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UBC Theses and Dissertations

Polarized proton induced exclusive pion production on ¹⁰B and ⁹Be for incident energies from 200 to 260… Ziegler, William Anthony 1985

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cC, Polarized Proton Induced Exclusive Pion Production on 1 0B and 9Be for Incident Energies from 200 to 260 MeV by William Anthony Zieg l e r B.Sc, University of Regina, 1980 M.Sc., University of B r i t i s h Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Physics We accept this thesis as conforming to the required standard A The University of B r i t i s h Columbia August 1985 © William Anthony Ziegler, 1985 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the- head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Q^c^. l3./?^£ i i Polarized Proton Induced Exclusive Plcm Production on i U B and ^Be  for Incident Energies from 200 to 260 Mev Abstract Two sets of experiments for studying polarized proton induced exclusive pion production were performed at TRIUMF. The"angular d i s t r i b u t i o n s of both the d i f f e r e n t i a l cross-sections and analyzing powers are presented for: 1 0 B<P'* +> l l Bg.s., 2." MeV 9 B e ( P ^ + ) 1 ° B e g . s . , 3 . 3 7 M e V a n d 9Be(p,u o o U w ,r f ° r incident proton energies between 200 and 260 g.s.,°. j n tiev MeV. The results indicate a number of new trends i n pion production. The preference for populating high-spin f i n a l states (at least i n these reactions) seems to be primarily a spin s t a t i s t i c a l weighting e f f e c t . The dependence of the shape of the angular d i s t r i b u t i o n s on incident proton energy i s linked with the quantum numbers of the f i n a l state nucleus. Transitions to a f i n a l state that can be described as s i n g l e - p a r t i c l e have a strong and s i m i l a r energy dependence. In contrast transitions to the other states, those that cannot be described as having a s i n g l e - p a r t i c l e nature show a weak energy dependence. The d i f f r a c t i v e forward angle peaking which i s common to most pion production d i f f e r e n t i a l cross-section d i s t r i b u t i o n s was found to have exponential slopes consistent with production r a d i i of the single nucleon size rather than that of the nucleus. F i n a l l y , the results are compared to predictions of current t h e o r e t i c a l models of pion production. Failures of these models are discussed and suggestions for improvements offered. i i i Table of Contents Abstract 11 L i s t of Tables i v L i s t of Figures v I. Introduction 1 II. Experimental Setup 10 2.1 TRIUMF and Beamline IB 10 2.2 Beam Monitoring 12 2.3 The "Resolution" Spectrograph System 20 II I . Data Analysis 25 3.1 Background Discrimination 25 3.2 Focal Plane and Dispersion Relation 31 3.3 Lineshape 35 3.4 E f f e c t i v e S o l i d Angle 37 IV. Results 40 V. Trends i n the Data 57 5.1 Choice of Variables ....57 5.2 S t a t i s t i c a l Weighting 58 5.3 Incident Energy Dependence 63 5.4 D i f f r a c t i o n Peak Structure 86 5.5 n~ Production 89 VI. Status of the Theoretical Models 90 6.1 R e l a t i v i s t i c One Nucleon Model 90 6.2 Two Nucleon Model 92 6.3 Comparison of Models • 95 6.4 Comparisons of Models with Experiment 97 VII. Conclusion 104 References 108 i v L i s t of Tables I. Areal Thickness of Targets 24 II. D i f f e r e n t i a l Cross-sections and Analyzing Powers for  l0*G.*+) ll*gmB.t2ml2 „ e V 42 I I I . D i f f e r e n t i a l Cross-sections and Analyzing Powers for 9 B e ( | , l t V ° B e g < s > > 3 > 3 7 M e V 43 IV. D i f f e r e n t i a l Cross-sections and Analyzing Powers for 9Be ( S,O 1 0C g > 3 t >3. 3 4 M f i V 44 V L i s t of Figures 1. D i f f e r e n t i a l Cross-sections for a variety of (p ,Tt) reactions 3 2. (p,u~) reaction Spectra 5 3. One nucleon model 6 4. Two nucleon model 7 5. Beamline IB at TRIUMF 11 6. Geometry and Electronics of the f i r s t Polarlmeter 14 7. Geometry and Electronics of the second Polarlmeter 15 8. Ratio of — T — for the second Polarlmeter 19 A H 9. The "Resolution" Spectrograph 21 10. Electronics for the "Resolution" system 22 11. Energy loss D i s t r i b u t i o n 26 12. Time-of-flight D i s t r i b u t i o n 28 13. Standard Deviation Distributions 29 14. Y target Extrapolation Distributions 30 15. 1 0 B ( p , u + ) u B Energy Spectrum 33 16. 9Be(p,n +) 1 0Be Energy Spectrum 34 17. D i f f e r e n t i a l Cross-sections for 1 0 B(p.u"1")11 B„ _ 45 18. D i f f e r e n t i a l Cross-sections for 1 0 B ( p 1 B 2 ^ 1 2 M e v 4 6 19. D i f f e r e n t i a l Cross-sections for 9 Be(p , i t + ) 1 0 Be„ „ 47 g • s. 20. D i f f e r e n t i a l Cross-sections for 9 B e ( p , u + ) 1 0 B e 3 < 3 7 M e V 48 21. D i f f e r e n t i a l Cross-sections for 9Be(p,u~) 1 0C„ _ 49 22. D i f f e r e n t i a l Cross-sections f or 9 Be(p . T I - ) 1 0 C 3 > 3 4 M g V 50 vi 23. Analyzing powers for 1 0 B ( p , T t + ) L 1 B„ a 51 g. s • 24. Analyzing powers for 1 0 B ( p , i r + ) 1 1 Bj 1 2 MeV 5 2 25. Analyzing powers for 9 B e ( p , i t + ) 1 0 Be„ _ 53 g. s. 26. Analyzing powers for 9 Be(p,n +) 1 0 Beg 3 7 M e V 54 27. Analyzing powers for 9 Be(p , u ~ ) 1 0 C 55 g. s. 28. Analyzing powers for 9 Be(p,u~) 1 0 Cj ^ 3 4 M g V 56 29. Matrix Element for 1 0 B ( p , u + ) 1 1 B 59 30. Average Matrix Element per f i n a l state for 1 0 B ( p , u + ) 1 1 B 60 31. Matrix Element for 9 Be(p , u + ) 1 0 Be 61 32. Average Matrix Element per f i n a l state for 9Be(p,u +) 1 0Be 62 33. Matrix Element for 1 2 C ( p , u + ) 1 3 C 64 34. Average Matrix Element per f i n a l state for 1 2 C ( p , i t + ) 1 3 C 65 35. Forward angle |M|2 for 9 B e ( p , 7 i + ) 1 0 B e g # g > 66 36. Analyzing powers as a function of t for 9 Be(p,u +) 1 0Be„ 67 g * s. 37. Exponential slopes of the 9 B e ( p , n + ) 1 0 B e _ _ D i f f e r e n t i a l Cross-sections ..68 38. Forward angle |M|2 for 9 B e ( p , u + ) 1 0 Be3 > 3 7 M e V 70 39. Analyzing powers as a function of t for 9 B e ( p , T i + ) 1 0 B e 3 3 7 M g V 71 40. Exponential slopes of the 9 Be(p,u +) 1 0 Be3 37*jeV D i f f e r e n t i a l Cross-sections '.. 72 41. Forward angle | M | 2 for I 2 C ( p , n + ) 1 3 Q, > 5 M g V 73 42. Analyzing powers as a function of t for 1 2 C ( p > i t + ) 1 3 C 9 5 M g V 74 43. Exponential slopes of the 1 2 C ( p , n + ) 1 3 C g _ 5 M e V D i f f e r e n t i a l Cross-sections 75 v i i 4 4 . Forward angle |M| 2 for 1 0 BCp.u"*")11 BCT c 7 6 g • 5 • 4 5 . Analyzing powers as a function of t for 1 0B(p,u +) 1 1B t t _ 7 7 g • s • 4 6 . Exponential slopes of the 1 0B(p,u +) nB„ 0 Differential Cross-sections . . 7 8 g • s • 4 7 . Forward angle |M| 2 for 1 0 B (P,TI +) 1 1 % > 1 2 M G V 8 0 4 8 . Analyzing powers as a function of t for 1 0 B ( P , T C + ) 1 1 B 2 1 2 M E V 8 1 4 9 . Exponential slopes of the 1 0 B(p,u+)11 B2 1 2 M e V Differential Cross-sections ! 8 2 5 0 . Forward angle | M ] 2 for 1 2C(p , 7 i + ) 1 3 Ca E 8 3 g • s • 5 1 . Analyzing powers as a function of t for 1 2C(p,u +) 1 3C„ „ 8 4 g. s. 5 2 . Exponential slopes of the 1 2C(p,u +) 1 3C„ Differential Cross-sections . . 8 5 g • a « 5 3 . Relativistic 0 N M comparison with 1 2 C ( p , i i + ) 1 3 C g > s Differential Cross-sections 98 5 4 . Relativistic 0 N M comparison with 12C(p.u"1-)13Ca Analyzing Powers 9 9 g * s • 5 5 . Relativistic O N M comparison with 1 0 B(p,it+)11B Differential Cross-sections 1 0 1 5 6 . Relativistic O N M comparison with 1 0 B(p,it +) 1 1 B„ Analyzing Powers 1 0 2 g • s • 5 7 . T N M comparison with 1 2 C ( p , T i + ) 1 3 Cg > 5 M g V Differential Cross-sections . . . . 1 0 3 1 I. Introduction H i s t o r i c a l l y , perhaps hastened by the f i r s t simple model calculations, i t was expected that proton-induced pion production reactions, A(p,u+)A+1 would constitute a useful spectroscopic tool for the investigation of high momentum components of the nuclear wave functions. In i t s simplest terms, the (p,it +) cross-section i s d i r e c t l y proportional to the Fourier transform of the bound state wave function of the captured neutron. For example i n the plane wave Born approximation (PWBA) using a simple s t a t i c model for the pion production operator (a • involving the nucleon spin and the gradient of the pion f i e l d ) the (p,it +) cross-section from a spin-zero target nucleus i n the centre-of-mass (cm) system i s given by 1 da E p 3E.E. - 2 2 p n A A+l f ( 2 J + 1 } ^ ( q ) | 2 dQ 8u 2p (E.+E ) 2 m2 f n r p A p 11 where E and p are the t o t a l energy and momentum of the various p a r t i c l e s , f 2/4u = 0.08 i s the uN coupling constant, m^  i s the pion mass, i s the spin of the f i n a l nucleus, and <J>n(<l) Is the Fourier transform of the neutron bound - > ' •+ state wave function with respect to the momentum transfer, q = p - p . As a result, reactions of this type have attracted considerable attention over the past several years. The reviews of Hoistad, 1 Measday and M i l l e r , 2 and Fearing 3 give excellent summaries. By the late seventies, i t had become clear that this description of the (p,rc) process i s too simple and Eqn. (1-1) most certainly i s wrong. At the very least a correct description of the process would involve the correct 2 use and understanding of four basic ingredients: the pion emission operator, proton d i s t o r t i o n , pion d i s t o r t i o n , and nuclear wave functions. Each of these basic ingredients has an inherent uncertainty as to how important or how accurately i t must be understood. A number of questions must be asked. What is the basic production operator? Does i t involve more than one nucleon? How important are r e l a t i v i s t i c and r e c o i l corrections? How does the A-resonance manifest i t s e l f ? Are usual pion o p t i c a l potentials adequate to reproduce the pion wave function inside the nucleus where i t i s required? For both pions and protons, can one describe d i s t o r t i o n effects via average o p t i c a l potentials or are more e x p l i c i t calculations required? How e f f e c t i v e are d i s t o r t i o n effects in sharing the momentum transfer? How sensitive are results to details of the nuclear wave functions? Thus the major interest i n the pion production process has shifted from using It as a spectroscopic t o o l , to understanding the fundamental physics of the reaction mechanism i t s e l f . In this energy region the pion-nucleon vertex i s the dominant term i n the microscope strong i n t e r a c t i o n . Since the pion production reaction represents one of the simplest testing grounds for the understanding of this vertex, we are compelled to endeavour to understand i t . In order to appreciate the true richness, and therefore complexity, of the pion production process, one need only look at a selection of the angular distributions of the d i f f e r e n t i a l cross-sections. Figure 1 taken from Fearing's review 3 i s a good i l l u s t r a t i o n . The angular distr i b u t i o n s are very dependent on the nature of both the target nucleus and the states of the residual nucleus which are involved. Some f a l l rapidly with increasing angle, others are more or less f l a t , and some have a very complex angular dependence with several minima. The more the pion production process is studied the more aware we are of i t s complexity. 