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UBC Theses and Dissertations

Simple pion-nucleon and pion-nucleus potentials with applications of the doorway model to pion-nucleus… Rowe, Glenn William 1977

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SIMPLE PION-NUCLEON AND PION-NUCLEUS POTENTIALS WITH APPLICATIONS OF THE DOORWAY MODEL TO PION-NUCLEUS SCATTERING by GLENN WILLIAM ROWE B. Sc., U n i v e r s i t y of B r i t i s h Columbia, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF • MASTER OF SCIENCE THE FACULTY OF GRADUATE STUDIES Dept. of Physics We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1977 ( c \ l e n n William Rowe, 1977 i n In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t ha t t he L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f PHYSICS The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date February 26,1977. ABSTRACT The pion-nucleon total cross section i s reproduced in the pion kinetic energy range of zero to 250 MeV by a simple Yukawa potential with the depth.and width as free parameters. This potential is then averaged over a nuclear matter distribution in an attempt to produce a pion-nucleus potential for pion- 1 2C scattering. The model i s ex-plored by varying several parameters in the potential and observing their effect on the differential cross section for pion- 1 2C scattering with the pion kinetic energy equal to 50 MeV. No completely satisfac-tory f i t to the data is found. It is suggested that treating this system as a pion-nucleon system interacting in the average f i e l d of the residual nucleus may give better results and some ideas are presented as to how this might be done. Applications of such a treatment of pion-nucleus interactions to the description of pion-nucleus "doorway" states (so far, apart from the N*, unobserved) are discussed. i i i CONTENTS I. Introduction 1 I I . Construction of a Pion-Nucleon P o t e n t i a l 1 I I I . Construction of a Pion-Nucleus P o t e n t i a l 9 IV. Motivation f o r a Doorway Model i n Pion-Nucleus Scat t e r i n g 26 Bibliography 40 iv TABLES I. Parameters Used i n Phase S h i f t C a l c u l a t i o n 7 I I . Cross Section Minima 21 FIGURES 1. Comparison of Yukawa Well Cross Section with Data 8 2. Pion-nucleus Cross Sections Predicted by Standard Potentials . 11 3. Nuclear Density (Eq. (20)) 13 4. Pion-nucleus Potential (Eq. (22)) 15 5. Effect of Absorption on Cross Section 19 6. Effect of Nuclear Radius on Cross Section 20 7. Effect of Nuclear Skin Thickness on Cross Section 22 8. Effect of Potential Depth on Cross Section 24 9. Low Resolution Nucleon-Nucleus Cross Section 34 10. Higher Resolution Nucleon-Nucleus Cross Section 34 11. Symbolic Doorway Model Picn - N u c l e u 3 Interaction 37 12. More Symbolic Pion-Nucleus Interactions 37 13. More Symbolic Pion-Nucleus Interactions 38 ACKNOWLEDGEMENT I would l i k e to express my thanks to Dr. A. W. Thomas and to Dr. D. S. Beder f o r h e l p f u l comments, ideas and c r i t i c i s m s . My greatest debt, however, i s to Dr. E r i c h Vogt, who provided the motivation f o r t h i s t hesis and whose patient supervision during every stage of the project was instrumental i n i t s completion. The depth of h i s i n s i g h t s i n t o the ways of the world, not only i n physics but i n many other areas as w e l l , have had a profound impact on my l i f e and have been g r e a t l y appreciated. (But I s t i l l reserve the r i g h t to sue him f o r my broken ankle.) 1 I. INTRODUCTION Most of the work that has been done i n the t h e o r e t i c a l t r e a t -ment of pion-nucleus s c a t t e r i n g i s characterized by a formal, math-ematical approach. Many approximations are made, and one i s l e f t wi.th an uneasy f e e l i n g because i t i s very d i f f i c u l t to p i c t u r e i n simple terms.what i s a c t u a l l y happening i n the maze of equations. What we intend to do here i s to present a simple p i c t u r e of pion-nucleus scat-t e r i n g , and to see what one can learn while avoiding as much formalism as pos s i b l e . We w i l l f i n d that we cannot f i t the data, which i s not s u r p r i s i n g , but we wish to point out that the formal approaches to t h i s problem often f a r e no better than we.1 II. CONSTRUCTION OF A PION-NUCLEON POTENTIAL We w i l l f i r s t review the method of obtaining cross sections from a given p o t e n t i a l . The wave equation which describes the pion's i n t e r a c t i o n with a nucleon i s the Klein-Gordon equation with Coulomb and nuclear p o t e n t i a l s added. The equation i s derived by replacing the momentum p_ of the par-t i c l e - by the quantum mechanical operator -iTiV_ In the equation (1) E 2 = p 2 c 2 + m 2^ The Coulomb p o t e n t i a l may be included in. a covariant manner by the su b s t i t u t i o n s (2) E• -*- E - e<£ and , cp •+ cp - eA 2 Here e is the charge of the particle, i s the electrostatic potential and A i s the vector potential. We w i l l take A = 0 here. To include a strong interaction potential i n (1) we should know i t s Lorentz trans-formation properties. However, since these are not known we have no way of determining which part of (1) i s the one into which the potential should be placed. It i s a common assumption that the strong potential, which we c a l l v, can be included with the energy term: (3) E -* E - e<j> - v The Klein-Gordon equation i s thus (4) {--fi 2c 2V 2 + m2^}^ = (E - etj) - V)2IJJ The terms 2e<j>v and v 2 are usually dropped from the right-hand side of (4) . This cannot be j u s t i f i e d on any physical grounds (unless E » e<f> >> v which i s not the case i n what follows) but since the location of the potential i s already somewhat arbitrary we w i l l accept this approx-imation. Equation (4) then becomes 2-r2 (5) {-•n2c2V2 + ~$r- + m2ct,>^ = {E 2 - 2e<J>E + e2<}>2 - 2EvH where V 2 i s the radial part of the Laplacian: and L?2 is the square of the orbital angular momentum operator. 