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Investigation of electric quadrupole strength in ¹³N using the ¹²C(p,Ύ₀)¹³N reaction Helmer, Richard Lloyd 1977

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INVESTIGATION OF ELECTRIC „QUADRUPOLE STRENGTH IN i * N USING THE i 2 C ( p Y ) 1 3 N REACTION #  0  by RICHARD LLOYD HELM EH B . A . S c , U n i v e r s i t y of B r i t i s h Columbia, 1966 H, A. Sc., O n i v e r s i t y o f B r i t i s h Columbia, 1969  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF r  DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES {Department of Physics)  We accept t h i s t h e s i s as conforming to the r e g u i r e d standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1977 © R i c h a r d Lloyd  Helmer, 1977  In  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  f u r t h e r agree  for  scholarly  by h i s of  this  written  thesis at  the U n i v e r s i t y  make  that  it  purposes  for  may  financial  is  of  British  2075 Wesbrook Place Vancouver, Canada V6T 1W5  W  f  t  1111  British  by  for  gain  Columbia  shall  the  that  not  requirements  Columbia,  I  agree  r e f e r e n c e and copying  t h e Head o f  understood  of  The U n i v e r s i t y  of  for extensive  be g r a n t e d  It  fulfilment of  available  permission.  Department  Date  freely  permission  representatives. thesis  in p a r t i a l  of  or  that  study.  this  thesis  my D e p a r t m e n t  copying  for  or  publication  be a l l o w e d w i t h o u t  my  ABSTRACT  The  E2 c r o s s s e c t i o n f o r  been measured from  10 MeV  the  to 17 Me?  l 2  C{p,Y )i3H  A  total  E2 capture  was  energy-weighted (10.3  ± 4.0)%.  capture  model  c r o s s s e c t i o n s were of the order of 0.2  sum  rule  Calculations provide  depleted based  capture.  The  in  this a  ybarns the  E2  energy range i s semi-direct  a good d e s c r i p t i o n of the  experimental  E2  on  amount of  The  direct  r e s u l t s by i n c l u d i n g only d i r e c t c o l l e c t i v e E1  plastic  used t o d e t e c t the gamma r a y s .  and no resonance e f f e c t s were observed.  by  beams of p o l a r i z e d  10 i n . , * x 10 i n , N a l ( T l ) d e t e c t o r with a  anti-coincidence shield  has  in the l a b o r a t o r y system  bombarding an e n r i c h e d carbon-12 t a r g e t with protons.  reaction  Q  capture  and  direct  plus  iii  TABLE OF CONTENTS  ABSTRACT .......................  i i  TABLE OF CONTENTS  i i i  LIST OF TABLES  V  LIST OF FIGURES  vi  ACKNOWLEDGEMENTS  CHAPTER I  .viii  Introduction  ,  1  1.1  General I n t r o d u c t i o n  1  1.2  Review  9  1.3  Present Work  CHAPTER I I  of P r e v i o u s Work .....  10  Experimental Apparatus and Procedure ......... 13  2.1  General Experimental Arrangement  2.2  Gamma Ray Spectrometer  2.3  Gamma Spectrometer E l e c t r o n i c s ...............18  2.4  P a r t i c l e Detection  24  2.5  Targets  27  2.6  Current Integration  29  2.7  Polarization  30  2.8  Data Accumulation ............................ 30  Measurements  13 ...17  CHAPTER I I I Data A n a l y s i s and R e s u l t s .................... 33 3.1  Gamma Ray Spectra A n a l y s i s ................... 33  3.2  P a r t i c l e Spectra Analysis  3.3  R e s u l t s from the P a r t i c l e A n a l y s i s  .......37 38  .3.4  R e s u l t s from t h e Gamma Ray A n a l y s i s  3.5  Beam P o l a r i z a t i o n Measurements ............... 46  3.6  Angular D i s t r i b u t i o n F u n c t i o n s ................ 48  3.7  Results o f t h e Angular D i s t r i b u t i o n F i t s  3.8  E x t r a c t i o n of the T-matrix Elements  3.9  Determination o f the Cross S e c t i o n s .......... 73  CHAPTER IV  The  40  54  .......... 58  DSD Model, Sum Rules and Comparison with  Other Experiments  77  4.1  The DSD Model  78  4.2  The O p t i c a l Model and C a l c u l a t i o n o f the Have Functions  .86  4.3  C a l c u l a t i o n o f t h e D i r e c t Semi-Direct c a p t u r e  4.4  Sum Rules  4.5  Comparison with Other Work  CHAPTER V  Summary  96  ...........................102 ......103  and C o n c l u s i o n s  .....109  BIBLIOGRAPHY  .... ...113  APPENDIX A  The Energy Weighted Sum Rule  APPENDIX B  T-matrix  APPENDIX C  P o l a r i z e d Proton Beam Asymmetries  Elements  .,  ...121  ...................124  ..128  V  LIST OF TABLES  Table I I I - 1  Summary o f Beam P o l a r i z a t i o n  Table I I I - 2  R e l a t i o n between the Angular D i s t r i b u t i o n Coefficients  Table I I I - 3 Table I I I - 4  Solid l 2  C (p, Y ) 13 Q  N  Table I I I - 7  .................................  54  I 2  C(p,^ )  1 3  Q  N  57  Angular  - S o l u t i o n I ...................61  T-matrix Element F i t s t o Distributions  53  Angular D i s t r i b u t i o n  T-matrix Element F i t s t o Distributions  Table I I I - 6  and t h e R e a c t i o n Matrix Elements  Angle C o r r e c t i o n F a c t o r s ...............  Coefficients Table I I I - 5  Measurements ..,.,47  1 2  C(p,Y ) Q  1 3  H Angular  - Solution II  .,..,62  E1 Amplitude Ratio s/d and Parameters Related t o E2 Capture f o r S o l u t i o n I .................  65  Table I I I - 8  second S o l u t i o n s t o t h e T-matrix Element F i t s  69  Table IV-1  Optical  90  Table IV-2  GDR Parameters Dsed t o Reproduce the T o t a l  Model Parameters .....................  Cross S e c t i o n Using the WSS P o t e n t i a l Table B-1  ........100  R e l a t i o n between the Angular D i s t r i b u t i o n Coefficients  and the Reduced T-matrix  Elements T a b l e B-2  125  R e l a t i o n s between the Angular D i s t r i b u t i o n Coefficients  and t h e Reaction Amplitudes .,,..126  vi  LIST OF FIGURES  F i g u r e II-1  Schematic view of the beam l i n e  Figure I I - 2  Block  ............. 14  diagram o f the gamma spectrometer  e l e c t r o n i c s ................................. 20 Figure I I - 3  C{p,  i a  y Q  )  1 3  N  gamma ray spectrum at  E+ = 11.2 HeV  23  p  F i g u r e II-4  Block  diagram of the p a r t i c l e  electronics  detector  .................................26  Figure I I - 5  Particle  spectrum at E£ = 11.2 MeV .......... 28  Figure III-1  Reject  Figure I I I - 2  P o l a r i z e d proton  t o accept  r a t i o s f o r E^ = 10 MeV ..... 36 beam asymmetries a t  E-* = 15 MeV and 16 MeV  39  P  Figure  III-3  The complete angular  distribution  measurements a t E+ = 10 MeV Figure III-4  Figure III-5  Distribution  of x  2  43  f o r the C ( p ^ ) l 2  r  0  l 3  N  y i e l d s and asymmetries ...................... 45 C ( p , Y ) i 3 j * normalized d i f f e r e n t i a l c r o s s  1 2  0  sections Figure I I I - 6  12  C-tp,  55 l 3  N angular  distributions  asymmetries Figure III-7 Figure III-8  1  2  C{p,Y )i3fj 0  f o r the ............56  normalized  angular  distribution  coefficients  59  E1 amplitudes and r e l a t i v e phase  63  vii  F i g u r e 111-9  The amplitude r a t i o and phases r e l a t e d t o E2 capture  Figure I I I - 1 0  66  P r o j e c t i o n o f the multidimensional  x ~surface 2  onto the E 2 s t r e n g t h a x i s F i g u r e I I I - 1 1 The normalized  a  and  . 6 8 angular  distribution 71  coefficients F i g u r e 'III-12 The E 2 c r o s s s e c t i o n s Figure I V - 1  Schematic r e p r e s e n t a t i o n o f d i r e c t and semidirect  F i g u r e IV-2  . 7 4  1 2  80  processes  C(P,p ) Q  1 2  C  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  a n a l y z i n g power at E + = 1 6 . 9 6 4 MeV . . . . . . . . . . F i g u r e IV-3  Comparison of the o p t i c a l model analyses some experimental  93  with  p a r t i a l reaction cross  s e c t i o n s .................................... 94 F i g u r e IV-4  The r e a l  p a r t s o f the r a d i a l wave f u n c t i o n s  c a l c u l a t e d with t h e NEW p o t e n t i a l Figure I V - 5  F i t s to the  1 2  C ( p , Y ) i 3 N t o t a l cross section 0  97 98  viii  ACKNOWLEDGEMENTS  I  owe  Hasinoff,  sincere  appreciation  to my s u p e r v i s o r . Dr.  f o r the j u d i c i o u s prodding which has  the  end of my graduate student days.  was  made necessary by my many d r i n k i n g  Hotel,  sailing  friends  at  brought  Refocussing friends  of  British  me  to  my a t t e n t i o n  at  the  Cecil  the K i t s i l a n o Yacht Club, hockey  f r i e n d s a t the Winter Sports Centre and school f r i e n d s Universities  Mike  Columbia  and  Washington  at t h e who a l l  d i s t r a c t e d me from my s t u d i e s . I  wish  Laboratory  to of  f r i e n d l y and  thank  the  the  staff  University  helpful  at  of  assistance  the  Nuclear  Washington  during  the  Physics  for their course  very  o f the  experimental measurements. I and  wish e s p e c i a l l y t o thank John B u s s o l e t t i , Katsu Ebisawa  Dr.  discussions  Kurt  Snover  about  up with me during  f o r many  helpful  t h i s work, and Dr.  and  stimulationg  P h i l Dickey f o r p u t t i n g  my stays i n S e a t t l e .  I would have thanked Lee at t h i s point f o r a l l the and  encouragement  she  has  given  me  and f o r the tremendous  amount of work she d i d i n h e l p i n g to prepare t h i s she  p r e f e r r e d cash  instead!  love  t h e s i s , but  1  Chapter I  INTRO DDCTIQN  1.1 General  Introduction  Photonuclear r e a c t i o n s are obtain The  some  of  simplicity  which  mediates  perturbation describe  because  the  interaction  electromagnetic  can  effects  be  of  operator  acguired  relatively  the  with  the Giant first  be  to  studied  exist  c h a r a c t e r i z e d by t h r e e exists  strength  operator  confidence  to  In a d d i t i o n , t h e  understood,  so  that  the  of  the  f o r c e need be assumed. types  of  photonuclear  has been the e x c i t a t i o n and decay o f  in  basic  This  the  resonance,  late  properties  forties  which  was  (BA 47), i s  (FU 73) .  First, i t  i n a l l n u c l e i a t an e x c i t a t i o n energy which v a r i e s from  approximately 80A~*A MeV f o r t h e heavy n u c l e i t o .5OA-0  to  i s d i r e c t , i n the sense that no a p r i o r i  Dipole Resonance (GDR).  shown  some  interaction.  i s well  One of the most f r u i t f u l to  way  i s r e l a t i v e l y weak, so t h a t  used  the  electromagnetic  knowledge o f the l e s s w e l l known nuclear  reactions  simple  d e t a i l s of the s t r u c t u r e o f the nucleus.  arises  theory  the  information  the  a  HeV in  f o r the that  lighter  i t exhausts  c l a s s i c a l d i p o l e sum r u l e .  nuclei. slightly  approximately  Second, in  i t has great  excess  T h i s sum r u l e was f i r s t  of the  derived f o r  2  nuclei  by Levinger and Bethe {LE 50), and g i v e s a c o n s e r v a t i o n  law f o r the i n t e g r a t e d a b s o r p t i o n exchange  and  velocity  cross  section.  Neglecting  dependent f o r c e s , the sum r u l e i s given  by MeV-barns  1-1  F i n a l l y , the d i p o l e s t r e n g t h i s concentrated narrow  energy  region,  two  properties  a  relatively  the width v a r y i n g from about 3 MeV f o r  c l o s e d s h e l l n u c l e i t o about last  in  9 MeV f o r deformed n u c l e i .  combine  to  give  These  the GDR i t s resonance  shape. The GDR i s viewed oscillation  of  in  the  language,  incoming  {GO 48 and  t h e resonance  this  41A / -1  3  MeV)  which  ST 50).  bulk  In  shell  momentum  i s too  low.  Brown  and  behind  process must be s t r o n g l y c o r r e l a t e d i n angle with the  and p a r i t y  GDR i s c o n s t r u c t e d particle-hole  applicability refinements  of 1 . _  from  states.  descriptions  experiment  o f t h e GDH should be  (BR 59a) pointed out that the hole that i s l e f t  e x c i t e d nucleon because t h e two  model  a  i s formed by the a c t i o n of the  T h i s would imply t h a t the energy  {or about  Bosterli in  as  gamma r a i s i n g a nucleon t o the next higher major s h e l l  {WI 56). 1 -hoi  model  a l l t h e protons i n the nucleus moving a g a i n s t  a l l t h e neutrons i n t h e nucleus model  collective  of  (SP 69), to  improve  a r e coupled  to  an  angular  These authors then showed t h a t t h e  a  coherent  Both  the  the  GDR  and the  superposition  collective have  of  these  model and s h e l l  their  limits  of  both have had many e x t e n s i o n s and agreement  between  (see, f o r example, DA 65 and SP 69).  theory  and  3  In  the past few years, evidence o f a resonance other than  the GDR has come to l i g h t . electron  scattering  data  T h i s was f i r s t  seen  inelastic  (PI 71) and i n reexamination  of e a r l i e r i n e l a s t i c proton s c a t t e r i n g data which  in  showed a new resonance  (TY 5 8 ) ,  (LE 72) both  of  l o c a t e d 2 to 3 HeV below t h e GDR.  These e a r l y s t u d i e s , as w e l l  as  more  recent  s t r o n g l y suggest t h a t t h i s new resonance  ones  (BE 76a),  i s electric  guadrupole  (E2) i n c h a r a c t e r , and hence i t has come t o be c a l l e d the G i a n t Quadrupole time  Resonance  before  (GQR).  The GQR had been expected f o r some  i t s d i s c o v e r y , s i n c e the e f f e c t i v e charges needed  to e x p l a i n e l e c t r i c quadrupole depended  explicitly  transition  rates  and  moments  on some of the E2 s t r e n g t h to be l y i n g at  high e x c i t a t i o n e n e r g i e s (BO 69a). The s h e l l model d e p i c t s t h e particle-hole  states  one  (isoscalar)  down  from  Mottelson have expected  one  Because  a t t r a c t i v e i n the pulled  a  superposition  Two modes  of  coherent  motion  energy  in  which  are  mode,  t h e expected  of  move  out  of  phase  the i n t e r a c t i o n between the nucleons i s  isoscalar  shown,  they  on  quite  the value general  resonance of  2  energy  nto. .  grounds,  the i s o s c a l a r GQR i s 5 8A~V  3  that  the  HeV (BO 69b) , There  should a l s o be an i s o v e c t o r part t o the GQR, which, because repulsive  nature  of  is  Bohr and  and t h i s i s approximately the observed resonance energy.  the  of  i n which the neutrons and protons move i n phase  and  fisovector).  as  i n which the p a r t i c l e s have been e x c i t e d  through two major s h e l l s . possible,  GQR  of  the i n t e r a c t i o n between nucleons i n  t h i s mode, i s expected t o l i e a t  higher  excitation  energies.  Finally,  excitation  of  particle-hole  shell contribute t o c o l l e c t i v e , states  correspond  to  states  i s o s c a l a r E2  the l o w - l y i n g  2  +  w i t h i n a major  strength.  