n o p t i c a l m o d e l is one t ha t descr ibes the effects of the s t r o n g i n t e r a c t i o n b y means of a n i n t e r a c t i o n p o t e n t i a l . T h i s p o t e n t i a l is i ndependen t of the coo rd ina te s of the i n d i v i d u a l nuc leons a n d depends o n l y o n the c oo rd i n a t e o f the i nc iden t p i o n w i t h respect to the nuc leus as a who le [Kis55]. T h i s s em i - phenomeno l o g i c a l 1 Thi s coefficient is analogous to the index of refraction of some medium through which photons propagate. Here, the wavefunction of the pion is modified by its propagation through nuclear matter and is denoted by a complex index of refraction. 7 approach is based on the elementary pion-nucleon interactions 7T + N -+ 7T + N (elastic scattering) (7) and TT + N + N—tN + N (absorption). (8) The model should successfully predict pion-nucleus scattering data in the energy region between 0 — 50 MeV. Here, the effects of the A 3 3 resonance do not cause too strong an absorption. Also, the optical parameters in the model, should vary uniformly with no sudden fluctuations over this energy region. There are many forms of this potential, among which one of the most successful is the M S U potential [SMC79,SCH80,CMSB82] that follows the Ericson-Ericson potential for pionic atoms [EE66]. Both potentials are based on the Kisslinger potential [Kis55]. In general most potentials obtain good fits to data since they incorporate many similar features useful in describing 7r-nuclear interactions. In the next section we discuss the optical potential following the notation of Friedman and Gal [FG80]. II.3 Outline of the 7r-Nucleus Potential The standard pion-nucleus optical potential which is inserted into the Klein-Gordon equation (eq. 1) has the form [EE66,KE69] where to is the total pion centre-of-mass energy, q(r) is the local momentum-independent part arising from pion-nucleon s-wave interaction. The local potential is written in terms of neutron (pn) and proton (pp) density distributions as K ( r ) = ~ f a ( r ) + V • a(r)V] (9) 8 where m is the nucleon mass ( « 9 3 1 MeV). The constants 60 and b\ represent effective 7r-nucleon s-wave scattering lengths through isoscalar and isovector channels respectively. The constant B0 is complex. Its imaginary part is related to the absorption of a pion on a pair of nucleons. Absorption on a single nucleon is improbable due to conservation of energy and momentum, and so the process is most likely to occur on proton-neutron pairs. The real part of BQ describes dispersion effects. The term V • a(r)V is the p-wave momentum-dependent part of the potential where cv(r) is decomposed as follows with a x = 4TT f^ l + ^ [c0(pn + pp) + ci(p n - pp)\ , (12) and cv2 = 4TT + 4C0PnpP • (13) Here the constants c 0 and c\ represent effective p-wave scattering volumes through isoscalar and isovector channels respectively. The constant Co is a complex parameter in analogy with Bo. The modification of single nucleon terms due to nuclear correlations is represented by the so-called Lorentz-Lorenz (LL) effect2, and its strength depends on the value of £. An extra term transforming the 7r-nucleon c m . to 7r-nucleus cm. , by which p-wave 7r-nucleon interaction gives rise to terms of order 00/m, is included. 2Nuclear pair correlations produce an effect similar to that caused by the scattering of electro-magnetic waves in a dense, polarizable medium. 9 This local term has the form (14) The nucleon density distributions pn and pp appearing in the above equations are determined using a phenomenological form for the distribution. The proton and charge3 density distribution parameters were obtained from the fits of electron scattering data 4 to a three parameter Fermi function5 [Jac70] where po is a normalization constant to a specified number of nucleons, c is the radius at half the nuclear density, and the diffuseness a is related to the nuclear skin thickness. The parameter w is referred to as a wine boitle parameter which, depending on its value being either positive or negative, generates a hump or depression near the origin. The neutron density distribution parameters were obtained by similar fits from proton or a-particle scattering data (or Hartree-Fock calculations). The values obtained from these fits are shown in table I. II.4 Potential Model Calculations The differential cross sections for elastically scattered low energy pions from nuclei are calculated using GLBKISS [Fri88a], a computer program incorporating an optical model potential of the Ericson-Ericson M S U type. The program determines the solution to the Klein-Gordon equation with the potential in equation 9 and it is compared to the solution for only Coulomb scattering6 at a point well outside 3 Not shown above, but used in determining the Coulomb potential in target nuclei. 4 After correcting for the finite size of the proton charge. d i s t r ibut ion forms such as this are not deduced from first principles but are phenomenological, which are found by experience to lead to agreement with a wide selection of data. 6These are well known wavefunctions. (15) 10 Target Nuclei Distribution (fm) a (fm) w (fm) 1 2 C proton 2.343 0.386 0 neutron 2.343 0.386 0 charge 2.343 0.450 0 40Ca proton 3.808 0.512 -0.166 neutron 3.748 0.512 -0.166 charge 3.808 0.586 -0.166 Table I: Fermi three parameter fit values for neutron, proton, and charge density distributions in 1 2 C and 4 0 C a taken from Batty et al [BFG83]. 11 the nuclear radius where the effect of the strong force is sufficiently small. From the comparisons of these solutions, the phase shifts7 (eq. 4) are obtained and the differential cross section calculated (eq. 5,6). The optical potential parameters were initially determined from fits to strong interaction level shifts and widths in pionic atoms [EE66,KE69,BBF +79,FG80,BFG83]. This served as the basis for the determination of the parameters for normal states. Fits to anomalous states were also made [OBB+78,KPK + 79,vEBD + 84,TSS + 84,TvEB + 85,LTD + 85,OFM+85] and the parameters determined. Several attempts to modify the Ericson-Ericson 7r-nucleus potential to account for these anomalous effects proved to be unsatisfactory [FG80,BFG83,Sek82,OTK84]. The normal and anomalous potential parameters, for a scattering energy of 20 MeV pions, used in G L B K I S S were extrapolated from the pionic atom data parameters (see table II) and are shown in table III. These parameter sets were used for calculations to compare with our elastic scattering data as described in Chapter IV. The errors associated with these parameters are given in [Fri88b]. The large error in £ is a result of the insensitivity8 of this parameter during the fit to pionic atom and elastic scattering data [MFA+88]. Outside the nucleus the wavefunction of the pion has a constant complex phase shift compared with the wavefunction of a purely Coulomb scattered pion. It is this shift that contains the infor-mation about the interaction which happened well inside the nucleus. 