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A theoretical study of the (π,πn) knock-out reaction Shrimpton, Neil Douglas 1981

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A THEORETICAL STUDY OF THE Crr.TTn) KNOCK-OUT REACTION by NEIL DOUGLAS SHRIMPTON B . S c , The U n i v e r s i t y of V i c t o r i a , 1 9 7 8 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of Physics) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October 1 9 8 1 (c) N e i l Douglas Shrimpton, 1 9 8 1 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a llowed without my w r i t t e n p e r m i s s i o n . . Department o f The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date OCf If/ tfp  DE-6 (2/79) - i i -Abstract The (Tr,Trn) reaction i s studied theoretic a l l y t the s p e c i f i c case examined being 0"*"^(Tr + , TT+p)N^^. Calculations of the d i f f e r e n t i a l cross sections f o r lP-jy 2 a n c* "*"P3/2 P 1 7 0^ 0 1 1 3 are made at various incident pion energies. The remaining kinematic variables are s p e c i f i e d by a geometry which emphasizes the behavior of the two-body pion nucleon i n t e r -action. The aim i s to examine the influence of the nucleus on the two-body in t e r a c t i o n . In p a r t i c u l a r , the influences of Pauli exclusion of the nucleon, the o f f - s h e l l e f f e c t , and the e f f e c t i v e p o l a r i z a t i o n of the nucleon are examined. The computation i s performed using the fa c t o r i z e d d i s t o r t e d wave impulse approximation. By evaluating the distorted wave matrix element i n coordinate space the l o c a l i z a t i o n of the knock-out reaction i n the nucleus i s determined. The Pauli exclusion of the nucleon i s found to have the largest influence on the d i f f e r e n t i a l cross section at incident pion energies of 1 1 6 MeV. At higher energies the knock-out reaction occurs at the extreme edge of the nucleus and the eff e c t of Pauli exclusion i s minimal. At lower energies, the two-body i n t e r a c t i o n i t s e l f i s less sensitive to Pauli exclusion. O f f - s h e l l effects were found to be very small. The p o l a r i z a t i o n of the proton was found to have a large influence on the cross sections. Furthermore, i t i s - i i i -noted t h a t comparing the c r o s s s e c t i o n s f o r 1 P ^ £ a n ( * l p 3 / 2 protons w i l l i n d i c a t e the e f f e c t i v e p o l a r i z a t i o n of the p r o t o n i n the nucleus. - i v -Table of Contents Abstract i i L i s t of Tables v L i s t of Figures v i Acknowledgement v i i i I. Introduction 1 I I . Theory 8 I I I . Two-Body t Matrix 12 IV. Distorted Wave Matrix Element 1 6 V. Results 33 VI. Discussion 36 Comparison of DWIA and PWIA Calculations 36 O f f - S h e l l E f f e c t s 38 Pauli Exclusion E f f e c t s 38 E f f e c t i v e P o l a r i z a t i o n 40 Comparison with the Results of Levin and Eisenberg 4 1 VII. Conclusion 4 4 Bibliography 4 6 Appendix 4 8 Figure Captions 4 8 Figures 5 5 - V -L i s t of Tables 1 £> 1. Wood-Saxon Bound State Potential Parameters f o r 0 21 1 fi 2. Potential Parameters for Proton Scattering from 0 22 1 £> 3. Potential Parameters f o r Pion Scattering from 0 25 k. P o l a r i z a t i o n Weight Factors f o r g^ when 1 = 1 31 v i -L i s t of Figures Figure 1 55 Figure 2 56 Figure 3 57 Figure 4 58 Figure 5 59 Figure 6 60 Figure 7 6l Figure 8 62 Figure 9 63 Figure 10 64-Figure 11 65 Figure 12 66 Figure 13 67 Figure 1^ _ 68 Figure 15 69 Figure 16 70 Figure 17 71 Figure 18 72 Figure 19 73 Figure 20 7^  Figure 21 75 Figure 22 76 Figure 23 ' 77 - v i i -F i g u r e 2k 78 F i g u r e 25 79 F i g u r e 26 80 - v i i i -Acknowledgement I would l i k e to thank my Supervisor, Dr. A. W. Thomas, f o r giving me the guidance and patient attention without which t h i s thesis would not have been possible. I would also l i k e to thank Andy M i l l e r f o r providing the outline and some of the subroutines used i n the c a l c u l a t i o n and Daphne Jackson f o r suggesting examination of the r a d i a l behavior of the DWME to obtain the fermi momentum cut-off. F i n a l l y , I would l i k e to thank my wife, Miriam, f o r her assistance i n the preparation of the manuscript. Introduction The knock-out reaction, which has been an extremely-useful t o o l i n nuclear physics, involves an incident p a r t i c l e s t r i k i n g a nucleus and emerging afte r having knocked a nucleon out. Early work i n t h i s area involved an examination of the energies of the p a r t i c l e s i n the f i n a l state which allowed the binding energy of the struck nucleon to be calculated. These binding energies were found to c l u s t e r around the well-defined values predicted by the nuclear s h e l l model. Further-more, the angular d i s t r i b u t i o n s of the knocked out nucleons made i t possible to d i s t i n g u i s h between p a r t i c l e s with zero and non-zero angular momenta. The r e s u l t i n g assignments were i n complete agreement with the predictions of the simple s h e l l model.* These binding energies, coupled with the charge d i s t r i -butions of the protons determined from e l a s t i c electron scattering, have allowed s i n g l e - p a r t i c l e potentials to be p determined f o r these s h e l l s . As computers became more powerful, c a l c u l a t i o n of the eff e c t of the nucleus on the incident and outgoing p a r t i c l e s became f e a s i b l e and tests of o p t i c a l model potentials could be performed. The (p,2p) reaction was well-suited to t h i s type of analysis as the symmetries present i n the reaction - 2 -a l l o w e d f o r much s i m p l i f i c a t i o n o f the c a l c u l a t i o n . - ^ The (p,2p) r e a c t i o n i s a l s o e x t r e m e l y s e n s i t i v e t o t h e e f f e c t s o f n u c l e a r d i s t o r t i o n t h a t t h e n u c l e u s has on t h e p r o t o n wave f u n c t i o n s because t h e r e a r e two o u t g o i n g p r o t o n s and t h e i r 14. e n e r g i e s v a r y w i t h a n g l e . Modern computers have made c a l c u l a t i o n s o f knock-out c r o s s s e c t i o n s f o r more g e n e r a l c a s e s p o s s i b l e . S t u d i e s o f t h e ( p , p n ) ^ and ( TT , - r r n ) ^ r e a c t i o n s have been r e p o r t e d . W i t h c u r r e n t s t u d i e s of the ( i r »"rrn) knock - o u t r e a c t i o n , t h a t have the o u t g o i n g p i o n and n u c l e o n measured i n c o i n c i d e n c e , now underway a t LAMPF and SIN, d e t a i l e d c a l c u l a t i o n s o f t h i s r e a c t i o n have become d e s i r a b l e . As s t a t e d e a r l i e r , t h e s e c a l c u l a t i o n s w i l l p r o v i d e a good t e s t o f o p t i c a l model p o t e n t i a l s . F u r t h e r m o r e , t h e q u a s i - f r e e n a t u r e o f t h e r e a c t i o n w i l l a l l o w an e x a m i n a t i o n o f t h e two-7 body i n t e r a c t i o n between t h e p i o n and t h e s t r u c k n u c l e o n . As t h i s i n t e r a c t i o n o c c u r s w i t h i n t h e n u c l e u s , i t i s d i f f e r e n t from t h e f r e e - s p a c e r e a c t i o n i n two a s p e c t s . F i r s t l y , s i n c e t h e n u c l e o n r e s i d e s i n t h e f e r m i s e a of t h e n u c l e u s , i t i s p r e v e n t e d , by t h e P a u l i E x c l u s i o n P r i n c i p l e , from h a v i n g a f r e e range o f e n e r g i e s . S e c o n d l y , because t h e n u c l e o n i s bound w i t h i n t h e n u c l e u s , t h e energy r e q u i r e d t o f r e e i t o causes t h e two-body r e a c t i o n t o be o f f - s h e l l . The purpose o f t h i s t h e s i s i s , t h e r e f o r e , t o p r e d i c t t h e b e h a v i o r o f t h e s e e f f e c t s on t h 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 o f t h e knock-out r e a c t i o n . The t h e o r y o f t h e knock-out r e a c t i o n w i l l be d i s c u s s e d i n S e c t i o n I I . S e c t i o n s I I I - 3 -and IV w i l l give a detailed examination of the components of the c a l c u l a t i o n , namely, the Two-Body t Matrix and the Distorted Wave Matrix Element. The f i n a l sections w i l l focus on the res u l t s of the knock-out reaction c a l c u l a t i o n . The formal framework used i n t h i s analysis of the knock-out reaction i s the Distorted Wave Impulse Approximation (DWIA). This approximation considers the reaction to he a sequential process. The p a r t i c l e incident on a nucleon i n the nucleus has i t s wave function d i s t o r t e d by the nucleus. The incident p a r t i c l e undergoes a two-body i n t e r a c t i o n with the bound nucleon. These two p a r t i c l e s then leave the nucleus, which acts upon them to d i s t o r t t h e i r respective wave functions. The general form of the T matrix describing t h i s process i s : ( l . D where t 7 r n i s the