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UBC Theses and Dissertations

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

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cC,  P o l a r i z e d Proton Induced E x c l u s i v e  Incident  P i o n P r o d u c t i o n on  1 0  B  and B e f o r 9  E n e r g i e s from 200 to 260 MeV  by William B.Sc,  Anthony Z i e g l e r  U n i v e r s i t y of Regina, 1980  M.Sc., U n i v e r s i t y  of B r i t i s h Columbia, 1983  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of P h y s i c s  We accept t h i s t h e s i s as conforming to the r e q u i r e d  standard  A  The  U n i v e r s i t y of B r i t i s h Columbia August 1985  © William  Anthony Z i e g l e r , 1985  In presenting this thesis in partial fulfilment  of the  requirements for an advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the- head of my department  or  by  his or  her  representatives.  It  is  understood that  copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  Physics  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  Q^c^.  l3./?^£  ii  P o l a r i z e d Proton Induced E x c l u s i v e Plcm P r o d u c t i o n on f o r Incident Energies  from 200  to 260  i U  B  and  ^Be  Mev  Abstract  Two  s e t s of experiments f o r s t u d y i n g p o l a r i z e d proton  p i o n p r o d u c t i o n were performed at TRIUMF. the d i f f e r e n t i a l c r o s s - s e c t i o n s and 1 0 B  9  <P'* > +  l l B  Be(p,u  g . s . , . " MeV 2  o o g.s.,°.  j nU  9 B e (  w ,r f ° tiev  r  P^  + ) 1  induced  The"angular d i s t r i b u t i o n s of  a n a l y z i n g powers are presented °  B e  exclusive  g.s.,3.37  M  eV  a  n  both  for:  d  i n c i d e n t proton energies between 200  and  260  MeV. The  results  preference  i n d i c a t e a number of new  trends  i n pion p r o d u c t i o n .  f o r p o p u l a t i n g h i g h - s p i n f i n a l s t a t e s (at l e a s t  seems to be p r i m a r i l y a s p i n s t a t i s t i c a l weighting the shape of the angular with  i n these  effect.  The  diffractive  a s i n g l e - p a r t i c l e nature forward  angle peaking  final  similar  In c o n t r a s t t r a n s i t i o n s to the other s t a t e s , those  cannot be d e s c r i b e d as having The  dependence of  T r a n s i t i o n s to a  s t a t e that can be d e s c r i b e d as s i n g l e - p a r t i c l e have a s t r o n g and  dependence.  reactions)  d i s t r i b u t i o n s on i n c i d e n t proton energy i s l i n k e d  the quantum numbers of the f i n a l s t a t e nucleus.  energy dependence.  The  that  show a weak energy  which i s common to most  p i o n p r o d u c t i o n d i f f e r e n t i a l c r o s s - s e c t i o n d i s t r i b u t i o n s was  found  to have  e x p o n e n t i a l slopes c o n s i s t e n t with p r o d u c t i o n r a d i i of the s i n g l e nucleon r a t h e r than that of the Finally,  nucleus.  the r e s u l t s are compared to p r e d i c t i o n s of c u r r e n t  models of p i o n p r o d u c t i o n . suggestions  size  theoretical  F a i l u r e s of these models are d i s c u s s e d  f o r improvements o f f e r e d .  and  iii  Table of Contents  Abstract L i s t of Tables L i s t of F i g u r e s I. Introduction II. Experimental Setup 2.1 TRIUMF and Beamline IB 2.2 Beam M o n i t o r i n g 2.3 The " R e s o l u t i o n " Spectrograph System I I I . Data A n a l y s i s 3.1 Background D i s c r i m i n a t i o n 3.2 F o c a l Plane and D i s p e r s i o n R e l a t i o n 3.3 Lineshape 3.4 E f f e c t i v e S o l i d Angle IV. Results V. Trends i n the Data 5.1 Choice of V a r i a b l e s 5.2 S t a t i s t i c a l Weighting 5.3 Incident Energy Dependence 5.4 D i f f r a c t i o n Peak S t r u c t u r e 5.5 n~ P r o d u c t i o n VI. Status of the T h e o r e t i c a l Models 6.1 R e l a t i v i s t i c One Nucleon Model 6.2 Two Nucleon Model 6.3 Comparison of Models 6.4 Comparisons of Models with Experiment VII. Conclusion References  •  11 iv v 1 10 10 12 20 25 25 31 35 37 40 57 ....57 58 63 86 89 90 90 92 95 97 104 108  iv  L i s t of Tables  I.  A r e a l Thickness of Targets  II.  D i f f e r e n t i a l C r o s s - s e c t i o n s and Analyzing Powers  for *G.* ) * . l0  III.  +  ll  „  gmB t2ml2  e  V  42  D i f f e r e n t i a l C r o s s - s e c t i o n s and Analyzing Powers  for B e ( | , 9  IV.  24  l t  V°Be  Differential for B e ( S , O 9  g < s > > 3 > 3 7  M  e  V  43  C r o s s - s e c t i o n s and Analyzing Powers 1 0  C  g > 3 t >  3.  3 4  M f i V  44  V  L i s t of Figures  1.  D i f f e r e n t i a l Cross-sections  f o r a v a r i e t y of (p,Tt) r e a c t i o n s  3  2.  (p,u~) r e a c t i o n Spectra  5  3.  One nucleon model  6  4.  Two nucleon model  7  5.  Beamline IB at TRIUMF  6.  Geometry and E l e c t r o n i c s of the f i r s t  7.  Geometry  8.  Ratio of — T — f o r the second Polarlmeter H  19  The " R e s o l u t i o n " Spectrograph  21  11 Polarlmeter  14  and E l e c t r o n i c s of the second Polarlmeter  15  A  9.  10. E l e c t r o n i c s f o r the " R e s o l u t i o n " system  22  11. Energy l o s s D i s t r i b u t i o n  26  12. T i m e - o f - f l i g h t D i s t r i b u t i o n  28  13. Standard D e v i a t i o n D i s t r i b u t i o n s  29  14. Y target E x t r a p o l a t i o n D i s t r i b u t i o n s  30  15.  1 0  B ( p , u ) B Energy Spectrum +  33  u  16. B e ( p , n ) B e Energy Spectrum 9  +  34  1 0  17. D i f f e r e n t i a l C r o s s - s e c t i o n s  for  18. D i f f e r e n t i a l C r o s s - s e c t i o n s  for  19. D i f f e r e n t i a l C r o s s - s e c t i o n s  f o r Be(p , i t )  20. D i f f e r e n t i a l C r o s s - s e c t i o n s  for B e ( p , u )  21. D i f f e r e n t i a l C r o s s - s e c t i o n s  for Be(p,u~) C„ _  22. D i f f e r e n t i a l  f or  Cross-sections  1 0  1  9  0  B(p.u" ") B„ _ 1  B ( p  1  B +  9  ^  2  1  2  M  v  e  4  Be„ „  1 0  1 0  Be  3 < 3 7  M  e  48  V  49  1 0  Be(p . T I ) -  10  C  3 >3 4  M  6  47  g • s.  +  9  9  45  11  g  V  50  vi 23. Analyzing powers f o r  B(p,Tt )  1 0  24. Analyzing powers f o r 1  +  L 1  B(p,ir )  0  +  1 0  26. Analyzing powers f o r B e ( p , n )  1 0  9  +  9  +  9  9  29. M a t r i x Element f o r B ( p , u ) 1 0  +  1 1  Bj  5  MeV  1 2  Be„ _ g. s. Beg 3 7  27. Analyzing powers f o r Be(p , u ~ ) 28. Analyzing powers f o r B e ( p , u ~ )  51  a  1 1  25. Analyzing powers f o r B e ( p , i t )  B„ g. s •  C  1 0  53  e  54  V  55  g. s.  Cj ^  10  M  3 4  M  g  56  V  B  59  30. Average M a t r i x Element per f i n a l s t a t e f o r B ( p , u ) 1 0  31. M a t r i x Element f o r Be(p , u ) 9  +  1 0  2  +  1 1  B  60  Be  61  32. Average M a t r i x Element per f i n a l s t a t e f o r B e ( p , u ) B e  62  33. Matrix Element f o r C ( p , u )  64  9  1 2  +  1 3  +  1 0  C  34. Average M a t r i x Element per f i n a l s t a t e f o r C ( p , i t ) 1 2  35. Forward angle  |M|  for  2  9  Be(p,7i ) +  Be  1 0  +  1 3  C  65 66  g # g >  36. A n a l y z i n g powers as a f u n c t i o n of t f o r B e ( p , u ) B e „ g * s. 9  37. Exponential slopes of the B e ( p , n + ) 9  38. Forward angle  |M|  forBe(p,u )  2  9  +  10  Be_  Be3  1 0  +  > 3 7  _ D i f f e r e n t i a l C r o s s - s e c t i o n s ..68 M  e  70  V  39. Analyzing powers as a f u n c t i o n of t f o r B e ( p , T i ) B e 40. E x p o n e n t i a l slopes of the B e ( p , u ) Be 37*j V 9  9  +  +  3 7  M  +  V  Q,  1 3  > 5  M  g  73  V  1 2  +  >  +  71  72  42. A n a l y z i n g powers as a f u n c t i o n of t f o r C ( p i t ) C 43. E x p o n e n t i a l slopes of the C ( p , n ) C g _ Differential sections 1 2  g  '..  | M | for C ( p , n ) I 2  3  e  D i f f e r e n t i a l Cross-sections 2  1 0  1 0  3  41. Forward angle  67  10  1 3  5  M  e  V  74  1 3  9 5  M  g  V  Cross75  vii  4 4 . Forward angle | M | for 2  1 0  BCp.u"*") B  76  11  CT  c  g•5• 4 5 . Analyzing powers as a function of t for B(p,u ) B  _ 77 g•s• 4 6 . Exponential slopes of the B ( p , u ) B „ Differential Cross-sections . . 7 8 g• s• 10  10  +  +  11  tt  n  0  B(P,TI )  %  47.  Forward angle | M | for  48.  Analyzing powers as a function of t for  2  1 0  +  11  >  1  2  M  G  80  V  B(P,TC )  1 0  +  1 1  B  2  1  2  M  E  V  81  4 9 . Exponential slopes of the B(p,u ) B2 Differential Cross-sections ! 82 5 0 . Forward angle | M ] for C ( p , 7 i ) Ca 83 g•s• 5 1 . Analyzing powers as a function of t for C(p,u ) C„ „ 84 g. s. 5 2 . Exponential slopes of the C(p,u ) C„ Differential Cross-sections . . 8 5 g•a « 5 3 . Relativistic 0 N M comparison with C ( p , i i ) C Differential Cross-sections 98 5 4 . Relativistic 0 N M comparison with C(p.u" ) C a Analyzing Powers 99 g* s• 5 5 . Relativistic O N M comparison with B(p,it ) B Differential Cross-sections 101 1 0  +  11  1 2  2  +  12  M e V  13  E  12  12  +  +  13  13  1 2  +  1 3  g > s  12  1 0  5 6 . Relativistic O N M comparison with 57.  TNM  comparison with  1 2  C(p,Ti ) +  1 3  1 0  Cg  > 5  1-  13  +  11  B(p,it ) B„ Analyzing Powers g•s• Differential Cross-sections +  M g V  11  102 ....103  1  I.  Introduction  H i s t o r i c a l l y , perhaps hastened by the f i r s t i t was expected that proton-induced  simple model c a l c u l a t i o n s ,  p i o n production  r e a c t i o n s , A(p,u )A+1 +  would c o n s t i t u t e a u s e f u l s p e c t r o s c o p i c t o o l f o r the i n v e s t i g a t i o n of high momentum components of the n u c l e a r wave f u n c t i o n s .  In i t s simplest  terms,  the ( p , i t ) c r o s s - s e c t i o n i s d i r e c t l y p r o p o r t i o n a l to the F o u r i e r transform of +  the bound s t a t e wave f u n c t i o n of the captured  neutron.  For example i n the  plane wave Born approximation (PWBA) u s i n g a simple  s t a t i c model f o r the pion  production operator  s p i n and the g r a d i e n t of  (a •  i n v o l v i n g the nucleon  the pion f i e l d ) the ( p , i t ) c r o s s - s e c t i o n from a spin-zero t a r g e t nucleus i n +  the centre-of-mass (cm) system i s given by 1  da dQ  E p E.E. p n A A+l  -  3  8u p 2  r  p  (E.+E ) A p  f 2  2  +  ( 2 J  m  1 }  f  2  11  ^  ( q ) |  2  2  n  where E and p are the t o t a l energy and momentum of the various f /4u 2  particles,  = 0.08 i s the uN c o u p l i n g constant, m^ i s the pion mass,  of the f i n a l nucleus,  i s the s p i n  and <J>(<l) Is the F o u r i e r transform of the neutron bound n  •+  ->'  s t a t e wave f u n c t i o n with respect to the momentum t r a n s f e r , q = p  - p .  As a  r e s u l t , r e a c t i o n s of t h i s type have a t t r a c t e d c o n s i d e r a b l e a t t e n t i o n over the past s e v e r a l years. The reviews of H o i s t a d ,  1  Measday and M i l l e r ,  2  and F e a r i n g  give e x c e l l e n t summaries. By the l a t e s e v e n t i e s , i t had become c l e a r that t h i s d e s c r i p t i o n of the (p,rc) process  i s too simple  and Eqn. (1-1) most c e r t a i n l y i s wrong.  At the  very l e a s t a c o r r e c t d e s c r i p t i o n of the process would i n v o l v e the c o r r e c t  3  2  use and understanding  of four b a s i c i n g r e d i e n t s :  proton d i s t o r t i o n , pion d i s t o r t i o n , and b a s i c i n g r e d i e n t s has  the pion emission  nuclear wave f u n c t i o n s .  an inherent u n c e r t a i n t y as to how  operator,  Each of  important  or  how  a c c u r a t e l y i t must be understood.  A number of questions must be asked.  i s the b a s i c p r o d u c t i o n operator?  Does i t i n v o l v e more than one  important manifest  are r e l a t i v i s t i c and itself?  r e c o i l corrections?  How  protons, can one  c a l c u l a t i o n s required?  r e s u l t s to d e t a i l s of the n u c l e a r wave f u n c t i o n s ? the pion p r o d u c t i o n process has to understanding  shifted  For both  pions  How How  effective  sensitive  are  are  Thus the major i n t e r e s t i n  from u s i n g I t as a s p e c t r o s c o p i c t o o l ,  the fundamental physics of the r e a c t i o n mechanism i t s e l f .  t h i s energy r e g i o n the pion-nucleon microscope strong i n t e r a c t i o n .  In  vertex i s the dominant term i n the  Since the pion production r e a c t i o n represents  of the simplest t e s t i n g grounds f o r the understanding  of t h i s v e r t e x ,  we  to endeavour to understand i t .  In order to a p p r e c i a t e the true r i c h n e s s , and the pion p r o d u c t i o n process,  one  Fearing's review  3  i s a good i l l u s t r a t i o n .  dependent on the nature  of both  with s e v e r a l minima.  The  The  F i g u r e 1 taken  angular  the t a r g e t nucleus  r e s i d u a l nucleus which are i n v o l v e d . others are more or l e s s f l a t ,  t h e r e f o r e complexity,  need only look at a s e l e c t i o n of the  d i s t r i b u t i o n s of the d i f f e r e n t i a l c r o s s - s e c t i o n s .  aware we  How  d e s c r i b e d i s t o r t i o n e f f e c t s v i a average o p t i c a l  d i s t o r t i o n e f f e c t s i n s h a r i n g the momentum t r a n s f e r ?  are compelled  nucleon?  