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Low energy π± interactions with S-D shell nuclei Tacik, Roman 1984

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LOW ENERGY ir* INTERACTIONS WITH S-D SHELL NUCLEI by ROMAN TACIK B.Sc, McGill U n i v e r s i t y , 1978 M.Sc, M c G i l l U n i v e r s i t y , 1980 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Physics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Ju l y 1984 © Roman Tacik, 1984 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i 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 an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f PHYSICS  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main M a l l V a n c o u v e r , Canada V6T 1Y3 Date J u l Y 26, 1984 Abstract A magnetic spectrometer has been commissioned for use i n performing low energy pion s c a t t e r i n g experiments. It has been used to measure the d i f f e r e n t i a l cross sections for the e l a s t i c s c a t t e r i n g of 50 MeV ir* from 1 2 C , 1 8 0 , and 2 6Mg targets, as well as the i n e l a s t i c s c a t t e r i n g to the 2 ^ states of these n u c l e i . The analysis of the e l a s t i c s c a t t e r i n g data has been performed with an o p t i c a l p o t e n t i a l model. It i s found that the ir* e l a s t i c cross sec-tions f o r 1 2 C and 1 8 0 are well described by c a l c u l a t i o n s employing a g l o -b a l parameter set (CMS 82), which has been previously shown to reproduce i r + e l a s t i c s c a t t e r i n g data f o r a range of n u c l e i . The TT* e l a s t i c data f o r 2^Mg require a v a r i a t i o n of the p-wave p o t e n t i a l parameters. A si m i -l a r requirement i s seen with ir* - 3 2 S e l a s t i c data (Sob 84a). Distor t e d wave c a l c u l a t i o n s , inputting separate neutron and proton t r a n s i t i o n d e n s i t i e s , have been performed f o r the analysis of the i n e l a s -t i c data. These c a l c u l a t i o n s demonstrate a strong s e n s i t i v i t y of i r + to protons, and ir" to neutrons, present i n the nucleus. By varying the strengths of the neutron and proton t r a n s i t i o n densi-t i e s , i n order to reproduce the TT* i n e l a s t i c cross sections, i t i s pos-s i b l e to determine the r a t i o of neutron to proton matrix elements (Mn/Mp) for the given 2 + s t a t e s . It i s shown that the value of Mn/Mp obtained i n t h i s way i s independent of the o p t i c a l p o t e n t i a l parameters used i n the c a l c u l a t i o n . The extracted values of Mn/Mp are: 0.97 ± 0.08, 1.81 ± 0.15, and 0.90 ± 0.07 f o r 1 2 C , 1 8 0 , and 2 6Mg r e s p e c t i v e l y . These values are i n agreement with those obtained by other means. - i i i -Table of Contents Abstract i i Table of Contents i i i L i s t of Figures v L i s t of Tables v i i i Acknowledgements x Chapter I INTRODUCTION 1 1.1 Nuclear Properties and the S h e l l Model 3 1.2 Scattering Experiments 7 1.3 Pion-Nucleon Scattering 9 1.4 Experimental Considerations 14 Chapter I I EXPERIMENTAL DETAILS 18 2.1 M13 Beamline 19 2.2 QQD Spectrometer 24 2.2.1 Detectors 24 2.2.2 E l e c t r o n i c s and Data A c q u i s i t i o n 30 2.2.3 Momentum Determination 33 2.2.4 Resolution 36 2.3 Targets 40 Chapter I I I EXPERIMENTAL ANALYSIS . ... .42 3.1 C a l c u l a t i o n of Cross Sections 3.1.1 Beam Normalization 44 3.1.2 Target Thickness. 49 3.1.3 E f f i c i e n c y 50 3.1.4 Peak F i t t i n g 53 3.1.5 Spectrometer Acceptence 54 3.2 Results 60 - i v -Chapter IV THEORETICAL DETAILS 67 4.1 E l a s t i c Scattering 4.1.1 The MSU Opt i c a l P o t e n t i a l 68 4.1.2 Further Discussion 71 4.2 I n e l a s t i c Scattering 4.2.1 C a l c u l a t i o n of Cross Sections 76 4.2.2 T r a n s i t i o n Densities 78 4.2.3 Neutron and Proton Matrix Elements 81 4.2.4 Isovector S e n s i t i v i t y of I T * I n e l a s t i c Scattering.. .83 Chapter V RESULTS AMD CONCLUSIONS 92 5.1 E l a s t i c Scattering 95 5.2 I n e l a s t i c Scattering 5.2.1 Results 103 5.2.2 Model Dependence 109 5.2.3 Comparison with Other Experiments 113 References • 117 Appendix A D e t a i l s of E l a s t i c C a l c u l a t i o n 123 Appendix B D e t a i l s of I n e l a s t i c C a l c u l a t i o n 126 Appendix C Nuclear Density Models.. ...129 Appendix D TT •*• uv Kinematics 130 - v -L i s t of Figures F i g . 1.1 S h e l l Model Structure of 1 8 0 and 2 6Mg 4 F i g . 1.2 Energy Level Schemes for 1 2 C , 1 8 0 , and 2 6Mg 6 F i g . 1.3 ir—p T o t a l Cross Sections 10 F i g . 1.4 i r ±p D i f f e r e n t i a l Cross Sections at 50 and 160 Mev 12 F i g . 1.5 Ratio of T r +p / i r~p D i f f e r e n t i a l Cross Sections at Several Energies 13 F i g . 2.1 M13 Beamline and QQD Spectrometer 20 F i g . 2.2 QQD Spectrometer with Associated Detectors 25 F i g . 2.3 Configuration of E l e c t r o n i c s 31 F i g . 3.1 T y p i c a l T T + and TT~ Time of F l i g h t Spectra 46 F i g . 3.2 Pion Flux Correction Factors 48 F i g . 3.3 T y p i c a l 1 2 C Energy Spectrum 55 F i g . 3.4 T y p i c a l 1 8 0 Energy Spectrum... 56 F i g . 3.5 T y p i c a l 2 6Mg Energy Spectrum 57 F i g . 3.6 REVMOC C a l c u l a t i o n of V a r i a t i o n of QQD S o l i d Angle over Scattering Target.. 59 - v i -F i g . 4.1 50 MeV T f 1 D i f f e r e n t i a l Cross Sections for 2 6Mg(2 1+) with 8 n = Bp - 0.50 84 F i g . 4.2 Ratio of I T ~ / T T + D i f f e r e n t i a l Cross Sections f o r 2 6 M g ( 2 1 + ) for Several Values of 6 85 F i g . 4.3 50 MeV T T * D i f f e r e n t i a l Cross Sections f o r 2 6 M g ( 2 1 + ) for Several Values of 6 87 Fi g . 4.4 /c+ vs 8 for 50 MeV I T 1 Scattering to 2 6 M g ( 2 1 + ) 89 F i g . 4.5 a+/a~ vs B p/B n for 50 MeV T T 1 Scattering to 2 6 M g ( 2 1 + ) 91 F i g . 5.1 1 2 C and 1 8 0 E l a s t i c Cross Sections C a l c u l a t i o n with Set E Parameters 96 F i g . 5.2 2 6Mg and 3 2 S E l a s t i c Cross Sections C a l c u l a t i o n with Set E Parameters 97 F i g . 5.3 2 6Mg and 3 2 S E l a s t i c Cross Sections C a l c u l a t i o n with 'Best F i t ' Parameters 101 F i g . 5.4 T y p i c a l x 2 Plot f o r F i t of 8 n and B p to 2 6Mg I n e l a s t i c Cross Sections 104 F i g . 5.5 1 2 C ( T T , T T ' ) 1 2 C * ( 2 1 + ) I n e l a s t i c Cross Sections 106 F i g . 5.6 1 8 0 ( T T , T T ' ) 1 8 0 * ( 2 1 + ) I n e l a s t i c Cross Sections 107 F i g . 5.7 2 6 M g ( i r , T T l ) 2 6 M g * ( 2 1 + ) I n e l a s t i c Cross Sections 108 - v i i -F i g . 5.8 V a r i a t i o n of 'Best F i t ' B's with Changes i n Rec Q 110 F i g . 5.9 x2 Contour Plot Using 'Best F i t ' O p t i c a l P o t e n t i a l Parameters I l l F i g . 5.10 Comparison of Mn/Mp as Obtained Using D i f f e r e n t Probes ....115 F i g . D.l TT uv Decay i n Center of Mass and Laboratory Frames 131 F i g . D.2 E u vs <|>u for 50 MeV I T ' S 133 F i g . D.3 Decay Muon Angular D i s t r i b u t i o n 134 - v i i i -L i s t of Tables Table 2.1 S p e c i f i c a t i o n s for QQD MWPCs 27 Table 2.2 T y p i c a l Set of QQD Transfer C o e f f i c i e n t s for WC4 P o s i t i o n 37 Table 2.3 T y p i c a l Set of QQD Transfer C o e f f i c i e n t s for WC5 P o s i t i o n 38 Table 3.1 Experimental Target Thicknesses 51 Table 3.2 Measured Cross Sections f or I 2 C ( T T , T T ) 1 2 C 61 Table 3.3 Measured Cross Sections f or 1 2 C ( i r , T r ' ) 1 2C*(2+,4.44) 62 Table 3.4 Measured Cross Sections f o r 1 8 0 ( i r , T r ) 1 8 0 63 Table 3.5 Measured Cross Sections for 1 8 0 ( T r , T r ' ) 1 80*(2+, 1.98) 64 Table 3.6 Measured Cross Sections f or 2 6 M g ( T r , T r ) 2 6 M g 65 Table 3.7 Measured Cross Sections f or 2 6 M g ( T r , T r ' ) 2 6 M g * ( 2 + , 1.81) 66 Table 4.1 Set E O p t i c a l P o t e n t i a l Parameters 72 Table 5.1 Nuclear Density D i s t r i b u t i o n Parameters 93 Table 5.2 Best F i t Values for O p t i c a l P o t e n t i a l Parameters 99 - ix -Table 5.3 Calculated T o t a l Reaction Cross Sections 102 Table 5.4 F i n a l Results for Mn/Mp Derived from Analysis of I n e l a s t i c Pion Scattering Data 105 Table 5.5 Comparison of Present Results with Those Obtained Using Other Techniques 114 - x -Acknowledgements F i r s t and foremost, I suppose I should thank the man with the funny h a i r c u t and yellow framed glasses, my research supervisor, Dick Johnson. Also, both for the i n c r e d i b l e amount of time he spent on the spectrometer, and for saying, at four o'clock one morning, a f t e r a week of night s h i f t s , that he did physics because he enjoyed i t , I thank Dave G i l l . B i l l Gyles and Bruce Barnett were graduate students with the group before I joined i t . Randy Sobie got involved not long afterward. Every-one knows that i t i s r e a l l y the graduate students that do a l l the hard work which enables experiments to get done. Of course, t h i s i s not meant to underplay the valuable contributions of Karl Erdman, our d r i f t chamber expert; Sig Martin and Chris Wiedner, beam op t i c i a n s extraordinaires, our German contingent; Hans Roser, our Swiss research associate; and Tom Drake, our man i n Toronto. I musn't neglect to thank our technicians. Robert Oppenshaw for his mastery of the art of wire chamber maintenence; Doug Maas for his know-ledge of e l e c t r o n i c s ; and Grant Scheffer f o r a whole host of things. I wish I could thank the engineer who worked with us for a while, but the memory of the unnecessarily heavy s t a i n l e s s s t e e l housings for the back wire chambers i s s t i l l too strong. Maybe some day. Unfortunately, I did a l l the typing and diagrams f o r the thesis myself, so I can't thank anyone for helping me there, but I must acknow-ledge the support provided by friends who would have helped, had I asked. - 1 -CHAPTER I INTRODUCTION In t h i s t h e s i s , the re s u l t s of experiments i n v o l v i n g the s c a t t e r i n g of 50 MeV pions from several nuclei are presented and discussed. The aim of these experiments was to provide data from which i n s i g h t s i n t o the d e t a i l s of the pion-nucleus i n t e r a c t i o n could be gained, and nuclear structure information extracted. The pion-nucleus system i s s u f f i c i e n t l y complex, that there is no one d e f i n i t i v e experiment which can be performed, which w i l l provide the answer to a l l questions. Knowledge i n this f i e l d i s gained through the examination of large quantities of diverse data. The experiments discus-sed i n t h i s t h e s i s represent a s i g n i f i c a n t contribution i n th i s d i r e c t i o n i n several respects: We have measured the e l a s t i c s c a t t e r i n g of 50 MeV pions of both charge states, from a range of n u c l e i , and found that a l l the data can be described co n s i s t e n t l y within the framework of a single model. We have measured, for the f i r s t time, the i n e l a s t i c s c a t t e r i n g of 50 MeV pions of both charges to low-lying excited states of several n u c l e i , s p e c i f i c a l l y 1 8 0 and 2 6Mg. The shapes of the measured angular d i s t r i b u t i o n s are reproduced reasonably well within the context of the same model used f o r the e l a s t i c s c a t t e r i n g . The r e l a t i v e contributions of the neutrons and protons i n these n u c l e i to the excited states can be extracted from the data, i n a manner which shows l i t t l e dependence on the actua l model used. In an e f f o r t to put these r e s u l t s into a more global context, the present introductory chapter of th i s thesis begins with a short, general - 2 -d i s c u s s i o n of nuclear properties, and t h e i r d e s c r i p t i o n within the s h e l l model, which provides a picture of nu c l e i i n both t h e i r ground and excited s t a t e s . In se c t i o n 1.2, the discussion continues with a general consider-a t i o n of how nuclear properties can be investigated by s c a t t e r i n g various p a r t i c l e s from n u c l e i . In section 1.3, a discussion of the pion- nucleon i n t e r a c t i o n i s undertaken. It i s pointed out that i n low energy scat-t e r i n g the T T ~ ( T T + ) i s much more s e n s i t i v e to neutrons(protons) than to protons(neutrons). This low energy s e n s i t i v i t y , l a r g e l y ignored or un-noticed by p h y s i c i s t s i n the f i e l d , provides a strong motivation for i n -v e s t i g a t i n g pion-nucleus s c a t t e r i n g . In se c t i o n 1.4 then, the d i f f e r e n t p o s s i b l e experimental techniques f o r performing pion s c a t t e r i n g experi-ments are explored, and reasons given for the use of the apparatus which has i n fac t been employed i n the se r i e s of experiments which are the sub-jec t of t h i s t h e s i s . In Chapter I I , a more detai l e d d e s c r i p t i o n of the equipment and methods used i n performing the present experiments i s given. Chapter I II contains a q u a n t i t a t i v e discussion of how the raw data r e s u l t i n g from these measurements i s analysed, and concludes with a tabulated presenta-t i o n of the d i f f e r e n t i a l cross sections. In Chapter IV, the t h e o r e t i c a l o p t i c a l model c a l c u l a t i o n s for d i f f e r e n t i a l cross sections which have been performed are o u t l i n e d . Some time i s spent i n the discussion of the pre-d i c t i o n s of the i n e l a s t i c c a l c u l a t i o n s , since s i m i l a r ones have not been reported previ o u s l y . Connection i s made with the experimental r e s u l t s i n Chapter V, which contains an examination of the conclusions which may be drawn from the present r e s u l t s ; the impli c a t i o n s f o r , and l i m i t a t i o n s of the t h e o r e t i c a l model; and a comparison with other experimental r e s u l t s . - 3 -1.1 Nuclear Properties and the S h e l l Model The simplest conceptual picture of atomic n u c l e i i s one i n which they are viewed as conglomerations of i n d i v i d u a l protons and neutrons, i n r e l a t i v e motion with respect to each other. Although t h i s p i c t u r e i s i n f a c t too simple i n many instances, i t does provide a q u a l i t a t i v e d e s c r i p t i o n of many basic nuclear properties, such as charge and mass. A more quantitative d e s c r i p t i o n can only be provided by a mathematical model. The basic one Is known as the s h e l l model, i n which each i n d i v i d u a l nucleon i s viewed as moving i n a p o t e n t i a l generated by a l l the others, i n much the same way as an electron moves i n the Coulomb f i e l d of the proton i n a hydrogen atom. As i n the case of the hydrogen atom, the s h e l l model p o t e n t i a l permits only c e r t a i n allowed o r b i t s or energy l e v e l s . These are i l l u s t r a t e d i n F i g . 1.1, as occupied by protons and neutrons i n 1 8 0 and 2 6Mg. Unlike the case of the hydrogen atom, however, where the form of the Coulomb p o t e n t i a l i s well known from f i r s t p r i n c i p l e s , the nuclear p o t e n t i a l i s not. In some Instances, the shape of the nuclear potential may be approximated, or calculated i n a phenomenological, s e l f - c o n s i s t e n t way. But the most s a t i s f a c t o r y derivations are those b u i l t up from the basic nucleon-nucleon p o t e n t i a l . The best modern N-N p o t e n t i a l s (e.g. the Pa r i s p o t e n t i a l (Vin 78)) are based on the exchange of pions between nucleons. Thus, there i s some motivation for the i n t e r e s t i n pion-nucleus i n t e r a c t i o n s . A l s o , i t has been pointed out by Gyles (Gyl 84) that low energy TT~-nucleus e l a s t i c s c a t t e r i n g i s perhaps the best way of studying the neutron d i s t r i b u t i o n s of nuclear ground s t a t e s . This i s c e r t a i n l y one of the nuclear properties one would hope to c a l c u l a t e with the use of some A E F i g . 1.1 Sh e l l Model Structure of 1 8 0 and 2 6 Mg - 5 -mathematical model. Within the framework of the s h e l l model, nuclear excited states are formed by promoting nucleons occupying low-lying s h e l l model l e v e l s to those higher i n energy. In general, there are many possible combinations which w i l l r e s u l t i n excited nuclei with i d e n t i c a l spins and p a r i t i e s , and s i m i l a r energies. Thus, excited nuclear wavefunctions are l i n e a r combinations of the many possible nucleon e x c i t a t i o n s . It w i l l be shown i n Chapter IV, that i n low energy ir-nucleus i n e l a s t i c s c a t t e r i n g , T T - , S are more s e n s i t i v e to neutrons, and i r + , s to protons then vi c e - v e r s a . Thus, one of the d i r e c t aims of the experiments discussed i n t h i s t h e s i s was to use a combination of low energy ir*-nucleus I n e l a s t i c s c a t t e r i n g to separate the neutron and proton contributions to several nuclear excited s t a t e s . It i s worth noting that when the number of d i f f e r e n t single or multiple nucleon e x c i t a t i o n s contributing to a nuclear excited state become large, i t i s sometimes more convenient to describe the nuclear e x c i t a t i o n In terms of a c o l l e c t i v e model, rather than the s h e l l model. In c o l l e c t i v e models, e x c i t a t i o n s are assumed to a r i s e as a r e s u l t of rotations or v i b r a t i o n s of the nucleus as a whole. Note also, that within the s h e l l model, the number of l e v e l s to which an i n d i v i d u a l nucleon may be promoted i s quite high. In order to f a c i l i t a t e c a l c u l a t i o n s , the s h e l l model space i s usually truncated, and compensated f or by assigning neutrons and protons e f f e c t i v e charges, d i f f e r e n t from t h e i r r e a l ones. The energy l e v e l s of 1 2 C , 1 8 0 , and 2 6Mg, up to =7.5 MeV e x c i t a t i o n energy, are given i n F i g . 1.2 (LS 78). 7.65 4,44 7.12 (o-i s. ee . ,3" « * 6.40 r 6.20 2" 5.53 ^5.26 5.?^  —* 3-1- 4.46 2* 3.92 3.63, c c 5^  -(2r 4* ar E-4«,.-3* IL2L _&2i ML 5.47 4.97 4.90 4.351 n / •»•? _2Ji 359 16 O 26 Mg 2.94 1.98 1.81 F i g . 