two-body ( TT n) t matrix /*/ i s the f i n a l nucleon distorted wave function K fi i s the f i n a l pion distorted wave function K i r ^Kir ^ S i n c ; L C * e n t P i ° n distorted wave function % i s the nucleon bound state wave function K i s a kinematic factor. The DWIA has been shown to be the f i r s t term i n a serie s . Subsequent terms include nucleus r e c o i l e f f e c ts and multiple s c a t t e r i n g . ^ Comparisons of the DWIA with exact calculations on l i g h t nuclei show that the shape of the DWIA i s i n quite good - 4 -11 agreement, although the normalization varies widely. There-fore, one quite often considers only the r e l a t i v e "behavior of the DWIA c a l c u l a t i o n when making comparisons with experimental data. To further simplify the c a l c u l a t i o n , the two-body t matrix has been removed from the integration involved with equation (1.1). To i l l u s t r a t e t h i s , consider the momentum space 12 representation of equation (1.1): t f • J « * h A r d ' » r d , < i n^ ( P»W' e V^ ^0P») ^ H f ) ' ^A„KpJt(Ol P o f f> where K x indicates the asymptotic momentum of p a r t i c l e "x" p i s the i n i t i a l r e l a t i v e momentum i n the two-body i r n i n t e r a c t i o n p i s the f i n a l r e l a t i v e momentum i n t h i s i n t e r a c t i o n I Or) £ o n i s the energy of the interaction.® At energies above 50 MeV, the dist o r t e d wave functions become strongly peaked about the asymptotic value of t h e i r respective 12 p a r t i c l e ' s momentum and away from the (3»3) resonance, the 13 t matrix i s a slowly varying function of energy. J The t matrix, therefore, has l i t t l e influence on the integration and can be evaluated at the asymptotic momentum values and o factored out. Thus, r i f - < p M | t ( O l R f f > < ^ ^ ; l ^ « > " - a This approximation can be further improved by making correc-tions to P o n i £ o n » a n <* P Q f f f o r "kne effects of the nucleus 12 on the momentum of the incident and outgoing p a r t i c l e s . - 5 -Since t h i s adds s i g n i f i c a n t l y to the complexity of the ca l c u l a t i o n , t h i s has not been done here. As indicated i n the (p,2p) reaction, consideration of the speci a l case where the protons emerge at equal but opposite angles greatly s i m p l i f i e s the c a l c u l a t i o n , while r e t a i n i n g i t s s e n s i t i v i t y to d i f f e r e n t o p t i c a l model potentials. S i m i l a r l y , s e l e c t i n g a spe c i a l geometry w i l l reveal the s e n s i t i v i t y of s p e c i f i c aspects of the knock-out reaction; i n th i s case, s e l e c t i n g a geometry that w i l l emphasize the behavior of the two-body t matrix i s desired. I f one makes the Plane Wave Impulse Approximation that the nucleus does not d i s t o r t the wave functions, the Distorted i Wave Matrix Element (DWME) becomes: Note Kn'K'v ( 1 . 5 ) where K n i s the i n i t i a l nucleon momentum i n the nucleus. Therefore, i n t h i s approximation the DWME depends only on the i n i t i a l nucleon momentum. Furthermore, since the d i f f e r e n t i a l cross section i s summed over a l l possible orientations of the bound nucleon, the r e s u l t i n g spherical symmetry causes the cross section to be dependent only on the magnitude of the momentum. The momentum of the nucleus i s assumed not to change i n the knock-out reaction, and since the t o t a l nucleus and nucleon i n i t i a l system i s stationary i n the lab frame, the i n i t i a l nucleon momentum i s equal to and opposite the f i n a l - 6 -momentum of the nucleus. For these reasons, i t i s expected that a geometry which f i x e s a l l possible variables except the nuclear r e c o i l angle w i l l emphasize the behavior of the two-body t matrix i n the f u l l d istorted wave c a l c u l a t i o n of the d i f f e r e n t i a l cross section. As the distorted wave functions destroy the spherical symmetry present i n the plane wave 25 approximation, J the DWME i s expected to cause some v a r i a t i o n i n the d i f f e r e n t i a l cross section even i n such a geometry. However, the factors which produce t h i s v a r i a t i o n can be minimized. For a given bound state nucleon, the DWME i s a function of the incident and outgoing p a r t i c l e momentum only. I f the p a r t i c l e energies are held constant, or at most, vary only slowly with the r e c o i l angle of the nucleus, then the variations i n the DWME w i l l be due mainly to the changing orientations of the p a r t i c l e s . This w i l l simplify the behavior of the DWME and w i l l allow i t s influence on the d i f f e r e n t i a l cross section to be more predictable. A geometry that w i l l s a t i s f y these conditions s p e c i f i e s o the following parameters: Ei i s the incident pion k i n e t i c energy i s the binding energy of the nucleon 6r i s the r e c o i l momentum of the nucleus i s the r e c o i l angle of the nucleus A ( 1 . 6 ) where T$ i s the f i n a l pion k i n e t i c energy i s the f i n a l nucleon k i n e t i c energy. By varying 0 R i t i s expected that the behavior of the two-body t matrix w i l l be the dominant feature of (equation (1.1)). However, s e l e c t i n g t h i s geometry does not mean that information cannot be obtained from the DWME. This matrix element i s calculated i n coordinate space and involves an in t e g r a l i n r\ As w i l l be shown i n Section I I I , one can determine the region i n the nucleus that has the most influence on the reaction by examining t h i s i n t e g r a l . The two-body t matrix i s influenced by the nucleus due to the Pauli Exclusion Principles therefore, knowing the region where the reaction i s l o c a l i z e d w i l l enable one to choose a suitable value f o r the fermi momentum. - 8 -Section II Theory The knock-out reaction can be described as a three body reaction. The general form of the n o n - r e l a t i v i s t i c three body t r a n s i t i o n operator "U", which describes the reaction when p a r t i c l e "a" i s incident on the bound pa i r "be" r e s u l t i n g i n the bound pa i r "ac" and the free p a r t i c l e "b" i s : 9 Uab(z) » -(.-«Sab)(Ho-z) + V-Va-Vb40bVb-VQGU)VB ( 2 > 1 ) where Vx i s the i n t e r a c t i o n between the two p a r t i c l e s other than p a r t i c l e "x" v„-v-vx Z, i s the t o t a l energy of the system. For the case where p a r t i c l e "a" knocks the bound pair "be" a p a r t , 1 0 U o a = V a - V Q V a ( 2 . 2 ) Since the f i n a l state i s three free p a r t i c l e s , z = H . o For the s p e c i f i c case of a pion " i r " knocking a nucleon "n" out of the nucleus "N": UTTOM* tfir-VCHz)^ ( 2 . 3 ) where V T * + V™ - 9 -VTn i s the pion nucleon pote n t i a l i s the pion nucleus pote n t i a l VnNj i s the nucleon nucleus potential Z i s the t o t a l energy without i n t e r a c t i o n s E i s the t o t a l energy U i s the sum of the interactions within the re s i d u a l nucleus. Following the derivation of Kazaks and Koshel,"'"0 i t i s shown that n t r + ^vfSfr^ (2.4) The f i r s t term i n t h i s expansion corresponds to the Distorted Wave Impulse Approximation (DWIA) fo r the knock-out reaction. .CL__ acts on the f i n a l "TT" and "n" plane waves and d i s t o r t s TT n them by the p o t e n t i a l N and V n N respectively. -Tin- acts on the i n i t i a l pion plane wave and d i s t o r t s i t by the pote n t i a l VTTN* "two-body t matrix describing the TTn reaction. The second term describes multiple scattering effects and the t h i r d term describes nucleus r e c o i l e f f e c t s . These subsequent terms have been considered and are found to leave 12 the r e l a t i v e shape of the d i s t r i b u t i o n unchanged. Thus the knock-out reaction can be quite accurately described by the matrix M • <<; I nJJ f l J^ f > ( 2 . 5 ) - 10 -where i s i n i t i a l pion plane wave i s the nucleon hound state wave function i s "the f i n a l pion plane wave i s "the f i n a l nucleon plane wave. Using the operators n and X l ^ r e s u l t s i n where i s the plane wave distorted "by the potential v a N » a As stated i n the Introduction, the matrix element has been fa c t o r i z e d to simp l i f y the c a l c u l a t i o n . Therefore, where t n n i s the Trn two-body t matrix, and i s the Distorted Wave Matrix Element (DWME). ir n n The two-body t matrix and the ef f e c t that the nucleus has on i t w i l l be considered i n Section I I I , while the Distorted Wave Matrix Element w i l l be considered i n Section IV. The d i f f e r e n t i a l cross section f o r a three p a r t i c l e f i n a l state with momenta of p a r t i c l e s "1" , "2", and "3" i n the range "d 3k 1", "d 3k 2", and "d 3k 3" respectively i s : 1 ^ For the knock-out of p a r t i c l e "2" from "3" by "1" , t h i s has 1 1 been shown to have the form: J a<r z — k|k**Ez£^ M1** <2-9) dE.dn.dna k eE 2 i [k,E3 + E2(k2-k0cos821 k2cos(e,-S2))J dsi - 11 -where Et| E2, E3 are the f i n a l energies of p a r t i c l e s "1", " 2 " , and "3" respectively £ 2- i s the i n i t i a l energy of the bound p a r t i c l e &t,Qi are the f i n a l angles of the outgoing p a r t i c l e s "1" and " 2 " with respect to the incident p a r t i c l e "1" 0 i s the DWME <C/^K'J I Yi\ ^ 4^ - i s the irn two-body d i f f e r e n t i a l cross section. oil The cross sections 6<r and cto d(E,-ez)dxi.,daz 6Bzdd,da2 can be obtained from equation ( 2 . 