8 B y changing the value of £, it was required to adjust other parameters in the model to obtain similar fits, indicating that there are some correlations between the parameters. These fits did not substantially reduce the value of x 2 per point. 12 Optical Potential Parameters Normal Anomalous -0.009 -0.009 M m ; 1 ) -0.094 -0.094 -0.115+i0.055 -0.112+^0.030 c0(m;3) 0.23 0.23 c i (m; 3 ) 0.15 0.15 0.051+i0.053 -O.lO+zO.074 1.8 0.8 Table II: Optical potential parameters obtained from fits to normal and anomalous pionic atom data [Fri88b]. 13 Optical Potential Parameters Normal Anomalous -0.0123 -0.0123 -0.094 -0.094 -0.098+»0.035 -0.095+i0.019 c 0 (m; 3 ) 0.23+i0.02 0.23+i0.02 c i ( m - 3 ) 0.15 0.15 C o ( m ; 6 ) 0.051+i0.053 -O.IO-H'0.074 1.8 0.8 Table III: Optical potential parameters extrapolated to 20MeV from fits to elastic scattering data at higher energies. The values of B0 and C0 used in the program are multiplied by a factor of four. Note that only b0 and Bo vary significantly from their pionic atom values. 14 C h a p t e r I I I T h e E x p e r i m e n t This low energy elastic scattering experiment was performed in the summer of 1986 at T R I U M F ( T R I University Meson Facility). T R I U M F ' s cyclotron is capable of producing a primary 500 MeV unpolarized proton beam at a current of 140//A. This high energy proton beam strikes a series of targets, usually beryllium or carbon, for the production of pions. These pions are then channeled to various secondary beam lines in the experimental areas. There are two such production targets in the Meson Hall. The first of these, called 1AT1, produced the pions for this experiment. III.l TRIUMF's M13 Low Energy TT - fi Channel The facility's M 1 3 beam line was used for this experiment. This low energy pion channel is capable of delivering pion beams with a momentum range of 20-150 MeV/c [OWMD81]. It views the 1AT1 production target at 135° to the primary proton beam. A series of dipoles, quadrupoles, and sextupoles achromatically focus the beam to the nuclear scattering target located 86 cm beyond the last quadrupole. This can be seen in figure 2. The first two quadrupoles (Ql and Q2), and initial dipole (BI), are adjusted to create a dispersive focus at FI. The momentum of the beam is defined at FI by suitable horizontal and vertical slit positions and widths. The momentum dispersion of the pion beam has been measured experimentally to be 1.22 c m / % ^ [OWMD81]. For a horizontal slit width of 30 mm (dispersion of 2.5% ^ ) the 7r+ flux was « 1.2 x 106 particles/sec. 15 Figure 2: Schematic layout of the TRIUMF M13 pion channel and spectrometer. BL1A is the primary beam line and T l is the pion production target; Q1-Q7, SX1-SX2, and B1-B2 are the channel quadrupole,sextupole and dipole magnets, respec-tively; F l and F2 are the intermediate focus points; QT1-QT2 and B T are the spectrometer quadrupole and dipole magnets, respectively. 16 F o r the s ame se t t i ng , the ir~ f l u x was « 3.2 x 1 0 5 par t i c les/sec. A l s o s i t u a t e d at F I is a CH2 ab so rbe r u sed t o s top heav ier pa r t i c l e s e m e r g i n g f r o m 1 A T 1 such as ct's a n d p ro ton s . F o l l o w i n g th i s , a set of quad rupo l e s (Q3 ,Q4 ,Q5 ) , a n d sextupo les ( S X 1 , S X 2 ) , are a d j u s t e d to p r o d u c e a second d i sper s i ve focus at F 2 . A n o t h e r set o f m o m e n t u m de f i n i ng s l i t s are l o c a t e d at th i s p o s i t i o n . F i n a l l y a second d i po l e ( B2 ) , a n d two quad rupo l e s (Q6 ,Q7) b r i n g the b e a m to a t h i r d a n d f i n a l a c h r o m a t i c focus at the s c a t t e r i n g targets center. T h e b e a m l i ne a n d s pec t r omete r magne t s were set a n d m o n i t o r e d t h r o u g h o u t the e xpe r imen t u s i n g P B X , a p r o g r a m to r e m o t e l y select cu r ren t values fo r each i n d i v i d u a l magne t . T h e a c t u a l m a g n e t i c fields were mea su red by N M R probes fo r the d ipo le s , a n d H a l l p robes fo r the quad rupo l e s a n d sextupoles . T w o i n - b e a m counters cons i s t i ng of N E 1 1 0 1 p l a s t i c s c i n t i l l a t o r were used. T h e first counte r , B I , was s i t u a t ed p r e ced i n g t he s c a t t e r i n g ta rget w h i l s t t he second, B 2 , was p l a c e d b e h i n d the target. T h e co inc idence of B 1 - B 2 mea su r ed t he abso lu te b e a m flux t r ave r s i ng the target . T w o pa i r s o f m u o n counter s were l o c a t e d o n e i the r s ide of the M 1 3 b e a m p i pe at « 15° t o the i n c i den t p i o n b e a m 2 . The se counter s were u sed t o measure muon s f r o m the decay o f p i on s i n f l i gh t . T h e co inc idence fil • fj,2 a n d //3 • fi4 are u sed as r e l a t i ve p i o n flux mon i t o r s . T h i s is e x t r e m e l y u se fu l s ince at ve ry f o r w a r d angles B 2 c anno t be p l a c e d b e h i n d the ta rget as t he spec t r omete r p h y s i c a l l y b l ock s the b e a m . A s m e n t i o n e d ear l ie r , t he b e a m t r a v e l i n g d o w n the M 1 3 channe l conta in s a ' s , p 's, 7r 's, / / ' S a n d e's. T h e CH2 ab so rber at F I b l ock s the heav ie r a ' s a n d p's w h i l e t he l i g h te r //'s a n d e's are sepa ra ted b y t he i r flight t imes t h r o u g h the channe l . A c apac i t i v e p r obe ( T C A P ) l o c a t e d near the p r o d u c t i o n target a n d the 1 These scintillators are fast signaling plastic detectors able to handle large flux and are manu-factured by Nuclear Enterprises™. 