Are u s u a l p i o n o p t i c a l p o t e n t i a l s adequate to reproduce the  p o t e n t i a l s or are more e x p l i c i t  one  What  does the A-resonance  pion wave f u n c t i o n i n s i d e the nucleus where i t i s required? and  these  and  angular  from  d i s t r i b u t i o n s are the s t a t e s of  very  the  Some f a l l r a p i d l y with i n c r e a s i n g angle,  and some have a very complex angular  more the pion production process  are of i t s complexity.  of  dependence  i s s t u d i e d the more  3  30  60  90  e  v  F i g u r e 1.  120  150  160  (deg)  Schematic r e p r e s e n t a t i o n of the angular d i s t r i b u t i o n data f o r a v a r i e t y of (p,n) r e a c t i o n s . The curves correspond t o : (a)  1 2  C(p,* ) +  1 3  C  (c) B e ( p , 7 i ) B e 9  (e)  1 6  (g)  1 3  +  0(p,n ) +  C(p,O  g > g >  ,  l 0  1 7  1 4  0 0  3  3  > 3 7  > 8 5  200 MeV , 185 MeV  , 185 MeV , 185 MeV  (b) " • C a ( p , u ) ' * C a 0  +  1  (d) B e ( p , n ) l ° B e 9  +  (f) B e ( p . O 9  1 0  (h) B e ( p , O 9  C  1 0  g > g >  3 > 3 w  C  g>s#  ,  ,  200 MeV  185 MeV  , 185 MeV  g > s >  ,  185 MeV.  "double-charge exchange" ( p , T i ) r e a c t i o n has  The  share of s u r p r i s e s .  Initially,  r e l a t i v e l y i s o t r o p i c and  also contributed i t s  the d i f f e r e n t i a l c r o s s - s e c t i o n s appeared to  be  s m a l l compared to that of the ( p , i i ) r e a c t i o n s . +  This suppression  of u  (1-1)  ) r e a c t i o n s cannot proceed unless a more complex process i s  s i n c e (p,n  production  i s c o n s i s t e n t with the PWBA model of  included.  Measurements at IUCF * (Indiana U n i v e r s i t y C y c l o t r o n  discovered  that (p,n  1  ) r e a c t i o n s l e a d i n g to c e r t a i n h i g h - s p i n  t w o - p a r t i c l e one-hole f i n a l s t a t e s had  Eqn.  Facility) stretched  c r o s s - s e c t i o n s of the same order  as  (p,u ) cross-sections.  An i l l u s t r a t i o n of the e x c i t a t i o n of such s t a t e s i s  shown i n F i g u r e  of the strongest  +  ll  *C(p,it )  1 5  0  7  3  2. M  e  One  nb/sr over the angular d i s t r i b u t i o n . ~  50 to 1 nb/sr.  of these s t a t e s i s  with 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 range of - 800  V  the  "* Ca(p,ii ) ^ T i ^ ^ MeV> 8  This s e l e c t e d sample of data o b v i o u s l y  h a S  a  to r a n  8  200 e  shows that the  °f pion  process has many f a c e t s to i t . The  t h e o r e t i c a l approaches to t h i s r e a c t i o n can be d i v i d e d i n t o  c l a s s e s ; one  nucleon mechanisms (0NM)  d i s t i n c t i o n r e f e r s to the use  and  two  two  nucleon mechanisms (TNM).  of the production  operator  (H.  This  ) r a t h e r than to  int  the f u l l mechanism s i n c e i n f a c t almost a l l models are r e a l l y  multi-nucleon.  The  once  ONM  (illustrated  i n Figure  i n t e r a c t i o n s with other  operator  to i n v o l v e two  i s used three  operator  and  nucleons have been i n c l u d e d i n some average way  optical potential distortions. constructed  3) uses the production  The  TNM  (illustrated  nucleons e x p l i c i t l y .  i n Figure  That i s , the  via  4) i s production  times, making i t p o s s i b l e to i n v o l v e the A-resonant  interactions e x p l i c i t l y .  Again other nucleons, i f i n c l u d e d at a l l ,  i n c l u d e d only as an average e f f e c t .  are  Both c l a s s e s of models have advantages  5  Figure 2.  Spectra for (p,n~) reaction on several targets at 9 i ^ • 30° ( 2 8 ° for the C target), showing selective excitation of one or a few low-lying states. a  l l 4  igure 3.  One nucleon model of plon production.  7  ~ -D  \  \  /  Figure A.  O  Two nucleon model of pion production. In the two nucleon case there are four'possible types of diagrams: ( a ) p r o j e c t i l e post-emission (b) target post-emission (c) p r o j e c t i l e pre-emission (d) target pre-emission. The spectator nucleons In the nucleus A have not been shown explicitly.  8  and disadvantages  though n e i t h e r of them can c l a i m any t r u e success.  The most  advanced model i n each c l a s s w i l l be d i s c u s s e d i n Chapter VI along with some comparisons to the experimental  data.  E x p e r i m e n t a l l y , much e f f o r t has been devoted  to s e a r c h i n g f o r s y s t e m a t i c  trends i n the data i n order to uncover c l u e s f o r understanding r e a c t i o n mechanism.  the b a s i c  Now that much data e x i s t s , a number of g e n e r a l f e a t u r e s  can be e x t r a c t e d but as yet no complete understanding of the r e a c t i o n has been developed.  I n the case of the s i m p l e s t of n u c l e i (hydrogen),  some success.  One technique, i n t r o d u c e d by the H e l s i n k i group,  Green and N i s k a n e n ) i n v o l v e d the development of a 28  technique.  Such c o u p l i n g of the NN and NA channels  exchange i m p l i c i t l y . predictions  2 9  there has been  F o r the f i r s t  (principally  coupled-channels i n c l u d e d both n and p  time, theory was able to provide  which compared w e l l with experimental measurement (not only of  the d i f f e r e n t i a l and t o t a l c r o s s - s e c t i o n s , but a l s o of a v a r i e t y of polarization observables). the many nucleon problem.  U n f o r t u n a t e l y t h i s success does not continue t o Thus the purpose of t h i s t h e s i s i s to present i n a  systematic way the data obtained from two major sets of experiments point out new trends suggested  by t h i s data.  and to  These new trends not only g i v e  d i r e c t i o n to f u r t h e r experimental work, but p o i n t out as w e l l new t h e o r e t i c a l approaches that should be i n v e s t i g a t e d . The  t h e s i s begins with a b r i e f d e s c r i p t i o n of the experimental setup and  data a n a l y s i s (which has been d e s c r i b e d e l s e w h e r e ) . 5  Next the angular  d i s t r i b u t i o n s of both the d i f f e r e n t i a l c r o s s - s e c t i o n s and a n a l y z i n g powers are presented  f o r the f o l l o w i n g r e a c t i o n s :  9  g.s., 2.12 MeV  9  Be(p,u ) Be +  1 0  g-s.,  9Be(p,n")l°C  3.37 MeV  3.34 MeV  f o r i n c i d e n t proton energies ranging from 200 to 260 MeV. Chapter V continues with suggested energy  trends i n the n u c l e a r data i n t h i s  range and p o i n t s out new general d i r e c t i o n s f o r both experimental and  t h e o r e t i c a l work. approaches,  Chapter  VI looks at the s t a t u s of the t h e o r e t i c a l  and Chapter VII concludes  of some s p e c i f i c  recommendations.  the t h e s i s with a summary and d i s c u s s i o n  10  II.  2.1  Experimental Setup  TRIUMF and Beamllne IB  The TRIUMF  6  f a c i l i t y has both polarized and unpolarlzed Ion sources which  the cyclotron can accelerate to an energy range from 200 to 520 MeV. The maximum beam Intensity available depends on the energy as well as the type of Ion source.  For example, at maximum energy a current of 140 uA (unpolarlzed)  or about 500 nA (polarized) can be extracted. A special feature of the TRIUMF design i s the acceleration of H  ions.  Extraction of a proton beam Is thus  accomplished by Intercepting the negative ions with a thin f o i l , which strips the two electrons from the H ion leaving protons which then curve out of the cyclotron f i e l d .  The proton beam Is periodic and consists of pulses of  roughly 2 nsec duration occurring every 43 nsec.  The separation of the pulses  corresponds to the period of the accelerating radio frequency power (RF) which Is the f i f t h harmonic of the cyclotron resonance frequency. The "Resolution" spectrograph system  5  used In these experiments was  situated on beamllne IB, i l l u s t r a t e d In Figure 5. By the use of dipole and quadrupole magnetic elements the proton beam was transported through the beamllne to the target location (1BT1). At each beam energy special care was taken i n tuning the elements of the beam line and the extraction parameters of the cyclotron.  A low background i n  the experimental area had to be maintained i n order for multiwlre proportional chambers at the exit of the spectrograph to operate s a t i s f a c t o r i l y , as there was only limited l o c a l shielding around the focal plane array. With a poor beam tune, there was sufficient beam loss along the line that the background  Figure  5.  B e a m l i n e IB a t TRIUMF.  12  in the experimental area Increased to the point where the chambers could not handle the singles rates (even without a target at 1BT1).  Rather than  constructing the massive shielding that would be needed to protect the chambers, high quality beam tunes were developed and special procedures were followed to handle them.  F i r s t the cyclotron was maintained at a good tune  such that the extracted beam was monoenergetic with very l i t t l e halo.  Next  using monitors for indicating the position and profile of the beam at various points along the beamline, special care was taken to steer the beam down the center of the beamline.  The readings of the s p i l l monitors along the outside  of  the beamline were kept at a minimum.  In addition, the p r o f i l e and position  of  the beam was monitored regularly either by remotely viewing a s c i n t i l l a t i n g  target with a video monitor or measuring the profile with a wire chamber that could be inserted at the target location.  A small beam spot (~ 5x2 mm) was  maintained at 1BT1. Another important reason for maintaining the good tune was the effect of beam misalignment on the polarization measurements.  In order to obtain a  correct measure of the beam polarization, i t was necessary to keep the beam centered i n position and p a r a l l e l while passing through the polarimeter. Such instrumental asymmetries  recorded for unpolarized beam were monitored at five  minute intervals to assure that no changes occurred i n the beam tune.  2.2  Beam Monitoring  For  monitoring both the intensity and polarization of the incident proton  beam a four-arm polarimeter was situated immediately downstream from the beam dipole 1BVB2 (see Figure 5). the  Two different polarimeters were employed during  course of these experiments.  The f i r s t  7  was used for a l l the Be work and 9  13  a small amount of the  1 U  B.  A schematic  of the geometry of the  together with I t s e l e c t r o n i c s i s shown i n F i g u r e 6. used f o r most of the  1 0  B  work Is shown i n F i g u r e  Both p o l a r i m e t e r s were based on monitoring target protons target.  The  second  pp e l a s t i c 2  t h i n CH  the s c a t t e r e d protons  background ( t y p i c a l l y  count corresponded  5%)  2  the  (polyethylene) by  Nevertheless,  from q u a s i - e l a s t i c s c a t t e r i n g of  to be taken i n t o account.  5  i n a two-arm system  s i t u a t e d at the angles a p p r o p r i a t e f o r f r e e pp s c a t t e r i n g .  w i t h i n the carbon had  s c a t t e r i n g with  from the carbon were d i s c r i m i n a t e d a g a i n s t  c o i n c i d e n c e d e t e c t i o n of both  significant  polarlmeter  7.  provided by the hydrogen i n a 5 mg/cm  Background events  polarlmeter  For the f i r s t  a  protons  polarlmeter a  to the c o i n c i d e n t d e t e c t i o n of a proton s c a t t e r e d at 26° to  the r i g h t ( l e f t ) with respect to the beam d i r e c t i o n together with a backward s c a t t e r e d proton detected at 60° to the l e f t angles  (right).  f o r the second p o l a r l m e t e r were 17° and  f a c i l i t a t e a more r i g i d pp-elastic left-right  support  71.3°  The  corresponding  ( r e s p e c t i v e l y ) to  s t r u c t u r e as w e l l as to Increase  asymmetry by ~  both  the  15%.  I f a p o l a r i z e d proton beam i s i n c i d e n t upon an u n p o l a r l z e d t a r g e t , the differential  c r o s s - s e c t i o n da/dQ can be w r i t t e n i n terms of u n p o l a r l z e d  p o l a r i z e d components, that Is;  da  da  ->  ir i r =  where:  da —— au  + p  •*•  da  * *r n  i s the u n p o l a r l z e d d i f f e r e n t i a l c r o s s - s e c t i o n .  da i s the p o l a r i z e d d i f f e r e n t i a l c r o s s - s e c t i o n .  and  Ik Recoil P (right) \ '  F i g u r e 6.  Fwd. P(left)  A schematic of the geometry and e l e c t r o n i c s of the f i r s t p o l a r i m e t e r (used f o r the Be work). 9  15  Figure 7.  A schematic of the geometry and e l e c t r o n i c s of the second p o l a r i m e t e r (used f o r the B work). 1 0  16  -»•  P  and  n  i s t h e i n c i d e n t p r o t o n beam p o l a r i z a t i o n , i s a unit vector  normal t o the s c a t t e r i n g plane i n the  direction  f  * 1t ( t h e M a d i s o n  do Therefore  '  ir dr  da„ R  ^ —  da  1 6  da  =  3  where:  do  Convention ).  + p  o  4r  _  da  (2  p  i s the d i f f e r e n t i a l cross-section of the l e f t  °R is dQ  -  2)  , _ „.  scattered  protons.  d  and  P  the d i f f e r e n t i a l  c r o s s - s e c t i o n of the r i g h t scattered  protons.  i s t h e m a g n i t u d e o f t h e i n c i d e n t p r o t o n beam p o l a r i z a t i o n .  A d d i n g Eqns. ( 2 - 2 ) and ( 2 - 3 ) l e a v e s  da  T h u s , t h e sum o f a l l l e f t  scattered  .  