3 30 60 90 120 150 ev (deg) 160 Figure 1. Schematic representation of the angular d i s t r i b u t i o n data for a variety of (p,n) reactions. The curves correspond to: (a) 1 2 C ( p , * + ) 1 3 C g > g > , 200 MeV (b) "• 0Ca(p,u +)'* 1Ca g > s #, 200 MeV (c) 9 B e ( p , 7 i + ) l 0 B e 3 > 3 7 , 185 MeV (d) 9 B e ( p , n + ) l ° B e g > g > , 185 MeV (e) 1 6 0 ( p , n + ) 1 7 0 3 > 8 5 , 185 MeV (f) 9 B e ( p . O 1 0 C 3 > 3 w , 185 MeV (g) 1 3 C ( p , O 1 4 0 , 185 MeV (h) 9 B e ( p , O 1 0 C g > s > , 185 MeV. The "double-charge exchange" (p , T i ) reaction has also contributed i t s share of surprises. I n i t i a l l y , the d i f f e r e n t i a l cross-sections appeared to be r e l a t i v e l y i s o t r o p i c and small compared to that of the (p,ii +) reactions. This suppression of u production i s consistent with the PWBA model of Eqn. (1-1) since (p,n ) reactions cannot proceed unless a more complex process i s included. Measurements at IUCF1* (Indiana University Cyclotron F a c i l i t y ) discovered that (p,n ) reactions leading to certain high-spin stretched two-particle one-hole f i n a l states had cross-sections of the same order as the (p,u +) cross-sections. An i l l u s t r a t i o n of the excitation of such states i s shown i n Figure 2. One of the strongest of these states i s l l*C(p,it ) 1 5 0 7 3 M e V with a d i f f e r e n t i a l cross-section range of - 800 to 200 nb/sr over the angular d i s t r i b u t i o n . "* 8Ca(p,ii ) ^ T i ^ ^ MeV> h a S a r a n 8 e °f ~ 50 to 1 nb/sr. This selected sample of data obviously shows that the pion process has many facets to i t . The t h e o r e t i c a l approaches to this reaction can be divided into two classes; one nucleon mechanisms (0NM) and two nucleon mechanisms (TNM). This d i s t i n c t i o n refers to the use of the production operator (H. ) rather than to i n t the f u l l mechanism since i n fact almost a l l models are r e a l l y multi-nucleon. The ONM ( i l l u s t r a t e d i n Figure 3) uses the production operator once and interactions with other nucleons have been included i n some average way v i a o p t i c a l potential d i s t o r t i o n s . The TNM ( i l l u s t r a t e d i n Figure 4) i s constructed to involve two nucleons e x p l i c i t l y . That i s , the production operator i s used three times, making i t possible to involve the A-resonant interactions e x p l i c i t l y . Again other nucleons, i f included at a l l , are included only as an average e f f e c t . Both classes of models have advantages 5 Figure 2. Spectra for (p,n~) reaction on several targets at 9 i a ^ • 30° (28° for the l l 4 C target), showing selective excitation of one or a few low-lying states. igure 3. One nucleon model of plon production. 7 ~ \ - D \ / O Figure A. Two nucleon model of pion production. In the two nucleon case there are four'possible types of diagrams: ( a ) projectile post-emission (b) target post-emission (c) projectile pre-emission (d) target pre-emission. The spectator nucleons In the nucleus A have not been shown expl ic i t ly . 8 and disadvantages though neither of them can claim any true success. The most advanced model in each class w i l l be discussed i n Chapter VI along with some comparisons to the experimental data. Experimentally, much e f f o r t has been devoted to searching for systematic trends i n the data i n order to uncover clues for understanding the basic reaction mechanism. Now that much data exists, a number of general features can be extracted but as yet no complete understanding of the reaction has been developed. In the case of the simplest of nuclei (hydrogen), there has been some success. One technique, introduced by the Helsinki group, ( p r i n c i p a l l y Green and Niskanen 2 8) involved the development of a coupled-channels technique. Such coupling of the NN and NA channels included both n and p exchange i m p l i c i t l y . For the f i r s t time, theory was able to provide p r e d i c t i o n s 2 9 which compared well with experimental measurement (not only of the d i f f e r e n t i a l and t o t a l cross-sections, but also of a variety of polarization observables). Unfortunately this success does not continue to the many nucleon problem. Thus the purpose of this thesis i s to present i n a systematic way the data obtained from two major sets of experiments and to point out new trends suggested by this data. These new trends not only give dire c t i o n to further experimental work, but point out as well new theo r e t i c a l approaches that should be investigated. The thesis begins with a br i e f description of the experimental setup and data analysis (which has been described elsewhere 5). Next the angular distributions of both the d i f f e r e n t i a l cross-sections and analyzing powers are presented for the following reactions: 9 g.s., 2.12 MeV 9Be(p,u +) 1 0Be g-s., 3.37 MeV 9Be(p,n")l°C 3.34 MeV for incident proton energies ranging from 200 to 260 MeV. Chapter V continues with suggested trends in the nuclear data i n this energy range and points out new general directions for both experimental and theoretical work. Chapter VI looks at the status of the theoretical approaches, and Chapter VII concludes the thesis with a summary and discussion of some s p e c i f i c recommendations. 10 II. Experimental Setup 2.1 TRIUMF and Beamllne IB The TRIUMF6 f a c i l i t y has both polarized and unpolarlzed Ion sources which the cyclotron can accelerate to an energy range from 200 to 520 MeV. The maximum beam Intensity available depends on the energy as well as the type of Ion source. For example, at maximum energy a current of 140 uA (unpolarlzed) or about 500 nA (polarized) can be extracted. A special feature of the TRIUMF design is the acceleration of H ions. Extraction of a proton beam Is thus accomplished by Intercepting the negative ions with a thin f o i l , which strips the two electrons from the H ion leaving protons which then curve out of the cyclotron f i e l d . The proton beam Is periodic and consists of pulses of roughly 2 nsec duration occurring every 43 nsec. The separation of the pulses corresponds to the period of the accelerating radio frequency power (RF) which Is the f i f t h harmonic of the cyclotron resonance frequency. The "Resolution" spectrograph system5 used In these experiments was situated on beamllne IB, illustrated In Figure 5. By the use of dipole and quadrupole magnetic elements the proton beam was transported through the beamllne to the target location (1BT1). At each beam energy special care was taken in tuning the elements of the beam line and the extraction parameters of the cyclotron. A low background in the experimental area had to be maintained in order for multiwlre proportional chambers at the exit of the spectrograph to operate satisfactorily, as there was only limited local shielding around the focal plane array. With a poor beam tune, there was sufficient beam loss along the line that the background F i g u r e 5. Beamline IB a t TRIUMF. 12 in the experimental area Increased to the point where the chambers could not handle the singles rates (even without a target at 1BT1). Rather than constructing the massive shielding that would be needed to protect the chambers, high quality beam tunes were developed and special procedures were followed to handle them. First the cyclotron was maintained at a good tune such that the extracted beam was monoenergetic with very l i t t l e halo. Next using monitors for indicating the position and profile of the beam at various points along the beamline, special care was taken to steer the beam down the center of the beamline. The readings of the s p i l l monitors along the outside of the beamline were kept at a minimum. In addition, the profile and position of the beam was monitored regularly either by remotely viewing a scintillating target with a video monitor or measuring the profile with a wire chamber that could be inserted at the target location. A small beam spot (~ 5x2 mm) was maintained at 1BT1. Another important reason for maintaining the good tune was the effect of beam misalignment on the polarization measurements. In order to obtain a correct measure of the beam polarization, i t was necessary to keep the beam centered in position and parallel while passing through the polarimeter. Such instrumental asymmetries recorded for unpolarized beam were monitored at five minute intervals to assure that no changes occurred in the beam tune. 2.2 Beam Monitoring For monitoring both the intensity and polarization of the incident proton beam a four-arm polarimeter was situated immediately downstream from the beam dipole 1BVB2 (see Figure 5). Two different polarimeters were employed during the course of these experiments. The f i r s t 7 was used for a l l the 9Be work and 13 a small amount of the 1 U B . A schematic of the geometry of the polarlmeter together with Its electronics i s shown i n Figure 6. The second polarlmeter 5 used for most of the 1 0 B work Is shown i n Figure 7. Both polarimeters were based on monitoring pp e l a s t i c scattering with the target protons provided by the hydrogen i n a 5 mg/cm2 thin CH2 (polyethylene) target. Background events from the carbon were discriminated against by coincidence detection of both the scattered protons in a two-arm system situated at the angles appropriate for free pp scattering. Nevertheless, a s i g n i f i c a n t background ( t y p i c a l l y 5%) from quasi-elastic scattering of protons within the carbon had to be taken into account. For the f i r s t polarlmeter a count corresponded to the coincident detection of a proton scattered at 26° to the right ( l e f t ) with respect to the beam dire c t i o n together with a backward scattered proton detected at 60° to the l e f t ( r i g h t ) . The corresponding angles for the second polarlmeter were 17° and 71.3° (respectively) to both f a c i l i t a t e a more r i g i d support structure as well as to Increase the pp-elastic l e f t - r i g h t asymmetry by ~ 15%. If a polarized proton beam i s incident upon an unpolarlzed target, the d i f f e r e n t i a l cross-section da/dQ can be written i n terms of unpolarlzed and polarized components, that Is; da da -> •*• da ir = i r + p * n *r da where: — — i s the unpolarlzed d i f f e r e n t i a l cross-section. au da i s the polarized d i f f e r e n t i a l cross-section. Ik Recoil P (right) \ ' Fwd. P(left) Figure 6. A schematic of the geometry and electronics of the f i r s t polarimeter (used for the 9 Be work). 15 Figure 7. A schematic of the geometry and electronics of the second polarimeter (used for the 1 0 B work). 