2 The optimum position of the muon counters is half the maximum muon cone angle since, at this position, the detectors are not too sensitive to fluctuations in beam position. 17 in-beam scintillator B l supply the T D C (time-to-digital converter) stop and start, respectively, for this measurement. The result of this measurement is shown in figure 8 which indicates that a relatively clean pion beam was obtained. III.2 QQD Spectrometer The Q Q D (Quadrupole Quadrupole Dipole) low energy pion spectrometer was used to measure the elastically scattered pions for this experiment [SDB+84]. The two quadrupoles serve as a lens for enlarging the spectrometer's solid angle to « 1 6 msr. The dipole is used as a momentum analyser for the elastically scattered pions, as it deflects them 70° to the left in the horizontal plane. These pions are brought to a dispersive focus beyond the final wire chamber at an angle of 72° to the central momentum trajectory. The optical layout of the spectrometer can be seen in figure 3. Only the second quadrupole (QT2) which focuses in the vertical direction and dipole (BT) were used in this experiment. The first quadrupole (QT1), which focuses in the horizontal direction, was not used as it would increase the solid angle slightly (?»5 %) in exchange for decreasing the resolution in the target traceback. Four delay-line multi-wire proportional chambers ( D L M W P C ) utilized by the Q Q D give position information for the elastically scattered pion. The first (WCl) and second (WC3) chamber positions are located ahead of QT1 and after QT2 respectively. There is room between the two quadrupoles for a second chamber but its use would not sufficiently improve first order position and angular resolution in the target traceback. The third (WC4) and fourth (WC5) chambers are positioned after B T . The momentum of the pion is determined by using the position information from the four chambers. Details of position and momentum calibration are given in sections III.3 and and III.4 respectively. At the rear of the spectrometer there are three large plastic scintillator 18 Figure 3: Schematic view of the QQD spectrometer. The symbols W C l -W C 5 are locations in which wire chambers are positioned; E l -E3, B1-B2, fj,l-fi4 are the various scintillators associated with the spectrometer and beamline. The dashed line represents the central momentum trajectory of the scattered pion. 19 detectors, E l , E 2 , and E 3 respectively. Only E l and E 2 were used for this experiment since a substantial fraction of pions would not make it beyond the first two scintillators at this low energy. A spectrometer event required the coincidence of B1-E1-E2 as this would most probably occur as a result of a particle traversing the QQD. Hardware coincidences and electronics setup are discussed with greater detail in section III.6. III.3 Wire Chamber Design and Calibration The D L M W P C ' s employed by the QQD were constructed at the University of Carleton Workshop [BPRW71,SDB+84]. Each chamber consists of 3 planes of equally spaced gold-tungsten wires. The anode plane at high positive potential, is sandwiched between two cathode planes at ground potential. The anode wires are oriented horizontally at with spacing of 2 mm to create a uniform electric field throughout the active area of the chamber. The cathode planes have a wire spacing of 1 mm and are oriented parallel and perpendicular to the anode wires. The horizontal cathode wires give position information in the ^-direction while the vertical cathode wires give this information in the z-direction. The front 2 chambers (WCl,WC3) are filled with a helium-isobutane gas mixture to minimize multiple scattering3 while the back chambers (WC4,WC5) are filled with an argon-isobutane mixture. As the pion traverses the chamber it ionizes the gas inside producing electron-ion pairs. The electrons cascade towards and induce a signal on the closest anode which in turn capacitively couples to the neighbouring cathodes, producing signals in both x and y directions. One end of each cathode wire is connected to a printed circuit delay-line strip with a delay of 0.55 nsec between consecutive wires. The two ends of the delay-line are each connected to 3Multiple scattering in the spectrometer due to mylar windows (for wire chambers) and gases (throughout QQD and wire chambers) limit the spectrometer resolution to « 1 MeV. 20 amplifiers which are then fed to discriminators and time-to-digital converters (TDC's). Calibration of these chambers by converting T D C values to a position in millimeters is made convenient by the arrangement of these chambers. The conversion is defined by X = m-t + b (16) where X is the position in mm, m is the slope converting T D C values to mm, t is the T D C difference value, and b is the offset in mm. to define the center of the chamber. The determination of the conversion factor m, in the front chambers is achieved by observing the 'picket fence' structure obtained from the y T D C difference data. This is shown in figure 4(a). The ^-direction cathode wires and anode wires are parallel. Thus, a signal induced in any one anode will produce a strong signal in the adjacent cathode wire, and a relatively weak signal in any neighbouring cathode. The x-direction cathode wires run perpendicular to the anode wires thus the anode signal will distribute a strong signal to one or more cathode wires resulting in a smooth spectrum. This can be seen in figure 4(b). Given that the physical wire spacing of the anodes is 2 mm, we may obtain the conversion factor from T D C differences to distance, i.e. 2 = m • T D C d t / / + 6 (17) assuming 6=0 initially, we have m m Tiki, (18) where TDCdiff is the difference between two consecutive pickets. The slope can be used for both x and y directions since each plane is identical except for their 21 90 2 mm anode spacing -400 -240 - 80 80 240 400 WC3Y (TDC d l„ data) WC3X (TDCdif, data) Figure 4: ( a ) Picket fence structure obtained from y T D C difference data used to determine wire chamber coefficients, ( b ) Posi-tion information obtained from x T D C difference data. Note the smoother structure compared with y T D C difference data. 22 orientation. Once m is found, 6 is adjusted to center the beam at the center of the chamber. For the back chambers, the same method can be used for the y-direction information. However, these chambers are much larger than the front chambers and are segmented in x to reduce dispersion and attenuation of signals in the delay-line. Wire chambers 4 and 5 each have 3 sections in the x-direction. Each section is 203 mm wide. The right-middle and middle-left edges overlap, so a particle passing through these points are likely to fire both sections. By examining the positions of double-hits in the central segment one can determine the conversion factor m, i.e. 203 = m • T D C * / / + 6 (19) again assuming 6=0 we have where T D C * / / is the difference between the two double-hit peaks. Since all sections in any particular chamber are identical, the same conversion factor is assumed. The central offset is adjusted to position the double-hit peaks at ± 1 0 1 . 