These  s t a t e s o f even-even  nuclei. In  n u c l e i h e a v i e r than *°Ca, t h e GQR seems t o be l o c a l i z e d  enough t o appear as a resonance, with about 80 o r 90  per cent  of  sum  t h e Gell-Mann-Telegdi  (EWSR) being d e p l e t e d .  (GE 53)  energy-weighted  rule  T h i s sum r u l e i s a c o n s e r v a t i o n law f o r  i s o s c a l a r E2 t r a n s i t i o n s t h a t i s e s s e n t i a l l y model  independent.  The EWSR i s given by  1-2 where  <r >  i s t h e mean  z  centre-of-mass and  E  sguared  displacement  from  the  of a nucleon i n the ground s t a t e of the nucleus,  i s t h e energy.  The  expression  1-2 i s developed i n  appendix A. The peak energy MeV  of the GQR l i e s  63A / -1  f o r n u c l e i i n t h e range 40<A<120, p o s s i b l y a l i t t l e  than t h i s f o r n u c l e i with higher lower of  a t approximately  mass  numbers  the resonance i s  decreases heavier.  from  region  the f a c t that  study  shell  30% o f t h e EWSR i s exhausted  f o r n u c l e i with A<40.  of  little  The width  nuclei,  and  3 MeV f o r t h e i n the g i a n t  T h i s i s p a r t l y due t o  more o f the s t r e n g t h a p p a r e n t l y  l o w ^ l y i n g bound s t a t e s o f l i g h t A  f o r closed  7 HeV f o r the l i g h t e r n u c l e i to  Only about  resonance  smallest  higher  and a  f o r n u c l e i with lower mass numbers (BE 76a).  3  resides  i n the  n u c l e i than heavy n u c l e i .  i n e l a s t i c e l e c t r o n s c a t t e r i n g on *°Ca showed  that the E2 s t r e n g t h was r a t h e r u n i f o r m l y spread between 10 and  5  20  MeV  been  <TO  73).  seen  in  (HO 74), MeV.  ;  Spreading o f the guadrupole s t r e n g t h inelastic  where 43% of the  Further  (Hfl 73) ,  on and  considerable MeV,  and  the f i r s t  *2  other  2 -state  about  30$  76)  E  x  even-even  1 2  EWSR.  C  (KN  about t h i s energy, but  which was  because of c o u p l i n g  (KN 76).  Thus  now  C,  the resonance has  68),  show  3*Ar  that  appears  of the  a 3  (RU  74)  of  However, and  a  several  GQR  which alpha of  a  EwSR was  seen near  with a r e c e n t  continuum  which p r e d i c t e d  a  GQR  expected to be g u i t e broad  that  MeV,  nuclei).  showed no evidence  to l o w - l y i n g  p e r s i s t s perhaps down t o A = 16 l 2  76)  (BI 75)  (r>>5 MeV)  it  capture  A similar inelastic  This r e s u l t i s consistent  s h e l l model c a l c u l a t i o n f o r at  *°Ca  evidence  although a s m a l l amount (6 ± 2%) = 27 MeV.  20  -1  show  the  s c a t t e r i n g measurement on GQR  =  x  i s spread from t h i s energy down to  do  of  a  (ME  2®Si  (KD* 74) ,  alpha s c a t t e r i n g s t u d i e s on (KN  E  **0  r u l e i s exhausted below 63A /  (in  +  other l i g h t n u c l e i  on  below  example,  and  nuclei  f r a c t i o n of the sum  excited  found  For  2*i26ftg  t h a t the s t r e n g t h  inelastic  exhausts  reactions.  several  studies  also  of t h i s e f f e c t have been seen i n many  (SN 74),  C  scattering  E8SR l i m i t was  examples  r a d i a t i v e alpha c a p t u r e reactions  electron  has  a  collective  states  resonance s t r u c t u r e  but f o r n u c l e i as l i g h t  e i t h e r disappeared or  has  been  as  washed  out because of broadening. Good i^o were  resolution  (~150  keV)  alpha p a r t i c l e s c a t t e r i n g from  showed a number o f peaks i n the assigned  to  L=2  exhausted about 401 observed  with  of  transfer the  inelastic  expected (HA  7 6) ,  GQR the  E8SR.  Similar  proton  scattering  region sum  peaks in  that  o f which have the  been giant  6  resonance r e g i o n  of  l 2  C  (GE 75), although these  not confirmed i n the i n e l a s t i c Knopfle  et a l . (K» 76)..  the guadrupole s t r e n g t h for  EWSR  is  and  alpha s c a t t e r i n g measurements of  Thus, i n a d d i t i o n to the spreading  may  T h i s makes  experimental  help t o e x p l a i n  n u c l e i , only zoapb seems  fine structure In  (MO  why  observation  so much l e s s of  the  more  to  exhibit  any  similar  76) .  general,  theories  than  distributions.  of  total  For  the  giant  cross  example,  resonances p r e d i c t  sections  and  Bosterli  {BR  no more than d e s c r i b e the gross shape of the  GDR.  complicated  to  configurations  strength  c a l c u l a t i o n s based on the bound  p a r t i c l e - h o l e e x c i t a t i o n s of Brown and  are  calculations  can  When describe  the  difficult  and  the p h y s i c a l e f f e c t s tend to become obscured.  is  necessary  they have not  Wang and  Shakin  describe  the  (WA  72)  met  Such c a l c u l a t i o n s  with t o t a l success.  included  both  intermediate  photodisintegration phenomenological  of  E2  * 0.  amplitudes of  the  found  predicted.  Thus  It  that the  the  E2  For  seen that  to  been  example,  effects  to  in  the  fairly  large  were then r e g u i r e d to f i t the  Cole  et a l . (CO 69).  measurements of p o l a r i z e d proton capture i n found  more  have  above  structure  I t was  6  neutron p o l a r i z a t i o n data  (HA 74a)  much  to i n c l u d e the e f f e c t s of the continuum  d e s c r i b e the s i t u a t i o n p r o p e r l y . but  become  do  more  structure,  also  the  introduced  59a)  intermediate  done,  of  observed i n l i g h t n u c l e i than i n heavy n u c l e i . , among  the heavier  little  were  i n l i g h t n u c l e i , there i s some evidence  i t to be fragmented.  more d i f f i c u l t ,  findings  amplitudes  theoretical  1S  N  However,  by Hanna et a l .  were much l e s s than  description  of  the  giant  7  resonances i s not  l3  N  The  present  via  the  capture t o the  on one  yet i n a s a t i s f a c t o r y s t a t e . work  inverse 1 2  C.  The  reaction, inverse  s t u d i e d by  expression  i s a study of guadrupole a b s o r p t i o n  the  relating  radiative  N(Y,p )  t 3  o  p r i n c i p l e of  the two  1 2  polarized  proton  C reaction i s related  detailed  balance.  that  p h y s i c a l l y independent i n f o r m a t i o n  reaction  (GL 73).,  a v a i l a b l e to help Studies  i s obtained  less  intense  difficult  capture  First, than  more  also  be  on t h e  part  in  constraints  from  several  ground  obtained,  on  state  it  consideration  therefore  of  it  is  concerning  the  residual  although i n i s o l a t e d cases  to o b t a i n  low-lying  Third,  and  Second, i n f o r m a t i o n  possible  g i a n t resonances b u i l t nucleus.  suffer  radiation,  to observe d i r e c t l y .  should  are  the  guadrupole r a d i a t i o n i s 10 to 10 0 times  nucleus i s a l l t h a t can be  residual  reactions  dipole  g i a n t resonances b u i l t  it  there  on  e x t r a c t the parameters o f i n t e r e s t .  with  disadvantages.  Thus  A.  protons  i n t e r f e r e n c e between the v a r i o u s p a r t i a l waves t a k i n g the  An  r e a c t i o n s i s given i n Appendix  The reason f o r using p o l a r i z e d r a t h e r than u n p o l a r i z e d is  in  may  information  excited  about  states  of  happen that the GQR  nucleus  under  particle  decays  excited  s t a t e s i n the r e s i d u a l n u c l e i , so very  to  of  the the the  high-lying  little  strength  w i l l appear i n the ground s t a t e channels. However, the extracted  with  E2 great  angular d i s t r i b u t i o n well  developed.  strength  In  that  confidence,  and  is  seen  can  often  s i n c e the i n t e r p r e t a t i o n of  polarization  measurements  has  a d d i t i o n , the backgrounds u n d e r l y i n g  peaks of i n t e r e s t i n the y - r a y  be  s p e c t r a are much l e s s severe  been the and  8  much b e t t e r understood than the continuum u n d e r l y i n g in  inelastic  scattering spectra.  case being c o n s i d e r e d , of in  13  about  10 minutes  N  be  can  13  N  beta decays t o  (AJ 70)  studied  Finally,  and  i 3  directly  only  peaks  f o r the p a r t i c u l a r  C  therefore  the  with a h a l f the g i a n t  via  life  resonances  radiative  capture  reactions. Most  previous  Convention  {BA  70)  (p,' )  are used  have  c o n c e n t r a t e d on  first  such measurement was  studied  the  1 J  for  example *He  (GL  73)  adoptions  throughout  B(p,Y )tzc  Stanford  (GL 73),  «ozr  strength the  mentioned  group, who with ls  the  {HA  73a),  studies  et  The  thesis)  studies  a l . {GL  GDR  zo^e  0  reaction  have  The  who  i n other n u c l e i , (GL 73)  and  *asi  by  the  of  E2  r e s u l t s of t h e i r work  already  found evidence f o r a GQR o f the  The  72)  were a l l c a r r i e d out  Y  1 6  which exhausted 30%  Madison  using t h i s t e c h n i q u e .  (p, ) reaction.  N (p,"f ) 0 They  the  a l s o made the f i r s t e x t e n s i v e study  0  briefly.  this  made by G l a v i s h reaction.  Q  above  of  l e a r n i n g more d e t a i l s about the GDR.  have a l s o been i n v e s t i g a t e d  The  on  {the  Y  been  i n the  EWSR between E  =20  discussed  (Y,p ) channel o  MeV  and  26.5  x  MeV.  Recently,  have been s t u d i e d BU 76a).  This  this at work  reaction the will  Oniversity be  chapters where comparisons to the made. .  and  the of  described present  l 4  C(p,Y ) 0  l s  N  Washington  reaction (AD  77,  more f u l l y i n l a t e r experiment  will  be  9  1.2 Review  of Previous Work  Previous  experimental  work  on  1 2  C(p,Y )i u  i n the giant  3  0  resonance r e g i o n has centred on e x t r a c t i n g the d e t a i l s GDH.  The  radiation very  p  measurements  energy  range  et a l . (FI 63)  the  ground  by warburton  later  = 11 MeV to 39 MeV.  beam  of  i n the g i a n t resonance r e g i o n were  limited  Fisher E  first  obtained  and Funsten  investigated  the  effort  (ME 65a) at d e s c r i b i n g t h e  i t s excitation  was  c a l c u l a t i o n s o f Barker In  resonance  measurements, E  resolution.  seen  = 8.6  p  Thus  at  E„ ~14  at  MeV  they  to  from  resolution;  involved  t o the s h e l l model  (EA 62).  16.0  that  p  =24.4  the  main  MeV  from  yield E  p  = 3  by  strength  channel was c e n t r e d a t E a d d i t i o n , the  MeV  in  the  earlier  measured the 90* y i e l d with  much  improved  were able to observe some i n t e r e s t i n g  10.6 MeV and 13.1 MeV.  extended to E  region  a  (wA 62).  region  mechanisms  MeV  e t a l . (ME 73a)  features i n t h i s region, including  found  over  were e l u c i d a t e d and  made by comparison  (BA 61) and E a s l e a  Measday  from  dips  gamma  order t o l e a r n more about the d e t a i l s of the low energy  "pygmy"  curve  the  Both measurements were hampered by poor  n e v e r t h e l e s s , the g r o s s f e a t u r e s of the GDR  in  state  energy r e s o l u t i o n and by poor detector energy  a first  of  x  two  in  interference  The 90® y i e l d curve was Berghofer  the  later  e t - a l * ,(BE 76b)  of the GDR  = 20.8 MeV  curve  dramatic  seen i n the {p,  with a width of 4 MeV. experimentally  who Y 0  )  In  difficult  MeV t o 9 MeV has been measured by Johnson  (JO 74) . Angular d i s t r i b u t i o n s were measured  at  several  incident  10  proton  energies  between  Legendre polynomial  10 and  expansion  2H MeV  of  by Berghofer et a l .  the  angular  A  distributions  r e g u i r e d the presence o f odd terms t o f i t the data, and i t w i l l be  shown  in  radiation. E2  Chapter  III  Thus t h e r e was  r a d i a t i o n was  that  the  odd terms a r i s e from  evidence i n these measurements  present and  was  E2  that  i n t e r f e r i n g with the dominant  E1 r a d i a t i o n , but no g u a n t i t a t i v e estimates could be made. Evidence o f E2 r a d i a t i o n i n the g i a n t resonance the  stable  mirror  nucleus  4 3  C  was  seen  in  region  the  inelastic  e l e c t r o n s c a t t e r i n g data o f Shin et a l . (SH 71), but  again  g u a n t i t a t i v e e s t i m a t e s were made.  T h i s l a t t e r measurement  be d i s c u s s e d more f u l l y i n Chapter  IV.  Further evidence of E2 s t r e n g t h i n Drake  and  Halpern  neutron capture by  1 2  (AH 75). C  MeV,  and found non-zero  the  angular  1 3  C  was  of  no will  seen by A r t h u r ,  These authors s t u d i e d r a d i a t i v e  at an e x c i t a t i o n energy  in  1 3  C  of  18  odd Legendre polynomial c o e f f i c i e n t s i n  distribution.  It  can  be  shown t h a t  radiative  neutron capture i s more s e n s i t i v e than r a d i a t i v e proton capture to c o l l e c t i v e E2 s t r e n g t h (HA 73b), so t h i s strong  evidence f o r a p o s s i b l e GQB  c l e a r whether t h i s  strength  was  in  1 3  C  measurement  gives  (although i t was  not  isoscalar  or  isovector  in  character).  1.3  Present Work It  can  be  seen  from  the  previous  section  that  a  measurement of the E2 s t r e n g t h i n the g i a n t resonance r e g i o n i s  11  a n a t u r a l extension of usually  possible  to  the  work  extract  already  done.  It  the E2 amplitudes  particular  instance  of  polarized  i n c i d e n t on a s p i n 0 (or s p i n 1/2) here,  these amplitudes  (provided M1 of  i 2  can,  However,  spin  nucleus,  1/2  as  C ( p , Y ) 13{j i s u s e f u l because i t may 0  mentioned e a r l i e r where the  E2  is  the  a  be one of those can  be  case  obtained  Therefore,  strength  for  particles  i n p r i n c i p l e , be uniquely  r a d i a t i o n can be n e g l e c t e d ) .  not  unambiguously  even from a p o l a r i z e d proton capture experiment. the  is  study cases  confidently  extracted. Recently, in  the  the  c o n s i d e r a b l e ambiguity  interpretation  (BU 76b). to  however,  These E2  difficulty,  cross and  of  even  these  has been found  simple  experiments  a m b i g u i t i e s i n c l u d e f i n d i n g double s o l u t i o n s sections  derived  others, w i l l  from  the  data.  This  be d i s c u s s e d more f u l l y  in later  chapters where comparisons w i l l be made to the present  results.  The f a c t t h a t i t might be p o s s i b l e to  reliably  determine  the guadrupole s t r e n g t h p r o v i d e s a second independent reason study  this  reaction.  It  was  mentioned  introduction that d i f f i c u l t i e s  are  encountered  earlier when  in  to the  attempts  are made t o c a l c u l a t e t h e o r e t i c a l l y the p r o p e r t i e s o f t h e g i a n t resonances.  