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Calculated cross-sections of pion production by 450-mev protons on various nuclei. McMillin, Douglas John 1968

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CALCULATED CROSS-SECTIONS OF PION PRODUCTION  BY 450-ME? PROTONS ON VARIOUS NUCLEI by DOUGLAS JOHN MCMILLIN A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e Department o f PHYSICS We a c c e p t t h i s t h e s i s as co n f o r m i n g t o t r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 1963 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 o f t h e r e q u i r e m e n t s , f o r an a d v a n c e d d e g r e e ' a t .the- Uni v e r s i t y o f B r i t i s h C o l u m b i a , 1 a g r e e : t h a t ; t h e . L i b r a r y , s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r b y hits r e p r e s e n t a t i v e s . i t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t . m y w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f P h y s i c s 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 V a n c o u v e r 8, .Canada 2.6 Sep tember , 1968 - i i -ABSTRACT We construct a model to explain the production of pions i n the bombardment by protons of various nuclei and we use the model to calculate r e l a t i v e cross-sections f o r the process. The model assumes that the incident proton interacts with the target nucleons i n d i v i d u a l l y and that the proton-nucleon cross-section can be used as a free parameter. The model accounts f o r many important nuclear e f f e c t s , some for the f i r s t time i n explaining the A-dependence of the pion y i e l d . The eff e c t s included are those due to proton and pion absorption, to the back-ground nuclear potentials and to the struck-nucleon momentum and density d i s t r i b u t i o n s . We compute d i f f e r e n t i a l cross-sections i n several s p e c i a l cases and compare them with experimental data at 450 MeV. Agreement i s only moderate, but i t i s as good as any previously obtained and, unlike the e a r l i e r r e s u l t s , i t does not depend on the assumption of an absorbing neutron blanket. Our agreement depends instead on the use of a modern nuclear radius and a reasonable treatment of pion absorption. In t h i s respect our r e s u l t s confirm what e a r l i e r workers had assumed, that absorption i s the dominant factor c o n t r o l l i n g the proton-nucleus production of pions. Also important i s the proton-nucleon production rate, a reasonable value of which we assume. Potential e f f e c t s are important because the basic production rate and pion absorption are both very energy dependent. The effects of struck-nucleon momentum and density d i s t r i b u t i o n , as we calculate them, are small at the energy considered. - i i i -TABLE OF CONTENTS *>age ABSTRACT i i LIST OF FIGURES i v ACKNOWLEDGMENTS v CHAPTER I - INTRODUCTION 1.1 P i o n p r o d u c t i o n and the TRIUMF p r o j e c t 1 1.2 The b a s i c i n t e r a c t i o n 4 1.3 E x i s t i n g models 7 1.4 P r e s e n t program 10 CHAPTER I I - THE MODEL 2.1 Theory 13 2.2 An e s s e n t i a l s i m p l i f i c a t i o n . . . . 18 2.3 Parameters chosen 22 CHAPTER I I I - THE CALCULATION 3.1 The i n t e g r a t i o n 27 3.2 R e s u l t s compared t o experiment 28 3.3 R e s u l t s w i t h parameters v a r i e d . . . . . . . . . . 3 5 3.4 C o n c l u s i o n 37 REFERENCES 39 APPENDIX A E x p e r i m e n t s on p r o t o n - n u c l e o n p i o n p r o d u c t i o n . . 43 APPENDIX B E x p e r i m e n t s on p r o t o n - n u c l e u s p i o n p r o d u c t i o n . . 45 APPENDIX C D e r i v a t i o n o f p a t h l e n g t h s i n a n u c l e u s 47 APPENDIX D T r a n s f o r m a t i o n o f energy from l a b o r a t o r y frame t o r e s t frame o f a s t r u c k n u c l e o n . . . . . . . . 50 APPENDIX E D e r i v a t i o n o f an e x p r e s s i o n used t o compute p i o n a b s o r p t i o n c o e f f i c i e n t s . 51 APPENDIX F O u t l i n e o f computer program PIPROD 53 - i v -LIST OF FIGURES page FIGURE 1 Model o f n u c l e u s showing p a t h o f i n c i d e n t p r o t o n and emerging p i o n 15 FIGURE 2 C r o s s - s e c t i o n s f o r t h e p r o d u c t i o n o f £3-MeV p o s i t i v e p i o n s a t 2 1 . 5 ° by 450-MeV p r o t o n s on v a r i o u s n u c l e i 2 9 FIGURE 3 More c r o s s - s e c t i o n s f o r p o s i t i v e p i o n p r o d u c t i o n . . 32 FIGURE 4 Some c r o s s - s e c t i o n s f o r n e g a t i v e p i o n p r o d u c t i o n . . 34 FIGURE 5 C r o s s - s e c t i o n s o b t a i n e d by v a r y i n g parameters . . . 36 FIGURE 6 Geometry o f p a t h l e n g t h s i n a n u c l e u s 48 ACKNOWLEDGMENTS The author wishes to thank, above a l l , Professor E. W. Vogt who suggested t h i s investigation and provided invaluable guidance through a l l the phases of i t s completion. Dr. Vogt's continued in t e r e s t and patience i n the matter are g r a t e f u l l y acknowledged. He also thanks L. Lam who sorted out many aspects of the problem and Dr. P. C. Bhargava who assisted i n the early stages of the c a l c u l a t i o n . F i n a n c i a l assistance provided by the National Research Council of Canada i s also acknowledged. - 1 -CHAPTER I - INTRODUCTION 1.1 Pion production and the TRIUMF project I t i s now possible to produce pions i n large quantity. The most important production process brings a beam of protons from an accelerator into c o l l i s i o n with other nucleons which may be free (a hydrogen target) or bound together as a nucleus (in a carbon target f o r example). The beam energy at which pion production can begin depends strongly on the kind of target being used, but the energies at which pion production i s at a l l e f f i c i e n t are generally i n excess of 3 50 MeV. The recent f e a s i b i l i t y of high beam currents at these energies of e f f i c i e n t production has led to the proposed construction of meson " f a c t o r i e s " able to turn out pions and t h e i r decay products at an unprecedented rate. I t was one of these proposals, namely the TRIUMF proposal^^^, which stimulated the present investigation into the pion production process. The heart of the TRIUMF project i s a sector-focussed cyclotron designed to accelerate H - ions rather than protons. A continuous proton beam i s extracted by the simple and highly e f f i c i e n t process of mechanically s t r i p p i n g the two electrons from each H"~ ion as i t gains a predetermined o r b i t . In addition to providing an intense beam of r e l a t i v e l y high energy protons the design features permit easy energy v a r i a b i l i t y . This v a r i a b i l i t y (though of prime inter e s t i n proton experiments).^ along with the freedom of choice of a pion-producing target - 2 -p r o v i d e s v a r i e t y i n t h e n a t u r e o f t h e p i o n y i e l d . A way t o maximize t h e p i o n y i e l d , i m p o r t a n t t o many e x p e r i m e n t s , i s a l s o p r o v i d e d . J u s t how t h i s i s p o s s i b l e we soon make c l e a r . To s e l e c t an upper l i m i t t o put on t h e p r o t o n beam energy t h e d e s i g n e r s o f TRIUMF b a l a n c e d t h e two i m p o r t a n t f a c t o r s o f machine c o s t , w h i c h f a v o u r s low energy, and t h e p i o n p r o d u c t i o n r a t e , w h i c h f a v o u r s h i g h energy. I n a car b o n t a r g e t , f o r example, t h e p r o d u c t i o n r a t e a t s m a l l a n g l e s (where t h e y i e l d i s always g r e a t e s t ) i n c r e a s e s s h a r p l y w i t h p r o t o n energy from 350 t o 450 MeV, w h i l e beyond 450 MeV i t i s r e l a t i v e l y c o n s t a n t . T h e r e f o r e , w h i l e the mean p i o n energy may become ever g r e a t e r w i t h i n c r e a s i n g p r o t o n energy, t h e r e i s not a v e r y l a r g e g a i n i n th e number o f p i o n s produced a t p r o t o n e n e r g i e s above 450 MeV. T h i s o f co u r s e assumes t h a t t h e beam c u r r e n t i s c o n s t a n t w i t h c hanging p r o t o n energy. S e e k i n g a maximum p i o n y i e l d r a t h e r t h a n a h i g h mean energy and a t the same t i m e a l l o w i n g a margin f o r c o mfort t h e TRIUMF d e s i g n e r s chose t h e v a l u e o f 500 MeV f o r th e p l a n n e d maximum p r o t o n beam energy. The c h o i c e o f an o p e r a t i n g beam energy and a p i o n -p r o d u c i n g t a r g e t i s made c o m p l i c a t e d by s e v e r a l a d d i t i o n a l f a c t o r s . Because t h e e l e c t r i c d i s s o c i a t i o n o f an H~ i o n i s l e s s p r o b a b l e a t lower energy t h e a c c e l e r a t o r can t o l e r a t e l a r g e r beam c u r r e n t s a t lo w e r energy. Hence the p i o n y i e l d , p r o p o r t i o n a l t o bo t h t h e beam c u r r e n t and t h e p r o d u c t i o n r a t e , can be g r e a t e r a t 450 MeV t h a n a t t h e maximum beam energy o f 500 MeV. The y i e l d may become s t i l l g r e a t e r i f the p r o t o n energy - 3 -i s further reduced. Moreover, the pion y i e l d may be enhanced by the use of a heavier target material since a heavier nucleus i s r i c h e r i n nucleons having high i n t e r n a l momentum which can r a i s e the amount of energy available to the production reaction. This enhancement i s e s p e c i a l l y noticed at proton energies near the pion production threhhold. A competing process however i s the one i n which either the incident proton or the emerging pion i s absorbed or scattered by the nucleus hosting the production event and i n t h i s respect a heavier host nucleus i s c l e a r l y not favoured. Supporting t h i s trend we note that a heavy nucleus also provides a large Coulomb poten t i a l which serves to lower the incident proton energy. Determining the balance of these "nuclear e f f e c t s " and i t s dependence on both nuclear mass and beam energy i s the concern of t h i s t h e s i s , but the reason f o r wanting to be able to do so i s at once evident: i t w i l l permit the e f f i c i e n t adjustment of machine parameters to maximize the TRIUMF pion y i e l d . Because of the nature of the TRIUMF project we are interested i n the production of charged pions at small angles i n the bombardment of various nuclei by protons i n the energy range around 450 MeV. In the next chapter we introduce a simple model fo r the production process, one which keeps separate i t s more important physical aspects, and i n the chapter a f t e r that we describe calculations made on the basis of t h i s model. We compute r e l a t i v e production cross-sections which agree with available experimental cross-sections to such an extent as to afford a f a i r degree of confidence i n the assumptions we make i n - 4 -constructing our model. We are also able to discover which of the physical aspects of the process dominate at the energies considered. We hope that subsequently refined versions of our model can then be used to predict cross-sections where none are yet ava i l a b l e experimentally and thereby become an aid to more e f f i c i e n t planning i n the TRIUMF or s i m i l a r projects. 