9 ) by multiplying by the appropriate Jacobian. i - 12 -Section III Two-Body t Matrix The two-body t matrix describing the pion nucleon i n t e r a c t i o n has the same general form as that describing free space scattering. This i s then corrected f o r the effects of Pauli exclusion i n the nuclear medium and f o r the fact that the nuclear binding energy causes the reaction to become o f f - s h e l l . The general form of the free space d i f f e r e n t i a l cross section at a center of mass momentum "k" and scattering angle "6 " i s : l 6 2SzL--i Trace (M/>M+) d a (3.1) where p i s the density matrix M i s the scattering matrix element, M - f(0) + Lq(e)cr-n and g(6) i s the spin f l i p s cattering amplitude f (0) i s the non-spin f l i p s cattering amplitude C? i s the Pauli spin operator 7T i s a vector perpendicular to the plane of scattering. For a pion nucleon system, isospin and t o t a l angular momentum considerations lead to the following forms for 17 f ( e ) and g ( # ) ( w i t h s- and p-waves) $ ' (a) f o r Tr +p —> TT+p f (6) ' Fj + (2.^ 33 * F3|) COS 6 + cou lomb correc+ions g(e)'(F3r F33) sine (b) f o r xr"p — * TT"p f (©) r f [F 3 + 2 F , + ( 2 F 3 3 1 F 3 1 + 4 F , 3 + £F],)cos© + coulomb correc+ions] gCe) 1 !^- F 3i - 2F.3- 21=;.)s^ e] ( 3 # 3 ) (c) f o r Tr'p -T> TT„ f ( 0 ) ' f f a - F , • (ZF58 - 2F, 3 • F 3 J - F „ )cos e] ^>f [(fij-FIj + F i i - F i i ) ^ fl] where F * -and o c = 2 ( i s o s p i n ) , 2 ( t o t a l a n g u l a r momentum) a r e phase 1 ? s h i f t s d e t e r m i n e d by t h e f i t o f Salomon. J To c o r r e c t f o r o f f - s h e l l e f f e c t s , t h e s e p a r a b l e p o t e n t i a l s 18 o f Thomas have been used. B r i e f l y , i f one c o n s i d e r s e x p a n s i o n o f t h e p o t e n t i a l i n t o s p h e r i c a l harmonics, frf\*t V ^ W ^ W (3..) 19 and assumes \ W ) ' W P r (3.5) 1 8 t h e n , t h e s p h e r i c a l harmonic e x p a n s i o n o f t h e t m a t r i x becomes: ti jWp;E)^,<pf)^(E) 9 y(p) 0 . 6 ) W h i l e t h e s e p a r a b l e p o t e n t i a l s were f i t t e d q u i t e a c c u r a t e l y t o Salomon's phase s h i f t f i t s , t o a v o i d d i s c r e p a n c i e s i n t h e o n - s h e l l c r o s s s e c t i o n s , t h e "g, ." form f a c t o r s a r e used t o - I n -correct , rather than cal c u l a t e , the o f f - s h e l l t matrices: (3.7) where the f u l l y on-shell t, .(E) are obtained from Salomon's J phase s h i f t f i t s : f I r ~ F « u(k) • m r A k (3.8) To correct f o r the eff e c t s of Pauli exclusion, i t i s i l l u s t r a t i v e to consider the form of the free scattered t matrix: t - V + V Q a t (3.9) or, p i c t o r i a l l y : TT, TT The nucleus r e s t r i c t s the nucleon from having a free range of momentum. I f the nucleus i s considered to be a fermi gas, t h i s has the ef f e c t of preventing the nucleon from having a momentum les s than k f, the momentum of the sea. P i c t o r i a l l y , t h i s can be represented by: n or by: where t * V W G 0 Q p o u l i t Qpouii 0 when k < kt I when k > kt (3.10) - 1 5 -Thus, when the t matrix i s evaluated between i n i t i a l and f i n a l states, i t has the form ^ < p l i . . l P W . (P P) J ^ ' W ^ W V?-> E> Using the separable potentials (equation ( 3.5))» V V P ) 0 y ( e V p , ) (3.U) where ? /,*•j » * , «\ 00 > ^ ' J E-ECp-) ( 3 - 1 2 ) To simplify t h i s i n t e g r a l , Q p a u i i i s expanded i n spherical 20 harmonics and only the f i r s t term i s retained. To avoid small on-shell differences, Dn.p* * ' * ) ' " g - ( 3 . 1 3 ) where ( / > u ( « ) r ™ n ( k + m i r J With these corrections applied to the t matrices, the corrected F^ i n equation ( 3 . 3 ) are: R** -TTuCkJtflt ( 3 . 1 4 ) S e c t i o n IV D i s t o r t e d Wave M a t r i x Element W i t h t h e f a c t o r i z e d two-body t m a t r i x p r e v i o u s l y d e s c r i b e d , t h e t o t a l knock-out r e a c t i o n m a t r i x a l s o c o n t a i n s a D i s t o r t e d Wave M a t r i x Element (DWME) w h i c h must now be e v a l u a t e d . ( 4 . 1 ) T h i s i s not e a s i l y done i n c o o r d i n a t e space. The c o o r d i n a t e s used a r e as f o l l o w s : (a) i n i t i a l l y , where r Q goes t o t h e c e n t e r o f mass o f n and N. Thus, 8 r*T>*\**H+**f\lri (b) F i n a l l y , S i n c e t h e i r n i n t e r a c t i o n i s so s h o r t - r a n g e , t h e s i m p l i f i c a t i o n c an be made: and r* • ~r\ - 17 -Thus, DWME •JdV^WPCcMC') <V) (4.2) where a -To evaluate t h i s i n t e g r a l , consider the distorted wave 7*(r). A p a r t i c l e incident on a nucleus w i l l f i n d i t s free space behavior (described by the Hamiltonian H ) altered by the ef f e c t of the nuclear forces. This e f f e c t may be described by an e f f e c t i v e one-body o p t i c a l p o t e n t i a l V ( r ) , and the behavior of the p a r t i c l e near the nucleus can be described by H = H Q + V ( r ) . This Hamiltonian may be separated into i r a d i a l and angular coordinates y i e l d i n g the general solution: K \ r ) 'M*rT Z i V ^ M Yf (nr)\f Uh) (4.3, where £lft£l^ are the directions of r and the incident wave vector k i s a spherical harmonic U(kf) ^ s ^ h e s o l u " f c i ° n " t o t n e r a d i a l p a r t i a l d i f f e r e n t i a l equation and w i l l be discussed l a t e r i s the coulomb phase s h i f t . In the single p a r t i c l e s h e l l model, a bound nucleon may also be described by an e f f e c t i v e one-particle Hamiltonian, H. (This i s c l o s e l y related to the o p t i c a l p o t e n t i a l developed above, except that at negative energies, there are no open reaction channels.) The wave function of a nucleon i n the - 18 -s h e l l model state (l,m,n) i s written* (4.4) Thus, the wave functions i n the DWME are described by-expansions i n spherical harmonics. Using the d i r e c t i o n of the incident pion to determine the orientation of the system A . =0, equation (4.3) reduces to: K o ;£'.4wfc«rrZ lV*U,U..«r)Y,(nr) fig- ( 4 . 5 , and the DWME has the form: J r * 2 1 Using the spherical harmonic i d e n t i t y , (4.7) the t o t a l angular i n t e g r a l has the form: J Y U \ V T » 4 T T ( 2 t , 0 <Ll 0 00|Ll e lO><l\m,m,l l l Um> *<U 1 oo|v, l v l 0 ><V^.^IVUU> ( ^ 8 ) where m = m1 + m2> (Note that the Clebsh Gordon c o e f f i c i e n t s are non-zero only when L + 1 q + l + l i s even.) Combining equations (4.6) and (4.8) gives the f i n a l - 19 -form of the DWME: DWME • ^ J r* (21 * >) 0 K^l 0 00|Ll clO^Ll 0rwO|L\ plm><L lU<?0|l,UlO><l lUm , rn 1|l Il llm> (4.9) where L + 1 + 1> + 1 0 i s even and m = m. + m0. o l c i d . The i n t e g r a l (4.10) involves just the r a d i a l parts of- the wave functions f o r the bound state and scattering equations (4.3) and (4.4). In the case of the nucleon, t h i s wave function i s the solution of the Schroedinger equation since the mass i s so large and the energies involved are r e l a t i v e l y small: V>(r)*^r[E-V(r)]?(r)*0 1 The separated r a d i a l equation i s : (4.11) d r 1 (4.12) For the i n i t i a l bound state, E i s the binding energy which i s negative. For the f i n a l , unbound state, E i s just the asymptotic energy; i n t h i s case using the de Broglie wave number, • W - 20 -e q u a t i o n (4.12) may be w r i t t e n as: d 2 (4.13) As u s u a l , t h e o p t i c a l p o t e n t i a l f o r t h e n u c l e o n i s t a k e n 4 t o have a Wood-Saxon shape w i t h a Thomas s p i n - o r b i t term: ( 4 . 1 4 ) r-fti V(r)SVeCr)-URf(r)-ifwvSv(r)*W030V-)]+(^)*(u5o*iW4e) where - f ( j » ) r ( | + c R R r a - d i u s o f t h e n u c l e u s / T r V » ^ v ( r ) s v * ^ / R I i s t h e r a d i u s o f a b s o r p t i o n q p C r ) * * ^ (4.15) VI + « •% / r , \ / ^av^N*1 Ro i s "the r a d i u s o f t h e s u r f a c e e f f e c t s Vc(^)s r n r^^c R c ^ h e c n a r S e r a d i u s F o r t h e bound n u c l e o n t h e a b s o r p t i o n terms a r e z e r o , hence Wy = WD = W s o = 0. The o t h e r parameters f o r t h e bound s t a t e have been f i t by E l t o n and S w i f t and a r e g i v e n i n T a b l e 1. Note t h a t t h e r a d i u s p arameters and U R a r e s p e c i a l l y s e t by the program t o f i t t h e RMS r a d i u s and s e p a r a t i o n o f t h e bound n u c l e o n . F o r t h e unbound n u c l e o n , t h e parameters have been 4 d e t e r m i n e d by J a c k s o n and t h e v a l u e s a r e g i v e n i n T a b l e 2. - 2 1 -Table 1 Wood-Saxon Bound State Potential Parameters f o r 0 R c 3. 