In  order  to  t e m p o r a r i l y , i t i s necessary help  distinguish  models provide a states  concerned  among  the  connection and  a m e l i o r a t e the s i t u a t i o n , a t l e a s t t o r e s o r t to  to  help  between  the  parameters  the e x p e r i m e n t a l l y observed  understand  models  possible alternatives.  Such a r e a c t i o n model i s being developed (SN 75)  reaction  to  Reaction of  the  quantities.  by Snover and  Ebisawa  the E2 s t r e n g t h seen i n r a d i a t i v e  12  capture  reactions.  semi-direct  This  capture  model  (DSD)  is  model  based  first  e x p l a i n E1 c r o s s s e c t i o n s near the GDR  on  the  direct  proposed by Brown t o  (BR 64).  The  reaction  * C ( p , Y ) i 3 { j s h o u l d p r o v i d e a good t e s t of t h i s r e a c t i o n model. 2  0  The  model  and  comparisons  t o the data presented here w i l l  be  d e s c r i b e d i n Chapter IV. The energy range covered i n t h i s experiment i s from to 17 MeV  incident  approximately  proton  energy.  This  section  This  the  resonance  C(Y,n  Koch  also  )««C r e a c t i o n  and  Thies  appears  off  in  (KO 76).  Below The  (ME 73a,  BE 76b).  1 3  C{Y,n) C  10 MeV,  guadrupole  data  1 2  of  the d i p o l e s t r e n g t h strength  presumably  even more r a p i d l y , except f o r the p o s s i b l e presence  of i s o l a t e d narrow  s t a t e s . , The  upper  d i c t a t e d by the maximum beam energy An  energies  90° y i e l d curve of the  (JO 77), and i n the  begins to f a l l r a p i d l y . falls  of  MeV  covers the "pygmy" resonance observed i n the 90°  y i e l d curve and i n the t o t a l c r o s s  13  range  10  account  of  the  limit  of  17  MeV  was  available.  apparatus and measurement t e c h n i q u e s  used i n t h i s experiment i s given i n Chapter I I . The methods o f data a n a l y s e s and the r e s u l t s a r e presented i n Chapter I I I . Comparison  of the r e s u l t s f o r the  1 2  C(p,Y )* N 3  0  reaction to  other experiments and to the EWSR are g i v e n i n Chapter addition the DSD  IV,  in  t o a d e s c r i p t i o n of the attempts t o f i t the data with model.  Chapter V c o n t a i n s conclusions.  a  summary  of  the  results  and  the  13  Chapter II  EXPERIMENTAL APPARATUS AND  The  main  o b j e c t i v e o f t h i s experiment was  E2 c r o s s s e c t i o n as a * C (p, Y ) N. 2  In  13  Q  accurately  as  distribution  quantities the  possible,  of it  from of  was  polarized various  waves t a k i n g part  model  energy  in  and  the  reaction  d e s i r a b l e to measure, as and  unpolarized  angular  other parameters r e l a t e d the  reaction.  the data c o u l d  Snover  to measure the  for  All  the  then be compared t o  Ebisawa.  a d e s c r i p t i o n of the eguipment and  c o l l e c t the  2.1  addition,  extracted  reaction  contains  function  c o e f f i c i e n t s and  to the p a r t i a l  PROCEDURE  This  chapter  procedures used t o  data.,  General Experimental Arrangement A schematic diagram of the experimental set-up i s given  F i g u r e IT-1.  The  i n c i d e n t p o l a r i z e d proton beam  was  produced  by the U n i v e r s i t y of Washington Lamb-shift P o l a r i z e d Ion (FA 71).  The  Washington FN 90°  analyzing  line  (30°)  focussed target.  beam  was  tandem Van magnet  de Graaff and  by a s w i t c h i n g through  a  A f t e r passing  then a c c e l e r a t e d  Source  by the U n i v e r s i t y  of  a c c e l e r a t o r , bent through  a  d i r e c t e d down the a p p r o p r i a t e  magnet.  collimator  in  The and  beam skimmer  through the t a r g e t , the  was  beam  magnetically  system onto beam  the  travelled  15  another  7  m  f u r t h e r downstream u n t i l  i t reached the beamstop  l o c a t e d behind c o n c r e t e and wax s h i e l d i n g .  The long downstream  beam tube served as t h e Faraday cup. The  beam l i n e was o p t i c a l l y a l i g n e d  telescope  mounted  at  cross-hair located  the  downstream  a t the e x i t of the  by viewing  through  end and f o c u s s i n g on a switching  magnet.  c o l l i m a t o r and skimmer, which a r e l o c a t e d i n a separate of  beam  section  line, the  were  necessary  mounted i n t h e c e n t r e moved  until  the same  the  time,  cross-wire  then a c c u r a t e l y  was  amount.  cross-wire  a  plumb  The  section  by s h i f t i n g t h i s  a  cross-wire  was  chamber and the chamber was  was centred  bob  vertically  centred  Finally,  o f the t a r g e t  a  was  i n the beam l i n e .  used  to  above the c e n t r e  check  At  that  the  a x i s o f the gamma  ray angular d i s t r i b u t i o n t a b l e . The changed  s p i n o r i e n t a t i o n of by  reversing  the  the  polarized  one  beam  guench and argon f i e l d s .  c o n t r o l l e d by means of a f l i p p e r described (AD 73).  proton  The p o l a r i z a t i o n c o u l d  was  T h i s was  by Adelberger jgt aJL.  be f l i p p e d a u t o m a t i c a l l y  from  t o ten times a second, o r i t could be f l i p p e d manually when  desired.  The  flipper  also  provided  a l o g i c routing  which was used t o route other s i g n a l s according  signal  t o whether  the  proton s p i n was up o r down. The  amount  of beam on t a r g e t  r a t e i n the N a l ( T l ) namps  were  detector.  satisfactory,  Currents from  depending  beam s t r i k i n g the c o l l i m a t o r s was was t y p i c a l l y  0.2 narap.  was l i m i t e d by the c o u n t i n g 30  namps  to  on the beam energy.  continuously  monitored  I f the c o l l i m a t o r c u r r e n t  as 2 namps, the experiment was stopped and the beam  60 The and  rose as high refocussed.  16  This  was  necessary  backgrounds  t o prevent t h e appearance  i n the spectrum.  The t a r g e t h o l d e r was a could  be mounted.  ladder  on  t o any  which  three  targets  One o f these was an aluminum blank with the  same diameter hole as the a c t u a l t a r g e t . rotated  o f troublesome  orientation  The l a d d e r  about a v e r t i c a l  could  axis.  be  T h i s was  u s e f u l i n t h a t i t allowed the t a r g e t h o l d e r and l a d d e r frame t o be turned out of t h e l i n e o f s i g h t o f t h e gamma d e t e c t o r . The t a r g e t h o l d e r was which  was  maintained  at  surrounded liguid  by  a  copper  cylinder  n i t r o g e n temperature.  helped t o reduce any b u i l d u p o f beam l i n e contaminants target.  A  slot  This  on t h e  was cut out of the middle o f the c y l i n d e r t o  a l l o w the beam and s c a t t e r e d protons t o pass through f r e e l y . copper s t r i p 0 . 0 0 2  inches thick  where  rays  the gamma  virtually  soldered  frame  seemed  t o be  T h i s produced  or  can a r i s e from beam  small  striking  the copper c o l d - t r a p .  enough  of  diameter  t o prevent t h i s  aluminum  blank  i n the target  holder.  s c a t t e r e d protons were observed i n t h e p a r t i c l e these c o n d i t i o n s .  the  However, a 1/4  happening.  Checks were made a t v a r i o u s times by p a s s i n g the beam the  slot  but d i d improve t h e  c o l l i m a t o r o f diameter 3/16 i n c h and skimmer inch  the  v i c i n i t y o f the t a r g e t .  U n d e s i r a b l e backgrounds target  over  passed t o t h e d e t e c t o r .  no a t t e n u a t i o n o f t h e gamma f l u x ,  vacuum i n the immediate  aluminum  was  A  through  No e l a s t i c a l l y spectra  under  17  2.2 Gamma Hay Spectrometer The  most important instrument used i n t h i s experiment was  the gamma Nal(Tl)  spectrometer.  I t consisted  of  a  large  central  d e t e c t o r surrounded by a p l a s t i c a n t i - c o i n c i d e n c e (AC)  shield.  This  elsewhere  spectrometer  (HA  has been  described  in  detail  74b), and g e n e r a l c o n s i d e r a t i o n s f o r t h e design  o f such spectrometers have been given by P a u l (PA 7 4 ) , so the  salient  features  will  be d e s c r i b e d here.  only  A view o f the  spectrometer i s shown i n F i g u r e I I - 1 . The c e n t r a l c r y s t a l i s i n the form of a c y l i n d e r in  diameter  by 25.4 cm long.  25.4 cm  I t i s viewed by seven EMI 9758B  photonultipliers. The surrounding a n t i - c o i n c i d e n c e s h i e l d , manufactured the p l a s t i c s c i n t i l l a t o r NE 110, i s 10.8 cm t h i c k . the  phototubes The  phototubes  and t h e f r o n t  This  helps  to  The s i d e s o f t h e e n t i r e of  lead  central crystal. This detector  provision  to  reduce  The f r o n t  reduces from  of  reduce  neutron c a p t u r e i n t h e c e n t r a l  the  plastic  space between the two d e t e c t o r s i s f i l l e d  (LI 75).  inches  The c y l i n d e r by two  (HCA 8055).  t h i c k s e l f - s u p p o r t i n g mixture  lead.  I t covers  s i d e s and f r o n t face of t h e c e n t r a l c r y s t a l .  i s viewed by s i x  from  lithium  with a 1 cm  carbonate  the background  and wax  due t o slow  crystal. assembly  a r e surrounded  by  4  the cosmic ray f l u x r e a c h i n g t h e i s also shielded  by  4  inches of  the low energy gamma background r e a c h i n g the target.  The  front  shielding  f o r d i f f e r e n t s i z e c o l l i m a t o r s t o be i n s e r t e d .  has The  18  resolution  o f t h e spectrometer  collimators, the  i s improved  with  smaller  but s i n c e t h i s was not of paramount importance i n  present experiment, t h e i n s e r t i o n h o l e , with a diameter  of  6 i n c h e s , was l e f t c o m p l e t e l y open. The  gamma  the  beam l i n e  to  the front  spectrometer was l o c a t e d s u f f i c i e n t l y f a r from  - about face  16 i n c h e s from the centre of the t a r g e t of  t h e l e a d s h i e l d i n g - t h a t i t could be  swung through angles from 43° t o desirable,  of  course,  to  137°.  I t would  have  consequent  meant  having  the detector  the the  However,  further  this  back with a  decrease i n c o u n t i n g r a t e .  The constancy o f the d i s t a n c e from centre  been  have had a l a r g e r angular range i n  order t o reduce t h e e r r o r s i n t h e experiment. would  have  the d e t e c t o r  t o the  o f t h e chamber was checked by mounting a RaTh source i n  t a r g e t h o l d e r and measuring  the "angular  distribution"  as  d e t e c t o r was swung through i t s range., The counts recorded  were i s o t r o p i c t o w i t h i n 1/4%.  This  test  also  ensured  that  a b s o r p t i o n through t h e chamber w a l l s was uniform.  2.3  Gamma Spectrometer A  fairly  processing  Electronics  complex, but now b a s i c a l l y s t a n d a r d , system f o r  the signals  from  t h e gamma  spectrometer  was  u t i l i z e d . , The fundamental i d e a behind the e l e c t r o n i c system i s to  veto  events  which  a r e a f f e c t e d by p i l e - u p , and t o r e j e c t  cosmic r a y events and events escapes  f o r which  from the c e n t r a l c r y s t a l .  some  of  the  energy  A l l of these e f f e c t s worsen  19  the d e t e c t o r r e s o l u t i o n and o f t e n make i t d i f f i c u l t  to  extract  from t h e spectrum the number o f t r u e events a s s o c i a t e d reaction given  being  i n Figure .11-2 The  signals  detector one  studied.  A block  with the  diagram of the e l e c t r o n i c s i s  and a d e s c r i p t i o n f o l l o w s .  from  the  seven  phototubes  are a c t i v e l y summed, then sent  branch, t h e l i n e a r s i g n a l ,  is  on t h e Nal(Tl)  to a fan-out from which  amplified  and  sent  to a  l i n e a r gate.  A second branch from the fan-out i s c a b l e c l i p p e d  to  of  a  width  50  nsec,  fraction discriminator (HLD).  The  bias  amplified  called  the  and  High  sent t o a c o n s t a n t  Level  Discriminator  on t h e d i s c r i m i n a t o r i s s e t j u s t f a r enough  below the r e g i o n of i n t e r e s t i n the spectrum t h a t any t h r e s h o l d e f f e c t s o f the HLD have disappeared. two the  low l e v e l  pulses  spectrum.  No  i s effectively attempt  In t h i s way,  pile-up  of  prevented from appearing i n  i s made  to d i s c r i m i n a t e  against  high-low p i l e - u p . One linear  output of t h e HLD opens the l i n e a r gate, a l l o w i n g signal  t o pass t o an a n a l o g - t o - d i g i t a l c o n v e r t e r  i n t e r f a c e d to the Nuclear P h y s i c s Laboratory SDS 930 The  signal  HLD  (ADC)  computer.  i s shaped c o r r e c t l y and delayed a p p r o p r i a t e l y f o r  the ADC by a l i n e a r gate and s t r e t c h e r . the  the  i s f e d to  (dead t i m e - l e s s )  two c o i n c i d e n c e  A second  output  from  c i r c u i t s v i a an updating  d i s c r i m i n a t o r t o check f o r c o i n c i d e n c e s  with  the AC channel. The  signals  from  a l l eight  s c i n t i l l a t o r s are a c t i v e l y summed.  phototubes  on the p l a s t i c  The r e s u l t a n t pulse i s then  a m p l i f i e d , c a b l e c l i p p e d t o a width of 80 nsec, a m p l i f i e d and  passed to an updating d i s c r i m i n a t o r .  The  bias  again  level  on  TIMING FILTER AMP  FAST AMP  FAST COINC  UPDATE DISC  UPDATE . DISC  GATE AND DELAY  FAST ANTICOINC  GATE AND DELAY  ADC GATE (REJECT)  PLASTIC  M I (Tl)  FAN OUT  FAST AMP  LINEAR AMP  CONSTANT FRACTION DISC  UPDATE DISC  ADC GATE (ACCEPT)  SHIELD UPDATE DISC  TIMING FILTER \ AMP  LINEAR GATE  F i g . I I - 2 : Block diagram o f the gamma spectrometer  LINEAR GATE STRETCHER  electronics.  o  21  this  discriminator  is  set  just  around 100 keV,  and t h e output  with the output  of the HLD  k  here  coincidence  above  the n o i s e , somewhere  i s fed i n t o a  fast  coincidence  { a f t e r s u i t a b l e d e l a y s - not shown).  i m p l i e s that energy has been deposited i n  both the c e n t r a l c r y s t a l and  the surrounding  shield.  the l i n e a r s i g n a l i s routed i n t o a p o r t i o n of l a b e l l e d " r e j e c t " , because these events  Therefore  computer  are normally  memory  discarded,  a c t u a l l y , i n t h i s experiment, the " r e j e c t " s p e c t r a were used i n the subseguent a n a l y s i s . If  there i s no c o i n c i d e n c e between the two  i n p r i n c i p l e no energy was Hence  linear  signals  lost  for  from  these  "accept" p o r t i o n of memory.  the  events  channels,  central  then  detector.  are routed i n t o  T h i s i s accomplished  by the second  c o i n c i d e n c e t e s t , which r e q u i r e s a c o i n c i d e n c e between the output  and  a  null  output  from the f i r s t  the  coincidence  HLD  (again  after suitable delays). The r e j e c t and accept r o u t i n g according  to  Nal(Tl)  the  detector.  