3 If we define (7) UN s * 2 " (£/h) 2 u he E 2 - m2ch E h c ; we can write (5) f o r a state of d e f i n i t e £ as (8) u£ + {k2 + u 2 - 2c(u c + uN) - A C L + _ 0 } U £ = Equation (8) i s integrated outward from the o r i g i n to some cut-o f f radius R, where U N can be set equal to zero for a l l r>R. Thus f o r r>R, u^ w i l l be a s o l u t i o n to the pure Coulomb s c a t t e r i n g problem which, f o r n o n - r e l a t i v i s t i c kinematics, i s a linear' combination of the i r r e g u l a r and regular Coulomb wave functions and G 0: (9) u£ = exp(i<$£) (F £cos 6 £ + G £ s i n 6 £) = 3s(exp(2i5 ) + 1 ) F £ + ~ ( e x p ( 2 i 6 £ ) - 1)G £ r > R For a d e s c r i p t i o n of the properties of F £ and G £ see reference 2. Th quantity 6 £ i s the phase s h i f t due to U N. Then the c o n t i n u i t y requi ment together with equation (9) give re-do) exp(2i6 £) = De (Re + JG g) - (FJ + i G » ( F £ - i G £ ) - D £ ( F £ - i G £ ) r=R where D £ ( r ) = u £ ( r ) / u £ ( r ) f o r r£R. The use of n o n - r e l a t i v i s t i c i n place of r e l a t i v i s t i c kinematics g r e a t l y s i m p l i f i e s the analysis of the Coulomb wave functions without introducing s i g n i f i c a n t errors, as has been checked 4 by E i s e n s t e i n and M i l l e r . 3 The pion nucleon i n t e r a c t i o n depends on the t o t a l angular momentum J and the t o t a l i s o s p i n I of the pion-nucleon system as we l l as on £. Since the pion has spin zero and the nucleon has spin J can have only two values f o r each £: J = l±h. The pion i s o s p i n i s 1^ =1 and the nucleon i s o s p i n i s I J J = % so I can be either \ or 3/2. The complete expression f o r the d i f f e r e n t i a l cross s e c t i o n a(9) i s 4 a(6) = | f ( 6 ) | 2 + |g(6)| 2 where f(6) = f (6) +£exp(2iaJ{(£+l)f + £f }P (cos 9) £=0 Jt- • Jt g(6) = I e x p ( 2 i a J l ) { f £ + - f ^ J s i n 6P^(cos 6) The Coulomb amplitude fc(6) i s £ c ( 6 ) = 2k sin^(6/2) exp{2i(o 0-n £n(sin6/2))} 7 7 m n = ffzjf"; *Pi = arg r ( £ + i + i n ) where Z^ i s the pion charge, the nucleon charge, the r e l a t i v i s t i c center of mass frame pion mass, 6 the center of mass s c a t t e r i n g angle. The nuclear amplitudes f are defined i n terms of i n d i v i d u a l amp-l i t u d e s f ^ I for each isospin' channel. For the case of s c a t t e r i n g i n an I = 3/2 channel (e.g. T T + P or i; _n) we have f n j _ = f 3 , while for T T~P or '£± £± 7r +n s c a t t e r i n g f = (fjj + 2fJ.)/3. Jt± £± £± 21 21 The f are defined i n terms of a phase s h i f t 6 „ ^ as follows: * — £ ± f£± = 2ik ( e x P 2 i 6 " " 1) 5 The phase s h i f t s are determined from equation (10) where must now be considered as depending on J and I as well as £, i . e . a d i f f e r e n t p o t e n t i a l v i s required i n eqn.- (5) f o r each combination of £, J and I. Since thej,= l , J=I=3/2 resonance (the (3,3) resonance) dominates pion-nucleon s c a t t e r i n g at low energies, we w i l l assume that the corres-ponding p o t e n t i a l v^^^ i s the only one that i s non-zero here. This s i m p l i f i e s the cross s e c t i o n formula considerably: f(6) = f c ( 6 ) + 2exp(2ia 1)fJ P (cos 9) g(6) = e x p ( 2 i o 1 ) f 3 + s i n e Pj(cos 9) We s h a l l consider iT~n s c a t t e r i n g so that a l l Coulomb e f f e c t s d i s -appear: f c=o^=0. Then, using P^(cos9)=cos 9 and P|=l we have a(6) = | f ( 6 ) | 2 + |g(9)| 2 = | f 3 + | 2 ( 3 c o s 2 e + 1 ) The t o t a l cross s e c t i o n i s then (12) cr x = 2TT IT 'a(9)sin9 de = -^(l-cos263 ) 0 k , + I f Coulomb s c a t t e r i n g i s present, as i n a pion-nucleus i n t e r a c t i o n , the t o t a l cross section becomes i n f i n i t e . D i f f e r e n t i a l cross sections can s t i l l be c a l culated and compared with data, however. The relevant formulas a r e 5 (for a spin-zero, isospin-zero nucleus such as 1 2 C ) : 0(6) = | f c ( 9 ) f f N ( 9 ) | 2 OO f N ( 6 ) = f k E (2£+l)exp(2ia £)(exp(2i6 £) - l ) P £ ( c o s 6 ) where the Coulomb amplitude f and Coulomb phase s h i f t s are defined i n (11) 6 Let us now try to find a simple potential which reproduces the observed pion-nucleon cross section at low energies. For the pion-nucleon potential vp33^ r) (henceforth called v(r)) we w i l l assume a Yukawa interaction with two free parameters: (14) v(r) = -V E X P ( 7 / r ^ o r / r n The depth parameter V 0 is positive and has dimensions of energy; the width parameter r n has dimensions of length. We w i l l attempt to adjust V D and r n to f i t the existing (3,3) resonance data for cross sections. Because v(r) i s real, 6 i s real and JS | = 1. What this means physically is that the scattering i s purely elastic. This i s a very good approximation for the (3,3) resonance since the branching ratio for decay into the elastic channel i s > 0.99.6 In order to calculate values of V Q and r n which give an acceptable f i t to the irN scattering data the program PIRK3 was modified to calculate the phase shift for the potential v(r) in (14). The quantity O"J./4TT was calculated from the phase shift S 3 + using (12) for various values of V Q and r n in the energy range O^T.^350 MeV where is the lab .kinetic energ; of the incoming pion. In order to compare the calculation with the data, an analysis of irN scattering data by Rowe, Salomon and Landau7 was used. We w i l l sum-marize the analysis here. The TTN S-matrix i s factorized into a non-resonant part (b for background) and a resonant part S r: S.= S^S^ Using a r e l a t i v i s t i c Breit-Wigner form for the resonant part and an effective range approx-imation for the non-resonant part one finds the equation 7 (15) Y , ) l ( q ) - - a + b q 2 + cq" + X F ^ ^ 2 ' " 1 " o Here q i s the center of mass pion or nucleon momentum, x i s the branch-ing r a t i o f o r the decay of a TTN resonance i n t o the e l a s t i c channel (x depends on £ and I ) , tx>Q i s the t o t a l center of mass energy at resonance, q Q i s the center of mass momentum of the pion at resonance, and u i s the t o t a l center of mass energy corresponding to momentum q: (16) u = / q z + m^  + / q z + m^ The parameters a, b and c are obtained by f i t t i n g (15) to experimentally determined phase s h i f t s . For our purposes only the J=3/2, 1=3/2 P-wave phase s h i f t (denoted by $1+) i s needed. The values of a, b and c obtained i n the l e a s t squares f i t (15) together with the resonance parameters are given i n Table I. TABLE I PARAMETERS USED IN PHASE SHIFT CALCULATION a = 4.03 x 10~ 8 (MeV/c) - 3 x = 0.99 b - -2.79 x 1 0 - 1 3 (MeV/c) - 5 ' r o = 116 MeV c = 7.26 x 1 0 - 1 9 (MeV/c)- 7 u Q = 1233 MeV It was found that V Q = 2110 MeV and