16 -»• P i s the i n c i d e n t p r o t o n beam p o l a r i z a t i o n , and n i s a u n i t v e c t o r normal t o the s c a t t e r i n g p l a n e i n the d i r e c t i o n * 1t f ( t h e Madison C o n v e n t i o n 1 6 ) . do do da T h e r e f o r e' i r = d r + p 4 r (2-2) da„ da da 3 R o _ p , _ „. where: ^ — i s the d i f f e r e n t i a l c r o s s - s e c t i o n of the l e f t s c a t t e r e d p r o t o n s . d°R i s the d i f f e r e n t i a l c r o s s - s e c t i o n of the r i g h t s c a t t e r e d p r o t o n s . dQ and P i s the magnitude of the i n c i d e n t p r o t o n beam p o l a r i z a t i o n . Adding Eqns. (2-2) and (2-3) l e a v e s da . da. d a D Thus, the sum of a l l l e f t s c a t t e r e d (L) and r i g h t s c a t t e r e d (R) counts i s independent of the p o l a r i z a t i o n of the beam, and t h e r e f o r e p r o v i d e s a measure 17 of the beam current. The c a l i b r a t i o n for the f i r s t polarlmeter 7 for the sum of L and R counts per nanocoulomb was: L + - R = 106.7 + 9.95 x IO - 2 T (2-5) nC p and for the second polarlmeter 5 L * R = 235.7 + 0.128 T + 4.0 x 10 _ 4T 2 (2-6) nC p p where T^ Is the incident proton energy i n MeV and the CH2 target thickness i s 5.0 mg/cm2 i n each case. The systematic uncertainty associated with both these f i t s to the measured values i s ~ ± 5%. The c a l i b r a t i o n of the e f f e c t i v e s o l i d angle of the spectrograph (Sec. 3.4) was accomplished by using these same polarimeters. In this way any absolute normalization of the beam current cancelled since a l l measurements were r e l a t i v e to the known1-1*,l5 pp •*• d u + cross-sections. The Incident proton beam polariz a t i o n , P, was obtained using Eqns. (2-2) and (2-3) along with the d e f i n i t i o n of the analyzing power. do do daL d°R Assuming L and R are d i r e c t l y proportional to ^ — and , respectively, 18 Because of the presence of quasi-elastic scattering from the carbon, a correction was made to the analyzing power. Therefore, P = (2-9) As more pp-elastic data has recently become available, the values for A^^ used i n Refs. 5 and 7 are now out of date. A phase s h i f t analysis using the most recent data (SAID 8 data set SM84) was used to define A^. In the case of the f i r s t polarimeter, the carbon component was corrected for as i n Ref. 9. The least squares f i t to the A^^ i s : A„„ = 8.38 x IO - 2 + 9.74 x 10^T - 7.29 x 10 _ 7T 2 (2-10) p P with a systematic uncertainty of ~ ± 5%. T^ i s again the incident proton energy i n MeV. In the case of the second polarimeter, the ratios of A^ H /A^ were measured and are shown In Figure 8 ( r e f . 10). The least squares f i t to this r a t i o i s : A CH T T T T~ = 1-01 + ( i ^ O 3 6 - (TJO)('°36 " (TOO-) - 0 0 4 8 7)] <2-L1> whereas the least squares f i t to the best A^ (at this time) i s : A - -5.82 x 10 - 3 + 1.92 x 10 - 3T - 1.85 x l O ^ T 2 H p p (2-12) 1 9 20 The o v e r a l l systematic uncertainty i n A i s estimated as ~ ± 2% for the proton energy range of interest i n this study. 2.3 The "Resolution" Spectrograph System The basic instrument was a 65.0 cm Browne-Buechner11 magnetic spectrograph. The p a r t i c l e detection system consisted of a counter teles.cope composed of three s c i n t i l l a t i o n counters together with three h e l i c a l l y wound delay line multiwire proportional chambers (MWPC).12 Helium boxes were inserted between the three MWPC's to reduce multiple scattering within the chamber system. The layout of the spectrograph system i s shown i n Figure 9 with i t s associated electronics in Figure 10. A detailed description of the experimental arrangement can be found in Ref. 5. The three s c i n t i l l a t i o n counters (CE, C l , C2) provided the event d e f i n i t i o n as well as timing and energy loss information. The MWPC's (multiwire proportional chambers), on the other hand, provided p o s i t i o n information from which the exit trajectories of each p a r t i c l e were determined. The intersection of the p a r t i c l e t r a j e c t o r i e s with the focal plane determined the momentum of the p a r t i c l e . The coordinate system used i s shown i n Figure 9. Z i s the v e r t i c a l , X i s horizontal i n the bend plane, and Y i s horizontal i n the non-bend plane. Thus the X position information was collected by the h e l i c a l cathode of the MWPC's and the Y position from the anode wires which were connected to the delay l i n e . The difference i n time for a pulse to travel to the two ends of the h e l i c a l cathode (or the delay l i n e ) was d i r e c t l y proportional to the X (or Y) position. The timing " s t a r t " of a l l the TDC's (time-to-digital converters) was triggered by the output of the mean-timer of 21 MWPC 3 He BOX 2 MWPC 2 He BOX I MWPC I CE 3cm Pb SHIELD TARGET DRIVE, 65cm MAGNET WITH VACUUM BOX MAGNET TROLLEY P BEAM scoie:-t 40 cm. t Figure 9. The "Resolution" Spectrograph. 22 CIL CIR C2L C2R ® n _ L ® L L ®=L_L ®-L_L ® n M JJ TRUE RAND EVENT EVENT' s)—ZL OR. ay IT) START STROBE f, | 4 0 n t SIGN CONFIGURATION OFF DOWN UP BUSt r COMPUTER BUSY M.W. P. C . L O G I C IKI I K 2 IAI I A 2 2 K I 2 K 2 2 A I 2 A 2 3 K I 3 K 2 3 A I 3 A 2 R F L I ! © © © © © © © © © 5 © © © DISCRIMINATOR LRS 621 TYPE MT )M E A N T | M E R LRS 624 DELAY CAMAC S ^ ) KINETIC SCALER S 3615 ^L-CAMAC ADC V*JLRS 2249 , COINCIDENCE (2 FOLD) LRS 622 JILL COINCIDENCE \TJ <4 FOLD) III LRS 465 LOGIC LRS 4 2 9 •C}. PATTERN UNIT ^ E G G C2I2 6 CAMAC TDC LRS 2228 Figure 10. The overall electronics for the "Resolution" system. 23 Cl (see Figure 10). This time difference (position information) unit of a TDC bin represents 0.2 nsec which corresponds to ~ 0.55 mm. P a r a l l e l event-defining logic (see Figure 10) was u t i l i z e d providing both true and random event d e f i n i t i o n s . The timing gate of CE was wide enough to allow for both true and random event coincidences. The random events proved to be i n s i g n i f i c a n t ; much less than 1% of the true events for any measurement. The true event coincidence, though, consisted of a mixture of electrons (or positrons), protons and pions. The energy loss and time-of-flight information allowed for the elimination of the protons and electron-like events. Tests on the c o - l i n e a r i t y of the p a r t i c l e ' s path through the chamber system and extrapolated target position were also applied (Sec. 3.1). The angular range accessible to the measurements was r e s t r i c t e d to 46" -135° due to the geometry of the magnet yoke. In addition, pion k i n e t i c energies could be measured over the range of 30 to 110 MeV. The lower l i m i t was defined by the energy loss of the pion t r a v e l l i n g through the system. Pions of less than 30 MeV had Inadequate penetrating power to traverse the whole system. The upper l i m i t of 110 MeV was caused by the maximum magnetic f i e l d strength achievable. The magnetic f i e l d was monitored by a nuclear magnet resonance magnetometer (NMR)13 positioned i n the magnet gap (see Figure 9). The f i e l d varied by less than IO - 4 tesla over the time of any run. The targets for this system were mounted at either 45° or 135° with respect to the incoming beam depending on whether the spectrograph was at a forward or backward scattering angle. The areal thickness of a l l targets used were measured to an accuracy of 1%. The actual target thicknesses are given i n Table I. 24 Table [ Areal Thickness of Targets Target Areal Thickness Scattering (mg/cm2 ) Angles 9 Be 100.8 Forward 46.8 Backward 1 0B(92% enriched) 99.3 Forward 101.6 Forward 92.3 Backward 300.0 Forward 100.0 Backward 25 I I I . Data Analysis 3.1 Background Discrimination During data taking, the p a r t i c l e s which triggered the event coincidence (and thus traversed the f o c a l plane) consisted of a mixture of electrons (or positrons), protons and pions. The f i n a l elimination of the background protons and electrons was accomplished by subjecting each event to a series of tests. A detailed description of this process i s described i n Ref. 5. There were bas i c a l l y four tests applied to each event: energy loss, time-of-flight, c o - l i n e a r l t y , and target extrapolation. The pulse heights from the three s c i n t i l l a t i o n counters provided the energy loss information. Separation of pions from protons was proven possible at a l l pion energies of i n t e r e s t . The background protons were predominantly low-energy multiply-scattered protons of large dE/dx. The separation of pions and electron-like events improved s l i g h t l y with decreasing pion energy. Figure 11 i l l u s t r a t e s a t y p i c a l energy loss d i s t r i b u t i o n . The protons are actually off the right-hand side of the plot while the electrons constitute the f i r s t peak. Clearly energy-loss, while adequate to separate the protons, was inadequate for electrons. The counters also provided tlme-of-flight measurements. The timing " s t a r t " was triggered by the output of the mean-timer of C l (see Figure 10). Since the path distances between CE, C l , and C2 were small, the time-of-flight separation associated with the C2 signal or the appropriately delayed CE signal was too small to be useful. The time i n t e r v a l between C l and the cyclotron RF was the measure of the time from a r r i v a l of the beam proton at the production target to the reaction p a r t i c l e traversing C l . Thus the raw RF 26 fiiifffisiiiiifiiiiifiiiiiiicisiiiii*52" Figure 11. A typical energy loss distribution. The cut to reject the protons is shown. 27 defined time-of-flight d i s t r i b u t i o n (shown i n Figure 12) showed the largest separation between pions and electron-like events. Because of the p e r i o d i c i t y of the RF timing signal (43 nsec), these time differences are not unique, with "wrap-around" di s t r i b u t i o n s a r i s i n g 5 (see, for example, the electron contribution at each end of the 43 nsec window shown i n Figure 12). The c o - l i n e a r l t y test determined the extent to which the p a r t i c l e motion through the three MWPC's constituted a straight l i n e trajectory, thus discriminating against pion decay or multiply-scattered events occurring within the chamber system. The c o - l i n e a r l t y test was performed by accumulating d i s t r i b u t i o n s of both the X and Y standard deviations characterizing straight l i n e least squares f i t s through the chambers on an event-by-event basis. Figure 13 i l l u s t r a t e s the t y p i c a l standard deviation d i s t r i b u t i o n s . Since there i s no focussing i n the Y direction (see Figure 9), except for small edge focussing e f f e c t s , an extrapolation to the Y position at the target was readily performed. By i n s i s t i n g that the p a r t i c l e s originate in the illuminated region of the target, some of the p a r t i c l e s a r i s i n g from pion decay or those which had been multiply-scattered somewhere on route to the focal plane detectors could be distinguished. A t y p i c a l Y target d i s t r i b u t i o n i s shown i n Figure 14. The events i n the central peak correspond to "undisturbed" t r a j e c t o r i e s , whereas those in the t a i l s are associated with multiply-scattered or decayed events. The cuts on this test were chosen to give the best momentum resolution at the focal plane. For example, in the case of 9Be(p,ix +) 1 0Be, the peak to valley r a t i o for the ground state peak went from ~ 7 with no cuts to ~ 15 with the Y target cuts. The f o c a l plane spectrum of pions taken from the t a i l region of the Y target d i s t r i b u t i o n 28 Figure 12. A t y p i c a l RF defined time-of-flight d i s t r i b u t i o n . The cuts to reject the electrons and protons are shown. 29 19*0 1 3 « 0 1 3 9 0 > l * 0 t>oo XViO I M C 1 » 3 0 I M O •too 1 T 4 0 m o l « 3 0 I H O 1 » O 0 1 4 4 0 I M O 1 1 3 0 I 3 » 0 1 3 O 0 1 1 4 0 •ooo 1 0 1 0 •oo •oo • ' 0 T » 0 T 3 0 M O • O O H O « » 0 4 3 0 M O 1 0 0 1 4 0 O O oo • 0 (a) I M I t l I I I I I * * I I I ! T i l l 9 I I I I I H U M • I I I I I I I I I I I I I I I I I I I I I I I I • I I I I I I I I I I I I I t • I I I I I I I I I I I I I I I } I I I I I I I I I I I I K I I I I I I I I I I I I I I T I I I I I I I I I I I I I I I I I I I I } I I I I I I I I I I I I I I I I I I I I I I K I I I I I I I I I I 4 I I I I I I I I I I I I I I I ! I l l l H I I t I l l l l l l l l l l l l t ("-3mm) Typical Cut m « t < i I I I I I I I ; : I I I I I I I I K I M C 3 4 3 I I I I 1 I I I I I I I I I I I I I 4 B I f 1 I 1 I I 1 I I I I I I I I I I 4 m i n t X I I I I I I I I I i i i i i i i i i i i i i i i n t I I I I I I I I I I I I I I I I I I I I I I I I I I I I IIIIIIKMC34 I I I I I I I I I I I I I I I I I I I I I I 19**TMT«T4ST«4M4t4$4943S4MS34443594944344444»4930«3S<«4344S44444 n H H 1 I H 1 f l U t 1 1 1 H 1 t 1 1 « I I I H I H M I M n i H l i n n *(1lt33]373>l)34444«99S4MMMYTTTTtttB«n*t>OOOOOl (1'1273733333344444TTTTS8BSI99999 03<W7'(«034f^7<«BC?4«^!