1.2 The basic i n t e r a c t i o n The production of pions by proton-nucleus c o l l i s i o n s and by proton-nucleon c o l l i s i o n s are related processes. At energies above the pion production threshhold both the incident proton wavelength and the proton-nucleon cross-section are small enough to permit the assumption that the proton interacts with but a single target nucleon to produce a pion and not with the target nucleus as a whole. In t h i s approximation the proton-nucleus cross-section i s proportional to a corresponding proton-nucleon cross-section and to the number of those nucleons i n the target nucleus, although the nature of the proportionality becomes complex as one admits the operation of nuclear effects such as those mentioned i n the previous section. The two-nucleon c o l l i s i o n then i s at the root of the pion production process. The two-nucleon reactions that begin with at least one proton and produce a p o s i t i v e pion are as follows: P + P p + p n + p - 5 -There i s only one such reaction producing a negative pion, v i z : n + p rr- + p + p The threshhold energy i n the laboratory system i s 294 Me? f o r the reaction y i e l d i n g deuterium and 253 MeV for the others. The d i f f e r e n t i a l cross-sections f o r the reactions are poorly known experimentally (experiments(2-11) a r e summarized i n Appendix A) and the existing theories(12-16) ^-^ch m i g h t be used to compute cross-sections are either l a r g e l y phenomenological or incomplete. Consequently we are able to calculate only r e l a t i v e proton-nucleus cross-sections, using the proton-nucleon cross-sections as free parameters to adjust our r e s u l t s . I t should be noted however that the combined cross-section for the f i r s t two reactions l i s t e d above i s approximatly ten times the cross-section f o r the t h i r d . This i s the correct r e s u l t ( ^ * f o r pion production proceeding through an intermediate 3-3 resonance state. Hence we assume that f o r the production of p o s i t i v e l y -charged pions i t i s the protons of the target nucleus which are the most important. The nuclear e f f e c t s which influence the nature of the proport i o n a l i t y between the proton-nucleus cross-section and the number of target nucleons are: 1) the momentum eff e c t wherein the momentum of a struck nucleon within i t s nucleus a l t e r s the amount of energy available to the production reaction and both the energy and exit angle of the pion produced. Just how the pion y i e l d i s affected I - 6 -depends on the nature of the va r i a t i o n of the proton-nucleon cross-section with proton and pion energy and pion angle i n the regions of i n t e r e s t . Internal momentum i s espe c i a l l y important near the pion production threshhold which, because of i n t e r n a l momentum, i n a nucleus may be as low as 170 Mev"; 2) the absorption-scattering e f f e c t wherein the non-p a r t i c i p a t i n g nucleons of the struck nucleus, mere spectators i n the production event, provide a background o p t i c a l p o t e n t i a l which a l t e r s the pion y i e l d by allowing f o r absorption within the nucleus of either the incoming proton or the outgoing pion or, i n the case o f d i f f e r e n t i a l cross-sections, f o r scattering o f pions away from the energy and angle of i n t e r e s t . Experimental r e s u l t s on the proton-nucleus production of pions as a function of increasing nuclear mass (the experiments(1^~30) a r e summarized i n Appendix B) show a decrease i n production e f f i c i e n c y , i . e . , i n the number of pions produced per target nucleon, consistent with the operation of an absorption e f f e c t . The nuclear cross-: sections are found to increase more and more slowly with A u n t i l often there i s no increase at a l l ; 3) p o t e n t i a l effects wherein the amount of energy avail a b l e f o r pion production and the energy of the pion as i t i s seen outside the nucleus are altered by the presence of back-ground Coulomb and nuclear potentials, i . e . , by the r e a l part of the o p t i c a l p o t e n t i a l . These potentials also a f f e c t the process of absorption, which i s usually energy dependent, and change the paths taken by incoming and outgoing p a r t i c l e s ( r e f r a c t i o n ) ; - 7 -4) d e n s i t y e f f e c t s w h e r e i n t h e momentum, a b s o r p t i o n -s c a t t e r i n g and p o t e n t i a l e f f e c t s , a s w e l l as t h e s i m p l e p r o b a b i l i t y o f f i n d i n g a n u c l e o n t o s t r i k e , each become f u n c t i o n s o f p o s i t i o n w i t h i n t h e t a r g e t n u c l e u s ; and f i n a l l y 5) m i s c e l l a n e o u s e f f e c t s w h i c h as f a r as ou r work i s c o n c e r n e d a r e r e l a t i v e l y u n i m p o r t a n t and w i l l no t be d i s c u s s e d f u r t h e r . I n c l u d e d i n t h e s e a r e the e f f e c t s o f c o r r e l a t i o n among t h e t a r g e t n u c l e o n s ( i m p o r t a n t i n v e r y l i g h t n u c l e i ) and t h e P a u l i i n h i b i t i o n o f r e a c t i o n s w h i c h w o u l d l e a v e a n u c l e o n i n an a l r e a d y o c c u p i e d s t a t e ( i m p o r t a n t a t l o w e n e r g i e s and l a r g e s c a t t e r i n g a n g l e s ) . 1.3 E x i s t i n g mode l s V a r i o u s models t a k i n g n u c l e a r e f f e c t s i n t o a c c o u n t have been u s e d t o e x p l a i n t h e o b s e r v e d A-dependence o f t h e p r o t o n - n u c l e u s p r o d u c t i o n o f p i o n s . To compute r e l a t i v e p i o n y i e l d s t h e y a l l assume c o r r e s p o n d i n g p r o t o n - n u c l e o n c r o s s -s e c t i o n s . The e a r l i e s t work o f i n t e r e s t t o us i s t h a t o f G a s i o r o w i c z ^ ^ who e x p l a i n s t h e p r o d u c t i o n o f l o w - e n e r g y p i o n s b y 2 4 0 - and 340-MeV p r o t o n s . He c o n s i d e r s , i n an a p p r o x i m a t e f a s h i o n , t h e a b s o r p t i o n e f f e c t and assumes , f u r t h e r m o r e , t h a t t h e r e e x i s t s a t t h e n u c l e a r s u r f a c e a s h e l l o r b l a n k e t o f t h e e x c e s s n e u t r o n s w h i c h , s i n c e t h e m a j o r i t y o f p o s i t i v e p i o n s a r e p r o d u c e d i n p r o t o n - p r o t o n c o l l i s i o n s , e n c l o s e s t h e e f f e c t i v e p r o d u c t i o n volume and r e d u c e s t h e p i o n y i e l d o b t a i n e d f rom a h e a v y n u c l e u s r e l a t i v e t o t h a t o b t a i n e d f rom a l i g h t e r o n e . H i s - 8 -c a l c u l a t e d c r o s s - s e c t i o n s agree w e l l w i t h t h e t h e n - a v a i l a b l e e x p e r i m e n t a l ones, but because o f u n c e r t a i n t y i n h i s chosen parameters he can draw no c o n c l u s i o n about the e x i s t e n c e o r t h e n o n - e x i s t e n c e o f a n e u t r o n b l a n k e t . G a s i o r o w i c z d i s c u s s e s t h e e f f e c t o f t h e Coulomb p o t e n t i a l , b u t o n l y on t h e r e l a t i v e shapes o f p o s i t i v e and n e g a t i v e p i o n energy s p e c t r a . M e r r i t t and H a m l i n ^ ' 2 ) f o r m u l a t e an A-dependent a t t e n u a t i o n f a c t o r u s i n g t h e n u c l e a r r a d i u s and energy-dependent p r o t o n and p i o n mean f r e e p a t h s as parameters w h i c h t h e y a d j u s t t o o b t a i n approximate agreement w i t h t h e i r e x p e r i m a n t a l c r o s s -s e c t i o n s f o r p o s i t i v e p i o n p r o d u c t i o n By 335-MeV p r o t o n s . As f o r G a s i o r o w i c z t h i s agreement i s i n c o n c l u s i v e because t h e v a l u e s o f t h e parameters used a r e u n c e r t a i n and t h e d a t a p o i n t s a r e few. They c o n s i d e r o n l y t h e a b s o r p t i o n e f f e c t . H e n l e y ^ 3 ) d i s c u s s e s and t h e n n e g l e c t s t h e e f f e c t o f p i o n a b s o r p t i o n on p i o n y i e l d . H i s main c o n c e r n i s u s i n g t h e e f f e c t o f s t r u c k - n u c l e o n momentum t o e x p l a i n e x p e r i m e n t a l p i o n energy s p e c t r a f r o m ca r b o n . S i m i l a r use o f t h e momentum e f f e c t has been made b y o t h e r s , e.g. R o s e n f e l d ^ . > Imhof e t a l . ( ^ ) use t h e model o f G a s i o r o w i c z t o e x p l a i n t h e i r e x p e r i m e n t a l r e s u l t s a t 340 MeV and c o n s i d e r t h e momentum e f f e c t t o e x p l a i n t h e p i o n y i e l d i n c r e a s i n g from l i t h i u m t o c a r b o n . A l l c a l c u l a t i o n s a r e cru d e . Ansel'm and S h e k h t e r ^ ' ' ) d e r i v e an e x p l i c i t e x p r e s s i o n f o r an A-dependent a t t e n u a t i o n f a c t o r assuming t h a t p r o t o n and - 9 -pion absorption i s energy independent. They are able to f i t curves to most of the then-available experimental data, but they do not consider what values f o r the absorption parameters might be r e a l i s t i c . They consider no e f f e c t other than absorption. The most extensive and successful of the previous calculations i s that of L i l l e t h u n ^ - ^ who combines an improved energy-dependent treatment of the absorption e f f e c t with the concept of a neutron blanket as proposed by Gasiorowicz. Pion absorption c o e f f i c i e n t s are calculated f o r various nuclei from o p t i c a l potentials given by Frank et a l . ^ ^ J . L i l l e t h u n ' s parameters are i n general quite reasonable and his calculated pion y i e l d s agree w e l l with a large amount of experimental data at 450 MeV and t h i s agreement, as L i l l e t h u n shows, depends very strongly on the neutron blanket assumption. L i l l e t h u n also treats the Coulomb and nuclear potentials, ignoring r e f r a c t i o n . In summary then the absorption e f f e c t has been most often used to obtain agreement between theory and experiment, but t h i s agreement i s never conclusive without the assumption concerning the existence of an absorbing neutron blanket. The ef f e c t of i n t e r n a l nucleon momentum has not been discussed i n conection with the general problem of A-dependence; neither has the effect of using a nucleon density d i s t r i b u t i o n that does not have square edges. Potential e f f e c t s have been included i n only one model, that of L i l l e t h u n . We must conclude that the roles played by the important physical factors c o n t r o l l i n g the pion production process are not well understood. - 1 0 -In passing we note that models not of the type out-l i n e d above have been used by Serber^^^, Metropolis et a l . ^ 9 ) ^ and M a r g o l i s ^ ^ ^ and that these have met with only limited amounts of success. Therefore they are of l i t t l e i n t e r e s t to the present inves t i g a t i o n . 