r n = 0.52 fm gave good agreement with the least squares f i t f o r T7T<250 MeV. In Figure 1 the quantity (1 - cos26)/k 2 i s pl o t t e d against T ^ f o r both c a l c u l a t i o n s . In order to check that t h i s p o t e n t i a l (17) v ( r ) - -2110- e XPi: r /°-f f m )MeV r/0,52 f m 8 has the (3,3) resonance as i t s f i r s t resonance rather than some higher resonance the wave function u ^ r ) (as i n (8) with Z = 1) was ca l c u l a t e d fo r = 190 MeV, the resonance energy. It was found that there i s one node f o r r<0.52 fm (the one f o r r = 0) i n d i c a t i n g that t h i s i s indeed the f i r s t resonance. However, the p o t e n t i a l (17) does NOT give the correct phase s h i f t s f o r S-wave s c a t t e r i n g . There i s a broad S-wave resonance which would i f included, g r e a t l y increase the TTN cross s e c t i o n . I f one wished, one could attempt to f i n d a new set of parameters (V and r ) to produce a o n more accurate representation of the S-wave s c a t t e r i n g , but there i s l i t t l e value i n doing t h i s here since the act u a l S-wave cross s e c t i o n at t h i s energy i s much less than the P-wave. What the c a l c u l a t i o n shows i s that i f one t r i e s to reproduce TTN s c a t t e r i n g using a simple l o c a l p o t e n t i a l the p o t e n t i a l depends on %. 'XII. CONSTRUCTION OF A PION-NUCLEUS POTENTIAL The model f o r pion-nucleus s c a t t e r i n g that we w i l l t r y i s the follow-ing: we assume a pion nucleus p o t e n t i a l of form (18) V(r) = p(r')v(|r.-r' | ) d 3 r ' Here 'p(r) i s the density of nuclear matter i n the target nucleus and v i s the pion-nucleon p o t e n t i a l (17). This p o t e n t i a l makes several assump-tions : ( i ) It assumes that the nucleus i s a s t a t i c matter d i s t r i b u t i o n . T h i s i s not correct since the i n d i v i d u a l nucleons have a Fermi motion due to the exclusion p r i n c i p l e which, e s p e c i a l l y f o r low energy pion s c a t t e r i n g , can be greater than the energy of the pion. However, i f the pion-nucleus p o t e n t i a l i s not momentum dependent the Fermi motion does not matter, 10 since the nucleus as a whole i s at r e s t . The energy dependence of the cross section then depends e n t i r e l y on the p o s i t i o n s of the resonances i n a fi x e d p o t e n t i a l . ( i i ) The pion i s always a f r e e p a r t i c l e i n t h i s d e s c r i p t i o n . We w i l l see l a t e r that t h i s a bad approximation. It i s here that the doorway model can improve matters by t r e a t i n g the N* separately from the average pion-nucleus i n t e r a c t i o n . ( i i i ) Since V i s r e a l , only e l a s t i c s c a t t e r i n g i s permitted. Absorption i s known to occur w i t h i n the nucleus and t h i s requires the add i t i o n of an imaginary term to the p o t e n t i a l . This w i l l be done to show the e f f e c t of absorption on the d i f f e r e n t i a l cross s e c t i o n . ( i v ) v ( r ) was derived f o r the 1=1 p a r t i a l wave only and i s known to be i n c o r r e c t f o r Z = 0. A proper c a l c u l a t i o n of picn-nucleus s c a t t e r i n g requires the i n c l u s i o n of a l l I. We can attempt to compensate f o r t h i s by varying the depth of the p o t e n t i a l . The r a t i o n a l e f o r t h i s i s that to produce a minimum i n the S-wave cross s e c t i o n f o r pion-nucleon scat-t e r i n g we would have to vary the depth V Q of the p o t e n t i a l (14). Thus to account f o r the presence of both S and P waves i n the pion-nucleus p o t e n t i a l (18) we could adjust i t s depth. Given a l l these drawbacks one would hardly expect t h i s approach to reproduce the data w e l l and i n fac t i t does not as we^will see. But the i n t e r e s t i n g thing i s that the p o t e n t i a l (18) (with absorption added) gives q u a l i t a t i v e l y j u s t as good (or bad) a curve as several standard p o t e n t i a l s ( K i s s l i n g e r , Laplacian, e t c . ) . Figure 2 (obtained from r e f -erence 8) shows the cross sections obtained for T T + on 1 2 C at T^ - 50 MeV from three standard p o t e n t i a l s . The parameters used i n the p o t e n t i a l s were obtained from pion-nucleon phase s h i f t s . Using a l e a s t squares proc-i i i i i r O 2 0 4 0 60 80 100 120 140 F I G U R E 2 Pion-nucleus cross sections predicted by some standard potentials compared with the data from r e f -erence 8. (LMM = Loridergan-McVoy-Moniz) o For d e t a i l s of the c a l c u l a t i o n of these curves see the references contained i n reference 8. 12 edure, i t is possible to f i t the data with these potentials, but the resulting parameters are unphysical. Further details can be found in reference 8 and references contained therein. We now describe the calculation of (18). The pion-nucleon potential w i l l be M Q i w l r i-'h - -V 0exp(-/r^+r , z - 2 r r'cos6/r n) (19) v(|r-r |) - < V A-r^ - 2 r r'cose/r n where 6 is the angle between _r and jr'. For the nuclear density we w i l l take the Fermi form (20) P(r') 1 + exp((r'-c)/t) c and t are radius and thickness parameters for the target nucleus. A plot of the density functi-n i s given in Figure 3 for 1 2C. P is a norm-alization constant determined from the requirement that fp(r')d 3r' = A where A is the mass number of the nucleus. This gives for P: (21) P = J + C - ^ i - 2t3 j (-1)* 1 expt-nc/t)]" 1 —' -* i XI ^ n=l J 4ir Substituting (19) and (20) into (18) gives after doing the angular integration: V(r) = 2TTV0Pr2 r'/r Jo l+exp((r'-c ) / t ) exp(-|r-r'|/r n) -exp(-(r+r')/r n)Jdr' or 13 FIGURE 3 NUCLEAR DENSITY (EQ. (20)) 0.01 O.01Q h Parameters: P = 0.0176 (A = 10) c = 4.5 fm t = 1 fm (I i ; i i i _ _ r r=r—<— 1 1 o 2. 