«t»QJ*t4^ T D C BINS 1 1 3 0 X > 4 0 1 X 0 3 1 1 0 JlOO 3 7 3 0 3 « 4 0 1S«0 34to 3 4 0 0 3 3 3 0 1 3 4 0 1<«0 3O4J0 3 O 0 0 1 1 3 0 • • 4 0 I T « 0 '•to I I O O 1 » 3 0 1 4 4 0 1 1 * 0 1 3 > 0 1 3 O 0 1 1 3 0 I O 4 0 M O M O • 0 0 T 3 0 • 4 0 MO 4*0 4 0 0 1JO 140 1 * 0 ( b ) • 4 • •• I I I III« I • 11 • H I T I I I I I I I I I I I I I I I 4 • I I I I I I I I I I I I I I I I I I I I I I 1 1 I I I I I I I I I I I I I I • I I I I I I ? I I I I I I I I I I I I I I I I I I I I I I I I I I I I I H I t I I I I I I I I I I I I I I I I I I 4 I I I I I I I I I I I I I I I I I I I I I • I I I I I I H I I t 111111111111 • I I I I I I I I I I I I I I I I I I H I I I I I t I I I I I I I I I I I H I 3 (~ 5 mm) Typical Cut I I I I I I I I I I I H I 3 1 - ' ' I i n i i i l i i i t i i i i 9 I I I I I I I I I I I I I I I I I H 5 9 I • i I I I I i I i I I i i I I i i i i i 9 « « C T 4 M ' 3 3 t 4 43 M i l ) 113 11 1 i i i i i i i i ( i i n i i i i i i i i i i i i * i i i i i i i i i f n i i t i i i i i i i | i | i i i | — i - . - i i |.. IIIIIII141tIM9IIMtl»St8«T?atC(7«<4C94 444 4S594344 3333 3 3 33 3 11111111111111111111111113333333333333333333333)333333333333333333333333333 0 0 0 1 t33333<4<»c:<11Mlt*000< I111114445MMT7IMMOO01 I 333334 J <55111' T• 1999OO01 13333344<S»(<TT|tl99 0-11 ?•<>«• 3 1 0 4 a 3 ( C M • t o n ?to<a :to< • ?*o*a ?»o*a :ao ia a : « C M a 7 t Q 4 i : ( O J a?fr04a;to«a ; K M I ;tc*4 a;c TDC BINS Figure 13. A t y p i c a l (a) X and (b) Y standard deviation d i s t r i b u t i o n along with the t y p i c a l cut. 30 II' 3 o o o a. >» Si :- §1 * * », * • » m | t o 5 z 6 5 I o tf h • k «• • i i i : ! T r M i i i • 2 u = i 2 - tr Figure 14. A t y p i c a l Y target extrapolation d i s t r i b u t i o n along with the t y p i c a l cuts. 31 (outside the cut lev e l ) showed very l i t t l e structure, suggesting that mostly "disturbed" t r a j e c t o r i e s were being rejected. An estimate of the probability of the pion surviving a l l these tests was determined assuming they were independent of each other, and monitoring the effect of removing them one at a time. The estimate was approximately 70%. For the energy loss and time-of-flight tests, where the tests applied to the same p a r t i c l e s , this method i s reasonable. But i n the case of the co- l l n e a r i t y or especially the target extrapolation tests, where they are neither independent nor redundant, this method i s less j u s t i f i a b l e . A more detailed accounting of this s u r v i v a l probability was accomplished by lncludln the effect In the d e f i n i t i o n of the effe c t i v e s o l i d angle. 5 The i d e n t i c a l + test conditions were applied to the pp du data used to calibrate the effective s o l i d angle (Sec. 3.4). 3.2 Focal Plane and Dispersion Relation After selecting the good events by means of the cuts described i n Sectio 3.1, the momentum of each remaining plon was determined from i t s position on the focal plane (XFP) using the dispersion r e l a t i o n for the spectrograph. This equation relates the pion momentum to the measured XFP and the magnetic f i e l d of the magnet. The XFP was determined by the X component of the intercept of the pion trajectory with the foca l plane. The trajectory of the p a r t i c l e was determined from the MWPC position Information. The char a c t e r i s t i c s of the focal plane are described i n Ref. 5 and are summarized here. The plane i s defined by the equation 32 z a - bX (3-1) where a = 700 ± 20 mm b = 0.94 ± .03 mm/bin Z i s v e r t i c a l and X i s horizontal i n the bend plane (see Figure 9). The v e r t i c a l height was measured from the middle of MWPC1 (see Figure 9). The determination of the dispersion r e l a t i o n was also described i n Ref. 5. The relationship obtained i s : where the XFP are i n TDC time units. The momentum, P, i s in MeV/c whereas the magnetic f i e l d , B, i s i n t e s l a . The k i n e t i c energy of each p a r t i c l e was obtained from Eqn. (3.2) using the usual kinematic r e l a t i o n : • where m^  i s the mass of the pion, 139.57 MeV/c2. This was done in order to depict a pion spectrum that more closely resembles the excitation of the f i n a l nucleus. Figures 15 and 16 i l l u s t r a t e this technique with two such spectra. (3-2) 33 oo S1ND0D Figure 15. 1 0 B ( p , n + ) 1 1 B energy spectrum of n + produced at 50 ° m ftom 225 MeV incident protons with spin down.16 Lineshape f i t s for the f i r s t two states are shown by the s o l i d l i n e . 3^ O'O'O — o o ID S 1 N D 0 D Figure 16. 9Be(p,u ) 1 0Be energy spectrum of n produced at 50° from 200 MeV incident protons with mixed spin. Lineshape f i t s for the f i r s t t w o states are shown by the s o l i d l i n e . 35 3.3 Lineshape The c o - l i n e a r i t y and target extrapolation tests did not eliminate a l l the pions that had suffered s i g n i f i c a n t multiple scattering or that had decayed into muons. Such pions tended to produce a " t a i l " i n the momentum di s t r i b u t i o n of a single l i n e and proved to be an effect that had to be accounted for. These effects have been described i n Ref. 5 where both an experimental measurement and a Monte-Carlo simulation were completed for the pp •*• du + l i n e . The pole-face scattering i n the spectrograph was shown to be the major contributor to these " t a i l s " . The k i n e t i c energy spectra of the pp •*• d u + l i n e were best f i t 5 by the analytic form: The t o t a l number i n the lineshape was normalized to unity. The f i r s t term, a Gaussian type, characterized the peak component of the spectrum, whereas the second term, the exponential decay (with a Fermi-type cut off at the peak position, T^ = B) was used to describe the t a i l component. These parameters are energy dependent and could be expressed in terms of B, the centroid of the peak (MeV) by the following: 5 = A e (T -B)/F e it (3-4) 1 + . < V B > ' 6 D = (140 ± 6)/B 2 (3-5) F = (-.93 ± .40) + (.083 ± .007)B (3-6) G = (.13 ± .12) + (.019 ± .002)B (3-7) 36 The peak component parameters B and C were l e f t free when f i t t i n g the experimental energy d i s t r i b u t i o n s since B i s different for each line observed and C (related to the line's width) i s dependent on both the energy spread of the incoming beam and the kinematic broadening due to the r e c o i l of l i g h t targets. Parameter A was fixed such that the proper r a t i o of peak to t a i l components was maintained. Since the amplitude of the peak (A) must be allowed to vary as i t s width (C) changes, A was fixed by the following method. Since the lineshape i s normalized to unity, i . e . / F(T )dT = 1 (3-8) — oo Then substitution of Eqn. (3-4) along with the representation of x = T - B, u y i e l d s : oo 2 I 0 0 x/F / A e _ X dx +/ 5? dx . 1 (3_9) — oo — oo i + e The f i r s t term i s just a Gaussian and the second the area of the t a i l component (A,,,). Therefore Eqn. (3-9) becomes; A /Cu + (A T) = 1 (3-10) or 1 " (A T) /CTT" (3-11) A^ , was obtained by numerical Integration of the t a i l component (as defined by the appropriate D, F, and ,G parameters) for any given pion energy. A least 37 squares f i t to A^ , gave (.34 ± .02) - (2.4 ± .2) x 10 - 3 B (3-12) Thus the final shape of any line is dependent on only two parameters the centroid (B) and the width (C). Any pion kinetic energy spectrum consisting of a number of discrete lines can be f i t by: where M is the number of lines in the spectrum S, is the number of events in the ith line, and is the lineshape defined by Eqns. (3-4), (3-5), (3-6), (3-7), (3-11) and (3-12) with two parameters B^^ (centroid) and C (width) for any of the i lines. Since the width of a l l lines of a given spectrum was dominated by the energy spread of the incoming beam, C was forced to have the same value for a l l M lines. The' solid curves of Figures 15 and 16 illustrate the typical quality of such f i t s . Since only the f i r s t two states were treated in this work (well away from the continuum associated with three or more particles in the final state), no continuum correction to Eqn. (3-13) was needed. Typical values of the reduced x 2 ( x 2 P e r degree of freedom) for these fi t s ranged from .8 to 2.0. M S ( V = E N 1P 1(B 1,C) 1=1 (3-13) 3.4 Effective Solid Angle The effective solid angle of the spectrograph, AQ , is not just the 38 geometrical solid angle, but includes a l l the effects of decay, multiple scattering, and as well the effects of the cuts on the data. Depending on the energy of the pions as many as 20% could have decayed of which only a few percent of the decay muons ended up in the final spectrum. The " t a i l " component of a line (mainly multiply scattered pions) ranged from 5 to 10% of its total area. The efficiency of the cuts was estimated as approximately 70% (Sec. 3.1). Since a l l these effects are interdependent, i t proved to be more accurate and convenient to include a l l these inefficiencies in an energy dependent effective solid angle. The calculation of AQ is described in Ref. e 5 so only a brief summary is given here. The calibration of the AQg was performed by comparing the measured pp -+ drt+ to the known*1*'1,5 cross-sections. That is N AQe - N — (3-14) e N ( — ) Pt *2_ p MVT |cos9 | dQ where: is the number of pions defined by the lineshape f i t (Eqn.(3-13)) applied to pions from the pp * dn + reaction. ^ is the number of incident protons (Eqn. (2-5)) r "\ P t ["nrrJ i Q i Is the number of scattering centers in the target where: v MW; | c o s 9 t ( ° 6 NQ Is Avogadro's number n is the number of scattering centers per molecule MW is the molecular weight of the target material pt is the areal thickness of the target in mg/cm2 9 is the target angle with respect to the incoming beam 39 e is Che efficiency of the MWPC's which varied from one individual data collection run to another, thus making i t impossible to include in the effective solid angle (this efficiency was typically 60 to 70%) and ^2-are the known11*'15 differential cross-sections for the pp-*-du+ reaction. This calibration was completed at three pion kinetic energies; 50, 70, and 100 MeV. The uncertainty of each calibrated effective solid angle was ~ ± 5%, caused mainly by a combination of systematic uncertainties in the pp+du+ cross-sections together with uncertainties of the absolute beam current normalization. Upon examination of the effective solid angle values, the best f i t was given by a straight line. The least squares f i t to these three values yielded the following energy dependence for AQg: AQ = (1.15 ± .15) + (5.12 ± 2.0) x 10"3 T (3-15) e u where AQ is in msr and T is in MeV. Thus the systematic uncertainty of any e TI interpolated value of AQ Is ~ ± 15%. 40 IV. Results The analyzing power, A N Q ( Q ) t and the spin-averaged (unpolarized) differential cross-section, (9), were calculated using the relations: A (Q) - dg(+)/dB - d c ( + ) / d a and da ... P(t)da(0/dQ + P(Oda(+)/dQ dQ v ' P(t) + P(+) (4-2) where P is the magnitude of the beam polarization (Eqn. (2-9)) and ^  is the dQ spin-dependent differential cross-section. The arrows indicate the spin direction according to the Madison convention. 