1.4 Present program I t i s the objective of the present investigation to calculate some r e l a t i v e d i f f e r e n t i a l cross-sections f o r charged pion production i n the proton-nucleus reaction at energies around 450 MeV. We must assume that the free two-nucleon cross-section though not well known can be used as a parameter, with which we may normalize our re s u l t s to experimental values. The model we s h a l l use i s of the general kind we have been describing, the kind having the most success so f a r . I t makes the basic assumption of i n d i v i d u a l proton-nucleon i n t e r -action and, unlike those described, i t accounts f o r a l l the nuclear e f f e c t s which might be important at the energy of interest (the effects l i s t e d as l)-4) i n section 1.2). The ca l c u l a t i o n we make attempts to sort out those effects which a c t u a l l y are important. I t seems p a r t i c u l a r l y important to avoid the neutron-blanket assumption f o r two reasons. F i r s t , f a i r agreement with experimental re s u l t s has already been obtained by L i l l e t h u n who used t h i s assumption and,: second, we f e e l that while cert a i n - 11 -evidence (the isotopic spin term of the nuclear o p t i c a l model potential) may indicate an excess of neutrons at the nuclear sur face the actual d i s t r i b u t i o n i s nearer to the one with no excess than to the opposite extreme with complete proton and neutron separation. We therefore regard the hypothesis of Gasiorowicz and L i l l e t h u n as a r t i f i c i a l . In t r e a t i n g proton and pion absorption i t i s most convenient to use the same absorption c o e f f i c i e n t s as L i l l e t h u n , but important corrections must be made. We describe these corrections and apply them i n our c a l c u l a t i o n . The absorption c o e f f i c i e n t s were computed f o r protons from experimental proton-proton and proton-neutron cross-sections (about the same at our energy) and f o r pions from mean free paths f o r either absorption or i n e l a s t i c s cattering. By using these c o e f f i c i e n t s we there4-fore ignore the e f f e c t s of r e f r a c t i o n ( e l a s t i c scattering) and the p o s s i b i l i t y that pions may be scattered from other energies and angles into the region of i n t e r e s t . Multiple scattering e f f e c t s are s i m i l a r l y ignored. The ignored interactions have small cross-sections however. The pion mean free paths were i n turn computed from o p t i c a l model potentials obtained from two-body scattering phenomena assuming that two-body forces are not appreciably modified within the nucleus. Hence there i s some consistency i n method. Pion absorption gets s p e c i a l attention i n our ca l c u l a t i o n since i t i s both greater and more energy dependent than proton absorption and since, as i t turns out, the ef f e c t of absorption i t s e l f i s greater than the other e f f e c t s . - 12 -In dealing with the momentum e f f e c t we make the assumption that the momentum d i s t r i b u t i o n has a neg l i g i b l e e f f e c t on both the energy of the pions produced and on the angle of t h e i r emission. The j u s t i f i c a t i o n f o r t h i s seemingly bold assumption i s discussed i n d e t a i l at the time we make i t . I t i s a necessary one to make and i t leads us to suspect that the A-dependence of the pion y i e l d i s not as sensitive to the type of momentum d i s t r i b u t i o n as i t i s to the mean struck-nueleon k i n e t i c energy, at our value of proton energy anyway. In our work we therefore consider only one d i s t r i b u t i o n , that of a Fermi gas. The density e f f e c t s which we consider are those of using a Saxon-Woods shape on the density d i s t r i b u t i o n , i . e . , of giving the nucleus a dif f u s e edge, and that of reducing the basic nuclear radius from the larger values used by e a r l i e r workers to the value now generally accepted. The l a t t e r consideration ;. turns out to be an espe c i a l l y important one: i t l i b e r a t e s , as we show, the model from the need of a neutron blanket. This l i b e r a t i o n , together with the re l a t e d confirmation that absorp-t i o n i s the dominant nuclear e f f e c t i n proton-nucleus pion production at 450 MeV, constitutes the main r e s u l t of the inves-t i g a t i o n to which we now turn. CHAPTER II - THE MODEL 2 . 1 Theory The theory and assumptions underlying our model are stated most d i r e c t l y by a mathematical expression f o r the proton-nucleus cross-section f o r pion production. This cross-section i s a d i f f e r e n t i a l cross-section which i n the laboratory system depends on the energy E of the incident proton, on the angle ¥ JL of pion emission r e l a t i v e to the incident proton beam, and on the energy E^ of the pion emitted. The expression f o r i t i s : [ 1 J (7(E Ijf,^) - //Gf(T ¥,T„,k>exp(-/n ds -/n^ds). .#(r).f(k,r ) . d 3 r . d 3 k , where Of i s the pion production cross-section, i n the laboratory system, f o r protons on "free" nucleons, i . e . , i t i s what we have, been c a l l i n g the corresponding proton-nucleon cross-section. I t i s assumed f o r positve pions to be the proton-proton cross-section and f o r negative pions the proton-neutron cross-section. I f we ignore r e f r a c t i o n e f f e c t s Of depends on the same angle Ijf of pion emission as 0" depends on. Of also depends on proton k i n e t i c energy Tp and pion k i n e t i c energy T^ . inside the target nucleus, which d i f f e r from t h e i r energies outside by pot e n t i a l terms depending on position r i n the nucleus and the mass number A of the nucleus. F i n a l l y , Of depends on the momentum vector k of the p a r t i c u l a r nucleon struck since we are i n the laboratory frame. We have more to say about t h i s l a t e r . The other terms - 14 -i n the integrand of [1] account f o r the various nuclear effects mentioned i n the Introduction. The exponential term i n the integrand of [1] accounts f o r the absorption and scattering e f f e c t s . np and n,,. are respectively proton and pion absorption c o e f f i c i e n t s . They depend on the corresponding k i n e t i c energy, T p or T^, and on the l o c a l nucleon density jo at points s along Sp and s,,. which denote i n that order the paths followed by the incoming proton and the outgoing pion. Since we have ignored r e f r a c t i o n these paths are the straight l i n e s i l l u s t r a t e d i n Figure 1. I f we adopt a s p h e r i c a l l y polar coordinate reference (Figure 1) and l a b e l the point of proton-nucleon c o l l i s i o n by r = (r,Q,<{>) we can derive (Appendix C) e x p l i c i t expressions f o r the distances t r a v e l l e d by the proton and the pion inside the target nucleus.. Where R m a x i s the radius beyond which we assume n e g l i g i b l e absorption these expressions are respectively: [2] Sp = r.cosO + (R^ax ~ r 2sin 29)£ and, where X = cos9*cos!jF + sin9«sinty'Cos<f>, [3] s„ = - r-X + ( R 2 a x - r 2 ( l - X 2))£. In our model absorption ( n p and n n also account f o r certain kinds of scattering) i s averaged over these path lengths. The nucleon density d i s t r i b u t i o n /o[r) i s normalized to the t o t a l number of possible pion producers i n the target nucleus, which we assume are protons f o r positive pion production and neutrons f o r negative pion production. Thus y/b(r).d3r i s - 15 -FIGURE 1 Model o f n u c l e u s showing p a t h o f i n c i d e n t p r o t o n and emerging p i o n . P r o t o n e n t e r s n u c l e u s a t A, t r a v e l s d i s t a n c e Sp, and s t r i k e s a n u c l e o n a t r where i t c r e a t e s a p i o n . The p i o n t h e n t r a v e l s d i s t a n c e s f f and l e a v e s n u c l e u s a t a n g l e Ijl. P a t h s remain i n p l a n e p a r a l l e l t o t h a t o f p a p e r . C o o r d i n a t e system i s i n d i c a t e d . - 16 -p u t e q u a l t o e i t h e r Z o r N = A - Z a c c o r d i n g l y . The n u c l e o n momentum d i s t r i b u t i o n f ( k , r ) i s assumed t o depend on p o s i t i o n r t h r o u g h t h e l o c a l n u c l e o n d e n s i t y / o ( r ) . f ( k , r ) i s always;! n o r m a l i z e d t o u n i t y , i . e . , /f(k,r)»d3k = 1. B e f o r e d e s c r i b i n g a s i m p l i f i c a t i o n we make o f f l ] we r e v i e w t h e a p p r o x i m a t i o n s w h i c h d i s t i n g u i s h our model from a pr o p e r t h e o r y and comment on t h e e x t e n t t o wh i c h our model i s an improvement o v e r i t s e a r l i e r v e r s i o n s . The b a s i c a p p r o x i m a t i o n s i n h e r e n t t o [1] a r e t h o s e due t o : 1) t h e assumption t h a t p i o n s a r e produced i n s i n g l e