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 ~~«• 3 1 0 4 a 3 ( C M • t o n ?to» 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 ~~__ > > > 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 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* __

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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