1 6 As in Eqn. (3-14), the spin-dependent differential cross-sections were defined as: A N d ° " % — (4-3) dQ NQH E N (-^rr) i ^l-T-AQ p x MW |cos9 I e where is the number of events from the f i t of the spin-dependent spectrum. The areal thicknesses (pt) are given in Table I. 9 is either 45° or 135° (Sec. 2.3), thus |cos9j is 1// 2 . The effective solid angles (AQ ) are defined by Eqn. (3-15). In the case of the 1 0B target (i.e. targets enriched to 92% 1 0B), the background due to the 8% contamination of 1 1B was determined by also collecting data with the 1 1B targets under identical conditions to the 1 0B runs. For the results presented here, where only the ground and f i r s t excited kl states are considered, the 1 1B backgrounds in this region was found to contribute less than 1% to the two states for a l l measurements. The final results are tabulated in Tables II, III, and IV and shown in 17 18 Figures 17 through 28. The available data from IUCF ' and previous TRIUMF data7 for this energy range are also shown. The absolute normalization of a l l the sets of data agree within 10% (well within the systematic errors). Preliminary results of this data have already.been published. 3 4' 3 5 As well, the final results of the 1 0B(p,u +)HB reaction have recently been published. 3 6 Only the relative uncertainties are indicated in the tables and figures. In addition, there is an overall systematic uncertainty of ~ ±15% for the differential cross-sections. For the analyzing powers of 1 0 B ( p , T t + ) 1 ^ B , there is ~ ±2% systematic uncertainty, whereas for 9Be(p,u +) 1 °Be and 9Be(p,u ) 1 0C, i t is ~ ±5%. The relative error consists of both the counting statistics and the random fluctuation in the beam current measurements (mainly due to the wrinkling of the thin polarlmeter targets 7). The majority of the systematic uncertainty in the differential cross-section arises from the uncertainty in the calibration of the effective solid angle of the spectrograph (Sec. 3.4). The systematic uncertainty of the analyzing powers Is due to the uncertainty in the analyzing powers of the polarimeters (Sec. 2.2). k2 TABLE II ( M e V ) 10 • ) l l B , 6 c m ( D e g . ) ( n b / s ? 5 ^ 0 10 ( D e g . ) (nb / sH ^ 0 200 49.8 64.6 74.9 85.2 95.2 110.0 124.5 138.6 471.(30.) 339.(22.) 196.(13.) 91.4(6.6) 94.7(6.5) 50.7(3.5) 56.7(3.8) 47.3(3.2) -O.222(.032) -0.372(.034) -O.475(.032) -0.459(.046) -0.329(.041) -0.19K.040) -0.189(.036) -0.187(.039) 49.9 64.6 75.0 85.2 95.3 110.1 124.5 138.7 130.0(9.8) 104.3(8.1) 55.6(4.3) 38.5(3.3) 36.1(2.9) 25.5(1.9) 19.9(1.5) 18.1(1.4) -O.45K.059) -0.428(.061) -0.694C.056) -0.607(.067) -0.586(.061) -0.533(.051) -0.379C.059) -0.468C.060) 225 49.8 59.3 64.4 79.9 87.1 95.0 109.8 124.3 138.5 593.(37.) 348.(27.) 285.(18.) 122.8(8.1) 98.2(7.8) 75.0(4.9) 46.9(3.2) 30.0(2.0) 42.9(2.9) -0.305C.018) -O.252C.058) -0.368C.024) -0.095C.036) O.OO0C.O73) -O.04K.035) 0.053C.042) -O.094C.043) -0.327(.043) 49.8 59.3 64.4 79.9 87.1 95.1 109.9 124.3 138.5 172.(11.) 8 4 . 9 ( 9 . 6 ) 9 0 . 5 ( 6 . 1 ) 51.6(3.8) 5 5 . 1 ( 5 . 1 ) 3 9 . 5 ( 2 . 8 ) 3 2 . 4 ( 2 . 3 ) 19.2(1.4) 34.1C2.4) -0.432C.033) - 0 . 6 3 C . 1 0 ) -0.473C.042) -0.270C.054) -0.435C.094) -O.318C.048) -0.218C.050) - 0 . 2 9 5 C . 0 5 3 ) -0.457C.047) 250 49.6 57.0 64.4 74.7 84.9 95.0 109.8 124.2 130.4 138.5 539.(33.) 376.(24.) 257.(16.) 132.0(8.5) 73.2(5.9) 38.2(2.5) 14.2(1.4) 14.1(1.2) 12.1(1.2) 14.2(1.2) -0.02K.021) -0.029(.024) -0.004(.023) 0.082C.031) 0.287(.066) 0.534(.034) 0.386C.099) -0.289C.086) -0.505C.087) -O.506C.075) 49.7 57.1 64.4 74.7 84.9 95.0 109.8 124.3 130.5 138.5 114.0(7.9) 82.4C5.8) 62.2(4.3) 40.5C3.0) 29.1(3.1) 19.2C1.4) 10.1(1.1) 7.00C.77) 7.05(.83) 7.19C.77) -0.064C.046) -0.351C.049) -0.343C.045) -0.201C.056) 0.05C.11) 0.163C.052) 0.29C.12) 0.43C.12) -0.04C.13) -0.28C.11) 260 49.7 64.4 74.7 84.9 95.0 104.9 114.6 124.3 138.5 580.(39.) 252.(16.) 127.4(8.7) 82.5(5.4) 53.0(4.3) 27.8(2.4) 16.1(1.5) 15.1(1.5) 10.6(1.2) -0.033(.039) 0.048(.024) 0.262(.042) 0.377(.037) 0.525C.065) 0.493(.092) 0.16C.10) -0.36(.10) -0.47C.12) 49.7 64.4 74.7 85.0 95.0 104.9 114.7 124.3 138.5 130.(11.) 61.9(4.3) 38.6(3.2) 43.3(3.1) 24.0(2.4) 16.9(1.7) 8.23(.97) 8.5(1.0) 8.8(1.1) -0.079C.082) -0.225C.048) -0.322C.076) 0.106C.053) 0.541C.096) 0.47C.12) 0.51C.13) 0.41C.14) -0.43C.13) A l i s t of the values for the d i f f e r e n t i a l cross-sections and analyzing powers for 1 0B(p,u +) reaction leading to 1 1B and l lB, , - „ „ states. g.s. MeV TABLE III 9 B e ^ , 7 t + ) 1 0 B eg.s. 9Be(p\u +) 1 0Be 3 .37 MeV (MeV) e . cm „ ("eg.) da/dQ (nb/sr) AN0 ®cm „ (Deg.) do7dQ (nb/sr) AN0 200 49.8 64.6 82.4(5.7) 59.8(4.9) -0.736(.038) -1.036(.044) 49.9 64.7 204.(13.) 165.(11.) -0.314(.028) -0.752(.035) 225 49.7 59.2 69.6 79.9 95.0 114.7 138.5 75.8(6.3) 54.6(5.0) 43.8(3.9) 24.8(2.7) 11.7(1.5) 3.8(1.5) 3.8(1.2) -0.590(.073) -0.903(.072) -0.910(.073) -1.079(.084) -0.44(.15) -0.38(.55) -0.26(.40) 49.7 59.2 69.6 79.9 95.1 114.7 138.5 231.(16.) 159.(11.) 150.(10.) 117.4(8.5) 122.9(8.5) 69.0(7.5) 32.9(3.9) -0.135(.046) -0.645(.049) -0.767(.043) -0.679(.051) -0.175(.049) -0.07(.13) -0.23(.14) 250 49.6 59.1 69.6 79.8 95.0 109.8 124.2 138.5 65.7(5.5) 37.0(3.5) 25.7(2.5) 16.1(1.7) 9.4(1.1) 5.12(.95) 7.3(1.5) 4.60(.91) -0.483(.075) -0.61(.ll) -0.950(.067) -0.46(.ll) 0.11(.14) 0.65(.24) -0.38(.26) -0.37(.28) 49.7 59.2 69.6 79.9 95.0 109.8 124.3 138.5 209.(14.) 124.2(9.0) 97.8(6.9) 85.2(6.0) 60.8(4.3) 45.4(3.8) 19.3(2.5) 17.3(2.0) 0.196(.045) -0.297(.064) -0.709(.041) -0.496(.048) -0.165(.054) 0.222(.087) 0.06(.17) -0.60(.14) A l i s t of the values for the differential cross-sections and analyzing powers for 9Be(p,u +) reaction leading to 1 0Be g a n <* ^Bej 3 7 M e V states. TABLE IV 9 B e C t . O « C g . a . 9 B e ( ? f n ' ) " C 3 . 3 1 t M e V (MeV) 9 cm AN0 (Deg.) (nb/sr) 6 cm / o / * AN0 (Deg.) (nb/sr) 200 49.8 1.26(,22) -0.37(.24) 59.3 1.39(.34) -0.12(.32) 95.2 1.47(.64) -0.36(.48) 49.9 5.33(.54) -0.03(.12) 59.4 4.39(.65) 0.19(.18) 95.3 5.1(1.2) -0.31(.26). 225 49.7 2.15(.41) -0.81(.20) 69.6 1.26(.22) 0.08(.22) 95.0 0.90(.22) 0.19(.33) 138.5 1.47(.51) -0.22(.46) 49.7 6.58(.79) 0.07(.14) 69.6 5.09(.51) -0.30(.ll) 95.1 2.09(.35) 0.13(.22) 138.5 1.72(.55) -0.12(.43) 250 49.6 1.04(.28) -0.87(.26) 69.6 1.24(.34) -0.38(.39) 95.0 0.86(.21) 0.37(.33) 138.5 1.30(.39) 0.09(.43) 49.7 4.45(.62) -0.54(.16) 69.6 2.02(.44) -0.28(.31) 95.0 2.01(.33) 0.30(.22) 138.5 1.38(.40) 0.02(.42) A l i s t of the values for the differential cross-sections and analyzing powers for 9Be(p,u ) reaction leading to 1 0 C and 1 0 C 3 3 4 M g V states. 45 O a o o a O a O • • • a a 4* < < > v L Z O CM llll I l l I o o o o > w 2 O IT) CVJ • o o o > z tt CM CVJ • l l l l l l l l > a> 2 O O CVJ • a. t-u ~. 3 • o o CVJ mi i •un i i i O CO o <3-o CM o o E u CD O O O O O O O o o CO o CD O o (js/qu) 7JP •OP Figure 17. The differential cross-sections for the reaction 10B(^.n"*")11B The 200 MeV results of Ref. 17 are also shown. The statis t i c a l errors are less than the size of the symbol on the plot. 46 (iVQu) g£ Figure 18. The differential cross-sections for the reaction 10BC$,n )llB2.12MeV The st a t i s t i c a l errors are less than the size of the symbol on the plot. 47 > O in CVJ II Q. a < a < 2 5 s o ^ OCT w WI O " II I D O ro LLli O O 1 J | 1 1 1 ' \ \ l " l I I I , , N V II o o o ( J S / qu ) S E o o o in O ro E u C D Figure 19. The d i f f e r e n t i a l cross-sections for the r e a c t i o n ' B e t f , * * ) ! 0 Be The 200 MeV r e s u l t s of Ref. 7 are also shown. 8's" 48 > 2 O in C J II o. I-> in C M C J II a > OJ Z> o o CJ OJ rr a. — O O in O ro • < E CJ CD O <T> a < a < D <3 6 o o o in O o o IlLLI o o o o o o o o o ro (JS/qu) Figure 20. The d i f f e r e n t i a l cross-sections for the r e a c t i o n 9Be(p\it ) 1 0 B e 3 > 3 7 M e V ' The 200 MeV r e s u l t s of Ref. 7 are also shown. 49 > O in CM II CL > 0> in CVJ CM II Q. > — 5 -O O v CM CC O O lllu L\VJ o O o CM O O O CO E o CD O O CM (JVqu) ^ Figure 21. The differential cross-sections for the reaction 9Be(p,n~) 1 0 C The 200 MeV results of Ref. 18 are also shown. 50 > 5 O in CM II Q. > OJ ID CM CM II O. > OJ O O CM II Q. Il " i . . . o (js/qu) UP •op O o o CM o O o GO O CO o E o CD \\ I I o o ~ o .CM Figure 22. The differential cross-sections for the reaction 9Be("$,u -) 1 0 Cj 3 4 M e V 51 0 -0.5 T p « 2 0 0 MeV -1.0 0 -0.5 k N 0 "D D-Tp«225 MeV 1.0 0.5 0 -0.5 -1.0 Tp«250 MeV \ - > — \.Of Tp« 2 6 0 MeV 0.5 0 -0.5 > C • i ' J  20 40 60 80 100 120 140 9 cm Figure 23. Analyzing powers for the reaction 10BCp,* ) The lines serve only as a guide to the eye. 52 0 - 0 .5 -1.0. 0 - 0 . 5 'NO 0.5 0 •0.5 0.5 0 •0.5 1 1— T p » 2 0 0 MeV O ~£> r>— — — T p » 2 2 5 MeV T p « 250 MeV Tp « 2 6 0 MeV \ 4 0 60 80 100 120 140 e , cm Figure 24. Analyzing powers for the reaction 1 0 , u + ) l 1 B 2 1 2 M E V The lines serve only as a guide to the eye. 53 0 •0.5 -1.0 Tp = 2 6 0 MeV '("Ref. 7j j 1 NO - 0 . 5 -1.0 T p = 2 2 5 MeV — — — ? 9 J I L_ J L 0.5 -•0.5 -- 1 . 0 Tp = 2 5 0 MeV / \ i \ / / / J J I y I L J L 4 0 6 0 8 0 100 120 140 G, 'cm Figure 25. Analyzing powers f o r the rea c t i o n 9 B e ( | , i t + ) 1 0 B e g j . The 200 MeV r e s u l t s of Ref. 7 are also shown. The'lines serve only as a guide to the eye. 54 •0.5 •1.0 Tp = 2 0 0 MeV (»Ref . 7) NO -0 .5 -1.0 T p = 2 2 5 MeV o s ^ - 6 0.5 0 -0.5 -1.0 Tp = 250 MeV 4-•<5 cr \ fr 40 60 80 100 120 140 Figure 26. Analyzing powers f o r the r e a c t i o n 9Be(fc,n +)10Be 3 3 7 M e V . The 200 MeV r e s u l t s of Ref. 7 are also shown. The l i n e s 6erve only as a guide to the eye. 55 0.5 0 •0.5 Tp= 200 M e v ' o R e f . 18) 4 NO 0.5 0 -0.5 -1.0 0.5 0 -0.5 -1.0 Tp = 2 25 MeV - 4 -Tp = 2 5 0 MeV f X / / 4 0 60 80 100 120 140 160 cm Figure 27. Analyzing powers f o r the rea c t i o n 9 B e ( | , i t ~ ) 1 0 C The 200 MeV r e s u l t s of Ref. 18 are al90 shown. 8*fhe l i n e s serve only as a guide to the eye. 56 0.5 0 •0.5 -Tp = 2 0 0 MeV 'NO 0.5 0 •0.5 Tp= 2 2 5 MeV i — 4 — 0.5 0 -0.5 --1.0 Tp= 2 50 MeV f -4 0 60 80 100 120 140 8. cm Figure 28. Analyzing powers for the reaction 9Be(p\it") 1 0C 3 ^ M e y . The lines serve only as a guide to the eye. 57 V. Trends in the Data 5.1 Choice of Variables The example of Couvert 1 9> 2 0 and Nefkens33 was followed to look, for possible trends in the data. In terms of the Lorentz invariant matrix elements,|M| 2, the unpolarized differential cross-section is given by: da < - « ' k (he) 1_ _p_ 1 _.M|2 (5-1) dQ 64uz s k (2J A+l)(2S n+l) Tt A P where s is the square of the center-of-mass energy of the reaction, kp is the center-of-mass momentum of the incoming proton, k^ is the center-of-mass momentum of the outgoing pion, is the total angular momentum quantum number of the target (A), and Sp is the spin of the incoming proton. The use of 1M]2 rather than for these systematic studies has the benefit of separating out phase space and t r i v i a l kinematic factors explicitly. In this way a clearer view of the effects of the reaction itself is seen rather than the effects of kinematics. Comparison of the dependence of | M | 2 on 0 (scattering angle), q (three momentum transfer), and t (the square of the four momentum transfer), indicated that the best independent variable for displaying the trends was t. t is defined as: -t = 2E E - m - m - 2k k cos 9 p rt p u p Tt cm where E, m, and p are the total energy, mass, and momentum of the various particles. 58 5 . 