integrator  and  from  fan-out  the  split  The  a  pulse  generator  was  i n p a r a l l e l with s i g n a l s from  pulser  was  fired  by  the  fed the  current  hence gave a d i r e c t measure of the dead time i n  the d e t e c t o r e l e c t r o n i c s system. check  further  u s i n g the f l i p p e r r e f e r r e d to e a r l i e r .  In a d d i t i o n , a s i g n a l into  are  whether the proton beam i s s p i n up or s p i n down.  T h i s i s accomplished  directly  pulses  linearity  of  The  p u l s e r was  also  used  to  the e l e c t r o n i c s p r i o r to each set of  measurements. V a r i o u s outputs  were s c a l e d i n case i t became necessary  consider r e j e c t i n g a questionable  data  point.  This  is  to not  22  shown i n the f i g u r e . .. The s c a l e d outputs i n c l u d e d t h e number o f accept  and  the p l a s t i c deposited  reject  r o u t i n g p u l s e s and the number of counts i n  scintillators.,  In  addition,  a l l events  more than about 250 keV i n t h e N a l ( T l ) d e t e c t o r were  s c a l e d , and t h e c o u n t i n g r a t e f o r these events was 40  which  kHz.  The  kept  below  s i g n a l s f o r these low l e v e l events were d e r i v e d  from a s e p a r a t e branch of the fan-out. F i n a l l y , i t was necessary t o s t a b i l i z e the phototubes the  Nal (Tl)  counting  detector  rates.  This  against was  drifts  incurred  accomplished  by  on  variable  dynamically  by an  e x t e r n a l feedback system which adjusted t h e high v o l t a g e on the tubes i n such a way as t o keep c o n s t a n t the height o f the pulse from  some  peak i n t h e low energy part of t h e spectrum.  Gamma  r a y s from the i n e l a s t i c a l l y e x c i t e d l e v e l at 4.43 MeV were used f o r t h i s purpose. An example o f a spectrum o b t a i n e d with the spectrometer i s shown i n F i g u r e I I - 3 . MeV,  The HLD c u t o f f i s noted  around  y  = 7  and t h e part o f t h e spectrum below t h i s energy has been  omitted.  I t i s seen that t h e accept spectrum  improved  over  t h e combined  spectrum.  i s considerably  Removal of those events  a s s o c i a t e d with the l o s s of one o f the p a i r a n n i h i l a t i o n i s r e s p o n s i b l e f o r most o f (full  E  width  the improvement.  The  guanta  resolution  at h a l f maximum) of the accept part i s about 4.0%  compared t o 1,0% f o r the sum spectrum. The background above the peak r e s u l t s mainly rays,  although  there  is a  i s due  partly  cosmic  small excess over t h e background  expected from t h i s source i n t h e accept spectrum. background  from  This  excess  t o high energy gamma r a y s from t h e  23  300  PEAK WINDOW  -  j |  200 SINGLE ESCAPE PEAK  100 co  0  HLD CUTOFF  Z  •  ••••  .BACKGROUND WINDOW  O o u. o  c_  LU  00  0  JL_ .  L  J  1  I  !  200  25  Z3 Z  100  •  ACCEPT  e o e 0  e  50  _L  e  .\'V"" j—g.»oooo»o  e  •  e  "  REJECT  ^  8  10  12  14  16  E/MeV) Fig. I I - 3 :  1 2  C(p,Y ) 0  1 3  N  gamma ray spectrum at  = 11.2 MeV.  18  24  1  * N ( p , Y ) i s o r e a c t i o n , although only  trace  of  particle  yields  there  was  excess  background might a l s o a r i s e from p i l e - u p , o r  unidentified  a  the  contaminant  **N i n the t a r g e t .  i n the t a r g e t .  not prove to be a d i f f i c u l t l  the e x c e l l e n t  detector  would have merged with t h e C (p, Y ) isjj 1 2  Q  been  included  some i t did  around  E  y  = 11  r e s o l u t i o n , t h i s peak peak  i n t h e a n a l y s i s , although  been a s e r i o u s problem i n t h e present The  from  I n any event,  noted  7  Q  Without  Some of the  problem to handle.  A peak due t o *0{p, Y ) * F i s a l s o HeV.  indicated  and  would  have  t h i s would not have  case.  p u l s e r peak l i e s o f f - s c a l e a t an e g u i v a l e n t gamma r a y  energy o f about 30 MeV. The  window  regions  summed a r e a l s o shown.  i n which The  t h e number o f counts was  positioning  of  the windows i s  d i s c u s s e d i n Chapter I I I .  2.4  P a r t i c l e Detection Two  lithium  d r i f t e d s i l i c o n d e t e c t o r s were l o c a t e d i n the  s c a t t e r i n g chamber as shown i n F i g u r e I I - 1 . to  p r o v i d e a constant  monitor  T h e i r purpose  was  o f t h e beam p o l a r i z a t i o n v i a the  * 2 C ( p , p ) * C r e a c t i o n , and t o p r o v i d e a secondary 2  means of data  • o  normalization.  They were symmetrically  i n c i d e n t beam d i r e c t i o n . interfere it  placed a t 160° t o t h e  T h i s angle was f a r enough back not t o  with gamma r a y s going t o the gamma spectrometer  when  was l o c a t e d at back angles, but not so f a r back t o i n t e r f e r e  with the incoming  beam.  25  C o l l i m a t o r s c o n s i s t i n g of v e r t i c a l  s l i t s 0.125 i n c h e s wide  by 0.44 inches  high were mounted i n f r o n t  The  from t h e c e n t r e  distance  of  the  detectors.  of t h e t a r g e t to the c o l l i m a t o r s  was 4.25 i n c h e s . Signals  from  preamplifiers and  the  located  detectors immediately  thence to the counting  block  were  fed to  outside  Ortec  the t a r g e t chamber  room where they were  processed.  diagram of t h e e l e c t r o n i c s i s shown i n Figure  description  A  I I - 4 , and a  follows.  From the l i n e a r a m p l i f i e r , one branch was sent gate  109A  and  stretcher,  where  the  signals  to a l i n e a r  were  delayed  a p p r o p r i a t e l y and then passed t o a sum a m p l i f i e r and the ADC., The (SCA) low  l o g i c branch was sent  where  to  a  single  channel  a low l e v e l d i s c r i m i n a t o r was used t o c u t out the  energy pulses.  discriminator  The c o u n t i n g  threshold  rate  f o r pulses  was about 2 kHz,  to  route  appropriate The  linear  signals  present  above the  The output from the  SCA was mixed with t h e l o g i c s i g n a l s from the p u l s e r sent  analyzer  at  and  then  the ADC i n t o t h e  p o r t i o n o f memory.  logic signals  were  also  fed t o  an  "exclusive-or"  mixer.  The purpose o f t h e mixer was t o gate the ADC when there  was  logic  a  pulse present  from only one d e t e c t o r .  the ADC would not know from which d e t e c t o r come, and i n amplifier  any  would  event,  this  probably  be  linear a  t h e l i n e a r pulse had  signal  pile-up  Otherwise,  from  the sum  of p u l s e s  from both  detectors. The dead  p u l s e r was f i r e d  time  by the c u r r e n t  integrator  c o r r e c t i o n s could be made d i r e c t l y .  so  that  Unfortunately,  L PART DETECTOPJ  LINEAR GATE . STRETCHER  LINEAR AMP  PREAMP L TEST IN  TIMING SCA  OR  SCALER 1  ADC ROUTE  GATE AND h — SCALER DELAY  BIC"  DUAL DECADE ATTEN  PULSE GEN  OR  L TEST OUT  SCALER  R TEST OUT  ADC GATE  GATE AND DELAY  BIC'  TIMING SCA  OR  R TEST LN R PART PETECTOPJ  GATE AND DELAY  PREAMP  Fig.  LINEAR AMP  ADC  *\ SCALER  GATE • AND DELAY  SCALER ' ADG ROUTE  LINEAR GATE STRETCHER  : Block diagram o f t h e p a r t i c l e d e t e c t o r  electronics  to  27  the pulse generator l a t e r found  appeared t o be  faulty,  and  i t was  t h a t t h e r e s u l t s were more s e l f - c o n s i s t e n t i f the counts  were not dead time c o r r e c t e d . As with the gamma further  separated  electronics,  a l l linear  pulses  were  a c c o r d i n g t o whether t h e proton s p i n was up  or down. S e v e r a l o f t h e branches were s c a l e d .  These a r e  shown i n  the f i g u r e . An example o f a p a r t i c l e spectrum i s given i n F i g u r e I I - 5 . The  strongest  peaks  inelastic scattering  are from e l a s t i c s c a t t e r i n g o f f leaving  1 2  C in its first  excited  The peaks r e s u l t i n g from e l a s t i c s c a t t e r i n g o f f * ~ and l  also  clearly  seen, but note the l o g a r i t h m i c s c a l e .  peak l i e s below t h e t h r e s h o l d  1 2  C and  state. 1  6  0 are  The p u l s e r  f o r l i n e a r s i g n a l s so t h a t i t i s  i n a background-free r e g i o n of the spectrum..  2 . 5 Targets Three  different  experiment. was  used  targets  were used i n the course o f t h i s  A n a t u r a l carbon t a r g e t of  f o r t h e measurement at E J = 1 3 . 5 MeV.  t h a t gamma r a y s from the C ( p , Y ) 1 3  spectrum about to the  above  1.1% C ) . 1 3  subtract 1 3  C(p,Y)  certain.  thickness  a 1 4  1 4  background  mg/cm  2  i t was found  N r e a c t i o n contaminated  t h e peak o f i n t e r e s t Although i t would  1.9  the  ( n a t u r a l carbon c o n t a i n s  always have been  necessary  due t o cosmic r a y s , t h e presence o f  N gamma rays made the background  Thus i t was decided t o run with pure  subtraction 1 2  less  C targets. I t  28  12 C(  P»P|4.43 4 */  C  ,2  10'  PULSER  C(p,p ) C 0  ,2  J  CO *-  I0  :  o o y.  ,6  o  0C&p ) 0  I o  ,6  14 K V *  N(p, )"N  .1  A  1.0*  •* ••• •c • ». • • *— •• • • ••••  •  •  30  •  60  •  J 90  CHANNEL  •  Po  B • «• •  120  I  150  180  NUMBER  F i g . II-5 : P a r t i c l e spectrum at E£ = 11.2 HeV. Note the l o g a r i t h m i c s c a l e of the o r d i n a t e . The arrows 'A• and » B' d e f i n e the window region r e f e r r e d to i n the t e x t (Chapter I I I ) .  29  was  hoped t h a t only the much more c e r t a i n cosmic r a y background  s u b t r a c t i o n would then be necessary. A c c o r d i n g l y , two t a r g e t s were ordered Tech.*  The t a r g e t s  were  from  approximately  Penn  1 mg/cm  thick.  2  t h i c k n e s s was measured by comparing y i e l d s i n t h i s to  the e l a s t i c  (ME 76) (SW  scattering  A gamma spectrum  data  o f Swint  i s shown  s p e c t r a , i t has a l r e a d y been pointed out that from  nitrogen  and  p a r t i c l e y i e l d s i n t h i s experiment section  n i t r o g e n y i e l d t o the data nitrogen  oxygen.,  l 3  c i n these  there By  i s some  comparing t h e  with t h e d i f f e r e n t i a l  0.01 mg/cm . 2  of Hintz  comparison  (HI 57)  cross  content i s only 0.0004 mg/cm . 2  of the  indicates the  A l l runs except those  a t 13.5 MeV were taken with one or t h e other o f these C  al.  data o f Daehnick (DA 64), i t was found that t h e oxygen  content i n t h e t a r g e t was about  1 2  e_t  i n Figure I I - 5 .  there i s no evidence f o r the presence o f  contamination  experiment  from one o f the t a r g e t s i s shown i n  F i g u r e I I - 3 and a p a r t i c l e spectrum Although  The  s c a t t e r i n g c r o s s s e c t i o n data o f Meyer jet a l *  and t h e i n e l a s t i c  66).  Spectra  enriched  targets.,  2.6 C u r r e n t I n t e g r a t i o n The c u r r e n t c o l l e c t e d Brookhaven  1  Instruments  i n t h e Faraday  Corporation  Penn Spectra Tech 411 Bickmore Drive H a l l i n g f o r d , Pennsylvania 19086  cup was measured by a  (BIC) c u r r e n t i n t e g r a t o r .  30  The BIC d e l i v e r s a r o u t i n g pulse f o r a c e r t a i n amount o f charge collected.  These p u l s e s were d i v i d e d , as usual,  whether the proton s p i n was up or down. to  to  S c a l e r s were then used  r e c o r d the i n t e g r a t e d charge. The  the  according  BIC output pulse was a l s o used t o f i r e t h e p u l s e r s i n  d e t e c t o r e l e c t r o n i c c i r c u i t s t o keep t r a c k o f dead times.  2.7 P o l a r i z a t i o n Measurements The beam p o l a r i z a t i o n was the  runs by the C ( p , p l2  w e l l known (ME 76).  )  1 2  continuously  in  I n a d d i t i o n , the p o l a r i z a t i o n was measured  a s e p a r a t e beam l i n e .  placed a t 112.5° t o the incoming angle,  the  during  C r e a c t i o n whose a n a l y z i n g power i s  s e v e r a l times throughout the runs (BA 75)  monitored  analyzing  power  with  a  helium  polarimeter  The p a r t i c l e d e t e c t o r s were proton  beam  since  at  this  f o r the He{p,p )*He r e a c t i o n i s 4  o  c l o s e t o 1.0 f o r a l l the e n e r g i e s measured (SC 71).  2.8 Data  Accumulation  Three d i f f e r e n t runs were made i n t h i s experiment. first, In In  In the  data were o b t a i n e d only f o r a proton energy of 13.5 MeV.  the second, data were taken a t 12 MeV,  14 MeV  the f i n a l run, data were gathered a t 10 MeV,  MeV, 15 MeV and 17 excitation  energy,  MeV. there  Because  the  GQR  and  16  MeV.  11.2 MeV,  lies  at  a  12.8 high  should be a l a r g e number o f allowed  decay channels and hence t h e resonance  will  be  guite  broad.  31  Thus i t was f e l t  that  measurements i n approximately  1 MeV steps  would be adequate t o survey t h e r e g i o n . In  order  to  h e l p d e t e c t p o s s i b l e s y s t e m a t i c e r r o r s , the  data a t most energies were c o l l e c t e d with f o u r passes over the angles  measured.  At one energy  (17 MeV), only two passes were  made because of time c o n s t r a i n t s  and  at  two  other  energies  {11.2 MeV and 13.5 MeV) t h r e e passes were made. It  was  necessary t o have the face o f the t a r g e t  a t an angle g r e a t e r than 20° from t h e gamma d e t e c t o r avoid  absorption  through  the t a r g e t  angles 43°, 55°, 7 0 ° , 90°, and target  at  110°.  The  137° were  angles  the t a r g e t  t h a t t h e r e were target rotation. The  angles  at no  to  Therefore the  measured  with the  The end p o i n t s were  both o r i e n t a t i o n s i n each  systematic  effects  measured  pass to ensure  associated  with  the  None were observed. were  chosen  v a r i o u s Legendre p o l y n o m i a l s . criterion  angle  43°, 110°, 125°, and 137° were  measured with the t a r g e t a t 70°. with  holder.  pointing  t o be egual t o the zeros o f t h e There was  f o r the c h o i c e o f a n g l e s  no  other  reasonable  - f o r example, no angle i s  more s e n s i t i v e than another t o the presence of E2 r a d i a t i o n . When the gamma spectrometer was angles,  6  collimators radiation  inches and  of  lead  spectrometer  produced  by  was  located placed  at  between  collimator.  the beam s t r i k i n g  the  This  forward  t h e beam prevented  the c o l l i m a t o r s  from  reaching the N a l ( T l ) d e t e c t o r d i r e c t l y . Some runs were taken by c o l l e c t i n g 30 the  ucoul other.  the complete  charge  of  f i r s t with t h e proton s p i n i n one d i r e c t i o n , then i n Others  were  taken  with  the  spin  flipping  32  automatically  once  a  second.  