da.  da  D  ( L ) and r i g h t s c a t t e r e d  i n d e p e n d e n t o f t h e p o l a r i z a t i o n o f t h e beam, and t h e r e f o r e  (R) c o u n t s i s provides  a measure  17  of the beam c u r r e n t .  The c a l i b r a t i o n f o r the f i r s t  polarlmeter  7  f o r the sum  of L and R counts per nanocoulomb was:  L  and  f o r the second  +  nC  R  = 106.7 + 9.95 x I O  polarlmeter  L  * nC  R  = 235.7 + 0.128 T  (2-5)  p  + 4.0 x 1 0 T _ 4  p  i n each case.  these f i t s  to the measured values  (2-6)  2  p  energy i n MeV and the CH  5.0 mg/cm  2  target thickness i s  The systematic u n c e r t a i n t y a s s o c i a t e d with  s o l i d angle of the spectrograph same p o l a r i m e t e r s .  T  5  where T^ Is the i n c i d e n t proton 2  - 2  i s ~ ± 5%.  both  The c a l i b r a t i o n of the e f f e c t i v e  (Sec. 3.4) was accomplished by using  these  In t h i s way any absolute n o r m a l i z a t i o n of the beam c u r r e n t  c a n c e l l e d s i n c e a l l measurements were r e l a t i v e to the known - * 1  1  ,l5  pp •*• d u  +  cross-sections. The and  Incident proton beam p o l a r i z a t i o n , P, was obtained using Eqns. (2-2)  (2-3) along with the d e f i n i t i o n of the a n a l y z i n g power.  do  do  da  L  Assuming L and R are d i r e c t l y p r o p o r t i o n a l to ^ — and  d  °R , respectively,  18  Because of the presence of q u a s i - e l a s t i c s c a t t e r i n g from the carbon, a c o r r e c t i o n was made to the a n a l y z i n g power.  Therefore,  P =  (2-9)  As more p p - e l a s t i c data has r e c e n t l y become a v a i l a b l e , the values f o r A ^ ^ used i n Refs. 5 and 7 are now out of date. most recent data (SAID the f i r s t The  8  A phase s h i f t a n a l y s i s using the  data s e t SM84) was used to d e f i n e A^.  In the case o f  p o l a r i m e t e r , the carbon component was c o r r e c t e d f o r as i n Ref. 9.  l e a s t squares f i t t o the A ^ ^ i s :  A„„  = 8.38 x I O  - 2  + 9.74 x 1 0 ^ T  with a systematic u n c e r t a i n t y of ~ ± 5%.  - 7.29 x 1 0 T p P _ 7  2  (2-10)  T^ i s again the i n c i d e n t p r o t o n  energy i n MeV. In the case of the second p o l a r i m e t e r , the r a t i o s of A ^  H  /A^ were  measured and are shown In F i g u r e 8 ( r e f . 10). The l e a s t squares f i t to t h i s ratio i s : A  CH  T  T~  =  1-01 +  (i^O  T  3 6  T  - (TJO)'° " (TOO) (  36  -  00487  )] <-> 2  L 1  whereas the l e a s t squares f i t to the best A^ (at t h i s time) i s :  A  H  - -5.82 x 1 0  - 3  + 1.92 x 1 0 T - 3  p  - 1.85 x l O ^ T  2  p  (2-12)  19  20  The o v e r a l l systematic u n c e r t a i n t y i n A proton energy range of i n t e r e s t i n t h i s  2.3  The " R e s o l u t i o n " Spectrograph  The  i s estimated as ~ ± 2% f o r the study.  System  b a s i c instrument was a 65.0 cm Browne-Buechner  spectrograph.  11  magnetic  The p a r t i c l e d e t e c t i o n system c o n s i s t e d of a counter  teles.cope  composed of three s c i n t i l l a t i o n counters together with three h e l i c a l l y wound delay l i n e m u l t i w i r e p r o p o r t i o n a l chambers (MWPC).  12  Helium boxes were  i n s e r t e d between the three MWPC's to reduce m u l t i p l e s c a t t e r i n g w i t h i n the chamber system.  The layout of the spectrograph  system i s shown i n F i g u r e 9  with i t s a s s o c i a t e d e l e c t r o n i c s i n F i g u r e 10. A d e t a i l e d d e s c r i p t i o n of the experimental The  arrangement can be found  three s c i n t i l l a t i o n counters  i n Ref.  5.  (CE, C l , C2) provided the  d e f i n i t i o n as w e l l as timing and energy l o s s i n f o r m a t i o n .  event  The MWPC's  (multiwire p r o p o r t i o n a l chambers), on the other hand, provided  position  i n f o r m a t i o n from which the e x i t t r a j e c t o r i e s of each p a r t i c l e were  determined.  The  determined  i n t e r s e c t i o n of the p a r t i c l e t r a j e c t o r i e s with the f o c a l plane  the momentum of the p a r t i c l e . 9.  The coordinate system used i s shown i n F i g u r e  Z i s the v e r t i c a l , X i s h o r i z o n t a l i n the bend plane, and Y i s h o r i z o n t a l  i n the non-bend plane.  Thus the X p o s i t i o n i n f o r m a t i o n was c o l l e c t e d by the  h e l i c a l cathode of the MWPC's and the Y p o s i t i o n from the anode wires which were connected  to the delay l i n e .  The d i f f e r e n c e i n time f o r a pulse t o  t r a v e l to the two ends of the h e l i c a l cathode (or the delay l i n e ) was d i r e c t l y p r o p o r t i o n a l t o the X (or Y) p o s i t i o n .  The timing " s t a r t " of a l l the TDC's  ( t i m e - t o - d i g i t a l c o n v e r t e r s ) was t r i g g e r e d by the output  of the mean-timer of  21  MWPC 3 He BOX 2 MWPC 2 He BOX I  3cm Pb SHIELD TARGET  DRIVE,  MWPC I CE  6 5 c m MAGNET  P BEAM  WITH VACUUM BOX  MAGNET TROLLEY scoie:t  F i g u r e 9.  The " R e s o l u t i o n " Spectrograph.  4 0 cm.  t  22  CIL  CIR  ®n_L  C2L  ® L L  ®=L_L  ®-L_L  C2R  ®nM  SIGN CONFIGURATION OFF DOWN UP BUSt  r  TRUE EVENT  s)—ZL  RAND EVENT'  f,  COMPUTER BUSY  |40nt  OR.  JJ  ay STROBE IT) START  M.W. P. C . L O G I C IKI  I K 2 IAI  IA2 2KI 2 K 2 2 A I 2 A 2 3KI 3 K 2 3AI 3 A 2  L  I  R F  !  ©©©©©©©©©5©©© DISCRIMINATOR LRS 621 TYPE  , COINCIDENCE (2 FOLD) LRS 622  MT )  M  E  A  N  T  |  M  E  R  LRS 624  JILL COINCIDENCE \TJ  III  <4 FOLD) LRS 465  DELAY  LOGIC LRS 4 2 9  CAMAC SSCALER ^ ) KINETIC S 3615  •C}. PATTERN UNIT ^ E G G C2I2  ^L-CAMAC ADC V*JLRS  2249  6  CAMAC TDC LRS  Figure 10. The overall electronics for the "Resolution" system.  2228  23  Cl bin  (see F i g u r e 10).  T h i s time d i f f e r e n c e ( p o s i t i o n i n f o r m a t i o n ) u n i t of a TDC  represents 0.2 nsec which corresponds to ~ 0.55  mm.  P a r a l l e l e v e n t - d e f i n i n g l o g i c (see F i g u r e 10) was u t i l i z e d p r o v i d i n g both true and random event d e f i n i t i o n s .  The timing gate of CE was wide enough to  allow f o r both true and random event c o i n c i d e n c e s . The random events proved to  be i n s i g n i f i c a n t ; much l e s s than 1% of the true events f o r any measurement.  The true event c o i n c i d e n c e , though, c o n s i s t e d of a mixture of e l e c t r o n s (or p o s i t r o n s ) , protons and p i o n s .  The energy l o s s and t i m e - o f - f l i g h t i n f o r m a t i o n  allowed f o r the e l i m i n a t i o n of the protons and e l e c t r o n - l i k e events. the  T e s t s on  c o - l i n e a r i t y of the p a r t i c l e ' s path through the chamber system and  e x t r a p o l a t e d target p o s i t i o n were a l s o a p p l i e d (Sec. 3.1). The angular range a c c e s s i b l e to the measurements was r e s t r i c t e d to 46" 135° due to the geometry of the magnet yoke.  I n a d d i t i o n , pion k i n e t i c  energies could be measured over the range of 30 to 110 MeV.  The lower  limit  was defined by the energy l o s s of the pion t r a v e l l i n g through the system. Pions of l e s s than 30 MeV had Inadequate whole system. field  The upper l i m i t of 110 MeV was caused by the maximum magnetic  strength a c h i e v a b l e .  The magnetic  magnet resonance magnetometer (NMR) 9).  The f i e l d  p e n e t r a t i n g power to t r a v e r s e the  f i e l d was monitored by a n u c l e a r  p o s i t i o n e d i n the magnet gap (see F i g u r e  13  v a r i e d by l e s s than I O  - 4  t e s l a over the time of any run.  The t a r g e t s f o r t h i s system were mounted at e i t h e r 45° or 135° w i t h respect to the incoming beam depending on whether the spectrograph was at a forward or backward s c a t t e r i n g angle. were measured to an accuracy of 1%. i n Table I .  The a r e a l t h i c k n e s s of a l l t a r g e t s used  The a c t u a l target t h i c k n e s s e s are given  24  Table  Target  [  A r e a l Thickness of Targets  A r e a l Thickness  Scattering  (mg/cm )  Angles  2  9  1 0  Be  B ( 9 2 % enriched)  100.8  Forward  46.8  Backward  99.3  Forward  101.6  Forward  92.3  Backward  300.0  Forward  100.0  Backward  25  III.  3.1  Data A n a l y s i s  Background D i s c r i m i n a t i o n  During data t a k i n g , the p a r t i c l e s which t r i g g e r e d the event (and  thus t r a v e r s e d  the f o c a l plane) c o n s i s t e d of a mixture of e l e c t r o n s  p o s i t r o n s ) , protons and protons and tests.  pions.  e l e c t r o n s was  The  accomplished by s u b j e c t i n g each event to a s e r i e s of  were b a s i c a l l y four t e s t s a p p l i e d to each event:  The  target  pulse heights  at a l l pion energies  Separation  of i n t e r e s t .  of pions from protons was  The  a c t u a l l y o f f the right-hand  was  peak.  the  proven p o s s i b l e  background protons were predominantly The  11 i l l u s t r a t e s a t y p i c a l energy l o s s d i s t r i b u t i o n .  the f i r s t  There  extrapolation.  e l e c t r o n - l i k e events improved s l i g h t l y with decreasing  Figure  5.  energy l o s s , t i m e - o f - f l i g h t ,  low-energy m u l t i p l y - s c a t t e r e d protons of l a r g e dE/dx. and  i n Ref.  from the three s c i n t i l l a t i o n counters provided  energy l o s s i n f o r m a t i o n .  (or  f i n a l e l i m i n a t i o n of the background  A d e t a i l e d d e s c r i p t i o n of t h i s process i s described  c o - l i n e a r l t y , and  coincidence  separation  of pions  pion energy. The  protons  are  s i d e of the p l o t while the e l e c t r o n s c o n s t i t u t e  C l e a r l y energy-loss,  while adequate to separate the  protons,  inadequate f o r e l e c t r o n s . The  counters a l s o provided  " s t a r t " was Since  t r i g g e r e d by  the path d i s t a n c e s  t l m e - o f - f l i g h t measurements.  between CE,  too s m a l l to be u s e f u l .  c y c l o t r o n RF was the production  timing  the output of the mean-timer of C l (see F i g u r e  s e p a r a t i o n a s s o c i a t e d with the C2 s i g n a l was  The  C l , and  C2 were s m a l l , the t i m e - o f - f l i g h t  s i g n a l or the a p p r o p r i a t e l y The  10).  delayed  time i n t e r v a l between C l and  CE the  the measure of the time from a r r i v a l of the beam proton at target to the r e a c t i o n p a r t i c l e t r a v e r s i n g C l .  Thus the raw  RF  26  fiiifffisiiiiifiiiiifiiiiiiicisiiiii*" 52  Figure 11. A t y p i c a l energy loss d i s t r i b u t i o n . protons is shown.  The cut to reject the  27  defined t i m e - o f - f l i g h t  distribution  s e p a r a t i o n between pions and of  the RF  timing s i g n a l  (43  electron-like  arising  contribution  the  The  co-linearlty  through the  5  (see,  t e s t determined the  are  f o r example, the  not  periodicity unique, with  electron  extent to which the  a straight  largest  12). p a r t i c l e motion  line trajectory,  thus  a g a i n s t pion decay or m u l t i p l y - s c a t t e r e d events o c c u r r i n g  chamber system.  The  accumulating d i s t r i b u t i o n s characterizing  Because of the  43 nsec window shown i n F i g u r e  three MWPC's c o n s t i t u t e d  discriminating w i t h i n the  of  events.  showed the  n s e c ) , these time d i f f e r e n c e s  "wrap-around" d i s t r i b u t i o n s at each end  (shown i n F i g u r e 12)  straight  event-by-event b a s i s .  co-linearlty  of both the X and  t e s t was  performed  Y standard d e v i a t i o n s  l i n e l e a s t squares f i t s through the F i g u r e 13  by  i l l u s t r a t e s the  chambers on  t y p i c a l standard  an  deviation  distributions. Since there i s no  f o c u s s i n g i n the Y d i r e c t i o n  (see  F i g u r e 9),  s m a l l edge f o c u s s i n g e f f e c t s , an e x t r a p o l a t i o n to the Y p o s i t i o n was  r e a d i l y performed.  illuminated  r e g i o n of  By the  decay or those which had  i n s i s t i n g that the t a r g e t , some of  particles originate  The  distinguished.  events i n the  c e n t r a l peak correspond  give the  best momentum r e s o l u t i o n  case of B e ( p , i x ) B e , the 9  +  1 0  went from ~ 7 with no  The  at the  peak to v a l l e y  cuts to ~  spectrum of pions taken from the  target  the  route to  A t y p i c a l Y target  "undisturbed" t r a j e c t o r i e s , whereas those i n the m u l t i p l y - s c a t t e r e d or decayed events.  in  the  for  p a r t i c l e s a r i s i n g from p i o n  been m u l t i p l y - s c a t t e r e d somewhere on  f o c a l plane d e t e c t o r s could be i s shown i n F i g u r e 14.  the  at  except  t a i l s are  cuts on  the  distribution to  a s s o c i a t e d with  t h i s t e s t were chosen to  f o c a l plane. r a t i o f o r the  For  example, i n  ground s t a t e  15 with the Y target c u t s . t a i l r e g i o n of the Y target  The  the  peak  f o c a l plane  distribution  28  Figure 12.  A t y p i c a l RF defined t i m e - o f - f l i g h t d i s t r i b u t i o n . r e j e c t the e l e c t r o n s and protons are shown.  