15 fm RR = 3. 15 fm R s o = 3. 15 fm aR = 0. 65 fm a = so 0. 65 fm V S Q = -9. MeV 1 P ^ 2 separation energy = 1 2 . 1 MeV U R = 56.970 MeV RMS radius = 2 . 6 7 5 fm 1 P ^ 2 separation energy = 18 .45 MeV U R = 5 8 . 4 4 2 MeV RMS radius = 2 . 7 4 2 fm Experimental RMS radius = 2 . 6 7 4 fm - 22 -Table 2 P o t e n t i a l Parameters f o r Proton S c a t t e r i n g from 0 UR = 24 . 1 3 (1 - 0 . 0 0 3 5 9 E ( = 2 7 . 0 5 wD 0 . 0 U s o - - 2 . 0 6 4 W s o = 0 . 0 RR = 3.664 a R = 0 . 5 5 ^ R I = 2 . 3 4 8 aj = 0 . 6 1 2 R s = 2 . 3 8 9 a s = 0 . 4 9 2 R c = 3.820 - 23 -These parameters g i v e good f i t t o d a t a f o r p r o t o n e l a s t i c s c a t t e r i n g from 0 i n t h e energy range 65 t o 156 MeV. I n t h e case of t h e p i o n wave f u n c t i o n , t h e a p p r o p r i a t e 22 s c a t t e r i n g e q u a t i o n i s a l i n e a r i z e d K l e i n - G o r d o n e q u a t i o n , where k i s the a s y m p t o t i c p i o n momentum and u3 i s t h e r e d u c e d p i o n energy i n t h e p i o n n u c l e u s c e n t e r o f mass. The o p t i c a l 22 p o t e n t i a l used i s n o n - l o c a l , t h e g e n e r a l form b e i n g : EcDUopt'qCr)*7-*(r)\7 T ' v / v ( 4 . 1 8 ) Because o f t h e g r a d i e n t t erms, t h e s u b s t i t u t i o n ' ' (i-a)''* i s r e q u i r e d t o s o l v e e q u a t i o n ( 4 . 1 ? ) . The r e s u l t a n t form i s : ( 4 . 1 9 ) T h i s e q u a t i o n c a n be s e p a r a t e d i n t o r a d i a l and a n g u l a r wave f u n c t i o n s , t h e e q u a t i o n f o r t h e r a d i a l wave f u n c t i o n d e f i n e d as: where ( 4 . 2 0 ) ( 4 . 2 1 ) and where t h e r a d i a l wave f u n c t i o n o f t h e p i o n i s : (J r ( 4 . 2 2 ) - 24 -(4.23) The form of the o p t i c a l p o t e n t i a l used i n t h i s c a l c u -l a t i o n was developed hy S t r i e k e r , McManus, and Carr: 2-^ where b(r) • /Qt(bo/9(r)- E^b.^Cr)) (4.24) CCr) * ^  (C p /»(r) - EWC, c>Cr)) ( 4 . 2 5 ) l-W-h¥(V)cCr)r (4.26) and«^» ( r * ) y^Cr )C r * ) i s the d i f f e r e n c e between the neutron and proton d e n s i t i e s . (4.2?) F i n a l l y , we note t h a t yO^ and y©2 are k i n e m a t i c a l f a c t o r s , E r i s "the charge of the pi o n , and V (r) i s the coulomb p o t e n t i a l . The c o e f f i c i e n t s b Q and c Q can be c a l c u l a t e d from pion nucleon e l a s t i c s c a t t e r i n g data, and C Q takes i n t o account a b s o r p t i o n i n the nucleus. The E r i c s o n - E r i c s o n f a c t o r , L ( r ) , takes i n t o account higher order e f f e c t s of m u l t i p l e s c a t t e r i n g . The f a c t o r A determines the degree to which p-wave s c a t t e r i n g i s reduced. The values of these 2 parameters are given i n Table 3. The d i f f e r e n t i a l equations f o r 1^ (equations (4.12) and ?4 (4.20)) are solved n u m e r i c a l l y using the Numerov technique. This technique determines the value of the r a d i a l wave f u n c t i o n at s e l e c t e d p o i n t s , the step s i z e "h" between each p o i n t being f i x e d . The e r r o r due to t h i s method i s of order h-^, so tha t i f the step s i z e i s halved, the e r r o r i s reduced by 2^. Table 3 Potential Parameters for Pion Scattering from 0 30 MeV 40 MeV 50 MeV 116 MeV 180 MeV 220 MeV b o (fm) -0.038ti0.003 -0.038*-i0.004 -0.038+i0.006 -0.09+ 10.018 -0.12+ iO.032 -0.13 + 10.042 (fm) -0.11 - iO.OOl -0.11 -iO.OOl -0.11 -iO. 002 -0.12+ iO.OOl -0.12 + i0.008 • -0.12+ 10.012 B o (fm 4) -0.18 + iO.18 -0.18 +i0.l8 -0.18*10.18 0.0 "+ i0.22 0.0 +i0.26 0.0 + i0.26 c o (fm 3) 0.68 + 10.007 0.68 + i0.015 0.68+ iO.029 0.81 + 10.31 0.12 + i0.70 -0.23i-i0.52 °1 (fm 3) 0.62 +-i0.004 0.62 + i0.007 0.62 +i0.0l4 0.44+ 10.16 0.08+ iO.35 -0.11+ i0.26 ' C 0 (fm 6) -0.41+ i0.4l -0.4l + i0.4l -0.41 +10.41 0.0 + il .35 0.0 + ii .29 0.0 + iO.65 For energies between those described above , values for the parameters have been linearly interpolated. - 26 -The r a d i a l wave functions are thus integrated using LL 24 Simpson's method, which has an error of order h ; Since the smaller the step si z e , the longer the computation time, there i s a compromise between accuracy and e f f i c i e n c y . Consideration of various step sizes resulted i n the optimal value h = 0.2 fm "being adopted. Reducing t h i s value by h a l f only resulted i n changes i n the t h i r d s i g n i f i c a n t figure, suggesting that the value chosen i s s u f f i c i e n t l y accurate. Another consideration i s the upper bound on the i n t e -gration variable. Past a c e r t a i n radius, the integrand w i l l be n e g l i g i b l e , so that c a l c u l a t i o n beyond t h i s point w i l l not s i g n i f i c a n t l y improve the r e s u l t . This radius i s chosen as being the point at which the amplitude of the bound state r a d i a l wave function f a l l s below 0.1 percent of i t s maximum value. This i s generally i n the range of 12 fm. A l l r a d i a l wave functions are calculated out to t h i s point and then matched v i a logarithmic derivatives to the asymptotic coulomb wave functions so they may be properly normalized. As noted i n equation (4.9), the DWME involves a summation of the contributions of each p a r t i a l wave i n the expansion of the respective unbound wave functions. As there are i n f i n i t e numbers of p a r t i a l waves i n each expansion, and one i s l i m i t e d by computation time as to how many terms may be considered, i t i s important to determine at which point further terms i n the series expansion have a n e g l i g i b l e influence on the ca l c u l a t i o n . Each p a r t i a l wave corresponds to a tr a j e c t o r y having an angular momentum "1" about the nucleus. For large values of - 2 ? -"1" , the p a r t i a l waves correspond to t r a j e c t o r i e s which completely miss the nucleus and therefore have l i t t l e influence on a reaction in v o l v i n g the nucleus. Since the region pertinent to t h i s c a l c u l a t i o n i s fix e d by the bound state wave function, which goes r a p i d l y to zero outside the nuclear region, p a r t i a l waves of high angular momentum can be neglected. The computer program, therefore, evaluates the eff e c t of each successive p a r t i a l wave on the c a l c u l a t i o n and when the contribution of the " l " t h p a r t i a l wave i s less than 0.5 percent of the running t o t a l , the series i s terminated. The discussion to t h i s point has considered a pion incident on a s p e c i f i c nucleon i n the nucleus. Experimentally, the only c r i t e r i a f o r s e l e c t i n g which nucleon has been struck i s the binding energy; one cannot be more precise than s t a t i n g which nuclear s h e l l the nucleon came from. As there may be more than one nucleon present i n t h i s s h e l l , one must consider the effect of t h i s m u l t i p l i c i t y . Assuming a l l other nucleons not i n the given s h e l l as being part of an i n e r t core, the t o t a l i n i t i a l wave function i s : ^ ' C ^ t f U . , «,,....«**) ( 4 . 2 8 ) where C i s the normalization constant i s the pion wave function i. A .f are "the coordinates of the nucleons i n the sp e c i f i e d s h e l l R i s the coordinate of the remaining nucleus. Since the nucleons i n the s h e l l are i d e n t i c a l , we should anti-symmetrize i n ^  the coordinates (%^,...4n)« The - 28 -i n i t i a l wave f u n c t i o n t h e n t a k e s the form: where N i s the t o t a l number o f n u c l e o n s Q i s t h e p e r m u t a t i o n o p e r a t o r Cq^  g i v e s t h e s i g n based on f e r m i a n s t a t i s t i c s . Note t h a t s i n c e the p i o n i s t o t a l l y d i s t i n g u i s h a b l e , i t has no e f f e c t on t h e s t a t i s t i c s . S i n c e t h e "n" n u c l e o n s i n t h e s p e c i f i e d s h e l l a r e i d e n t i f i a b l e by t h e i r b i n d i n g energy, t h e n o r m a l i z a t i o n c o n s t a n t i s : S i m i l a r l y , t h e f i n a l s t a t e i s : (4.30) where i s the p i o n wave f u n c t i o n z^p i s the unbound n u c l e o n d i s t o r t e d wave G' i s the n o r m a l i z a t i o n c o n s t a n t £ t * are t h e c o o r d i n a t e s o f t h e r e m a i n i n g n u c l e o n s i n t h e s p e c i f i e d s h e l l R i s t h e g e n e r a l i z e d c o o r d i n a t e d e s c r i b i n g a l l t he o t h e r n u c l e o n s . I f we n e g l e c t t h e exchange term i n v o l v i n g t h e o u t g o i n g p a r t i c l e , w h i c h as u s u a l , s h o u l d be s m a l l , t h e a n t i - s y m m e t r i z e d wave f u n c t i o n i s : rT N i ^ p P ' (4.31) S i n c e t h e r e a re ( n - 1 ) n u c l e o n s i n t h e s p e c i f i e d s h e l l , (N - n) i n t h e i n e r t c o r e , and one f r e e n u c l e o n , t h e no r m a l -- 29 -i z a t i o n constant i s : N » CW Cn-i)!