1.2 Energy Level Schemes for 1 2 C, 1 8 0, and 2 6 Mg - 7 -1.2 Sc a t t e r i n g Experiments Perhaps the most common way of measuring physical observables i s through s c a t t e r i n g experiments. An example that springs to mind r i g h t away i s that of the s c a t t e r i n g of l i g h t i n t o the human eye, which forms the basis of v i s i o n . The existence of atomic n u c l e i was suggested by Rutherford as a r e s u l t of experiments inv o l v i n g the s c a t t e r i n g of alpha p a r t i c l e s by f o i l s of d i f f e r e n t materials. Mathematically, the d i f f e r e n t i a l cross section f o r s c a t t e r i n g a p a r t i c l e from a p o t e n t i a l U(r) can be obtained from Fermi's Golden Rule. That i s , doVdt. = |<?f |u(r) | Y±> | 2 , where ¥^ and f f are the i n i t i a l and f i n a l p a r t i c l e wavefunctions. I f these can be taken to be plane waves, i . e . „, i k - r r i k f r 4*i = e 1 and Tf = e 1 , then <¥ f|U(r)|f i> = / e i q * r U(r) d 3 r = U(q) , where q = k-j-kf. And so da/dfi = |U(q) | 2 . For the case of p a r t i c l e s c a t t e r i n g from a nucleus, i f the i n t e r a c t i o n between the p a r t i c l e and a point nucleon i s given by a p o t e n t i a l V ( r ) , then U(r) = / V(r-r') p(r') d 3 r ' , where p(r') i s the density d i s t r i b u t i o n of nucleons i n the nucleus. Then <Tf|U(r)|?i> = / e i q # r d 3 r / V ( r - r ' ) p(r') d 3 r ' =V(q ) p(q) , - 8 -and da/dQ. « |V(q)p(q) | 2 . From the above, i t can be seen that i n general, p a r t i c l e s c a t t e r i n g from n u c l e i w i l l depend both on the particle-nucleon i n t e r a c t i o n , and the nucleon d i s t r i b u t i o n s within the n u c l e i . In the case of electron s c a t t e r i n g , where the i n t e r a c t i o n i s the f a m i l i a r electromagnetic one, nuclear structure information can be extracted from the data quite e a s i l y . However, because the i n t e r a c t i o n i s electromagnetic, the nuclear property one i s s e n s i t i v e to i s the charge d i s t r i b u t i o n . From t h i s , proton matter d i s t r i b u t i o n s can be obtained. No information i s obtained about the neutrons present in the nucleus. In order to study these, i t i s necessary to employ hadronic probes, such as nucleons, alpha p a r t i c l e s , or pions. Usually, the hadron-nucleus In t e r a c t i o n i s described through the use of an o p t i c a l p o t e n t i a l , which i s independent of the coordinates of the i n d i v i d u a l nucleons within the nucleus. The o p t i c a l p o t e n t i a l s have r e a l and imaginary parts, the l a t t e r accounting for various i n e l a s t i c channels. The o p t i c a l p o t e n t i a l used i n the present a n a l y s i s of low energy pion s c a t t e r i n g i s discussed i n section 4.1.1. It w i l l be pointed out i n section 1.3, and then i n section 4.2.4, that low energy t r - l s are e s p e c i a l l y s e n s i t i v e to neutrons. This fact can be used, i n combination with techniques such as s c a t t e r i n g from adjacent isotopes, or simultaneous f i t s to T T * data, to reduce the dependence of the nuclear structure Information extracted from pion s c a t t e r i n g data on the d e t a i l s of the pion-nucleus i n t e r a c t i o n . It i s not evident that the same i s true of other hadronic probes. - 9 -1.3 Pion-Nucleon Scattering F i g . 1.3 i l l u s t r a t e s the ir* - proton t o t a l s c a t t e r i n g cross sections, as ca l c u l a t e d with the phase s h i f t s of Arndt and Roper (AR 82). These phase s h i f t s are the result of a g l o b a l f i t to a l l the a v a i l a b l e u^p s c a t t e r i n g data, at a l l energies. The most outstanding feature of t h i s plot i s the peak at approximately 180 MeV incoming pion energy. This resonance i n the cross section i s interpreted as being due to the formation of a new p a r t i c l e , the A. Its c h a r a c t e r i s t i c quantum numbers are spin = 3/2 and i s o s p i n = 3/2, thus i t i s frequently r e f e r r e d to as the 3-3 resonance. To explain the dif f e r e n c e i n T T + and ir~ t o t a l cross sections, one may consider u ±p s c a t t e r i n g i n terms of a p a r t i a l wave expansion. The d i f f e r e n t i a l cross section for the sc a t t e r i n g of a pion of e i t h e r charge st a t e , which may be integrated to y i e l d the t o t a l cross section, can be written: da/an = | g I ( e ) | 2 + |h!(e) | 2 . h(9) and g(9) are the s p i n - f l i p and non s p i n - f l i p irp s c a t t e r i n g amplitudes. They are related to sums of phase s h i f t s and Legendre polynomials, over the various contributing p a r t i a l waves. I i s an i s o s p i n index. Tr+p can only exist i n a pure i s o s p i n = 3/2 sta t e , while ir~p can be a mixture of 3/2 and 1/2 states. Thus, da/dSl ( T T + P) = | g 3 / 2 ( 6 ) | 2 + | h 3 / 2 ( 0 ) | 2 , 1 2 1 2 and do7dJ2 (ir"p) = | j g 3 / 2 + y g 1 / 2 | 2 + | j h 3 / 2 + - h J / 2 | 2 . - 10 -- 11 -Near 180 MeV incident pion energy, because of A production, the 3/2 amplitudes are much larger than the 1/2. Thus, i t i s evident from the above that the r a t i o of T T + to ir~ t o t a l cross sections, and d i f f e r e n t i a l cross sections independent of angle, w i l l be 9. At lower incident pion energies, t h i s w i l l no longer be the case. However, i t can be seen that i f the 1/2 and 3/2 amplitudes are s i m i l a r i n magnitude, but d i f f e r i n sign, then there i s a p o s s i b i l i t y of c a n c e l l a t i o n . This i s i n fact the s i t u a t i o n at approximately 50 MeV in c i d e n t pion energy. F i g . 1.4 i l l u s t r a t e s the n^p d i f f e r e n t i a l cross sections for both 50 MeV and 160 MeV pions. As noted above, the i r + to ir~ r a t i o i s constant at the higher energy, but increase with angle at the lower energy. F i g . 1.5 i l l u s t r a t e s this r a t i o for several incident pion energies. For 50 MeV i t r i s e s to 800 at 180°. The average value i s 20. Thus, i t i s seen that low energy n+'s i n t e r a c t more strongly with protons, and i r - ,s with neutrons, than v i c e - v e r s a . Of course i t i s not cle a r that t h i s s e n s i t i v i t y w i l l also be present i n the case of pion-nucleus s c a t t e r i n g . D i f f e r e n t combinations of p a r t i a l waves could give r i s e to d i f f e r e n t c a n c e l l a t i o n e f f e c t s . Coulomb e f f e c t s w i l l be more important out to further angles. But the pion-nucleon s e n s i t i v i t y provides a strong motivation for i n v e s t i g a t i n g nuclear s c a t t e r i n g . In f a c t , t h i s isovector s e n s i t i v i t y Is present i n the pion-nucleus system. Evidence i n favor of such a conclusion w i l l be presented i n section 4.2.4. - 13 -F i g . 1.5 Ratio of T t + p / i r ~ p D i f f e r e n t i a l Cross Sections at Several Energies - 14 -1.4 Experimental Considerations In the past, pion s c a t t e r i n g experiments have been performed with: a) Nal detectors, b) Ge detectors, and c) p l a s t i c s c i n t i l l a t o r telescopes. Each of these presents i t s own problems. a) In a Nal detector, a stopping pion loses energy through i n t e r a c t i o n s with the electrons present i n the detector c r y s t a l . Light emitted as a r e s u l t of the deexcitation of the electrons i s c o l l e c t e d and converted Into an e l e c t r o n i c s i g n a l . The s i z e of the s i g n a l i s proportional to the amount of l i g h t , which i n turn i s p r o p o r t i o n a l to the amount of energy deposited by the pion. But the nature of the whole process i s such that the best energy r e s o l u t i o n obtainable with Nal detectors, for incident pions with 50 MeV k i n e t i c energy, i s approximately 1.5 MeV. This i s too large to separate the ground and excited states i n a l l but a few n u c l e i , and thus must be improved upon i n order to undertake a s e r i e s of i n e l a s t i c s c a t t e r i n g experiments. Furthermore, Nal detectors cannot be used for T T - , S . This i s due to the fa c t that, unlike i r +'s, stopping TT~'S get captured i n atomic o r b i t s , and i n t e r a c t d i r e c t l y with the n u c l e i present i n the detector c r y s t a l , causing the n u c l e i to d i s i n t e g r a t e . The pion r e s t mass energy i s transformed i n t o the k i n e t i c energy of the fragments of the d i s i n t e g r a t i n g nucleus. In order to measure the k i n e t i c energy of the incoming pion accurately, one would have to know exactly how the nucleus had broken up, i n order to account for the various binding energy c o r r e c t i o n s . On the other hand, one would also have to ensure that a l l the fragments stopped i n the detector, i n order to measure the t o t a l energy. These two requirements are mutually incompatible. - 15 -b) In a Ge detector, the electrons with which the stopping pions i n t e r a c t are gathered by means of an e l e c t r i c f i e l d present within the detector c r y s t a l , and produce a current pulse d i r e c t l y . Thus, obtainable energy resolutions are much lower than with Nal's; approximately 300 keV fo r 50 MeV pions being t y p i c a l . This i s c e r t a i n l y adequate f o r many i n e l a s t i c s c a t t e r i n g experiments. Unfortunately, Ge detectors cannot be used f o r T T - , S e i t h e r , f o r the same reasons as outlined above. c) Of the detectors mentioned thus f a r , only the p l a s t i c s c i n t i l l a t o r telescope can be used to detect i r - , s , as well as T T + , S . Such a telescope consists of a series of thi n p l a s t i c s c i n t i l l a t o r s , each of which operates i n much the same way as the Nal detector, converting energy deposited by passing pions into l i g h t , but with much poorer i n t r i n s i c energy r e s o l u t i o n . Pions are i d e n t i f i e d , and t h e i r energies obtained, however, not by summing the heights of the s c i n t i l l a t o r s i g n a l s , but by measuring t h e i r range i n the telescope. That i s , by noting i n which s c i n t i l l a t o r of the se r i e s they stop. Unfortunately, the s t a t i s t i c a l nature of the stopping process means that the best r e s o l u t i o n obtainable with such detectors i s of the order of 3 MeV. This i s unacceptably high f o r most a p p l i c a t i o n s . The detector chosen f o r the experiments discussed i n t h i s t hesis i s the magnetic spectrometer. Such devices have been used i n nuclear physics, i n many a p p l i c a t i o n s , f o r many years, and are capable of providing excellent energy re s o l u t i o n s . In p r i n c i p l e , t h e i r operation i s based on the f a c t that charged p a r t i c l e s , moving through a magnetic f i e l d , - 16 -experience a force proportional to t h e i r momenta. In the simplest case of a p a r t i c l e moving i n vacuum, with i t s v e l o c i t y perpendicular to the d i r e c t i o n of a constant, uniform dipole f i e l d , the p a r t i c l e t r a j e c t o r y i s a c i r c l e , whose radius i s r = p/qB. Thus, a measurement of the curvature of the p a r t i c l e ' s path, obtained by detecting i t s p o s i t i o n before and a f t e r i t s passage through a dipole f i e l d , gives i t s momentum, and thus i t s k i n e t i c energy. The operation of p o s i t i o n s e n s i t i v e detectors w i l l be described in s e c t i o n 2.2. One notes, however, that since they do not measure the p a r t i c l e ' s t o t a l energy, they do not require the p a r t i c l e to come to r e s t , and thus can be used with equal e f f i c i e n c y for detecting both i r - and T T + . Although the use of the spectrometer i s straightforward i n p r i n c i p l e , i n p r a c t i c e the analysis of the data i s quite complicated, and w i l l be discussed further i n section 2.2.3. Part of the complication a r i s e s from the f a c t that the magnetic f i e l d s produced experimentally are not i n f i n i t e i n extent, and one must account for the non-uniform f i e l d shapes occuring at the entrances and e x i t s of r e a l magnets. Fortunately, techniques f o r designing (Ste 65), (Ban 66), and constructing magnets which produce f i e l d s suited for s p e c i f i c a p p l i c a t i o n s are well known. In p a r t i c u l a r , one can construct quadrupole magnets which have the e f f e c t of focussing beams of charged p a r t i c l e s , as lenses do for l i g h t . Apart from good energy r e s o l u t i o n , there are several other requirements for the detection system to be used for a low energy pion s c a t t e r i n g experiment. One of these i s a large acceptance, to compensate for the f a c t that incoming pion fluxes are r e l a t i v e l y low. In the present case, t h i s requirement was met by i n s t a l l i n g a p a i r of quadrupole magnets - 17 -before the bending magnet. These serve to focus the beam, mainly i n the non-bend plane, where the spectrometer dipole gap l i m i t s the acceptence, and thus increase the s o l i d angle into which pions can scatter and s t i l l be observed. The name of the spectrometer, the QQD, i s taken from the magnet co n f i g u r a t i o n : quadrupole-quadrupole-dipole. - 18 -CHAPTER I I EXPERIMENTAL DETAILS The purpose of the present chapter Is to provide d e t a i l s of the apparatus and techniques used to perform the s c a t t e r i n g experiments which are the subject of t h i s t h e s i s . In p r i n c i p l e , carrying out these experi-ments i s straightforward: one simply counts the number of pions which sc a t t e r from a given nuclear target i n a given d i r e c t i o n . There are, however, many p r a c t i c a l considerations which must be taken into account. The f i r s t of these i s the source of incident pions. At TRIUMF, pions are created i n the i n t e r a c t i o n of 500 MeV protons with the n u c l e i contained i n some production t a r g e t . Through the use of a ser i e s of magnets, c a l l e d a channel or beamline, the pions emitted i n a p a r t i c u l a r d i r e c t i o n can be c o l l e c t e d and d i r e c t e d toward a nuclear s c a t t e r i n g target. The present experiments were performed with pions from TRIUMF's M13 channel, which i s the subject of s e c t i o n 2.1. Af t e r t h e i r i n t e r a c t i o n with target n u c l e i , scattered pions must be detected, and t h e i r energies or momenta determined. Section 2.2 deals with the spectrometer u t i l i z e d for t h i s purpose i n the present experi-ments. It s t a r t s with a consideration of the detectors used i n conjunc-t i o n with the spectrometer magnets, and the configuration of e l e c t r o n i c c i r c u i t s which processed the signals from these detectors. There follows a d i s c u s s i o n of how t h i s information i s used to c a l c u l a t e the scattered pions' momenta, and the various l i m i t a t i o n s of the whole system. Section 2.3 contains a d e s c r i p t i o n of the preparation and compo-s i t i o n of the s c a t t e r i n g targets. - 19 -2.1 M13 Beamllne The M13 beamline, along with the QQD spectrometer, i s i l l u s t r a t e d i n F i g . 2.1. Protons accelerated to 500 MeV with the TRIUMF cyc l o t r o n pass through a production target, T l , where some of them i n t e r a c t and produce pions. Those IT'S emitted at 135° with respect to the incoming proton beam are focussed by the Ql and Q2 quadrupole magnets, de f l e c t e d by the Bl dipole magnet, and come to a dispersed focus at F l . A quadrupole t r i p l e t , Q3, Q4, and Q5, serves to produce another dispersed focus at F2. F i n a l l y , the pion t r a j e c t o r i e s are bent by B2, and focussed by Q6 and Q7, to form an achromatic beam spot at the pivot point of the spectrometer, where the nuclear s c a t t e r i n g targets are placed. Pions are not the only p a r t i c l e s to come down the M13 beamline. Other charged p a r t i c l e s of the same momentum coming from the T l production target w i l l a lso be present. These include protons with i r + , and y's and e's with both i r + and ir~. The u's come from the decay of charged TT'S near T l , while the e's come from the decay of ir^'s. These can a l l be accounted for by measuring the p a r t i c l e s ' time of f l i g h t from T l , which i s accomplished with the use of a capacitive probe i n the main proton beamline, just before T l . This probe outputs a s i g n a l coincident with the passage of a burst of protons, which i n turn occurs every 43.5 ns, corresponding to the RF frequency of the TRIUMF cy c l o t r o n . This topic w i l l be addressed again i n section 3.1.1, which deals with the normalization of the incident pion f l u x . For many experiments, the beamline may be thought of simply as a source of pions, with no consideration given to the q u a l i t y of the beam. This i s not so i n the present case. There are three beam properties which F i g . 2.1 M13 Beamline and QQD Spectrometer - 21 -have a d i r e c t influence on QQD experiments: a) The o v e r a l l energy r e s o l u t i o n which may be achieved with the spectrometer depends on the energy spread of the incoming beam, which thus must be small, b) Because i n e l a s t i c s c a t t e r i n g cross sections are small (usually < 1. mb/sr), pion fluxes must be high enough to ensure that experiments can be completed i n reasonable periods of time, c) Due to the l i m i t e d q u a n t i t i e s of separated isotope target m a t e r i a l a v a i l a b l e , and the non-uniformity of the spectrometer acceptence, the f i n a l beam spot should be as small as pos s i b l e . These three considerations are i n fact coupled, and i n general cannot be optimized Independently. The t o t a l momentum acceptence of the M13 channel i s Ap/p = 6.7%. For 50 MeV pions, 1% Ap/p i s equivalent to 870 keV spread i n energy, and so the momentum acceptence of the channel must somehow be reduced. This i s accomplished by means of mechanical s l i t s , positioned at the dispersed f o c i F l and F2. The dispersion of the beam at these points i s 1.25 cm/%. Unfortunately, the t i l t of the f o c a l planes i s such (81°) that pions with a momentum spread greater than 1% w i l l pass through a 1.25 cm s l i t opening. Two sextupole magnets, SX1 and SX2, were i n s t a l l e d i n the channel i n order to straighten the f o c a l plane at F2. But these have not as yet proven u s e f u l because of other not yet f u l l y understood e f f e c t s present i n the channel. One should note also that there i s a l i m i t to how t i g h t l y the s l i t s can be closed, and s t i l l reduce the momentum spread of the pion beam. There i s a magnification factor of = 1. from the T l target to the F2 focus. In e f f e c t , t h i s means that even a monochromatic beam of pions, emitted from T l , would not come to a point focus, but would be spread out - 22 -over a space equivalent to the s i z e of the production t a r g e t . Thus, one would l i k e to use as thi n a production target as p o s s i b l e . But t h i s would reduce the pion f l u x , as does c l o s i n g down the s l i t s . One would also l i k e to use a target with as high a Z as possible, to increase pion production. But t h i s cannot be done because of the e f f e c t i t would have on the q u a l i t y of the main proton beam downstream of T l . For the experiments described i n t h i s t h e s i s , the following c o n f i g u r a t i o n was adopted: the T l target was 10 mm graphite; the F l and F2 s l i t s were set- to 0.5 % Ap/p. The r e s u l t i n g pion fluxes were: = 2 x l 0 6 i r +/s, and = 3 x l 0 5 ir~/s. The energy spread of the incoming beam, as measured with a Ge detector, was - 750 keV. The beam spot at the nuclear s c a t t e r i n g target was » 20 mm FWHM i n both h o r i z o n t a l and v e r t i c a l d i r e c t i o n s . H a l l probes were i n s t a l l e d i n a l l M13 quadrupole magnets, and nuclear magnetic resonance (NMR) probes i n the dipoles Bl and B2, i n order to monitor magnet s t a b i l i t y , and ensure r e p r o d u c i b i l i t y of the magnet s e t t i n g s . The energy of the pion beam was determined by s c a l i n g to the B l magnetic f i e l d . This had been c a l i b r a t e d with an a-source, at the time M13 was o r i g i n a l l y commissioned (0ra+ 81). Unfortunately, subsequent changes i n the channel mean that this c a l i b r a t i o n cannot be trusted to better than ±2.