r. p r o t o n - n u c l e o n c o l l i s i o n s ( t h e h i g h energy o f t h e p r o t o n and t h e s m a l l c r o s s - s e c t i o n s f o r p r o t o n - n u c l e o n i n t e r a c t i o n a t t h i s e n e r gy j u s t i f y t h i s a p p r o x i m a t i o n ) ; 2) t h e assumption t h a t o n l y t h e p r o t o n s o f a t a r g e t n u c l e u s can c o n t r i b u t e t o t h e p r o d u c t i o n o f p o s i t i v e p i o n s ( t h i s we assume because t h e n e u t r o n c o n t r i b u t i o n i s known t o be down by one o r d e r o f ma g n i t u d e ) ; 3) t h e n e g l e c t o f n u c l e a r d e t a i l w h i c h we assume when we use t a r g e t - n u c l e o n momentum and d e n s i t y d i s t r i b u t i o n s (many s i m i l a r c a l c u l a t i o n s , e.g. on e l e c t r o n s c a t t e r i n g , meet w i t h a rem a r k a b l e amount o f s u c c e s s u s i n g t h i s t r e a t m e n t ) ; 4) t h e n e g l e c t o f in c o m i n g and o u t g o i n g p a r t i c l e r e f r a c t i o n w h i c h we assume when we t a k e Sp and s^ t o be s t r a i g h t l i n e s and when we i g n o r e e l a s t i c s c a t t e r i n g i n computing np and n,r ( t h i s a p p r o x i m a t i o n i s p a r t i a l l y j u s t i f i e d by the h i g h energy o f b o t h t h e p r o t o n and t h e p i o n and by t h e s m a l l c r o s s - s e c t i o n - 17 -f o r e l a s t i c s c a t t e r i n g ) ; 5) t h e a s s u m p t i o n t h a t 0~f i s a known f u n c t i o n o f T _ , f , T J J , and k (we do n o t i n f a c t know t h i s f u n c t i o n , b u t i n t h e n e x t s e c t i o n we d e s c r i b e how, by mak ing s e v e r a l a p p r o x i m a t i o n s n o t l i s t e d h e r e , we can move (T f o u t s i d e t h e i n t e g r a l s i g n i n [ 1 ] and t r e a t t h e c a l c u l a t i o n p h e n o m e n o l o g i c a l l y ) ; and 6 ) t h e n e g l e c t o f s u c h t h i n g s as m u l t i p l e s c a t t e r i n g and t h o s e e f f e c t s m e n t i o n e d a s m i s c e l l a n e o u s i n s e c t i o n 1.2 (we s i m p l y c o n s i d e r t h e s e u n i m p o r t a n t t o o u r c a l c u l a t i o n ) . Our r e s u l t s w i l l i n d i c a t e t h a t a more a c c u r a t e t r e a t -ment may n o t be w a r r a n t e d . The p i o n p r o d u c t i o n p r o c e s s , as o u r r e s u l t s c o n f i r m , i s d o m i n a t e d b y p i o n a b s o r p t i o n and t h e b a s i c p r o d u c t i o n r a t e , b o t h o f w h i c h a r e a c c o u n t e d f o r by o u r m o d e l . The e a r l i e r mode l s ( d e s c r i b e d i n s e c t i o n 1 .3) make e s s e n t i a l l y t h e same a p p r o x i m a t i o n s as we l i s t a b o v e . The most d e v e l o p e d o f t h e e a r l i e r mode l s i s t h a t o f L i l l e t h u n who i g n o r e d t h e momentum o f t h e s t r u c k n u c l e o n s (we can do t h i s by p u t t i n g f ( k ) = 0*3(0) i n t o [ 1 ] ) ' . He l i m i t e d t h e i r d e n s i t y d i s t r i b u t i o n t o h a v i n g a s q u a r e shape (he p u t />(r) e q u a l t o a c o n s t a n t up t o a n u c l e a r r a d i u s and t o z e r o beyond) and chose t o use a s q u a r e -shaped n e u t r o n b l a n k e t w h i c h i s n o t a p a r t o f o u r m o d e l . I n a number o f o t h e r c a s e s h i s c h o i c e o f p a r a m e t e r s d i f f e r e d q u i t e s i g n i f i c a n t l y f rom o u r s , w h i c h we d i s c u s s i n a l a t e r s e c t i o n . - 13 -2.2 An es s e n t i a l s i m p l i f i c a t i o n Our model as expressed by [1] cannot be used to calculate a cross-section u n t i l the dependence of Of on Tp, If, T^, and k i s known. I t i s because t h i s dependence i s not known that we must make the following s i m p l i f i c a t i o n . Consider the momentum i n t e g r a l of [1], v i z : C4] CTk(r) = /0 f(T p,f,T f r,k).f(k).d3k. In addition to angle the free cross-section, Of, depends on the ki n e t i c energies T p and T,,. and on the momentum k, a l l of which we assumed were measured r e l a t i v e to a laboratory frame. A l t e r -natively, the free cross-section depends on an angle ljP and on k i n e t i c energies T p and T^, a l l measured i n the rest frame of the struck nucleon. We can write [5] C T f f T p ^ T ^ k ) : 0 f ( T p , i p , T ; , 0 ) , showing that the free cross-section does not, i n the moving frame, depend e x p l i c i t l y on momentum since by d e f i n i t i o n k T = 0 . This does not help us integrate however since there i s an i m p l i c i t In-dependence i n the k i n e t i c energies T p and T£, i . e . , i n the trans-formation back to the laboratory frame. The transformation i s given by the following r e l a t i v i s t i c r e l a t i o n s : [ 6 ] T p - (T pT - kp.kj/m + T. + T p and [7] T» z ( V - k^.kj/m + (u/m)T + T„, - 19 -w h i c h a r e d e r i v e d i n Appendix D. T i s t h e s t r u c k - n u c l e o n k i n e t i c energy c o r r e s p o n d i n g t o k and kj^ i s t h e momentum w h i c h c o r r e s p o n d s t o T^, i = p o r I T , All unprimed q u a n t i t i e s a r e i n t h e l a b o r a t o r y system; m i s t h e n u c l e o n r e s t mass (933 MeV) and u i s t h e p i o n r e s t mass (140 MeV). L e t us i n t r o d u c e two more a p p r o x i m a t i o n s : 1) t h e n e g l e c t o f t h e e f f e c t o f t h e s t r u c k - n u c l e o n momentum, k, on t h e a n g l e o f p i o n e m i s s i o n , w h i c h l e t s us w r i t e U f = U ( t h i s a p p r o x i m a t i o n i s a t l e a s t p a r t i a l l y j u s t i f i e d by t h e h i g h i n c i d e n t p r o t o n energy, w h i c h keeps s m a l l .the.rianguiarJspread caused by ch a n g i n g k, and by t h e f r e e c r o s s - s e c t i o n , w h i c h v a r i e s o n l y m o d e r a t e l y w i t h p r o d u c t i o n a n g l e ) ; and 2) t h e n e g l e c t o f t h e e f f e c t o f t h e s t r u c k - n u c l e o n momentum, k, on t h e energy o f t h e p i o n e m i t t e d , w h i c h l e t s us put T£ = T f T ( t h i s a p p r o x i m a t i o n has a j u s t i f i c a t i o n s i m i l a r t o t h a t o f th e f i r s t ) . The second a p p r o x i m a t i o n s h o u l d be d i s c u s s e d . The s p r e a d i n ' p i o n k i n e t i c energy caused by changing k i s not t o o l a r g e i f compared t o the c o r r e s p o n d i n g s p r e a d caused i n e f f e c t i v e p r o t o n energy. I t can be seen (by u s i n g numbers) t h a t f o r t h e h i g h p r o t o n energy b e i n g c o n s i d e r e d and f o r a r e a s o n a b l y h i g h p i o n energy (30 MeV say) T^ i s always b e t t e r a p p r o x i m a t e d by T„ ( e q u a t i o n [7]) t h a n Tp* i s by T p ( e q u a t i o n [6]) even though i n t h e extreme cases o f head-on and t a i l - o n c o l l i s i o n a t h i g h T (where n e i t h e r a p p r o x i m a t i o n i s v e r y good) t h e d i f f e r e n c e i s not always g r e a t . The d i f f e r e n c e becomes more i m p o r t a n t however when we note t h a t t h e f r e e c r o s s - s e c t i o n f o r - 2 0 -pion production i s , but not without exception, more sensitive to a small change i n T p than i t i s to a si m i l a r one i n T^. Thus our model w i l l , by using the above approximations, account f o r the momentum ef f e c t only as i t a l t e r s the amount of energy available to the production reaction. The two approximations l i s t e d i n t h i s section do not a f f e c t the model i n any other way. Putting f = f , = T f f, and expanding [5] by a Taylor ser i e s i n about the point T p gives us C*] ^ ( T p . f . V k ) = 0 X ^ , 1 ^ , 0 ) = OfCTp.^.O) «. + d f f f C T p ^ T ^ O M T p - T p) * + ... A l t e r n a t i v e l y , we can expand [5] i n T^, ¥', and T^ and, i n the approximation described above, drop terms containing dOf/df, (¥' - ¥ ) , dOf/dT^, or (T,» - T^). The r e s u l t i s the same. I f we drop high-order terms and put [8j into [ 4 ] , using [6] to note C9) / (T p - T p)-f(k)-d>k r rs, where y i s the f a c t o r (T + m)/m and T i s the l o c a l mean value of the struck-nucleon k i n e t i c energy, we get the r e s u l t that UOJ Q k(r) z (J f(T . y . T J + d f f f t T p . f . T j ^ . dT v P We have stopped indicating e x p l i c i t l y the fact that k' = 0 . As our l a s t approximation we remove the r-dependence from a l l terms - 2 1 -i n [ 1 0 ] e x c e p t T by assuming t h a t , f o r t h e purpose o f computing Tp and 1n f r o m g i v e n v a l u e s o f E p and E„. o n l y , t h e n u c l e a r p o t e n t i a l s depend j u s t on mass number A . We f i n a l l y f i n d t h a t [I] can be w r i t t e n i n t h e f o l l o w i n g manner: [II] ( n E p . V . E j = C f(T p . Y . T j . l ! • d0-f(T ^ T ^ . T l z , d T p where I± z f e x p ( - ^ h p d s -^n^ds) «o(r).d3r and 12 = f T ( r ) .exp(-/n_ds -/n^ds) «>o(r) «d3r. °p" °1T T h i s i s t h e e x p r e s s i o n on w h i c h our c a l c u l a t i o n s are; based. The p r o t o n - n u c l e u s c r o s s - s e c t i o n f o r p i o n p r o d u c t i o n i s now e x p r e s s e d i n terms o f mean s t r u c k - n u c l e o n k i n e t i c energy r a t h e r t h a n i n terms o f a p a r t i c u l a r momentum d i s t r i b u t i o n . T h i s s i m p l i f i e d v e r s i o n o f our model must be used u n t i l such t i m e as we have more i n f o r m a t i o n about t h e b a s i c p r o t o n - n u c l e o n p r o d u c t i o n p r o c e s s . I n [ l l ] t h e p r o t o n - n u c l e o n c r o s s - s e c t i o n f o r p i o n p r o d u c t i o n and i t s f i r s t d e r i v a t i v e w i t h r e s p e c t t o p r o t o n e n ergy appear as c o e f f i c i e n t s t o i n t e g r a l s w h i c h can be e a s i l y e v a l u a t e d . I n our c a l c u l a t i o n we e v a l u a t e b o t h I ^ and I2 and use t h e c o e f f i c i e n t s t o a d j u s t r e s u l t s t o e x p e r i m e n t a l d a t a on t h e p r o t o n - n u c l e u s p r o d u c t i o n p f _ p i o n & The A-dependence o f our c a l c u l a t e d c r o s s - s e c t i o n s e n t e r s o n l y t h r o u g h 1-^  and I2: we cannot t a k e i n t o a c c o u n t t h e A-dependence e n t e r i n g t h r o u g h the: c o e f f i c i e n t s a p p e a r i n g w i t h them. We cannot t a k e i n t o a c c o u n t any dependence on i n c i d e n t p r o t o n energy s i n c e i t e n t e r s o n l y t h r o u g h t h e c o e f f i c i e n t s . Our l a c k o f knowledge c o n c e r n i n g t h e - 22 -free production process also keeps us from knowing just when the assumptions we make i n going from [1] to [11] might be i n v a l i d . I f we wish to ignore the momentum e f f e c t , as well as the approximations made i n t h i s section, we need only set T = 0. The r e s u l t w i l l be the same model we would get by putting f ( k ^ = o^'(0). into Cl]. With the appropriate choice of yo(r) we would have the model of L i l l e t h u n , but without the neutron blanket. 