2 Statistical Weighting In the definition of given in Eqn. ( 5 - 1 ) , the matrix element | M | 2 has been averaged over the i n i t i a l spin states. A sum over the final spin states, however, Is s t i l l implied in | M | 2 . This statistical factor can be treated similarly by assuming that each final state contributes the same amount on average to | M | 2 . That i s , | M | 2 = I | m f | 2 - ( 2 J A + 1 + 1 ) (ZS^+Dlml 2 ( 5 - 2 ) where J ^ + ^ Is the total angular momentum quantum number of the final nucleus (A+l). S^  is the spin of the outgoing pion, which of course is zero, and |m|2 is the average matrix element per final state. By removing this st a t i s t i c a l factor from | M | 2 , the data from the different states of the A(p , ix +)A+1 reactions s t i l l showed distinct differences. Thus as expected, the pion production mechanism does show dependence on the angular momentum (J) of the final state. However, by plotting the data versus t (as shown in Figures 29-34) where the areas marked by horizontal and vertical lines represent the scatter of points for both the angular and incident energy range, the |m|2 plots show much more overlap compared to the | M | 2 plots. For example, by comparing I 0 B ( p , i t + ) 1 1 B g ( y - ) to 1 0B(p, 1t +) 1 1B 2 > 1 2 ), one can see very l i t t l e overlap of the two bands in the plot of | M | 2 VS. t (Figure 29). Figure 30 compares the same reactions with |tn|2 vs. t. There is clearly much more overlap. Figure 31 and 32 compare 9Be(p,it +) 1 0Be (0 +) to g • s • 9Be(p,n +) 1 0Be 3 3 7 v ( 2 + ) . In this case the difference between | M | 2 and |m|2 59 Figure 29. |M|2 as a function of t for the two final states of l lB in the 1 0B(p,Ti +) l lB reaction. 60 61 62 Figure 32. Jm|2 as a function of t for the two final states of 1 0Be In the 'Be(p,n +) 1 0Be reaction. 63 is even more impressive. The comparison of the carbon data^ 1, 1 2C(p,u +) 1 3 C g # g > ( | - " ) to 1 2C(p,u +) 1 3 Q,^ M e V ( | + ) (Figures 33 and 34) is the best illustration of this effect. The overlap seems to imply that though the production mechanism does depend on the spin state of the final nucleus, the magnitude of the differential cross-section basically reflects a simple statistical weighting. Although this statistical weighting does not always account for a l l the difference seen in some of the (p,u~) work4 (Figure 2), i t should be taken into account prior to investigating enhancements of high-spin states by the reaction mechanism. 5.3 Incident Energy Dependence The ( p , T i + ) differential cross-section distributions for the incident proton energy range discussed in this work have qualitatively the same structure. For 4r vs. 9 plots, there is an almost exact exponential dQ cm r decrease at forward angles (G C M<90°) followed by a relatively flat backward angle region. The slope of the exponential drop off in the forward angle was found to be energy dependent for some reactions but energy independent In others. In addition, i t was found that where energy dependence was found, a strong energy dependence was also observed in the shape of the analyzing power distributions. In contrast, for those cases yielding energy independent slopes, only a weak energy dependence was observed for the analyzing power distributions. In the 9Be(p,u +) 1 0Be reaction (shown in Figure 35), the slopes of 64 Figure 33. W •» a function of t for the two f i n a l states of " C i n the 1 2 C ( p , i t + ) 1 3 C reaction. 6 5 Figure 34. Jm|2 as a function of t for the two final states of 1 3C in the 1 2C(p,n +) 1 3C reaction. 66 Figure 35. The forvard angle data of the 9 B e ( $ , n + ) 1 0 B e . The s o l i d l i n e s least squares f i t s to the data. 67 CVJ O (Ref. > CU MeV MeV O m OJ in CM CM O O CM II II a t— a. r— Q. • 4 • -ID - in 0) : / \ \ i A — a — \ / V V "I j I I I I I I I I I j I I I O IP — a / '111II f I IT 1111111111 o in a n 11 j 11111 a o i o z < Figure 36. The analyzing powers as a function of t f o r 9 B e ( $ , T t + ) 1 0 B e g > s > . The l i n e s serve only as a guide to the eye. 6 8 I I I I I I I ! I I I I I I 11 I I I I I I I I I I I!I I I I I I I I I I I I I | | | | I O -to CM O • in CM • CM CM CL r -o •CM CM O CM O I— O CM • CO CO CD o CM „—. , CM CO ^ CO U LO C L _o CU O F i g u r e 37. The i n c i d e n t p r o t o n energy dependence of the e x p o n e n t i a l s l o p e s of the 9 B e ( " p . i t + ) i Q B e d i f f e r e n t i a l c r o s s - s e c t i o n data as a f u n c t i o n of t . 69 the | M | 2 VS. t semilog plot are the same within the experimental error. In fact the least squares f i t s of these sets of data almost l i e on a single line. In addition the analyzing powers (shown in Figure 36) as well show l i t t l e dependence on incident proton energy. Figure 37 shows a plot of the fitted slopes, —-J—'—, versus the incident proton energy, T^. A constant value of 5.5 (GeV/c)"2 is consistent with the f i t s . For the 9Be(p,rc+) 1 0Be 3 3 7 M e V reaction (Figures 38 - 4 0 ) , a similar characteristic behaviour Is observed. In this case, the fitted slopes are consistent with a constant value of 3.5 (GeV/c)~2. The next case of weak dependence on incident proton energy is that of 1 2 C ( O , T C + ) 1 3 C Q = „ „, 2 1 (Figures 41 - 4 3 ) . In this case, the semilog plot of 7 . J M ev |M|2 vs. t is not as energy independent as the previous two cases. Though the slopes are generally the same (approximately 4.9 (GeV/c) - 2), the f i t s do not l i e on a single line. The reason for this difference is not known. The analyzing power distributions do show the same general trend as for the previous cases, however. In contrast to the situation encountered with the next three reactions to be discussed where a strong dependence on Incident proton energy occurs, i t is more appropriate to designate this reaction energy Independent or at least weakly dependent. In the reaction 1 0B(p,u +) 1 1B both the exponential slopes and the g. s • analyzing powers are strongly energy dependent. Figure 44 shows the | M | 2 VS t plot. As the incident proton energy increases, the exponential slope decreases. In the case of the analyzing powers, as the incident energy increases there is a large increase in the lower t region (shown in Figure 4 5 ) . Figure 46 illustrates the continually decreasing slope with Increasing 70 Figure 33. The forward angle data of the 9Be(p\n ) 1 0 B e 3 > 3 7 M e y . The solid lines are l e a s t squares f i t s to the data. 71 CM O or > > ? OJ OJ OJ 2 2 5 O <o O i n CJ o CJ CJ CJ H • • H o r * CD Q . C L i — r— / \ \ •L / / / \ o i n n n 111 > 1111 .(N \ I I I I I I IIII 11 I I I I I I I I I I I I I I I 11 IM CO CD o o o o o Figure 3 9 . The analyzing power, as a f u n c t i o n of t for »Bert.ii+)»B«j . 3 7 M e V -The l i n e s serve only as a guide to the eye. 72 r- CD = ro CM CL > o <u (/I •—* o ID CM Q in CM a CM o .cn o • CM CM a CM o l-o CM i i i i i i i | i i i i i i i i i | i i i i i i i i H i i i i i i i i ' | ' " " " " CO ro co a .cn 73 CM CC > > > > <u CO CU 2 5 2 o r- in 10 m ro CVJ CVJ CVJ CVJ CVJ II II II 11 Q. Q- Q. ca \— f— 1— • • < • 1111 i i i r •01 1111 i i " i — r • in in -in • in CM - i n cn .-9 in cn 3*° CO O Figure A l . The forward angle data of the 1 2 C ( p , n + ) 1 3 C 9 > 5 M e V . The solid lines are least squares f i t s to the data. 74 CM QJ o I—in CVJ 0) or MeV MeV MeV in CVJ CO CVJ CVJ CVJ It 11 II CL r— CL f-Q. r— • « • \ . A (0 / \ \ \ l-rn \ CO • CM n 1111111 CD Q 11111111111 p I1 Ti M 111 j 11111111111111111 * O t CO o z a l Figure 42. The analyzing powers as a functi o n of t f o r 1 2 C ( p , u + ) 1 3 C 9 < 5 M e V< The l i n e s serve only as a guide to the eye. 75 o L-CD t CM O r - i n CM O • f CM I I I I I I | I I I CT LO I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I O •CO — CM > a • CM CM O CM O l-o CM • 0) LO o> ^ ""~ ^ CL ^ O -gj o Figure 43. The incident proton energy dependence of the exponential slopes of the 1 2 C ( p , i t + ) 1 3 C 9 > 5 M g V differential cross-section data as a function of t. 76 Figure U . The forward angle data of the «B(t.« +>»B g.... The s o l i d l i n e s are l e a s t squares f i t s to the data. 77 > > > > CU cu <u 2 5 2 2 O O U") O ID m CJ o CJ CJ CJ CJ II II II n Q . C L C L C L \— H— r- h-• + • ^ t > cu O \ CD [-IP 'O SIP - I / I'-Jl I I \ 00 ten \ b CD t-fM O f - C M X Y \ T TI 111 I M M 11 I | M 1 I 1 1 I 1 CD f) O o o ^ o TiTll I n fl) CD l' l' Figure 45. The analyzing powers are shown for the r e a c t i o n 1 0 B C ^ , u ) u 3 The l i n e s serve only as a guide to the eye. 78 o I-in CM O CN 1111111111 r-' 1 1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I O • ro > CM a> O • CM CM a I" CM o I— o CM O .cn =• CD a> C L O to IT: LO > CU O F i g u r e 46. The i n c i d e n t p r o t o n energy dependence of the e x p o n e n t i a l slopes of the B g . s . d i f f e r e n t i a l c r o s s - s e c t i o n data as a f u n c t i o n of t . 79 T . It is unreasonable to approximate this curve by a constant value. The 1 0 B ( p , T t + ) 12 Mev r e a c t i o n displays much the same pattern as the i 0B(p,u +) n B reaction (illustrated ln Figures 47 - 49). Also the g. s. 1 2C(p,u +) 1 3C reaction 2 1 behaves in much the same way (Figures 50 - 52). g. s • Another important trend that must be pointed out, other than the fact that the exponential slopes and analyzing powers are both either weakly or strongly dependent on the incident proton energy, is the fact that their dependence is very similar. The pattern of the changing exponential slope and of the increasing analyzing powers seem to be followed in the last three cases. The f i r s t three reactions as well seem to contain approximately the same features. One possible interpretation of this energy dependence in some reactions and not others might be that of specific effects associated with single-particle final states. 2 2 It seems plausible that final states not described in terms of single particle ( 1 0Be , 1 0Be, , 7 „ „, 1 3 C Q K w „) ° g.s. •>.37 MeV 9 .5 MeV could be candidates for an averaging effect and thus exhibit a "smoothed-out" energy dependence. On the other hand, those states with good single-particle nature (like 1 3C ) could be expected to manifest a strong energy g.s. dependence. The ^B state, a single-hole state, would be expected to act g.s. like a single-particle state. The 1 1B 2 i 2 MeV s t a t e » a two-hole one-particle state, however also shows this strong energy dependence. Since particles (including holes) like to couple to zero spin, i t would not be unreasonable to expect the 1 1B 2 12 MeV state to act as an effective single-pa?uicle state. The other three cases ( 1 0Be g g , l uBe 3 3 7 M g V , L 3 C 9 5 M g V ) are unlikely to couple to form an effective single-particle state. In the case of 80 Figure 4 7 . The forward angle data of the B f l i , i t + ) 1 1 B 2 ^ 1 2 M g V . The solid lines are least squares f i ts to the data. 8 1 > > > > cu CU a» 2 O O m O CO m CM O CM CM CM CM II II 11 II Q. a. a. Q. r— t- r- \-• 4 m Figure 48. The analyzing powers as a function of t for "\K*.*+)»h.l2 MeV The l i n e s serve only as a guide to the eye. 82 111111111111111111111111 m 111111111111111 LP to LP LO CU CL _o in IP LP TT O • O CM O • LO CM O . CM O • CO CM O • CM CM O CM O - o CM o LO > cu Figure 49. The incident proton energy dependence of the exponential slopes of the B 2 . 1 2 MeV differential cross-section data as a function of t. 83 Figure 50. The forward angle data of the 1 * cCp,« +) 1 3C g. 9. • The s o l i d l i n e 9 l e a s t squares f i t s to the data. 84 Figure 51. The analyzing powers as a function of t for l 2 C ( p \ * + ) 1 3 C g < g > • The l i n e s serve only as a guide to the eye. 85 o r o r F r r r o CM r o r > r a j l C M r Psj r r o CM O • O CM r o r 71 • 111111111111111 M 111111111111111111111111111 r _ (0 f\| CO o CM a cu . ;*,<-> o cu LP Figure 52. The incident proton energy dependence of the exponential slopes of the * 2 C ( o . i i + ) 1 3 C differential cross-section data as a function of t. 86 1 0Be , a two-hole state, the two holes probably do couple to zero spin, g.s. This leaves the state as an effective closed shell. 