5 mm. The left and right offsets are adjusted such that their respective edges with the central segment edges overlap. III.4 Momentum Calibrations Two sets of coefficients are needed to completely define the trajectory and momentum of the pion through the spectrometer. These include the front end wire chamber coefficients for ray-tracing back to the target and a set of magnet transfer coefficients which are essential to calibrate the pion's momentum through the spectrometer's dipole B T . 23 III.4.1 Target Traceback Coefficients A set of coefficients is needed to trace the pion's path through the two front end wire chambers back to the scattering target. These coefficients essentially define the pion's initial trajectory towards B T . The quadrupole magnet QT2, is positioned between W C l and WC3, and thus the traceback is not simply a linear one. However, since QT2 is not a dipole, it is assumed the there will not be a large momentum dependence in the traceback. The coordinates required are the pion's position at the scattering target in the x and y (XO and FO). This results in Two special targets constructed of nichrome strips 3 mm. wide set about 10 mm. apart, one with horizontal strips, one with vertical strips, are placed at the scattering target position and spectrometer data is taken. From the analysis of this data the coefficients 01,03,61, and 6 3 are adjusted to obtain the correct strip positions. III.4.2 Magnet Transfer Coefficients The passing of multi-energetic pions through the spectrometer's dipole results in the emergence of those pions at different positions in the back end wire chambers. The corresponding positions can then be converted to pion energies which can be expressed in terms of 6, where 6 = (23) Po Here, po is the central momentum of the spectrometer and p is the pion's momentum after scattering. The positions in the back chambers alone will not X0 = a1-X1+a3- X3 (21) and FO = • Yx + 63 • F 3 (22) 24 clearly indicate the energy spectrum of the pions since the focal plane of the QQD lies at 72° to the central ray and is located beyond WC5 (see figure 3). We must therefore incorporate the information from the front end wire chambers to produce a good spectrum. The x and y positions in WC1 and WC3 can be employed directly to obtain the transfer coefficients in the software package Q Q D M P [Bar85]. It is these coefficients which relate the position information in the back chambers to a value 6 (one for each chamber, £ 4 , 65). The routine Q Q D M P assumes that the back-end wire chamber coordinates may each be written in terms of a polynomial of the front-end wire chamber coordinates and the momentum parameter 6, WC4X = A + B • 64 + C • 6\ (24) with A being a polynomial of order mo in front-end coordinates, B being a polynomial of order mi in front-end coordinates, and C being a polynomial of order m 2 in front-end coordinates. The above expression may be meaningfully inverted to determine 64 if the coefficients A, B, and C are known. Similarly S5 can be obtained. The corresponding expression for WC4Y or W C 5 Y will have little or no dependence and therefore may not be used for 6 determination. The data used to determine these transfer coefficients were obtained by elastically scattering 20 MeV pions from 1 2 C at 0%, ± 5 % , and ± 1 0 % QQD central momentum settings. In the program Q Q D M P , one defines the elastic peak locations and attempts to minimize the peak widths. Once this is done, a multiple linear regression is performed to determine the coefficients A, B, and C. The values of 84 and 65 are then calculated and analysed further. 25 The analysis routine in Q Q D M P called Q Q D A N A was used to optimize the data used for determining transfer coefficients. Cuts on the data were imposed to eliminate background events triggered in the spectrometer. The primary source of these events come from the decay of pions via 7r —• fie. Ideally, one would expect S4 = <55. However, if a pion decays into a muon, its path will deviate from the pion's trajectory by an angle limited by kinematics to [30° [Gre], resulting in substantially different 6's. The muon will either hit some appendage within the spectrometer and stop or trigger an event. These events are eliminated by placing a cut on DDIF which is the difference between 64 and S5. If the difference was greater than 1.5%, the event was cut. Another useful cut is the A N G L cut. Using the value of 64, the trajectory to WC5 is predicted, and a polar angle between the actual trajectory and the predicted one is calculated. If this value is too large, the event is cut. The DDIF and A N G L cuts overlap to a large extent, however it is useful to employ them both. A sample of the DDIF cut is shown in figure 5. III.5 Targets Two targets, one 1 2 C , and one 4 0 C a , were used in this experiment. Both solid targets were held in plexi-glass target holders at beam height by a target ladder situated at the pivoting point of the QQD. The target holders were designed so that they do not interfere with the pion beam at any of the angles used in the experiment. A description of this holder is shown in figure 6. The carbon target used was a 76 mm 2 graphite slate which should have been large enough to intercept the entire beam, however, this was not the case (see Appendix A). The calcium target was made of three self supporting plates of metallic natural calcium (97% 4 0 Ca) of size 51 mm by 39 mm held in the target holder by a thin nylon