The  cuts  to  29  19*0 13«0 1390 >l*0 t>oo XViO IMC 1»30 IMO •too 1T40 mo l«30 IHO 1»O0 1440 IMO 1130 I3»0 13O0 1140  •ooo 1010 •oo •oo •'0 T»0 T30 M O •OO H O «»0 4 30 M O 100 140 OO  oo •0  (a) I M Itl III II** III! T i l l9 IIIII H U M •IIIII IIIIII IIIIII IIIIIII •IIIIII IIIIIIIt •IIIIIII IIIIIIII} IIIIIIIII IIIKIIIII IIIIIIIIIT IIIIIIIIII IIIIIIIIII} IIIIIIIIIII IIIIIIIIIII KIIIIIIIIII4 IIIIIIIIIIII III! I l l l H I I t I l l l l l l l l l l l l t  ("-3mm)  I1II1IIIIIIIIII4 m i n t XIIIIIIIII i i i i i i i i i i i i imi « i ntt< i IIIIIIIIIIIII IIIIIII;: I I I I I I I I I I I I I IIIIIIIIIIIIIIIKIIKMMCC3344 3 I I I I I I I I I I I I I I IIIIIII1IIIIIIIIII I I I I I I I I 4 B 19**TMT«T4ST«4M4t4$4943S4MS34443594944344444»4930«3S<«4344S44444  Typical  I  f  Cut  1  n H H 1 I H 1 f l U t 1 1 1 H 1 t 1 1 « I I I H I H M I M n i H l i n n *(1lt33]373>l)34444«99S4MMMYTTTTtttB«n*t>OOOOOl ( 1 ' 1 2 7 3 7 3 3 3 3 3 3 4 4 4 4 4 T T T T S 8 B S I 9 9 9 9 9 03<W7'(«034f^7<«BC?4«^!«t»QJ*t4^  TDC  1130 X>40 1 X 0 3110 JlOO 3730 3«40 1S«0 34to 3400 3330 1340 1<«0 3O4J0 3O00 1130 • •40 IT«0 '•to IIOO 1»30 1440 11*0 13>0 13O0 1130 IO40 M O M O •00 T30 • 40 MO 4*0 400 1JO 140 1*0  BINS  (b)  •4  • ••  III III« I • 11 • HIT IIIII IIIII IIIII4 •IIIII IIIIII IIIIII I IIII11 IIIIIII IIIIIII •IIIIII? IIIIIIII IIIIIIII IIIIIIII IIIIIHIt IIIIIIIII IIIIIIIII4 IIIIIIIIII IIIIIIIIIII • IIIIIIHIIt 111111111111 •IIIIIIIIIIII IIIIIIHIIIIIt II II II II II II II II II II II HH II 33 1  (~ 5 mm) Typical Cut  ' ' I i n i i i l i i i t i i i i 9 I IIIIIIIIIIIIIIIIH59 I • i I I I I i I i I I i i I I i i i i i 9 « « C T 4 M ' 3 3 t 4 43 M i l ) 113 11 1 i i i i i i i i ( i i n i i i i i i i i i i i i * i i i i i i i i i f n i i t iI iI iI I iI i Ii Ii 1| 4i 1| ti I M 9 I I M t l » S t 8 « T ? a t C ( 7 « < 4 C 9 4  i  i  444 4S594344 3333 3 3 33 3  | — i1-1.1 -1 1i 1 1 1 1 1 1 1 1 1 1i1 1 1 1 1 1 1 1 1|.. 13333333333333333333333)333333333333333333333333333  0 0 0 1 t33333<4<»c:<11Mlt*000< I111114445MMT7IMMOO01 I 333334 J <55111' T• 1999OO01 13333344<S»(<TT|tl99 0 - 1 1 ?•<>«• 3 1 0 4 a 3 ( C M •  t o n ?to<a :to< • ?*o*a ?»o*a : a o i a  TDC  a : « C M a 7 t Q 4 i : ( O J a?fr04a;to«a ; K M I ;tc*4 a;c  BINS  Figure 13. A t y p i c a l (a) X and (b) Y standard d e v i a t i o n d i s t r i b u t i o n the t y p i c a l c u t .  along with  30  II' Si :- §1  3  o o o  *  |  * », * • » m  5 6  a.  >»  to  z  5  I o  tf «•h• •i k  ii: !  T r M  iii  • 2  Figure  u=i 2-  tr  14. A t y p i c a l Y target e x t r a p o l a t i o n d i s t r i b u t i o n cuts.  along with the  typical  31  (outside  the cut l e v e l ) showed very l i t t l e  " d i s t u r b e d " t r a j e c t o r i e s were being  structure,  suggesting that  rejected.  An estimate of the p r o b a b i l i t y of the pion s u r v i v i n g determined assuming they were independent e f f e c t of removing them one at a time. For  the energy  The estimate was approximately 70%.  l o s s and t i m e - o f - f l i g h t  c o - l l n e a r i t y or e s p e c i a l l y the target independent  nor redundant,  a l l these t e s t s was  of each other, and monitoring the  t e s t s , where the t e s t s a p p l i e d  same p a r t i c l e s , t h i s method i s reasonable.  neither  mostly  to the  But i n the case of the  extrapolation  t e s t s , where they a r e  t h i s method i s l e s s j u s t i f i a b l e .  A more  d e t a i l e d accounting of t h i s s u r v i v a l p r o b a b i l i t y was accomplished by l n c l u d l n the e f f e c t In the d e f i n i t i o n of the e f f e c t i v e s o l i d a n g l e .  5  The i d e n t i c a l  + t e s t c o n d i t i o n s were a p p l i e d e f f e c t i v e s o l i d angle (Sec. 3.2  du  data used to c a l i b r a t e t h e  3.4).  F o c a l Plane and D i s p e r s i o n R e l a t i o n  After selecting 3.1,  to the pp  the good events by means of the cuts d e s c r i b e d i n S e c t i o  the momentum of each remaining plon was determined  the f o c a l plane (XFP) u s i n g the d i s p e r s i o n T h i s equation r e l a t e s  from i t s p o s i t i o n on  r e l a t i o n f o r the spectrograph.  the pion momentum to the measured XFP and the magnetic  f i e l d of the magnet. The XFP was determined by the X component of the i n t e r c e p t t r a j e c t o r y with the f o c a l plane. determined  The t r a j e c t o r y of the p a r t i c l e was  from the MWPC p o s i t i o n Information.  The c h a r a c t e r i s t i c s of the  f o c a l plane are d e s c r i b e d i n Ref. 5 and are summarized here. defined by the equation  of the pion  The plane i s  32  z  a - bX  (3-1)  where a = 700 ± 20 mm b = 0.94 ±  .03 mm/bin  Z i s v e r t i c a l and X i s h o r i z o n t a l i n the bend plane The  v e r t i c a l height was measured from the middle of MWPC1 (see F i g u r e 9). The  5.  (see F i g u r e 9 ) .  determination  of the d i s p e r s i o n r e l a t i o n was a l s o d e s c r i b e d i n Ref.  The r e l a t i o n s h i p obtained i s :  (3-2)  where the XFP are i n TDC time u n i t s .  The momentum, P, i s i n MeV/c whereas the  magnetic f i e l d , B, i s i n t e s l a . The k i n e t i c energy of each p a r t i c l e was obtained from Eqn. (3.2) u s i n g the usual kinematic  relation:  • where m^ i s the mass of the pion, 139.57 MeV/c . 2  This was done i n order to  d e p i c t a pion spectrum that more c l o s e l y resembles the e x c i t a t i o n of the f i n a l nucleus.  F i g u r e s 15 and 16 i l l u s t r a t e t h i s technique with two such s p e c t r a .  33  oo  S1ND0D  F i g u r e 15. B ( p , n ) B energy spectrum of n produced at 5 0 ° ftom 225 MeV i n c i d e n t protons with s p i n down. Lineshape f i t s f o r the f i r s t two s t a t e s are shown by the s o l i d l i n e . 1 0  +  1 1  +  m  16  3^  O'O'O  —  o o ID  S 1 N D 0 D F i g u r e 16. B e ( p , u ) B e energy spectrum of n produced at 50° from 200 MeV i n c i d e n t protons with mixed s p i n . Lineshape f i t s f o r the f i r s t t w o s t a t e s are shown by the s o l i d l i n e . 9  1 0  35  3.3  Lineshape  The c o - l i n e a r i t y and t a r g e t e x t r a p o l a t i o n t e s t s d i d not e l i m i n a t e a l l the pions that had s u f f e r e d s i g n i f i c a n t m u l t i p l e s c a t t e r i n g or that had decayed i n t o muons.  Such pions tended to produce a " t a i l " i n the momentum  d i s t r i b u t i o n of a s i n g l e l i n e and proved to be an e f f e c t  that had to be  accounted f o r . These e f f e c t s have been described i n Ref. 5 where both an experimental measurement and a Monte-Carlo s i m u l a t i o n were completed f o r the pp •*• d u the  +  line.  The p o l e - f a c e s c a t t e r i n g i n the spectrograph was  shown to be  major c o n t r i b u t o r to these " t a i l s " . The k i n e t i c energy s p e c t r a of the pp •*• d u  l i n e were best f i t  +  5  by the  a n a l y t i c form:  (T -B)/F e it  = A e  1+  .<V >' B  (3-4) 6  The t o t a l number i n the l i n e s h a p e was normalized to u n i t y . Gaussian type, c h a r a c t e r i z e d  The f i r s t  term, a  the peak component of the spectrum, whereas the  second term, the e x p o n e n t i a l decay (with a Fermi-type cut o f f at the peak p o s i t i o n , T^ = B) was are  energy dependent  used to d e s c r i b e the t a i l component.  These  parameters  and could be expressed i n terms of B, the c e n t r o i d of the  peak (MeV) by the f o l l o w i n g :  5  D = (140 ± 6)/B F = (-.93  (3-5)  2  ± .40) + (.083  G = (.13 ± .12) + (.019  ± .007)B ± .002)B  (3-6) (3-7)  36  The peak component parameters B and C were l e f t f r e e when f i t t i n g the experimental energy  d i s t r i b u t i o n s since B i s d i f f e r e n t  f o r each l i n e  and C ( r e l a t e d to the l i n e ' s width) i s dependent on both the energy the incoming beam and the kinematic broadening targets.  observed spread of  due to the r e c o i l of l i g h t  Parameter A was f i x e d such that the proper r a t i o of peak to t a i l  components was maintained.  Since the amplitude  allowed to vary as i t s width  of the peak (A) must be  (C) changes, A was f i x e d by the f o l l o w i n g method.  Since the l i n e s h a p e i s normalized  i.e.  /  to u n i t y ,  F(T )dT  = 1  (3-8)  — oo  Then s u b s t i t u t i o n of Eqn. (3-4) along with the r e p r e s e n t a t i o n of x = T  - B, u  yields: 2  oo  /  A e  _ X  dx  first  5?  +/  — oo  The  x/F  00  I  — oo  i + e  term i s j u s t a Gaussian and the second  component (A,,,).  dx  .  1  _  (3  9)  the area of the t a i l  T h e r e f o r e Eqn. (3-9) becomes;  A /Cu  + (A ) = 1 T  (3-10)  1 " (A ) T  or  /CTT"  (3-11)  A^, was obtained by numerical I n t e g r a t i o n of the t a i l component (as d e f i n e d by the a p p r o p r i a t e D, F, and ,G parameters) f o r any given pion energy.  A least  37  squares f i t to A^, gave  (.34 ± .02) - (2.4 ± .2) x 10  -3  B  (3-12)  Thus the f i n a l shape of any line i s dependent on only two parameters the centroid (B) and the width (C). Any pion kinetic energy spectrum consisting of a number of discrete lines can be f i t by:  M S (  V  =  E N P (B ,C) 1=1 1  1  (3-13)  1  where M i s the number of lines i n the spectrum S, in the i t h line, and  i s the number of events  i s the lineshape defined by Eqns. (3-4), (3-5), (3-6),  (3-7), (3-11) and (3-12) with two parameters B^^ (centroid) and C (width) for any of the i lines.  Since the width of a l l lines of a given spectrum was  dominated by the energy spread of the incoming beam, C was forced to have the same value for a l l M l i n e s . the  The' solid curves of Figures 15 and 16 i l l u s t r a t e  typical quality of such f i t s .  Since only the f i r s t two states were  treated i n this work (well away from the continuum associated with three or more particles i n the f i n a l state), no continuum correction to Eqn. was needed.  Typical values of the reduced x 2 ( x 2 P  e r  (3-13)  degree of freedom) for  these f i t s ranged from .8 to 2.0.  3.4  Effective Solid Angle The effective s o l i d angle of the spectrograph, AQ , i s not just the  38  geometrical solid angle, but includes a l l the effects of decay, multiple scattering, and as well the effects of the cuts on the data.  Depending on the  energy of the pions as many as 20% could have decayed of which only a few percent of the decay muons ended up i n the f i n a l spectrum.  The " t a i l "  component of a line (mainly multiply scattered pions) ranged from 5 to 10% of i t s total area. (Sec. 3.1).  The efficiency of the cuts was estimated as approximately 70%  Since a l l these effects are interdependent, i t proved to be more  accurate and convenient to include a l l these inefficiencies i n an energy dependent effective s o l i d angle.  The calculation of AQ i s described i n Ref. e  5 so only a brief summary i s given here.  The calibration of the AQ was g  performed by comparing the measured pp -+ drt to the known* *' +  1  1,5  cross-sections.  That i s N AQ  e  e  where:  N  N — (—) Pt *2_ p MVT |cos9 | dQ  (3-14)  i s the number of pions defined by the lineshape f i t (Eqn.(3-13))  applied to pions from the pp * d n reaction. +  ^  i s the number of incident protons (Eqn. (2-5))  " \ Pt ["nrrJ i i Is the number of scattering centers i n the target where: r  MW  v  N  ;  Q  |cos9 ( Q  t  °  6  Is Avogadro's number  n i s the number of scattering centers per molecule MW i s the molecular weight of the target material pt i s the areal thickness of the target i n mg/cm  2  9  i s the target angle with respect to the incoming beam  39  e i s Che efficiency of the MWPC's which varied from one individual data collection run to another, thus making i t impossible to include i n the effective solid angle (this efficiency was typically 60 to 70%) and ^2-are the known *' 11  15  d i f f e r e n t i a l cross-sections for the pp-*-du  +  reaction. This calibration was completed at three pion kinetic energies; and 100 MeV.  50, 70,  The uncertainty of each calibrated effective solid angle was ~  ± 5%, caused mainly by a combination of systematic uncertainties i n the pp+du  +  cross-sections together with uncertainties of the absolute beam current normalization.  Upon examination of the effective solid angle values, the best  f i t was given by a straight l i n e .  The least squares f i t to these three values  yielded the following energy dependence for AQ : g  AQ  where AQ  = (1.15 ± .15) + (5.12 ± 2.0) x 10" T u 3  e  i s i n msr and T i s i n MeV. e TI interpolated value of AQ Is ~ ± 15%.  (3-15)  Thus the systematic uncertainty of any  40  IV.  The analyzing power,  A N Q  (Q)t  d i f f e r e n t i a l cross-section,  A  Results  and the spin-averaged  (unpolarized)  (9), were calculated using the relations:  (Q) -  dg(+)/dB  -  dc(+)/da  and  da ... dQ  P(t)da(0/dQ + P(Oda(+)/dQ P(t) + P(+)  '  v  (4-2)  where P i s the magnitude of the beam polarization (Eqn. (2-9)) and ^  i s the  dQ  spin-dependent d i f f e r e n t i a l cross-section.  The arrows indicate the spin  direction according to the Madison convention.  16  As i n Eqn. (3-14), the  spin-dependent d i f f e r e n t i a l cross-sections were defined as:  A  N  ° " dQ d  NQH E  N  (-^rr) i  p  where  —  %  x  (4-3) ^l-T-AQ  |cos9 I  MW  e  i s the number of events from the f i t of the spin-dependent spectrum.  