2.3 The parameters chosen To calculate a proton-nucleus cross-section f o r pion production using [11] we must specify absorption c o e f f i c i e n t s , a density d i s t r i b u t i o n , a momentum d i s t r i b u t i o n (a mean k i n e t i c energy at l e a s t ) , and a free proton-nucleon cross-section. The free cross-section we cannot specify exactly hence we calculate only r e l a t i v e proton-nucleus cross-sections using the free cross-section as a normalizing parameter. When discussing our r e s u l t s we comment on how the value we must assume f o r the free cross-section compares with:the poorly known experimental one. The density d i s t r i b u t i o n that we choose has the f a m i l i a r Saxon-Woods form, v i z : [12] Mr) - >0 o.(l + exp((r - R ) / a ) ) - l , i n which the nuclear radius R (for nucleon density and the o p t i c a l potentials o f protons and pions) i s the usual o p t i c a l model radius R = r ^ A 1 / 3 with r Q = 1.25 fermis. This i s the - 23 -value used i n our work. L i l l e t h u n and e a r l i e r workers used a larger value ( r 0 = 1.35 fm.) and because the pion production rate i n a heavy nucleus i s l a r g e l y controlled by pion absorption the difference i s s i g n i f i c a n t . Our r e s u l t s show t h i s . R i s the radius where the density i s 50% of i t s central value and i s not to be confused with R m ax> t n e radius beyond which absorption i s assumed n e g l i g i b l e . The two coincide only i n a square-edge nucleus. For our diffuse-edge nucleus we a r b i t r a r i l y set R max to be the radius where the density has f a l l e n to 10% of i t s central value so that R m a x and R together define the value S = 2» •(Rmax - R) which i n turn defines the surface thickness constant a = S / ( 4 » l n 3 ) . For a we have chosen a standard value of 0 . 5 5 fm. Our r e s u l t s are not very sensitve to a. The normalization constant o 0 i s varied according to the context of i t s use. For use i n the integrands of [111 f>0 t a k e s the form o Q = /o0»Z or >o0 = £ > 0 » N depending on whether posit i v e or negative pion produc-t i o n i s being considered, where >o0 has the standard value [13] >60 = ( 3 / ( 4 f r R 3 ) ) . ( l + t r 2 a 2 / R 2 ) . When used alone /o£ normalizes the density d i s t r i b u t i o n to unity. For use with o p t i c a l potentials and absorption c o e f f i c i e n t s , i n equations l i k e V(r) - /o(r).V 0 and n(r) = /o(r).n Q, /oQ takes the form fiQ - >° 0 , v°l> where Vol i s the nuclear volume ( 4 / 3JtR 3. For the l o c a l mean k i n e t i c energy of the target nucleons we choose a value corresponding to the momentum d i s t r i -bution of a Fermi gas appropriate to the density>o(r), v i z : - 24 -[14] T(r) = 0 . 6 T f ( r ) , where T f ( r ) = i O A f r ) ) 2 / 3 i s t h e F e r m i energy a t r and p{r) i s n o r m a l i z e d t o e i t h e r Z o r N, The p r o t o n a b s o r p t i o n i s r e l a t i v e l y weak a t t h e e n e r g i e s o f i n t e r e s t t o us: we have chosen n t o be 0 . 1 8 2 fm.~^ i n a l l our c a l c u l a t i o n s . T h i s i s a l s o t h e v a l u e used by L i l l e t h u n . I t was computed by him f rom t o t a l p r o t o n - p r o t o n and p r o t o n - n e u t r o n c r o s s - s e c t i o n s . P i o n a b s o r p t i o n on the o t h e r hand i s s t r o n g and h i g h l y energy dependent. P i o n a b s o r p t i o n c o e f f i c i e n t s depend on p i o n k i n e t i c e nergy i n s i d e t h e t a r g e t n u c l e u s . At any p o i n t r i n s i d e t h e n u c l e u s the p i o n k i n e t i c energy, T ^ , i s r e l a t e d t o i t s t o t a l e n e r gy, i . e . , i t s k i n e t i c energy o u t s i d e t h e n u c l e u s , B,,., by [ 1 5 ] Eff = 1„ + V c ( r ) + V r ( r , T f f ) , where V c ( r ) i s t h e Coulomb p o t e n t i a l and v" r(r , T n . ) i s t h e r e a l p a r t o f t h e p i o n o p t i c a l p o t e n t i a l , depending on p i o n k i n e t i c energy. I n o u r c a l c u l a t i o n V c ( r ) i s computed under t h e assump-t i o n t h a t Z p r o t o n s a r e d i s t r i b u t e d u n i f o r m l y i n s i d e t h e charge r a d i u s R c = 1 . 0 7 A V 3 f m . we t h e r e f o r e t a k e [ 16 ] V c ( r ) ' = Ze£._J^.(3 - r2/R2) f o r r < R and -v"c(r.) = Z § £ . 1 f o r r > R „ , 4rre 0 r c where e /(4ffC 0) has t h e n u m e r i c a l v a l u e o f 1 . 4 4 . - 25 -Values of V ^ r , ^ ) vs. T f f have been computed by Frank et a l . We use t h e i r values corrected by a factor of ( r 0/ r 0)^> where r£ i s the basic nuclear radius assumed by Frank et a l . and r Q i s our value of 1 .25 fm. Frank et a l . i m p l i c i t l y assume a value of 1.4-1 fm. f o r r£ when they set A = 1 ( c f . the caption to t h e i r Table I I ) . A i s defined by the r e l a t i o n (their equation (3)) R s A* A A"^ 3 , where X' i s the Compton wavelength f o r pions (1.41 fm.). The correction which we make to V r i s appropriate since A enters into the expression giving V r as an inverse cube (c f . Frank et a l . , t h e i r equation (13)). We also apply a correction f o r l o c a l nucleon density, i o ( r ) , normalized to nuclear volume ( c f . t h e i r equation (1)) and by manipulating [15] compute the pion k i n e t i c energy, T^, at r from the given pion t o t a l energy, E^. Having thus counted T^ at a point r we can i n t e r -polate for n f f using the values of n„. vs. T , , . also given by Frank et a l . (they give values of pion mean free path vs. Hn, but these are just reciprocals of the absorption c o e f f i c i e n t s ) . Three corrections must be applied to the interpolated value: 1) a correction f o r units, inverse Compton wavelengths to inverse fermis ( c f . Frank et a l . , t h e i r Table I ) ; 2) a correction for l o c a l nucleon density, /o(r), normalized to volume; and 3) the correction mentioned i n connection with Vr concerning the basic nuclear radius, r Q . The l a s t correction, applied to n^, i s again one of (r^/r^)-? obtained from an inspection of Frank et a l . , equations ( 6 ) , (7) and ( 9 ) . Their equation ( 9 ) , giving n^ ., i s - 26 -i s derived using a model of Brueckner et a l . v ^ x ' which assumes that pion absorption by nuclear matter i s proportional to the pion capture cross-section of a deuteron. We re-derive t h i s equation i n Appendix B to show i t s inverse cube dependence on A, not shown e x p l i c i t l y by Frank et a l . A l t e r n a t i v e l y , we have taken f o r our pion absorption c o e f f i c i e n t s those values l i s t e d by L i l l e t h u n ( i n his Table V), who also uses the data of Frank et a l . , but without applying the corrections described above. L i l l e t h u n uses an interpolation technique which seems to be d i f f e r e n t from ours since, i f f o r no other reason, h i s computed values of n f f are not at a l l smooth functions of A and we think they should be. Nuclear size i s important to the absorption e f f e c t i n two ways: 1) the distance a p a r t i c l e t r a v e l s inside a nucleus depends d i r e c t l y on the nuclear size; and 2) the absorption c o e f f i c i e n t depends on l o c a l nucleon density, which i n "turn depends on the nuclear s i z e . The f i r s t dependence i s roughly l i n e a r i n the nuclear radius and the second i s one involving the inverse cube of the nuclear radius. Hence, since t o t a l absorp-t i o n depends on the product of distance t r a v e l l e d and absorption c o e f f i c i e n t , we have the seemingly curious r e s u l t that the smaller of two equally massive nuclei i s the stronger absorber. I t must not be forgotten however that the smaller nucleus i s also the smaller target, i . e . , that the integrations of [ l l ] are over a smaller volume. The net balance of these and other effects on pion y i e l d we calculate i n a special case. - 2 7 -CHAPTER I I I - THE CALCULATION 3 . 1 The i n t e g r a t i o n Our c a l c u l a t i o n s r e q u i r e d t h e n u m e r i c a l e v a l u a t i o n o f th e i n t e g r a l s 1 ^ and I 2 o f e q u a t i o n [ 1 1 ] . T h i s we d i d u s i n g t h e U n i v e r s i t y o f B r i t i s h Columbia's IBM 7 0 4 4 computer and a FORTRAN IV program c a l l e d PIPROD, t h e l o g i c o f w h i c h i s o u t l i n e d i n Appendix F. I n t h i s program t h e i n t e g r a n d s o f 1 ^ and I 2 a r e each e v a l u a t e d t h r o u g h o u t t h e n u c l e u s o v e r a network o f p o i n t s , none o f wh i c h i s s e p a r a t e d f rom i t s n e a r e s t n e i g h b o u r by more t h a n a f r a c i o n o f a f e r m i ( 1 0 ~ 1 3 cm.). The i n t e g r a t i o n i s completed by an a p p r o p r i a t e number o f Simpson's r u l e a p p r o x i -m a t i o n s . The e r r o r i n t r o d u c e d by each a p p r o x i m a t i o n i s e s t i -mated by r e - c a l c u l a t i n g t h e p a r t i c u l a r i n t e g r a l o v er a c o a r s e r network o f p o i n t s o b t a i n e d by d o u b l i n g the s e p a r a t i o n d i s t a n c e . I n o u r c a l c u l a t i o n t h e c o a r s e and f i n e i n t e g r a l s d i f f e r e d by an amount t h a t was u s u a l l y l e s s t h a n two p e r c e n t and v e r y seldom more t h a n f i v e p e r c e n t o f th e f i r s t , i . e . , t h e f i n e i n t e g r a l . The t r u e e r r o r , i . e . , t h e d i f f e r e n c e between t h e f i n e and an i n f i n i t e l y f i n e i n t e g r a l , a t each s t a g e s h o u l d always be much s m a l l e r t h a n t h e one e s t i m a t e d i n t h e above manner. At each p o i n t o f t h e network t h e paths s p and s ^ have t o be de t e r m i n e d . A t each o f s e v e r a l p o i n t s ( s e p a r a t e d by l e s s t h a n f e r m i ) a l o n g sn "the p i o n o p t i c a l p o t e n t i a l } V r , and t h e p i o n a b s o r p t i o n c o e f f i c i e n t , n , r , have t o be de t e r m i n e d . T h i s we d i d by f i t t i n g a t h i r d o r d e r p o l y n o m i a l t o t h e known d a t a o f - 28 -Frank et a l . (di f f e r e n t at each point since we include a density-correction) and interpolating at the pion k i n e t i c energy of in t e r e s t (which also varies from point to point) The averaging of nn- along Sfr, l i k e the integration of 1^ and J-2, was done i n the Simpson's ru l e approximation with an error estimation. Variations run on the PIPROD program are not d i s -cussed i n t h i s t h e s i s . Variations were run however and t h e i r outcomes constitute the subject matter of section 3 . 