thread. The nylon thread is thin enough that the 26 -120 -40 40 DDIF (Ap/p % x 10) 200 Figure 5: The ^-difference (DDIF) cut. This represents the difference between 64 and 65 calculated from the x position information in WC4 and WC5 respectively. background it contributes is insignificant, thus eliminating the need for an "empty" target measurement. Specifications of these targets are shown in table IV. III.6 Data Acquisition This single arm elastic scattering experiment utilizing the QQD spectrometer was set up using the standard electronic arrangement described in previous theses from the PISCAT group [Roz85,Hes85,Bar85] without the use of the F2 counter4. The online data acquisition programs were run on a PDP11/34 computer using the S T A R [Smi86] system to acquire experimental data and record it to magnetic tape. This data, in the form of A D C (analog-to-digital converter), T D C , scaler values and bit patterns, was produced by various modules in a C A M A C 5 4The F2 counter is a fast readout wire chamber placed at the second dispersive focus in the M13 channel. This chamber, when operating efficiently, gives additional momentum information of the incident pions. 5The CAMAC system is an interface system which digitizes analog signals from the hardware 27 <—-1.0 > Figure 6: Target holder used for the Calcium target. The nylon line is shown as a dotted line threaded through holes in the target frame and anchored to the frame at the outermost holes. The solid Calcium target is interlaced through the nylon thread. Target Nucleus Mass Density mg/cm 2 cm 2 graphite 1 2 C 378 1.90 x 10 2 2 calcium 4 0 C a 414 6.23 x 10 2 1 Table IV: Target mass densities and scattering center densities of targets used in Exp. 373 28 crate. A specific CAMAC module, the C212 unit, generated LAM (look-at-me) signals enabling the PDP11/34 to read data created by a spectrometer hardware event. In this experiment, a hardware event was defined as the coincidence B1-E1-E2 with the start signal given by BI and the stop given by E l . The back scintillators E l and E2 each have two phototubes whose output was measured in a meantiming circuit to make the event timing position independent. A schematic diagram depicting the electronic logic used is shown in figure 7. electronics and supplies this information to the PDP11/34 computer. 29 Figure 7: The experimental electronic logic. 30 C h a p t e r I V D a t a A n a l y s i s T h e of f - l ine ana l y s i s u t i l i z i n g t he R E P Q Q D [OH87] sof tware package is d i scussed i n t h i s c hap te r a l ong w i t h the c a l c u l a t i o n of ab so lu te d i f f e ren t i a l cross sect ions a n d t h e i r errors. T h e cons t ra in t s a n d n o r m a l i z a t i o n o f the d a t a are a lso p resented. F i n a l l y , t he resu l t s are fitted u s i n g the o p t i c a l p o t e n t i a l m o d e l d i scussed i n C h a p t e r II. IV. 1 Off-Line Analysis Program Tapes c o n t a i n i n g e x p e r i m e n t a l d a t a were ana l y s ed of f - l ine u s i n g R E P Q Q D , a sof tware package s i m i l a r t o the on - l i ne S T A R s y s t em b u t spec i f i ca l l y f a b r i c a t e d t o ana ly se QQD s pec t r omete r da t a . T h i s ana ly s i s r o u t i n e a l lows t he user t o r ead events a n d scalers f r o m tape , create a n d ca l cu l a te new pa ramete r s , a p p l y so f tware cons t ra in t s fo r the r e m o v a l of b a c k g r o u n d a n d ' b a d ' events, a n d con s t r u c t one or t w o d i m e n s i o n a l h i s t o g r ams o f these pa ramete r s . T h e h i s t og r ams c a n t h e n be saved f o r l a t e r use such as peak fitting. IV.2 Event Definitions T h e r e are two types of events w r i t t e n t o t ape ; s pec t r omete r events a n d b e a m s a m p l e events. T h e d e f i n i t i o n o f a g o o d spec t romete r event is a n M 1 3 p i o n s c a t t e r i n g f r o m a spec i f ic ta rget l o c a t i o n , firing a l l s pec t r omete r w i r e chamber s a n d s c i n t i l l a t o r s w i t h o u t decay i ng i n flight. W e requ i re t he e l ec t r on i c co i nc idence B 1 E 1 - E 2 . B e a m s amp le events are def ined by the co inc idence B 1 - B 2 . The se 31 events are se lected at r a n d o m th r oughou t t he r u n a n d u sed fo r d e t e r m i n i n g the p i o n f r a c t i o n o f the i n c o m i n g b e a m . B o t h types o f events are essent ia l i n the d e t e r m i n a t i o n o f t he d i f fe rent i a l cross sect ions. T h e y are d i scussed be l ow, a l ong w i t h t he i r va r i ous cons t ra in t s . IV.2.1 Spectrometer Events T h e r e are severa l cut s a n d tests t o be passed before a p i o n c an be cons ide red a t r ue event. F o r e xamp le , t he event mus t i n i t i a l l y sat i s fy t he s p e c t r o m e t e r h a r d w a r e event de f ined above. F o l l o w i n g th i s , t he p a r t i c l e m u s t l i e w i t h i n t i m i n g l i m i t s fo r p i on s i m p o s e d i n the b e a m t ime-o f - f l i gh t s p e c t r u m . T h i s s p e c t r u m is gene ra ted f r o m the occu r rence of s pec t r omete r events a n d is t he t i m e i t takes the p a r t i c l e t o t r a v e l f r o m TCAP t o B l (see f i gu re 8 (a)). A n o t h e r s i m i l a r con s t r a i n t is o b t a i n e d f r o m the t ime-o f - f l i gh t s p e c t r u m t h r o u g h the QQD ( E 1 T ) . T h i s cut is use fu l fo r the r e m o v a l of la rge amoun t s o f b a c k g r o u n d events caused by p i on s decay i n g t o m u o n s w i t h i n the spect rometer . T h i s c an be seen i n figure 8 (b) . Mo reove r , a l l 4 w i r e chamber anodes are r equ i r ed to have fired a n d reasonab le T D C va lues o b t a i n e d . The se va lues are checked fo r v a l i d i t y b y r e q u i r i n g t he s u m of the T D C va lues fo r b o t h ends of the de lay l i ne of each sec t i on o f each chambe r t o l i e w i t h i n spec i f ied l i m i t s . T h e back chamber s have a n a d d i t i o n a l r equ i r ement t h a t b o t h r i ght a n d left sect ions i n x not fire s imu l taneous l y . T h e s e tests are usefu l w h e n the chambe r s exper ience noise. C u t s are a lso p l a c e d o n the t r a ceback t o the target . T h i s is e x p l a i n e d i n de t a i l i n A p p e n d i x A . A t 20 M