%. Several methods have been employed i n an attempt to improve on t h i s uncertainty, but as yet none has been s u c c e s s f u l . It may be of i n t e r e s t to note that improvements to the system have been made on an ongoing basis since the experiments presently under consideration have been performed. A 3 mm synthetic diamond target has been i n s t a l l e d i n place of the 10 mm graphite target at T l . Because of - 23 -the higher density of the diamond, pion fluxes w i l l not decrease, but the i n t r i n s i c r e s o l u t i o n of the channel w i l l improve. P o s i t i o n s e n s i t i v e detectors have been i n s t a l l e d i n place of the mechanical s l i t s at F l and F2. Thus, the e n t i r e pion f l u x can be used for experiments, since the r e l a t i v e momenta of the incoming pions can be determined and so compensated f o r . The s i z e of the beam spot at the experimental target has been reduced considerably, by optimizing the settings of the Q3-Q5 magnets. Further advances w i l l continue to be made. - 24 -2.2 QQD Spectrometer The QQD Spectrometer i s shown i n place at the end of the M13 beamline i n F i g . 2.1. The spectrometer consists of two quadrupole magnets, QT1 and QT2, and one dipole magnet BT. QTl i s h o r i z o n t a l l y focussing, while QT2 focusses i n the v e r t i c a l d i r e c t i o n . BT i s designed to bend pions by 70° to the l e f t , i n the h o r i z o n t a l d i r e c t i o n . The d i r e c t i o n i s such as to allow eventual d i s p e r s i o n matching with the M13 beam. In order to keep losses from the decay of pions passing through the spectrometer small, i t was desirable to make the system as short as p o s s i b l e . The present length i s 2.38 m, from the s c a t t e r i n g target to the center of the l a s t wire chamber. The spectrometer f o c a l plane, however, i s beyond the l a s t wire chamber, and t i l t e d at an angle of 72°. The spectrometer may be used from 0° with respect to the incoming pion beam d i r e c t i o n , to approximately 137°. The system was designed to have an acceptence of 18 msr, and a momentum acceptence of ±20%. The experimentally determined values w i l l be discussed i n s e c t i o n 3.1.5. 2.2.1 Detectors An overview of the QQD spectrometer, i l l u s t r a t i n g a l l the detectors used i n data a c q u i s i t i o n , i s given i n F i g . 2.2. The detectors B l , B2, u l , u2, E l , E2, and E3 are p l a s t i c s c i n t i l l a t o r s (NE110), f i t t e d with RC8575R phototubes. A l l are 6.4 mm thick, except B l , which i s 0.8 mm, and E3, which i s 12.7 mm. B l , B2, y l , and u2 were used to monitor the incident pion f l u x , which i s discussed further i n section 3.1.1. Coincident sig n a l s from E l , E2, and E3 were used to i d e n t i f y p a r t i c l e s passing through the spectrometer. Three detectors were used i n order to reduce F i g . 2.2 QQD Spectrometer with Associated Det ectors - 26 -the incidence of random and background events. Such events were not a problem i n the o f f l i n e analysis, where they could be i d e n t i f i e d e a s i l y , but t h e i r on l i n e e l i m i n a t i o n was desirable, i n order to save space on the magnetic tapes on which a l l events were recorded. The detectors WC1, WC3, WC4, and WC5 are p o s i t i o n s e n s i t i v e counters. They are commonly referred to at TRIUMF as Wire Chambers, but are more properly c a l l e d Multi Wire Proportional Counters (MWPCs). The r e s p o n s i b i l i t y f or the operation and maintenence of these detectors was the major t e c h n i c a l contribution of the author to the present experiments. The chambers themselves were constructed at the Workshop of the U n i v e r s i t y of Carleton ( B i r t 71), based on a design adopted from the U n i v e r s i t y of A l b e r t a ( G i l 84). The exact s p e c i f i c a t i o n s are given i n Table 2.1. They consist of three planes of equally spaced, p a r a l l e l t h i n wires. The middle plane (anode) i s kept at a p o s i t i v e high voltage, while the other two wire planes (cathodes), one of which has i t s wires p a r a l l e l to the anode wires ( y - d i r e c t i o n ) , and the other perpendicular ( x - d i r e c t i o n ) , are grounded. The chambers are f i l l e d with a s p e c i a l l y prepared gas mixture, commonly referred to as magic gas, c o n s i s t i n g of 69.7% argon, 30% isobutane, and 0.3% freon. A charged p a r t i c l e passing through the chamber Ionizes the gas. The l i b e r a t e d electrons accelerate toward the nearest anode wire, creating an avalanche. The r e s u l t i n g p o s i t i v e ions, i n moving away from the anode wire, Induce a pulse on the nearest cathode wires. A l l the wires from each cathode plane are soldered onto a printed c i r c u i t type delay l i n e (Bos+ 75). A f t e r reaching i t , the induced pulse s p l i t s , and travels to both ends of the delay l i n e . The - 27 -WC1 and WC3 WC4 and WC5 Anode Wire Thickness 20.3 um 20.3 um Cathode Wire Thickness 63.5 um 63.5 um Wire Plane Separation 4.76 mm 6.35 mm Anode Wire Separation 1.0 mm 1.0 mm Cathode Wire Separation 2.0 mm 2.0 mm Number of Anode Wires ( y - d i r e c t i o n ) 169 308 Number of Cathode Wires ( x - d i r e c t i o n ) 169 3 x 203 Operating Voltage (with Magic Gas) 4.3 kV 5.4 kV Table 2.1 S p e c i f i c a t i o n s f o r the QQD Mult i Wire Proportional Counters - 28 -time d i f f e r e n c e between the a r r i v a l of the pulse at e i t h e r end i s used to locate the p o s i t i o n of the i n i t i a l charged p a r t i c l e ' s passage. It i s to be noted that the electron avalanche occurs at a p a r t i c u l a r , f a i r l y l o c a l i z e d p o s i t i o n around one anode wire. Thus, while the x - d i r e c t i o n p o s i t i o n spectra w i l l be continuous, the y - d i r e c t i o n spectra w i l l have a 'picket-fence' s t r u c t u r e . Each picket or peak corresponds to the l o c a t i o n of an anode wire. Some e f f o r t was put into determining the best method for c a l i b r a t i n g the conversion of time differences i n t o p o s i t i o n s i n mm. One method investigated Involved the soldering of ' f i d u c i a l ' wires onto known lo c a t i o n s on the delay l i n e s , along which s i g n a l s simulating the passage of charged p a r t i c l e s through the chambers could be sent. This was not s u c c e s s f u l , however, because the i n t r o d u c t i o n of the f i d u c i a l wires a l t e r e d the c h a r a c t e r i s t i c transmission properties of the delay l i n e s . The method f i n a l l y adopted involved a combination of the use of electron sources and the known distances between peaks i n the y - d i r e c t i o n 'picket-fence* spectra i n order to convert time dif f e r e n c e s into r e l a t i v e p o s i t i o n measurements. Absolute positions were determined by f u l l y i l l u m i n a t i n g the chambers with the pion beam, and l o c a t i n g the known positio n s of the chamber edges. At each end of each delay l i n e , s i g n a l s passed through a d i f f e r e n t i a l a m p l i f i e r . The amplifier was b u i l t based on a design o r i g i n a l l y developed at Los Alamos (Stu 74), but modified somewhat to better s u i t the present a p p l i c a t i o n . The a m p l i f i e r gain was chosen to be =100, producing chamber pulses >300 mV i n most cases. The a m p l i f i e r i s capable of producing output pulses with 5 ns r i s e times, but the actual - 29 -s i g n a l s from the chambers had =15 ns r i s e times due to d i s p e r s i o n of the o r i g i n a l induced pulses i n the delay l i n e s . It i s t h i s f a c t o r which l i m i t s the length of the delay l i n e s which may be used. In the x - d i r e c t i o n , the WC4 and WC5 cathode planes, which were a t o t a l of 600 mm wide, were connected to three separate delay l i n e s . The operating voltages for the chambers were determined by measuring t h e i r e f f i c i e n c i e s at various voltages with the use of a 1 0 6 R u electron source, and two p l a s t i c s c i n t i l l a t o r s . Under experimental conditions, a l l chambers operated with >96% e f f i c i e n c y . A more d e t a i l e d discussion of the o v e r a l l spectrometer e f f i c i e n c y w i l l be presented i n section 3.1.3. It should be noted that space charge e f f e c t s produce sparking i n WC1 at spectrometer angles <50°, where background s c a t t e r i n g events r a i s e the f l u x through WC1 to >101+ per second. The i n t r i n s i c p o s i t i o n r e s o l u t i o n of WC1 and WC3 was approximately 0.6 mm i n both the x and y d i r e c t i o n s . For WC4 and WC5, the resolutions were approximately 1.5 and 2.5 mm i n the two d i r e c t i o n s . This increase i s due to several f a c t o r s , the major cont r i b u t i o n coming from the dispersion i n the longer lengths of the delay l i n e s used. The i n t r i n s i c r e s o l u t i o n may be estimated i n several ways: a) with the use of highly collimated e l e c t r o n sources; b) from the widths of the peaks i n the y - d i r e c t i o n 'picket-fence' spectra; and c) from the width of the peak i n a spectrum of the sum of the times from each end of a delay l i n e . Note that t h i s l a s t peak must be corrected for the i n i t i a l d r i f t time of electrons to the anode wire. This may be as long as 25 ns, depending on the l o c a t i o n of the o r i g i n a l charged p a r t i c l e ' s passage. It can be measured by observing induced pulses on the anode wires d i r e c t l y , through a c a p a c i t i v e f i l t e r . - 30 -2.2.2 E l e c t r o n i c s and Data A c q u i s i t i o n A PDP 11/34 computer, running under the RSX operating system, was used to record event by event spectrometer data onto magnetic tape f o r l a t e r d e t a i l e d o f f l i n e a n a l y s i s . This was done using the standard TRIUMF data a c q u i s i t i o n program DA ( M i l 84), which responds to an in t e r r u p t (LAM) from a predetermined CAMAC crate module, i n the present case a C212 b i t - p a t t e r n u n i t , by reading the information stored i n the other modules i n the crate, TDCs, ADCs, and s c a l e r s . Some immediate on l i n e data a n a l y s i s was also performed using a modified version of the program MULTI (Fer 79). A schematic diagram of the e l e c t r o n i c modules and l o g i c used i n the experiments i s given i n F i g . 2.3. Two d i s t i n c t types of LAMs could be generated; those i d e n t i f i e d as spectrometer events, or those i d e n t i f i e d as beam sample events. A spectrometer event consisted of the coincidence E1*E2*E3*B1. Note that the pulses from these detectors were set In such a way as to always take the absolute timing from the leading edge of the E l mean time. This was done so as to allow for the elimination of B l from the coincidence, should i t prove not to be functioning e f f i c i e n t l y at high incident pion f l u x e s . This did not occur. Note also that because of t h e i r length (1. m), E l , E2, and E3 were f i t t e d with phototubes at both ends. The signals from these phototubes were passed through mean timers, i n order to make them independent of the l o c a t i o n of a p a r t i c l e ' s passage through the s c i n t i l l a t o r . The spectrometer event signal was fanned out to provide s t a r t s for TDCs whose stops came from the ends of the wire chamber delay l i n e s . This - 31 -E1R E1L E2R E2L E3R. E3R-B1 B2 Un 2 MT MT D ED OSCRIMINATOR AND MEAN TIMER Un 2 1 E3L- > E3L- | \ MT MT M 00 BATE GENARATOR OR B I T O lin 2 SPECTROMETER EVENT O—• rDlf BEAM SAMPLE EVENT LAM TDC starts '—I GG BIT1 T1 ION T1 C T1 CAP PROBE MWPC SIGNALS Is O CAM AC ADC O CAMAC TDC STOP • ADC gdtes inhibit scalers ^ CAMAC output rogistsr (comp busy) VISUAL AND CAMAC SCALERS CAMAC BIT PATTERN UNIT F i g . 2.3 Configuration of E l e c t r o n i c s - 32 -TDC i n f o r m a t i o n was u s e d to d e t e r m i n e t h e p o s i t i o n , i n t h e MWPCs, of the p a r t i c l e w h i c h had p a s s e d t h r o u g h the s p e c t r o m e t e r . Note t h a t t h e h e i g h t s of t h e i n d i v i d u a l E l , E2,. and E3 s i g n a l s were a l s o r e c o r d e d i n ADCs. These s i g n a l s c o u l d be summed t o p r o v i d e a measure o f t h e energy l o s s o f the p a r t i c l e s p a s s i n g t h r o u g h t h e s c i n t i l l a t o r s , w h i c h c o u l d be used t o i d e n t i f y p i o n s w h i c h had decayed t o muons i n t h e s p e c t r o m e t e r . N ote a l s o t h a t t h e i n d i v i d u a l B l s i g n a l was r e c o r d e d i n a TDC. T h i s p r o v i d e d a d i r e c t measure of the f r a c t i o n o f the ti m e p i o n s were produced i n c o n s e c u t i v e p r o t o n beam b u r s t s , and t h u s , the p r o b a b i l i t y o f h a v i n g two p i o n s i n t h e same beam b u r s t , f o r w h i c h the i n c i d e n t f l u x must be c o r r e c t e d . The p u r p o s e o f t h e beam sample c i r c u i t was t o measure t h e ti m e o f f l i g h t o f p a r t i c l e s down the M13 c h a n n e l , f r o m w h i c h t h e p i o n f r a c t i o n i n t h e i n c i d e n t beam c o u l d be d e t e r m i n e d . A beam sample e v e n t c o n s i s t e d o f a c o i n c i d e n c e between B1«B2 and t h e o u t p u t o f a g a t e g e n e r a t o r , whose w i d t h was s e t t o =1. s, t h e r a t e a t w h i c h the i n c i d e n t beam was sampled. Such a c o i n c i d e n c e would p r o v i d e a s t a r t f o r a TDC, t h e s t o p coming from a c a p a c i t i v e probe i n t h e main p r o t o n beam l i n e . Note t h a t as soon as a LAM o f e i t h e r t y p e was g e n e r a t e d , a c i r c u i t c o n s i s t i n g o f a LOGICAL OR u n i t and g a t e g e n e r a t o r was used t o i n h i b i t a d d i t i o n a l e v e n t s , u n t i l t he c u r r e n t one had been p r o c e s s e d . The same s i g n a l was used t o i n h i b i t a l l s c a l e r s , t h u s e n s u r i n g t h a t no a d d i t i o n a l d e ad-time c o r r e c t i o n s need t o be a p p l i e d t o t h e v a r i o u s beam m o n i t o r s . - 33 -2.2.3 Momentum Determination In the consideration of the passage of charged p a r t i c l e s through a system of magnets, i t i s common pract i c e (see e.g. (Ban 66),(Ste 65)) to s t a r t by de f i n i n g a c e n t r a l t r a j e c t o r y through the system. The c h a r a c t e r i s t i c s of the beam are then described by the displacements and divergences of the outermost rays away from the c e n t r a l one, and the changes i n these c h a r a c t e r i s t i c s given by 'transfer matrices'. For example, consider the two dimensional case of a beam moving i n the z - d i r e c t i o n from point 0 to point 1. The beam i s described at these two points by the column vectors (x 0,dx Q/dz) and ( x ^ d x j / d z ) . They are r e l a t e d by where L i s the distance between z Q and Z j . S i m i l a r l y , the trans f e r matrix for a beam passing through a focussing quadrupole magnet i s where kq = ( B Q / a ) ( l / B p Q ) , L i s the e f f e c t i v e length of the quadrupole magnet, B Q i s the f i e l d at radius a, and (Bp Q) i s the magnetic r i g i d i t y of the c e n t r a l t r a j e c t o r y . In a more general, three dimensional case, one must consider both h o r i z o n t a l (x,dx/dz=9) and v e r t i c a l (y,dy/dz=<J>) displacements and divergenges. Also, the momentum spread of the beam, <5=Ap/pc where p c - 34 -i s the momentum of the c e n t r a l ray. Thus, for the QQD spectrometer system for example, the beam c h a r a c t e r i s t i c s at the p o s i t i o n of WC5 can be re l a t e d to those at the sca t t e r i n g target: / x 5 \ / R l l R12 R13 R14 R16 \ / x Q \ / 6 5 \ / R21 R22 R23 R24 R26 \ / e Q \ y 5 -I R31 R32 R33 R34 R36 y. (2.1) \ *5 / \ R*l R^2 R43 R44 R46 J\ t)Q J \ 6 5/ \ R61 R62 R63 R64 R66 / \ 6 „ / There are seve r a l computer programs a v a i l a b l e , most notably TRANSPORT (Bro+ 80), which w i l l c a l c u l a t e t r a n s f e r c o e f f i c i e n t s ( i . e . R's i n (2.1)) for a system, given the magnet co n f i g u r a t i o n s , f i e l d strengths, and distances involved. Once these t r a n s f e r c o e f f i c i e n t s are known, and x 0 , 9Q , y Q , and <J>0 determined from p o s i t i o n s i n WC1 and WC3, (2.1) can be inverted to f i n d 6Q, the momentum of the pion scattered from the nuclear target. Unfortunately, (2.1) represents only the f i r s t order beam t r a n s f e r . If i t i s rewritten as = f(r j ( j ) , then the more general s i t u a t i o n i s r i 5 = f ( r j n ) + 8(rj0*k0) + h< rj0rk0 rfc0) + ••• > or, i n more conventional notation, r i = I [ r i l r j l r j + I [r±|r-jr k]rjr k + £ [r± ' r ^ r ^ r jrkr£, + ... j jk jk£ where [ r j _ | r j ] , [ r i | r j r k ] , [ri|rjr kr£], ... are f i r s t , second, t h i r d , ... order t r a n s f e r c o e f f i c i e n t s . The program TRANSPORT can only calculate f i r s t and second order c o e f f i c i e n t s , while i n general, higher order c o e f f i c i e n t s may be equally, or even more important. In any case, one cannot r e l y on c a l c u l a t e d values of the c o e f f i c i e n t s , since these are only - 35 -as good as the input data, and cannot compensate for minor misalignments and i r r e g u l a r i t i e s i n magnetic f i e l d s which are i n v a r i a b l y present i n any system. Thus the transfer c o e f f i c i e n t s must be determined experimentally. There have been two d i f f e r e n t approaches to t h i s problem adopted by the group of people involved with the QQD experiments. A l l i n d i c a t i o n s are that the two y i e l d s i m i l a r r e s u l t s . One approach ((Sob 84a) and (Sob 84b)), which i s perhaps the more straightforward, involves measuring the v a r i a t i o n i n one coordinate while varying another, d i r e c t l y . But t h i s involves taking a large amount of data with the spectrometer. For example, one could determine the c o e f f i c i e n t s [x|x] , [ x | x 2 ] , [ x | x 3 ] , etc. by p l o t t i n g x 5 vs x Q , with 9 Q, y Q , <J>Q , and 6 Q a l l held constant at zero, and f i t t i n g the r e s u l t i n g curve with a high order polynomial. The d i f f i c u l t y with this approach l i e s i n getti n g s u f f i c i e n t s t a t i s t i c a l accuracy while ensuring that a l l coordinates other than x Q are i n fac t zero. The a l t e r n a t e approach ((Gyl 84) and (Bar 84)), and that adopted by the author i s the following: Using the pion s c a t t e r i n g data from a CH 2 target, one i n i t i a l l y constructs an energy or momentum spectrum using the TRANSPORT pre d i c t i o n s for the transfer c o e f f i c i e n t s . One then runs a computer program, which uses a multiple regression technique to vary the values of the c o e f f i c i e n t s , to any desired order, to simultaneously minimize the widths of the three peaks i n the spectrum, i . e . the 1 2 C ground state, f i r s t excited state, and proton peaks. A pr e s e l e c t i o n of data may be performed i n order to eliminate background, and equalize the number of counts i n each peak, to ensure proper s t a t i s t i c a l weighting. Also, a p a r t i a l F te s t i s applied to each c o e f f i c i e n t , so that those which - 36 -are s t a t i s t i c a l l y i n s i g n i f i c a n t are set to zero. A t y p i c a l set of c o e f f i c i e n t s found i n this way are presented i n Tables 2.2 and 2.3. Note that the c o e f f i c i e n t s r e l a t e the coordinates of WC4 and WC5 d i r e c t l y to those of WCl and WC3, rather than those of the i n i t i a l target p o s i t i o n . It was found that d i f f e r e n t sets of c o e f f i c i e n t s could produce i d e n t i c a l r e s u l t s , i n d i c a t i n g that they are c o r r e l a t e d . There i s a c e r t a i n minimum peak width which cannot be improved upon. The choice of c o e f f i c i e n t s does a f f e c t the shapes of the r e s u l t i n g peaks, however. 