(N-n)'.(i)! (4.32) Considering the overlap of the i n i t i a l and f i n a l wave functions, , ' (^.33) However, ^ ,. CC' = N ^ n!(N-n)( and since only permutations which change nucleons i n the sp e c i f i e d s h e l l and/or nucleons i n the in e r t core among them-selves give non-zero overlaps, there are only n!(N - n)! non-zero terms i n the series. Hence, where P^p i s the wave function of the f i n a l unbound nucleon i s the wave function of the struck bound nucleon 7°^ i s the pion wave function. The magnitude i s not the only information that can be gleaned from the DWME. I t i s noted that to t h i s point, the matrix element i s calculated f o r a s p e c i f i c orientation "m" (in the scattering plane) of the angular momentum "1" of the bound state wave function, i . e . , The t o t a l magnitude of the DWME influence on the t o t a l scattering cross section i s simply the weighted sum i n "m" of the magnitudes J s i l ^ ' T^e weighting factors given i n - 30 -Table 4 are just the p r o b a b i l i t i e s that an unpolarized nucleon w i l l have a given orientation. The ef f e c t i v e p o l a r i z a t i o n of the nucleon i n the nucleus can be determined by r o t a t i n g the g™ matrix elements to be perpendicular to the scattering plane, then summing i n "m" the magnitudes g™J2 with the weight factors f o r a spin up nucleon and then by summing the weight factors f o r a spin down nucleon. The eff e c t i v e p o l a r i z a t i o n i s just the normalized difference between these two, and i t i s used i n the two-body t matrix c a l c u l a t i o n . Since the plane wave matrix element i s symmetric i n spin orientation, analysis of the e f f e c t i v e p o l a r i z a t i o n w i l l provide a good i n d i c a t i o n of the influence of nuclear OK d i s t o r t i o n on the wave functions. J Additional information which can be obtained from the c a l c u l a t i o n of the DWME i s the region of the nucleus i n which the reaction i s l o c a l i z e d . This i s useful i n determining an appropriate value f o r the fermi momentum cut-off used i n the two-body t matrix c a l c u l a t i o n . The c a l c u l a t i o n of the DWME involves an integration of the r a d i a l wave functions. To the computer, t h i s i s simply a summation of a sequence of terms evaluated at increasing r a d i i . By considering these terms, one can determine the degree of influence each has on the t o t a l matrix element, the degree of influence of the "r " t h term beings (4.36) where W m i s the orient a t i o n weight factor q i s the DWME calculated without the " r " t h term. - 31 -T a b l e 4 P o l a r i z a t i o n Weight F a c t o r s f o r g^ When 1 = 1 m = -1 m= 0 m = l s p i n up 0.333 0.167 0.0 1 P l / 2 ' s p i n down 0.0 0.167 O.333 IP3/2 s p i n u p 0.083 O.I67 0.250 s p i n down O.250 0.16? 0.083 - 32 -Notice that A r w i l l be a maximum f o r the term that has the most influence on the c a l c u l a t i o n , thus i n d i c a t i n g the lo c a t i o n i n the nucleus where the reaction predominantly occurs. - 33 -Section V Results The knock-out reaction O l 6 ( i r + tT^)N'1"^ was considered and calculations were made of the cross sections 6<Fr-rn)dntdA- (equation (2.9)). As noted i n the Introduction, the geometry selected to specify the kinematics varies the nucleus r e c o i l angle while keeping a l l other kinematic variables constant. Since t h i s keeps the plane wave matrix element constant, i t was expected that the DWME should be slowly varying. To examine the v a r i a t i o n of the DWME, the r a t i o of DWIA to PWIA was plotted as a function of nucleus r e c o i l angle f o r incident pion energies of 60, 116, 180, and 220 MeV on protons i n the 1P^2 a n d 1 P3/2 s h e l l s > The cross sections were also plotted as a function of nucleus r e c o i l angle ( a l l other kinematic variables held constant) f o r the above mentioned energies and proton s h e l l s . To t e s t the o f f - s h e l l and Pauli exclusion e f f e c t s on the t o t a l cross section, the two-body t matrix was calculated on-shell and o f f - s h e l l , at f i v e d i f f e r e n t values of fermi momentum cut-off. The fermi momenta sp e c i f i e d correspond to: free space scattering (0.0 fm ), scattering at the edge of the nucleus (0.7 fm ), scatt e r i n g at the RMS radius (1.0 fm ), and scattering at the center of the nucleus (I .36 fm ). Also, _ 3^ -through an examination of the r a d i a l c h a r a c t e r of the D//ME, an e f f e c t i v e fermi momentum was c a l c u l a t e d . This corresponds t o the l o c a t i o n i n the nucleus where the i n t e r a c t i o n i s most predominant, and should be the c u t - o f f momentum f e l t by the majo r i t y of the two-body i n t e r a c t i o n s . As noted i n S e c t i o n IV, the DM.E can be c a l c u l a t e d f o r s p i n up and s p i n down o r i e n t a t i o n s of the proton. The normalized d i f f e r e n c e of these two r e s u l t s give an e f f e c t i v e p o l a r i z a t i o n of the proton. The e f f e c t of t h i s p o l a r i z a t i o n was a l s o examined i n combination with the o f f - s h e l l and P a u l i e x c l u s i o n e f f e c t s by c a l c u l a t i n g the two-body i n t e r a c t i o n w i t h and without the e f f e c t i v e proton p o l a r i z a t i o n (equation (3.2)). To f a c i l i t a t e examination of the r a d i a l c h a r a c t e r of the DWME f o r each angle of nuclear r e c o i l , three dimensional p l o t s of A values (equation (4.36)) were made f o r i n c i d e n t pion energies of 60, 116, 180, and 220 MeV w i t h protons i n the l P 1 / > 2 a n d 1 P 3/2 s n e l l s > A s "the 1 P 3/2 P i 0 ' * ' 3 were very s i m i l a r to the lP-jy2 P i 0 " * - 3 ^ o r each energy, only the p l o t s f o r !P-jy 2 P1"0^0*1 s c a t t e r i n g have been presented. The cross s e c t i o n s , (equation (2.9)) were then c a l c u l a t e d using the geometry of L e v i n and Eisenberg.^ This geometry r e q u i r e s the outgoing p i o n energy and angle to be s p e c i f i e d ; the cross s e c t i o n s were p l o t t e d as a f u n c t i o n of the f i n a l proton angle. Curves were a l s o p l o t t e d f o r s c a t t e r i n g from a lP-jy2 a n c * a ^^3/2 P r o ' t o n when - 35 -the pions are incident with energy 130 MeV and leave with energies of 51.2 or 61 MeV at an angle of 150 or 120 degrees respectively. For comparison, Levin and Eisenberg's curves of DWIA and fa c t o r i z e d DWIA calcu l a t i o n s were plotted. - 36 -Section VI Discussion Comparison of DWIA and PWIA Calculations As indicated i n the Introduction, the geometry used was selected to minimize the v a r i a t i o n of the DWME. What i s immediately apparent from the r a t i o of PWIA to DWIA ( f i g . 1-4) i s that t h i s r a t i o i s f a r from constant; further-more, the r a t i o i s f a r from unity, reaching 220 i n some cases ( f i g . 3). The size of the PWIA to DWIA r a t i o does indicate that absorption by the nucleus i s quite s i g n i f i c a n t . Examination of the three dimensional plots ( f i g . 5-8) reveals a general s i m i l a r i t y f o r s c a t t e r i n g at 180 and 220 MeV ( f i g . 7 and 8 r e s p e c t i v e l y ) . That the pion i s strongly absorbed at energies of 180 and 220 MeV i s born out by these plots. The curves peak at r a d i i of 3.6 and 4.0 fm respectively; thus, 1 6 given that the RMS radius of 0 X D i s 2.675 fm, the knock-out reaction occurs at the extreme edge of the nucleus. Analysis of the angular nature of figures 7 and 8 shows that the DWME increases with increasing scattering angle. This reveals the eff e c t that absorption has on the cross section. Pions incident on the nucleus at these energies w i l l be so r e a d i l y absorbed that the number available f o r a knock-out reaction w i l l be heavily reduced as the pions penetrate the nucleus. Furthermore, pions that are scattered - 37 -i n a knock-out reaction w i l l have a greater chance of emerging i f the path they take does not traverse much nuclear medium. Thus, pions that leave at large angles from interactions at the front of the nucleus w i l l he more p l e n t i f u l than any others. In t h i s geometry, the angle of the scattered .pion changes as the nucleus r e c o i l angle i s varied, confirming the behavior i l l u s t r a t e d i n figures 3 and 4. For pion scattering at 116 MeV the curves of the PWIA to DWIA r a t i o ( f i g . 