The areal thicknesses  (pt) are given i n Table I. 9  (Sec. 2.3), thus |cos9j  i s either 45° or 135°  i s 1// 2 . The effective solid angles (AQ ) are  defined by Eqn. (3-15). In the case of the B target ( i . e . targets enriched to 92% B ) , the 1 0  1 0  background due to the 8% contamination of B was determined by also 1 1  collecting data with the B 1 1  runs.  targets under identical conditions to the  1 0  B  For the results presented here, where only the ground and f i r s t excited  kl  states are considered, the B 1 1  backgrounds i n this region was found to  contribute less than 1% to the two states for a l l measurements. The f i n a l results are tabulated i n Tables II, I I I , and IV and shown i n 17  Figures 17 through 28. data  7  18  The available data from IUCF  for this energy range are also shown.  '  and previous TRIUMF  The absolute normalization of a l l  the sets of data agree within 10% (well within the systematic errors). Preliminary results of this data have already.been  published. ' 34  As well,  35  the f i n a l results of the B ( p , u ) H B reaction have recently been 10  published.  +  36  Only the relative uncertainties are indicated i n the tables and figures. In addition, there i s an overall systematic uncertainty of ~ ±15% for the d i f f e r e n t i a l cross-sections.  For the analyzing powers of  1 0  B(p,Tt ) ^B, +  1  there  is ~ ±2% systematic uncertainty, whereas for B e ( p , u ) °Be and Be(p,u ) C , 9  +  1  9  1 0  i t is ~ ±5%. The relative error consists of both the counting s t a t i s t i c s and the random fluctuation i n the beam current measurements (mainly due to the wrinkling of the thin polarlmeter t a r g e t s ) . The majority of the systematic 7  uncertainty i n the d i f f e r e n t i a l cross-section arises from the uncertainty i n the calibration of the effective solid angle of the spectrograph  (Sec. 3.4).  The systematic uncertainty of the analyzing powers Is due to the uncertainty in the analyzing powers of the polarimeters (Sec. 2.2).  k2  TABLE I I  10  cm (Deg.) 6  (MeV)  10  •) B, ll  (nb/s?5  ^0  (Deg.)  (nb/sH  ^0  200  49.8 64.6 74.9 85.2 95.2 110.0 124.5 138.6  471.(30.) 339.(22.) 196.(13.) 91.4(6.6) 94.7(6.5) 50.7(3.5) 56.7(3.8) 47.3(3.2)  -O.222(.032) -0.372(.034) -O.475(.032) -0.459(.046) -0.329(.041) -0.19K.040) -0.189(.036) -0.187(.039)  49.9 64.6 75.0 85.2 95.3 110.1 124.5 138.7  130.0(9.8) 104.3(8.1) 55.6(4.3) 38.5(3.3) 36.1(2.9) 25.5(1.9) 19.9(1.5) 18.1(1.4)  -O.45K.059) -0.428(.061) -0.694C.056) -0.607(.067) -0.586(.061) -0.533(.051) -0.379C.059) -0.468C.060)  225  49.8 59.3 64.4 79.9 87.1 95.0 109.8 124.3 138.5  593.(37.) 348.(27.) 285.(18.) 122.8(8.1) 98.2(7.8) 75.0(4.9) 46.9(3.2) 30.0(2.0) 42.9(2.9)  -0.305C.018) -O.252C.058) -0.368C.024) -0.095C.036) O.OO0C.O73) -O.04K.035) 0.053C.042) -O.094C.043) -0.327(.043)  49.8 59.3 64.4 79.9 87.1 95.1 109.9 124.3 138.5  172.(11.) 84.9(9.6) 90.5(6.1) 51.6(3.8) 55.1(5.1) 39.5(2.8) 32.4(2.3) 19.2(1.4) 34.1C2.4)  -0.432C.033) -0.63C.10) -0.473C.042) -0.270C.054) -0.435C.094) -O.318C.048) -0.218C.050) -0.295C.053) -0.457C.047)  250  49.6 57.0 64.4 74.7 84.9 95.0 109.8 124.2 130.4 138.5  539.(33.) 376.(24.) 257.(16.) 132.0(8.5) 73.2(5.9) 38.2(2.5) 14.2(1.4) 14.1(1.2) 12.1(1.2) 14.2(1.2)  -0.02K.021) -0.029(.024) -0.004(.023) 0.082C.031) 0.287(.066) 0.534(.034) 0.386C.099) -0.289C.086) -0.505C.087) -O.506C.075)  49.7 57.1 64.4 74.7 84.9 95.0 109.8 124.3 130.5 138.5  114.0(7.9) 82.4C5.8) 62.2(4.3) 40.5C3.0) 29.1(3.1) 19.2C1.4) 10.1(1.1) 7.00C.77) 7.05(.83) 7.19C.77)  -0.064C.046) -0.351C.049) -0.343C.045) -0.201C.056) 0.05C.11) 0.163C.052) 0.29C.12) 0.43C.12) -0.04C.13) -0.28C.11)  260  49.7 64.4 74.7 84.9 95.0 104.9 114.6 124.3 138.5  580.(39.) 252.(16.) 127.4(8.7) 82.5(5.4) 53.0(4.3) 27.8(2.4) 16.1(1.5) 15.1(1.5) 10.6(1.2)  -0.033(.039) 0.048(.024) 0.262(.042) 0.377(.037) 0.525C.065) 0.493(.092) 0.16C.10) -0.36(.10) -0.47C.12)  49.7 64.4 74.7 85.0 95.0 104.9 114.7 124.3 138.5  130.(11.) 61.9(4.3) 38.6(3.2) 43.3(3.1) 24.0(2.4) 16.9(1.7) 8.23(.97) 8.5(1.0) 8.8(1.1)  -0.079C.082) -0.225C.048) -0.322C.076) 0.106C.053) 0.541C.096) 0.47C.12) 0.51C.13) 0.41C.14) -0.43C.13)  A list  of the values  f o r the d i f f e r e n t i a l c r o s s - s e c t i o n s and a n a l y z i n g  powers f o r B ( p , u ) r e a c t i o n 1 0  +  leading  to B 1 1  and B , g.s. l l  , - „ „ states. MeV  TABLE III  9  (MeV)  B e ^ , 7 t  +  )  9 1  0  e da/dQ . cm „ ("eg.) (nb/sr)  B e  g.s. A  B e ( p \ u ) B e 3 .37 MeV +  10  do7dQ  N0  ®cm „ (Deg.) (nb/sr)  200  49.8 64.6  82.4(5.7) 59.8(4.9)  -0.736(.038) -1.036(.044)  49.9 64.7  225  49.7 59.2 69.6 79.9 95.0 114.7 138.5  75.8(6.3) 54.6(5.0) 43.8(3.9) 24.8(2.7) 11.7(1.5) 3.8(1.5) 3.8(1.2)  -0.590(.073) -0.903(.072) -0.910(.073) -1.079(.084) -0.44(.15) -0.38(.55) -0.26(.40)  250  49.6 59.1 69.6 79.8 95.0 109.8 124.2 138.5  65.7(5.5) 37.0(3.5) 25.7(2.5) 16.1(1.7) 9.4(1.1) 5.12(.95) 7.3(1.5) 4.60(.91)  -0.483(.075) -0.61(.ll) -0.950(.067) -0.46(.ll) 0.11(.14) 0.65(.24) -0.38(.26) -0.37(.28)  A  N0  204.(13.) 165.(11.)  -0.314(.028) -0.752(.035)  49.7 59.2 69.6 79.9 95.1 114.7 138.5  231.(16.) 159.(11.) 150.(10.) 117.4(8.5) 122.9(8.5) 69.0(7.5) 32.9(3.9)  -0.135(.046) -0.645(.049) -0.767(.043) -0.679(.051) -0.175(.049) -0.07(.13) -0.23(.14)  49.7 59.2 69.6 79.9 95.0 109.8 124.3 138.5  209.(14.) 124.2(9.0) 97.8(6.9) 85.2(6.0) 60.8(4.3) 45.4(3.8) 19.3(2.5) 17.3(2.0)  0.196(.045) -0.297(.064) -0.709(.041) -0.496(.048) -0.165(.054) 0.222(.087) 0.06(.17) -0.60(.14)  A l i s t of the values for the d i f f e r e n t i a l cross-sections and analyzing powers for Be(p,u ) reaction leading to B e * ^Bej states. 9  +  10  an<  g  3 7  M  e  V  TABLE IV 9  B e C t . O « C  cm (Deg.) (nb/sr) 9  (MeV)  g  .  a  .  9  A  N0  Be(?fn')"C3.  cm /o/* (Deg.) (nb/sr) 6  3 1 t  A  M  e  V  N0  200  49.8 59.3 95.2  1.26(,22) 1.39(.34) 1.47(.64)  -0.37(.24) -0.12(.32) -0.36(.48)  49.9 59.4 95.3  5.33(.54) 4.39(.65) 5.1(1.2)  -0.03(.12) 0.19(.18) -0.31(.26).  225  49.7 69.6 95.0 138.5  2.15(.41) 1.26(.22) 0.90(.22) 1.47(.51)  -0.81(.20) 0.08(.22) 0.19(.33) -0.22(.46)  49.7 69.6 95.1 138.5  6.58(.79) 5.09(.51) 2.09(.35) 1.72(.55)  0.07(.14) -0.30(.ll) 0.13(.22) -0.12(.43)  250  49.6 69.6 95.0 138.5  1.04(.28) 1.24(.34) 0.86(.21) 1.30(.39)  -0.87(.26) -0.38(.39) 0.37(.33) 0.09(.43)  49.7 69.6 95.0 138.5  4.45(.62) 2.02(.44) 2.01(.33) 1.38(.40)  -0.54(.16) -0.28(.31) 0.30(.22) 0.02(.42)  A l i s t of the values for the d i f f e r e n t i a l cross-sections and analyzing powers for Be(p,u ) reaction leading to C and C states. 9  1 0  1 0  3  3 4  M  g  V  45  O  CO  O  o  <3-  a  o o  a  o  CM  O  a  O  o o  4*  •  E  •  u CD  <  o  •  CO  < a  u ~. 3 •  a  > 2  O IT) CVJ •  O CM  o o o  llll I l l  I  2  o O o O CVJ CVJ • a. t-  tt CM CVJ • llll  o o o  a>  > z  w  LZ  CD  >  o  >v  o  O O O  l l l l  mi i  O O O  (js/qu)  •un i i i  O  o  O o  7JP •OP  Figure 17. The d i f f e r e n t i a l cross-sections for the reaction B(^.n"*") B The 200 MeV results of Ref. 17 are also shown. The s t a t i s t i c a l errors are less than the size of the symbol on the plot. 10  11  46  (iVQu) g£ Figure 18. The d i f f e r e n t i a l cross-sections for the reaction BC$,n ) 2.12MeV The s t a t i s t i c a l errors are less than the size of the symbol on the plot. 10  llB  47  > O in  ^  CVJ II  w  Q.  o  5s  2  "  OCT WI O ID  II  O ro  E u  o  CD  o a  <  a  LLli 1J|  O Oo  o  <  1 1  1  ' \\  l"l I  in  I I  ,  II  o ( /qu) J S  F i g u r e 19. The The  ,NV  o  O ro  SE  d i f f e r e n t i a l c r o s s - s e c t i o n s f o r the r e a c t i o n ' B e t f , * * ) ! Be 200 MeV r e s u l t s of Ref. 7 are a l s o shown. 8' " 0  s  48  >  >  2  O in  >  in  Z> o o  CM  CJ  CJ  II  o. I-  O  OJ  CJ  II  a  OJ  rr  O in  a. —  O ro  E CJ  CD  <  •  O  <T>  a  <  a  O  o o  6  <  D  o o o  o  o  <3  IlLLI  o  in  o o  o o o  o ro  (JS/qu)  F i g u r e 20. The The  d i f f e r e n t i a l cross-sections 200 MeV  f o r the r e a c t i o n B e ( p \ i t )  r e s u l t s of Ref. 7 are a l s o shown.  9  1 0  Be  3 > 3 7  M e V  '  49  >  > 0>  O in  in  CM II  CL  >  —  5  -  O O  v  CM CC  o  CVJ CM II  Q.  O  o  CM  O O E  O  o CD  CO  O  O  O O  lllu  L  ( Vqu) J  \V  J  CM  ^  Figure 21. The d i f f e r e n t i a l cross-sections for the reaction Be(p,n~) C The 200 MeV results of Ref. 18 are also shown. 9  10  50  >  > OJ  5  O in  CM II  Q.  ID CM CM  O  II  O.  o > OJ  o  O O  CM  CM II  Q.  o  O  o  CD  E o  GO  O  CO  o  Il "  o  i .  .  \\ I  .  o  o  I  o  ~.CM  (js/qu) UP •op  Figure 22. The d i f f e r e n t i a l cross-sections for the reaction Be("$,u ) Cj 9  -  10  3 4  MeV  51  0  T p « 2 0 0 MeV  "D  D-  -0.5 -1.0 Tp«225 MeV  0 -0.5 k  N0  1.0  Tp«250 MeV  0.5  \ 0 -0.5 -1.0 - > —  \.Of  Tp« 2 6 0 M e V  0.5  0 -0.5 > C 20  •  40  60 9  i  '  80  100  cm  Figure 23. Analyzing powers for the reaction BCp,* ) The lines serve only as a guide to the eye. 10  J  120  140  52  0  1 Tp»200  1— MeV  O  -0.5  ~£> r>— — —  -1.0. Tp»225  MeV  T p « 250  MeV  0 -0.5  'NO  0.5  0 •0.5 Tp « 2 6 0  MeV  0.5 \  0 •0.5  40  60  80 e,  100  120  140  cm  Figure 24. Analyzing powers for the reaction  1  0  ,u )l1B +  The lines serve only as a guide to the eye.  2  1 2  M E V  53  Tp = 2 6 0  0  MeV  '("Ref. 7j  •0.5  -1.0  j Tp = 2 2 5  1  MeV  NO  —  —  — ?  -0.5  -1.0 J  Tp = 2 5 0  0.5  9  I  J  L_  L  MeV  -  \  /  i •0.5  /  \  -1.0  J  I  40  60  J  /  /  y  I  L  80 G,  100  J  L  120  140  'cm  F i g u r e 25. A n a l y z i n g powers f o r the r e a c t i o n B e ( | , i t ) B e . The 200 MeV r e s u l t s of R e f . 7 are a l s o shown. T h e ' l i n e s serve as a guide t o the eye. 9  +  1 0  g j  only  54  Tp = 2 0 0  MeV  Tp=225  MeV  ( » R e f . 7)  •0.5 •1.0  NO  -0.5  o ^ s  6  -1.0  Tp = 250  MeV  0.5  4-  0  •<5  -0.5  \  cr  fr  -1.0  40  60  80  100  120  F i g u r e 26. A n a l y z i n g powers f o r the r e a c t i o n B e ( f c , n ) 1 0 B e 9  140  +  The 200 MeV r e s u l t s of Ref. 7 are a l s o shown. as a guide t o the eye.  3  3 7  M e V  .  The l i n e s  6erve  only  55  Tp= 200  0.5 0  M e v ' o R e f . 18)  4  •0.5  Tp = 2 25  MeV  -4-  0.5 0 NO  -0.5 -1.0  0.5  Tp = 2 5 0  MeV  0 -0.5 -1.0 40  f  /  60  X  /  80  100  120  140  160  cm F i g u r e 27. A n a l y z i n g powers f o r the r e a c t i o n B e ( | , i t ~ ) C The 200 MeV r e s u l t s of Ref. 18 are al90 shown. *fhe l i n e s serve only as a guide t o the eye. 9  1 0  8  56  Tp = 2 0 0  0.5  MeV  0  •0.5  -  Tp=  225  MeV  0.5  4—  i —  0 'NO  •0.5  Tp= 2 5 0  MeV  0.5  f -  0  -0.5 -1.0  -  40  60  80 8.  100  120  140  cm  Figure 28. Analyzing powers for the reaction B e ( p \ i t " ) C ^ The lines serve only as a guide to the eye. 9  10  3  M e y  .  57  V.  5.1  Trends i n the Data  Choice of Variables The example of C o u v e r t > 19  possible trends i n the data.  20  and Nefkens  was followed to look, for  33  In terms of the Lorentz invariant matrix  elements,|M| , the unpolarized d i f f e r e n t i a l cross-section i s given by: 2  da dQ  <-« ' (he) 1_ _p_ 1 _. |2 64u s k (2J +l)(2S +l) Tt A P k  M  (5-1)  z  A  n  where s i s the square of the center-of-mass energy of the reaction, kp i s the center-of-mass momentum of the incoming proton, k^ i s the center-of-mass momentum of the outgoing pion, is the t o t a l angular momentum quantum number of the target (A), and  Sp i s the spin of the incoming proton.  The use of 1M] rather than 2  for these systematic studies has the benefit of  separating out phase space and t r i v i a l kinematic factors e x p l i c i t l y .  In this  way a clearer view of the effects of the reaction i t s e l f i s seen rather than the  effects of kinematics. Comparison of the dependence of | M | on 0 2  (scattering angle), q (three  momentum transfer), and t (the square of the four momentum transfer), indicated that the best independent variable for displaying the trends was t. t i s defined as: -t = 2E E - m - m - 2k k cos 9 p rt p u p Tt cm where E, m, and p are the t o t a l energy, mass, and momentum of the various particles.  58  5.2  S t a t i s t i c a l Weighting In the d e f i n i t i o n of  given in Eqn.  been averaged over the i n i t i a l spin states. however, Is s t i l l implied i n | M | .  the matrix element | M | has  (5-1),  2  A sum over the f i n a l spin states,  This s t a t i s t i c a l factor can be treated  2  similarly by assuming that each f i n a l state contributes average to | M | . 2  the same amount on  That i s , |M|  =  2  I  |m | f  2  -  (2J  A  +  1  +1)  (ZS^+Dlml  (5-2)  2  where J ^ ^ Is the total angular momentum quantum number of the f i n a l nucleus +  (A+l).  S^ i s the spin of the outgoing pion, which of course is zero, and  is the average matrix element per f i n a l state.  2  By removing this s t a t i s t i c a l  factor from | M | , the data from the different states of the 2  reactions s t i l l showed distinct differences.  |m|  A(p,ix )A+1 +  Thus as expected, the pion  production mechanism does show dependence on the angular momentum (J) of the f i n a l state.  However, by plotting the data versus t (as shown i n  Figures 29-34) where the areas marked by horizontal and v e r t i c a l lines represent the scatter of points for both the angular and incident energy plots show much more overlap compared to the | M | plots.  