2.2.4 Resolution The narrowest peak widths i n energy spectra accumulated during the present experiments are =1.1 MeV FWHM. The c o n t r i b u t i o n of the energy spread of the i n c i d e n t pion beam from the M13 channel i s =750 keV. Assuming the channel and spectrometer contributions combine i n quadrature, t h i s implies that the r e s o l u t i o n of the QQD spectrometer was =800 keV. From the use of computer programs which simulate the passage of pions through the system, including TRIUMF's REVMOC (KR 83), i t can be estimated that the major contribution to t h i s r e s o l u t i o n comes from the e f f e c t of multiple Coulomb scattering of the pions i n the m a t e r i a l present i n the spectrometer. At the time the experiments were performed, t h i s material consisted of 12.7 um t h i c h mylar windows on WCl and WC3, 50.8 um thick kapton windows on WC4 and WC5, and the magic gas mixture i n the chambers. The a i r i n s i d e the spectrometer was pumped out, and replaced with He gas before undertaking data c o l l e c t i o n . Future experiments w i l l be conducted with the spectrometer t o t a l l y - 37 -i j k I m c o e f f i c i e n t 0 0 0 0 0 4.906907 0 0 0 1 0 0.010630 0 0 1 0 0 1.203300 1 0 0 0 0 -0.348225 0 0 0 2 0 0.004083 0 0 2 0 0 -0.005827 0 1 0 1 0 -0.007299 0 1 1 0 0 -0.001407 0 2 0 0 0 0.002154 1 0 0 1 0 -0.002407 1 1 0 0 0 0.009717 0 0 0 0 1 -6.124691 0 0 0 1 1 -0.002581 0 0 1 0 1 0.043025 1 0 0 0 1 -0.011264 0 0 0 2 1 0.000119 0 0 1 1 1 -0.000003 0 0 2 0 1 -0.000019 0 1 0 1 1 -0.000189 0 2 0 0 1 0.000051 1 0 1 0 1 -0.000096 1 1 0 0 1 -0.000263 2 0 0 0 1 0.001364 0 0 0 0 2 0.025394 0 0 0 1 2 -0.000183 0 0 1 0 2 -0.000699 0 1 0 0 2 0.000169 1 0 0 0 2 0.000360 Table 2.2 Typical Set of QQD Transfer C o e f f i c i e n t s f o r WC4 P o s i t i o n . Notation i s : = £ coeff i c i e n t ' X j i ' y j J *x 3^» J^^'Sg 1 1 1 - 38 -i j k I m c o e f f i c i e n t 0 0 0 0 0 1.634761 0 0 0 1 0 -0.006957 0 0 1 0 0 0.564156 1 0 0 0 0 -0.290266 0 0 0 2 0 0.005915 0 0 2 0 0 -0.008948 0 1 0 1 0 -0.006424 0 1 1 0 0 -0.001811 1 0 1 0 0 0.001299 1 1 0 0 0 0.011112 0 0 0 0 1 -9.188643 0 0 0 1 1 0.000021 0 0 1 0 1 0.059224 0 0 0 2 1 0.000193 0 0 2 0 1 -0.000117 0 1 0 1 1 -0.000259 0 1 1 0 1 0.000087 0 2 0 0 1 0.000199 1 1 0 0 1 -0.000489 2 0 0 0 1 0.002178 0 0 0 0 2 0.020447 0 0 0 1 2 0.000119 0 0 1 0 2 -0.001601 0 1 0 0 2 0.000259 1 0 0 0 2 0.002622 Table 2.3 Typ i c a l Set of QQD Transfer C o e f f i c i e n t s for WC5 P o s i t i o n . Notation i s : Xg = \ c o e f f i c i e n t ' X j i » y j J ' X g k . y3*"So m - 39 -evacuated. This w i l l be accomplished by replacing WCl and WC3 with a d r i f t chamber, whose housing w i l l be able to withstand the pressure d i f f e r e n t i a l . Note that some e f f o r t was expended i n conducting tests of the present MWPCs with reduced gas pressures. Also, experiments were performed (Gyl 84), i n which the standard magic gas mixture was replaced by a mixture c o n s i s t i n g of 40% methane, 25% isobutane, 34.8% argon, and 0.2% freon. A l l i n order to reduce the density and average atomic number of the gas, and therefore the multiple s c a t t e r i n g . Subsequently, i t was found that the MWPCs could be operated e f f i c i e n t l y with a magic gas mixture i n which a l l the argon was replaced by helium. The best combined spectrometer-channel r e s o l u t i o n seen to date (May 1984), under act u a l experimental conditions, i s 850 keV. - 4 0 -2.3 Target8 The 2 6Mg target was a piece of r o l l e d metal, with dimensions 43 X 25.6 mm, and a uniform thickness of 0.300 g/cm2. It i s on loan from the Max Planck I n s t i t u t fur Kernphysik, Heidelberg, West Germany. It i s 99.5% i s o t o p i c a l l y pure. The same target has been used on previous occasions at TRIUMF (Gyl 84). The self-supporting nature of the target made i t unnecessary to mount i t i n a holder from which pions could s c a t t e r . Thus no background runs or subtractions were needed. The 1 8 0 target was purchased i n the form of H 20 from Los Alamos (Batch no. P4-H20-17). Its i s o t o p i c composition was quoted to be 95.3% 1 8 0 , 2.8% 1 7 0 , and 1.9% 1 6 0 . This has subsequently been v e r i f i e d by independent analyses (Bar 84). The * 80 target was prepared as follows: a quantity of target material was heated, and mixed with a small amount of Agar (1.4% by weight). While s t i l l hot, the s o l u t i o n was i n j e c t e d into a prepared target holder. Upon cooling, the s o l u t i o n forms a g e l , which i s more l i k e l y to maintain a constant uniform thickness than a l i q u i d . The thickness of the g e l i n the target holder was 0.348 g/cm2. The target holder consisted of a frame of 6.35 mm thick aluminum. 25.4 |im thick mylar windows were glued, under tension, over the area to be occupied by the target material, which was i n j e c t e d through a small hole at the top of the frame, which was l a t e r sealed with epoxy. After the target m a terial was i n place, a d d i t i o n a l 12.7 um thick aluminum windows were glued to the holder, i n order to prevent exchange of 1 6 0 i n the atmosphere with 1 8 0 i n the target through the mylar windows. Background runs were taken with an empty target holder, at a l l angles, i n order to separate the c o n t r i b u t i o n of the window material from that of the 1 8 0 - 41 -target i n the accumulated spectra. The 1 2 C target was i n the form of two sheets of polythene (CH 2), each of which was 0.160 g/cm2 t h i c k . Runs were taken with this target for several reasons: a) the spectrometer s o l i d angle was normalized with respect to the Tf +p cross sections; b) the CH 2 targets were l a r g e r than the pion beam spot, while the other targets were not. Thus, by in t e r s p e r s i n g CH 2 runs with other target runs, the siz e of the beam spot could be measured, and accounted f o r . c) Because the f i r s t excited state of 1 2 C i s at 4.4 MeV, the ground state was c l e a r l y separated, and could be used to determine the peak shape, which was then used i n f i t t i n g the data taken with other t a r g e t s , d) The CH2 data was used to determine QQD transfer c o e f f i c i e n t s , as described i n section 2.2.3. A l l targets were mounted on a remotely c o n t r o l l a b l e target ladder (Gyl 84), which moved i n s i d e the spectrometer s c a t t e r i n g chamber. - 42 -CHAPTER I I I EXPERIMENTAL ANALYSIS In s e c t i o n 2.2.3, the method used for determining the energy of a pion passing through the QQD spectrometer was o u t l i n e d . A f t e r a s u f f i c i e n t number of pions have passed through the spectrometer, one may construct a pion energy spectrum. Each peak i n the spectrum corresponds to a pion having scattered, and l e f t the target nucleus i n a p a r t i c u l a r s t a t e . It i s the number of counts i n each peak that one wants to determine. In order to compare the r e s u l t s of the present experiments with others, however, or with t h e o r e t i c a l c a l c u l a t i o n s , they must be properly normalized. The normalized p r o b a b i l i t y for a p a r t i c l e to scatte r i n t o a given element of s o l i d angle i s c a l l e d the d i f f e r e n t i a l cross s e c t i o n , and i s given by: = N s c a t  dft N i n c » N a t o m s » d J 2 » e f f ' where N s c a t i s the number of pions scattered through the spectrometer and detected, having l e f t the target nucleus i n a p a r t i c u l a r energy st a t e ; Nine * s t n e t o t a l number of pions incident on the target; N a t o m s i s the number of target atoms per cm2, as seen by the incoming pions; dfi i s the element of s o l i d angle subtended by the spectrometer, into which the pions can sc a t t e r ; and e f f i s the e f f i c i e n c y of the spectrometer system f o r detecting pions. - 43 -In section 3.1, each of the above factors i s considered i n turn; with reference to the experimentally determined values, and t h e i r associated u n c e r t a i n t i e s . In section 3.2, the tabulated r e s u l t s f or the d i f f e r e n t i a l cross sections measured i n the experiments under consideration are presented. - 44 -3.1 C a l c u l a t i o n of Cross Sections 3.1.1 Beam Normalization There were four separate incident pion monitors. The main one was a) two p l a s t i c s c i n t i l l a t o r s (Bl and B2) positioned d i r e c t l y i n l i n e with the incoming beam on either side of the nuclear s c a t t e r i n g target, which were used to measure the absolute number of p a r t i c l e s passing through the t a r g e t . Then, b) two small p l a s t i c s c i n t i l l a t o r s (pl and u2) separated by 300 mm, whose centers were aligned at an angle of approximately 9° with respect to the incoming beam. Such a 'muon telescope' may be used to measure the absolute number of pions emerging from the beamline, by counting the number of muons from pion decay which pass through the two counters (Wad 76). But i n the present case, i t was used as a r e l a t i v e monitor only, c) An ion chamber and d) a Cerenkov counter around the T l production target were also used as r e l a t i v e monitors. For most runs, the absolute monitor Bl«B2 was used e x c l u s i v e l y . The r e l a t i v e monitors were ca l i b r a t e d against Bl«B2, and for these runs, the r a t i o s of the various monitors were found to remain constant to within ±2.%. However, at c e r t a i n spectrometer angles, <50°, the edges of QTl and the WC1 housing intercepted part of the i n c i d e n t pion beam, a f t e r i t had passed through the s c a t t e r i n g target, but before i t had passed through the B2 counter. In these cases, B1»B2 was no longer r e l i a b l e , and one of the other monitors had to be used instead. There i s also the p o s s i b i l i t y that Bl or B2 would not operate e f f i c i e n t l y at high incident pion f l u x e s . But t h i s problem was not i n fact encountered. The t o t a l number of B1»B2 coincidences must be corrected for several e f f e c t s before i t i s a measure of the absolute number of pions passing - 45 -through the s c a t t e r i n g target: i ) One must account for the fact that e's and u's emerge from the beamline, as w e l l as I T ' S . As mentioned in sections 2.1 and 2.2.2, t h i s i s accomplished by measuring the time of f l i g h t of the p a r t i c l e s from the T l production target. F i g . 3.1 i l l u s t r a t e s t y p i c a l time of f l i g h t spectra, and demonstrates the clean separation of T T ' S , U ' S , and e's. T T + and T T ~ f r a c t i o n s are =94% and =91% r e s p e c t i v e l y . Note that t h i s i s not a true measure of the absolute f r a c t i o n s . Some e's have already been eliminated by r a i s i n g the d i s c r i m i n a t o r threshold on B l , and r e l y i n g on the f a c t that e's lose r e l a t i v e l y l i t t l e energy i n that counter. i i ) One must account for the f a c t that several pions may be created i n a s i n g l e main proton beam burst. Two or more p a r t i c l e s passing through Bl and B2 within a few ps of each other w i l l not be d i s t i n g u i s h e d , and w i l l be counted as one. The method used to correct f o r t h i s i s outlined below. It r e l i e s on the f a c t that with the c o n f i g u r a t i o n of e l e c t r o n i c s used i n data a c q u i s i t i o n , i f pions were produced i n two successive proton beam bursts (43.5 ns apart), the f i r s t one passing through the target without i n t e r a c t i n g , and the second s c a t t e r i n g into the spectrometer, the recorded Bl time would be that of the f i r s t pion. The r a t i o of the areas of the two r e s u l t i n g peaks i n the Bl time spectrum i s a d i r e c t measure of the p r o b a b i l i t y of observing pions i n successive beam bursts. More s p e c i f i c a l l y , for a given A = (pion f l u x ) / ( c y c l o t r o n RF frequency) , the p r o b a b i l i t y of observing n pions per beamburst i s : P = (xne-*)/n! , so that the p r o b a b i l i t y of observing 1 or more pions In two successive - 46 -lOOOh 10 20 TIME (ns) 30 F i g . 3.1 T y p i c a l T T + and TT " Time of F l i g h t Spectra - 47 -beam bursts i s : S = ( I X n e" x / n! ) 2 = ( 1 - e~* ) 2 . n-1 This i s the number measured experimentally, and can be solved f o r X = l n ( l - / S ) . The c o r r e c t i o n factor that needs to be applied to the measured B1«B2 i s : I n X n e~ x I n! F = ° = 1 I X n e -* / n! n-1 A p l o t of S and F versus X i s given i n F i g . 3.2. T y p i c a l experimental values for TT+ are: S = 0.039 -»• F = 1.11 , and for ir" : S = 0.011 + F = 1.06 . This i s an important c o r r e c t i o n to make. Because the T T + fluxes obtained for present experiments were approximately 6 times higher than the ir" flux e s , t h i s c o r r e c t i o n a f f e c t s the r e l a t i v e T T + / T T - normalization. i i i ) It i s possible for a pion emerging from the M13 B2 dipole magnet to decay before reaching the nuclear s c a t t e r i n g target. Pion decay kinematics are presented in Appendix D. Roughly one h a l f of the muons from 50 MeV pion decay are emitted at angles between 15° and 18° with respect to the i n c i d e n t pion d i r e c t i o n ; the other half at smaller angles. Thus, some f r a c t i o n of the muons w i l l pass through the Bl and B2 counters, and be mistaken f o r pions. It has been estimated (Bar 84), from Monte Carlo simulations, that the t o t a l e f f e c t on the f l u x , as measured by B1»B2 may be as high as 7%. Because of the u n c e r t a i n t i e s involved i n making such an estimation, t h i s c orrection has not been applied to the present data. In any case, the e f f e c t i s constant f o r a l l experimental runs, and w i l l be the same for T T + and T T ~ . It i s e f f e c t i v e l y included In the spectrometer s o l i d angle. - 48 -F i g . 3.2 Pion Flux Correction Factors - 49 -i v ) In the cases of both 1 8 0 and 2 6Mg, the incident pion beam spots at the target p o s i t i o n s were larger than the targets themselves. For the purpose of e x t r a c t i n g cross sections, the relevent pion f l u x i s that through the targets, not that emerging from the M13 beamline. The d i f -ference between these two numbers was measured by int e r s p e r s i n g runs with CH 2 targets between the others. The CH 2 targets were larger than the pion beam spots, and s c a t t e r i n g data taken with them could be used to recon-s t r u c t incident beam p r o f i l e s , which could then be integrated over the areas occupied by the smaller targets. This technique r e l i e s on the beam p r o f i l e s not changing from one run to the next. In f a c t , r e s u l t s show that the beam spots remained constant throughout the e n t i r e experimental beam time. 3.1.2 Target Thickness The number required i n the expression f o r the d i f f e r e n t i a l cross se c t i o n i s the number of sca t t e r i n g centers per cm2 i n the target, as seen by the incoming pions. A l l the present experiments were performed i n a transmission geometry. For a given spectrometer angle 26, the sc a t t e r i n g target normals were i n c l i n e d to the d i r e c t i o n of the beam by an angle 6, i n order to equalize the amount of target material through which a scat-tered pion passed. The number of sca t t e r i n g centers i s thus given by N/cos0 , where N i s the number of scatterers per cm2 i n a target whose normal i s p a r a l l e l to the incident beam d i r e c t i o n . The f r a c t i o n a l error i n the target thickness due to the uncertainty i n the target angle i s given by (tan9)A6, with A6 = ±1.°. The number of scatt e r e r s per cm2 for each of the targets used In the - 50 -present experiments i s given i n Table 3.1. For the H 20 target, the con-t r i b u t i o n of the window material was taken i n t o account by subtracting the background measured i n the empty target frame runs. For the purpose of accounting for the i s o t o p i c impurity of the target; the 1 8 0 , 1 7 0 , and 1 6 0 e l a s t i c cross sections were a l l assumed to be of s i m i l a r magnitude, so that no co r r e c t i o n was applied; for the i n e l a s t i c cross sections, the f i n a l 1 8 0 r e s u l t scaled by l./(0.953 ± 0.02) . 