2) are s i m i l a r to the curves at 180 and 220 MeV. Given the strong absorption of the pion at 116 MeV, i t would be expected that the v a r i a t i o n of DWME with r e c o i l angle should follow the same form as scattering at 180 and 220 MeV. An examination of the three dimensional plot ( f i g . 6) does reveal much the same form as the plots at higher energies. At 60 MeV ( f i g . 1), the DWME peaks at angles where the pion i s only s l i g h t l y deflected, and decreases as the pion sca t t e r i n g angle i s increased. The r a t i o of DWIA to PWIA i s much smaller, and the three dimensional plot ( f i g . 5) of the DWME shows that the matrix element i s peaked well within the nucleus (2.8 - 3.0 fm). As the absorption at low energies i s small and the pot e n t i a l i s a t t r a c t i v e , the pion wave functions w i l l form a focus at the back of the nucleus.* Interactions w i l l therefore occur at t h i s focus as well as at the front of the nucleus. This focusing e f f e c t i s more pronounced f o r outgoing pions as they have a lower energy and are absorbed to a lesser degree. Pions leaving with small angles of def l e c t i o n w i l l originate either at the out-going wave function - 38 -focus, at the front of the nucleus, or from the incident wave function focus at the hack of the nucleus. As the scattering angle of the pion i s increased, the out-going focus w i l l move away from the front of the nucleus, and fewer scattered pions w i l l he detected. This agrees with what i s observed i n the behavior of the DWME ( f i g . 1). Off- S h e l l E f f e c t s g As noted previously, the o f f - s h e l l e f f e c t does not have much influence on the knock-out cross section. This e f f e c t only becomes apparent at 60 MeV ( f i g . 1), where a change i n the magnitude of the cross section of 10 to 15 per-cent i s found. Pauli Exclusion E f f e c t s 8 As found by Jackson et a l . , the Pauli exclusion ef f e c t i s most dramatic at higher energies (approximately 200 MeV). However, i t i s noted that the cross sections obtained with the program-calculated e f f e c t i v e fermi momentum l i e between the curves for free scattering and scattering at the extreme edge of the nucleus. In f a c t , f o r the curves at 220 MeV ( f i g . 18 - 21), the free s c a t t e r i n g curves are almost indistinguishable from the curves using the e f f e c t i v e fermi momentum. I t i s not u n t i l the energy f a l l s to 116 MeV that e f f e c t i v e momentum curves ( f i g . 11 - 13) are s i g n i f i c a n t l y d i f f e r e n t from free space scattering. To understand t h i s e f f e c t , i t i s i n s t r u c t i v e to examine the three dimensional plots ( f i g . 5 - 8 ) . What i s immediately apparent i n the plots f o r 180 and 220 MeV scattering ( f i g . 7, 8) i s the extreme - 39 -radius of the interaction. The DWME has a peak at J.6 fm and 4.0 fm respectively. This extreme behavior at these energies i s caused by the highly absorptive nature of the pion-nucleus i n t e r a c t i o n . I f the imaginary terms i n the pion o p t i c a l p o t e n t i a l (equation (4.23)) are removed, the DWME peaks at 2.8 fm, a value s l i g h t l y higher than the RMS radius of the nucleus (2.675 fm) because of the repulsive nature of the potential at these energies. For scattering at 116 MeV, the three dimensional plot ( f i g . 6) reveals a si m i l a r structure to figures 7 and 8 although the peak i s much more pronounced. The radius of the peak i s approximately 3.2 fm which corresponds to the radius of the Wood-Saxon pot e n t i a l of the bound state wave function (3.15 fm) (equation (4.15)). This was expected as the pion-nucleus i n t e r a c t i o n i s less absorptive at t h i s 26 energy and allows greater penetration of the nucleus. Since the nucleus has a skin thickness of 0.65 fm (Table 1), interactions occuring on the surface of the nucleus w i l l vary quite sharply with radius, a fac t indicated by the pronounced peak of the DWME ( f i g . 6). Analysis of the cross sections at 60 MeV reveals that the Pauli exclusion effect i s minimal, the greatest c o n t r i -bution being of the order of 10 percent. This i s unfortunate because the three dimensional plot ( f i g . 5) f o r t h i s energy indicates that the in t e r a c t i o n occurs within the nucleus and the e f f e c t i v e fermi momentum i s s i g n i f i c a n t (approximately 0.8 f m - 1 ) . - 40 -E f f e c t i v e P o l a r i z a t i o n As noted i n Section IV, the cross section can be constructed f o r spin up and spin down orientation of the proton. The normalized difference between these two re s u l t s gives the e f f e c t i v e p o l a r i z a t i o n of the proton. By varying the angle of nuclear r e c o i l , the p o l a r i z a t i o n was found to vary from -0.5 to +0.5 f o r scattering from the I P 3 / 2 P r o t o n and from approximately -1.0 to +1.0 f o r scattering from the 1 P l / 2 P r o' t o n (fig» 2 2 ) - This behavior did not change r a d i c a l l y as the incident pion energy was varied. The more pronounced influence of spin on the I P ^ ^ c a l c u l a t i o n was also noted by Levin and Eisenberg^ who ascribe t h i s to absorption effects and the spin o r b i t coupling of the out-going proton. The curves-plotted without the p o l a r i z a t i o n effect ( f i g . 9,10,12,13,15,17,19.21) show that at a given energy, the cross sections f o r the IP-jy^ 1 P 3/2 P r o" t o n scattering are quite s i m i l a r i n shape. This i s expected as the out-going p a r t i c l e energies and angles are almost i d e n t i c a l f o r scattering from these two proton s h e l l s , causing the large e f f e c t s due to pion absorption to be s i m i l a r . However, when the e f f e c t i v e p o l a r i z a t i o n i s turned on, t h i s s i m i l a r i t y i s no longer observed. Examination of figures 18 and 19 f o r 220 MeV scat t e r i n g from the l P ^ / / 2 Photon shows that p o l a r i z a t i o n emphasizes the peak at a nucleus r e c o i l angle of 75 degrees and reduces the structure at lower r e c o i l angles. Figures 20 and 21 f o r l"]?3/2 proton sc a t t e r i n g show that the p o l a r i z a t i o n has the reverse e f f e c t , emphasizing the peak at lower angles and - M -reducing the peak at 30 degrees. To understand t h i s behavior, i t i s i l l u s t r a t i v e to consider the behavior of the p o l a r i z a t i o n vector f o r these two cases. Figure 22 shows that the e f f e c t i v e p o l a r i z a t i o n f o r lP^/2 P r o" t o n scattering i s almost exactly-opposite and approximately a f a c t o r of two reduced from the corresponding 1P^£ Pro^on- case. This i s expected as the p o l a r i z i n g weight factors given i n Table 4 show; the spin orientation weight factors for l P j y 2 scattering are close to the opposite spin orientation factors f o r 1 P ^ 2 scattering, the IP3/2 f a c " t ° r s being more mixed. Analysis of the data at 180 Mev ( f i g . Ik - 17) reveals much the same behavior as described f o r data at 220 MeV. For 1 P l / 2 P 1" 0^ 0 1 1 scattering ( f i g . 14, 15) , the e f f e c t i v e polar-i z a t i o n emphasizes the peak at a nuclear r e c o i l angle of 60 degrees, whereas 1P^//2 P 1" 0^ 0 1 1 scattering ( f i g . 16, 17) shows that the peak at 20 degrees i s emphasized. At 116 MeV ( f i g . 11, 12), the effect i s most noticeable i n the cross sections of lP-]y 2 calculated without fermi momentum corrections. The peak at a nucleus scattering angle of 60 degrees becomes larger than the peak at 20 degrees. At 60 MeV, the spin f l i p term (equation (3.2)) i s quite small and the e f f e c t i v e p o l a r i z a t i o n causes less than 10 per-cent v a r i a t i o n i n the cross sections, thus, no curves were plotted. Comparison with the Results of Levin and Eisenberg^ Levin and Eisenberg have calculated the knock-out cross section, q t 7 (equation (2.9)) - 42 -usi n g the momentum space c a l c u l a t i o n (equation (1.2)). To evaluate the e f f e c t of the f a c t o r i z a t i o n approximation (equation ( 2 . 