3 . 3.2 Results compared to experiment A t y p i c a l r e s u l t of our main ca l c u l a t i o n i s i l l u s t -rated i n Figure 2, f o r the case i n which we have incident protons of 450 MeV producing pions of 83 MeV at an angle of 21.5° i n the laboratory system. The experimental cross-sections are those of L i l l e t h u n . The calculated curves have been normalized to the experimental value at 2 ? A 1 . The curve marked L i s L i l l e t h u n ' s published r e s u l t , calculated from his model with an absorbing neutron blanket. We were able to duplicate t h i s r e s u l t i n our work by using L i l l e t h u n ' s neutron blanket and parameters, these including his calculated absorption c o e f f i c i e n t s . The curve marked 1^ i n Figure 2 i s our r e s u l t , c a l -culated using our model as expressed by [11] and the parameters and corrections described i n section 2 .3. We have plotted the i n t e g r a l I]_ only, thereby ignoring the e f f e c t of a momentum d i s t r i b u t i o n . The i n t e g r a l I 2 , which allows us to take a - 29 -3 0 0 C T |JB/MEV/ST. £ ,+=83 MEV Ep =450MEV 2 0 0 0 100 -M A S S NO.. A 0 100 2 0 0 FIGURE 2 Cross-sections f o r the production of 33-MeV pos i t i v e pions at 21.5* by 450-MeV protons on various n u c l e i . - 3 0 -momentum e f f e c t into account, coincides almost exactly with the plotted curve, when normalized at 2 7 A 1 . The r a t i o I 2 / I 1 F O R the momentum d i s t r i b u t i o n of a Fermi gas i s shown as an inset to Figure 2 . . The ind i c a t i o n i s that the e f f e c t of a momentum d i s t r i b u t i o n on the r e l a t i v e pion y i e l d i s small, although the ef f e c t may be important when choosing normalizing factors 0~f and dOf/dTp to compare with experimental data on the proton-nucleon production of pions. Our model i s not yet able to treat pion production at proton energies near the production threshhold where the momentum eff e c t i s certain to be more important, but at the energy of 450 MeV i t suggests that we may neglect t h i s e f f e c t . We have thus ignored the e f f e c t of a momentum d i s t r i -bution i n p l o t t i n g our r e s u l t s . There are two other A-dependent e f f e c t s which we have ignored, one i n deriving [ 1 1 ] and one i n p l o t t i n g our curves. They are: 1) the e f f e c t of target-nucleus neutrons on p o s i t i v e pion production (we could estimate t h i s e f f e c t by including i n our r e s u l t the factor ( 1 0 « Z + N ) / 1 1.A«Z which removes the normalization imposed on >o(r) and weights the proton-proton and proton-neutron free cross-sections i n :,. accordance with pion production proceeding through an i n t e r -mediate 3 -3 resonnance state: i t would rais e a l l pion y i e l d s , but would e s p e c i a l l y favour high values of A, by 5% i n 2<-^Pb as shown i n Figure 2 f o r example); and there i s 2 ) the effect due to the co e f f i c i e n t s Of, dOf/dTp and 7 , which have an A-dependence that enters through the potential terms used to r e l a t e k i n e t i c energy i n the nucleus with t o t a l energy outside (this e f f e c t i s hard to - 31 -even estimate, but i t probably lowers s l i g h t l y the r e l a t i v e y i e l d from a heavy nucleus at our energies). The value f o r Of which we i m p l i c i t l y assume i n normalizing our r e s u l t of Figure 2 to the experimental cross-27 section of A l i s 31.0 microbarns/MeV/steradian. Putting the i n t e g r a l 12 into our r e s u l t we can lower t h i s value to nearly 25 ub/MeV/st. by taking f o r dOf/dT p/0f the value 0.015, which i s the value of the logarithmic derivative at 450 MeV taken from . data on the t o t a l cross-sections 0(pp-*rr+d) and 0"(pp-»- tr+pn) vs. proton energy as summarized by Mcllwain et a l . ^ 2 ^ . Pondrom^ uses 450-MeV protons to obtain d i f f e r e n t i a l cross-sections at 20°27» f o r the reaction pp-* tr+pn. His value i s almost 8 ub/MeV/ / s t . at the pion energy of 122 MeV, t h i s being the k i n e t i c energy needed by a p o s i t i v e pion inside an 2^A1 nucleus i f i t i s to be seen outside with an energy of 83 MeV. Gell-Mann and Watson^ x^ indicate that 70% of the positive pions coming from proton-nucleon reactions at 300-400 MeV are due to the reaction pp-** +d. I f we assume that t h i s i s also true at 450 MeV and that ten times the number that come from pn-*tr+nn come from the two pp reactions, then we can estimate a possible experimental value f o r Of of just over 31 ub/MeV/st. This l a s t value i s very uncertain, but i t i s nonetheless close to our assumed value and for that reason i t i s encouraging. I l l u s t r a t e d i n F i g u r e 3 a r e some r e s u l t s on p o s i t i v e p i o n p r o d u c t i o n a t 450 MeV, bu t a t o t h e r p i o n e n e r g i e s and p i o n s c a t t e r i n g a n g l e s as i n d i c a t e d . The e x p e r i m e n t a l p o i n t s and t h e - 32 -150 50 c o > LU 0 CO b 5 0 0 50 1 1 1 E P = 4 5 0 M E V i Y=21.5° E^F 166 MEV I I I o - o - L - ^ -1 Y = 2i.5° £^=236 MEV — — • ') — — ° r>. . Q I . ° ^ ^ x y - ^ o u *- — I I I I IjJ = 6 0 ° E^r 14 9 MEV ^ \ i l I 6 100 2 0 0 FIGURE 3 More cross-sections f o r positive pion production. - 33 -curves marked L are once again those of L i l l e t h u n . What we have said about the re s u l t i n Figure 2 can also i n general be said about the r e s u l t s i n Figure 3. Some r e s u l t s on negative pion production at 450 MeV are given i n Figure 4. Pion energies and angles are indicated and again the experimental points and the curves marked L are due to L i l l e t h u n . In the case of negative pions the remark we made in connection with the r e s u l t i n Figure 2 concerning a correction factor f o r the target-nucleus neutron contribution to the pion y i e l d no longer applies since the neutrons of the target, and only the neutrons, are considered . There i s no proton c o n t r i -bution whatever. The remark made about the A-dependence of the normalizing parameters s t i l l applies. Agreement between our calculated curves and the data of L i l l e t h u n i s seldom excellent and i t i s never r e a l l y poor. It i s often as good as any previously obtained. We show i n the next section how the absorption e f f e c t , e s p e c i a l l y pion absorp-t i o n , dominates the proton-nucleus production process. It therefore seems l i k e l y that the use of d i f f e r e n t , more modern data on pion absorption might lead to d i f f e r e n t and perhaps better agreement than we obtain. Hence agreement between our. calculated r e s u l t s and those of experiment should not be viewed as being c r i t i c a l to our model. The importance of our results i s of another kind: good agreement previously depended on the neutron blanket assumption and now i t does not. We show why i n the next section. - 34 -30 20 10 0 .10 1 : U E P = 4 5 0 M E V IjJ = 21.5 132 MEV =21.5 LU E^.1.66 MEV 10 CO D 0 10 0 10 l|l=2f.5°, E ^ 2 0 0 M E V Y = 60 99MEV FIGURE 4 Some c r o s s - s e c t i o n s f o r n e g a t i v e p i o n p r o d u c t i o n . - 3 5 -3 . 3 R e s u l t s w i t h parameters v a r i e d I n o r d e r t o det e r m i n e t h e manner i n which t h e v a r i o u s p h y s i c a l f a c t o r s c o n t r o l l i n g t h e p r o t o n - n u c l e u s p i o n p r o d u c t i o n p r o c e s s a f f e c t t h e A-dependence o f t h e p i o n y i e l d , we c a l c u l a t e d c r o s s - s e c t i o n s u s i n g parameters o t h e r t h a n t h o s e d e s c r i b e d i n s e c t i o n 2 . 3 . The r e s u l t s o f t h e s e c a l c u l a t i o n s a r e i l l u s t r a t e d i n F i g u r e 5. As f o r F i g u r e 2 we have c o n s i d e r e d 450-MeV p r o t o n s p r o d u c i n g 83-MeV p o s i t i v e p i o n s a t 2 1 . 5 ° and a g a i n c u r v e , L i s L i l l e t h u n 1 s c a l c u l a t e d r e s u l t , w h i c h we were a b l e t o d u p l i c a t e u s i n g a n e u t r o n b l a n k e t and h i s p a r a m e t e r s , t h e s e i n c l u d i n g h i s c a l c u l a t e d a b s o r p t i o n c o e f f i c i e n t s . Curve I i s t h e r e s u l t o f u s i n g a n e u t r o n b l a n k e t and L i l l e t h u n * s p a r a m e t e r s , b u t u s i n g o ur i n t e r p o l a t i o n t e c h n i q u e t o c a l c u l a t e p i o n a b s o r p t i o n c o e f f i c i e n t s . Curve I I shows t h e e f f e c t o f removing t h e n e u t r o n b l a n k e t , i . e . , t h e e f f e c t o f con-s i d e r i n g t h e p r o t o n s and n e u t r o n s u n i f o r m l y d i s t r i b u t e d w i t h i n t h e n u c l e a r r a d i u s o f L i l l e t h u n , u s i n g t h e same parameters and i n t e r p o l a t i o n t e c h n i q u e as we used f o r curve I . Curve I I I , on the o t h e r hand, was o b t a i n e d by u s i n g a v a l u e o f 1 . 2 5 fm. f o r r Q , a Saxon-Woods d e n s i t y d i s t r i b u t i o n , and our i n t e r p o l a t i o n t e c h -n i q u e t o compute p i o n a b s o r p t i o n . The c o r r e c t i o n s d e s c r i b e d i n s e c t i o n 2 . 