e l e c t r o n i c s was a u t o m a t i c a l l y  When  this  was t h e case, the  shut down f o r 1  msec  while  the  f i e l d s were r e v e r s i n g . The  data  f o r each measurement were s t o r e d d i r e c t l y  SDS 930 computer. carried being  A  preliminary  analysis  of  the  i n the  data  was  out at the end o f each run while t h e d e t e c t o r angle was changed.  The data  were a l s o w r i t t e n onto magnetic tape  for later o f f - l i n e analysis.  33  Chapter  III  DATA ANALYSIS AND  RESULTS  The main aim o f t h i s experiment small  E2  cross  "background". very  section  in  Thus i t was  carefully  to  r e s u l t s occurred.  the  was  presence  necessary  ensure  that  to  to  places  semi-direct  parameters used Chapter  3.1  throughout  model  in  the  Gamma Ray  are  fit all  described,  and  presented.  the c h a p t e r , comparison i s  results  and  calculation. calculations  essentially  the  peaks are s i t t i n g overlap.  peaks  the The  will  results model  be  two  of  and  described  a the in  One  ways to determine the area of  i s to use  standard  The  line-shapes  of i n t e r e s t . , T h i s i s u s e f u l when the  on l a r g e backgrounds or when two  or peaks of i n t e r e s t window.  be  Spectra A n a l y s i s  the peaks i n the s p e c t r a .  them  data  IV.  There  to  the  In t h i s chapter, the methods used to analyze  made between the experimental direct  very  no s y s t e m a t i c b i a s i n g of the  then the r e s u l t s of the a n a l y s i s w i l l be several  a  of a very l a r g e E1  scrutinize  the data and t o check i t s c o n s i s t e n c y w i l l  In  measure  or  more  of  other i s to d e f i n e a window around the peak and simply  sum  the  counts  within  this  In t h i s experiment, the s p e c t r a were reasonably c l e a n  34  and  the peaks were w e l l s e p a r a t e d except f o r a s m a l l number  high  energy background gamma r a y s from proton c a p t u r e on  and  nitrogen.  Thus i t was  a n a l y s i s , t a k i n g care above  the  peak  Q-value of 0.6 background  to place  from  MeV.  decided to use  the  I t was  1 6  oxygen  the second method of  the lower l i m i t  reaction  of  of  the  window  0 ( p , Y ) * F , which has  a  7  Q  a l s o necessary t o s u b t r a c t  a  small  which arose from the contaminants in the t a r g e t  from cosmic r a y s  which penetrated the l e a d  shield.  The  same  computer  program was  used to analyze the data both o n - l i n e  off-line  (BU  A  75a).  brief  description  of  the  and  and  analysis  procedure f o l l o w s . F i r s t , a window was a  fraction  (<1.0)  spectrum and was  made.  defined, new  and  successively the  channel. and  centroid  was  calculated.  For the  in  This  the  of  window was  procedure  this  window  window so  d e f i n e d and  continued  agreed to within 0.1  c e n t r o i d p o s i t i o n was  and  peak i n the  then c a l c u l a t e d f o r the  determined c e n t r o i d s error  (>1.0) above  of a strong  guess o f the c e n t r o i d  from t h i s c e n t r o i d a new  centroid  since  below the c e n t r o i d  an i n i t i a l  A new  d e f i n e d as a f r a c t i o n  until channel,  typically  0.1  purpose of d e f i n i n g the c e n t r o i d , the s p i n  s p i n down s p e c t r a ,  i n c l u d i n g both  the  accept  a  and  up  reject  p a r t s , were summed. When  the  centroid  had  been determined, the counts were  summed w i t h i n a second window, a l s o d e f i n e d centroid.  For the data at  l i n e s i n the s p e c t r a and the  15.1  MeV  centroid  and  17 MeV,  there  were  from i n e l a s t i c s c a t t e r i n g o f f the  l e v e l s of * C, 2  was  16 MeV  as f r a c t i o n s of  respectively.  At  determined from these strong  these  the  strong  12.7  MeV  energies,  peaks s i n c e  they  35  are l e s s s u s c e p t i b l e t o s h i f t s due  to  For  no peak stronger  all  other e n e r g i e s ,  t h e r e was  the s p e c t r a , hence t h i s l i n e An  example  Figure  used t o d e f i n e  of a window r e g i o n d e f i n e d  down,  accept  and  determined  above and window  by  i s shown i n  defined  above  the  The  as  simply  background The  The the  the  window  between the channels  counts i n  a  background  peak window, and  were then  subtracted  e r r o r of the counts i n each  statistical  the  peak  was  e r r o r ; t h a t i s , { t o t a l area  *  area) / . 1  number of  counts  corrected  for  the s p i n up and  number  up  peak window were summed f o r each of  2  y i e l d s were c o r r e c t e d  s p i n up and  of  background counts were normalized to  number of channels i n the from each peak sum.  The  spin  were then summed.  ends  linear extrapolation  below the window l i m i t .  the four s p e c t r a .  the  in  0  centroid.  individual  reject, spectra,  Y i e l d s i n the f r a c t i o n a l channels at the  given  than Y  the  in t h i s way  counts w i t h i n the window f o r the  spin  were  variations.  II-3.  The and  was  background  in  any  the  f o r dead time by d i v i d i n g by  pulser  peak.  This  automatically  d i f f e r e n c e s i n the charge c o l l e c t e d during  s p i n down p a r t s of the run.  a l t e r n a t i v e l y , the  s p i n down counts could be normalized of  the  counts  obtained  in  D i f f e r e n c e s between the n o r m a l i z a t i o n  the  according  particle  to  detectors.  methods w i l l be  discussed  i n s e c t i o n 3. 4. R a t i o s of counts i n the r e j e c t s p e c t r a to accept run.  spectra In  were c a l c u l a t e d f o r each s p i n up and  Figure I I I - 1 , these r a t i o s are  of angle f o r one  counts  energy.  I t i s seen t h a t  in  the  s p i n down  p l o t t e d as a f u n c t i o n the  average  values  36  1—3  hi  I  i  80  100  120  140  "0-|_ (Degrees) ab  F i g . I I I - 1 : Reject t o accept r a t i o s f o r E£ = 10 MeV. The p l u s s i g n s a r e the averages at the given anqles and spin orientations; the surrounding p o i n t s with error bars are the corresponding experimental measurements. See t e x t f o r d i s c u s s i o n .  37  fluctuate  a  fair  amount as a f u n c t i o n  f o r the s p i n down fluctuation  spectra.  about  the  In  addition,  seemed t o recorded  In be  some,  but  correlated  not  with  i n the AC s h i e l d .  I t was  necessary to i n c l u d e  analysis.  The  total  window.  particle  the  option  of  the  found 1 2  o  established,  peaks  showed  similar  fluctuations  number  reject  done  also  of  counts  spectra  compared  was  centre  1 2  in  the  to  to  using  to have the for  of  defined  peak  peak window.  agreed  helium  II-5  defined  background  within  polarimeter.  and  C,  and  14  •B*  N  within  In the were  one  initial  analyzed.  errors  over  1 2  on  beam p o l a r i z a t i o n measured  c a r r i e d out  (between arrows * A*  and  was  f i n a l a n a l y s i s was the  a  program s l i d e the windows  the  the  C reaction the  within  peak were read i n t o the computer  peaks shown i n Figure  using the  summed  peak window  i n a l l cases that t h e  C(p,P )  measurements  window  were  calculated  a n a l y s i s , the four  by the  peaks  above the  u n t i l the c e n t r o i d  I t was  Although  Analysis  Channel l o c a t i o n s f o r the  An  channel  fair  accept a n a l y s i s .  windows below and cards.  a  repeat measurements at each angle were found  P a r t i c l e Spectra The  is  because of these v a r i a t i o n s  be more s e l f - c o n s i s t e n t when t h i s was  3.2  energy  a l l , c a s e s , the the  t h a t i t was  only the  there  average v a l u e at each angle.  d i f f e r e n t i n d e t a i l s , the data at each variations.  of angle, p a r t i c u l a r l y  i*0  in Figure  with  the  After t h i s with  elastic 11-5} .  a  was  broad  scattering Summing a l l  38  three peaks  improved  the  statistical  accuracy  and  reduced  background u n c e r t a i n t i e s . The  program  calculated  the  charge  and  solid  asymmetries a s s o c i a t e d with the p o l a r i z e d beam, and monitored  throughout  c a l c u l a t e d and  the  monitored.  runs.  The  Expressions  angle  these  were  a n a l y z i n g power was  also  f o r these q u a n t i t i e s are  given i n Appendix C. ,  3.3  R e s u l t s from the P a r t i c l e The  Analysis  three asymmetries f o r the summed peaks at  are p l o t t e d i n the upper h a l f of Figure II1-2. typical results.  E£ =  In t h i s p a r t i c u l a r case, both the s o l i d  a n a l y z i n g power show a slow i n c r e a s e as  more  but  superimposed  on  not  general  trend.  results.  s t e e r i n g e f f e c t s which may non-uniformities.  or may  They  could  be  lines  and  correspond  charge  ratio  to measurements made  measurements  place  as  no  simple  beam  orientation. s o l i d angle  with  with  target  polarization The  dashed  asymmetry p l o t s  unpolarized  beam.  The  u n p o l a r i z e d beam w i l l  3.5.  I t i s i n general very beam s h i f t s and  is  coupled  real  of the a n a l y z i n g power with  be d i s c u s s e d i n s e c t i o n  taking  There  not be  spin  the  well  They could be due t o s m a l l beam  changes, p o s s i b l y f o r only one in  angle  s t a t i s t i c a l l y significant, fluctuations  this  e x p l a n a t i o n f o r these  MeV  These are q u i t e  asymmetry and rapid,  15  difficult  to  distinguish  between  p o l a r i z a t i o n changes u n l e s s there i s a r e a c t i o n in  the  t a r g e t f o r which the a n a l y z i n g power i s  39  1  0.69  1 1 1 ANALYZING POWER ASYMMETRIES EjJ = 15 MeV  "~  1  1 [  1  I  '  0.68 0.67 SOLID ANGLE ASYMMETRIES  1.15 1.10 1.05  "- I~I I* iIJ* 1  1.05 - J  n  f  l  * i•  l H f f  "i ^ - r r  1 1  - - - --  CHARGE RATIO ASYMMETRIES E £ = I 5 MeV  it  —  J  1.00  0.95 0.56  -  1^ £  j  . j*  0.54  ANALYZING POWER  —  ASYMMETRIES  / ' H I  E  = I6 MeV  P  0.52  SI  0.50  1  X  !  SOLID ANGLE ASYMMETRIES  1.05 1.00 " i l ' 0.95  "  511 1  "  1  1  5  f Ep = 16 MeV 5 *** 1 * * 1 1 1 1 1 1 1 , 1 ^ J l l  1  10  1  |5 RUN  Fig.  TII-2  :  Polarized proton a n d 16 MeV.  i  20  25  r  1  1  30  1  ! -  r  35  NUMBER  beam a s y m m e t r i e s  at  E± = p  15 MeV  40  close to zero. small  Such i s not the case here, hut t h e d r i f t s  i n any event.  I f they a r e due t o p o l a r i z a t i o n  are  changes,  then the change o f 1 o r 2% i s no more than the assumed e r r o r i n the p o l a r i z a t i o n , as w i l l be seen  later.  An e x c e p t i o n to these comments E + •= 16 MeV.  i n t h e data f o r  Shown i n the bottom part of F i g u r e I I I - 2 are t h e  beam p o l a r i z a t i o n energy.  occurs  and s o l i d  angle asymmetries measured a t  I t i s seen t h a t the p o l a r i z a t i o n takes a s u b s t a n t i a l  drop a t run 18, and then r e t u r n s t o the o r i g i n a l average in  two  solid  stages.  There  angle asymmetry, thus t h e r e  therefore  value  i s no corresponding v a r i a t i o n i n the i s fairly  t h a t t h e p o l a r i z a t i o n change i s r e a l . were  this  handled  slightly  strong  evidence  The gamma data at 16 MeV  d i f f e r e n t l y from the data a t  other e n e r g i e s and t h i s w i l l be d i s c u s s e d i n s e c t i o n 3.5.  3.4  R e s u l t s from the Gamma Ray A n a l y s i s There were f o u r p o s s i b l e f i n a l r e s u l t s f o r the gamma ray  analysis (ACC)  at each  or  the a c c e p t  normalized  to  either  p a r t i c l e s counted results  were  statistical the  angle, a c c o r d i n g t o whether the accept o n l y  results  plus  reject  t h e charge  (A*R) sums collected  were  (Qnorm)  (Pnorm) f o r each s p i n o r i e n t a t i o n .  punched  out on  cards  with  their  used, or the  A l l four respective  e r r o r s and analyzed by a computer program i n which were averaged  four p o s s i b i l i t i e s .  The  at each of the seven  output  from  this  angles f o r a l l  program  included  these averages and t h e i r r e s p e c t i v e e r r o r s transformed i n t o t h e  41  centre of mass frame. It  was  a f t e r t h i s averaging  that the A + R r e s u l t s were more results.  It  was  was done that i t was n o t i c e d  self-consistent  than  the  ACC  a l s o noted that the Qnorm r e s u l t s were more  c o n s i s t e n t than t h e Pnorm r e s u l t s . , The method  of  determining  these f a c t s was as f o l l o w s . . At  the  seven angles  measured f o r each energy, the number  of r e s u l t s t h a t were w i t h i n one standard average  was  counted,  deviations  (2a)  follow  normal  94.5%  a  should  expected can  t h e number between one and two counted,  etc.  distribution,  be w i t h i n 2a.  be found from  where  <1CT) of the  Assuming  67%  should  standard  these  numbers  be w i t h i n  1a, and  The number o f points t h a t could  t o l i e more than 2a away from the a p p r o p r i a t e  binomial  the  mean  and  standard  be  average  deviation  of  the  distribution  N i s the number of samples and n i s the number of events  that occur passes and,  was  deviation  with p r o b a b i l i t y , p.  over the angular  f o r p = .055, t h e  deviation  (^Np(1-p))  In the  average.  case  mean is  (Np)  1.4.  This  is  2.0  and  the  than  2a  away  it  for  Qnorm  points  standard  from  the  was always t r u e f o r the Qnorm A+R  A s m a l l i n c r e a s e i n the e r r o r s was r e g u i r e d the  four  Thus no more than two o r three  results. true  of  d i s t r i b u t i o n , there are 36 data  p o i n t s would be expected t o l i e more appropriate  normal  ACC r e s u l t s .  needed a s l i g h t i n c r e a s e i n the e r r o r s .  to  make  The Pnorm r e s u l t s a l s o The l a t t e r r e s u l t  can  be understood when i t i s r e c a l l e d t h a t the p a r t i c l e y i e l d s were  42  fluctuating  because  of  small  beam  shifts  v a r i a t i o n s , while the charge c o l l e c t e d these  changes.  There  may  also  was  have  or  polarization  not  been  sensitive  to  s m a l l dead time  v a r i a t i o n s i n the p a r t i c l e y i e l d s f o r which no c o r r e c t i o n s were made.  The  improvement of the  understood  from  reject/accept  the  over  fluctuations  self-consistent.  