7 ) ) , they evaluated the DWME i n momentum space. The geometry which L e v i n and Eisenberg s e l e c t e d examines the behavior of the cross s e c t i o n as the out-going proton angle i s v a r i e d . The other kinematic v a r i a b l e s are set so the i n i t i a l proton momentum approaches zero as the out-going proton angle i s v a r i e d . I n the Plane Wave Impulse Approximation the DWME becomes the momentum space r e p r e s e n t a t i o n of the bound s t a t e wave f u n c t i o n . Since the l p j y 2 and IP3/2 bound s t a t e wave f u n c t i o n s approach zero as the momentum approaches zero, the out-going angle g i v i n g minimum i n i t i a l proton momentum w i l l correspond t o a minimum of the d i f f e r e n t i a l cross s e c t i o n . As the nucleus causes d i s t o r t i o n i n the wave f u n c t i o n s , the value of t h i s minimum should be a l t e r e d . This provides a t e s t of the d i s t o r t e d wave f u n c t i o n s . I n t h i s study, curves c a l c u l a t e d to compare wi t h L e v i n and Eisenberg's r e s u l t s d i d not c o n t a i n the s p i n o r b i t term i n the proton o p t i c a l p o t e n t i a l (equation (4.14)). I n f i g u r e 24, the curve f o r the IP3/2 c r o s s s e c t i o n peaks at the same height as L e v i n and Eisenberg's f u l l c a l c u l a t i o n without s p i n o r b i t . The minimum corresponds t o L e v i n and Eisenberg's f a c t o r i z e d c a l c u l a t i o n , i n d i c a t i n g agreement w i t h a mix of the two e f f e c t s . The l p ^ 2 c u r v e ( f i g * 23) c a l c u l a t e d i n t h i s study maintained the same form as the IP3/2 a n c * not show the smoothed-out f e a t u r e s of L e v i n and Eisenberg's r e s u l t s . The peak at minimum i n i t i a l proton momentum i n d i c a t e d i n L e v i n and Eisenberg's f u l l DWIA r e s u l t d e f i n i t e l y was not - 4 3 -observed. Both lP-jy2 and ^3/2 c u r v e s f a l l °ff more r a p i d l y than Levin and Eisenberg's at higher proton scattering angles. This i s caused by the increasing e f f e c t of pion absorption and as Levin and Eisenberg's pion o p t i c a l potential does not include true absorption, agreement was not expected. Figures 25 and 26 show much the same behavior, the lP^/2 c u r v e calculated i n t h i s study i s not as smooth as the lP^/ £ c u r v e s from Levin and Eisenberg's study. _ 44 -Section VII Conclusion The knock-out reaction i s r i c h i n information about nuclear processes and, with s u f f i c i e n t data, provides a good test of many nuclear models. As indicated i n t h i s study, the knock-out reaction i s very sen s i t i v e to d i s t o r t i o n of the wave functions, s p e c i f i c a l l y , to the effect of pion absorption. Thus, data gathered on the knock-out reaction cross section allows one to evaluate o p t i c a l model potentials. The cross sections evaluated i n t h i s study proved to be ins e n s i t i v e to o f f - s h e l l e f f e c t s . However, the Pauli exclusion ef f e c t was noted to be s i g n i f i c a n t , and c a l c u l a t i o n of the distorted wave matrix element i n coordinate space was b e n e f i c i a l i n understanding t h i s e f f e c t . Examination of the r a d i a l behavior of the DWME allows one to determine the region of the nucleus i n which the knock-out reaction occurs. The region was noted to move to the extreme edge of the nucleus as the energy of the incident pion was increased. This indicates that data gathered at high energies (around 200 MeV) w i l l contain l i t t l e information about the Pauli e f f e c t , an unfortunate consequence since the two-body pion nucleon i n t e r a c t i o n i s most sensitive to the fermi momentum cut-off o at 200 MeV. However, at lower energies (around 116 MeV), the reaction w i l l occur further inside the nucleus ( f i g . 6), - 45 -and scattering data gathered at these energies w i l l provide information on the ef f e c t of the Pauli exclusion on the pion nucleon i n t e r a c t i o n ( f i g . 11, 13). The spin orientation of the proton was found to have a major influence on the knock-out cross section because the value of the e f f e c t i v e p o l a r i z a t i o n varied s i g n i f i c a n t l y . In some cases (IP-jy^ ^his value ranged from almost -1.0 to almost +1.0. The e f f e c t i v e p o l a r i z a t i o n f o r I P P h o t o n scattering i s almost exactly opposite and approximately a factor of two reduced from the corresponding 1 P ^ £ P r o ' t o n case. In the geometry used i n t h i s study, the 1 P ^ £ 1'P3/2 cross sections calculated without the e f f e c t i v e p o l a r i z a t i o n are s i m i l a r i n shape. Thus, consideration of the difference i n shape between scattering cross sections f o r the lPjy£ and IP3/2 P1'0"'1'01'13 measured under s i m i l a r kinematic conditions w i l l allow one to describe the e f f e c t i v e p o l a r i z a t i o n of the proton i n the nucleus, and therefore, the effect of proton spin can be observed. The cross sections calculated to compare with the r e s u l t s of Levin and Eisenberg did not contain the spin orbit term i n the outgoing proton o p t i c a l p o t e n t i a l . As Levin and Eisenberg claimed that the spin orbit term should strongly influence the knock-out cross section, a proper comparison between the r e s u l t s was not possible. However, i t should be noted that the magnitude and general behavior of the cross sections are i n f a i r agreement. - 46 -Bibliography 1. McCarthy, I. E. , Nuclear Reactions (Pergamon Press, London, 1970). 2. Elton, L. R. B. and A. Swift, Nucl. Phys. A%k, 52 (1967). 3. Jackson, D. F., Nuclear Reactions (Methurn & Co. Ltd., London, 1970). 4. A b d u l - J a l i l , I . and D. F. Jackson, private communication. 5. M i l l e r , A., private communication. 6. Levin, E. and J. M. Eisenberg, Ph.D. Thesis, TAUP872-80, (1980). 7. Jacob, G. and T. A. J. Maris, Rev. Mod. Phys. 45_, 6 ( 1 9 7 3 ) . 8. Jackson, D. F., Ioannides, A. A., and A. W. Thomas, Nucl. Phys. A322, 493 (1979). 9. A l t , E. 0., Grassberger, P., and W. Sandhas, Nucl. Phys. B2, 167 ( 1 9 6 7 ) . 10. Kazaks, P. A. and R. D. Koshel, Phys. Rev. C l , 1906 (1970). 11. Young, S. K. and E. F. Redish, Phys. Rev. C10, 4 9 8 ( 1 9 7 4 ) . 12. Redish, E. F., Phys. Rev. Letters 1 1 , 617 ( 1 9 7 3 ) . 13. Salomon, M., TRIUMF Report, TRI-74-2 ( 1 9 7 4 ) . 14. Goldberger, M. L. and K. M. Watson, C o l l i s i o n Theory (John Wiley & Sons, Inc., New York, 1 9 6 4 ) . 15. Jain, M., Roos, P. G., Pugh, H. G., and H. D. Holmgren, Nucl. Phys. A153, 49 (1970). 16. Merzbacher, E., Quantum Mechanics, 2nd ed. (John Wiley & Sons, Inc. New York, 1970). 17. Sakurai, J. J., Invariance P r i n c i p l e s and Elementary P a r t i c l e s (Princeton University Press, Princeton, New Jersey, 1 9 6 4 ) . 18. Thomas, A. W., Nucl. Phys. A 2 5 8 , 417 ( 1 9 7 6 ) . - 47 -1 9 . Lovelace, C., Phys. Rev. 135, B1225 ( 1 9 6 4 ) . 20. Landau, R. H. and A. W. Thomas, Nucl. Phys. A302, 4 6 1 ( 1 9 7 8 ) . 21. Messiah, A., Quantum Mechanics, Vol. II (John Wiley & Sons, Inc., New York). 22. K i s s l i n g e r , L. S. , Phys. Rev. .98. 761 ( 1 9 5 5 ) . 23. Strieker, K., McManus, H., and J. A. Carr, Phys. Rev. C 1 9 . 9 2 9 ( 1 9 7 9 ) . 24. Hamming, R. H., Numerical Methods f o r S c i e n t i s t s and Engineers (McGraw-Hill Book Co. Ltd., New York, 1962). 25. Jacob, G . , Maris, T. A. J., Schneider, C., and M. R. Teodoro, Nucl. Phys. A257, 517 ( 1 9 7 6 ) . 26. Sterheim, M. M. and R. R. S i l b a r , Ann. Rev. Nucl. S c i . 2 4 , 2 4 9 ( 1 9 7 4 ) . - 4 8 -Appendix Figure Captions F i g . 1 Ratio of the d i f f e r e n t i a l cross section f o r O l 6 ( i r + , • t r + p)N 1 5 calculated using the PWIA, to the d i f f e r e n t i a l cross section calculated using the DWIA. The r a t i o s are plotted as a function of the r e c o i l angle of the nucleus, 8 R . The incident pion energy, = 60 MeV, the r e c o i l momentum of the nucleus, q R = 0 . 3 fm , and the f i n a l energy r a t i o , A= 0 . 8 . The s o l i d l i n e gives the r a t i o s f o r knock-out of a 1 P ^ 2 P r o" f c o n a n c * ^he dashed l i n e gives the r a t i o s f o r knock-out of a I P 3 / 2 Pr°"t°n. F i g . 2 Ratio of PWIA to DMA d i f f e r e n t i a l cross sections f o r 0 1 6 ( T T + , t r + p ) N 1 ^ as a function of 6 R . The remaining kinematic variables are E^ = 1 1 6 MeV, q R = 0 . 6 fm , and A = 0 . 8 . The s o l i d l i n e gives the r a t i o s f o r the proton and the dashed l i n e gives the r a t i o s for a I P 3 / 2 Photon. Fi g . 3 Ratio of PWIA to DWIA d i f f e r e n t i a l cross sections f o r 0 l 6 ( t r + , TT +p)N 1^ as a function of 6R. The incident pion energy, E^ = 1 8 0 MeV. The remaining kinematic variables and explanation of l i n e s are the same as f i g . 2 . F i g . 4 Ratio of PWIA to DWIA d i f f e r e n t i a l cross sections _ 49 -f o r O l 6 ( T T + , T T +p)N 1^ as a function of 6 R . The incident pion energy, = 220 MeV. The remaining kinematic variables and explanation of l i n e s are the same as f i g . 2. Fi g . 5 Three dimensional plot of the r a d i a l character of 1 6 the DWME f o r a !P 1 // 2 P r o t o n i n 0 • T n e quantity A r (equation (4.36)) i s plotted as a function of distance from the center of the nucleus and 0 R . The remaining kinematic variables are the same as indicated i n f i g . 1. Fi g . 6 Three dimensional plot of the r a d i a l character of 1 6 the DWME f o r a lP.jy 2 P r o t o n i n 0 • T n e quantity A r i s plotted as a function of distance from the center of the nucleus and 0 R . The remaining kinematic variables are the same as indicated i n f i g . 2. F i g . 