3.1.3 E f f i c i e n c y The e f f i c i e n c y factor i n the expression on page 42 encompasses two d i f f e r e n t e f f e c t s : a) the inherent e f f i c i e n c y of the QQD MWPCs; and b) background events rejected i n the o f f - l i n e a n a l y s i s , before energy spectrum peaks are integrated. a) Only those events are analyzed, i n which v a l i d signals are obtained from a l l four MWPCs. Events i n which signals from only three of the MWPCs are obtained, are used as a measure of the i n e f f i c i e n c y of the fourth chamber. The product of the i n d i v i d u a l e f f i c i e n c i e s of the four wire chambers, as measured i n t h i s way, was t y p i c a l l y 96% for a l l runs, for both tr + and I T - . The i n t r i n s i c e f f i c i e n c y of a l l p l a s t i c s c i n t i l l a t o r s was assumed to be 100%. b) Some pions which scatter into the spectrometer w i l l decay into muons, before passing completely through the system. As explained i n Appendix D, those pions decaying before, or i n , the spectrometer dipole, w i l l probably not pass through WC4 and WC5. Those decaying a f t e r BT probably w i l l , but rather than f a l l i n t o a p a r t i c u l a r peak i n the energy spectrum, w i l l form part of a general background. Also, pions which - 51 -Target Nucleus N / cm2 error CH2 1.38 x 1 0 2 2 ± 1.0 % l H 2.75 x 1 0 2 2 ± 1.0 % 2 6 M g 2 6 M g 6.95 x IO 2* ± 1.0 % H 20 0 1.05 x 1 0 2 2 ± 2.0 % 2.10 x 1 0 2 2 ± 2.0 % Table 3.1 Thicknesses, In Scattering Centers per cm2, of Experimental Scat t e r i n g Targets - 5 2 -a c t u a l l y pass through i n d i v i d u a l wires of the MWPCs w i l l be deflected due to m u l t i p l e Coulomb s c a t t e r i n g . These may also end up contri b u t i n g to the background. It i s de s i r a b l e to eliminate such background events, which t y p i c a l l y represent some 12% of the data, i n order to obtain better f i t s to the peaks present i n the energy spectra. At the same time, these events must be accounted f o r properly because they do a r i s e from pions which scattered into the spectrometer. As was the case for f i n d i n g QQD tr a n s f e r c o e f f i -cients (see section 2.2.3), several d i f f e r e n t approaches have been adopted by the group of people involved i n the QQD experiments to eliminate back-ground events. The present data has been analyzed using a combination of a l l these methods, which are outlined below, and which are found to give s i m i l a r r e s u l t s . I) The sum of the E l , E2, and E3 ADC si g n a l s i s prop o r t i o n a l to the energy loss (AE) of the p a r t i c l e s passing through these counters. It i s possi b l e to construct an E-AE s c a t t e r p l o t of a l l events i n a given run, by using the AE for each event, as defined above, along with the energy as determined from the positions of the p a r t i c l e i n the four MWPCs. Pions w i l l form a d i s t i n c t band i n such a plot , while the background events w i l l be d i s t r i b u t e d more or less randomly. i i ) WC4 and WC5 may be used independently to determine the p a r t i c l e momentum 6 Q. If the two values of 6 Q so obtained are not i d e n t i c a l (or within 0.1% say) for a p a r t i c u l a r event, i t i s an i n d i c a t i o n that the event i s a background one. i i i ) Having determined 6 Q from WC4 and WC5, i t i s possible to take the average, and use t h i s number, along with the WCl and WC3 positions and - 53 -known transfer c o e f f i c i e n t s , to c a l c u l a t e the p a r t i c l e t r a j e c t o r y through WC4 and WC5. The polar angle between t h i s c a l c ulated t r a j e c t o r y and the one a c t u a l l y measured from the WC4 and WC5 positions can also be c a l c u l a -ted. Events for which this polar angle i s large (>3.5° say) are back-ground . i v ) Rather than solve the two equations H - f ( * i . y i»* 3»y3 » 6 o ) » x 5 = g ( x 1 , y 1 , x 3 , y 3 , 6 0 ) f o r 6 Q exactly, i t i s possible to define X 2 = K " f ) 2 + ( x 5 - g)2 , and determine the value of 6 Q which r e s u l t s i n the minimum x2^ Events for which the minimum value of x2 i s large can be i d e n t i f i e d as background. 3.1.4 Peak F i t t i n g The energy r e s o l u t i o n of the QQD spectrometer system during the present experiments (=1.15 MeV) was s u f f i c i e n t to allow for the unambi-guous i d e n t i f i c a t i o n of the peaks corresponding to s c a t t e r i n g to the ground and f i r s t excited states of 1 8 0 and 2 6Mg. It was i n s u f f i c i e n t , however, to enable the peak areas to be extracted from a simple summing of the number of counts i n a c e r t a i n energy region. The computer routine ZXMIN (IMSL 82) was used to f i t the peaks i n the energy spectra, and thus determine t h e i r i n t e g r a l s . It was found that a simple Gaussian d i s t r i b u t i o n was adequate to reproduce the shape of the i s o l a t e d 1 2 C ground state peak. Thus, t h i s form was also used for the other targets, with the peak widths held fixed at the values needed to f i t 1 2 C , and the r e l a t i v e energy separations - 5 4 -between ground and excited states fixed at t h e i r w ell known values. The s t a t i s t i c a l error associated with a peak area i s proportional to the square root of the t o t a l number of counts. An a d d i t i o n a l error corresponding to the q u a l i t y of the f i t was also included. Examples of t y p i c a l energy spectra f o r 1 2 C , 1 8 0 , and 2 6Mg are presented i n Figures 3.3, 3.4, and 3.5 r e s p e c t i v e l y . 3.1.5 Spectrometer Acceptence The QQD spectrometer laboratory s o l i d angle was obtained by using i r +p s c a t t e r i n g data, from CH2 targets, from several angles, i n the r e l a t i o n N s c a t  ~ Nine*N a t o m s.eff»(da/dfi) s where a l l the fa c t o r s have the meanings given on page 42, and discussed i n the previous sections of this chapter. (da/dJ2) s i s the i r +p d i f f e r e n t i a l cross section i n the laboratory frame, as c a l c u l a t e d i n the center of mass frame, at the appropriate energy, using the phase s h i f t s of the ElOD s o l u t i o n of Arndt and Roper (AR 82), and k i n e m a t i c a l l y transformed. No error has been associated with dJ2 due to the i n t r i n s i c uncertainty of the calculated i r + p cross s e c t i o n s . Note however, that over the angular region of Interest, the u +p cross sections change by roughly 3% per MeV change i n Incident pion energy. Thus, an uncertainty of ±3% i s associated with dQ because of the uncertainty i n the Incoming beam energy (see section 2.1). This has been added i n quadrature to the unce r t a i n t i e s due to the other f a c t o r s . Note a l s o that the spectrometer acceptence depends on the lo c a t i o n F i g . 3.3 Ty p i c a l 1 2 C Energy Spectrum F i g . 3.4 Ty p i c a l 1 8 0 Energy Spectrum F i g . 3.5 T y p i c a l 2 6Mg Energy Spectrum - 58 -on the s c a t t e r i n g target of the o r i g i n a l s c a t t e r i n g event. This i s i l -l u s t r a t e d i n F i g . 3.6, which i s based on a REVMOC (KR 83) c a l c u l a t i o n . Because of the v a r i a t i o n , the s o l i d angle as determined above Is i n f a c t averaged over the s i z e of the nuclear s c a t t e r i n g targets, weighed by the in c i d e n t beam p r o f i l e . The 1 8 0 and 2 6Mg targets were of s i m i l a r s i z e , and thus the same s o l i d angle was determined f o r both. This was done by accepting only those Tr+p events which originated from selected portions of the CH 2 target, corresponding to the size of the others. The r e s u l t i s dSl = 12.7 ± 0.8 msr. It w i l l be noted that t h i s number i s some 30% smaller than the REVMOC p r e d i c t i o n , as given i n F i g . 3.6. This d i f f e r e n c e a r i s e s from the f a c t that pions decaying i n the fr o n t end of the spectrometer do not make i t through to the back, and thus appear not to have been accepted i n i -t i a l l y . This e f f e c t and others (see e.g. page 47) have been incorporated into d£2, which thus plays the ro l e of an e f f e c t i v e spectrometer s o l i d angle, rather than a purely geometric one. The number of pions decaying i n the front end of the spectrometer w i l l vary with the scattered pion energy. This e f f e c t has been included i n the fa c t o r (da/dfi) s, since T f +p kinematics are such that the scattered pion energy changes r a p i d l y with the s c a t t e r i n g angle. This i s not the case for pion s c a t t e r i n g from heavier n u c l e i . F i n a l l y , note that comparisons made with other experimental data, and t h e o r e t i c a l c a l c u l a t i o n s are made i n the pion-nucleus center of mass frame. Thus, the laboratory value of dfi, as determined above, i s mu l t i -p l i e d by the Jacobian appropriate to the transformation from laboratory to center of mass frames for each i n d i v i d u a l nucleus. - 59 -SOLID ANGLE SUBTENDED 45 —1111111111111111111 j i II 111111 11111111111111111111111111111 --45 -30 "15 0 15 30 45 X DIRECTION (BID) F i g . 3.6 REVMOC C a l c u l a t i o n of V a r i a t i o n of QQD S o l i d Angle over Scattering Target. Contours l a b e l l e d i n msr. - 60 -3.2 Results The values of the measured d i f f e r e n t i a l cross sections are presented i n Tables 3.2 to 3.7. As mentioned i n section 3.1, the quoted errors Include the various uncertainties associated with beam normalization (=3%), target thickness (=2%), and s o l i d angle (=6%), as well as s t a t i s t i c s . For the e l a s t i c s c a t t e r i n g , the purely s t a t i s t i c a l error was t y p i c a l l y - 3.5 %, while for the i n e l a s t i c s c a t t e r i n g i t was = 10 %, except at the backwardmost angles. - 61 -G c m (deg) da/dfi (mb/sr) 49.0 4.18 + 0.43 59.1 2.74 + 0.28 69.2 2.35 + 0.30 79.2 3.77 + 0.32 92.7 6.07 + 0.55 102.7 6.58 + 0.57 112.6 6.86 + 0.61 122.6 7.28 + 0.63 132.3 6.11 + 0.56 49.0 9.23 + 1.03 64.1 2.32 + 0.35 79.2 3.85 + 0.44 89.2 5.82 + 0.56 i r - 92.7 6.41 + 0.64 102.6 8.19 + 0.78 112.6 8.03 + 0.76 122.6 8.56 + 0.79 132.3 7.55 + 0.71 Table 3.2 Measured Cross Sections f or 1 2 C ( T T , T T ) 1 2 C - 62 -©cm ( d e 8 ) da/dfl (mb/sr) 59.1 0.153 + 0.046 69.2 0.163 + 0.063 79.2 0.231 + 0.037 92.7 0.459 + 0.086 102.7 0.805 + 0.111 112.6 1.03 + 0.13 122.6 1.73 + 0.17 132.3 1.59 + 0.18 79.2 0.237 + 0.081 89.2 0.385 + 0.091 92.7 0.752 + 0.151 102.6 1.24 + 0.20 112.6 1.70 + 0.23 122.6 2.19 + 0.26 132.3 2.79 + 0.31 Table 3.3 Measured Cross Sections f o r 1 2 C ( T T , T T ' ) 1 2 C * ( 2 + , 4 . 4 4 ) - 63 -e c m (deg) do/dti (mb/sr) 43.7 9.37 + 0.77 47.2 6.77 + 0.54 48.7 6.60 + 0.53 58.8 3.81 + 0.32 68.8 3.66 ± 0.30 Tf+ 78.9 5.08 + 0.41 88.9 6.63 + 0.59 92.4 6.98 + 0.58 102.4 7.97 + 0.65 112.4 7.65 + 0.64 122.3 7.16 + 0.59 132.0 5.72 + 0.49 48.7 14.07 + 1.17 63.8 4.39 + 0.39 78.9 7.80 + 0.65 88.9 9.67 + 0.84 92.4 10.64 + 0.89 102.4 10.79 + 0.89 112.4 9.14 + 0.77 122.3 6.61 + 0.58 132.0 3.89 + 0.36 Table 3.4 Measured Cross Sections for 1 80(TT,T0 1 80 - 64 -0cm ( d ee) do/dfi (mb/sr) 48.7 0.150 + 0.026 58.8 0.117 + 0.019 68.8 0.182 + 0.023 78.9 0.194 + 0.022 88.9 0.282 + 0.048 92.4 0.439 + 0.058 102.4 0.667 + 0.073 112.4 0.685 + 0.081 122.3 1.06 + 0.10 132.0 1.18 + 0.11 63.8 0.258 + 0.051 78.9 0.653 + 0.084 88.9 1.26 + 0.16 i r - 92.4 1.32 + 0.16 102.4 1.56 + 0.16 112.4 2.16 ± 0.21 122.3 3.08 + 0.30 132.0 3.27 + 0.31 Table 3.5 Measured Cross Sections f o r 1 80(Tr ,T r') 1 80*(2+,1.98) - 65 -©cm ( d e 8 > dff/dfi (m b / s r ) 58.6 8.19 + 0.60 68.7 7.54 + 0.55 78.7 9.70 + 0.72 88.7 9.38 + 0.73 TT+ 92.2 9.14 + 0.70 100.2 7.89 + 0.61 108.2 6.07 + 0.49 112.2 5.52 + 0.46 116.2 5.18 + 0.42 72.2 8.82 + 0.66 82.2 11.78 + 0.87 TT — 92.2 9.88 + 0.75 100.2 7.97 + 0.62 108.2 5.71 + 0.46 116.2 2.88 + 0.25 T a b l e 3.6 Measured C r o s s S e c t i o n s f o r 2 6 M g ( T r , T r ) 2 6 M g - 66 -©cm <dee) da/dti (mb/sr) 58.6 0.34 + 0.04 68.7 0.50 + 0.05 78.7 0.82 + 0.07 88.7 1.29 + 0.13 1T+ 92.2 1.43 + 0.14 100.2 2.52 + 0.22 108.2 3.48 + 0.30 112.2 2.77 + 0.25 116.2 3.08 ± 0.26 72.2 0.49 + 0.06 82.2 1.00 + 0.10 IT — 92.2 1.98 + 0.18 100.2 2.36 + 0.21 108.2 1.97 + 0.18 116.2 2.57 + 0.23 Table 3.7 Measured Cross Sections for 2 6 M g ( TT.TT' ) 2 6Mg*(2 + > 1.81) - 67 -CHAPTER IV THEORETICAL DETAILS In t h i s chapter, some relevent d e t a i l s of the t h e o r e t i c a l c a l c u l a t i o n s which have been performed for the analysis of the experimental data are discussed. In section 4.1.1 the MSU o p t i c a l p o t e n t i a l , which has been used for a l l c a l c u l a t i o n s , i s presented. There i s no lengthy consideration of multiple s c a t t e r i n g formalism, as t h i s can be found i n any number of standard texts, e.g. (GW 64), (Tay 72). Rather, reference i s made to the i n d i v i d u a l terms included i n the p o t e n t i a l . In section 4.1.2 some discus s i o n of the l i m i t a t i o n s of the MSU form, and other formulations, i s undertaken. In s e c t i o n 4.2.1 the c a l c u l a t i o n of i n e l a s t i c d i f f e r e n t i a l cross sections i s described. The nuclear t r a n s i t i o n d e n s i t i e s , which are a v i t a l input to these c a l c u l a t i o n s , and t h e i r properties, are discussed i n sections 4.2.2 and 4.2.3. The r e s u l t s of the c a l c u l a t i o n s demonstrate the increased s e n s i t i v i t y of TT~ s c a t t e r i n g to neutron t r a n s i t i o n d e n s i t i e s , and TT + s c a t t e r i n g to proton t r a n s i t i o n d e n s i t i e s . These r e s u l t s are considered i n se c t i o n 4.2.4. - 68 -4.1 E l a s t i c S c a t t e r i n g 4.1.1 The MSU O p t i c a l P o t e n t i a l The following i s the general form of the o p t i c a l p o t e n t i a l used i n the present a n a l y s i s : 2EVn = Aip + A 2 v6p + A4P2 + V 2(A3P+A3 v6p+A 7p 2) / A 6p 2 + A 2p + A 2 v6p \ (4.1) + V \ 1 + (X/3)(A 6p 2+A 2p+A 2v'Sp) + A 5 P y 7 with Ai = -4Trbopi Aiv = 4ifebipi A 2 = 4 T T C O / P I a 2 V = - 4 T T £ C X / P I A3 = -4irco (p i-l ) /2p i A3 V = 4Treci(pi-l)/2pi A4 = - 4 T T B O P 2 a 5 = * T T C O / P 2 A 6 = 4 T T C 2 / P 2 Ay = - 4 T T ( C 0 + C 2 ) ( p 2 - l ) / 2 P 2 . p = p n + Pp, and 6p = Pn ~ Pp » where p n and pp are the neutron and proton ground state matter densities r e s p e c t i v e l y , e = ±1 f o r i r * . Pj and p 2 are kinematic factors given by _ (1+E/m) _ (l+E/2m) p l " (1+E/mA) a n d P2 ~ (l+E/2mA) ' A l l c o e f f i c i e n t s are complex. E l a s t i c pion-nucleus scattering cross sections are cal c u l a t e d with the computer program D W P I (EM 76), which has been modified to incorporate t h i s form of p o t e n t i a l . D W P I i s an extension of the e a r l i e r program P I R K (EM 74), and i s also used for c a l c u l a t i n g i n e l a s t i c cross sections. An ou t l i n e of the method of sol u t i o n , along with some mathematical d e t a i l s i s provided i n Appendix A. - 69 -The basic procedure involves a numerical i n t e g r a t i o n of the Klein-Gordon equation, which for a free p a r t i c l e i s written: (E 2-p 2)'P = m2Y . It i s common p r a c t i c e to include the e f f e c t s of the Coulomb and nuclear p o t e n t i a l s with the energy. That i s , one replaces E above by (E- V c - V n ) , so that the Klein-Gordon equation becomes ((E 2+V c 2+V n 2-2EV c-2EV n-V cV n-V nV c)-p 2)'P = m2,P . It i s also common pr a c t i c e to neglect the V n 2 , V cV n, and V nV c terms at t h i s p o i n t . Some c a l c u l a t i o n s i n which the l a t t e r two terms have been re-tained have been performed, however. These w i l l be discussed i n Chapter V. H i s t o r i c a l l y , the f i r s t pion-nucleus o p t i c a l p o t e n t i a l which had some success i n describing e l a s t i c s c a t t e r i n g data was introduced by K i s s l i n g e r i n 1955 (Kis 55). It incorporates the pion-nucleon s c a t t e r i n g amplitude: f = b Q + bjt-x + (c 0+c 1t»x)k«k* i n the standard m u l t i p l e s c a t t e r i n g formalism. To f i r s t order, the r e s u l t i n g o p t i c a l p o t e n t i a l may be written: 2EV n = b 0p + b x6p + V«(c 0p+c 16p)V . (4.2) The c o e f f i c i e n t s b Q and bj are ref e r r e d to as the i s o s c a l a r and isovector s-wave parameters, while CQ and Cj are c a l l e d the i s o s c a l a r and isove c t o r p-wave parameters. The o r i g i n of these names i s evident from the form of the pion-nucleon scattering amplitude. The values of the c o e f f i c i e n t s are obtained from pion-nucleon phase s h i f t s . - 70 -Pion absorption cannot occur on a single nucleon, except under very r e s t r i c t e d kinematic conditions. Its e f f e c t i s thus not e x p l i c i t e l y included i n (4.2). Absorption i s us u a l l y parametrized i n terms of the square of the nuclear density, assuming i t i s mainly due to two nucleon processes. The o p t i c a l p o t e n t i a l i s then written: 2EV n = b Qp + bjfip + B Qp 2 + V«(c 0p + c x6p + Cp 2)V . The new parameters B Q and C may be taken from t h e o r e t i c a l c a l c u l a t i o n s (CR 79), or extrapolated from f i t s to pionic atom widths (SCM 80) assuming some t h e o r e t i c a l model for the energy dependence. Ericson and Er i c s o n (EE 66) pointed out that another e f f e c t not included In (4.2) i s that of nuclear pair c o r r e l a t i o n s . They demonstrated that short range p a i r c o r r e l a t i o n s are important i n the mul t i p l e s c a t t e r i n g ; g i v i n g r i s e to a phenomenon analogous to one occuring i n the s c a t t e r i n g of electromagnetic waves i n dense media which i s c a l l e d the Lorentz-Lorenz e f f e c t . Q uantitatively, the influence on the pion-nucleus o p t i c a l p o t e n t i a l i s twofold. F i r s t , the value of the s-wave parameter b Q i s modified from that obtained from pion-nucleon phase s h i f t s . Second, the p-wave terms are m u l t i p l i e d by the f a c t o r : (l+(X/3)(c 0p+ C l6p+Cp 2))-l , which i s c a l l e d the Lorentz-Lorenz-Ericson-Ericson (LLEE) term. The parameter X determines the strength of the LLEE e f f e c t , and i t s value may be estimated t h e o r e t i c a l l y (see (SCM 80)). The question was raised by (SMC 79) as to whether the p-wave absorption term should be included within the LLEE f a c t o r or not. The general form of the o p t i c a l p o t e n t i a l used i n the present a n a l y s i s , eqn. (4.1), allows f o r both p o s s i b i l i t i e s . One may set C?