4 <o S 10 11 14-r (-fm) 14 V(r) =-2TTV0Pr2 ^ P ( : r / r n ) tr 2 r ' s i n h ( r ' / r n ) 0 l + e x p ( ( r ' - c ) / t ) d r + ( e x p ( 2 r / r n ) - l ) 0 r ' e x p ( - r ' / r n ) ' . l + e x p ( ( r ' - c ) / t ) a r The second i n t e g r a l can he given f i n i t e l i m i t s by the transformation x = 1/r' so the f i n a l expression for V(r) i s Y(r) - -2,V 0Pr2 S E P i z E Z l n i ' [ ^ r ' sinh(r '/r ) 1 1 r (J 0 l + e x p ( ( r ' - c ) / t ) (22) ( e x p ( 2 r / r n ) - l ) e x p ( - l / x r n ) dx  0 x 3 ( l + e x p ( l / x t - c/t)) Despite the apparent 1/r behaviour of V(r) for small r i t i s a c t u a l l y f i n i t e f o r r = 0. Note that the f i r s t i n t e g r a l vanishes f o r r = 0 while the second term behaves l i k e 2 r / r n m u l t i p l i e d by a constant (the i n t e g r a l ) . Thus (23) V(0) = -47TV nPr n f e x ^ - l / x r ^ ) dx ° n J 0 x 3 ( l + e x p ( l / x t - c / t ) ) A graph of V(r) i s given i n Figure 4 f o r a p a r t i c u l a r set of parameters Now that we have' the p o t e n t i a l we w i l l see how c l o s e to the data the r e s u l t i n g cross sections are. According to the arguments above, we can vary the depth of the p o t e n t i a l somewhat because of the omission of the TN S-wave component. We w i l l do t h i s by varying P, the normalization of the nuclear matter density. Since we have also ignored the i s o s p i n dep-endence of the pion-nucleus p o t e n t i a l , t h i s parameter i s already rather a r b i t r a r y . If we consider only 1=3/2 s c a t t e r i n g i n ~ s c a t t e r i n g then P should be normalized to include the protons plus only one t h i r d of the neutrons (due to Glebsch-Gordon algebra). I f we assume V(r) to be i s o s p i n 15 FIGURE 4 PION-NUCLEUS POTENTIAL (EQ. (22)) 2 0 Parameters » V D = 2110 MeV A = 10 nucleons (P r n = 0.52 fm c = 4.5 fm t = 1 f m = 0.0176 fm-3) 16 independent then P should be normalized to A. Thus by varying P we are using i t as a garbage dump to dispose of the un c e r t a i n t i e s i n V ( r ) . The reader may be f e e l i n g rather uneasy at t h i s point. We have made a huge number of approximations to get t h i s far—many of them p h y s i c a l l y u n r e a l i s t i c . But the point of what we are doing i s not to derive the pion-nucleus i n t e r a c t i o n from f i r s t p r i n c i p l e s — o n l y a theory of the strong i n t e r a c t i o n could do that—we are s t a r t i n g with a very simple idea and j u s t seeing how close to r e a l i t y we can make i t come. Many attempts have been made to f i t pion-nucleus data but none have been completely successful, and a l l attempts involve approximations that are very d i f f i c u l t to understand p h y s i c a l l y . This i s not to say that the approximations are w r o n g — i t i s j u s t that i t i s d i f f i c u l t to say exactly what part of the p h y s i c a l system we are ignoring when we make an approximation i n a math-ematical expression that i s several steps removed from p h y s i c a l i n t u i t i o n . What we present i n the next few fi g u r e s are several p l o t s of d i f f e r -e n t i a l cross sections f o r i r + - ^ 2 C e l a s t i c s c a t t e r i n g . In each graph we present three curves as w e l l as the data points (obtained from reference 8). The three curves i n each p l o t show the e f f e c t of varying one parameter i n (22) while a l l others are held f i x e d . In a l l curves the u + i s incident with a lab k i n e t i c energy of 50 MeV. A l l cross sections and sc a t t e r i n g angles are measured i n the center of momentum frame. In most of the p l o t s absorption has been added. This consists of adding a purely imaginary term to (22) of one of two forms: ( i ) The standard K i s s l i n g e r form (24) V a b s ( r ) - - A k 2 b o P ( r ) + Ab 1V.(p(r)V) ( i i ) The l o c a l Laplacian form 17 (25) V a b s ( r ) = -A^ a o + b ^ p C r ) - hto>iV2p(v) Here b Q and b^ are imaginary parameters. The c a l c u l a t i o n s were done by modifying the program PIRK to accept the p o t e n t i a l (22) as well as a Coulomb p o t e n t i a l appropriate to a charge d i s t r i b u t i o n of the form (20). For reference t h i s Coulomb p o t e n t i a l i s : (26) V c ( r ) = 4Tre2P_r , 2 d r ' 4TTe 2P 7r' dr' 0 l+exp((r ,-c)/t);•.' j l+ e x p ( ( r , - c ) / t ) For r<c (27) V c ( r ) = 4ire 2Pj T T2 t 2 v f f 2 t 3 t 2 1 L v ' l l n ^ 7 n^J & A P v " C r - c ) , c y n=l 2 t 3 ~ 3~exp(-nc/t) For r>c (28) V c ( r ) = 4ire 2Pj 'c3 . C T T 2 t 2 + 3r 3r n " 2 t 3 . t 2 < l n=l + n"r n ^ 7 2 t d E x p(n(c-r)/t) - ^ ^ e x p(-nc/t) where P z i s given by (21) with Z i n place of A. The cross sections were calculated using equations (12). The d e t a i l s of the c a l c u l a t i o n were as follows. The i n t e g r a t i o n of (8) was done using a step s i z e of 0.06 fm and the cutoff radius was 12 fin f o r a t o t a l of 200 i n t e g r a t i o n points. Two of the parameters i n the p o t e n t i a l (22) were the same i n a l l plots, 18 These are V Q = 2110 MeV and r n - 0.52 fm. In addition the sum in equation (12) for % ( 6 ) w a s calculated up to I - 7. In Figure 5 the effect of adding an imaginary (absorptive) term to the potential is shown. The parameters in (22) were A = 10 (used in (21)), c = 4.5 fm and t = lfm. The imaginary part is of form (24) with the values of b Q and b^ shown in the figure. The units of b D and bji are MeV-fm-5. Absorption is seen to reduce the cross section and smooth out the peaks and valleys. It also causes the minima to occur at smaller angles. The curve i n Figure 5 with b Q = -30i is about the best approximation to the data that was found. The minimum at scattering angle 60° agrees with the experimental angle and the qualitative shape of the curve is correct, but the cross section is too large by a factor of about three. Also, a l l curves rise much too steeply at small angles. In Figure 6 the effect of changing the radius of the nucleus i s shown. Here the parameters are the same as for the b D = -30i line i n Figure 5 however absorptive potential (25) was used this time, with b Q = -55i and b x = l l O i . Notice that the f i r s t minimum moves in as c is increased. This is the sort of behaviour that a diffraction pattern in optics shows: the larger the diffracting object the smaller the pattern. In fact B l a i r 9 derives an expression for the cross section for scattering from a black sphere of radius c, i.e. one which absorbs a l l particles which strike i t . This i s (29) a(6) = O a ^ z J j L i j p e } . kco This has minima (actually zeros) at the zeros of the Bessel function ; 19 F I G U R E 5 20 FIGURE 6 EFFECT OF NUCLEAR RADIUS ON CROSS SECTION 21 the f i r s t such case occurs when kc6 = 3.83. Now for = 50 MeV, k = 0.639 fm - 1 so that the f i r s t minimum should occur when (30) 0 = 6/c (radians) Fow our cases we have the results shown in Table II. TABLE II CROSS SECTION MINIMA C ( f m ) 9 B l a i r ( d e S > 6Rowe ( d e^ 3 115 80 4.5 75 65 6 55 50 Considering the crudeaess of both models even this much agreement is surprising, but i t illustrates that very simple notions can be import-ant in complex problems. Some of the discrepancies in Table II could be due to the facts that (i) the nucleus i s not a r i g i d sphere; i t has a fuzzy edge. This edge takes up a greater percentage of the nucleus for smaller values of c hence the discrepancy should be largest there, which i t i s . Also ( i i ) the nucleus i s not a perfect absorber. The f i r s t point i s illustrated in Figure 7 which shows the effect of varying t, the nuclear skin thickness. Here the absorption potential is (25) and a l l other parameters are as for the c = 4.5 line in Figure 6. Increasing t moves the minimum further in and also increases the cross section considerably. This could be due to the thinning out of the nuclear matter so that the effects of absorption are less pronounced. The case where t = 0.5 fm has a minimum at 9 = 70°, closer to the diffraction FIGURE 7 22 EFFECT OF NUCLEAR SKIN THICKNESS ON CROSS SECTION 0 C M (c leg) value of 75° than for t = 1 fm. The structure has l a r g e l y disappeared from the p l o t and the cross s e c t i o n i s generally smaller. Figure 8 shows the e f f e c t of changing the depth of the p o t e n t i a l . This was done by varying A i n (21). The absorption p o t e n t i a l i s (24) and a l l other parameters are the same as for Figure 5, b D = -30i l i n e . Varying the p o t e n t i a l depth does not have a s i g n i f i c a n t e f f e c t on the cross s e c t i o n curve over the range of A which we have chosen. As mentioned e a r l i e r , the only r e a l problem with a l l these cross sections i s that they are too l a r g e . We w i l l present here some pos s i b l e explanations of t h i s problem. In the medium and large angle region, the best c a l c u l a t e d cross s e c t i o n i s about a f a c t o r of three greater than the data. This i s most l i k e l y due to the contributions to the s c a t t e r i n g amplitude i n p a r t i a l waves other than £ = 1. R-:call that the irN p o t e n t i a l r e s u l t e d i n a large S-wave co n t r i b u t i o n to the cross s e c t i o n which was-ignored i n c a l c u l a t i n g the irN cross section but was included in'the pion-nucleus c a l c u l a t i o n s . One may think of expanding v ( r ) i n a p a r t i a l wave se r i e s before doing the i n t e g r a l (18) and i n c l u d i n g only the I = 1 term i n the i n t e g r a l . However, i t i s not p o s s i b l e to do t h i s and s t i l l have a pion-nucleus p o t e n t i a l which i s l o c a l . 1 " If one wishes to expend as much e f f o r t as would be required to carry through t h i s non-local p o t e n t i a l c a l c u l a t i o n one would be better advised to approach the problem from a d i f f e r e n t viewpoint; p o s s i b l y the doorway model described i n the next s e c t i o n . The huge cross sections at small angles are a different, matter, however. As can be i n f e r r e d from Figure 2, other t h e o r e t i c a l c a l c u l a t i o n s show a minimum at angles near 15° (since Coulomb e f f e c t s cause the cross s e c t i o n to increase as 8+0). A d e t a i l e d study of one p a r t i c u l a r pion-24 FIGURE 8 25 nucleus c a l c u l a t i o n at T u = 50 MeV by Thomas and Landau 1 0 shows that t h i s i s due to interference between the Coulomb and nuclear amplitudes and i t turned out that such interference does not e x i s t i n our c a l c -u l a t i o n . The reason, i t turns out, i s that our p o t e n t i a l has two bound pion-nucleus states (for A=10): one i n the S-vjave and one i n the P-wave. Thomas and Landau's p o t e n t i a l i s re p u l s i v e i n S-waves and possesses no P-wave bound states. There are two arguments to suggest that our p o t e n t i a l has bound st a t e s , the second of which also provides an explanation f o r the large cross sections at small angles. F i r s t l y , i f one approximates our p o t e n t i a l by a square well of depth 60 MeV and width 4.5 fm then one can show 1 1 that t h i s w e l l w i l l have one bound S-wave and one bound P-wave state but no others. Secondly, i f one c a l c u l a t e s - ( u s i n g PIRK) the S-matrix f o r very low energy pions (T = 0.5 MeV) one f i n d s that s i n 26 i s small and negative while cos 26 = 1 for £ = 0 and 1 but that s i n 26 i s p o s i t i v e f o r £ = 2 (for A=10). As one decreases A with T u constant, f i r s t the sign of s i n 26 changes f o r £ = 1 and, as A i s decreased f u r t h e r , s i n 26 f o r £ = 0 becomes p o s i t i v e . Now according to Levinson's theorem 1 2 ••«£py»o) - «.£CT;~> - v where n i s the number of bound states i n p a r t i a l wave £. Since 6 (<») i s taken to be zero i n the usual d e f i n i t i o n of phase s h i f t we see that 6 p (0) = TT f o r one bound state, but 6„(0) = 0 for no bound sta t e s . Thus 26 f o r a small T^, 5^ would be s l i g h t l y l e s s than n, explaining why s i n 26^ <0 here. As the w e l l depth i s decreased, the bound states become resonances, so f o r low T , 6„ would be small and p o s i t i v e , as would s i n 28. The presence of bound states w i l l therefore change the sign of the nuclear amplitudes so that they w i l l i n t e r f e r e c o n s t r u c t i v e l y rather than d e s t r u c t i v e l y with the Coulomb amplitude. I t should be noted that there i s no r e l i a b l e data at low angles so that i t i s unknown whether such Coulomb-nuclear interference a c t u a l l y takes place. IV. MOTIVATION FOR A DOORWAY MODEL IN PION-NUCLEUS SCATTERING In order to describe ."he pion-nucleus i n t e r a c t i o n one would, of course, l i k e to have an e x p l i c i t theory which describes the process, i . e . one would l i k e to have a strong i n t e r a c t i o n p o t e n t i a l . Since such a theory does not yet e x i s t , one must res o r t to some form of approximation to the true i n t e r a c t i o n i f one wishes to tackle the problem at a l l . In order to see what a reasonable.approximation might be, l e t us w r i t e down an equation describing pion-nucleus s c