e V , a p p r o x i m a t e l y 5 0 % of the p i on s decay d u r i n g t he i r flight t h r o u g h the spec t romete r . T o e l i m i n a t e these events, two separate cons t ra in t s are p l a c e d o n t he p o s i t i o n i n f o r m a t i o n g i ven b y the rea r w i r e chambers . The se are the D D I F (d i scussed i n sec t i on III.4) a n d the D Y 4 5 cuts . T h e l a t t e r is m a d e b y i m p o s i n g l i m i t s o n y p o s i t i o n d a t a i n W C 4 a n d W C 5 . If the p i o n decays to a m u o n , i t s t r a j e c t o r y dev iates f r o m the i n i t i a l p a t h of 32 4100 2100 1680 M1260 --i-> C P o O 840 420 -240 480 720 960 1200 TCAP ( t ime-o f - f l i gh t through M13) background muons from ir^/xv (b) elastic peak 240 480 720 960 E1T ( t ime-o f - f l i gh t through QQD) 1200 Figure 8: (a) Typical time-of-flight spectrum for particles travelling from 1AT1 production target to B l scintillator, (b) Time-of-flight of particles traversing the QQD spectrometer. Most of the background is easily removed by placing more stringent con-straints. 33 the pion by an angle limited by kinematics to < 30°. If the /^-difference was was applied to determine which events which were clearly true pions. This was possible by examining a two dimensional scatterplot of the time-of-flight through the spectrometer (E1T) vs. 64 (or S5) and eliminating inelastically scattered pion and background events. An example of this is shown in figure 9(b). The result is a relatively clean elastic scattering peak used in determining the differential cross sections. IV.2.2 Beam Sample Events As mentioned previously, the beam sample events are selected at random throughout each run. Defined by the coincidence B1-B2, these events are the measure of the flight time of particles traveling from the 1AT1 production target to the inbeam scintillator B l . From these spectra we may determine the actual pion fraction delivered to the scattering target. For positive polarity runs (see figure 10(a), 82% of the beam were 7 T + , whereas for negative polarity runs (see figure 10(b)) only 49% of the beam contained 7r~ particles. At a specific distance restricted by kinematics and the size of B l , a muon from a decaying pion may still intercept B l and be considered a pion. A small correction factor for this contamination was taken into account in determining the true pion flux. IV.3 Cross Sections The absolute differential cross section including normalization factors is defined as greater than 3 cm, the event was cut as shown in figure 9(a). Finally a box test da N, 1 1 1 • Jlab^CM (25) dQcM £dt •ecQQD • ewe • £dt Ni P-t-NAy A-cos 6tgt where Nscat is the number of scattered pions traversing the Q Q D 34 ure 9: (a) Difference in y position in WC4 and WC5. Scale is in mm x 10. (b) Scatterplot of S5 vs E1T. Here the muons are easily separated from true pion events. 35 2160 -240 480 720 960 time—of-flight (positive polarity) 1200 m 720 -240 480 720 960 time—of—flight (negative polarity) 1200 10: (a) Beam sample spectrum for positive polarity runs. Beam sample spectrum for negative polarity runs. Note increase in the number of electrons. 36 SdecQQD is the correction for 7r-decay through the spectrometer ewe is the normalizat ion due to wire chamber efficiencies en is the correction for computer dead time Ninc is the number of incident pions en is the pion fraction of the incoming beam £decB1 is the correction for 7r-decay before B l p • t is the target thickness i n g / c m 2 NAV is Avagadro's number (6.022 x 10 2 3 atoms/g-mole) A is the atomic weight of the scattering target digt is the target angle i n the lab frame AOjab is the spectrometer solid angle and Jiab-+CM is the Jacobian for the lab to C M . transformation T h e number of scattered pions (Nscat) was determined by fitting the sum of two Gaussian functions, one w i t h large w i d t h and one narrow, to the elastically scattered p i o n peak. F i t s to the data were performed using O P D A T A [BK87]. T h i s data manipulat ion program was employed once a l l background was effectively removed. A t this energy about 50% of the pions decay before reaching the back scintillators. Therefore, the values obtained by the fit have s t i l l to be corrected for p ion decay through the 2.38 m long spectrometer. T h i s factor is given by edeCQQD = exp - ^ p t c J (26) 37 where m f f , p, r , are the pion mass, momentum, and mean-life, and £QQD is the spectrometer decay length. Other corrections such as wire chamber efficiencies and computer dead time must be taken into account with this particular setup. With the initial wire chamber constraints discussed in section IV.2.1, we may define the efficiency of any particular chamber (WC3 for example) as WC1 • WC3 • WC4 • WC5 WC1 • WC4 • WC5 where W C m indicates a valid firing in both x and y sections of the mth chamber. The total efficiency is thus defined as ewe — Efftot = Eff! • Eff3 • Eff4 • Eff5 and was typically ~80%. During the processing of an event, the computer is busy and cannot recognize any forthcoming events until it is finished with the event at hand. A correction for this dead time is defined as Sd — ^LAM NSPECT + NBS where NLAM is the number of times the computer witnesses an event and N$PECT and NBS are the number of spectrometer and beam sample events respectively. The number of incoming particles was determined by the scaler monitoring the coincidence B1-B2. For runs where the Q Q D was at 50° or less, the incident flux was given by B l and then normalized to B1-B2 using values obtained for runs where B2 was used1. The beam sample spectrums mentioned in section IV.2.2 were used to determine what fraction of these particles were pions. This correction factor is given by N (27) JV, + J\L + N, formalizing to B1-B2 could also be done using scaler values from the coincidence fii • 1*2 and fJ-3 • M4 -38 where N*, N^, Ne, are the number of pions, muons and electrons2 respectively. As discussed earlier, pions decaying 43 mm , before B l 3 may still trigger the scintillator and be considered a pion. Although the likelihood of triggering B1-B2 is small, the possibility still exists and must be corrected for. This is especially true for runs where B2 cannot be put in coincidence. Similar to equation 26, this correction is given by edeCB1 = exp - i (28) V p-T-C ) where ls\ =43 mm upstream of B l and ITGT is the distance from B l to the scattering target. The value of £decBl was typically ~95%. The target thickness depends on the target's angle with respect to the Q Q D . To minimize straggling due to electromagnetic scattering, the target angle was chosen to be Otgt = l^QQD- (29) where 6tgt is measured from the beam axis to the normal to the target. At this angle, all pions scattered into the spectrometer travel the same distance through the target, no matter where the scattering occurs. The effective thickness the pion encounters is then *. / / = (30) COS 6tgt where ttgt is the actual target thickness in cm. The number of scatterers per unit area in the target is calculated from N w = ( 3 1 ) The values of N t f f t for both carbon and calcium targets are given in table IV. 2Discriminator thresholds reject many of the electrons. 