=0.0, Cn*0.0 to - 71 -take the absorptive p-wave term outside the LLEE f a c t o r , or set Cg=0.0, Cj^O.O to include i t . There i s some reason to believe that the l a t t e r procedure i s the more correct one (Car 84), and thus a l l the c a l c u l a t i o n s discussed i n Chapter V were performed with C Q set equal to zero. One f i n a l e f f e c t included i n the p o t e n t i a l (4.1) i s that a r i s i n g from the transformation of the k»k' factor i n the p-wave term of the pion-nucleon s c a t t e r i n g amplitude from the pion-nucleon to the pion-nucleus center of mass frames. This introduces terms proportional to 7 2p and V 2 p 2 . It was f i r s t pointed out by Thies (Thi 76). I t introduces no new parameters into the p o t e n t i a l , and improves the agreement with experimental data. 4.1.2 Further Discussion Table 4.1 presents the values of the glo b a l 'Set E' o p t i c a l p o t e n t i a l parameters of (CMS 82). These formed the s t a r t i n g point for the c a l c u l a t i o n performed to f i t the present data, to be described i n Chapter V. The values f o r these parameters were obtained as follows: ReB and ReC were taken from t h e o r e t i c a l c a l c u l a t i o n s , Imb and Ime were ca l c u l a t e d from phase s h i f t values, ImB and ImC were taken from f i t s to measured absorption cross sections (Nak+ 80), Reb and Rec were varied i n order to f i t the e x i s t i n g e l a s t i c s c a t t e r i n g cross sections. The following observations can be made about the Set E parameters: a) there may be some question (Jen 84) as to the v a l i d i t y of the approach adopted i n f i t t i n g ImB and ImC to absorption cross s e c t i o n s . (CMS 82) themselves warn that questions may a r i s e i n the i n t e r p r e t a t i o n of the r e s u l t i n g parameters. Also, the data of (Nak+ 80) seems to be i n - 72 -C o e f f i c i e n t Units Value Re b Q fm -0.061 Im b Q fm 0.006 Re b : fm -0.13 Im bj fm -0.002 Re c Q fm 3 0.70 Im c Q fm 3 0.028 Re c1 fm 3 0.46 Im C j fm 3 0.013 Re B Q fmk -0.02 Im B Q fm1* 0.11 Re C 2 fm 6 0.36 Im C 2 fm 6 0.54 \ 1.4 Table 4.1 (CMS 82) Set E O p t i c a l P o t e n t i a l Parameters - 73 -disagreement with the more recent r e s u l t s of (Nav+ 83). b) At the time the (CMS 82) f i t s were performed, the e x i s t i n g e l a s t i c cross sections consisted s o l e l y of i r + scattering data. As w i l l be discussed i n Chapter V, the i n c l u s i o n of i r - s c a t t e r i n g r e s u l t s may prove important. It has been pointed out (CMS 82) that an o p t i c a l p o t e n t i a l of the form (4.1) has too many parameters to be determined from experimental data alone. (SM 83) and (SMY 83) have established the i n s e n s i t i v i t y of the e l a s t i c s c a t t e r i n g data to the p o t e n t i a l s t r u c t u r e , which i s manifested i n the form of c o r r e l a t i o n s between the c o e f f i c i e n t s of the p and p 2 terms i n the p o t e n t i a l . In f a c t , several analyses of i r + e l a s t i c s c a t t e r i n g data have been performed with o p t i c a l potentials of the form (4.2) (Ama+ 81), and other few parameter forms (Pre+ 81). In these analyses, a l l c o e f f i c i e n t s were varied f r e e l y i n order to f i t the cross s e c t i o n s . The r e s u l t a n t best f i t parameters d i f f e r from nucleus to nucleus. An a l t e r n a t e approach has been adopted by Friedman ( F r i 83), who used an o p t i c a l p o t e n t i a l described by a Fourier-Bessel s e r i e s , i . e . U(r) = I a n j 0 ( n i r r / R c ) , and varied the c o e f f i c i e n t s a n to f i t a l l the e x i s t i n g e l a s t i c s c a t t e r i n g data. The aim was to extract as much information from the data as p o s s i b l e , without introducing any p o s s i b l y unnecessary assumptions. The conclusions included: a) that the c h a r a c t e r i s t i c shape which i s t y p i c a l of a -k 2p - (l/2)V 2p term i s indeed required by the data; and b) that the K i s s l i n g e r V»pV type term i s also necessary. In l i g h t of the above considerations, the viewpoint adopted by the author i s the following: A l l c a l c u l a t i o n s are performed with the - 74 -p o t e n t i a l (4.1). As indicated i n section 4.1.1 t h i s form of p o t e n t i a l includes a l l p h y s i c a l e f f e c t s thought to be of importance i n the pion-nucleus problem. This includes the angle transformation terms and the LLEE e f f e c t . Also, as indicated by the r e s u l t s of ( F r i 83), the general form of the p o t e n t i a l i s that required by the e x i s t i n g data. The p o t e n t i a l i s capable of describing TT+ e l a s t i c s c a t t e r i n g cross sections on n u c l e i from 1 2 C to 2 0 8 P b with the same parameter set, which i s not true of p o t e n t i a l s with fewer parameters. The claim i s not made, that the p o t e n t i a l (4.1) i s based on unshakable t h e o r e t i c a l p r i n c i p l e s . C e r t a i n l y , the approximations made i n i n c l u d i n g higher order multiple s c a t t e r i n g e f f e c t s mean that i t i s semi-phenomenological. It would be argued, however, that faced with the lack of extensive experimental measurements of low energy pion-nucleus t o t a l reaction cross sections, and si n g l e and double charge exchange cross se c t i o n s , the p o t e n t i a l (4.1) i s the best one can do at the present time, and not much w i l l be gained i n considering other p o s s i b i l i t i e s . The other p o s s i b i l i t i e s include formulations of the pion-nucleus p o t e n t i a l i n both coordinate (JS 83) and momentum space (LT 78). The computer code LPOTT (Lan 82), using the l a t t e r p o t e n t i a l , has been run, and i s found not to reproduce e x i s t i n g e l a s t i c s c a t t e r i n g data for n u c l e i heavier than 1 2 C , to within factors of 2 or 3. Note that although the p o t e n t i a l has no free input parameters, i t does r e l y on p a r t i c u l a r models to determine o f f - s h e l l amplitudes, and include the e f f e c t s of true pion absorption and P a u l i exclusion. A formulation of the pion-nucleus p o t e n t i a l which has met with great success at resonance pion energies i s the A-hole model (HLY 77), - 75 -(Hir+ 79). There, nuclear medium e f f e c t s are parametrized i n terms of a phenomenological A-nucleus spreading p o t e n t i a l . It i s not c l e a r how important the A i s at 50 MeV incident pion energy. Already at 100 MeV there was some problem i n using the A-hole model to f i t the e l a s t i c and i n e l a s t i c s c a t t e r i n g cross sections f o r 1 2 C and 1 3 C (Ant+ 83). There i s some i n d i c a t i o n (Mon 84) that c a l c u l a t i o n s f o r 50 MeV pions w i l l not be forthcoming, u n t i l these problems are resolved. - 76 -4 . 2 I n e l a s t i c S c a t t e r i n g 4.2.1 C a l c u l a t i o n of Cross Sections The cross sections for sca t t e r i n g to d i s c r e t e nuclear excited states are r e l a t e d to sums of T-matrix elements. The DWPI computer code, which was used to perform pion-nucleus i n e l a s t i c s c a t t e r i n g c a l c u l a t i o n s , i s based on a standard d i s t o r t e d wave impulse approximation (DWIA) model f o r the T-matrix element ( t r a n s i t i o n operator). That i s , e l a s t i c s c a t t e r i n g i s included to a l l orders through the use of d i s t o r t e d waves, and the i n e l a s t i c t r a n s i t i o n i s treated to f i r s t order only. Strongly c o l l e c t i v e e x c i t a t i o n s , such as the f i r s t 2 + and 3 - states of many even-even n u c l e i , may be viewed as v i b r a t i o n s or r o t a t i o n s of the nucleus as a whole, where the radius i s a function of angle. The nuclear density may then be expanded about a s p h e r i c a l d i s t r i b u t i o n , the non-spherical part g i v i n g the coupling between the ground and excited s t a t e s . More s p e c i f i c a l l y , one can write: p = p 0 + Ap = p Q + I 8 x F x ( r ) Y x " ( n ) a X l J , (4.3) where a Xy i s a l i n e a r combination of nuclear e x c i t a t i o n c r e a t i o n and a n n i h i l a t i o n operators. The nuclear o p t i c a l p o t e n t i a l may then be expressed as: v . v (0) + v ( i ) m n v n n It i s Ap , the deformed part of the density, that contributes to the part of the o p t i c a l p o t e n t i a l giving r i s e to nuclear e x c i t a t i o n . - 77 -For a t r a n s i t i o n from a ground state of spin 0, to an excited state of angular momentum J with projection M, the T-matrix element i s rela t e d to / < LM | V ( 1 ) | 00 > f ( + ) d 3 r . (4.4) The "i"s are incoming and outgoing pion d i s t o r t e d waves, ca l c u l a t e d using the f u l l nuclear o p t i c a l p o t e n t i a l , as discussed i n s e c t i o n 4.1.1. Some mathematical d e t a i l s r e l a t i n g to the evaluation of (4.4) are presented i n Appendix B. For the expansion of the nuclear density presented i n (4.3), < LM | Ap | 00 > = I f 3 x F x Y x u < LM | a X y | 00 > . But < LM | a X y | 00 > = 6 L X « M _ y , so that < LM | Ap | 00 > = 3 LF L(r)Y°(JJ) . (4.5) The r a d i a l f a c t o r 3 L F L ( r ) i s c a l l e d the t r a n s i t i o n density P t r ^ r ^ » and i s discussed fur t h e r i n the next s e c t i o n . It has been mentioned previously, that one of the prime motivations for performing s c a t t e r i n g experiments with hadrons i n general, and pions i n p a r t i c u l a r , i s t h e i r s e n s i t i v i t y to both the neutrons and the protons present i n the nucleus. Thus, the o r i g i n a l DWPI program has been alt e r e d i n order to account f o r the neutron and proton contributions to the t r a n s i t i o n density separately. That i s , s t a r t i n g from the expansion P " Pp + Pn = (P0p + P0n> + <APp + APn> > one can repeat the arguments presented above, and end up with P_ ( r ) = (p_ ) + (p_ ) = g F + 8 F H t r v J v ^ t r ' p v ^ t r y n P P n n - 7 8 -4.2.2 T r a n s i t i o n Densities T r a n s i t i o n d e n s i t i e s are perhaps most commonly encountered i n the context of discussions of electron s c a t t e r i n g . The general expression f o r the d i f f e r e n t i a l cross section measured with i n e l a s t i c e l e c t r o n s c a t t e r i n g i s (HB 83): do . , dfi = M a^' i s the Mott cross section, appropriate to the s c a t t e r i n g of two point charged p a r t i c l e s , m u l t i p l i e d by a r e c o i l c o r r e c t i o n f a c t o r . The F x's are nuclear charge and current form f a c t o r s , which account for the d e t a i l s of nuclear st r u c t u r e . For low-lying c o l l e c t i v e excitations i n n u c l e i , the dominant co n t r i b u t i o n i s from the l o n g i t u d i n a l form f a c t o r F x C ( q ) , which can be expressed as the Fourier-Bessel transform of the nuclear charge t r a n s i t i o n density: F x C ( q ) = / p t £ ( r ) j x ( q r ) r 2 d r . This t r a n s i t i o n density can be defined (Hei 83) as the reduced matrix element of the charge operator between i n i t i a l and f i n a l nuclear sta t e s : P t * ( r ) = / < f f || p o p Y x || f ± > d 3 r , with p Q p = I e ± 5 ( r - r i ) . Thus, Pt r i s e s s e n t i a l l y a measure of the overlap between nuclear ground state and excited state wavefunctions. I t can be c a l c u l a t e d m i c r o s c o p i c a l l y i f these wavefunctions are known, or assumed, as i n the context of the s h e l l model for example. - 7 9 -It should be pointed out, that although the t r a n s i t i o n density was introduced i n the preceding section with reference to a macroscopic model of nuclear deformation, a microscopically c a l c u l a t e d t r a n s i t i o n density, parametrized i n the form of eqn. (4.5), could be used In the DWPI c a l c u l a t i o n , without a f f e c t i n g any of the d e t a i l s presented i n Appendix B. The h i s t o r i c a l j u s t i f i c a t i o n for the use of macroscopic models (BM 75), l i e s i n the fa c t that measurements of the l o n g i t u d i n a l form fa c t o r s for c o l l e c t i v e states suggest a strong peaking of the t r a n s i t i o n density at the nuclear surface, as i n the case of shape o s c i l l a t i o n s . The most frequently used macroscopic model i s that of Tassie (Tas 56), i n which the nucleus i s described as an incompressible, i r r o t a t i o n a l f l u i d , and gives the r e s u l t : p t * ( r ) = 6 X r X _ 1 3p 0/3r , where po Is the ground state charge d i s t r i b u t i o n . Note that for a Gaussian ground state density d i s t r i b u t i o n , or a two parameter Fermi (see Appendix C), t h i s form for the t r a n s i t i o n density, with X = 2, Is i d e n t i c a l to the one u t i l i z e d i n the o r i g i n a l DWPI program: P t r ( r ) = 6 c 8p/9c . As the q u a l i t y of I n e l a s t i c e lectron s c a t t e r i n g data improved, i t was found that t h i s model di d not adequately reproduce the measured form f a c t o r s . Thus, for many n u c l e i , the parameters of the density p, whose d e r i v a t i v e i s under consideration, are varied from t h e i r ground state values, and adjusted to f i t the experimental data. In t h i s form, the expression for p t r presented above i s no longer a nuclear model, - 80 -but merely a convenient parametrizations f o r a surface peaked shape (HB 83). It should be mentioned at t h i s point, that i n c e r t a i n macroscopic models (BM 75), the parameter g can be re l a t e d to an i n t r i n s i c deformation of the nuclear shape. In l i g h t of the f a c t that c e r t a i n other parameters of the models must be varied i n order to obtain agreement with experimental data, one may be j u s t i f i e d i n maintaining a c e r t a i n amount of scepticism as to the v a l i d i t y of such a correspondence. As electron s c a t t e r i n g measurements were extended to higher values of momentum t r a n s f e r , i t was found that even the a n a l y t i c form deduced from the macroscopic models was too l i m i t i n g (Hei 81). The common pra c t i c e now i s to express the t r a n s i t i o n density as a Fourier-Bessel s e r i e s : P t r ( r ) = I V ^ j ^ X ^ r / R ) e(R-r) , where X n i s the n zero of the s p h e r i c a l Bessel function J x_^(x) , and R i s the radius beyond which p t r ( r ) i s assumed to be zero. For the ana l y s i s of the pion s c a t t e r i n g data on 1 8 0 and 2 6Mg, which were taken over a small range of momentum t r a n s f e r , the macroscopic form 8c9p/8c was used e x c l u s i v e l y . This shape i s consistent with the a v a i l a b l e e l e c t r o n s c a t t e r i n g data (Nor+ 82) and (Lee+ 74) for these n u c l e i . - 8 1 -4.2.3 Neutron and Proton Matrix Elements The p r o b a b i l i t y for a nucleus to undergo an electromagnetic t r a n s i t i o n of m u l t i p o l a r i t y X i s proportional to the square of the matrix element of the X-multipole t r a n s i t i o n operator, taken between i n i t i a l and f i n a l nuclear s t a t e s . This matrix element i s refered to as the 'proton matrix element'. It i s given by: Mp = O f | | 0 p X | \J±> , Z with 0 P * = I r j * Y X U ( % ) . i An equivalent (Hei 81) statement i s : MP - / (Ptr)p r X + 2 d r ' In the context of electromagnetic properties of n u c l e i , (ptr^)p i s the proton t r a n s i t i o n charge density. One can also discuss proton matrix elements, however, i n r e l a t i o n to hadron s c a t t e r i n g . In that case, nuclear e x c i t a t i o n s are pri m a r i l y due to the strong i n t e r a c t i o n , and (p t£)p should be taken as the proton t r a n s i t i o n matter density. This d i s t i n c t i o n i s frequently not made i n various analyses, and may make a s i g n i f i c a n t d i f f e r e n c e . Extending t h i s idea a b i t further, one can define, i n complete analogy with the above, the 'neutron matrix element': ^ - < J i M C M J i > • N with 0 n* = I r ± * Y X " ( % ) , i or M n = / ( P t * ) n r X + 2 dr . - 8 2 -The complete expression for the electromagnetic t r a n s i t i o n p r o b a b i l i t y i s : This can be re l a t e d to the t r a n s i t i o n rate (BM 75): +1) / i o \ 2 * + 1 / l \ For a s p e c i f i c nuclear energy l e v e l , the t r a n s i t i o n rate i s related to the l e v e l width and l i f e t i m e by: In a c o l l e c t i v e model, based on nuclear shape o s c i l l a t i o n s , the t r a n s i t i o n p r o b a b i l i t y may also be rel a t e d (BM 75) to the t o t a l nuclear deformation parameter B x through: R i s an e f f e c t i v e nuclear radius, whose value depends on the d e t a i l s of the model assumed. Thus, one should not expect to be able to compare the value of the s c a l i n g parameter 6 (as defined through the a n a l y t i c forms fo r the t r a n s i t i o n d e n s i t i e s introduced i n se c t i o n 4.2.2) with the 6 i n the above formula i n a model-independent fashion; nor to be able to r e l a t e i t to a p h y s i c a l deformation of the nuclear density. y(X) = r/fi = l / x m - 83 -4.2 .4 Isovector S e n s i t i v i t y of ir* I n e l a s t i c Scattering As mentioned i n section 4 . 2 . 1 , the DWPI computer code has been a l t e r e d to accommodate the input of separate neutron and proton t r a n s i t i o n d e n s i t i e s i n the c a l c u l a t i o n of low energy i n e l a s t i c pion s c a t t e r i n g . Since such c a l c u l a t i o n s have not been performed elsewhere, i t i s of some i n t e r e s t to i n v e s t i g a t e the r e s u l t s , even without reference to experimental data. F i g . 4.1 i l l u s t r a t e s the d i f f e r e n t i a l cross sections calculated for 50 MeV ir— s c a t t e r i n g to the 2 ^ state of 2 6Mg. A l l subsequent figures i n t h i s section w i l l r e f e r to c a l c u l a t i o n s f o r t h i s same s t a t e . The form c9p Q/9c has been u t i l i z e d for both neutron and proton t r a n s i t i o n d e n s i t i e s . p Q i s the ground state density of 2 6Mg, of the Fermi form, with parameters as determined by (Gyl 84) . The general shape of the angular d i s t r i b u t i o n s presented i n F i g . 4.1 i s c h a r a c t e r i s t i c of the angular momentum tra n s f e r i n the r e a c t i o n . The drop of the TT- cross section at back angles i s due to the fac t that the negatively charged pion i s attracted by the Coulomb f i e l d of the nucleus, and thus the TT- i n t e r a c t i o n occurs at a s l i g h t l y higher energy than the 7 T + . The s u r p r i s i n g feature of F i g 4.1 i s the d i f f e r e n c e i n magnitudes between the cross sections for the two pion charge stat e s , even though the c a l c u l a t i o n has been performed with 8n = 6 p =0.50, so that the neutron and proton t r a n s i t i o n d e n s i t i e s are of s i m i l a r magnitude. F i g . 4.2 i l l u s t r a t e s the r a t i o of TT- to i r + d i f f e r e n t i a l cross sections, c a l c u l a t e d with several d i f f e r e n t values of 8 n and g p , i n - 84 -- 86 -order to determine what e f f e c t the height of the assumed t r a n s i t i o n d e n s i t i e s has. One can see that f or both 8n=8p=0.35, and 8n=8p=0.65, the angular d i s t r i b u t i o n s of the TT - to TT + r a t i o s are very s i m i l a r to each other, and to that noted i n F i g 4.1. The r a t i o reaches a maximum of approximately f i v e near 70 degrees i n the center of mass frame, then drops to unity at about 135 degrees. This Is an i n d i c a t i o n that the simultaneous a n a l y s i s of TT - and i r + i n e l a s t i c s c a t t e r i n g cross sections, measured under s i m i l a r conditions, w i l l prove r e l a t i v e l y i n s e n s i t i v e to any u n c e r t a i n t i e s i n the absolute normalizations of the data points. One also notes that i n the angular region > 80 degrees, a 15% change i n Bp, for f i x e d B n » causes the r a t i o to change by =200%. This i s an i n d i c a t i o n that with the use of TT* i n e l a s t i c s c a t t e r i n g , one may have the s e n s i t i v i t y to determine the r a t i o 3n/Bp to good accuracy. Fig 4.3 i l l u s t r a t e s the calculated angular d i s t r i b u t i o n s of the d i f f e r e n t i a l cross sections themselves, for several values of the B's. In Fig 4 .3(a), B n i s held constant at 0.50, while B p i s assumed to be ei t h e r 0.35 or 0.65. In F i g 4 .3(b), the r o l e s of B n and Bp are reversed. One notes that increasing Bn(Bp) while keeping Bp(Bn) f i x e d produces a large increase i n the T f - ( - r r + ) r e s u l t , but a much smaller decrease i n the i r + ( i r - ) curve. This i s an i n d i c a t i o n that negative ( p o s i t i v e ) pions are much more s e n s i t i v e to the neutrons (protons) i n n u c l e i , through t h e i r s e n s i t i v i t y to the height of the neutron (proton) t r a n s i t i o n density, than they are to the protons (neutrons). As discussed i n Chapter 1.3, a s i m i l a r s e n s i t i v i t y i s seen i n the case of pion-nucleon s c a t t e r i n g . The case of pion-nucleus s c a t t e r i n g i s s u f f i c i e n t l y more complex, however, that one could not a p r i o r i assume F i g . 4.3 50 MeV n* D i f f e r e n t i a l Cross Sections for 2 6 M g ( 2 j + ) for Several Values of 8 - 88 -that such s e n s i t i v i t y would be present. The c a l c u l a t i o n s reproduced in F i g . 