3 were a l l a p p l i e d . Hence curve I I I i s our r e s u l t i n F i g u r e 2 , t h e momentum e f f e c t s t i l l b e i n g i g n o r e d . Curve IV i s o b t a i n e d when, s t i l l i g n o r i n g t h e momentum e f f e c t , we remove t h e d i f f u s e edge from t h e model f o r curve I I I by s e t t i n g a = 0. - 36 -7 z FIGURE 5 C r o s s - s e c t i o n s o b t a i n e d by v a r y i n g p a r a m e t e r s . - 3 7 -O t h e r w i s e t h e models used f o r c u r v e s I I I and IV a r e i d e n t i c a l . The s u g g e s t i o n here i s t h a t , l i k e t h e e f f e c t o f a momentum d i s t r i b u t i o n , t h e e f f e c t o f a d e n s i t y d i s t r i b u t i o n i s secondary when compared t o t h a t o f a b s o r p t i o n . The importance o f t h e a b s o r p t i o n e f f e c t i s i n d i c a t e d by t h e f a c t t h a t w i t h o u t a b s o r p -t i o n t h e p i o n y i e l d becomes d i r e c t l y p r o p o r t i o n a l t o Z (c u r v e Z i n F i g u r e 5 ) . The importance o f n u c l e a r s i z e t o t h e a b s o r p t i o n e f f e c t i s i n d i c a t e d by t h e f a c t t h a t t h e r e d u c t i o n o f t h e b a s i c n u c l e a r r a d i u s , r Q , from 1 . 3 5 fm. (c u r v e I I ) t o 1 . 2 5 fm. ( c u r v e IV) l o w e r s t h e r e l a t i v e p i o n y i e l d i n a heavy n u c l e u s by a l m o s t e x a c t l y t h e same amount as does t h e i n t r o d u c t i o n o f an a b s o r b i n g n e u t r o n b l a n k e t ( c u r v e I ) . That our agreement w i t h experiment i s n o t as good as L i l l e t h u n ' s t h e r e f o r e seems t o be due o n l y t o t h e f a c t t h a t t h e method we use t o i n t e r p o l a t e f o r p i o n a b s o r p -t i o n c o e f f i c i e n t s i s d i f f e r e n t t h a n t h e one used by him. Were we t o use L i l l e t h u n ' s method i n c a l c u l a t i n g our curve I I I we would o b t a i n h i s r e s u l t . And w i t h o u t a n e u t r o n b l a n k e t . 3 . 4 C o n c l u s i o n We have c o n s t r u c t e d a model t o e x p l a i n t h e p r o d u c t i o n o f p i o n s i n t h e bombardment by p r o t o n s o f v a r i o u s n u c l e i and we have used t h e model t o c a l c u l a t e r e l a t i v e c r o s s - s e c t i o n s f o r t h e p r o c e s s . The model assumes, among o t h e r t h i n g s , t h a t t h e i n c i d e n t p r o t o n i n t e r a c t s w i t h but a s i n g l e t a r g e t n u c l e o n t o produce a p i o n and not w i t h t h e t a r g e t n u c l e u s as a whole. I t assumes t h a t t h e p r o t o n - n u c l e u s c r o s s - s e c t i o n f o r p i o n p r o d u c t i o n - 33 -can be c a l c u l a t e d knowing t h e b a s i c p r o t o n - n u c l e o n c r o s s - s e c t i o n f o r p i o n p r o d u c t i o n and t h e number o f t a r g e t n u c l e o n s , p r o v i d e d c e r t a i n i m p o r t a n t n u c l e a r e f f e c t s a r e t a k e n i n t o a c c o u n t . The n u c l e a r e f f e c t s a c c o u n t e d f o r by our model a r e t h o s e due t o p r o t o n and p i o n a b s o r p t i o n , t o n u c l e a r p o t e n t i a l s , and t o t h e s t r u c k - n u c l e o n d e n s i t y and momentum d i s t r i b u t i o n s . Because our knowledge o f t h e b a s i c p r o t o n - n u c l e o n p r o d u c t i o n r a t e i s l i m i t e d , we can o n l y c a l c u l a t e r e l a t i v e c r o s s -s e c t i o n s and use t h e b a s i c r a t e as a f r e e parameter. We do t h i s i n s e v e r a l s p e c i a l c a s e s and compare our r e s u l t s w i t h d a t a f rom e x p e r i m e n t a t 450 MeV. Agreement i s o n l y moderate, but i t i s as good as any p r e v i o u s l y o b t a i n e d .and, u n l i k e t h e e a r l i e r r e s u l t s , i t does not depend on t h e r a t h e r a r t i f i c i a l a s s u m p t i o n o f an a b s o r b i n g n e u t r o n b l a n k e t . Our agreement depends i n s t e a d on t h e use o f a modern n u c l e a r r a d i u s and on a r e a s o n a b l e t r e a t m e n t o f p i o n a b s o r p t i o n . I n t h i s r e s p e c t our r e s u l t s c o n f i r m what t h e e a r l i e r w o r k e r s had assumed, t h a t a b s o r p t i o n , e s p e c i a l l y p i o n a b s o r p t i o n , i s t h e dominant p h y s i c a l f a c t o r c o n t r o l l i n g t h e p r o t o n - n u c l e u s p r o d u c t i o n o f p i o n s . A t h i g h p r o t o n e n e r g i e s we f i n d t h a t t h e e f f e c t s o f a s t r u c k - n u c l e o n momentum d i s t r i b u t i o n a r e a l l but n e g l i g i b l e i n d e t e r m i n i n g t h e A-dependence o f t h e p i o n y i e l d , a l t h o u g h t h e y may become more i m p o r t a n t a t lo w e r e n e r g i e s near t h e p i o n p r o d u c t i o n t h r e s h h o l d where we hope t o exte n d t h e p r e s e n t i n v e s t i g a t i o n . D e n s i t y d i s t r i b u t i o n e f f e c t s we f i n d a r e l i k e w i s e a l m o s t n e g l i g i b l e a t h i g h e n e r g i e s . These are our c o n c l u s i o n s . - 39 -REFERENCES Reference ( 1 ) describes the TRIUMF (Tr i - U n i v e r s i t y Meson F a c i l i t y ) proposal, which we mention i n Chapter I. ( 1 ) TRIUMF Proposal and Cost Estimate, E. W. Vogt and J . J . Burgerjon, eds., University of B r i t i s h Columbia (1966), with annual supplements. References (2-11) describe experiments on the proton-nucleon production of pions, which we summarize i n Appendix A. (2) M. M. Block, S. Passman, and ¥. W. Havens, J r . , Phys. Rev. 88, 1239 (1952). (3) R. H. March, Phys. Rev. 120, 1874 (I960). (4) A. H. Rosenfeld, Phys. Rev. £6, 130 (1954). (5) L. G. Pondrom, Phys. Rev. 114, 1623 (1959). (6) Y. M. Kazarinov and Y. N. Simonov, Sov. J . Nucl. Phys. 4, 100, (1967). (7) E. Heer, W. H i r t , M. Martin, E. G. Michaelis, C. Serre, P. Skarek, and B. T. Wright, Proceedings of the Williamsburg Conference on Intermediate Energy Physics, February 1966. (8) H. Heifer, A. S. Kuznetsov, M. G. Mescheryakov, W. Swiat-kowski, and V. G. Vovchenko, Nuc. Phys 23, 353 (1961). (9) V. G. Vovchenko, Sov. J . Nucl. Phys. 3, 803 (1966). (10) Yale i n t e r n a l report Y-3, March 1964. (11) R. P. Haddock, M. Z e l l e r , and K. M. Crowe, UCLA (MPG) 64-2. References (12-16) describe attempts at explaining - 40 -the proton-nucleon production of pions. (12) K. M. Watson, Phys. Rev. 8 3 , 1163 ( 1 9 5 2 ) . (13) K. M. Watson and K. A. Brueckner, Phys. Rev. 33 , 1 ( 1 9 5 1 ) . (14) M. Gell-Mann and K. M. Watson, Annu. Rev. Nucl. S c i . 4_, 219 ( 1 9 5 4 ) . (15) S. Mandelstam, Proc. Roy. Soc. A 2 4 4 . 491 ( 1 9 5 8 ) . (16) D. S. Beder, CERN preprint TH. 371 ( 1 9 6 8 ) . (17) D. C. Peaslee, Phys. Rev. £5, 1580 ( 1 9 5 4 ) , f o r example. References ( 1 8 - 3 0 ) describe experiments on the proton-nucleus production of pions, which we summarize i n Appendix B. (13 (19 (20 (21 (22 (23 (24 (25 (26 (27 (23 D. L. Clark, Phys. Rev. 3 £ , 157 ( 1 9 5 2 ) . W. Imhof, H. T. Easterday, and V . Perez-Mendez, Phys. Rev. 1 0 5 . 1859 ( 1 9 5 7 ) . R. Sagane and W. F. Dudziak, Phys. Rev. £ 2 , 212 (1953) D. A. Hamlin, M. Jakobson, J . M e r r i t t , and A. Schulz, Phys. Rev. 84_, 857 ( 1 9 5 1 ) . J . M e r r i t t and D. A. Hamlin, Phys. Rev. ££, 1523 ( 1 9 5 5 ) . M. M. Block et a l . , see reference ( 2 ) . A. H. Rosenfeld, see reference ( 4 ) . E. L i l l e t h u n , Phys. Rev. 125, 665 ( 1 9 6 2 ) . E. Heer et a l . , see reference ( 7 ) . H. Heifer et a l . , see reference ( 8 ) . A. G. Meshkovskii, I. I. Shalamov, and V . A. Shebanov, Sov. Phys.-JETP 8 , 46 ( 1 9 5 9 ) ; Z , 987 ( 1 9 5 8 ) ; and 6 , 463 ( 1 9 5 8 ) . A. G. Meshkovskii, I. S. P l i g i n , I. I. Shalamov, and V. A. - 4-1 -Shebanov, Sov. Phys.-JETP /fc, 842 (1957) and 5, 1085 ( 1 9 5 7 ) . (29) M. G. Meshcheriakov, I . K. Vzorov, V. P. Zrelov, B. S. Neganov, and A. F. Shabudin, CERK Symposium, Vol. 2, 357 ( 1 9 5 6 ) . Also Sov. Phys.-JETP 4 , 79 (1957) and 4 , 60 ( 1 9 5 7 ) . (30) R. P. Haddock et a l . , see reference (11). References ( 31 -36 ) describe attempts at explaining the proton-nucleus production of pions using models , which we describe i n section 1 . 3 , s i m i l a r to ours. References ( 38 -40 ) attempt the same explanation using other models. ( 3 D S. Gasiorowicz, Phys. Rev. 21, 843 ( 1 9 5 4 ) . (32) J . M e r r i t t and D. A. Hamlin, see reference (22). (33) E. M. Henley, Phys. Rev. 85_, 204 ( 1 9 5 2 ) . (34) W. Imhof et a l . , see reference ( 1 9 ) . (35) A. A. Ansel'm and V. M. Shekhter, Sov. Phys.-JETP 6 , 376 ( 1 9 5 8 ) . (36) E. L i l l e t h u n , see reference (25). (37) R. M. Frank, J . L. Gammel, and K. M. Watson, Phys. Rev. 1 0 1 . 391 ( 1 9 5 6 ) . (33) R. Serber, Phys. Rev. 2 2 , 1114 ( 1 9 4 7 ) . (39) M. Metropolis, R. B i v i n s , M. Storm, J . M. M i l l e r , G. F r i e d -lander, and A. Turkevich, Phys. Rev. 1 1 0 , 204 ( 1 9 5 3 ) . (40) B. Margolis, CERK preprint TH. 8 6 5 , ( 1 9 6 7 ) . (41) K. A. Brueckner, R. Serber, and K. M. Watson, Phys. Rev. 8 4 , 253 ( 1 9 5 1 ) . - 42 -(42) A . H. R o s e n f e l d , Phys. Rev. £ 6 , 140 ( 1 9 5 4 ) . (43) R. L. M c l l w a i n , K. H. D e a h l , M. D e r r i c k , J . G. F e t k o v i c h , and T. H. F i e l d s , P hys. Rev. 12_Z, 239 ( 1 9 6 2 ) . The c a l c u l a t i o n s d e s c r i b e d i n t h e t e x t o f t h i s t h e s i s a r e d i s c u s s e d i n two o t h e r works, w h i c h we have n ot mentioned. They a r e t h e f o l l o w i n g : (44) D. J . M c M i l l i n , P. G. Bhargava, L. Lam, and E. W. V o g t , Can. J . Phys. 4.6, 1141 ( 1 9 6 8 ) ; and (45) L. Lam, M. S c . T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia ( 1 9 6 8 ) . APPENDIX A EXPERIMENTS ON PROTON-NUCLEON: PION PRODUCTION We l i s t below t h e v a l u e s o f E p , ¥, and E^ (as d e f i n e d i n t h e t e x t ) a t w h i c h a d i f f e r e n t i a l c r o s s - s e c t i o n f o r a r e a c t i o n l i k e (p + n u c l e o n tr- +• n u c l e o n s ) has been measured. A l l v a l u e s a r e i n t h e l a b o r a t o r y frame u n l e s s o t h e r w i s e i n d i c a t e d . A u t h o r ( s ) Year R e f . R e a c t i o n En MeV ? deg. Err MeV B l o c k e t a l . 1952 (2 ) PP 3 8 1 90 0 - 9 5 March I 9 6 0 (3 ) PP 420 65 2 3 - 7 6 R o s e n f e l d 1954 (4) PP fT- 440 55 5 0 - 7 5 * tt tt 90 1 5 - 5 8 tt n 124 12-23 Pondrom 1959 (5 ) PP -»• tr+pn 450 3 0 ° 1 4 f 4 2 - 1 3 5 tt tt 20°27« 4 0 - 1 4 7 n n 1 3 ° 1 4 ' 4 5 - 1 5 2 K a z a r i n o v and Simonov 1967 (6 ) pn -* rr+»- 600 Heer e t e a l . 