3 A t this distance upstream from B l a muon from a decayed pion may still trigger B l . 39 The spectrometer solid angle used was A f i = 0.016 steradians. This value is based on previous group work [Bar85,Hes85]. Since Aft is constant4 throughout the cross section calculations, any associated error can be absorbed in the overall normalization of the data. I V . 4 C r o s s S e c t i o n E r r o r s The cross section error may be determined by summing the individual fractional errors of the contributing terms in quadrature. These terms do not include errors that effect all points equally, since they are absorbed in the normalization. Hence the error is written as ' A (*fc)V / I N / A W C V / A X 0 t r a c e 6 a c f c \ 2 / A f l u x \ 2 / A 7 r r f e c a , \ 2 S^ 2 - ) \ N s c a t ) + \ W C J ~U X 0 t r a c e b a c k J + V flux J ^ 7Tdecay ) ' (32) The wire chamber error has been estimated by Rozon [Roz85] where Eff is the fraction of events that pass the cut and N is the number of events in the cut region. This error is typically 1% or less. The target traceback error is the error estimated in the fitting procedure discussed in Appendix A. The contribution here ranges from a maximum of 8% or less for both positive and negative polarity runs. The error in beam flux is brought about from the normalization of B 1 - B 2 for runs where B 2 could not be used. It is an estimate taken to be 1%, the stability limit of the muon counters. A major contribution to background events is from the decay of pions. The error associated with this correction is estimated to be no more than 4%. This is because the majority of muons decay at large angles with respect to the initial pion. Those that do decay 4The spectrometer solid angle remains reasonably constant for the range of pion momentum measured in this experiment (see [SDB+84]). 40 at very forward angles (within 5°) either have too little or to great a momentum to be detected within the limits of the elastic scattering peak in the back wire chambers of the QQD. IV.5 Results and Comparisons The differential cross sections and their errors for 1 2 C and 4 0 C a are shown in tables V , VI, VII, and VIII. The beam energy is given as 20 MeV. This is the average pion energy at the target's center after allowing for energy losses in the mylar windows, scintillator, and target. The errors shown are statistical. The 7T+ 1 2 C results agree well with the data of [OBG+83]. At backward angles, it would appear that the TRIUMF data is consistently lower but well within the statistical limits, except for the 125° data point which disagrees substantially. Figure 11 shows the comparison of these results. Similarly the 7 r ~ 1 2 C data agree nicely with [WBB +87]. This can be seen in figure 12. A comparison of the 7 r + 4 0 C a and 7r~ 4 0 C a data are shown in figures 13 and 14, respectively. Here, the disagreement beyond 60° is obvious, especially at very back angles. The current TRIUMF data lies below the the data of [WMR+88]. An observation of the 7 T _ 4 0 C a data shows that the Coulomb-nuclear interference minimum lies at 65°. It is the location of this minimum which is critical for the determination of the anomaly. These two sets of data for 1 2 C and 4 0 C a constitute all available 20 MeV experimental data to date. The fact that there were two experiments ([WBB +87], [WMR +88]) to collect low energy negative pion data could be a factor in the discrepancy between the published data and the TRIUMF data. Since the TRIUMF cross sections were calculated from data collected in one experiment, the positive and negative scattering comparison may be more reliable. 41 Polarity &CM da dQcM A d a dUcM (deg) (mb/sr) (mb/sr) 45.6 7.77 ± 0.13 50.6 5.01 ± 0.24 55.7 4.09 ± 0.18 60.7 3.75 ± 0.14 65.7 3.48 ± 0.20 7T+ 69.8 3.63 ± 0.23 75.8 3.30 ± 0.26 82.9 3.78 ± 0.25 90.8 3.99 ± 0.28 105.8 4.56 ± 0.33 125.8 4.78 ± 0.33 Table V: Measured 7r+ differential cross sections for 1 2 C at 20 MeV. 42 Polarity &CM dc A d a dUcM (deg) (mb/sr) (mb/sr) 45.6 8.56 ± 0.32 50.6 4.65 ± 0.18 55.7 2.20 ± 0.08 60.7 1.31 ± 0.11 65.7 0.77 ± 0.07 69.8 0.75 ± 0.12 75.8 0.85 ± 0.07 82.9 1.42 ± 0.12 90.8 2.46 ± 0.19 105.8 4.33 ± 0.31 125.7 5.93 ± 0.40 Table VI: Measured TT differential cross sections for 1 2 C at 20 MeV. 43 Polarity 0CM der d Q c M (deg) (mb/sr) (mb/sr) 45.2 82.80 ± 3.21 50.2 45.21 ± 2.52 55.2 38.43 ± 2.03 60.2 34.45 ± 1.85 65.2 27.91 ± 1.53 69.5 24.57 ± 1.35 75.2 20.15 ± 1.61 82.7 19.16 ± 1.21 90.3 16.97 ± 0.93 105.2 14.86 ± 0.79 125.2 10.91 ± 0.60 Table VII: Measured 7 r + differential cross sections for 4 0 C a at 20 MeV. 44 Polarity ®CM da dUcM A d a dQCM (deg) (mb/sr) (mb/sr) 45.2 63.59 ± 2.91 50.2 34.69 ± 1.38 55.2 21.78 ± 0.95 60.2 18.52 ± 0.87 65.2 14.66 ± 1.01 69.5 15.19 ± 0.70 75.2 16.66 ± 1.35 82.7 25.26 ± 1.41 90.3 28.69 ± 1.64 105.2 26.18 ± 1.63 125.2 19.98 ± 1.51 Table VIII: Measured TT differential cross sections for 4 0 C a at 20 MeV. 45 10 G b 10v 0 30 60 90 120 150 180 °cm (degrees) Figure 11: A comparison between [OBG+83] and TRIUMF TT+ 1 2 C data. The stars are the TRIUMF data. 46 10 Ul 104 a b T3 10 -1 I I I L J I I I I L 2 J-2 7d 1 2 c ( 7 r , 7 T ) 1 2 c " i — i — I — r " i — i — i — i — i i i i i r 0 30 60 90 120 150 180 c^m (degrees) Figure 12: A comparison between [WBB+87] and TRIUMF ir~ 1 2 C data. The stars are the TRIUMF data. 47 102 c b 10 I I I I I I l I I I I I I I I L 4°Ca(7T+,7T +) 