4.3 i n d i c a t e that i t i s . In order to quantify t h i s s e n s i t i v i t y , one can proceed as follows. F i r s t , by following the d e t a i l s of the i n e l a s t i c c a l c u l a t i o n , as presented i n Appendix B, one notes that i n the o r i g i n a l DWPI code, the parameter B appeared as a m u l t i p l i c a t i v e factor i n every term c o n t r i b u t i n g to the T-matrix. Thus the d i f f e r e n t i a l cross sections were d i r e c t l y proportional to 6 2. In the a l t e r e d code, by introducing P t r = OpFp + M n ) » one would expect da/dn = (r3pZf(6) + 8 nNg(8)) 2 , where f and g represent the angular dependencies of the proton and neutron s c a t t e r i n g amplitudes, which are not i d e n t i c a l . One can now integrate the d i f f e r e n t i a l cross sections, to eliminate t h i s angular dependence, and write the t o t a l TT+ i n e l a s t i c cross section for s c a t t e r i n g to a s p e c i f i c state as a+ « (BpZb p+ + B nNb n+) 2 , ( 4 . 6 ) and s i m i l a r l y f o r TT~. b D + and b n + provide an i n d i c a t i o n of the sens i -t i v i t y of the Tr + Is to protons and neutrons r e s p e c t i v e l y . F i g . 4 . 4(a) i s a plot of \/a+ vs B p, as cal c u l a t e d with the alte r e d DWPI code, with 3 n held constant. F i g . 4 .4(b) i l l u s t r a t e s / c r + vs B n, with Bp held constant. The slopes of the curves i n F i g . 4 . 4(a) and 4 . 4(b) (note the d i f f e r e n t scales) are prop o r t i o n a l to bp + and b n + r e s p e c t i v e l y . It i s evident that b p +>>b n +. Also, the decrease of a + with increasing B n indicates that b n + < 0 . S i m i l a r curves and conclu-sions may be drawn for the case of ir~, with the roles of protons and neutrons interchanged. - 90 -The curves i n F i g 4.4(a) for 8p<0.3, and i n 4.4(b) f or a l l values of 8 n, are not s t r a i g h t l i n e s , i n d i c a t i n g that the approximations made i n w r i t i n g eqn (4.6) are not exact. In any case, because (4.6) i s only a p r o p o r t i o n a l i t y one could not use i t to obtain numerical values for the b's. One can however combine the r e l a t i o n s for TT+ and TT- to obtain: In F i g 4.5 curves r e s u l t i n g from the use of t h i s r e l a t i o n for various values of b n -/bp~, along with a shaded region which representing the r e s u l t s of the a l t e r e d DWPI c a l c u l a t i o n have been p l o t t e d . It i s evident that bn~/bp~=bp-,7br+=-20. This i s a new r e s u l t . One can compare i t to values of b n / b p for other nuclear probes. These values have been computed by (BBM 81), using a formulation s i m i l a r to the one presented above. They are: 0. for ele c t r o n s c a t t e r i n g ; 1.0 for a's; 0.83 and 0.95 for 800 MeV and 1 GeV proton s c a t t e r i n g r e s p e c t i v e l y ; 3 (1/3) for low energy protons (neutrons) and resonance energy ( i . e . =180 MeV) T f ~ ( T r + ) . The value f o r low energy pions i s c l e a r l y unique. I t may be shown that the pion s e n s i t i v i t y does not g r e a t l y depend on the d e t a i l s of the o p t i c a l p o t e n t i a l used i n the c a l c u l a t i o n s . Such considerations would not be meaningful, however, u n t i l some reference i s made to experimental data. Thus further d i s c u s s i o n i s postponed to Chapter V. L e t t i n g b p +=b n and b n +=b p~ , one can rewrite t h i s as: F i g . 4.5 a +/a~ vs B p/B n for 50 MeV it* S c a t t e r i n g t o 2 6 M g ( 2 1 + ) - 92 -CHAPTER V RESULTS AND CONCLUSIONS In th i s chapter , the o p t i c a l p o t e n t i a l c a l c u l a t i o n s discussed i n Chapter IV are app l ied to the measured d i f f e r e n t i a l cross s e c t i o n s . Sec t ion 5.1 deals with e l a s t i c s c a t t e r i n g . 50 MeV ir + s c a t t e r ing on 1 8 0 and 2 6 M g targets has not been measured p r e v i o u s l y . There are no publ i shed 50 MeV T T - absolute e l a s t i c cross s e c t i o n s , although s c a t t e r i n g data has been taken both at LAMPF, and at TRIUMF, p r i o r to the present experiments. LAMPF r e s u l t s (Daw 81) are a v a i l a b l e only i n an unpublished Master ' s t h e s i s . TRIUMF data (Joh+ 79), (Gyl 84) were taken as part of an ongoing program to use low energy IT - e l a s t i c s c a t t e r i n g to determine nuclear neutron d i s t r i b u t i o n s . Part of the s trength of t h i s program l i e s i n the fact that u n c e r t a i n t i e s due to the pion-nucleus i n t e r a c t i o n can be e l imina ted by comparing s c a t t e r i n g from adjacent isotopes ( e . g . ^ » l ^ O , 24,26^^ 32,34,36s) m Since only the r a t i o s of cross sec t ions are used i n t h i s type of a n a l y s i s , however, r e l a t i v e l y l i t t l e emphasis has been placed on determining absolute normal iza t ions , u n t i l experiments with the QQD spectrometer were undertaken. Note that the neutron dens i ty d i s t r i b u t i o n s used In the present ana lys i s are those determined with TT - s c a t t e r i n g at TRIUMF. The dens i ty parameters are given i n Table 5.1. Sect ion 5.2 deals with i n e l a s t i c s c a t t e r i n g . The measured d i f f e r e n t i a l cross sect ions are f i t by vary ing the magnitudes of the neutron and proton t r a n s i t i o n d e n s i t i e s ( i . e . 8 n and 8p) , which are input to the o p t i c a l model c a l c u l a t i o n s . The r e s u l t s of these f i t s are presented i n s e c t i o n 5.2.1. I t i s found that the r a t i o 8 n/B p - 93 -Charge Proton Neutron Nucleus Density c t or a c t or a c t or a 12c Gauss 1.687 1.067 1.595 1.067 1.595 1.067 18 0 Gauss 1.881 1.544 1.754 1.540 1.900 1.540 2 6 M g Fermi 3.050 .5234 2.896 .5238 2.967 .5491 3 2 S Fermi 3.262 .5407 3.153 .5407 3.153 .5407 Table 5.1 Density D i s t r i b u t i o n Parameters Used i n the Present Analysis. See Appendix C f o r Further D e t a i l s . - 94 -extracted i n t h i s way i s independent of the o p t i c a l p o t e n t i a l parameters used. A more complete discussion of the model dependence of the present r e s u l t s i s given i n section 5.2.2. A comparison of the values of Mn/Mp deduced from the present re s u l t s with those from other experiments i s made i n section 5.2.3. - 95 -5.1 E l a s t i c S c a t t e r i n g The measured d i f f e r e n t i a l cross sections f o r TT* e l a s t i c s c a t t e r i n g from 1 2 C and 1 8 0 are plotted i n F i g . 5.1, and those f o r TT* e l a s t i c s c a t t e r i n g from 2 6Mg and 3 2 S i n F i g . 5.2. Note that the 3 2 S data were taken with the QQD spectrometer, and analyzed by Sobie (Sob 84a). They are included here because of th e i r relevence to the d i s c u s s i o n . The s o l i d curves i n both f i g u r e s are based on the o p t i c a l model c a l c u l a t i o n s described i n Chapter IV, with the Set E parameters of (CMS 82). It i s evident that f o r 1 2 C and 1 8 0 , the c a l c u l a t i o n s provide a reasonable de s c i p t i o n of both i r + and TT - cross sections, while f o r 2 6Mg and 3 2 S , the i r + data appears to be reproduced much better than the TT -. It has been pointed out e a r l i e r , that part of the d e r i v a t i o n of the Set E parameters included a global f i t to a l l e x i s t i n g 50 MeV e l a s t i c s c a t t e r i n g data, and that that data consisted s o l e l y of it+ r e s u l t s . Thus some problem with f i t t i n g i r - r e s u l t s may be a n t i c i p a t e d . Before pursuing t h i s question further, however, i t may be worthwhile to consider the p o s s i b i l i t y that the differe n c e i n the q u a l i t y of the f i t s to the TT + and TT - cross sections i s due to some e f f e c t of the Coulomb i n t e r a c t i o n . The dashed curves i n Figures 5.1 and 5.2 are the r e s u l t of c a l c u l a t i o n s i n which the Coulomb nuclear cross terms have not been dropped from the Klein-Gordon equation (see p69). It i s evident that agreement with the data i s not improved. Ca l c u l a t i o n s have been performed (Bar 84) i n which r e l a t i v i s t i c Coulomb wave fuctions are used instead of the n o n r e l a t i v i s t i c ones included with the DWPI code. These do not d i f f e r appreciably from the ones presented i n Figures 5.1 and 5.2. F i g . 5.1 1 2 C and 1 8 0 E l a s t i c Cross Sections C a l c u l a t i o n with Set E Parameters F i g . 5.2 2 6Mg and 3 2 S E l a s t i c Cross Sections C a l c u l a t i o n with Set E Parameters - 98 -Recently measured energy s h i f t s and l e v e l widths i n high Z pionic atoms have been found to be several times smaller than expected. Ericson and Tauscher (ET 82) suggest that a l i k e l y explanation arises from a consideration of the gauge invariance of the electromagnetic i n t e r a c t i o n . This generates an a d d i t i o n a l electromagnetic p o t e n t i a l from the energy dependence of the ir-nucleus strong i n t e r a c t i o n p o t e n t i a l , given by 4ir a (1+u/M) p(r) V c ( r ) , with a = -0.044 T%~^« Calculations performed with the i n c l u s i o n of such a term i n the present o p t i c a l p o t e n t i a l are v i r t u a l l y i d e n t i c a l to the curves presented i n figures 5.1 and 5.2. Calculations have also been performed i n which the value of a was a r b i t r a r i l y v a r i e d . S i g n i f i c a n t d i f f e r e n c e s i n the c a l c u l a t i o n s were not obtained u n t i l a had been increased by a f a c t o r of « 20, and then the change was not such as to improve the f i t to the measured e l a s t i c cross sections. It i s possible that the r e l a t i v e l y poor f i t to the 2 6Mg and 3 2 S TT-data i s due to some as yet unconsidered e f f e c t , not included i n the present o p t i c a l p o t e n t i a l . Lacking any t h e o r e t i c a l input, however, one i s forced to consider v a r i a t i o n s of the p o t e n t i a l parameters themselves. Rather than adopt a complicated scheme for choosing which combination of c o e f f i c i e n t s to vary, i t was decided to vary each of them, except the isovector parameters bj and c^, one at a time, for each nucleus, leaving the unvaried ones f i x e d at t h e i r Set E values. The r e s u l t s of t h i s procedure are presented i n Table 5.2, which gives the value of each c o e f f i c i e n t required to minimize x 2/v, and the minimum x 2 / v achieved, x2 i s given by - 99 -Best F i t Values for Parameters Set E 12 c 1 8 0 2 6 M g 3 2 s Re b Q - . 0 6 1 - . 0 5 6 - . 0 6 2 - . 0 6 0 - . 0 5 8 Im b Q . 0 0 6 . 0 0 5 . 0 0 1 . 0 1 9 . 0 2 5 Re c Q . 7 0 0 . 6 7 2 . 6 8 4 . 6 0 1 . 5 3 7 Im c Q . 0 2 8 . 0 5 4 . 0 2 8 . 1 5 8 . 1 7 3 Re B Q - . 0 2 0 . 0 1 9 - . 0 3 5 . 0 2 0 . 0 7 0 Im B Q . 1 1 0 . 0 9 8 . 0 6 6 . 1 7 4 . 2 8 5 Re C 2 . 3 6 0 . 1 7 6 . 2 9 9 - . 3 3 0 - . 6 3 8 Im C 2 . 5 4 0 . 6 4 7 . 4 4 0 1 . 7 0 3 1 . 7 2 0 . X 1 . 4 0 1 . 5 4 6 1 . 4 6 0 1 . 9 7 3 2 . 2 2 5 Best F i t Value for x 2 12 C 18 0 2 6 M g 32 S R e b o 0 . 8 0 5 . 5 1 7 . 1 3 5 . 4 Im b Q 1 . 1 3 4 . 2 1 1 . 7 1 8 . 8 Re c Q 0 . 7 6 5 . 2 2 . 4 6 . 0 Im c Q 1 . 0 2 5 . 5 2 . 4 2 . 0 Re B Q 0 . 8 1 5 . 4 1 6 . 5 3 2 . 5 Im B 0 1 . 1 1 5 . 4 1 6 . 4 2 3 . 8 Re C 2 0 . 8 8 5 . 4 2 . 6 5 . 0 Im C 2 1 . 1 0 5 . 4 2 . 5 3 . 9 X 0 . 8 8 5 . 3 2 . 4 8 . 3 SET E 1 . 1 4 5 . 5 1 7 . 1 3 7 . 4 Table 5 . 2 Best F i t Values for O p t i c a l P o t e n t i a l Parameters, Varied One at a Time, and Resulting Minimum x 2 / v Values. Unvaried C o e f f i c i e n t s Are Left Fixed at t h e i r Set E Values. - 100 -/ d g / d n c a l c - d a / d ^ m e a s \ 2 / d o / d a c a l c - d q / d f t m e a s \ 2 (5.!) l>\ A da/dfi m e a s + ^ \ A da/dj^eag J T T and v i s the number of degrees of freedom. It i s evident from Table 5.2 that the q u a l i t y of the f i t s to the 1 2 C and 1 8 0 data i s not s i g n i f i c a n t l y improved by varying any one o p t i c a l p o t e n t i a l parameter. For the cases of 2 6Mg and 3 2 S however, s i g n i f i c a n t improvement can be obtained by decreasing the strengths of the p-wave parameters Re c Q or Re C 2, or increasing the values of Im c Q or Im C 2• In p a r t i c u l a r , i t appears that the choice of Im c Q = 0.165, or Im C 2 = 1.711 w i l l y i e l d a p o t e n t i a l which describes 2 6Mg and 3 2 S TT* e l a s t i c s c a t t e r i n g simultaneously f a i r l y w e l l . The r e s u l t s of such c a l c u l a t i o n s are shown i n F i g . 5.3. Table 5.3 presents the t o t a l reaction cross sections calculated using some of the parameter values from Table 5.2. Of p a r t i c u l a r i n t e r e s t i s the increase of f o r 2 6Mg and 3 2 S corresponding to the increased values of Im c Q and Im C 2 mentioned above. It has been pointed out by ( F r i 83) that i n c l u s i o n of measurements of aj with e l a s t i c s c a t t e r i n g data provides a noticeable improvement i n the accuracy of the model-independently determined p o t e n t i a l s . Such measurements would be of considerable i n t e r e s t i n the present case as w e l l . F i g . 5.3 2 6Mg and 3 2 S E l a s t i c Cross Sections C a l c u l a t i o n with 'Best F i t ' Parameters - 102 -Nucleus Parameters o*£ TT + (mb) aj TT - (mb) Set E 116 156 Re b 0 -.056 120 162 Re c Q .672 114 152 12 C Re B Q .019 120 162 Re C 2 .176 114 151 X 1.546 108 144 Set E 169 246 18 0 Im b Q .001 148 227 Im B Q .066 155 234 Set E 242 383 Re c Q .601 220 341 2 6 M g Im c Q .158 362 516 Re C 2 -.330 220 340 Im C 2 1.703 338 485 X 1.973 194 296 Set E 275 498 Re c Q .537 239 412 3 2 S Im c Q .173 417 655 Re C 2 -.638 242 417 Im C 2 1.720 383 610 Table 5.3 T o t a l Reaction Cross Sections as Calculated Using Some of the Best F i t Values for the C o e f f i c i e n t s . - 103 ~ 5.2 Inelastic Scattering 5.2.1 Results The c a l c u l a t i o n of i n e l a s t i c s c a t t e r i n g cross sections has been ou t l i n e d i n se c t i o n 4.2. The present analysis uses the macroscopic forms 8 nc3p n/3c = -8 n r 3 p n / 3 r and 8 pc3pp/3c = - 8pr3pp/3r f o r the neutron and proton t r a n s i t i o n d e n s i t i e s r e s p e c t i v e l y . p n and Pp are the neutron and proton ground state d e n s i t i e s (see Table 5.1). The values of 8 n and 8 p were varied independently i n order to obtain a best f i t to the TT+ and TT- i n e l a s t i c s c a t t e r i n g data simultaneously. A l l other p o t e n t i a l parame-ters were held constant at the values used to describe e l a s t i c s c a t t e r i n g . The best f i t was taken to be that r e s u l t i n g i n a minimum value of x 2 / v » with x 2/v defined as for the f i t s to the e l a s t i c s c a t t e r i n g (see eqn. 5.1). F i g . 5.4 i l l u s t r a t e s a t y p i c a l x2 contour p l o t for a f i t to the 2 j + state of 2 6Mg. Each contour represents an increase of 1 i n the value of X 2/v. The minimum value of x2 A* l i e s i n the middle of the centermost contour, whose boundaries provide an i n d i c a t i o n of the uncertainty a s s o c i -ated with the present r e s u l t s . The s t r a i g h t s o l i d l i n e s l i e along con-stant values of 8 n/3p« It w i l l be noted that the contours are elonga-ted i n the d i r e c t i o n of constant 8 n/3p« This i s an i n d i c a t i o n that the r a t i o 8 n/6p i s determined to a better accuracy than the i n d i v i d u a l values of the 8's. Table 5.4 gives the best f i t values of 8n» 3p» and B n/3p for 1 2 C , 1 8 0 , and 2 5Mg determined as described above. The measured d i f f e r e n -t i a l cross s e c t i o n s , and the re s u l t s of c a l c u l a t i o n s using the best f i t values of g n and 8 P are plotted i n Figures 5.5, 5.6 and 5.7 for these - 104 -F i g . 5.4 T y p i c a l x 2 P l o t for F i t of 8 n and B p t o 2 6 M g I n e l a s t i c Cross Sections - 105 -3 n Mn/Mp 0.522 0.540 0.97 ± 0.08 0.97 ± 0.08 18 0 0.478 0.406 1.18 ± 0.10 1.81 ± 0.15 2 6 M g 0.465 0.655 0.72 ± 0.06 0.90 ± 0.07 Table 5.4 Values of 8 n, g p, 8 n/8 p, and Mn/Mp as Determined by F i t t i n g I n e l a s t i c Scattering Data. - 106 -- 107 -F i g . 5.6 1 8 0 ( I T , T T , ) 1 8 0 * ( 2 1 + ) I n e l a s t i c Cross Sections - 108 -- 1 0 9 -three n u c l e i . Table 5.4 also l i s t s the values of the r a t i o of neutron to proton matrix elements (Mn/Mp) for the 2 + states under consideration. As explained i n se c t i o n 4.2.3, these are given by Ln / (Ptr)n r > t d r / (Ptr)p r l * d r For the case of 1 2 C , with N = Z, and p n = p p, Mn/Mp = B n/B p - 1. , as expected. For 1 8 0 and 2 6Mg, due to differences i n the other f a c t o r s , Mn/Mp i s only proportional to B n/Bp. 5.2.2 Model Dependence Note that the c a l c u l a t i o n s used to generate the curves i n Figures 5.5, 5.6 and 5.7 a l l used the Set E o p t i c a l p o t e n t i a l parameters. It has been shown i n sec t i o n 5.1, that c a l c u l a t i o n s with d i f f e r e n t parameters describe the TT* - 2 6Mg e l a s t i c cross sections b e t t e r . It w i l l be shown i n t h i s s e c t i o n , that the best f i t value f o r the r a t i o B n/8p i s indepen-dent of the o p t i c a l p o t e n t i a l parameters used to perform the c a l c u l a t i o n . F i g . 5.8 shows the best f i t values of B n and Bp for 2 6Mg, as determined by making use of the method outlined above, but using d i f f e r e n t values of the o p t i c a l p o t e n t i a l parameter Re c Q i n performing the c a l c u l a -t i o n s . It can be seen that the i n d i v i d u a l best f i t values of B n and Bp d i f f e r with v a r i a t i o n s i n Re c Q , but i n such a way that the r a t i o B n/8p remains approximately constant. Similar graphs can be produced for v a r i a t i o n s of the other o p t i c a l p o t e n t i a l parameters. F i g . 5.9 shows a x 2 contour p l o t , s i m i l a r to F i g . 5.4, but i n which - 110 -F i g . 5.8 V a r i a t i o n of 'Best F i t ' 6's with Changes i n Recj, F i g . 5.9 X 2 Contour Plot Using 'Best F i t ' O p t i c a l P o t e n t i a l Parameters - 112 -only the centermost contours are reproduced, f o r the four sets of o p t i c a l p o t e n t i a l parameters which gave good f i t s to the 2 6Mg e l a s t i c s c a t t e r i n g cross s e c t i o n s . I t i s evident that the best f i t values of B n/8p, as determined with c a l c u l a t i o n s using any of these parameter sets, are consistent with each other,and with the r e s u l t s of the c a l c u l a t i o n s using the Set E parameters. This i s not meant to imply that the present r e s u l t s are e n t i r e l y free of any model dependence. But i t does seem to in d i c a t e that the e f f e c t of the d e t a i l s of the pion nucleus i n t e r a c t i o n on the deduced values of Bn/Bp> and thus Mn/Mp, are minimal when the s c a t t e r i n g of i r + and i r - i s considered simultaneously, as has been done at present. It should be pointed out that the assumption has been made i n the present a n a l y s i s , that the shapes of the neutron t r a n s i t i o n d e n s i t i e s can be adequately described by the macroscopic form c3p/8c. I t may be worthwhile at some point to investigate what e f f e c t d i f f e r e n t assumptions for the shape would have on the c a l c u l a t i o n s , or the deduced values of Mn/Mp. Also, the ana l y s i s of proton s c a t t e r i n g data on 2 6Mg (Bla+ 82), (Alo+ 81) has demonstrated the importance of coupled channel e f f e c t s . There i s reason to believe (DeT 84) that such e f f e c t s would not a l t e r the value of Mn/Mp deduced from the present data, but t h i s should be v e r i f i e d . - 113 -5.2.3 Comparison with Other Results A summary of the most recently determined values f o r Mn/Mp, for the 2 ^ states of 1 8 0 and 2 6Mg i s given i n Table 5.5, and the values p l o t t e d i n F i g . 