range, the |m|  2  For  2  example, by comparing  I 0  B(p,it ) B +  1 1  g  (y-)  to  1 0  B(p, t ) B +  1  ), one  1 1  2 > 1 2  can see very l i t t l e overlap of the two bands in the plot of | M | VS. t (Figure 2  29).  Figure 30 compares the same reactions with |tn| vs. t. 2  There i s  clearly much more overlap. Figure 31 and 32 compare B e ( p , i t ) B e 9  9  Be(p,n ) B e +  1 0  (2 ). +  3  3 7  v  +  10  (0 ) to g•s• +  In this case the difference between | M | and 2  |m|  2  59  Figure 29. |M| as a function of t for the two f i n a l states of B B ( p , T i ) B reaction. 2  10  ll  +  ll  i n the  60  61  62  Figure 32. Jm| as a function of t for the two f i n a l states of B e 'Be(p,n ) Be reaction. 2  10  +  10  In the  63  is even more impressive. 12  C(p,u ) +  1 3  C  The comparison of the carbon data^ , 1  ( | - " ) to C ( p , u ) 12  g # g >  +  best i l l u s t r a t i o n of this effect.  1 3  Q,^  ( | ) (Figures 33 and 34) +  M e V  i s the  The overlap seems to imply that though the  production mechanism does depend on the spin state of the f i n a l nucleus, the magnitude of the d i f f e r e n t i a l cross-section basically reflects a simple s t a t i s t i c a l weighting.  Although  this s t a t i s t i c a l weighting does not always  account for a l l the difference seen i n some of the (p,u~) work  4  (Figure 2),  it  should be taken into account prior to investigating enhancements of high-spin states by the reaction mechanism.  5.3  Incident Energy Dependence  The  (p,Ti ) +  d i f f e r e n t i a l cross-section distributions for the incident  proton energy range discussed in this work have qualitatively the same structure.  vs. 9  For 4r dQ  decrease at forward angles angle region.  plots, there is an almost exact exponential  cm  r  (G <90°) C M  followed by a r e l a t i v e l y f l a t backward  The slope of the exponential drop off in the forward angle  was  found to be energy dependent for some reactions but energy independent In others.  In addition, i t was  found that where energy dependence was  strong energy dependence was also observed distributions.  found, a  in the shape of the analyzing power  In contrast, for those cases yielding energy independent  slopes, only a weak energy dependence was observed  for the analyzing power  distributions. In the Be(p,u ) B e 9  +  10  reaction (shown i n Figure 35),  the slopes of  64  Figure  33. W 1 2  •» f u n c t i o n of t f o r the two C ( p , i t ) C reaction. a  +  1 3  f i n a l s t a t e s of " C  in  the  65  Figure 34. Jm| as a function of t for the two f i n a l states of C C ( p , n ) C reaction. 2  1 2  1 3  +  1 3  i n the  66  F i g u r e 35. The f o r v a r d angle data of the B e ( $ , n ) B e l e a s t squares f i t s to the data. 9  +  1 0  .  The  solid  lines  67  \  -ID  \  CVJ  MeV  > CU  MeV (Ref.  O  V -in  i  A  0)  —a—\ /  O in O m CM O  OJ CM CM II  II  a  • t—  V  a. Q. r—  4  : /  •  /  "I  j O —  jIII n 11 j 11111 '111II f I IT 1111111111 IP a o in a o a i o z  I I I I I I I I I  <  F i g u r e 36. The a n a l y z i n g powers as a f u n c t i o n of t f o r B e ( $ , T t ) l i n e s serve o n l y as a guide t o the eye. 9  +  1 0  Be  g > s >  .  The  68  O  -to  CM  O  • in CM  •  CM  CM CL  r-  o  •CM CM  O  CM  O  — I  O  CM  I I I I I I I ! I I I I I I 11 I I I I I I I I I I I!I I I I I I I I I I I I I | | | | I  CO  CD „—.  CO  ^ CL  _o Figure  37.  o  • CO  CM ,  CM  CO  U  LO  CU  O  The i n c i d e n t p r o t o n e n e r g y d e p e n d e n c e o f t h e e x p o n e n t i a l s l o p e s o f Be("p.it ) Be d i f f e r e n t i a l c r o s s - s e c t i o n d a t a as a f u n c t i o n o f  9  +  i Q  the t.  69  the | M | VS. t semilog plot are the same within the experimental error.  In  2  fact the least squares f i t s of these sets of data almost l i e on a single l i n e . In addition the analyzing powers (shown i n Figure 3 6 ) as well show l i t t l e dependence on incident proton energy.  Figure 37 shows a plot of the f i t t e d  —-J—'—, versus the incident proton energy, T^.  slopes,  of 5.5 (GeV/c)"  A constant value  i s consistent with the f i t s .  2  For the Be(p,rc ) 9  +  1 0  Be  3  3 7  M  e  V  reaction (Figures  characteristic behaviour Is observed.  38  -  a similar  40),  In this case, the f i t t e d slopes are  consistent with a constant value of 3.5 (GeV/c)~ . 2  The next case of weak dependence on incident proton energy i s that of 1 2  C(O,TC ) +  |M|  2  1 3  vs.  C  Q  7  = „ „, M ev  .  21  J  (Figures  41  -  In this case, the semilog plot of  43).  t i s not as energy independent  as the previous two cases.  Though  the slopes are generally the same (approximately 4.9 (GeV/c) ), the f i t s do -2  not l i e on a single l i n e .  The reason for this difference i s not known.  The  analyzing power distributions do show the same general trend as for the previous cases, however.  In contrast to the situation encountered with the  next three reactions to be discussed where a strong dependence on Incident proton energy occurs, i t i s more appropriate to designate this reaction energy Independent or at least weakly dependent. In the reaction B ( p , u ) B 10  +  1 1  both the exponential slopes and the g. s •  analyzing powers are strongly energy dependent. plot.  Figure 44 shows the | M |  2  VS t  As the incident proton energy increases, the exponential slope  decreases.  In the case of the analyzing powers, as the incident energy  increases there i s a large increase i n the lower t region (shown i n Figure 45).  Figure 46 i l l u s t r a t e s the continually decreasing slope with Increasing  70  F i g u r e 33. The forward angle data o f the B e ( p \ n ) are l e a s t squares f i t s t o the d a t a . 9  1 0  Be  3 > 3 7  M  e  y  . The s o l i d lines  71  CM  o  O \  r* or  > > ? OJ  OJ  OJ  <o  O o  2  2  in  CJ CJ  O  CJ H  CD  5  CJ H  ••  Q.  /  CL  i—  r—  / •L  /  \  /  o  \ .(N  \ in n n  o  111 > 1111  I I I I I I IIII 11 I I I I I I I I I I I I I I I 11 IM  o  CO  CD  o  o  o  F i g u r e 3 9 . The a n a l y z i n g power, as a f u n c t i o n o f t f o r »Bert.ii )»B«j . The l i n e s serve o n l y as a guide t o the eye. +  3 7  M e V  -  72  o  ID CM  Q  in CM  a CM  o  .cn  o  • CM CM a CM  o l-o CM  iiiiiii|iiiiiiiii|iiiiiiiiHiiiiiiii'|'"""" ro  CD  r-  CO  co  CM  ro  =  CL  >  o  <u  (/I  •—*  a .cn  73  • in  in -in  • in  CM  -in  3*° CO  O cn .-9  CM CC  > ><u >CO >CU 2  5  o m  rro  CVJ II  CVJ II  2  in  CVJ CVJ II  10 CVJ  in  11  Q. Q- Q. ca  \— f—  ••  <  1— •  cn  1111 i i i r  •01  1111 i i  "i—r  Figure A l . The forward angle data of the C ( p , n ) are least squares f i t s to the data. 1 2  +  1 3  C  9 > 5  M e V  .  The s o l i d  lines  74  CM  \  QJ  I—in  o  CVJ  0)  MeV  CVJ  CVJ CVJ 11  CVJ  CL  Q.  in It  CL  •«  r—  f-  MeV  MeV  or (0  CO  . A  II  r— •  \  /  \ \  l-rn  \ CO • CM  n 1111111 11111111111 p I Ti M 111 j 11111111111111111 1  CD  Q  *  O  o z  t  CO a  l  F i g u r e 42. The a n a l y z i n g powers as a f u n c t i o n of t f o r C ( p , u ) The l i n e s s e r v e o n l y as a guide t o the eye. 1 2  +  1 3  C  9 < 5  M e V  <  75  o L-CD t CM  O  r-in  CM  O •  f  CM  O  •CO CM  — >  a • CM CM  O  CM  O  l-o CM  I I I I I I | I I I  • 0)  I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I  o  CT LO  LO  o> CL O  ^  ""~ ^  ^  -gj  Figure 43. The incident proton energy dependence of the exponential slopes of the C(p,it ) C d i f f e r e n t i a l cross-section data as a function of t. 1 2  +  1 3  9 > 5  M g V  76  F i g u r e U . The forward angle data of the « B ( t . « > » B . . . . l e a s t squares f i t s t o the d a t a . +  g  The s o l i d l i n e s are  77 \  CD [-IP  >CU > >cu ><u  ^ t >  5 2 2 O U") O ID m CJ o CJ CJ CJ CJ  2 O  II Q.  II CL  \— H—  •  II CL  r-  +  cu O  n  - I  CL  h-  'O SIP  •  I'-Jl I  /  00  ten  I \ b  \  CD  t-fM  O  X  f-CM  Y  \ T TI  111 I M  M  11 I  | M  1  TiTll I n  I  11  CD  f)  O  o  o  ^ o  I  1  fl) l'  CD l'  F i g u r e 45. The a n a l y z i n g powers are shown f o r the r e a c t i o n The l i n e s serve o n l y as a guide t o the eye.  10  BC^,u  ) 3 u  78  o  I-in CM  O  CN  O  • ro CM  > a>  O  • CM CM  a I" CM  I—  o o  CM  O  1111111111  1 1 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII I  .cn  IT: =•  r-'  a> CL  O to  Figure  46. The  incident B  proton g. . s  LO  CD > CU  O  e n e r g y dependence  differential  of the e x p o n e n t i a l  cross-section  data  slopes  as a f u n c t i o n  of the of t .  79  T . 1 0  It i s unreasonable to approximate this curve by a constant value. 12 M v  B(p,Tt ) +  r e a c t  e  i o n displays much the same pattern as the  i0  B(p,u ) B  reaction ( i l l u s t r a t e d ln Figures 47 - 49). g. s.  12  C(p,u ) C  reaction g. s •  +  +  n  13  The  21  behaves in much the same way  Also the  (Figures 50 - 52).  Another important trend that must be pointed out, other than the fact that the exponential  slopes and analyzing powers are both either weakly or  strongly dependent on the incident proton energy, is the fact that their dependence is very similar.  The pattern of the changing exponential  of the increasing analyzing powers seem to be followed i n the last cases.  slope  and  three  The f i r s t three reactions as well seem to contain approximately the  same features. One  possible interpretation of this energy dependence in some reactions  and not others might be that of specific effects associated with single-particle f i n a l s t a t e s .  It seems plausible that f i n a l states not  22  described in terms of single p a r t i c l e ( Be , Be, , „ „, C „) ° g.s. •>. MeV .5 MeV could be candidates for an averaging effect and thus exhibit a "smoothed-out" 10  10  1 3  7  Q  37  energy dependence. nature (like dependence.  13  C  K  w  9  On the other hand, those states with good single-particle  ) could be expected to manifest a strong energy g.s.  The ^ B  state, a single-hole state, would be expected to act g.s.  like a single-particle state.  The  1 1  B  2  MeV  i 2  s t a t e  »  two-hole one-particle  a  state, however also shows this strong energy dependence.  Since p a r t i c l e s  (including holes) l i k e to couple to zero spin, i t would not be unreasonable to expect the  1 1  B  2  12 MeV  state to act as an effective single-pa?uicle state.  The other three cases ( B e 1 0  , Be l u  g g  3  3 7  M g V  ,  L 3  C  9  to couple to form an effective single-particle state.  5  M g V  )  are unlikely  In the case of  80  Figure 4 7 . The forward angle data of the Bfli,it ) B ^ are least squares f i t s to the data. +  1 1  2  1  2  M  g  V  .  The s o l i d lines  81  > >cu >CU >a» O  2 O  CM  CM  II  II  CO m  m  CM CM 11  O O  CM II  Q. a. a. Q. r— t - r - \m 4  •  F i g u r e 48. The a n a l y z i n g powers as a f u n c t i o n of t f o r "\K*.*+)»h.l2 The l i n e s serve o n l y as a guide to the eye.  MeV  82  O •O  CM  O  • LO CM  O .  CM  O  • CO CM  > cu  O  • CM CM  O  CM  O - o  CM  o 111111111111111111111111 m  LP  LP  to  LO  111111111111111 IP  LP  TT LO  CU  CL _o in  Figure 49. The incident proton energy dependence of the exponential slopes of the 2 . 1 2 MeV d i f f e r e n t i a l cross-section data as a function of t. B  83  F i g u r e 50. The forward angle data o f the 1 * c C p , « ) C . . • l e a s t squares f i t s t o the d a t a . +  13  g  9  The s o l i d  line  9  84  F i g u r e 51. The a n a l y z i n g powers as a f u n c t i o n of t f o r The l i n e s serve o n l y as a guide to the eye.  l 2  C(p\* ) +  1 3  C  g < g >  •  85  o  r  o  r F r r r  o CM  r  o  >  r  a  r  jlCM r Psj  r  r  o  CM  O •O CM  r r  o  71  • 111111111111111 M 111111111111111111111111111 r _ (0 f\| CO o  CM  a  cu  o  .  ;*,<->  cu  LP  Figure 52. The incident proton energy dependence of the exponential slopes of the * C(o.ii ) C d i f f e r e n t i a l cross-section data as a function of t. 2  +  1 3  86  10  Be  g.s.  , a two-hole state, the two holes probably do couple to zero spin,  This leaves the state as an effective closed s h e l l . which i s consistent with this picture. three-hole  The  1 0  Be  3  3 7  10  is a 0  Be  state g.s. state, mainly a  M e V  +  one-particle state, again cannot couple to form an e f f e c t i v e 9+ i s a y state,  11  single-particle state.  The two-particle one-hole of  i 0  C  9  5  therefore the two p a r t i c l e s can not possibly couple to zero spin.  The  3coupling would leave a ^- state, which is not consistent with the 9.5 state.  MeV  In fact the state i s best described by having one particle i n the  level and the other in the D  5/2  P  1 / 2  level.  To investigate this effect further, additional nuclei should be In particular, measurements of the  1 6  studied.  0(p,Tt )^ 0 and *°Ca(p,u ) Ca reactions +  7  t  +  l+1  leading to low lying states should exhibit this single-particle behaviour.  In  fact since a l l the reactions compared in this study basically involve only  the  p-shell, i t would be b e n e f i c i a l to study higher shells such as in and 5.4  1+0  Ca(p,TC ) Ca +  l+1  1 6  0(p .n*) - 0 1  7  to determine i f i t i s only a 3L=l e f f e c t .  