1966 (7 ) PP • 600 0 100-max tt tt 20 100-max pn a - 4 tt 0 100-max tt 20 100-max H e i f e r e t a l . 1961 (3) PP 654 5.6 a l l PP •* ft+pn^ tt 56 a l l PP •* TT+d^  tt 56 a l l pn n 56 a l l - 44 -Bp ¥ Err Author(s) Year Ref. Reaction MeV deg. MeV Vovchenko 1966 (9) pp -* rr+ 655 Yale report 1964 (10) pp + «+ 660 19.5 Haddock et a l . 1964 (11) pp •* "+ 725 0 @ 62% polarized protons $. centre-of-mass system $ from a C H 2 - C subtraction & from a CD 2-CH 2 subtraction # i n H 2 and D 2 both % i n D 2 Some early references are l i s t e d by Gell-Mann and Watson^ 1 / f^ and by Mandelstam^ 1-^. - 45 -APPENDIX B EXPERIMENTS ON PROTON-NUCLEUS PION PRODUCTION We l i s t below t h e v a l u e s o f E p , ¥, and E^ (as d e f i n e d i n t h e t e x t ) a t w h i c h a d i f f e r e n t i a l c r o s s - s e c t i o n f o r a r e a c t i o n l i k e (p + n u c l e u s rr± +. n u c l e u s ' ) has been measured. A l l v a l u e s a r e i n t h e l a b o r a t o r y frame u n l e s s o t h e r w i s e i n d i c a t e d . A u t h o r ( s ) Y e a r R e f . T a r g e t ( s ) MeV ? deg. tr + Err MeV C l a r k 1952 (13) Be t o Pb 240 135® + 40 n tt 35* + 40 Imhof e t a l . 1957 (19) L i t o Pb 340 135 + 36 L i t o C n 135 + 63 Sagane and Dudz i a k 1953 (20) Be t o Pb 340 90 + 12.5-33 Hamlin e t a l . 1951 (21) Be t o Pb 340 0 + 53 M e r r i t t e t a l . 1955 (22) Be t o Bb 335 0 + 34-129 Tt 0 + 52-147 B l o c k e t a l . 1952 (23) D t o Pb 331 90 + 20-120 R o s e n f e l d 1954 (24) C o n l y 440 90 + 30-125 L i l l e t h u n 1961 (25) Be t o U 450 21.5 + 83-236 21.5 - 132-200 C o n l y 21.5 + 44-max Be t o U 60 + 149 C t o U 60 - 99 C o n l y » 60 + 44-max - 46 -E p rr Err Author(s) Year Ref. Reaction MeV deg. MeV Heer et a l . 1966 (26) Be to Pb 600 0 + 100-max tt tt 21 .5 100-max Heifer et a l . 1961 (27) C only 654 56 a l l Meshkovskii et 1959 , (28) C only 660 1 9 . 5 + 100-max a l . 195S , 1957 L i to Cu tt 45 + 70-max n tt 45 - 90-max Ag, Pb 45 +• 153 Meshcheriakov 1957 , et a l . 1956 (29) C 660 24 + 50-max Haddock et a l . 1964 (30) G 725 0 + 50-max @ everything between 130 and 150 <fr everything between 30 and 50 ( 2 5 ) Summaries are also given by L i l l e t h u n v J ' and by Heer (7) et a l , u ' . A rather detailed reference l i s t appears i n the Yale r e p o r t < 1 0 ) . - 47 -APPENDIX C DERIVATION OF PATH LENGTHS IN A NUCLEUS In Figure 6 A i s the point of proton entry and B = r i s the point of pion production. The o r i g i n 0 i s at the nuclear center. Hence OA = R m a x, OB = r , and AB = s p and we have [C.l] S p = ~ r.cosO + R m a x.cosoc , where oc i s the angle AOD. From the t r i a n g l e AOD, since AD = r.sinO, we get (C.2] R m a x = ( r . s i n O ) 2 + (R m a x.cosoo ) 2 . Solving [ C 2 ] f o r Rmax , c o s o 6 , (positive root) and putting the re s u l t into [C.l] then gives us [C.3] s p s r-cosO + (Rmax " r 2 s i n 2 0 ) ^ . I f we denote the angle COE by jQ, then CE i s given by [C . 4 ] R m a x ' s i V r» sinO* eos<j> + s ^ s i n f . Since OF = r»cosO + Sn-'cosf t r i a n g l e FOE establishes 2 2 2 [C . 5 ] (Rmax*003/9) = (r»sinO* sin^)) + (r«cosO Sn-• cosy). Squaring [C . 4 ] and adding the r e s u l t to [C . 5 ] gives us, since 2 2 sin/9 + cos /S r 1, a quadratic equation i n Srr having the roots [ C 6 ] s„ = - r-X ± ( R L x - - r 2 ( l - X 2 ) ) 1 , where X = cosO*cosf + sinO *sinI)F •cos4>. - 48 -Geometry o f p a t h l e n g t h s i n a n u c l e u s . - 49 -It i s e a s i l y shown that only the root i n [C.6J using the + sign always gives s„ ^ 0 . With t h i s i n mind we have the desired r e s u l t s ([2] and [3] of the text) i n [C.3] and [C . 6 ] . - 50 -APPENDIX B TRANSFORMATION OF ENERGY FROM LABORATORY FRAME TO REST FRAME OF A STRUCK NUCLEON I f we l e t P = (p_, iEp) and K = (k, i E k ) represent the four-momenta of p a r t i c l e s having rest masses of mp and mk respec-t i v e l y , then we can write [D.l] (P + K ) 2 = (£ + k ) 2 - (B p + E k ) 2 = (j2 + k ) 2 - (T p + T k + (mp + m k ) ) 2 , where T p and T k are k i n e t i c energies. Assume that the quantities on the right-hand side of [D.l] are measured in-the laboratory frame. Then i f primed quantities are measured i n the rest frame of the p a r t i c l e described by K we can also write [D.2] (P + K ) 2 = ( p _ ' + k ' ) 2 - ( E p + E k ) 2 = £** ~ <Tp + (»p • mk>>2> since by d e f i n i t i o n k' = T k = 0. I f we equate [D.l] and [D.2] and expand terms using 2 2 standard r e l a t i o n s l i k e p_ = T p + 2m pT p we get d i r e c t l y that [D.3] T p = T p + ( T p T k - £.k)/m k + (m p/m k)T p, which has [6] and [7 ] of the text as special cases. - 51 -APPENDIX B DERIVATION OF AN EXPRESSION USED TO COMPUTE PION ABSORPTION COEFFICIENTS To explain the absorption of pions i n nuclear matter Brueckner et a l . ^ D assume that the absorption per nucleon i n any nucleus i s proportional to the known capture cross-section of pions by deuterons. Letting P be the proportionality constant we can then write the nuclear cross-section f o r absorption as [ E . l ] (J = n-P-ad, where n i s the number of absorbing nucleons i n the nucleus and 0^ i s the deuteron pion-capture cross-section. I f we assume (with Brueckner et al . ) that the basic pion absorption reactions are just the inverses of the basic pion production reactions (some of which we l i s t i n the text, section 1.2) then f o r ,positive pions we have 0"^  i CT(TT+d * pp) and n f'N, i . e . , we have absorption by neutrons mostly. For negative pions we have 0"^  ^  0"(rr-d » nn) and n ± Z, absorption by protons'mostly. i Using the model described above we can write the pion absorption c o e f f i c i e n t , defined as absorption per unit volume, as (E.2) n a = n-?.(7d . (4/3)"R 3 To use [E.2] i n computing values of n a, Frank et a l . ^ ? ) take f o r (J^ (cf. t h e i r footnote (13)) the values - 52 -CE.3] 0"d = (4.45/q)-(0.14 + q 2 ) , where q i s t h e maximum ce n t e r - o f - m a s s momentum a v a i l a b l e t o the p i o n , i n u n i t s o f u«c. T h i s t h e y o b t a i n f rom the s e m i - e m p i r i c a l f o r m u l a o f Gel l - M a n n and W a t s o n ^ . I t can a l s o be o b t a i n e d f r om R o s e n f e l d ' s ^ ^  e x p r e s s i o n f o r 0~10 = 0(pp + n-+d) and the p r i n c i p l e o f d e t a i l e d b a l a n c e i n a low energy a p p r o x i m a t i o n . F r ank e t a l . assume, i n ac c o r d a n c e w i t h B r u e c k n e r e t a l . , t h a t f = 4 and t h e y have used ( t h e i r e q u a t i o n (13)) f o r t h e n u c l e a r r a d i u s R = (fi/(u«c))'A'A1/-^. P u t t i n g t h e above e x p r e s s i o n s f o r 0"ci and R i n t o (E.2) g i v e s us an e x p r e s s i o n f o r t h e p i o n a b s o r p -t i o n c o e f f i c i e n t , v i z . [E.4] n a = 1 = u - c . l 0.107.(0.14 + q 2 ) . A A -h A q w h i c h i s t h e F r a n k e t a l . e x p r e s s i o n ( t h e i r e q u a t i o n (9)) w i t h t h e 1/x^ f a c t o r made e x p l i c i t . Frank e t a l . i m p l i c i t l y s e t A z 1 when t h e y compute v a l u e s o f n„. = n a + n s , where n s i s t h e c o e f f i c i e n t f o r i n e l a s t i c p i o n s c a t t e r i n g . ( a l s o depending on a f a c t o r o f 1/X5). I n d e r i v i n g [E.4l we c o n s i d e r e d o n l y p o s i t i v e p i o n a b s o r p t i o n . B r u e c k n e r e t a l . quote r e f e r e n c e s w h i c h e x p l a i n why we may ex p e c t t h e a b s o r p t i o n o f n e g a t i v e p i o n s t o be s i m i l a r . - 53 -APPENDIX F OUTLINE OF COMPUTER PROGRAM PIPROD The following i s a sketch of a FORTRAN IV program written to numerically integrate equation [11] of the text. The symbols used below are defined i n the text. PIPROD Read constants E f f, 1|J, np, r 0 , a, etc. Read Frank data on mfp vs. T„. and V r vs. T„. f o r a range of T„.. 3 Correct Frank data for units (mfp) and f o r ( r 0 / r r (mfp, V r ) . Calculate n f f = 1/mfp vs. Tn f o r range of T„.. DO fo r each A available: Read A and Z. Calculate R, R m a x, R c, and / D Q . Calculate number of r-steps and exact step Ar. Set r - 0. DO fo r each r up to R m a x: Calculate number of 9-steps and exact A9. Set 9 = 0 . DO fo r each 9 up to it; Calculate number of $>-steps and exact A<t>. Set <j) = 0. DO fo r each <f> up to rr : Calculate s^. Calculate number of s-steps and exact As. Set s = 0. - 5 4 -1 2 3 4 DG f o r each s up to s^ .: Calculate r , <o, and V c. Calculate E„. = T,, + V r (T!„)*/o •  •• V c vs. T,,. f o r range of T^. Interpolate f o r T n at E„. of i n t e r e s t . Interpolate f o r n^ . at t h i s T n. Correct t h i s n,,. by /O. Store corrected n^ . as, say, A vs. s. L Change s by As. Integrate /A(s)ds. Check error and write i f c r i t i c a l . Calculate exp(/A(s)ds) and store as, say, B vs. <J>. L- Change cj> by A<1>. Integrate /B(<J>) d<b. Check error and write i f c r i t i c a l . Calculate Sp. Calculate 2 s i n 0 «exp(-sp »np) • /B{ty)dty.and Store as, say, C vs. 0 . L Change G by AG. Integrate /C(Q)d6. Check error and write i f c r i t i c a l . Calculate o, T. Calculate r2/>./C(Q)dG and r2>o./C(G)dG.f and Store as, say, D and E vs. r . L Change r by Ar. Integrate I i = /D(r)dr and I 2 = / E ( r ) d r . Galaulate I 2 A 1 . Store each of 1 ^ , I 2 , and I 2 / I i vs. A. - 5 5 -1 C a l c u l a t e n o r m a l i z a t i o n c o n s t a n t s i f A a p p r o p r i a t e . - Change A. N o r m a l i z e 1^, I 2 . W r i t e 1-^ , I 2 , and n o r m a l i z e d 1^, I 2 v s . A. W r i t e E,,., U, n o r m a l i z a t i o n c o n s t a n t s , and o t h e r d a t a o f i n t e r e s t . END S u b r o u t i n e s : SIMPLE - Simpson's r u l e i n t e g r a t i o n . TINT - t h i r d o r d e r p o l y n o m i a l i n t e r p o l a t i o n . Comments: 1) F o r t h e f i r s t i n t e r p o l a t i o n ( f o r T f f i n t h e s - l o o p ) check t h a t T„. v s . E^. i s s i n g l e - v a l u e d . I t i s f o r t h e Frank e t a l . d a t a . 2) The <J>-integration i s o n l y between 0 and tt because o f symmetry i n Q> (hence th e f a c t o r 2 i n C ( 0 ) ) . 3) E r r o r i n i n t e g r a l s i s e s t i m a t e d by r e - i n t e g r a t i o n u s i n g a d o u b l e d s t e p - l e n g t h . 

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