1 0Be is a 0 + state g.s. which is consistent with this picture. The 1 0Be 3 3 7 M e V state, mainly a three-hole one-particle state, again cannot couple to form an effective 11 9+ single-particle state. The two-particle one-hole of i 0 C 9 5 is a y state, therefore the two particles can not possibly couple to zero spin. The 3-coupling would leave a -^ state, which is not consistent with the 9.5 MeV state. In fact the state is best described by having one particle in the P 1 / 2 level and the other in the D 5 / 2 level. To investigate this effect further, additional nuclei should be studied. In particular, measurements of the 1 6 0(p ,Tt + )^ 7 0 and t*°Ca(p,u+)l+1Ca reactions leading to low lying states should exhibit this single-particle behaviour. In fact since a l l the reactions compared in this study basically involve only the p-shell, i t would be beneficial to study higher shells such as in 1 6 0(p .n*)1-7 0 and 1 + 0Ca(p ,TC +) l + 1Ca to determine i f i t is only a 3L=l effect. 5.4 Diffraction Peak Structure The exponential decrease of the differential cross-sections for forward angles has some similarity to the shape of the cross-section corresponding to "diffractive" effects characterizing elastic scattering. In fact i t can be shown23 that the Fraunhofer approximation of diffraction scattering and an exponential drop off are equivalent in the small angle approximation. In the Fraunhofer approximation the scattering amplitude i s : Jx(RksinG) f ( 9 ) - 1 k 6 R2 R k s l n Q (5-3) 87 where k is the incoming momentum, S is the absorption coefficient of the scattering center, R is the radius of the scattering center, Q is the scattering angle, and is the first-order Bessel function; i.e. J x(z) = j _ (y) 3/2! + ... (5-4) In the small angle approximation (sin 0 = 0 ) to order 0 2, F ( 9 ) -M*L[I- rn?] ( 5-5) Now let us turn to an exponentially decreasing cross-section, g.. A e ' ^ l = | f ( 9 ) | 2 (5-6) For small angles the momentum transfer is approximately, t = k 2sin20 = k 20 2 (5-7) Therefore Eqn.(5-6) becomes, = |f(0)| 2 exp(-bk202) (5-8) 8 8 h k 2 ^ 2 or f(9) = f(0) exp (- (5-9) where A = |f(0)| 2. Expanding the exponential to order G 2 leaves f(9) = f(0) [1- ^1 ] (5-10) By comparing Eqns. (5-5) and (5-10), f(0) = 1 k ^ R 2 (5-11) and R2 = 4b. (5-12) For elastic scattering from protons (i.e. it~p,n;+'p,K~p,K+p,pp and pp), the exponential slope b varies between 3 and 13 (GeV/c) - 2 (Ref. 23). This is in reasonable agreement with the proton radius of approximately 1 x 10 - 1 3 cm. The diffractive pattern created by light scattered from a black disk Is the same as that created by a coherent light source the same size as the disk. Thus i f one assumes that the peak structure seen in pion production is due to diffraction, one would expect source radii of the order of 2 x 10 - 1 3 cm for the nuclei discussed in this study (assuming R follows 2 1* <R2>1/2 = r Q A 1 / 3 (5-13) where r Q is equal to 1 x 10 - 1 3 cm). In fact, the exponential slopes vary from 3 to 13 (GeV/c) - 2. These values are very similar to the particle physics 89 values discussed above. This may be an indication that the production and rescattering effects are associated with individual nucleons rather than a collective effect. Of course further studies, both theoretical and experimental, must be made before any firm conclusions can be made. 5.5 u~ Production The (p,u ) data measured at IUCF4 showed preferred high-spin states (see Figure 2). It was interpreted that the preference was due to the creation of stretched two-particle one-hole final states that allowed for angular momentum matching. The 9Be(p,ic ) 1 0C results presented here are not substantively different from the early (p,it ) data1, obtained before the IUCF discoveries. There was no preferred final state. The differential cross-sections are relatively flat and small compared to the (p,u+) data. The analyzing powers, as well, tend to show much less structure than for the (p,it +) reaction. This does not conflict with the IUCF findings. 1 0C does not have any stretched two-particle one-hole excited states. Thus our results do not contradict the IUCF interpretation of enhanced cross-sections associated with stretched two-particle one-hole final states. 90 VI. Status of the Theoretical Model9 In the introduction i t was mentioned that at least four basic ingredient must be included in any description of the pion production process: the pion emission operator, proton distortion, pion distortion, and nuclear wave functions. In this chapter each of these four ingredients w i l l be examined i the context of an example of each class of models. 6.1 Relativistic One Nucleon Model The most complete ONM calculation has been carried out by Cooper,25 who treated the distorted wave Born approximation (DWBA) calculation fully r e l a t i v i s t i c a l l y . For the pion emission operator both pseudoscalar and pseudovector couplings were examined. In the case of pseudoscalar, the interaction Hamiltonian (see Figure 3) is: H l n t(x) = l / T g ^ Y5 ^ ( x ) (6-1) where g^ is the TtN coupling constant and <t>n(x) Is the pion field operator. For pseudovector coupling the Y5 is replaced by ^  ^ — where N P - - I h - y ^ 0 0 , (6-2) TJL^  is the mass of a nucleon and b^1 ^ is a derivative acting on the pion wave function. Cooper claims this derivative coupling of the pseudovector gives better agreement with the data than pseudoscalar coupling. 91 The proton distorted wave functions were generated by an optical potential of the form: U(r) = U (r) + U (r) + U , v ' v s coul (6-3) where U , is the Coulomb potential characterizing a uniform charge COUl V a b distribution. The vector potential was of Woods-Saxon form: V i W V " • r ^ R — - + T T T • <6"4> 1+ e x p f - j ^ 1+ exp { - ^ vl v2 and the scalar potential was similar: i.e. V i W 3 S V r ) = r-R r-R 1 + e x p ( ^ - ) 1+ exp ( 6 _ 5 ) where V s and W's are the strength of the real and imaginary potentials, respectively, R's are the radial parameters, and a's are the diffuseness parameters. In order to make the calculation simpler, only vector and scalar potentials were used. This was shown2'' to be a good approximation as long as the nucleus has doubly closed shells (both isospln and spin equal to zero). The twelve parameters were f i t to proton elastic scattering data for the appropriate nucleus and incident energy. For the pion distortion, the optical potential of Striker, McManus and Carr 2 7 was used. This potential not only fi t s the pion elastic scattering in the energy range of 0 to 50 MeV, but also provides f i t s to the level shifts and widths of the pionlc atoms. The single particle nuclear bound state wave functions were obtained In terms of Woods-Saxon potentials made up of only vector and scalar components, 9 2 v U (r) = v v r-R (6-6) V s r-R (6-7) 1+ exp( s a s Implying again restriction to isospin and spin zero nucleus. The geometric parameters in these potentials were defined by fitti n g proton-nucleus elastic scattering data. The strengths, V g and V^, were f i t to the binding energy using the mean field theory of nuclear matter. 6.2. Two Nucleon Model One of the TNM's involving the fewest approximations is the calculation of Iqbal. 2 6 At this point in time only the four diagrams of Figure 4 are included, those which involve the A-resonant (T = 3/2, J = 3/2) part of the pion production amplitude. For the incident proton energies in the range 200 to 400 MeV, the A-resonant amplitude is believed to be the dominant term. In this model the delta-isobar propagates via delta-nucleus Interactions described by a local nuclear density dependent potential. In addition, Iqbal concentrated on two-particle one-hole final states which cannot be easily reached via ONM's. In this model the pion emission operators are the static form of the itNN and nAN interaction Hamiltonian obtained by the Foldy-Wouthuysen30 non-relatlvistic reduction of the relativistic pseudoscalar or pseudovector coupling. That i s : 93 \ m " < W o ' ^ x • • (6-8) and * HrcAN = <frc '\> S • \ T • • <6"9> where: and f are the TCN and itA coupling constants, respectively, ra i s the pion mass, is the derivative operator acting on the pion wave function a and x are the matrix representations of the operators connecting the two-component spinors i n spin and isospin spaces, respectively, and S and T are the matrix representations of the operators connecting spin and isospin 3/2 states with spin and isospin 1/2 states, respectively. Note that in this model there are three vertices in each diagram, two TCA and one uN (see Figure 4). The proton distortions were generated by an o p t i c a l potential of the form: 2 d f3 d f . U ( r ) " V c o u l + V l + 1 W2 f2 ~ 7 <V3 S T + 1 W* d r - ) ( * ' 0 ) ( 6- 1 0> where v c o u ^ is the Coulomb potential characterizing a uniform charge d i s t r i b u t i o n . and W2 represent the strengths of the real and imaginary parts of the central potential, whereas V3 and W4 are the corresponding strengths for the spin-orbit part of the o p t i c a l potential. The f's are the form factors and are taken to be of the Woods-Saxon form, 94 f i ( r ) " W <6"n> l + e x p ( ^ - i ) The twelve parameters i n v o l v e d were f i t to p r o t o n e l a s t i c s c a t t e r i n g d a t a i n a s i m i l a r way as f o r the ONM c a s e . The p i o n d i s t o r t i o n was i n c o r p o r a t e d by the use of a m o d i f i e d K i s s l i n g e r p o t e n t i a l 3 1 of the form: U ( r ) = - Z b Q k 2 p ( r ) + Zt^V • p ( r ) • V - Z/2 [ (T + m ) / M ] b 1 V 2 p ( r ) (6-12) where: Z i s the a t o m i c number of the n u c l e u s , M i s the n u c l e o n mass, m i s the p i o n mass, k i s the wave number of the p i o n i n the p i o n - n u c l e u s c e n t r e - o f - m a s s , T^ i s the p i o n k i n e t i c energy, p ( r ) i s the n u c l e a r d e n s i t y , and b Q and b^ a r e the complex parmeters r e l a t e d t o the S- and P-wave p i o n - n u c l e o n phase s h i f t s . I q b a l examined both harmonic o s c i l l a t o r and Woods-Saxon wave f u n c t i o n s f o r the bound s t a t e n u c l e o n . The a n g u l a r 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 c r o s s - s e c t i o n s were found t o be i n s e n s i t i v e t o the type of bound s t a t e wave f u n c t i o n assumed. As p o i n t e d out i n the next s e c t i o n , t h i s i n s e n s i t i v i t y i s due to the momentum t r a n s f e r s h a r i n g i n the TNM models. 