D i f f r a c t i o n Peak Structure  The exponential  decrease of the d i f f e r e n t i a l cross-sections for forward  angles has some s i m i l a r i t y to the shape of the cross-section corresponding to " d i f f r a c t i v e " effects characterizing e l a s t i c scattering. shown  23  In fact i t can be  that the Fraunhofer approximation of d i f f r a c t i o n scattering and  exponential  drop off are equivalent  i n the small angle approximation.  an In the  Fraunhofer approximation the scattering amplitude i s : J (RksinG) x  f(9)  - 1 k 6 R  2 R  k  s  l  n  Q  (5-3)  87  where  k i s the incoming momentum, S i s the absorption coefficient of the scattering center, R i s the radius of the scattering center, Q i s the scattering angle,  and  i s the f i r s t - o r d e r Bessel function;  i.e.  J ( z ) = j _ ( y ) / 2 ! + ...  (5-4)  3  x  In the small angle approximation  F ( 9 )  (sin 0 = 0 ) to order 0 , 2  -M*L[I-  rn?]  (5  -5)  Now l e t us turn to an exponentially decreasing cross-section,  g.. A e ' ^ l  = |f( )|2 9  ( -6) 5  For small angles the momentum transfer i s approximately,  t = k sin20 = k 0 2  2  (5-7)  2  Therefore Eqn.(5-6) becomes,  = |f(0)|  2  exp(-bk 0 ) 2  2  (5-8)  88  hk ^ 2  or  2  f(9) = f(0) exp (-  where A = | f ( 0 ) | . 2  (5-9)  Expanding the exponential to order G  f(9) = f(0) [1- ^1  2  leaves  ]  (5-10)  By comparing Eqns. (5-5) and (5-10),  f(0) =  and R  2  1  k  ^  (5-11)  R 2  = 4b.  (5-12)  For e l a s t i c scattering from protons ( i . e . it~p,n; 'p,K~p,K p,pp and pp), the +  exponential slope b varies between 3 and 13 (GeV/c)  +  -2  (Ref. 23). This i s i n  reasonable agreement with the proton radius of approximately  1 x 10  - 1 3  cm.  The d i f f r a c t i v e pattern created by light scattered from a black disk Is the same as that created by a coherent light source the same size as the disk.  Thus i f one assumes that the peak structure seen i n pion production i s  due to d i f f r a c t i o n , one would expect source r a d i i of the order of 2 x 1 0  - 1 3  cm  for the nuclei discussed i n this study (assuming R follows * 21  <R > 2  where r  Q  i s equal to 1 x 1 0  3 to 13 (GeV/c) . -2  - 1 3  1/2  = r  Q  A  1 / 3  (5-13)  cm). In fact, the exponential slopes vary from  These values are very similar to the particle physics  89  values discussed above.  This may be an indication that the production and  rescattering effects are associated with individual nucleons rather than a collective effect.  Of course further studies, both theoretical and  experimental, must be made before any firm conclusions can be made.  5.5  u~ Production The (p,u ) data measured at IUCF  4  Figure 2).  showed preferred high-spin states (see  I t was interpreted that the preference was due to the creation of  stretched two-particle one-hole f i n a l states that allowed for angular momentum matching.  The Be(p,ic ) 9  1 0  C results presented here are not substantively  different from the early (p,it ) data There was no preferred f i n a l state.  obtained before the IUCF discoveries.  1,  The d i f f e r e n t i a l cross-sections are  relatively f l a t and small compared to the (p,u ) data. +  The analyzing powers,  as well, tend to show much less structure than for the (p,it ) reaction. +  does not c o n f l i c t with the IUCF findings. two-particle one-hole excited states.  1 0  C does not have any stretched  Thus our results do not contradict the  IUCF interpretation of enhanced cross-sections associated with stretched two-particle one-hole f i n a l states.  This  90  VI.  Status of the Theoretical  Model9  In the introduction i t was mentioned that at least four basic ingredient must be included i n any description of the pion production process:  the pion  emission operator, proton distortion, pion distortion, and nuclear wave functions.  In this chapter each of these four ingredients w i l l be examined i  the context of an example of each class of models.  6.1  R e l a t i v i s t i c One Nucleon Model  The most complete ONM calculation has been carried out by Cooper,  25  who  treated the distorted wave Born approximation (DWBA) calculation f u l l y relativistically.  For the pion emission operator both pseudoscalar and  pseudovector couplings were examined.  In the case of pseudoscalar, the  interaction Hamiltonian (see Figure 3) i s :  H  l n t  (x) = l / T g ^ Y  5  ^(x)  (6-1)  where g^ i s the TtN coupling constant and <t>(x) Is the pion f i e l d operator. n  For pseudovector coupling the Y 5 i s replaced by ^  ^ — where N  P  - - I h-y^  0 0  ,  (6-2)  TJL^ i s the mass of a nucleon and b^ ^ i s a derivative acting on the pion wave 1  function.  Cooper claims this derivative coupling of the pseudovector gives  better agreement with the data than pseudoscalar coupling.  91  The proton distorted wave functions were generated by an o p t i c a l potential of the form: U(r) = U (r) + U (r) + U , ' v s coul  (6-3)  v  where U  , i s the Coulomb potential characterizing a uniform charge COUl  a  V  distribution.  b  The vector potential was of Woods-Saxon form: V  V "  •  r^R—-  iW  T T T  +  1+ e x p f - j ^ vl  1+ exp  •  <"> 6  4  { - ^ v2  and the scalar potential was similar: i.e. V  V  r  )  1 +  where  iW  3  =  r-R exp(^-)  S  r-R  1+ exp  ( 6  _  5 )  V s and W's are the strength of the real and imaginary potentials, respectively, R's are the r a d i a l parameters, and a's are the diffuseness  parameters.  In order to make the calculation simpler, only vector and scalar potentials were used.  This was shown '' to be a good approximation as long as the nucleus 2  has doubly closed shells (both isospln and spin equal to zero).  The twelve  parameters were f i t to proton e l a s t i c scattering data for the appropriate nucleus and incident energy. For the pion d i s t o r t i o n , the optical potential of Striker, McManus and Carr  27  was used.  This potential not only f i t s the pion e l a s t i c scattering i n  the energy range of 0 to 50 MeV, but also provides f i t s to the level s h i f t s and widths of the pionlc atoms. The single p a r t i c l e nuclear bound state wave functions were obtained In terms of Woods-Saxon potentials made up of only vector and scalar components,  92  v  v r-R  U (r) = v  (6-6)  V 1+ exp(  s r-R a  (6-7)  s  s  Implying again r e s t r i c t i o n to isospin and spin zero nucleus.  The geometric  parameters i n these potentials were defined by f i t t i n g proton-nucleus scattering data.  The strengths, V  g  elastic  and V^, were f i t to the binding energy  using the mean f i e l d theory of nuclear matter.  6.2.  Two Nucleon Model  One of the TNM's involving the fewest approximations of I q b a l .  26  i s the calculation  At this point i n time only the four diagrams of Figure 4 are  included, those which involve the A-resonant pion production amplitude.  (T = 3/2, J = 3/2) part of the  For the incident proton energies i n the range 200  to 400 MeV, the A-resonant amplitude i s believed to be the dominant term. In this model the delta-isobar propagates via delta-nucleus Interactions described by a l o c a l nuclear density dependent potential.  In addition, Iqbal  concentrated on two-particle one-hole f i n a l states which cannot be easily reached via ONM's. In this model the pion emission operators are the s t a t i c form of the itNN and nAN interaction Hamiltonian obtained by the Foldy-Wouthuysen non-relatlvistic reduction of the r e l a t i v i s t i c coupling.  That i s :  30  pseudoscalar or pseudovector  93  " <W  \ m  o ' ^  (6-8)  x • •  and  * H  where:  and ra  is  f  are  rcAN  =  < rc '\> f  • \  S  • •  T  the TCN and itA coupling  <"> 6  constants,  respectively,  the pion mass,  i s the d e r i v a t i v e operator a c t i n g on the p i o n wave  a  and  the and  x are  T are  s p i n and  function  the matrix r e p r e s e n t a t i o n s of the operators connecting  two-component spinors  S and  9  i n s p i n and  i s o s p i n spaces,  respectively,  the matrix r e p r e s e n t a t i o n s of the operators connecting  i s o s p i n 3/2  s t a t e s with spin and  i s o s p i n 1/2  states,  respectively.  Note that one  uN  i n t h i s model there are  (see  The  Figure  three v e r t i c e s i n each diagram, two  TCA  and  4).  proton d i s t o r t i o n s were generated by an o p t i c a l p o t e n t i a l of  the  form:  " coul  U ( r )  where  V  v c  o  u  ^  V l  +  1  W  2 2 f  2 ~ 7 <3 V  df  ST  3 +  and  W  2  W  *  dr-  strengths f o r the s p i n - o r b i t part are  ) (  3  and  W  4  0 )  ( 6  -  1 0  >  a uniform charge  represent the strengths of the  of the c e n t r a l p o t e n t i a l , whereas V  form f a c t o r s and  . * '  d f  1  i s the Coulomb p o t e n t i a l c h a r a c t e r i z i n g  distribution. parts  +  are  r e a l and  the  corresponding  of the o p t i c a l p o t e n t i a l .  taken to be of the Woods-Saxon form,  imaginary  The  f's are  the  94  i  f  (  r  l +  The  twelve parameters  s i m i l a r way The  3 1  as f o r t h e ONM  of the  W  < " > 6 n  exp(^-i)  s c a t t e r i n g data i n a  case. i n c o r p o r a t e d by  t h e use  of a m o d i f i e d  Kisslinger  form:  U(r) = - Z b k  p ( r ) + Z t ^ V • p ( r ) • V - Z/2  2  Q  where:  "  i n v o l v e d were f i t t o p r o t o n e l a s t i c  p i o n d i s t o r t i o n was  potential  )  Z i s the atomic  number o f t h e  M i s the n u c l e o n  [ (T  + m)/M]b V p(r) 1  2  (6-12)  nucleus,  mass,  m i s t h e p i o n mass, k i s t h e wave number o f t h e p i o n i n t h e p i o n - n u c l e u s T^  i s the p i o n k i n e t i c  centre-of-mass,  energy,  p ( r ) i s the n u c l e a r d e n s i t y , and  b  Q  and  b^  p i o n - n u c l e o n phase  are the complex parmeters  related  t h e bound s t a t e n u c l e o n .  c r o s s - s e c t i o n s were f o u n d f u n c t i o n assumed. due  P-wave  shifts.  I q b a l examined both harmonic o s c i l l a t o r for  t o t h e S- and  As  The  t o be  t o t h e momentum t r a n s f e r  a n g u l a r d i s t r i b u t i o n s of the  insensitive  p o i n t e d out  and Woods-Saxon wave f u n c t i o n s differential  t o t h e t y p e o f bound s t a t e wave  i n the next  section,  s h a r i n g i n t h e TNM  models.  this  insensitivity  is  95  6.3  Comparison of Models  The major difference between the ONM of Cooper and other ONM's i s the inclusion by Cooper of r e l a t i v i s t i c effects.  A l l other ONM's use a  non-relativistlc reduction of the Interaction Hamiltonian such as the that of Foldy-Wouthuysen . 30  g H  int  = -  Zxa.  (a»V f <{>) +  2  ) ( a » P f<t>) + higher order terms '•m +1  (6-13)  This reduction creates an ambiguity by introducing an undefined parameter X. By an appropriate choice of \, one can approximate H^ f i r s t term (static form).  i n terms of only the  Alternatively, one can include the second term to  get the Galilean invariant form. desired.  nt  It has been shown  25  Even higher order terms could be retained i f  that conventional s t a t i c forms of H, int  do not  give the same results as the r e l a t i v i s t i c one. The difference between the TNM of Iqbal and the other TNM's i s the treatment of the A-resonance.  Although most TNM's involve the A-resonance  only Iqbal's allows the A to propagate via a delta-nucleus Interaction of more r e a l i s t i c form than the usual delta function.  In his TNM, Iqbal permits the  delta to propagate through a l o c a l nuclear density dependent potential. In comparing the two types of models, one Is struck by how much easier i t i s to use the ONM's.  This of course i s the major advantage of such models  over the more r e a l i s t i c TNM's and is probably the main reason why the r e l a t i v i s t i c formulation has only been incorporated i n the ONM case.  The  major disadvantages of the ONM as compared to TNM is Its treatment of u  96  production, A-resonant states.  Although  Intermediate e f f e c t s , and t w o - p a r t i c l e one-hole  final  a l l three of these e f f e c t s can be t r e a t e d c o n s i s t e n t l y In  terms of the f r e e N-N i n t e r a c t i o n l n the TNM, they can only be Incorporated i n terms of complicated m u l t i - s t e p processes in f;h;i c o n t e x t of the ONM. Another advantage claimed by the ONM proponents the d i s t o r t i n g p o t e n t i a l s . incoming  I n the case of the ONM,  i n v o l v e s the i n c l u s i o n of l t Is c l e a r that the  proton should be a f f e c t e d by a p o t e n t i a l a s c r i b i n g the f u l l  nucleus.  In  the TNM case where only one of the nuclear nucleons  In  the i n t e r a c t i o n o p e r a t o r , i t i s not c l e a r that a p o t e n t i a l of the f u l l  nucleus should be used (A-l)  i s involved e x p l i c i t l y  f o r d i s t o r t i n g the wave f u n c t i o n s .  What i s needed i s a  nuclear p o t e n t i a l i n the presence of a s p e c t a t o r nucleon.  l i k e t h i s cannot  be e a s i l y parameterized  A potential  i n terms of experimental data.  Thus,  a l l TNM's i n v o l v e a d i s t o r t i n g p o t e n t i a l d e s c r i b i n g the whole nucleus. However, TNM's are g e n e r a l l y much l e s s s e n s i t i v e to the d i s t o r t i o n than are ONM's, so t h i s e f f e c t may be s m a l l .  effects  A l s o , even though the o p t i c a l  p o t e n t i a l s f i t e l a s t i c s c a t t e r i n g data, i t does not mean they are a p p r o p r i a t e as d i s t o r t i n g p o t e n t i a l s .  