In  a n a l y s i s of the raw addition,  being used  reasons,  analyses. were  the  Disagreement  i n the  because  analysis,  shown  Qnorm  using  with  example  A+R  data was a  the  larger A+S  in  Figure  y i e l d s . . The is  i n the  the  most  number of  results  had  For the above  r e s u l t s were used  in a l l further  the  other  t h e Qnorm A+R  of  an  three  sets  plot  is  o r d i n a t e of  the  The  the  distribution  asymmetry,  ordinate  sum  the  of  1 2  of  as  angular  polarization,  and  reaction.  plus s i g n s are the average  The  polarized  C(p,Y )» N is 3  0  the  angular  distribution  the d i f f e r e n c e of these  q  D  data.  the s p i n up and s p i n down  polarized  defined  and  y i e l d s d i v i d e d by 2 ^ A , where P i s the magnitude A  raw  observed.  angular  III-3. ,  of  r e s u l t s occurred o n l y r a r e l y ,  d i s t r i b u t i o n obtained f o r the r e a c t i o n  distribution  plot  earlier  is  results.  and no s y s t e m a t i c e f f e c t s were  angular  results  At some e n e r g i e s , the v a r i o u s parameters of i n t e r e s t  extracted  An  ACC  mentioned  s m a l l e r s t a t i s t i c a l e r r o r s than the ACC two  the  ratios.  Thus the Qnorm A+R  counts was  A+R  of  the  beam  i s r e l a t e d t o t h e t o t a l s t r e n g t h of the  the surrounding p o i n t s with e r r o r bars  values at each angle,  are  the  actual  data.  The c o n s i s t e n c y of the s e p a r a t e measurements i s seen t o be very good.  43  Fig. III-3  : The complete angular d i s t r i b u t i o n measurements a t E£ = 10 MeV. The o r d i n a t e o f the upper p l o t i s the sum o f the s p i n up and s p i n down y i e l d s . The ordinate o f the lower p l o t i s the d i f f e r e n c e of these y i e l d s d i v i d e d by 2<Ph (see t e x t ) . The plus s i g n s a r e t h e averages of the measurements a t a given angle; the surrounding p o i n t s with e r r o r bars are the a c t u a l measurements at t h a t a n g l e . 0  44  One  other i n t e r e s t i n g  t e s t of the c o n s i s t e n c y of the data  sas made.  For every energy,  analyzing  power  at  the results  f o r the  each angle were averaged  The  2  averaged  in  pairs  measured f o r a given angle a  given  angle  were  ( i . e . the f i r s t  angle  distributions  procedure  at  to  increase the  The r e s u l t i n g c h i - s g u a r e s f o r a l l  were a t o t a l o f 128 v a l u e s f o r both This  two measurements  averaged)  and angles were then counted  power.  were  t o g e t h e r and then the next two  were  number o f c h i - s g u a r e v a l u e s . energies  data  i n the order i n which they were  averaged  measurements a t t h a t  and  and a c h i - s g u a r e  { X ) f o r t h e averaging process was c a l c u l a t e d . actually  yield  i n 0.1 wide b i n s .  the  yield  would be expected  with one degree of freedom.  and  There  analyzinq  to y i e l d chi-sguare Plotted  in  Figure  IXI-4 a r e the r e s u l t i n g histograms - t h e s o l i d curved l i n e s a r e the  expected  behaviour  results.  There appears  t o be no n o n - s t a t i s t i c a l  i n t h e a n a l y z i n g power histogram;  s m a l l excess o f p o i n t s between x  2  there i s p o s s i b l y a  = 0.7 and 2.1  in  the  Not shown i n t h i s f i g u r e a r e p o i n t s with x  histogram.  There were 6 o f these i n t h e  yield  and  5  in  2  yield > 5.0.  the a n a l y z i n g  power. ., The number o f c h i - s g u a r e values expected t o be g r e a t e r than 5.0 can be found The  probability,  freedom.  p,  from the binomial d i s t r i b u t i o n as before. of  x  2  > 5.0  is  .025  1  degree  of  Then t h e mean {with N = 128) i s 3.2 and the standard  d e v i a t i o n i s 1.8.  Thus between 1 and 5  are  be  expected  appears  for  to  greater  than  values  5.0.  of  O v e r a l l then, there  t o be no s t r o n g evidence f o r n o n - s t a t i s t i c a l  i n any of the data.  chi-sguare  behaviour  F i g . I I I - 4 : D i s t r i b u t i o n of x f o r the C ( p , Y ) i 3 N y i e l d s and asymmetries. The chi-sguares were obtained by averaging the data i n p a i r s (see t e x t ) . The s o l i d curves r e p r e s e n t the expected x - d i s t r i b u t i c n f o r 1 degree of freedom. z  t 2  0  2  46  3.5 Beam P o l a r i z a t i o n The  spectra  Measurements  obtained  from  the  measurements  helium  p o l a r i m e t e r were p r i n t e d out channel  final  analysis  measurements polarizations The l a t t e r energy. taken  was c a r r i e d out by hand.  are  shown  as  in  Table  determined  from  by channel and  1 2  along  C(pVP )  analyzing  powers  with  the  C reaction.  l 2  0  values are the averages of the runs The  the  The r e s u l t s o f these  II1-1 the  using the  for  the  given  f o r t h e *He measurements were  from the data of Schwandt e t a l . , (SC 71), and f o r the  measurements  from  results  given  are  Meyer  et  for  the  a l . (ME 76) .  No  1 2  C  polarization  10 Me? and 11.2 MeV  * C(p,p )* C 2  2  o  data.  Reference  t o t h e data o f Meyer and P l a t t n e r (ME 73b) and  T e r r e l l e t a l . (TE 68) 1 C (p, p ) *.C 2  2  o  is  shows  varying  that  rapidly  the at  analyzing  would  affect  the  analyzing  O v e r a l l , the agreement between the two is  very  from power  measured  significantly. measurements  good., S i n c e the helium p o l a r i m e t e r measurements were  power i s very c l o s e t o 1.0 throughout were  polarization throughout An  the  different  e s s e n t i a l l y f r e e from u n c e r t a i n backgrounds and  values  for  these e n e r g i e s , so t h a t  s m a l l d e v i a t i o n s o f the a c t u a l beam energy energy  power  used was  in  the  assumed  to  analyzing  the r e g i o n , t h e i r average  subseguent be  the  analysis,  constant  at  each  and  the  energy  each s e r i e s of runs.  exception  to  this  was necessary  f o r the 16 MeV  data  where, as has already been noted, a s u b s t a n t i a l change  in  the  polarization  of  the  occurred.  Here  the  p o l a r i z a t i o n , as measured f o r each angle  average  value  throughout  the  run,  Table Summary of Beam P o l a r i z a t i o n Reaction 1 2  C(p,P ) O  Polarization  12C  •He(p,p ) •He o C ( P , P ) 12C  1 2  O  Comments  .4601.020 .730±.017  P before  .7251.024  l2  C(p,p J  12C  .7301.020  1 2  C(P,P 5  »2C  .7361.018  o  Measurements  14.0 MeV  E  P  after  E  P  He (P,P ) •He  .7211.013  before  E  P  E  P  E  P  4  D  i2C(p,p ) o  I2  C  *He{p,p ) •He Q  1 2  C(P,P )  .7201.015  C(P,P )  -.0011.004  O  1 2  O  4  4  .7301.026 .737+.010  He{p,p ) •He o He{p,p ) •He o  after  .7311.008  after  -.0011.007  after  E  E  P  He(p,p ) •He o He(p,p ) •He  .7391.013  after  .0121.006  a f t e r E-> P  C(p,p ) 12C o  .7101.030  0  12  12. 8 MeV  = 12.8 MeV 15.0 MeV  E  E  15.0 MeV  P  -  4  12.8 MeV  15.0 Mev  12C(p,P ) 12C 4  =  E> P  P  O  16.0 MeV  15.0 MeV  E  O  =  E-> P  -  C{p,P )  16.0 MeV  P  *2C  12  12.0 MeV  P-  E  .7301.015  o  13.5 MeV 12.0 MeV  P  *He<p p ) •He r  =  ? P  ( c o i l s off)  (coils off)  10.0 MeV  =  11.2 MeV 11.2 MeV  =  11.2 MeV 17.0 MeV  ( c o i l s off)  HQ  was  used. A l s o shown i n Table I I I - 1 are the r e s u l t s of measuring the  polarization "coils  with  off".  u n p o l a r i z e d beam.  The  purpose  of  there  establish  whether  or  systematic  effects  contributing  were  indications  of  angular d i s t r i b u t i o n of gamma r a d i a t i o n f o l l o w i n g  the  I t can be seen t h a t any insignificant.  Functions  unpolarized  s e v e r a l authors  the  to  zero  of  to  any  was  d e v i a t i o n s frcm  beam.  Angular D i s t r i b u t i o n  capture  measurements  with  are u s u a l l y s m a l l and  The  these  as  asymmetries  polarized  3.6  not  These are i n d e n t i f i e d  projectiles  ( f o r example,  has  BI 60,  HO  been 67,  developed  BL 52),  and  by is  given by LW (9)> u  where  R ,R , t  a  r  e  t  C(t,t',k) Re(R R*,) P ( c o s 9) t,t' ,k ^  t  reduced  elements)  reaction  k  matrix elements  corresponding  to  different  III-2 (T-matrix channels  t,t» C(t,t*,k)  r e p r e s e n t s a sum  over angular momentum c o u p l i n g  coefficients and P {cos e ) are Legendre polynomials. The  maximum  value  of  k i s given by w e l l known theorems  l i m i t i n g the complexity of angular  distributions.  E x p e r i m e n t a l l y , the measured angular d i s t r i b u t i o n represented by  can  be  49  w .(e) ^ ^ ^ ( c o s  in-3  e)  u  k  where  the  Q, c o r r e c t k  are g i v e n by Rose (RO Thus the be  related  C(t,t',k),  f o r the  f i n i t e s i z e o f the  detector,  53) .  e x p e r i m e n t a l l y determined c o e f f i c i e n t s . to  as  the  and  T-matrix elements through the  A, ,  can  coefficients  follows, A Q k  k  Y. C(t,.t',k) R e ( R R * , )  =  III-4  t  t,t'  Methods f o r c a l c u l a t i n g these c o e f f i c i e n t s have been Sharp e t a l . (SH It  is  following  54)  shown  the  of p a r t i a l l y  111-2  must be  t  modified by  i s the  normal t o the Madison  work of Devons and Satchler spin  (SA 55),  1/2  making the  Goldfarb that  p a r t i c l e s , the  (DE  57),  for  the  expression  replacement  —  k  Re(R  Here P  i n the  polarized  Re(R R*,) P ( c o s 9)  by  among o t h e r s .  development by  case  given  R* ) t  P ( c o s 9) k  incident reaction  Convention  Legendre f u n c t i o n s ,  Im^R*,)  f ( t , t ' ) fi-fi P*(cos k  beam p o l a r i z a t i o n , plane ( i n the (BA  and  +  the  70)),  n  i s a unit  d i r e c t i o n defined  the  P (cos e) k  are  f a c t o r f ( t , t * ) i s given  9)  vector by  the  associated by*  k  1  Snover and Ebisawa (SN 75) have found t h a t f , { t , t * ) d i f f e r s by an o v e r a l l s i g n from that given by Devons and G o l d f a r b . ,  50  f  f t  V * ' '  t  n  where j and the  "  ;  -i'Ct'+i) + I(i+D - .1C.1+D - rCi'+D  j ' are t o t a l angular momentum  incident  momenta i  I I I  k(k+i)  projectile  and V,  quantum  correspondinq  to  numbers  orbital  _  5  of  angular  respectively.  Thus, equation I.II-2 becomes W  (8) ^  C ( t , t ' , k ) [ R e ( R R * ) P ( c o s 6) t  t  k  +  t,t\k  III-6 Im-(R  R  j) f ( t , t ' ) /P-n P ( c o s k  k  6)  For the case i n which the proton s p i n i s p e r p e n d i c u l a r the  reaction  plane,  the  results  of  measuring  the angular  d i s t r i b u t i o n of the gamma rays can be expressed as the sum difference  and  of t h e y i e l d s obtained with the proton s p i n up  and s p i n down  .  The sum  gives  angular d i s t r i b u t i o n , ® iQ)t  the  familiar  to  {•»+)  unpolarized  where  u  k  and  the  d i f f e r e n c e can be expressed i n terms of the a n a l y z i n g  power A (0) , as W (9)A(6) *  W  u  t  (  9  ) 2  -/+  (  6  )  =  I  k  Comparison of I I I - 7 and I I I - 8 experimentally elements C(t,t ,k) 1  by as  measured the  A  angular  before,  can  e x p e r i m e n t a l l y determined  k  k  to be  momentum  but now,  III-8  B Q P ( c o s G)  II.1-6  shows  related coupling  to the  that  T-matrix  coefficients  with a p o l a r i z e d beam, the  guantities B  the  can a l s o be r e l a t e d  new to  51  the f  T-matrix elements through t h e simple  multiplicative factor  j t , t ' ) as f o l l o w s , \Q  k  Ic(t,t',k)  =  Im(R R*,) f (t,t')  III-9  k  t  t,t'  I t i s convenient  t o f a c t o r out A , which i s a  measure  of  o  the  overall  above.  strength  of  Then I I I - 7 and  the  r e a c t i o n , from the  III-8 become  Wi(9) + W4-(9) = A [ l + 2  £a Q P (cos )]  Q  V a r i a t i o n s i n avariations overall  and  b  k  k  111-10  6  with energy  are  caused  appropriate  case of s p i n 1/2  T-matrix elements are determined by the  reaction.  by  and  to being only El and these  E2.  parity  conservation,  capture  which lead t o E1 r a d i a t i o n and  By angular  momentum and  p  ,  and  se  i<j>„  amplitude and The  , pe  !((>„•  , de  , and  great  s 1  d .  capture  5/2  Thus the T-matrix elements can  X<J>J  phase f o r the  and  f ,  3/2  labelled  leading  radiation  are, i n j j c o u p l i n g , s  l e a d to E2 r a d i a t i o n .  the  a gamma ray, as i s the case  only f o u r p a r t i a l waves can c o n t r i b u t e , i f the  i s restricted  noting  For  p a r t i c l e s i n c i d e n t on a s p i n 0 nucleus  a f i n a l s t a t e with s p i n 1/2  which  only  i n the T-matrix elements, and not by changes i n the  the p a r t i a l waves which take p a r t i n  here,  k  strength.  The  to  expressions  ^  fe 2  i<j>  P  f  , where s  a r t  ial  and  <j> s  are  be the  wave, e t c .  advantage i n using p o l a r i z e d beams now  becomes  52  apparent.  There a r e seven T-matrix element  parameters  to  determined - four amplitudes and three r e l a t i v e phases. highest  multipolarity  of  unpolarized  I f the  Y - r a d i a t i o n i s two then the maximum  value of k i n the summations I I I - 7 and I I I - 8 with  i s four.  Hence  beam, only f i v e c o e f f i c i e n t s can be measured  e x p e r i m e n t a l l y , but with p o l a r i z e d beam, n i n e c o e f f i c i e n t s be measured.  be  can  Thus measurment with a p o l a r i z e d beam e n a b l e s , i n  p r i n c i p l e , t h e d e t e r m i n a t i o n o f a l l seven T-matrix elements. The  relation  between  t h e A, and B k  T-matrix  parameters  are  amplitudes have been squares  of  the  listed  c o e f f i c i e n t s and the  k  i n Table III-2.  renormalized  so  that  the  It  sum  amplitudes w i l l be egual t o A .  of the  The r e l a t i o n  q  between these r e a c t i o n matrix elements and the matrix elements  The capture  actual  reduced  (R^R^,) i s given i n Appendix B.  i s seen  from  this  table  that  A2  i s dominated by  e l e c t r i c d i p o l e terms, with only i n c o h e r e n t c o n t r i b u t i o n s the much weaker guadrupole terms. electric  guadrupole  very s m a l l , while interference.  A  The  term  same  A^ i s a pure  and can t h e r e f o r e be expected to be  and  a  The c o e f f i c i e n t  from  A  3  result  from  considerations  dipole-guadrupole  hold  f o r the B, s, 1  k  although, as shown i n Table I I I - 2 , these are pure terms  { i . e . , no  and B  2  terms  interference  such as s , p , d , or f 2  2  2  2  are present)  w i l l not n e c e s s a r i l y be dominated by d i p o l e r a d i a t i o n i f  the s,d phase d i f f e r e n c e i s near 0° or 180°. Thus i t i s that  the  dominant  effect  of  the  r a d i a t i o n i s i t s i n t e r f e r e n c e with the the  consequent  presence dipole  of  guadrupole  radiation,  appearance of Legendre and a s s o c i a t e d  f u n c t i o n s o f odd degree.,  seen  and  Legendre  53 Table I I I - 2 R e l a t i o n between the Angular D i s t r i b u t i o n C o e f f i c i e n t s and the Reaction Matrix Elements * A  o  = s  2  * p2 + d  A. = 2 . 