a t t e r i n g symbolically and ask how much we know about each part of t h i s equation. We w i l l use n o n - r e l a t i v i s t i c quantum mechanics. This i s not accurate f o r a d e s c r i p t i o n of pion-nucleus s c a t t e r i n g near the energy of the N* resonance (190 MeV) because the pion here i s moving at r e l a t i v i s t i c speeds. The g e n e r a l i z a t i o n to r e l a t i v i t y can be made l a t e r , a f t e r we understand the p h y s i c a l ideas of the model. The Schrodinger equation describing the pion-nucleus system as a whole i s 27 (31) H * A - V x S H = l T ± + lvo±+ 1 v i 1 1=0 i=l i>j=l J Here p a r t i c l e 0 i s the pion, p a r t i c l e s 1 to A are the nucleons i n the target nucleus, T i s the k i n e t i c energy, i s the pion-nucleon i n t e r -a c t i o n and v.. i s the nucleon-nucleon i n t e r a c t i o n . This form f o r H assumes that a l l forces are two body forces, i . e . that the presence of a t h i r d p a r t i c l e does not a f f e c t the force between two other p a r t i c l e s . It may seem odd to describe the pion-nucleus system by a set of eigenfunctions corresponding to d i s c r e t e energies, each energy corres-ponding to one of the resonance energies of the system. (The parameter X i n (31) has the values 0, 1, 2, ... with X = 0 corresponding to the ground state of the pion-nucleus system.) A f t e r a l l , a continuous range of energies i s allowed whenever the t o t a l energy of the system i s above a l l p o t e n t i a l b a r r i e r s . The r a t i o n a l e i s the following. We have assumed that the exact resonant eigenfunctions V form a A basis f o r the pion-nucleus space. Being a basis the exact eigenfunction f o r any energy can be expanded i n terms of the Y . That i s , i f represents a pion-nucleus state with some energy E which i s not a resonant energy we can write (32) . * E - K E * A A f o r some set of constants a . A c t u a l l y the a,„ are continuous functions XL\ A h of the energy. They must s a t i s f y the condition that when E = E^, the resonant energy corresponding to the state ¥ , then a^v = 6.. . V X p Ay It would be possible to use the e n t i r e set of a l l pion-nucleus state functions to describe the pion-nucleus space, i n which case the expansions 28 (32) would not be needed. I t i s a l i t t l e easier f o r purposes of v i s u a l -i z a t i o n however, to consider only the set of ^  as the pion-nucleus bas i s . Equation (31) may not appear to h e l p — a f t e r a l l , we don't know the e x p l i c i t forms of e i t h e r v . or v... But i f we are only i n t e r e s t e d i n o i IJ } what the pion does, then v . i s a l l that matters since v. . doesn't even o i l j mention the pion. So l e t ' s t r y replacing H by another hamiltonian H Q i n which the v .'s are replaced by a p o t e n t i a l V + iW where V and W are o i r e a l and depend only on the pion's co-ordinates A A (33) H = I T, + V -KiW + Y v., = H.+ V + iW - J v . = H-H' ° i - 0 1 i>j l j i = l 0 1 What does a l l t h i s mean? We know that H gives a l l the pos s i b l e states of the pier, i n the f i e l d of A nucleons. H however, contains o only one p o t e n t i a l V + iW connecting the pion ..with the nucleus. In other words, H Q describes what happens when a pion encounters a p o t e n t i a l obtained by tr e a t i n g the nucleus as one p a r t i c l e — n o t n e c e s s a r i l y a point p a r t i c l e , but a " p a r t i c l e " which represents the average of A smaller p a r t i c l e s , the nucleons. Suppose H Q s a t i s f i e s (34) H ^ n = e n4- n n = 0, 1, 2, ... Then the ground state i]>0 of H Q should have the same energy cQ as the average over nucleon co-ordinates of H i n the state That i s (35) < * 0 | H | V = e D where the operation <I(> 0 | A|^ 0> means an average of the operator A over 29 a l l nucleon and pion co-ordinates. From (33) and (34), (35) implies (36) <* 0|H ' |V = 0 which, i f the v . are taken to be r e a l gives o i ° (37) < * 0 M V = < * o l I v o . ^ 0 > ; < ^ 0 I W I V = 0 To see what the function W represents we consider f i r s t HQ-iW. Suppose t h i s hamiltonian has a resonance of the Breit-Wigner form 1 2 E - E' - i r/2 with E as the resonance energy where a reaches i t s maximum and r i s the f u l l width at half-maximum (assuming T to be independent of E which i t u s u a l l y i s n ' t , but t h i s i s only a q u a l i t a t i v e argument). I t would seem reasonable to assume that adding an ^ imaginary part iW to the poten-t i a l would a f f e c t the width but not p o s i t i o n of a resonance; s p e c i f i c a l l y we might expect the width to be increased by about 2W (since the imag-inary part of the resonant energy i n (38) i s h a l f the width, not the whole width). But what i s W p h y s i c a l l y ? I t i s c e r t a i n l y not the same type of l i n e broadening as T since T would appear using even the exact hamiltonian H. W appears because we are using an approximate hamiltonian. Consider the case of a resonance due to H. Since a resonance i s not a stable p a r t i c l e , that i s , i t s l i f e t i m e can be l o c a l i z e d to within a f i n i t e time i n t e r v a l , i t s energy cannot be a p r e c i s e l y determined quantity. The time-energy uncertainty p r i n c i p l e t e l l s us that AtAE=»fi. Thus given that At<°°, AE must be non-zero. We can therefore only measure the p r o b a b i l i t y that the energy of the resonance i s any p a r t i c u l a r value. (38) TOT 30 A graph of t h i s p r o b a b i l i t y versus energy i s , i n f a c t , what a cross section plot represents. One way of v i s u a l i z i n g t h i s i s as follows. Suppose we had a machine that produced a constant current of p a r t -i c l e s , say 100 per second. Suppose also that the beam's energy could be continuously v a r i e d from zero to i n f i n i t y . We project t h i s beam onto a target and measure how many p a r t i c l e s pass through. We assume that any p a r t i c l e which i n t e r a c t s with the target i s never detected. We can now i r r a d i a t e the target f o r one second at each energy from zero to i n f i n i t y ( i f we have l o t s of beam time) and draw a gr.aph of (100 - number of p a r t i c l e s detected at energy E) versus E. If we observe