7 Three dimensional plot of the r a d i a l character of 1 ft the DWME fo r a !P 1 // 2 P r o" t o n i n 0 • T n © quantity A r i s plotted as a function of distance from the center of the nucleus and (9"R. The remaining kinematic variables are the same as indicated i n f i g . 3. F i g , 8 Three dimensional p l o t of the r a d i a l character of 1 6 the DWME f o r a l P 1 y 2 proton i n 0 . The quantity A r i s plotted as a function of distance from the center of the nucleus and 8 R . The remaining kinematic variables are the same as indicated i n f i g . 4. - 50 -F i g . 9 F i v e - f o l d d i f f e r e n t i a l cross section of the knock-1 £> out of a lP^/2 P r o ' t o n from 0 as a function of 6 R . The s o l i d l i n e gives r calculated with the o f f - s h e l l e f f e c t . The dashed l i n e i s the d i f f e r e n t i a l cross section calculated without the o f f - s h e l l e f f e c t . Both curves are calculated with a fermi momentum, = 0.8 fm and exclude the e f f e c t i v e p o l a r i z a t i o n of the nucleon. The kinematic s i t u a t i o n i s the same as indicated i n f i g . 1. F i g . 10 F i v e - f o l d d i f f e r e n t i a l cross section of the knock-1 f> out of a 1 P3 //2 P r c r t o n f r o m 0 as a function of 0 R . The explanation of the l i n e s i s the same as i n f i g . 9. The kinematic s i t u a t i o n i s the same as indicated i n f i g . 1. F i g . 11 F i v e - f o l d d i f f e r e n t i a l cross section of the knock-1 ft out of a lP-]y2 proton from 0 as a function of 9 R. The l i n e s give the cross sections calculated f o r various fermi momenta, and include the o f f - s h e l l e f f e c t and the e f f e c t i v e p o l a r i z a t i o n of the nucleon. The s o l i d l i n e gives the cross sections calculated with the fermi momentum determined by the l o c a l i z a t i o n of the knock-out reaction. In t h i s case, , , * i l o c a l 0.7 fm . The dotted l i n e gives the cross sections calculated with = 0.0, the dashed l i n e with K f = 0.7 fm" 1, the dash-dot l i n e with K f = 1.0 fm" 1 and the dash-two dot l i n e with K f = I.36 fm" 1. The - 51 -kinematic s i t u a t i o n i s the same as indicated f o r f i g . 2. Fi g . 12 F i v e - f o l d d i f f e r e n t i a l cross sections calculated without the e f f e c t i v e p o l a r i z a t i o n of the nucleon. Except f o r the p o l a r i z a t i o n , the explanation of the l i n e s and kinematic s i t u a t i o n are the same as f i g . 11. F i g . 13 F i v e - f o l d d i f f e r e n t i a l cross sections calculated 1 f f o r the knock-out of a ^^/Z P r o - t o n f r o m 0 . The cross sections have been calculated o f f - s h e l l and exclude the e f f e c t i v e p o l a r i z a t i o n of the nucleus. The explanation of the l i n e s i s the same as f i g . 11 and the kinematic s i t u a t i o n i s the same as indicated f o r f i g . 2. Fi g . 14 F i v e - f o l d d i f f e r e n t i a l cross sections calculated 1 f> f o r the knock-out of a l P j y ^ P r o t o n f r°m 0 . The cross sections are calculated at various fermi momenta and include the o f f - s h e l l e f f e c t and the ef f e c t i v e p o l a r i z a t i o n of the nucleon. The s o l i d l i n e gives the cross sections calculated with the fermi momentum determined by the l o c a l i z a t i o n of the reaction. In t h i s case, K f l o c a l * 0 , 4 f m _ 1 . The dotted l i n e gives the cross sections with -1 Kj = 0,0 fm , the dashed l i n e gives the cross _ i sections with = 0,7 fm . The kinematic s i t u a t i o n i s the same as indicated i n f i g . 3. F i g , 15 F i v e - f o l d d i f f e r e n t i a l cross sections calculated without the e f f e c t i v e p o l a r i z a t i o n of the nucleon. - 52 -Except f o r the p o l a r i z a t i o n , the explanation of the l i n e s and kinematic s i t u a t i o n i s the same as f i g . 14. F i g . 16 F i v e - f o l d d i f f e r e n t i a l cross sections calculated f o r an e f f e c t i v e l y polarized IP3/2 P*"0^011* T h e kinematic s i t u a t i o n i s the same as indicated f o r f i g . 3. The explanation of the l i n e s i s the same as f i g . 14, except that the altered l o c a l i z a t i o n of the reaction causes K« , ^ » 0.5 fm - 1. f l o c a l F i g . 17 F i v e - f o l d d i f f e r e n t i a l cross sections calculated without the e f f e c t i v e p o l a r i z a t i o n of the nucleon. Except f o r the p o l a r i z a t i o n , the explanation of the l i n e s and kinematic s i t u a t i o n i s the same as f i g . 16. F i g . 18 F i v e - f o l d d i f f e r e n t i a l cross sections calculated f o r an e f f e c t i v e l y polarized l P ] y 2 proton. The kinematic s i t u a t i o n i s the same as indicated i n f i g . 4. The explanation of the l i n e s i s the same as f i g . 14, except that the increased pion energy a l t e r s the l o c a l i z a t i o n of the reaction and t h i s changes K f l 0 c a l " °-3 ^ Fi g . 19 F i v e - f o l d d i f f e r e n t i a l cross sections calculated without the e f f e c t i v e p o l a r i z a t i o n of the nucleon. Except f o r the p o l a r i z a t i o n of the nucleon, the explanation of the l i n e s and the kinematic s i t u a t i o n i s the same as f i g . 18. Fig. 20 F i v e - f o l d d i f f e r e n t i a l cross sections calculated f o r an e f f e c t i v e l y polarized l p 3 / / 2 P1"0^011* T h e - 53 -explanation of the kinematic s i t u a t i o n i s the same as indicated i n f i g . 4. The explanation of the l i n e s i s the same as f i g . 16, except that the increased pion energy a l t e r s the l o c a l i z a t i o n of the reaction and t h i s changes K„ , , a O.1! fm ° f l o c a l F i g . 21 F i v e - f o l d d i f f e r e n t i a l cross sections calculated without the ef f e c t i v e p o l a r i z a t i o n of the nucleon. Except f o r the p o l a r i z a t i o n of the nucleon, the explanation of the l i n e s and the kinematic s i t u a t i o n i s the same as f i g . 20. F i g . 22 E f f e c t i v e p o l a r i z a t i o n of the hound proton as a function of 6 R . The s o l i d l i n e i s the e f f e c t i v e p o l a r i z a t i o n of a IV^^ proton. The dashed l i n e i s the e f f e c t i v e p o l a r i z a t i o n of a l p ^ / 2 Proton. The kinematic s i t u a t i o n i s the same as indicated i n f i g . h. F i g . 23 F i v e - f o l d d i f f e r e n t i a l cross sections der f o r a lP^^/2 P r o"t° n a s a function of out-going proton angle. The kinematic conditions are incident pion energy, E ^ = 130 MeV, out-going pion energy, E'^ = 51.2 MeV, and out-going pion angle 0^. = 150°. For the s o l i d l i n e , the cross section i s calculated on-shell and includes the e f f e c t i v e p o l a r i z a t i o n of the nucleon. The Pauli exclusion effect and the spin orbit term i n the out-going nucleon o p t i c a l model pot e n t i a l have not been included. Other l i n e s are taken from Levin and Eisenberg's r e s u l t s and _ 54 -the dashed l i n e i s the cross section calculated without the f a c t o r i z a t i o n of the two-body t matrix. The dash-dot l i n e i s the cross section calculated with the f a c t o r i z a t i o n approximation. The \ dotted l i n e i s the cross section calculated with-out the f a c t o r i z a t i o n approximation and without the spin orbit term i n the out-going nucleon o p t i c a l model po t e n t i a l . F i g . 24 F i v e - f o l d d i f f e r e n t i a l cross sections f o r the knock-1 6 out of a lp-y'2 P 1" 0^ 0 3 1 from 0 . The explanation of the l i n e s and the kinematic s i t u a t i o n i s the same as f i g . 23. F i g . 25 F i v e - f o l d d i f f e r e n t i a l cross sections f o r the knock-1 f out of a l p - iy2 proton from 0 . The explanation of the l i n e s i s the same as f i g . 23. The kinematic conditions are E T = 130 MeV, E\ = 61 MeV, and 6 T = 120°. F i g . 26 F i v e - f o l d d i f f e r e n t i a l cross sections f o r the knock-1 f out of a ^ r o J c o n from 0 . The explanation and kinematic s i t u a t i o n i s the same as f i g . 25. - 56 -F i g u r e 2 - 58 -F i g u r e 4 INCIDENT PION 60 MEV TARGET NUCLEON Pl/2 NUCLEUS RECOIL MOM. 0.30FM FINAL MOM. RATIO 0.80 RADIAL SCALE 1 GRID-.0.40FM INCIDENT PION 116MEV TARGET NUCLEON Pl/2 NUCLEUS RECOIL MOM. 0.60FM FINAL MOM. RATIO 0.80 RADIAL SCALE 1 GRID:0.40FM INCIDENT PION 180MEV TARGET NUCLEON PI/2 NUCLEUS RECOIL MOM. 0.60FM FINAL MOM. RATIO 0.80 RADIAL SCALE 1 GRID:0.40FM INCIDENT PION. 220MEV TRRGET NUCLEON P l/2 NUCLEUS RECOIL MOM. 0.60FM FINAL MOM. RATIO 0.80 RADIAL SCALE I GRID:0.40FM - 64 -F i g u r e 10 - 67 -F i g u r e 13 - 69 -F i g u r e 15 - 71 -F i g u r e 17 E F F E C T I V E POLAR IZAT ION i \ X 1 o ( " * V ° \ - \ \ \ 1 / / y / \ — j o - 77 -F i g u r e 23 - 78 -F i g u r e 24 - 79 -F i g u r e 25 -Q b |5 LU XJ C L 

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