4 5 s p c o s ( d > 1  A  0  = 0.5p  2  •f  2  ) - 0.  -<f>  s  p  - 0.5d  2  2  + 2 .55dfcos  -<j>.J  35pdcos(<j>  p  d  (<j>  -  <f>  )  d f-  + 0.57fz * 1.41sdcos ((f> -<f> ) s d  2  - 0. 35pf cos (4> -(f) ) P  A  Q 3  k  = 2.00sfcos{d, - d i . ) s r  = -0.57f  h  2  * 2.08pdcos(<f> -<j> ) - 1.13dfcos (d> -<f> ) p d d f  + 2.80pfcos  (<f> - f (  p  ) f  )  B  :  = 1.22spsin ( < | ) - < f ) ) + 0.69pdsin(*  B  2  =  B  3  s  p  -0.71sdsin<<l' -(t' s  = -0.67sfsin  (<(' -<f'  B^ = - 0 . 7 0 p f s i n  (<f> -<f>  s  p  f  d  f  d  )  + 0.2 9p.f s i n  ) - 0. 69pdsin  p  - 1. 27df s i n  -(f> ) d  (* -<l> d  p  (<f> -<f> p  f  d  + 0. 09df s i n  )  ((f> -<f> d  were  arrangement d e s c r i b e d  f  )  )  The s o l i d angle c o r r e c t i o n f a c t o r s Q They  )  (<f> -<!> )  * The r e l a t i o n between the r e a c t i o n matrix elements and of eguation I I I - 2 i s given i n Appendix B.,  III-3.  f  calculated  are l i s t e d i n T a b l e  f o r the p a r t i c u l a r  i n Chapter I I using  w r i t t e n f o r t h i s purpose  k  (LE 64).,  the R  a  computer  detector program  54 Table III-3 S o l i d Angle C o r r e c t i o n F a c t o r s Q  1.000  3.7  Q  l  Q 0  V  2  .995  .985  .970  R e s u l t s of the Angular D i s t r i b u t i o n The  averaged  data  at  each  ®h  .950  Fits  energy  were f i t by a l i n e a r  l e a s t sguares technique with t h e angular d i s t r i b u t i o n I I I - 7 and I I I - 8 . . T h i s f i t was two p a r t  computer  involved  are  analysis  linear  in  quite straightforward given by Bevington The  Figure I I I - 5 , and the  listed  in  (BO 76a).  Since  parameters,  closely  the  this calculation i s  follows  the  cross  and  asymmetries  and  fits  are  Table  III-4.  Note  2  are  occurs at 13.5  acceptable. there  the  are  acceptable  reasonable.  MeV,  level  in  of  are  freedom, I t i s seen  2  the  sense  that  the  The worst case f o r the asymmetry  where the x , o f 1.87 2  corresponds to  f o r 3 degrees of freedom,  The worst case f o r the y i e l d reduced  in  that the c h i - s g u a r e s t h a t are  thus reduced c h i - s g u a r e s ( x , ~ x / v ) •  t h a t a l l these f i t s  confidence  plotted  The e x t r a c t e d c o e f f i c i e n t s and t h e i r e r r o r s  are  chi-sguares  prescription  s e c t i o n s and f i t s a r e p l o t t e d i n  guoted have been d i v i d e d by the number of degrees v,  eguations  (BE 69a).  differential  Figure III-6.  performed as the f i r s t s t e p of a  the  and  functions  chi-sguare  of  4.24  c o n f i d e n c e l e v e l f o r 2 degrees of freedom. m a r g i n a l l y a c c e p t a b l e , nothing unusual was  a  13%  which i s c e r t a i n l y  occurs  at  14  corresponds to a While t h i s i s  MeV; 1.4% only  noted i n the f u r t h e r  55  i  ~i  45  J  i  I |——i  i  i  I I  90  135 6.  Fig. III-5 :  L  45 AT}  1  r  i  i  90  135  (DEGREES)  C (p, Y ) » 3 normalized differential cross sections. The s o l i d lines are from a least sguares f i t t o t h e data (see t e x t ) . Statistical errors are shown where they a r e l a r g e r than the spot s i z e .  1 2  D  N  56  i  J 45  1  I 90  1  L_  I I  135 e  Fig.  I  1—  T A R  1  1  r  I  L  L  45  90  135  (DEGREES)  III-6 : * 2 C ( p , Y ) i 3 N angular distributions for the asymmetries. The s o l i d l i n e s are l e a s t sguares fits to the data (see t e x t ) . S t a t i s t i c a l errors only are shown. 0  Table III-4: 4TTA  P (MeV)  o (ub) (c)  a  l  a  2  C(p,y ) a  3  N Angular D i s t r i b u t i o n Coefficients a  4  '  b  l  b  b  2  3  b  4  X  2 v  —  (a)  (b)  10.0  22.0  .120 ±.012  -.384 ±.039  -.100 ±.028  -.047 ±.045  .0118 ±.0088  .1916 ±.0055  .0248 ±.0053  .0018 ±.0064  1.02  1.54  11.2  21.0  .086 ±.012  -.620 ±.041  -.114 ±.029  .081 ±.045  -.0711 ±.0097  .1125 ±.0053  .0267 ±.0058  .0124 ±.0066  1.02  0.49  12.0  24.9  .218 ±.011  -.804 ±.038  -.118 ±.024  -.032 ±.041  -.0265 ±.0082  .1602 ±.0047  .0426 ±.0050  -.0009 ±.0056  0.66  0.90  12.8  26.4  .147 ±.010  -.641 ±.034  -.186 ±.024  .100 ±.036  -.1140 ±.0078  .1358 ±.0044  .0418 ±.0047  .0096 ±.0054  3.58  1.15  13.5  22.7  .167 ±.030  -.525 ±.092  -.106 ±.085  .104 ±.085  -.0571 ±.0224  .1860 ±.0179  .0334 ±.0165  -.0077 ±.0190  1.04  1.87  14.0  20.8  .270 ±.014  -.766 ±.050  -.134 • ±.034  . -.052 ±.055.  -.0871 ±.0117  .1713 ±.0064  .0498 ±.0069  .0011 ±.0079  4.24  0.51  15.0  17.8  .197 ±.014  -.713 ±.048  -.269 ±.033  -.023 ±.052  -.0341 ±.0109  .1023 ±.0059  .0394 ±.0064  '.0182 ±.0074  1.15  0.49  16.0  15.7  .245 ±.017  -.497 ±.055  -.148 ±.033  .018 ±.062  -.0359 ±.0132  .1522 ±.0076  .0538 ±.0080  .0186 ±.0092  0.93  0.13  17.0  10.7  .253 ±.037  -.230 ±.107  -.212 ±.077  -.045 ±.121  .0275 ±.0241  .1551 ±.0157  .0251 ±.0157  .0272 ±.0177  2.89  0.24  (a) Reduced chi-squared f o r f i t to y i e l d angular d i s t r i b u t i o n (b) Reduced chi-squared for f i t to asymmetry angular d i s t r i b u t i o n (c) A taken from Figure 6 of reference BE 76b Q  U l  58  a n a l y s i s of the 14 MeV data. Plots  of  the  a. and b k •  III-7.  c o e f f i c i e n t s are given i n F i g u r e k  Also shown i n t h i s f i g u r e a r e t h e  distribution  coefficients  (BE 76b). Only because  when  the the  a  and  fits  extended to i n c l u d e a  3  obtained  to  a  unpolarized  by  Berghofer  coefficients  2  angular  are  et a l . compared,  the data o f Berghofer e t a l . w e r e  and a^ terms, t h e e r r o r s i n  a  x  and  a  2  i n c r e a s e d to such an extent t h a t the o v e r a l l agreement with the present the  results  was obscured.  The s o l i d and dotted l i n e s are  r e s u l t s of c a l c u l a t i o n s with the DSD model, which  will  be  d i s c u s s e d i n Chapter IV. It and b  3  can be noted here t h a t the presence of the non-zero a c o e f f i c i e n t s throughout t h i s energy r e g i o n  implies e l e c t r i c dipole-guadrupole interference. bj  are  also  seen  to  be non-zero.  r a d i a t i o n , but i t can a l s o radiation detail  (see Appendix B ) .  result  3  unambiguously Both  a  x  and  T h i s could a r i s e from E2  from  the  presence  of  M1  T h i s problem i s d e a l t with i n more  later.  3.8 E x t r a c t i o n of the T-matrix  Elements  The capture amplitudes and phases were determined from the extracted  A,  and  k  the  coefficients  gradient  In t h i s part o f the a n a l y s i s , use expansion  a l g o r i t h m of Marquardt  perform a n o n - l i n e a r l e a s t squares Table  III-2.  as the second step i n the  k  computer a n a l y s i s . of  B,  f i t to  the  i s made (MA 63) t o  eguations  of  The f u l l e r r o r matrix i s r e t a i n e d from the f i r s t  59  -.1 -.2  I.  - A^Ao  -.3  +.r o  i  n  r  I  T  +.! 0  -.1 L - " B / A  0  -.2  :  +.1 0  +.05'  L  B / A  C  T  a/A 0 £ -.05  f 10  II  12  13  i  =5= 14  15  16  17  4° (MeV) b  Fig.  III-7  C(p,Y )»3N normalized angular distribution coefficients. Solid p o i n t s r e f e r t o the present data; open c i r c l e s r e f e r to the data of r e f e r e n c e ( B E 76b). The s o l i d and dotted l i n e s a r e from c a l c u l a t i o n s with the DSD model. 1 2  0  60  p a r t o f the a n a l y s i s , the fl^ and analyses  i n c l u d i n g t h e c o r r e l a t i o n terms among a l l  coefficients.  that  these  It  has  correlations  been in  some  v a l u e s obtained f o r the T-matrix elements, their uncertainties Two  (solution  one  in  previous  c a s e s a f f e c t the  and  always  affect  (BU 76a).  solutions  each energy,  found  with  acceptable  corresponding  I ) , and  to  c h i - s g u a r e s are found at dominant  d-wave  capture  the other corresponding t o dominant s-wave  capture  {solution I I ) .  phases  and  The e x t r a c t e d r e a c t i o n  associated errors  amplitudes  and  f o r the two s o l u t i o n s a r e l i s t e d  i n Tables I I I — 5 and I I I - 6 , along with the values of the reduced c h i - s g u a r e f o r each  fit.  Note  that  both  solution  s o l u t i o n I I occur a t e x a c t l y t h e same value of x * 1 ^ although  i t i s not  shown  explicitly  s o l u t i o n s occur a t e x a c t l y the same section  (a  E 2  Most  value  n  I  and  addition,  i n these t a b l e s , both f o r the  E2  cross  ) .  of  the  reduced c h i - s g u a r e s are c l e a r l y a c c e p t a b l e ;  the o n l y p o s s i b l e e x c e p t i o n i s f o r the f i t a t  12.8  MeV  where  the value o f 4.21 corresponds t o a c o n f i d e n c e l e v e l of 1.5% f o r 2 degrees The plotted the  of freedom. amplitudes  and  i n Figure III-8.  figure  correspond  relative The s o l i d  to  the  phase  f o r E1 capture a r e  l i n e s i n the upper p a r t  d-wave and s-wave amplitudes,  r e s p e c t i v e l y , c a l c u l a t e d with the DSD model and the s o l i d i n , the  bottom  part  of  the  f i g u r e corresponds  d i f f e r e n c e between t h e s-wave and d-wave, the  model.  The  of  also  line  t o the phase  calculated  by  c a l c u l a t e d d-wave amplitude i s seen t o agree  w e l l with the d-wave amplitude  from  solution  I.  Essentially  61  Table I I I - 5 T-matrix  Element F i t s t o "* CC((pp,,Y o )i* N Angular Solution I. 2  p  3  o )  s  E->  Y  a  P  f  <j>  -<j>  <f>  Distributions.  2  ~<f>  (MeV) 10.0  .292 ±.008  .040 .9546 ±.015 +.0035  .041 ±.006  -66° ±17°  65° ±12°  0.54  11.2  .240 ±.013  .072 +.022  .9666 ±.0048  .056 ±.010  -28° 134.7° 54<> ± 9° ±2.8° ±12°  2.85  12.0  .369 ±.016  .058 .9236 ±.012 ±.0078  .087 +.016  68° ±25°  0.87  12.8  .321 ±.015  .129 ±.023  .9322 ±.0087  .106 ±.012  -31° 137.5° ± 4° ±2.0°  51° ± 7°  4.21  13.5  .292 ±.028  .042 ±.038  .9533 ±.0108  .068 ±.025  -35° ±46°  103.8° ±5.0°  68° ±24°  0.17  14.0  .344 ±.017  .030 .9314 ±.014 ±.0071  .116 ±.015  41° ±59°  131.2° ±2.6°  106° ± 9°  0.43  15.0  .238 ±.015  .115 ±.024  .9635 ±.0057  .041 ±.031  16° ± 7°  140.0° ±3.0°  125° ±17°  2.69  16.0  .232 ±.012  .066 .9671 ±.016 ±.0046  .081 +.012  -12° 101.2° 101° ±20° ±4.3° ±10°  1.61  17.0  .282 ±.026  .067 ±.042  .104 -117° ±.017 ±21°  all  .9514 ±.0104  74.5° ±2.2°  138.2° 157° ±1.8° ± 9°  53.6° ±6.0°  59° ±12°  1.31  t h e o r e t i c a l models p r e d i c t that t h e dominant t r a n s i t i o n i n  the GDR w i l l be t h e one i n which t h e o r b i t a l  angular  momentum  o f the absorbing p a r t i c l e i s i n c r e a s e d by one u n i t and there i s no s p i n f l i p  (WI 56).  In t h e present case, t h i s corresponds to  62 Table III-6 T-matrix  E-> p  Element F i t s to C ( p , Y ) * 3 j j Solution I I . l 2  Q  s  p  d  angular  $ -<f>p  f  s  Distributions,  d  s  <t>f-<\> s  xv  2  c  (MeV) 10.0  .931 ±.004  .035 ±.008  .361 ±.009  .045 ±.014  13° ±16°  126.8° 142° ±1.6* ±14°  0.54  11.2  .857 ±.009  .059 ±.010  .507 ±.013  .070 ±.019  91° ±15°  157.7° 168° ±1.0° ±10°  2.85  12.0  .783 ±.013  .085 ±.020  .612 ±.015  .061 ±.014  -17® ± 8°  151.8° ±0.8°  88° ±21°  0.87  12.8  .803 ±.013  .111 ±.013  .572 ±.014  .125 ±.021  95° ± 9°  153.9° ±0.8°  173° ± 5°  4.21  13.5  .881 ±.017  .065 ±.024  .467 ±.026  .046 +.034  37° ±27°  139.0° ±3.3°  150° ±45°  0.17  14.0  .807 ±.013  .117 ±.018  .578 ±.017  .029 ±.022  22° ±10°  148.9° ±1.2°  132° ±44°  0.43  15.0  .849 ±.010  .027 ±.012  .514 ±.015  .119 ±.021  44° ±36°  160.3° 155° ±1.1° ± 7°  2.69  16.0  .900 ±.008  .075 ±.015  .423 ±.015  .072 ±.017  43° ±10°  144.7° 121° ±1.8° ±16°  1.61  17.0  .956 ±.011  .089 ±.024  .267 ±.025  .086 ±.039  -9° ±13°  122.1° ±6.5°  1.31  dominant  d-wave  capture,  and  so  agrees  167° ±16°  with  solution  T h e r e f o r e , most o f t h e f u t u r e d i s c u s s i o n w i l l be based on  I. this  solution. The  relative  phase  <|>- <f> d  g  shows  some  very  interesting  63  1.0  •6 » •« » °  •d-wave o s-wave 0.5  J5S0LUTI0N I  1.0 o  o  «  C  Q  o  •d-wave os-wave 0.5  s  200  SOLUTION 3T  c  o s !00  o  o  o  o  o  s  c  ©Solution I ©Solution IE 10  II  12  0(T0s _J L 13  14  15  16  17  EJ: (MeV) Qb  Fig.  III-8  E 1 amplitudes and r e l a t i v e phase. Errors are shown where they a r e l a r q e r than the p o i n t s i z e . The s o l i d and dotted curves r e p r e s e n t c a l c u l a t i o n s with the DSD model (Chapter IV) .  64  structure. this The  There appears t o be a broad  phase  difference,  and  overall  resonance  a s u b s t a n t i a l d i p near 13.5 HeV.  r e l a t i v e change i n the phase p o s s i b l y i n d i c a t e s  one  of  the  reaction  amplitudes  participates  The d i p might r e s u l t from i n t e r f e r e n c e  pygmy  some l e v e l near 13.5 MeV.  the l e v e l observed by Hasinoff in  that  in  resonance. and  to  only  t h e pygmy  between  the  The nearest candidate i s  et a l . (HA 72) at E  — —  = 14.04 MeV x  i 3 N (E+ = 13.12 MeV) with a width o f ^170 keV.  I t would  be  P  necessary  to  measure  in  finer  energy steps t o c l a r i f y  this  point. Parameters a s s o c i a t e d for  s o l u t i o n I are l i s t e d  Figure The  III-9  along  with the E2 r e a c t i o n i n Table I I I — 7 , and  with  are  plotted  approximate constancy o f t h e p,d phase d i f f e r e n c e  resonating,  unless  is  There  very  i t i s the s-wave phase t h a t  t h e p-wave and d-wave phases happen t o  both be undergoing the same phase changes, which would be surprising.  in  the r e s u l t s of the DSD c a l c u l a t i o n .  i n t e r e s t i n g i n that i t i m p l i e s t h a t is  matrix elements  very  appears t o be some f l u c t u a t i o n s i n the f , d  phase d i f f e r e n c e between E^ = 10 MeV and 14 MeV.  Here again i t  would i n t e r e s t i n g t o measure i n f i n e r steps to i n v e s t i g a t e t h i s structure The  more f u l l y . e r r o r s quoted f o r a l l of the parameters e x t r a c t e d  the data are s t a t i s t i c a l e r r o r s only, possible  errors  and do not  include  as beam s h i f t s on the t a r g e t o r e r r o r s  charge c o l l e c t i o n and beam p o l a r i z a t i o n measurements. already  been  shown  that  from such  i n the I t has  the data i s s e l f - c o n s i s t e n t without  i n c l u d i n g e r r o r s from these sources., The e f f e c t o f v a r y i n g t h e polarization  by  2%  caused  about  a  4%  change  in  the  B  v  65 Table III-7 E1 Amplitude Ratio s/d and Parameters Related t o E2 Capture f o r Solution I E+  s/d  P  p/f  + -* p  f  'E2  ^ f ' ^ j ^p'^d a  (MeV)  El+ E2 0  10.0  .306 ±.010  .99 ±.49  -131° ±13°  - 9° ±12°  -140° ±17°  .00328 ±.00081  11.2  .248 ±.014  1.28 ±.41  - 82° ±11°  -81° ±12°  -163° ± 9°  .0083 ±.0035  12.0  .399 ±.020  .67 ±.15  - 97° ±25°  19° ± 8°  • 78° ±25°  .0109 ±.0037  12.8  .344 ±.020  1. 