a peak i n t h i s graph we c a l l i t a resonance with energy E', which i s the energy at which the graph reaches i t s highest point, and width Y. Since t h i s graph i s a p r o b a b i l i t y gtaph we see that V i s analogous to a standard deviation of the resonance energy. F i s due to the f a c t that the resonance i s unstable: i f the resonance was stable, T would be zero. . Suppose now that we have a hamiltonian H which gives r i s e to a s e r i e s of stable states (or resonances with zero width i f you l i k e ) and that we decide to use an approximate hamiltonian H Q to describe one of these states, say VQ. In general the approximate state yp0 w i l l not equal ¥ Q , but can be expanded i n terms of the complete set of eigenstates ^ of H. In A other words ^ Q can be expanded i n terms of a set of functions each of which represents a d i f f e r e n t energy so that the approximation of VQ by ty0 i s a state with ncn-zero width. The p r o b a b i l i t y of f i n d i n g the approx-imate state at a given energy i s proportional to the square of the coef-f i c i e n t of the exact state that represents that energy. This i s the sort of width that W represents. When one's exact hamiltonian has eigenstates with non-zero width, then an approximate hamiltonian H Q w i l l have genuine 31 T-type widths as w e l l as W-type ones a r i s i n g from the approximation being made. A measure of the W-type width introduced by using H q i n place of H i s given by the second moment M 2 defined by (39) M 2 = l(Ex - e o ) 2 C 2 0 A where the are the eigenvalues of the exact states as i n (31) and e Q i s the eigenvalue of the approximate state The C are the expansion c o e f f i c i e n t s of • if) i n terms of the V. m A (40) if; = Jc. ; V = Ye, if) m f1 Xm X X L XmTm X m and have the properties (41) )C, C, - 6 : Yc. C ='6. : Ye2 = 1 " Am An mn' " A n p n Xu' Xm X n X In view of what has been said above i t is. reasonable to assume that (42) <*ol w 2lV = M2 I t i s also true that (43) M 2 = «J) 0|H , 2U 0> as we now see. H' 2i> Q> = (H-H 0)(H-H 0)|* o> = ( H - H 0 ) { H U >-e L >} o' o (H-H ){Jc. E |¥,>-e U >} o £ AO X1 X o 1 y o 32 r yc,.E2|?,> + e 2 U > -£ AO A' A o o AO Am A m1 m (• AO A A A ,m A = I C,nC, E2|tJ/ > - I C.ftC. E.e |ty > -, L AO Am A1 m , L AO Am A m|Tm A,m A ,m [C.„C. E.e U > + e 2 U > . L AO Am A o1 m o 1 o A ,m Using the orthonormality of | *J> > and (41) we obtain « P 0 | H , 2 | V = I C A 0 ( V £ o ) 2 = M2 Q ' E - D -X Now that we have explored the meaning of the replacement of H by H Q we should ask i f i t i s a reasonable approximation. When the pion i s out-side the nucleus we would expect the nucleus to appear as a smeared potential so here we are a l l right. But once we get inside the nucleus things are not so simple. We know that a pion interacts very strongly with a nucleon at energies of around 190 MeV so i t i s certainly l i k e l y that a pion-nucleon interaction would occur as the pion travels through the nucleus. To get some idea how l i k e l y , let us compare the lifetime of the N* to the mean time the pion spends between collisions with nucleons in nuclear matter. The N* lifetime i s roughly /1 § \ h 6.6 x 1Q~22 MeV-sec , n r i - 2 U j (44) - = n o - ^ = 6 x 1 0 seconds The mean free path for a pion in nuclear matter consisting of 60% neutrons and 40% protons at a total density of 1.7 x lO1*'* nucleons per cubic meter (this assumes a nuclear radius of 1.12 A 1/ 3 fm) can be found from exper-imental cross sections: c . = 200 mb at resonance while a + - 70 mb. Thus 33 (45) X = ^ = 0.49 x 10" 1 5 meters n na + + n na . n n n P iT^p If pions travel at v=c then the mean time between collisions i s X/c or roughly 1.6 x 10 2 4 sec. At resonance then, we expect the pion to spend more time as an N* than as a free particle. The simple model H Q is not appropriate inside the nucleus. Since a pion spends more of i t s time as an N* than as a free pion the next logical approximation i s to single out one of the nucleons from the nucleus (the nearest one to the pion since the pion-nucleon interaction i s known to be short range) and treat this interaction separately from the general pion-nucleus one. The method we w i l l suggest i s in analogy to one that has been tried successfully in nucleon-nucleus interactions. It has been found that in nucleon-nucleus interactions several levels of complexity can be observed in the scattering data. With crude energy resolution one obtains the so-called "single-particle giant resonances" which are shown schematically in Figure 9. These resonances are caused by the formation of a quasi-stable system consisting of the incident nucleon interacting with an average potential due to the entire target nucleus (hence the name "single-particle"). With better energy resolution one obtains what is shown in Figure 10. The finer peaks are caused by the incident nucleon interacting with one of the nucleons in the nucleus in particular; this two nucleon system is then treated as being in an average potential due to the rest of the nucleus. More detailed pictures are possible (e.g. a nucleon interacts with two or more nucleons) but i t turns out that the widths of the resonances from these interactions are greater than their spacing so that no structure i s observable. The ultimate level of detail i s that in which the energy Nucleon Kin e t i c Energy FIGURE 9 - An ove r s i m p l i f i e d nucleon-nucleus s c a t t e r i n g cross section with poor energy resolution.> This plot i l l u s t r a t e s the single particle, giant resonance. o Nucleon Kinetic Energy FIGURE 10 - The same cross section as i n Figure 9 with better energy r e s o l u t i o n . The f i n e r resonances may be interpreted as evidence for doorway states. 