The e l a s t i c s c a t t e r i n g i s an o n - s h e l l process  whereas that i n v o l v e d i n p i o n p r o d u c t i o n models i s very much o f f - s h e l l . is  There  no unambiguous way of determining what e f f e c t s should be Incorporated i n  the o p t i c a l p o t e n t i a l when going so f a r o f f - s h e l l . Another advantage of the TNM i s i t s i n s e n s l t i v l t y to bound s t a t e wave functions. i s concerned  Since the momentum t r a n s f e r i s shared by three wave f u n c t i o n s , one with p a r t i c l e wave f u n c t i o n s i n a r e g i o n of momentum t r a n s f e r  where the harmonic o s c i l l a t o r and Woods-Saxon wave f u n c t i o n s have very form and are reasonably w e l l known. little  similar  For the ONM, on the other hand, very  i s known of the wave f u n c t i o n s at the much l a r g e r momentum t r a n s f e r s  97  involved and i t has been shown  1  that the ONM i s very sensitive to the nature  of the bound state wave functions used i n the calculation.  6.4  Comparisons of Models with  Experiment  In Figures 53 and 54 Cooper's r e l a t i v i s t i c ONM calculation for 1 2  C(p,ir ) C +  1 3  g  g  at T  p  - 200 MeV i s compared with experiment.  Although the  shape of the theoretical d i f f e r e n t i a l cross-section i s q u a l i t a t i v e l y similar to experiment, the calculation i s low by an order of magnitude.  Since this  model does not include A-resonance effects an underestimate i s not unreasonable.  The analyzing powers resulting from this model also have the  right qualitative shape at forward angles, but predicts a change of sign for angles greater than about 90°, an effect which i s not observed experimentally. Certainly, given the lack of quantitative agreement, this model has severe limitations. Though the ONM was designed for transitions which start with a closed shell nucleus and end with a single-particle f i n a l state, we also investigated Its a p p l i c a b i l i t y to describe  1 0  B(p,u ) B  . B was approximated as a g. s. Since ^ B i s a single-hole state, i t g. s • +  closed shell even though i t i s not.  11  10  was treated as a single-particle f i n a l state i n Cooper's code. also encountered i n attempting to treat the distortions.  Problems were  Optical potentials  for the distortions are not available for boron (no e l a s t i c scattering data at this energy e x i s t s ) .  An approximate resolution of this problem involved use  of potentials appropriate to C ( p , u ) C 1 2  +  1 3  (scaled appropriately).  The  98  Figure  53. Cooper's r e l a t i v i s t i c ONM comparison to c r o s s - s e c t i o n s at T_ - 200 MeV.  1 2  C(p\Tt ) +  I 3  C  g # s >  differential  99  100  results are shown i n Figures 55 and 56.  Again the shape of the d i f f e r e n t i a l  cross-section curve i s quantitatively reasonable but the normalization i s not. The analyzing powers, however, agree remarkably well with experiment. Cooper's model appears to provide reasonable qualitative agreement on da/dQ and A^g for single-particle f i n a l states.  More comparisons are needed.  Next, Figure 57 compares the predictions of Iqbal's TNM with experiment. Comparisons of the C ( p , u ) C 1 2  +  1 3  g  5  MeV results at two incident energies, 200  and 250 MeV, are compared with Iqbal's predictions.  The overall shape again  is qualitatively similar to experiment, but shows too much incident energy dependence.  The prediction underestimates experiment but not to the same  extent as in the ONM case.  Again, an underestimate i s not unexpected since  only the A-resonance diagrams are included In this model.  At these energies  in addition to the A-resonance processes, non-resonant processes are expected to be important and these contributions should help to increase the cross-section.  Since analyzing  powers provide an additional stringent test  of theory, i t would be most interesting to have such a comparison. however, theoretical predictions are not available.  As yet,  101  Figure 55. Cooper's r e l a t i v i s t i c ONM comparison to B ( p , n c r o s s - s e c t i o n s at T_ » 200 MeV. 1 0  )  1 1  B  g > s >  differential  102  (\J  O  ^  (N  ID  o z  <  Figure 56. Cooper's r e l a t i v i s t i c ONM comparison to ° (P»'t ) g.g. analyzing powers at T = 200. MeV. 1  D  B  +  llB  103  104  VII.  Conclusion  The results of two major sets of experiments have been presented.  The  angular distributions of both the d i f f e r e n t i a l cross-section and the analyzing power have been measured for the following reactions:  1 0 B  9Be  ^ > +  l l B  g.s.,2.1  <P»* >  9 B e  +  ^~>  1 0 C  10Be  2  g.s.,3.37  g.s.,3.3*  over a range of incident proton energies from 200 to 260 MeV.  These  measurements extend the results previously obtained at Uppsala, IUCF and TRIUMF into a new energy region, one i n which the A-isobar Is expected a major role.  The measurements at T^= 200 MeV are In good agreement  to play with the  previous results, indicating consistency i n the absolute normalizations of a l l the results. These studies of nuclear pion production i n this energy region have helped to indicate new trends i n the data. (1)  S t a t i s t i c a l weighting of high-spin f i n a l states. It was demonstrated that when the s t a t i s t i c a l weighting of the f i n a l  spin i s removed from the matrix element, there i s much less dependence of the the resulting |m|  2  (vs. t) on f i n a l spin then i n the case for | M | . 2  This spin  independence implies that though the specific details of the reaction may depend on the spin state of the f i n a l nucleus, the dependence of the magnitude of the d i f f e r e n t i a l cross-section on the spin i s basically that of  105  simple s t a t i s t i c a l weighting.  Thus there seems to be no preferred f i n a l spin  state (at least for the reactions studied i n this work).  There i s no evidence  (in this work) that better momentum matching for higher spin states has  any  effect on the cross-section. (2)  Incident proton energy dependence. The energy dependence of both the exponential slope of the  t-dependence of the d i f f e r e n t i a l cross-sections at forward angles and analyzing powers are correlated.  the  It appears that the exponential slopes  and  analyzing powers are both either weakly or strongly dependent on the incident proton energy.  In addition, we have evidence that reactions involving f i n a l  states that have a strong single-particle nature (Including single-hole states) show a strong and similar energy dependence;  whereas those states  which are not strongly single-particle i n nature show only a weak energy dependence. (3)  D i f f r a c t l v e Peak Structure. A l l the d i f f e r e n t i a l cross-sections were strongly peak i n the  forward direction, with a shape very similar to that of d i f f r a c t l v e scattering.  Values of the exponential slopes of these peaks when plotted as a  function of t were shown to be similar to those expected for p r o j e c t i l e scattering from a single nucleon rather than the whole nucleus.  Is this  providing us an important clue Into the role of pion rescatterlng in such reactions? The status of the current level of theoretical understanding of pion production was  also discussed.  To date, no theoretical approach provides a  reasonable description of the experimental data.  Some specific  recommendations that follow from this thesis are: (1)  Modification to Iqbal's TNM  to handle single-particle f i n a l states.  It appears that f i n a l states which are not single-particle i n nature  106  have a "smoothed-out" energy dependence (see Figures 35-43).  Thus more  Information may be gained by comparing to single-particle f i n a l states where a strong dependence on incident proton energy has been seen (see Figures 44-52). (2)  The inclusion of non-resonant  diagrams into the TNM.  There i s reason to believe that non-resonant i n this energy region.  The comparison with experiment  significant contribution from non-resonant (3)  diagrams are important certainly implies a  diagrams.  The TNM code should be made r e l a t i v i s t i c . It i s known  25  that r e l a t i v i s t i c effects are important and therefore  should be included. Creating a r e l a t i v i s t i c formulation of a TNM code would be a major undertaking.  Therefore l n the meantime (In order to make the  computation easier), at least a better approximation to the r e l a t i v i s t i c interaction Hamiltonian should be used.  The s t a t i c form now used i s clearly  not a good approximation. (4)  Single nucleon pion rescatterlng. Since the forward angle structure i s similar to that of d i f f r a c t l v e  peaks associated with interaction regions of nucleon size, i t may not be appropriate to treat pion rescatterlng as an average effect described by an optical potential for the whole nucleus.  More e x p l i c i t calculations should be  Initiated to investigate pion rescatterlng effects for this process. As well, specific experimental recommendations also follow from this work.  (1)  16  0 ( p , r c ) 0 and ' C a < p , i i V C a +  17  t 0  1  It i s important to test the observation that single-particle f i n a l states are associated with a strong energy dependence for the reaction. For both  1 7  0 and ^ C a , a l l the low-lying excited states as well as the ground  states are single-particle i n nature.  A l l the single-particle f i n a l states  107  studied i n this work, have been those of the p-shell only.  Thus the study of  higher shells such as those describing single-particle states in 0 17  and  4 1  Ca  are needed to determine i f the features observed are truly general singleparticle effects or whether they depend instead on details of the spin structure of the reaction. (2)  Increased Incident proton energy. The next l o g i c a l step would be to increase the incident proton  energy.  Does the s t a t i s t i c a l weighting of high-spin f i n a l states continue to  dominate the spin dependence or does "angular momentum matching" become important at higher energies? change?  Does the dependence on incident proton energy  How w i l l the A-resonance, which i s known to play an important role i n  pion production, affect this energy dependence?  I would suggest that both  1 0  B ( p , u ) B and B e ( p , i t ) B e reactions as well as  1+0  Ca(p,u )  +  n  +  1+1  9  +  10  1 6  0 ( p , i t ) 0 and +  1 7  Ca reactions should be studied at a number of energies throughout  the A-resonant region to at least 400 MeV Measurements at 50 MeV  incident proton energy.  intervals would test whether the A-resonant effects  lead to an energy dependence at a l l similar to that seen In the pp •*• d n reaction.  +  108  References  1.  B. Hoistad, Adv. i n Nucl. Phys., Vol. LI, ed. J.W. (Plenum, New York, 1979) p. 135.  Negele and E. Vogt  2.  D.F. Measday and G.A. M i l l e r , Ann. Rev. Nucl. Part. S c i . 2_9, ed. J.D. Jackson, H.E. Gove and R.F. Schwitters (Annual Reviews Inc., California, 1979) p. 121.  3.  H.W. Fearing, Prog, i n Part, and Nucl. Phys., Vol. _7, ed. D. Wilkinson (Pergamon, New York, 1981) p. 113.  4.  S.E. Vigdor, T.G. Throwe, M.C. Green, W.W. Jacobs, R.D. Bent, J . J . Kehayias, W.K. P i t t s , T.E. Ward, Phys. Rev. Lett. 49, 1314 (1982).  5.  W. Ziegler, M.Sc. Thesis, University of B r i t i s h Columbia (1983).  6.  M.K. Craddock, K.L. Erdman and J.T. Sample, Nature 270, 671 (1977).  7.  E.L. Mathie, Ph.D. Thesis, University of B r i t i s h Columbia (1980).  8.  R.A. Arndt and L.D. Roper, "SAID", Virginia Polytechnic Institute and State University, V i r g i n i a , CAP3-80-3(rev.) (1982).  9.  D.V. Bugg, J.A. Edgington, C. Amsler, R.C. Brown, C.J. Oram, K. Shakarchi, N.M. Stewart, G.A. Ludgate, A.S. Clough, D. Axen, S. Jaccard, and J . Vavra, J . Phys. G4, 1025 (1978).  10. P.L. Walden, private communication (1984). 11. C P . Browne and W.W.  Buechner, Rev. S c i . Instr. 2_7, 899 (1959).  12. D.M. Lee, S.E. Sobottka, and H.A. Thiessen, Nucl. Instrum. Methods 120, 153 (1974). 13. G. Fremont, "Magnetometer", model CERN 9298 (1978). 14. G. Jones, Workshop on Pion Production and Absorption i n Nuclei, ed. R.D. Bent, AIPCP #79 (AIP, New York, 1982), p. 15. 15. G.L. Giles, Ph.D. Thesis, University of B r i t i s h Columbia (1985). 16. Proc. 3rd Int. Symp. on Polarization Phenomena i n Nuclear Reactions, Madison, Wisconsin, 1970, ed. H.H. Barschall and W. Haeberli (Univ. of Wisconsin, Madison, 1971). 17. P.H. P i l e , R.D. Bent, R.E. Pollock, P.T. Debevec, R.E. Marrs, M.C. T.P. Sjoreen, and F. Soga, Phys. Rev. Lett. 42, 1461 (1979).  Green,  18. T.P. Sjoreen, M.C. Green, W.W. Jacobs, R.E. Pollock, F. Soga, R.D. Bent, and T.E. Ward, Phys. Rev. Lett. 45, 1769 (1980).  109  19. P. Couvert, Workshop Studying Nuclei with Medium Energy Protons, ed. J.M. Greben (TRIUMF internal report TR1-83-3), 287 (1983). 20. P. Couvert, Intersections Between Particle and Nuclear Physics, ed. R.E. Mischke, AIPCP #123 (AIP, New York, 1984) p. 689. 21. G.J. Lolos, E.G. Auld, W.R. Falk, G.L. Giles, G. Jones, B.J. McParland, R.B. Taylor, and W. Ziegler, Phys. Rev. C30, 574 (1984). 22. M.J. Iqbal, private communication  (1984).  23. M.L. 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