95 6.3 Comparison of Models The major difference between the ONM of Cooper and other ONM's is the inclusion by Cooper of relativ i s t i c effects. A l l other ONM's use a non-relativistlc reduction of the Interaction Hamiltonian such as the that of Foldy-Wouthuysen30. g H = - (a»V f <{>) + 2 ) + 1(a»P f<t>) + higher order terms (6-13) int Zxa. '•m This reduction creates an ambiguity by introducing an undefined parameter X. By an appropriate choice of \, one can approximate H^nt in terms of only the fi r s t term (static form). Alternatively, one can include the second term to get the Galilean invariant form. Even higher order terms could be retained i f desired. It has been shown25 that conventional static forms of H, do not int give the same results as the rela t i v i s t i c one. The difference between the TNM of Iqbal and the other TNM's is the treatment of the A-resonance. Although most TNM's involve the A-resonance only Iqbal's allows the A to propagate via a delta-nucleus Interaction of more realistic form than the usual delta function. In his TNM, Iqbal permits the delta to propagate through a local nuclear density dependent potential. In comparing the two types of models, one Is struck by how much easier i t is to use the ONM's. This of course is the major advantage of such models over the more realistic TNM's and is probably the main reason why the relativi s t i c formulation has only been incorporated in the ONM case. The major disadvantages of the ONM as compared to TNM is Its treatment of u 96 production, A-resonant Intermediate effects, and two-particle one-hole f i n a l states. Although a l l three of these effects can be treated consistently In terms of the free N-N interaction l n the TNM, they can only be Incorporated i n terms of complicated multi-step processes in f;h;i c o n t e x t of the ONM. Another advantage claimed by the ONM proponents involves the inclusion of the d i s t o r t i n g potentials. In the case of the ONM, l t Is clear that the incoming proton should be affected by a potential ascribing the f u l l nucleus. In the TNM case where only one of the nuclear nucleons i s involved e x p l i c i t l y In the interaction operator, i t i s not clear that a potential of the f u l l nucleus should be used for d i s t o r t i n g the wave functions. What i s needed i s a (A-l) nuclear potential i n the presence of a spectator nucleon. A potential l i k e this cannot be e a s i l y parameterized i n terms of experimental data. Thus, a l l TNM's involve a d i s t o r t i n g potential describing the whole nucleus. However, TNM's are generally much less sensitive to the d i s t o r t i o n e f f e c t s than are ONM's, so this effect may be small. Also, even though the o p t i c a l potentials f i t e l a s t i c scattering data, i t does not mean they are appropriate as d i s t o r t i n g potentials. The e l a s t i c scattering i s an on-shell process whereas that involved i n pion production models i s very much o f f - s h e l l . There is no unambiguous way of determining what effects should be Incorporated i n the o p t i c a l potential when going so far o f f - s h e l l . Another advantage of the TNM i s i t s i n s e n s l t i v l t y to bound state wave functions. Since the momentum transfer i s shared by three wave functions, one is concerned with p a r t i c l e wave functions i n a region of momentum transfer where the harmonic o s c i l l a t o r and Woods-Saxon wave functions have very similar form and are reasonably well known. For the ONM, on the other hand, very l i t t l e i s known of the wave functions at the much larger momentum transfers 9 7 involved and i t has been shown1 that the ONM is very sensitive to the nature of the bound state wave functions used in the calculation. 6.4 Comparisons of Models with Experiment In Figures 53 and 54 Cooper's relati v i s t i c ONM calculation for 1 2 C ( p , i r + ) 1 3 C g g at T p - 200 MeV is compared with experiment. Although the shape of the theoretical differential cross-section is qualitatively similar to experiment, the calculation is low by an order of magnitude. Since this model does not include A-resonance effects an underestimate is not unreasonable. The analyzing powers resulting from this model also have the right qualitative shape at forward angles, but predicts a change of sign for angles greater than about 90°, an effect which is not observed experimentally. Certainly, given the lack of quantitative agreement, this model has severe limitations. Though the ONM was designed for transitions which start with a closed shell nucleus and end with a single-particle final state, we also investigated Its applicability to describe 1 0 B(p,u +) 1 1 B . 1 0B was approximated as a g. s. closed shell even though i t is not. Since ^ B is a single-hole state, i t g. s • was treated as a single-particle final state in Cooper's code. Problems were also encountered in attempting to treat the distortions. Optical potentials for the distortions are not available for boron (no elastic scattering data at this energy exists). An approximate resolution of this problem involved use of potentials appropriate to 1 2C(p,u +) 1 3C (scaled appropriately). The 98 Figure 53. Cooper's r e l a t i v i s t i c ONM comparison to 1 2 C ( p \ T t + ) I 3 C g # s > d i f f e r e n t i a l cross-sections at T_ - 200 MeV. 99 100 results are shown in Figures 55 and 56. Again the shape of the differential cross-section curve is quantitatively reasonable but the normalization is not. The analyzing powers, however, agree remarkably well with experiment. Cooper's model appears to provide reasonable qualitative agreement on da/dQ and A^g for single-particle final states. More comparisons are needed. Next, Figure 57 compares the predictions of Iqbal's TNM with experiment. Comparisons of the 1 2C(p,u +) 1 3C g 5 MeV results at two incident energies, 200 and 250 MeV, are compared with Iqbal's predictions. The overall shape again is qualitatively similar to experiment, but shows too much incident energy dependence. The prediction underestimates experiment but not to the same extent as in the ONM case. Again, an underestimate is not unexpected since only the A-resonance diagrams are included In this model. At these energies in addition to the A-resonance processes, non-resonant processes are expected to be important and these contributions should help to increase the cross-section. Since analyzing powers provide an additional stringent test of theory, i t would be most interesting to have such a comparison. As yet, however, theoretical predictions are not available. 101 Figure 55. Cooper's r e l a t i v i s t i c ONM comparison to 1 0B(p,n ) 1 1 B g > s > d i f f e r e n t i a l cross-sections at T_ » 200 MeV. 102 (\J O (N ^ ID o z < Figure 56. Cooper's r e l a t i v i s t i c ONM comparison to 1° B(P»'t +) l l Bg.g. analyzing powers at T D = 200. MeV. 103 104 VII. Conclusion The results of two major sets of experiments have been presented. The angular distributions of both the differential cross-section and the analyzing power have been measured for the following reactions: 1 0 B ^ + > l l B g . s . , 2 . 1 2  9 B e<P»* +> 1 0 B eg.s.,3.37 9 B e^~> 1 0 Cg.s.,3.3* over a range of incident proton energies from 200 to 260 MeV. These measurements extend the results previously obtained at Uppsala, IUCF and TRIUMF into a new energy region, one in which the A-isobar Is expected to play a major role. The measurements at T^ = 200 MeV are In good agreement with the previous results, indicating consistency in the absolute normalizations of a l l the results. These studies of nuclear pion production in this energy region have helped to indicate new trends in the data. (1) Statistical weighting of high-spin final states. It was demonstrated that when the statistical weighting of the final spin is removed from the matrix element, there is much less dependence of the the resulting |m|2 (vs. t) on final spin then in the case for | M | 2 . This spin independence implies that though the specific details of the reaction may depend on the spin state of the final nucleus, the dependence of the magnitude of the differential cross-section on the spin is basically that of 105 simple statistical weighting. Thus there seems to be no preferred final spin state (at least for the reactions studied in this work). There is no evidence (in this work) that better momentum matching for higher spin states has any effect on the cross-section. (2) Incident proton energy dependence. The energy dependence of both the exponential slope of the t-dependence of the differential cross-sections at forward angles and the analyzing powers are correlated. It appears that the exponential slopes and analyzing powers are both either weakly or strongly dependent on the incident proton energy. In addition, we have evidence that reactions involving final states that have a strong single-particle nature (Including single-hole states) show a strong and similar energy dependence; whereas those states which are not strongly single-particle in nature show only a weak energy dependence. (3) Diffractlve Peak Structure. A l l the differential cross-sections were strongly peak in the forward direction, with a shape very similar to that of diffractlve scattering. Values of the exponential slopes of these peaks when plotted as a function of t were shown to be similar to those expected for projectile scattering from a single nucleon rather than the whole nucleus. Is this providing us an important clue Into the role of pion rescatterlng in such reactions? The status of the current level of theoretical understanding of pion production was also discussed. To date, no theoretical approach provides a reasonable description of the experimental data. Some specific recommendations that follow from this thesis are: (1) Modification to Iqbal's TNM to handle single-particle final states. It appears that final states which are not single-particle in nature 106 have a "smoothed-out" energy dependence (see Figures 35-43). Thus more Information may be gained by comparing to single-particle final states where a strong dependence on incident proton energy has been seen (see Figures 44-52). (2) The inclusion of non-resonant diagrams into the TNM. There is reason to believe that non-resonant diagrams are important in this energy region. The comparison with experiment certainly implies a significant contribution from non-resonant diagrams. (3) The TNM code should be made rel a t i v i s t i c . It is known25 that re l a t i v i s t i c effects are important and therefore should be included. Creating a relati v i s t i c formulation of a TNM code would be a major undertaking. Therefore ln the meantime (In order to make the computation easier), at least a better approximation to the relativ i s t i c interaction Hamiltonian should be used. The static form now used is clearly not a good approximation. (4) Single nucleon pion rescatterlng. Since the forward angle structure is similar to that of diffractlve peaks associated with interaction regions of nucleon size, i t may not be appropriate to treat pion rescatterlng as an average effect described by an optical potential for the whole nucleus. More explicit calculations should be Initiated to investigate pion rescatterlng effects for this process. As well, specific experimental recommendations also follow from this work. (1) 1 60(p,rc +) 1 70 and ' t 0 C a < p , i i V 1 C a It is important to test the observation that single-particle final states are associated with a strong energy dependence for the reaction. For both 1 7 0 and ^ Ca, a l l the low-lying excited states as well as the ground states are single-particle in nature. A l l the single-particle final states 107 studied in this work, have been those of the p-shell only. Thus the study of higher shells such as those describing single-particle states in 1 70 and 4 1Ca are needed to determine i f the features observed are truly general single-particle effects or whether they depend instead on details of the spin structure of the reaction. (2) Increased Incident proton energy. The next logical step would be to increase the incident proton energy. Does the s t a t i s t i c a l weighting of high-spin final states continue to dominate the spin dependence or does "angular momentum matching" become important at higher energies? Does the dependence on incident proton energy change? How w i l l the A-resonance, which is known to play an important role in pion production, affect this energy dependence? 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