35 from the incident nucleon i s spread over a l l the nucleons i n the nucleus, r e s u l t i n g i n an enormously complicated diagram of resonances c a l l e d the compound nucleus resonances. In the pion-nucleus system one may describe the processes i n a s i m i l a r way. The simplest model i s one i n which the incident pion i n t e r a c t s with one of the nucleons i n the nucleus, forming the N* resonance. (There i s also a non-resonant pion-nucleus i n t e r a c t i o n at a l l energies but we are only considering resonances here.) The N* i s coupled to two channels: the entrance channel and the compound nucleus channel. In the f i r s t case, the N* decays i n t o a pion-nucleon system, the nucleon r e j o i n s the nucleus and the pion escapes. In the second case, the N* gives o f f i t s energy to the remaining nucleons with the pion having been absorbed i n the process. The simplest doorway hypothesis, which i s the one employed by reference 14, i s that these compound nucleus states can be-formed only through the N*, i . e . the N* system i s a doorway through which the pion-nucleus system must pass before compound nucleus states can be formed. In analogy with the nucleon system one asks i f there are other methods of coupling from entrance channel to compound nucleus. Such postulated a l t e r n a t e states may be c a l l e d hallway states and consist of a pion i n t e r -acting with more than one nucleon at a time (e.g. a pion-alpha p a r t i c l e i n t e r a c t i o n ) . At t h i s point one can assume p r a c t i c a l l y whatever one wishes. However, i n order f o r the hallway states to be of any importance they must s a t i s f y these two conditions: (i.) t h e i r widths must be le s s than t h e i r spacing i n order that they be observable and ( i i ) t h e i r postulated e x i s t -ence must not i n t e r f e r e with accepted s c a t t e r i n g data, i . e . f o r poor enough r e s o l u t i o n the hallway theory must reduce to the old "giant r e s -36 onance" cross section p r e d i c t i o n s . Symbolically reference 14 does what i s shown i n Figure 11 as a kind of flow chart. Two modifications to consider are i l l u s t r a t e d i n Figure 12. The f i r s t diagram here would consist of replacing the N* i n Figure 11 by a hallway state ( i . e . a d i r e c t coupling of the pion to more than one nucleon), while the second involves i n s e r t i n g an extra step i n t o the door-way process. The f i r s t would probably be easier to s t a r t with. In more conventional diagrams, Figure 13 shows two p o s s i b i l i t i e s that t h i s could lead to: e l a s t i c s c a t t e r i n g and nuclear breakup. The R* i s the analog of the N* (R f o r e i t h e r "Rowe" or "resonance" depending on the reader's i n t e r -p r e t a t i o n of the author's ego), i . e . i t i s the intermediate state c o n s i s t -ing of the pion and more than one nucleon. The mechanism f o r what happens i n s i d e the c i r c l e s i n Figure 13 i s anybody's guess at present. For the s p e c i a l case of the pion-nucleus doorway the argument may go something l i k e t h i s . I f we concentrate on nucleon n the new approx-, imate hamiltonian would be A (46) H D = Y T. + Y v.. + V. + iW. + v_ ° l . L., i i A A On i = l i>j^n J where v_ i s the pion nucleon i n t e r a c t i o n while V. + iW, represents the On A A average i n t e r a c t i o n of the center of mass of the N* with the r e s i d u a l (A-l)-nucleon nucleus. VQ ,^ i n exact form, would represent the i n t e r a c t i o n of the two quarks i n the pion with the three i n the nucleon (assuming, of course, that quarks e x i s t ) . This i n t e r a c t i o n would presumably give the experimental cross section exactly. Such a c a l c u l a t i o n i s not p r a c t i c a l at present however, as the quark-quark i n t e r a c t i o n s are not well known and five-body Free Pion Pion N* + y v. jf. Absorbed by Residual Nucleus Nucleus Nucleus Entrance Channel Doorway State Compound Nucleus State FIGURE 11 Free Pion Nucleus Entrance Channel R* + Residual Nucleus Hallway State Pion Absorbed by Nucleus Compound Nucleus State Free Pion + Nucleus N* + Residual Nucleus R* + Residual Nucleus Pion Absorbed by Nucleus v ? /-Entrance Channel Doorway State Hallway State Compound Nucleus State FIGURE 12 Residual Nucleus E l a s t i c Scattering 39 problems are very d i f f i c u l t to solve meaningfully. Rather we consider VQ to be a pion-nucleon o p t i c a l p o t e n t i a l : we average over the three nucleon quarks and the two pion quarks separately. If H Q has an eigenstate \pQ with resonance energy e 0 then we can expand ^ Q as i n (40) and use r e l a t i o n (36) (where H' i s now H - H Q with H Q given by (46)) to see the r e l a t i o n of eQ to the energies of the exact resonances: 0 = <^0|H-H0|^0> = <* 0|Hi> 0> - e Q = I C. nC .<>F, |E If > - en . L XO yO X1 y 1 y ° = I C._C n E 6. - e n ^ XO yO p Xy ° A ,y • = J C 2 n E , - e f XO X o or so that eQ i s the weighted average of the exact energies E^. Before we a c t u a l l y s t a r t w r i t i n g computer programs however, we should have some evidence that such intermediate states e x i s t , i . e . we should see some kind of fine' structure i n the pion-nucleus cross section analogous to Figure 10. Since t h i s has not yet been seen we must be content to stop here and await future developments. 40 BIBLIOGRAPHY 1. M. M. Sternheim and R. R. S i l b a r , Annual Review of Nuclear Science ' V o l . 24 (1974), p. 249. 2. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1965), Chapter 14. 3. R. A. E i s e n s t e i n and G. A. M i l l e r , PIRK: A Computer Program to  Calculate the E l a s t i c Scattering of Pions from N u c l e i , Carnegie-Mellon U n i v e r s i t y p u b l i c a t i o n COO-3244-24. 4. B. H. Bransden and R. G. Moorhouse, The P ion—Nucleon System, (Princeton, 1973), Section 2.2. 5. N. F. Mott and H.S.W. Massey, The Theory of Atomic C o l l i s i o n s , T h i r d E d i t i o n , (Oxford, 1965), Chapter I I I . 6. Review of P a r t i c l e P roperties, Reviews of Modern Physics 48, S181 (1976). 7. G. Rowe, M. Salomon and R. Landau, to be published. 8. J . F. Amann et a l . . Physics Review L e t t e r s 35_t 426 (1975) . 9. J. B l a i r i n : Lectures i n T h e o r e t i c a l Physics Volume 8C, (University of Colorado Press, 1966), Section 2b. 10. A. W. Thomas, p r i v a t e communication. 11. L. I. S c h i f f , Quantum Mechanics, Third E d i t i o n , (McGraw-Hill, New York, 1968), Section 15. 12. J . R. Taylor, Scattering Theory, (Wiley, 1972), Sections 12-e, 13-b. 13. A. Mekjian i n : Advances i n Nuclear Physics Volume 7, (M. Baranger and E. Vogt, e d i t o r s ) , (Plenum Press, New York, 1973). 14. L. S. K i s s l i n g e r and W. L. Wang, Annals of Physics 99, 374 (1976). 

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