4°Ca i i i i 2 i " i i i i i i i i — i i — i i — i — i — i — i — r 0 30 60 90 120 150 180 d c m (degrees) Figure 13: A comparison between [WMR+88] and T R I U M F ?r+ data. The stars are the T R I U M F data. 48 0 30 60 90 120 150 180 ecm (degrees) gure 14: A comparison between [WMR+88] and TRIUMF 7r~ data. The stars are the TRIUMF data. 49 IV.6 O p t i c a l P o t e n t i a l M o d e l C o m p a r i s o n s The optical potential discussed in Chapter II was used to produce theoretical curves for the elastic scattering of pion from 1 2 C and 4 0 C a at 20 MeV. Both normal and anomalous parameters were used and compared to experimental data. These comparisons are shown in figures 15, 16, 17, and 18. The anomaly was not expected to be overly noticeable for the carbon data. Even though both normal and anomalous curves differ minimally, it is clear that the normal fit is superior. It is again noted that at large back angles, theory and experiment disagree. The position of the Coulomb-nuclear interference minimum in the ir~ 1 2 C data lies at 70°, which according to theory indicates no anomaly. The 7 r + 1 2 C data shows little if any variation in this position. Qualitatively, both curves for the ir+ 4 0 C a data fit equally well. This is a direct indication that the anomalous effect is suppressed for positive pion data, as expected. The accentuation of this effect is clearly visible for the elastic scattering of negative pions from larger nuclei, as in the u~ 4 0 C a data. Here, the normal curve generally fits better at angles < 90°, especially in the region around the minimum 65°). At back angles however, the anomalous curve fits quite well as the data is lower than theoretically predicted. Overall, the indication of the data shows little or no support for the existence of the anomaly at positive pion energy values. Qualitatively this is seen in table IX. I V . 7 C o n c l u s i o n From the figures shown, it is obvious that there is no quantitative agreement between the anomalous curves generated by GLBKISS and the experimental data. However, the normal optical potential parameters [Fri88b] give rise to curves which are a closer fit to experimental data. Statistically, the normal optical 50 0 30 60 90 120 150 180 9cm (degrees) 15: N o r m a l a n d anoma lou s p r ed i c t i o n s c o m p a r e d t o 7 r + 1 2 C da t a . T h e so l i d l i ne represents the p r e d i c t i o n u s i n g n o r m a l o p t i c a l m o d e l pa rameter s . T h e da shed l i ne represents the anoma l ou s case. 51 0 30 60 90 120 150 180 dcm (degrees) F i g u r e 16: N o r m a l a n d anoma lou s p r ed i c t i on s c o m p a r e d t o 7r~ 1 2 C d a t a . T h e so l i d l i ne represents the p r e d i c t i o n u s i n g n o r m a l o p t i c a l m o d e l pa rameter s . T h e da shed l i ne represents t he a n o m a l o u s case. 5 2 17: Normal and anomalous predictions compared to TT+ 4 0 C a data. The solid line represents the prediction using normal optical model parameters. The dashed line represents the anomalous case. 53 Figure 18: Normal and anomalous predictions compared to TV~ 4 0 C a data. The solid line represents the prediction using normal optical model parameters. The dashed line represents the anomalous case. 54 Carbon Potential Parameter Type X2/pt 7T+ X2/Pt. TT Normal 1.7 6.7 Anomalous 8.5 35.9 Calcium Potential Parameter Type X 2/Pt 7T+ X 2/pt 7T-Normal 7.3 9.0 Anomalous 6.1 36.7 Table IX: The optical potential model fit summary for 1 2 C and 4 0 C a are shown for both normal and anomalous curves. potential model predictions are favoured, as shown in table IX. However, the ambiguities observed do not clearly rule out the effect of the anomaly. Also, it was found that the parameter £ in the optical model is too insensitive to definitely discard the existence of the anomaly. The difficulty lies in the gathering of high quality experimental data to test the anomaly. The Q Q D spectrometer was used for the first time at 20 MeV. At this energy, the resolution of the spectrometer decreases rapidly5. A much shorter spectrometer to reduce the number of decaying pions is required. The problem of the target ladder alignment should have been avoided. Extreme care must be taken when setting up an experiment to obtain data of such low energy. 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On the Interaction of Elementary Particles. I. Proceedings of the Physico-Mathematical Society of Japan, 17:48, 1935. 59 Appendix A Traceback Fitting The scattering location in the target is determined by projecting the pion's path back to the target from position information given by W C l and WC3 as discussed in section III.4.1 of Chapter II. During the analysis of the data it was found that the target chamber was out of alignment with respect to the beam. This was observed in the target traceback histograms. Since the target was in transmission mode (see equation 29), the traceback for 2 identical spectrometer angles on either side of the beam 1 should be similar. However, this was not the case. Target traceback histograms from data taken at QQD angles to the right of the beam displayed the target edges, as those on the left did not. The loss in scattering data resulted in substantially lower cross sections. This was compensated for by fitting a Gaussian to the beam profile (which remained constant throughout the experiment) as seen in the x direction by the traceback histograms. The traceback data in the y direction remained virtually unaffected. During the fitting procedure, the width ( c r ) of the Gaussian remained fixed while the area was varied. The ratio of the fitted area to the actual area was the factor used in modifying the cross section. The error of the fit was determined by varing a and the area under the function in order to obtain maximun and minimum values for which the fit remained reasonable. This method yielded errors typically ~8%. The assumption that the beam profile remained constant is based on the fact 1The physical size of the M13 experimental area prevents data to be taken at QQD angles i 70° to the left as measured downstream from the M13 channel. 60 that the beamline characteristics such as magnet currents and slit positions and widths remained unchanged. 61 

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