5.10. It can be seen that the present r e s u l t s are consistent with almost a l l the others. This statement may be s l i g h t l y misleading, however, since a l l of the other values are not i n agreement with each other. In p a r t i c u l a r , the mirror nucleus value seems to be high f o r both 1 8 0 and 2^Mg. The present r e s u l t has already been discussed i n the preceding s e c t i o n s . Perhaps i t would be appropriate at t h i s point to make a few comments on the other r e s u l t s . The a n a l y s i s of resonance energy pion s c a t t e r i n g i s not done i n a manner s i m i l a r to the present a n a l y s i s . Rather, the r a t i o O " ( T T ~ )/c ( T r + ) i s determined by i n t e g r a t i n g the laboratory cross sections; and Mn/Mp derived from It i s not immediately obvious how model dependent t h i s procedure i s , and how i t incorporates the d i f f e r e n t r a d i a l dependences of the neutron and proton t r a n s i t i o n d e n s i t i e s . (Bla+ 82) point out that while t h e i r a n a l y s i s of 800 MeV proton s c a t t e r i n g from 2**Mg and 2 6Mg y i e l d s a value of 0.74 ± 0.12 for Mn/Mp for the 2 X + state of 2 6Mg, i t also r e s u l t s i n a value of 0.81 ± 0.11 for the 2, + state of the N = Z nucleus 2l*Mg. - 114 -18 0 2 6 M g Present Result 1.81 ± 0.15 0.90 ± 0.07 : Resonance Energy Or,*') 1.67 ± 0.15 a 1.58 ± 0.13 b 1.86 ± 0.16 b 1.58 ± 0.15 c 0.62 ± 0.14 d 800 MeV (p,p') 0.74 ± 0.12 e 24 MeV (p,p') 0.69 ± 0.17 f 24 MeV (n,n') 1.02 ± 0.058 (<x,a')h 1.5 ± 0.50 0.80 ± 0.17 Mirror Nucleus1*- 2.07' ± 0.22 1.03 ± 0.09 TheoryJ 1.64 0.83 Table 5.5 Comparison of Presently Deduced Values of Mn/Mp with those from other probes. Data i s from a) (Ive+ 78), b) (Ive+ 79), c) (Lun+ 78), d) (Wie+ 80), e) (Bla+ 82), f ) (Alo+ 81), g) (Tai+ 83), h) (BBM 79), i ) (Ale+ 82), j ) (Bro+ 82). - 115 -1 1 r— I i i — r i . i i i i • present result — • — ! — • — ' 164-230 MeV • i . i (TT.TT') 9 ,— • - L — 1 (<*,<*') • — ! mirror nucleus — — * theory i i i i J i — • i J '.. i ).± i 1— 0 .4 .8 1,2 1.6 2.0 2.4 M n / M p for 1 8 0 - 1 1 1 r. i i i i i i - # - present result — • | 164 MeV (7r,7r') —w— I 800 MeV (p,p') • — j - j 24 MeV (p,p') J j (a,a') j -U- 24 MeV (n,n') I ( • mirror nucleus • I theory i i i l l i i i i i i 0 .4 .8 1.2 1.6 2.0 M „ / M p for 2 6 Mg 2.4 F i g . 5.10 Comparison of Mn/Mp as Obtained Using D i f f e r e n t Probes - 116 -It should be pointed out that the only r e s u l t which appears to come close to the mirror nucleus value i s that of (Tai+ 83). A close i n s p e c t i o n of the an a l y s i s , however, reveals that the r e s u l t quoted i s merely a r e f l e c t i o n of the EM value input i n t o the analysis i n the f i r s t p lace. As f o r the mirror nucleus r e s u l t i t s e l f , i t i s based on the idea that M n for a given nuclear state i s equal to Mp for the same state i n the mirror nucleus. For example, M n ( 1 8 0 ) _ M p( 1 8Ne) M p ( 1 8 0 ) = Mp(*°-0) * Mp f o r a given state i s derived from a measurement of i t s l i f e t i m e (see se c t i o n 4.2.3). S h e l l model c a l c u l a t i o n s (Bro+ 82), however, indi c a t e that M n ( 1 8 0 ) = (0.89) M p( 1 8Ne) , due to Coulomb i n t e r a c t i o n s . The mirror nucleus r e s u l t quoted i n Table 5.5 (Ale+ 82) already incorporates such a Coulomb c o r r e c t i o n . Unfortunately, the same c a l c u l a t i o n which i s r e l i e d upon to give the magnitude of the Coulomb c o r r e c t i o n predicts a value f o r Mn/Mp which i s inconsistent with the mirror nucleus r e s u l t . - 117 -REFERENCES Ale+ 82 T.K. Alexander, G.C. B a l l , W.G. Davies, J.S. F o r s t e r , I.V. M i t c h e l l , and H.B. Mak, Phys. L e t t . 113B (82) 132. Alo+ 81 P.W.F. Alons, H.P Blok, J.F.A. Van Hienen, and J . Blok, Nucl. Phys. A367 (81) 41. Ama+ 81 J.F. Amann, P.D. Barnes, K.G.R. Doss, S.A. Dytman, R.A. E i s e n s t e i n , J.D. Sherman, and W.R. Wharton, Phys. Rev. C23 (81) 1635. Ant+ 83 L.E. Antonuk, D. Bovet, E. Bovet, J.P. Egger, J.F. Germond, F. Goetz, P. G r e t i l l a t , C. Lunke, E. Swartz, K. Masutani, and T. Takaki, SIN Preprint PR-83-12, Ju l y 1983. AR 82 R.A. Arndt and L.D. Roper, SAID: Scattering Analysis and I n t e r a c t i v e D i a l - i n Program, Center for Analysis of P a r t i c l e S c a t t e r i n g , V i r g i n i a Polytechnic I n s t i t u t e and State U n i v e r s i t y I n t e r n a l Report CAPS-80-3 (1982). Ban 66 A.P. Banford, The Transport of Charged P a r t i c l e Beams, (E. & F.N. Spon Ltd., London, 1966). Bar 84 B.M. Barnett, private communication. BBM 79 A.M. Bernstein, V.R. Brown, and V.A. Madsen, Phys. Rev. L e t t . 42 (79) 425. BBM 81 A.M. Bers t e i n , V.R. Brown, and V.A. Madsen, Phys. L e t t . 103B (81) 255. Bir+ 71 L. B i r d , A. Peter, L. R a f f l e r , and J . Walters, Nucl. I n s t . Meth. 94 (71) 585. Bla+ 82 G.S. Blanpied, N.M. Hintz, G.S. Kyle, M.A. Franey, S.J. Seestrom-Morris, R.K. Owen, J.W. Palm, D. Denhard, M.L. B a r l e t t , C.J. Harvey, G.W. Hoffmann, J.A. M c G i l l , R.P. L i l j e s t r a n d , and L. Ray, Phys. Rev. C25 (82) 422. - 1 1 8 -BM 75 Bos+ 75 Bro+ 80 Bro+ 82 Car 84 CMS 82 CR 79 Daw 81 DeT 84 EE 66 EM 74 EM 76 ET 82 Fer 79 F r i 83 A. Bohr and B. Mottelson, Nuclear Structure, V o l . 2 (W.A. Benjamin, Inc., London, 1975). R. Bosshard, R.L. Chase, J . Fischer, S. Iwata, and V. Radeka, Proceedings of the 2 n d ISPRA Nuclear E l e c t r o n i c s Symposium, Stressa, I t a l y , May 20-23, 1975, p. 145. K.L.Brown, D.C. Carby, C H . I s e l i n , and F. Rothaker, TRANSPORT a Computer Program for Designing P a r t i c l e Beam Transport Systems, CERN report 80-04, 1980. B.A. Brown, B.H. Wildenthal, W. Chung, S.E. Massen, M. Bernas, A.M. Bernstein, R. Miskimen, V.R. Brown, and V.A. Madsen, Phys. Rev. C26_ (82) 2247. J.A. Carr, private communication. J.A. Carr, H. McManus, and K. Strieker-Bauer, Phys. Rev. C2_5 (82) 952. J . Chai and D.O. Riska, Nucl. Phys. A329 (79) 429. G.H. Daw, MSc Thesis, New Mexico State U n i v e r s i t y (1981). N. De Takacsy, private communication. M. Er i c s o n and T.E.O. Ericson, Ann. Phys. 36 (66) 323. R.A. E i s e n s t e i n and G.A. M i l l e r , Comp. Phys. Comm. 8_ (74) 130. R.A. E i s e n s t e i n and G.A. M i l l e r , Comp. Phys. Comm. 11 (76) 95. T.E.O. Ericson and L. Tauscher, Phys. L e t t . 112B (82) 425. Fermilab PN-97.5, M u l t i User's Guide, 1979. E. Friedman, Phys. Rev. C28 (83) 1264. - 119 -D.R. G i l l , p rivate communication. W. Gyles, PhD Thesis, U n i v e r s i t y of B r i t i s h Columbia (1984). M.L. Goldberger and K.M. Watson, C o l l i s i o n Theory, (John Wiley & Sons, Inc., New York, 1964). J. Heisenberg and H.P. Blok, Ann. Rev. Nucl. Part. S c i . 33 (83) 569. J. Heisenberg, Advances i n Nuclear Physics, V o l . 12, edited by W. Negele and E. Vogt (Plenum, New York, 1981), p61. M. H i r a t a , J.H. Koch, F. Lenz, and E.J. Moniz, Ann. of Phys. 120 (79) 205. M. H i r a t a , F. Lenz, and K. Yazaki, Ann. of Phys. 108 (79) 205. R. Hofstadter, Nuclear and Nucleon Structure, (W.A. Benjamin, New York, 1963). In t e r n a t i o n a l Mathematical & S t a t i s t i c a l L i b r a r i e s , Inc., Houston, Texas, 1982. S. Iversen, A. Obst, K.K. Seth, H.A. Thiessen, C L . Morris, N. Tanaka, E. Smith, J.F. Amann, R. Boudrie, G. Burleson, M. Devereux, L.W. Swenson, P. Varghese, K. Boyer, W.J. B r a i t h -waite, W. Cottingame, C.F. Moore, Phys. Rev. L e t t . 40 (78) 17. S. Iversen, H. Nann, A. Obst, K.K. Seth, N. Tanaka, C L . Morris H.A. Thiessen, K. Boyer, W. Cottingame, C F . Moore, R.L. Boudrie, D. Denhard, Phys. L e t t . 82B (79) 51. B.K. Jennings, private communication. R.R. Johnson, T. Masterson, B. Bassalleck, W. Gyles, T. Marks, K.L. Erdman, A.W. Thomas, D.R. G i l l , E. Rost, J . J . Kraushaar, J . A l s t e r , C. Sabev, J . Arvieux, and M. K r e l l , Phys. Rev. L e t t . 43 (79) 844. - 1 2 0 -JS 83 M.B. Johnson and E.R. S i c i l i a n o , Phys. Rev. C27 (83) 730. Kis 55 L.S. K i s s l i n g e r , Phys. Rev. 98 (55) 761. KR 83 C. Kost and P. Reeve, REVMOC: a Monte Carlo beam transport program, TRIUMF i n t e r n a l report, TRI-DN-83-28, 1983. Lan 82 R.H. Landau, Comp. Phys. Comm. 28 (82) 109. Lee+ 74 E.W. Lees, A. Johnston, S.W. Brain, C S . Curran, W.A. G i l l e s p i e , and R.P. Singhal, Jour. Phys. A7 (74) 936. LS 78 Table of Isotopes, Seventh E d i t i o n , edited by CM. Lederer and V.S. S h i r l e y , (John Wiley & Sons, Inc., New York, 1978). LT 78 R.H. Landau and A.W. Thomas, Nucl. Phys. A302 (78) 461. Lun+ 78 C. Lunke, R. Corfu, J.P. Egger, P. G r e t i l l a t , J . P i f f a r e t t i , E. Schwarz, J.Jansen, C. P e r r i n , B.M. Preedom, Phys. L e t t . 78B (78) 201. Mon 84 E.J. Moniz, at the Workshop on Low Energy Pion-Nucleus Sc a t t e r i n g , Los Alamos, Jan 1984. M i l 84 T. Miles, private communication. Nak+ 80 K. Nakai, T. Kobayashi, T. Numao, T.A. Shibata, J . Chiba, and K. Masutani, Phys. Rev. L e t t . 44 (80) 1446. Nav+ 83 I. Navon, D. Ashery, J . A l s t e r , G. Azuelos, B.M. Barnett, W. Gyles, R.R. Johnson, D.R. G i l l , and T.G. Masterson, Phys. Rev. C28 (83) 2548. Nor+ 82 B.E. Norum, M.V. Hynes, H. Miska, W. Be r t o z z i , J . K e l l y , S. Kowalski, F.N. Rad, C P . Sargent, T. Sasanuma, W. Turchinetz, and B.L. Berman, Phys. Rev. C25 (82) 1778. Ora+ 81 C J . Oram, J.B. Warren, G. Marshall, and J . Doornbos, Nucl. I n s t . Meth. 179 (81) 95. - 121 -Pre+ 81 B.M. Preedom, S.H. Dam, C.W. Darden I I I , R.D. Edge, D.J. Malbrough, T. Marks, R.L. Burman, M. Hamm, M.A. Moinester, R.P. Redwine, M.A. Yates, F.E. Bertrand, T.P. Cleary, E.E. Gross, N.W. H i l l , C.A. Ludemann, M. Blecher, K. Gotow, D. Jenkins, and F. Milder, Phys. Rev. C23 (81) 1134. SCM 80 K. S t r i e k e r , J.A. Carr, and H. McManus, Phys. Rev. C22 (80) 2043. SM 83 R. Seki and K. Masutani, Phys. Rev. C27 (83) 2799. SMC 79 K. S t r i e k e r , H. McManus, and J.A. Carr, Phys. Rev. C19 (79) 929. SMY 83 R. Seki, K. Masutani, and K. Yazaki, Phys. Rev. C27_ (83) 2817. Sob 84a R.J. Sobie, PluT Thesis, U n i v e r s i t y of Toronto, 1984. Sob 84b R.J. Sobie, T.E. Drake, B.M. Barnett, K.L. Erdman, W. Gyles, R.R. Johnson, H.W. Roser, R. Tacik, E.W. Blackmore, D.R. G i l l , S. Martin, C.A. Wiedner, T. Masterson, Nucl. I n s t . Meth. 219 (84) 501. Ste 65 K.G. Steffen, High Energy Beam Optics, ( J . Wiley and Sons, New York, 1965). Stu 74 J.K. Studebaker, Los Alamos Informal Report, LA-5749-MS, 1974. Tai+ 83 R.C. T a i l o r , J . Rapaport, R.W. F i n l a y , and G. Randers-Pehrson, Nucl. Phys. A401 (83) 237. Tas 56 L.J. Tassie, Aust. J . Phys. 9 (56) 407, Tay 72 J.R. Taylor, Scattering Theory, (John Wiley & Sons, Inc., New York, 1972) Thi 76 M. Thies, Phys. L e t t . 63B (76) 43. Vin 78 R. Vinh Mau, AIP Conference Proceedings #41, 1978, p. 140. - 122 -Wad 76 E.A. Wadlinger, Nucl. Inst. Meth. 134 (76) 243. Wie+ 80 C.A. Wiedner, K.R. C o r d e l l , W. Saathoff, S.T. Thornton, J . Bolger, E. Boschitz, G. Proebstle, and J . Zichy, Phys. L e t t . 97B (80) 37. - 123 -APPENDIX A DETAILS OF ELASTIC CALCULATION D i f f e r e n t i a l cross sections are calculated with the computer code DWPI i n the following way: one s t a r t s with the Klein-Gordon equation (E 2 + V"! - 2EV C- 2EV n- p 2 ) ? = m2>? (A.l) L e t t i n g p •»• ihV, and Y •*• U£<r) Y m(Q), ( A . l ) becomes /^-V2 + £ U + 1 ) _ ^ U £ Y m =^k 2 + V 2 - 2 E ( V C + V n ) ^ u j i Y m (A.2) Assuming a p a r t i c u l a r form f o r V n, one can rewrite (A.2) as Cju" + C 2u' = C 3u (A.3) where the primes denote derivatives with respect to r . (A.3) i s then integrated outward numerically to a predetermined matching radius at which point the phase s h i f t 6JJ, between the function u(r) and the Coulomb wave func t i o n c a l c u l a t e d without V n i s found, f o r each p a r t i a l wave. The d i f f e r e n t i a l cross section i s c a l c u l a t e d from da/da = | f c ( 9 ) + f n ( 9 ) | 2 where f c i s the Coulomb amplitude, and f n = (1/ik) I (2i+l) exp(2ia £) f £ P £(cos9) , where o"£ i s the point charge Coulomb phase s h i f t of the l t n p a r t i a l wave, and f £ = (l/2)(exp(2iS£)-l). The t o t a l r e a c t i o n cross section i s given by ^ r e a c t i o n = <*/k2) I ( 2 * + D C 1 " |exp(2i6 £) | 2 ) . - 124 -In the o r i g i n a l DWPI, the nuclear o p t i c a l p o t e n t i a l was taken to be 2EV n = A xp + A 2V»pV + A 3V 2p + A^V^p. The r e s u l t i n g c o e f f i c i e n t s i n (A.3) are: C : = A 2p - 1 C 2 = A 2p' C 3 - A 2p' + k 2 - 2EV C + V2, - A 3V 2p - Aj_p - A ^ p . r For the p o t e n t i a l used i n the present a n a l y s i s , given at the beginning of Chapter IV, the c o e f f i c i e n t s i n (A.3) are given by: C, = A«-p2 + —t—- - 1 1 5 (l+(XB /3)) C* = ( l + ( X B / 3 ) ) 2 + 2 A 5 P P ' C 3 = k 2 + V c 2 - 2EV C - A Lp - Alv<5p - A^p 2 - V 2 ( A 3 p + A 3 « p + A 7 p 2 ) + ( 2 A 5 p p ' + B ' X 1 £ ( £ + 1 ) r-2 ( l + ( X B / 3 ) ) 2 y r where B = AgP 2 + A2P + A^Sp and B' = 2A 6pp' + AgP ' + A 2 V ( 6 p ) * - 125 -As discussed i n the text, some c a l c u l a t i o n s have been performed i n which the Coulomb-nuclear cross terms (V cV n+V nV c) have been retained i n i n the Klein-Gordon equation. The r e s u l t i n g equation (A.3) then has the following c o e f f i c i e n t s : C l -C2 -E-V r E-V, _(l+(XB/3)) + A j 5 p 2 _ - 1 B ' , v c B 2 _ (l+(XB/3))2 + 2 A 5 P P J ~ ~E [ ( l+(XB/3)) + ^ _ C 3 = k 2 + V2. - 2EV C - E-V, (A 1p+A l v6p+A l tp 2+V 2(A 3p+A 3 V6p+A 7p 2)) where and + E-V, E V (l+(XB/3)) + A 5p2^- 1 £(£+1) E-V, Er [ ( l + ( X B / 3 ) ) 2 + 2A.PP' VI B' 2A 5pp' 2E [ ( l+(XB/3)) 2 B = Agp 2 + Ajp + Ajy^P B' = 2A 6pp' + Ajp' + A ^ o p ) ' V p JD 2E |_ ( l+(XB/3)) 5 Including the 'Ericson gauge term', mentioned i n Chapter V, i n a l l cases i s simple, i n v o l v i n g the add i t i o n of a term: 4ir3V cp to the c o e f f i c i e n t C g of (A.3). V c i n a l l cases i s calculated from: Vc(r') = / 0 p ( r ) d 3 r = , T, p ( r ) r 2 d r { Z p ( r ) r 2 d r - 126 -APPENDIX B DETAILS OF INELASTIC CALCULATION The d i f f e r e n t i a l cross sections for pion i n e l a s t i c s c a t t e r i n g from a nucleus with ground state spin 0 + to an excited state with s p i n and p r o j e c t i o n J and M are given by: — = )" aoP»(cos6) , with On = £ f ( J i ) | T | 2 , where f ( J i ) includes a number of Racah and Clebsch-Gordon c o e f f i c i e n t s which represent the d i f f e r e n t possible couplings of the intermediate angular momenta. As mentioned i n Chapter 4.2.1, the T-matrix element i s related to / »<-) < LM | VC1) | 00 > ?(+) d 3 r . Apart from some a d d i t i o n a l angular momentum coupling terms, t h i s expression for T can be written as the sum of three i n t e g r a l s , i . e . T « Ix + I 2 + I 3 , with I x = / uty* TD u £ + ) dr , ,(T) „ ( T K * /»„(+) .,(+) + T (X(X+l)-Jc.(JM-l )-rU'+l)) / ( — ) TDD ( — J dr and I 3 - / u^,)* DTD u £ + ) dr . The primed and unprimed variables refer to outgoing and incoming pion coordinates r e s p e c t i v e l y . - 127 -Coulomb e x c i t a t i o n i s accounted f o r by i n c l u d i n g an extra term: x - " ( 2 x + i ) R e ' H'* f ( r ) u* d r ' ( r / R c ) x r<R c ( R c / r ) x + 1 r>Rc where f ( r ) = and R c i s the radius of the s p h e r i c a l charge d i s t r i b u t i o n . In the o r i g i n a l DWPI, the nuclear o p t i c a l p o t e n t i a l was taken to be 2EV n = A :p + A^'pV + A 3V 2p . So that 2EV n 1^= A^Ap) + A 2V»(Ap)V + AgV 2(Ap) , and thus TD = Aj_BF , TDD = A 2BF , A a f „» 2 F ' & U + I ) F \ and DTD = A 3B f F" f2 ) » where BF i s the t r a n s i t i o n density, and the primes i n d i c a t e d e r i v i t i v e s with respect to r. For the p o t e n t i a l used i n the present a n a l y s i s , given i n Chapter 4.1.1, one has: 2EVH 1) = Ai(Ap) + Ai v(A(6p)) + 2A 4p(Ap) + V 2 ( A 3(Ap) + A 3 v ( A ( 6 p ) ) + 2A 7p(Ap) ) / A 2(Ap)+A 2 v(A(6p))+2A 6p(Ap) \ + V * ^ ( l f ( X / 3 ) ( A 2 P + A 2 v 6 P + A 6 p 2 ) ) 2 + 2 A 5 P ( A P ) J V , so that the f a c t o r s within the i n t e g r a l s are: - 128 -TD = AiF + A i v 6 F + 2A4pF , A 2 F + A 2 v 6 F + 2 A 6 P F  T D D = (1+(X/3)(A2P+A 2 v«p +A 6p^))2 + 2 A 5 P F • and DTD = (A3+2A7P)(F"+(2F'/r)-X(X+l)F/r 2) + A 3 v((6F)"+(2/r)(6F)'-X(X+1)6F/r 2) + 2A 7 ( ( 2 F p 7 r)+2F ,p ,+Fp") The notation i s : F = g n F n + g p F p and 6F = B n F n - g p F p , where 3n Fn a n d ^p Fp a r e t n e n e u t r o n a n d proton t r a n s i t i o n d e n s i t i e s r e s p e c t i v e l y . - 129 -APPENDIX C FORM OF NUCLEAR DENSITIES The following are the two a n a l y t i c forms f or nuclear d e n s i t i e s used i n the a n a l y s i s of the data presented i n t h i s t h e s i s : GAUSSIAN: - ( r / c ) 2 P(r) = —, T [1 + t x ( r / c ) 2 ] e (2+3a)(/Trc) 3 L FERMI: 1 + W(r/c) 2 P ( r ) = P ° 1 + exp((r-c)/t) where = 3K P ° 4Trc 3 [ l+(Trt /c) 2+(W/5)(3+10( 1 rt /c) 2+7(Trt /c) , + ] The normalizations are such that 4TT /p(r) d 3 r = K; where K = Z (the t o t a l number of protons) f o r pp, and K = N (the t o t a l number of neutrons) f or p n . - 130 -APPENDIX D ir + w KINEMATICS Y Y3 0 0 YS Y 0 0 0 0 1 0 0 0 0 1 Consider the decay of a pion into a muon and neutrino, as shown i n F i g . D . l . Four-momenta i n the lab and center of mass frames are related by a Lorentz boost, i . e . K^ = &P± (D.l) where A =| and y = » 6 = k^/cc , , In the center of mass frame, PT2 = ( P y + P v)2 , which implies xs^1 = my2 + 2 ( E U E V + p y 2 ) . (D.2) Using the f a c t that py 2 = E v 2 = E p 2 - mu2 , (D.2) can be solved to y i e l d PU = (mTT2 ~ my 2)/ 2 mTT > and Ey = (ra^2 + my_2)/2mir (D.l) y i e l d s the following: O J u = y ( E y + p ucos9) , kyCOS<|> = Y(8 Ey + PyCOS0) , and kysin<() = pysin6 131 -CENTER OF MASS FRAME PIT " ( 0,0,0 ) p u = ( Ey,pycos9,pysin8,0 ) P v = ( E v,-p ucos9,-pySinG,0 ) LABORATORY FRAME - ( <%,kTr»0,0 ) — K y = { o)y .kycos^j .kysinijij ,0 J M K v = ( a«v,kvcos<t>2 ,kvsin<j>2 ,0 ) F i g . D.l TT •»• pv Decay i n Center of Mass and Laboratory Frames - 132 -F i g . D.2 i s a plot of the muon lab energy versus the muon lab angle, for the decay of a 50 MeV pion, and demonstrates why few pions which decay before the QQD spectrometer dipole get through to the back wire chambers. It can be seen that muons emitted at angles < 5° have energies < 17 MeV, or > 68 MeV, and w i l l be bend severely by the dipole magnet. Muons with energies between 30 and 50 MeV are emitted at angles > 15°. In the center of mass frame, pion decay i s i s o t r o p i c . Using this f a c t , and the above r e l a t i o n s , F i g . D.3 can be generated. It i l l u s t r a t e s the r e l a t i v e number of muons emitted at a given lab angle, for the decay of a 50 MeV pion. This sort of picture may be used to estimate the number of pions which decay a f t e r the M13 E2 dipole magnet, but are s t i l l counted by the B1»B2 f l u x monitors. 20 | — I — i — i — i — | — i — i — i — i — | — i — i — i — r — | — r — i — i — i — | — i — i — r — i — | — i — i — i — i — | — i — i — r 10 20 30 40 50 60 70 80 Muon Lab Energy (MeV) F i g . D.2 E u vs <J>y for 50 MeV IT'S F i g . D.3 Decay Muon Angular D i s t r i b u t i o n 

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