22 +.20  - 82° ±7«>  - 87<> ±8°  •169° ± 50  .0279 ±.0076  13.5  .306 ±.033  .61 ±.73  -103° ±29°  -36° ±24°  -139° ±47°  .0063 ±.0025  14.0  .369 ±.021  .26 ±.14  - 65° ±52°  -25° ± 9°  - 90° ±59°  .0144 +.0030  15.0  .247 +.017  2.8 ±1.4  -142° ±13°  -15° ±17°  -157° ± 7°  .0149 +.0045  16.0  .240 + .013  .81 ±.30  •114° ±17°  0° ± 9°  •114° ±20°  .0109 ±.0012  17.0  .296 ±.030  .64 ±.50  -176° ±21°  5° +11°  -171° ±20°  .0153 ±.0036  coefficients,  but  this  usually  r e s u l t e d i n a change o f l e s s  than 1% i n the values o f t h e T-matrix elements. Because the e g u a t i o n s of Table I I I - 2 a r e n o n - l i n e a r , there i s no guarantee t h a t  additional solutions  do  not  exist.  An  F i g . I I I - 9 : The amplitude ratio and phases r e l a t e d to E2 capture. The s o l i d and dotted curves represent c a l c u l a t i o n s with the DSD model (Chapter I V ) .  67  attempt  to  made i n  the  arbitrary  locate  a t l e a s t some o f these other s o l u t i o n s was  following  value  way.  First, a  expressed  as  a  was  E 2  fixed  at  f r a c t i o n of the t o t a l c r o s s  s e c t i o n , and a l l t h e other parameters were allowed t o x •  minimize  Then a  2  was  some  stepped  to  vary  to  a new value and the  E2  process  was  repeated,  parameters a t each the  previous  with  the  step.  In  this  way,  the  cast  2  of  the  E 2 strength  the  The r e s u l t s o f t h i s s e a r c h a r e p l o t t e d i n F i g u r e I I I - 1 0 .  point.  The  first,  doubly degenerate described  ratio  minimum.  and  deepest,  The second  as  the  starting  minimum i s i n each case t h e  minimum i n each case  II  appears  to a s o l u t i o n which has a d i f f e r e n t value of the  than  is  obtained  f o r the s o l u t i o n s at the f i r s t  A t y p i c a l v a l u e o f s/d f o r s o l u t i o n I i s 0.3, while a  t y p i c a l value f o r the second that  I  s o l u t i o n corresponding t o s o l u t i o n s I and  previously.  correspond  s/d  f o r the  projection  onto  These r e s u l t s were obtained with s o l u t i o n  to  guesses  s u c c e s s i v e step being the values obtained i n  m u l t i - d i m e n s i o n a l x ~ s u r f a c e was axis.  starting  are  obtained  at  s o l u t i o n i s 0.7.  the  second  The  parameters  minima are l i s t e d i n Table  III-8.  At some, but not a l l , e n e r g i e s , the also  found  to  be  doubly  degenerate.  s o l u t i o n s , where they appeared, family  were found  t h a t begins with s o l u t i o n I I .  second  solution  These to  other belong  was  second t o the  No exhaustive search was  made t o f i n d them a l l . Several statistical  of  the  second  grounds.  c o n f i d e n c e l i m i t f o r each  Shown  solutions in  Figure  f i t ( x •= 9 . 2 ) . 2  can  be  III-10  excluded is  the  on 1%  A l l o f the s o l u t i o n s  68  1  1  I  l  I  I  0  1  I  I  1  1  I  1  I  1  I  1  10  1  l 0.08  I  I  1  l l 0.16  I  I I I ' I  I  1  I  1  I  1  I  1  I  I  1  MeV  i I i 0.24  I 1I •  °E2  Fig. Ill-10  1  0  I i I . I . I . I . I . I 0.08 0.16 0.24  TOTAL  : Projection of the multidimensional onto the E2 s t r e n g t h a x i s .  x -surface 2  69 Table I I I - 8 Second S o l u t i o n s t o t h e T-matrix Element s/d  E^ P  Fits J  p/f  p  T  f  V  f  *d  p  d  El+ E2  CT  {MeV)  E2 a  10.0  •790 ±. 022  * 73 3 ±. 026  135.8° ±1.4°  -179° ± 7°  -67.6° ± 2.6°  113° ± 5°  • 1509 ±. 0053  24.7  11.2  647 ±. 029  •719 ±.068  154.7° ±1.3°  - 133° ±13°  -91.1° 136° ± 5. 1° ± 8°  • 0849 ±. 0080  25.5  12.0  • 642  1. 11 ±. 16  148.2° ±1.2°  -167° ±21°  72.8° -95° ± 9. 4° ±12°  0402 0049 ±.  5.22  • 560 ±. 059  140.4° ±4. 8°  -202° ±38°  -59. 1° ±10.5°  99° ±27°  09 32 0194 ±.  3. 79  *  391 084  145.7° ±1.5°  -186° ±32°  -56.2° ± 7.2°  118° ±25°  0562 0071 ±.  2.43  «  806  158.8° ±1. 3°  -143° ±13°  -80.0° ± 5.7°  137° ± 8°  •0902 ±. 0078  13.3  *  035  *  12.8* 13.5  #  710  ±. 089  14.o  • 659  038  *  *  15.0  •767 034  16.0  • 833 ±. 040  • 727  07 2  146.1° ±1.8°  -146° ±12°  -75.7° ± 4.9°  138° ± 8°  • 1234 ±. 0089  11.6  • 787  941 ±. 102  136.5° ±4.8°  - 151° ±14°  -72.9° ± 5.5°  137° ± 9°  2165 ±. 0172  5.97  17.0  063  ±. 088  *  * No second s o l u t i o n could be found at 12.8 MeV {see t e x t ) .  at  the lowest v a l u e o f o  second  solutions,  17 MeV f a l l  E 2  fall  below t h i s l i m i t ,  but  of the  o n l y those a t 12 MeV, 13.5 MeV, 14 MeV, and  below the l i m i t .  Reference t o  Table  III-8  shows  t h a t t h e second s o l u t i o n s at 15 MeV and 16 MeV can o n l y j u s t be excluded on t h i s No MeV.  second  basis. solution  could  be found f o r t h e data a t 12.8  Searches s t a r t i n g from many d i f f e r e n t parameter s e t s were  70  made,  but  they  always  resulted  in  convergence  either  to  s o l u t i o n I or t o s o l u t i o n I I . ,. A  few  sections minima  other s o l u t i o n s were found which a l s o had E2 c r o s s  that were l a r g e r than  the  solution  I  values.  The  o f c h i - s g u a r e f o r t h e s e a d d i t i o n a l s o l u t i o n s always l a y  a t unacceptably high l e v e l s . The subject  e x t r a c t i o n o f a l l the s o l u t i o n s d e s c r i b e d  to the c o n d i t i o n t h a t t h e r e i s no M1 r a d i a t i o n  involved  al.  (HA 74a)  i n the r e a c t i o n . that  a  Aj and  most  check  on  out by Hanna this  to  the  can be made by  These c o e f f i c i e n t s  presence  of  M1  analysis,  the A  2  and B  l  N (p, y ) 16o  reaction,  Q  are  r a d i a t i o n (see  using t h e T-matrix elements r e s u l t i n g  from  such  c o e f f i c i e n t s can be c a l c u l a t e d and  compared t o the experimental values. 1 S  et  condition  from the a n a l y s i s .  sensitive  Appendix B). an  I t was pointed  consistency  excluding the  thus f a r are  Hanna  In t h e i r a n a l y s i s of  et  a l . found  c o e f f i c i e n t s were s a t i s f a c t o r i l y reproduced  that  from  an  the these  analysis  which excluded them. The  results  p l o t t e d i n Figure  of such an a n a l y s i s f o r the present data are III-11.  It  is  seen  that  the  calculated  c o e f f i c i e n t s do not agree with the e x p e r i m e n t a l c o e f f i c i e n t s i f only  the  experimental  e r r o r s i n the  e r r o r s are taken i n t o account.  calculated  coefficients  are  also  taken  account, then the agreement i s much more s a t i s f a c t o r y . of  the  apart,  points  l i e more than two combined standard  however, and t h i s could  that  there  this  region.  is  some  be taken  to  be  M1 r a d i a t i o n u n d e r l y i n g  an  I f the into  Several  deviations indication  the s t r u c t u r e i n  71  Et° (MeV) b  F i g . 111-11  The normalized k and B angular distribution coefficients. S o l i d p o i n t s correspond to angular distribution f i t s to the experimental d a t a ; open c i r c l e s correspond to and B coefficients l generated from a T-matrix element f i t to the other seven A, and B, c o e f f i c i e n t s . k k l  1  72  Two  of these  energies  of  offending  11.2  MeV  points  and  12.8  l i e at MeV,  incident  which  are  i n t e r f e r e n c e d i p s i n the 90° y i e l d curve observed al.  (ME 73a).  upper  the broad pygmy resonance. the  lower  dip,  and  observed  11.88  however,  corresponding  in  levels in  MeV. to  l 3  decay  the  11.88  with  M1  experiment  (WI 69).  and  of  the  MeV  The width of t h i s l e v e l i s 130 i s 3/2"  (HS 71).  r a d i a t i o n , the l a r g e between  reproduced  there  ambiguities  has  been  Fleming  mirror  an  present  keV  some  jet a l .  level  10.78  in  l 3  scattering  experiment,  this  level  10.76  MeV.  energy  of  (A3 70) , and i t s s p i n - p a r i t y  observed  at  E^> = 11.2  c o e f f i c i e n t and the Aj  1  C  inelastic  I f t h i s l e v e l indeed decays by  discrepancy  the experimental k  some  with  l e v e l has been observed t o be in  the  the  level interfering  region.  would be e x c i t e d at an i n c i d e n t proton  assignment  the  by Measday et  at e x c i t a t i o n e n e r g i e s of  radiation In  +  still  this  N  The  consistent  by a 3/2  There are  i n d i c a t i o n of M1 s t r e n g t h (PL 68)  near  I t seems t o be f a i r l y w e l l e s t a b l i s h e d t h a t  of these d i p s i s caused  about  proton  from t h e a n a l y s i s i n which  it  was  M1 MeV  coefficient  excluded  could  p o s s i b l y be e x p l a i n e d . The H1  decay  o f the 10.78  (or p o s s i b l y E2) r a d i a t i o n  MeV  level  (91-69).  e x c i t e d at an i n c i d e n t proton energy  i s a l s o c o n s i s t e n t with This  o f 9.58  level MeV  would  i n the  be  present  experiment. The  dominant  M1  strength  decay of the f i r s t T = 3/2  in  t h i s r e g i o n occurs i n the at  E  = 15.07  MeV  X  J  {DI 68).  state  in  However,  this  s t a t e i s very narrow, so i t s e f f e c t s  73  should not be f e l t more than a few tens of k i l o v o l t s on s i d e o f the resonance energy Of for  course,  a  MeV).  i t i s a l s o p o s s i b l e t h a t more s o l u t i o n s e x i s t  the a n a l y s i s with k  was  {E+ = 14.23  either  second  and 3^ excluded.  1  solution  different starting  found,  Only at  however,  one  energy  although  several  guesses were used at each energy.  There are t h r e e problems with the a n a l y s i s i n which A.j and B  1  are excluded.  very narrow to  the  The f i r s t  i s that the minima i n x  problem  just  mentioned.  guantities,  freedom  there  are  0  degrees  must i n f a c t v a n i s h at the  possible  a  The second i s that to seven  2  P ce  are  so the s o l u t i o n s are d i f f i c u l t t o f i n d , which l e a d s  seven T-matrix elements are being f i t t e d  so x  2 _ s  to  of  solution.  because  experimental in  Thus  the f i t , it  is  not  judge whether a s o l u t i o n i s a c c e p t a b l e based on a  chi-sguare c r i t e r i o n .  F i n a l l y , e x c l u s i o n of A  x  and B  existence  more  Conseguently, the e r r o r s on  the  the  other c o e f f i c i e n t s .  e x t r a c t e d E2 c r o s s s e c t i o n s are much l a r g e r  E2  puts the  onus of p r o v i d i n g evidence f o r the on  of  2  than  strength  when  and B^ are i n c l u d e d i n the a n a l y s i s .  3.9  Determination of the Cross S e c t i o n s It  to  is  seen from the above d i s c u s s i o n that i t i s p o s s i b l e  obtain at l e a s t t h r e e s o l u t i o n s t o the parameter  interest Figure  -  the  111-12.  statistically  E2 c r o s s s e c t i o n . The  energies  acceptable  at  of  central  The r e s u l t s are p l o t t e d i n which  there  are  two  s o l u t i o n s {at a ^% c o n f i d e n c e l i m i t )  7a  < r  E  Fig.  2  ITI-12  The E2 c r o s s s e c t i o n s . The s o l i d p o i n t s a r e from s o l u t i o n I (or I I ) and a d d i t i o n a l s o l u t i o n s which s a t i s f y a 1.0 % confidence limit (see text). Open c i r c l e s are from s o l u t i o n s o b t a i n e d with Aj and excluded from the T-matrix element f i t . The solid and dotted c u r v e s are from a DSD c a p t u r e model c a l c u l a t i o n (Chapter I V ) .  75  when a l l the c o e f f i c i e n t s are i n d i c a t e d by two open  points  s o l i d points  correspond  excluded from the by  comparing  included  the  at t h a t energy on  to  analysis the  are  graph.  The  B  s o l u t i o n s obtained with Aj and  analysis.  the  in  The  v a l u e s of a  f r a c t i o n of E2 s t r e n g t h  were  E 2  1  calculated  found i n the  present  a n a l y s i s to the t o t a l c r o s s s e c t i o n  r e s u l t s of Berghofer et a l .  (BE 76b) .  include  The  e r r o r s shown do  normalization  error  c o n s t r a i n t s , i t was the  detector;  from  not  their  the  analysis.  overall  (Because of time  not p o s s i b l e t o measure the  efficiency  the  detector  T h e r e f o r e , the from the  of  i . e . the r a t i o of the number of events recorded  i n the a n a l y s i s window to the t o t a l number of events in  ±20%  by  E2  Y-rays  from  cross sections  present a n a l y s i s ) .  r e s u l t s from a DSD  the  could  The  1 2  C  not  (p,Y ) Q  x3jj  reaction.  be determined  s o l i d and  c a l c u l a t i o n and  initiated  dotted  solely  curves  w i l l be d i s c u s s e d  are  further i n  the next chapter. It  i s seen t h a t  a c c e p t a b l e , and from  the  both the  the s o l u t i o n s obtained with A  a n a l y s i s l i e at higher values of  e r r o r s than the s o l u t i o n s values of  a  more  £2  of  a. E2  The  found  "high"  from the a n a l y s i s with A than  indicates  second s o l u t i o n s , where they  the that  "low" the  at values  2  and  values.  "low"  the  The  2  a„„  B  latter  a,  and  k  Figure  v a r i a t i o n of  III-7) a, E2  r a p i d l y varying  but  second  I)  solutions)  also fluctuate  result  possibly  values are the c o r r e c t ones, because  the observed smooth v a r i a t i o n of the (see  have l a r g e r  (solution  included  l  excluded  1  and  "low"  (the  B  and  are  would would  follow be  energy dependence.  naturally surprising  b,  coefficients  k  from if a  E2  a  smooth  showed  a  76  It a  E2  i s difficult  to know what t o make o f the f a c t t h a t the  l i e at s y s t e m a t i c a l l y higher values  excluded  than  when  they  when  are i n c l u d e d .  A  and  2  Some of the problems  a s s o c i a t e d with t h e a n a l y s i s when they are excluded discussed  in  coefficients.  reference  to  the  have  been  of  these  satisfactory  fits  reproducibility  I t should a l s o be noted  that  Bj a r e  are obtained under t h e assumption t h a t there i s no Ml r a d i a t i o n present.  Unfortunately,  the  only  way  unambiguously i f M1 r a d i a t i o n i s present plane  polarization  of  r e a c t i o n i n i t i a t e d by a  the  outgoing  polarized  i s to  photon  proton  to  ascertain  measure  produced  (BU 75b).  measurement i s not c u r r e n t l y e x p e r i m e n t a l l y p o s s i b l e .  the by  a  Such  a  77  Chapter IV  THE  DSD  MODEL, SUM  The  HOLES AND  COMPARISON BITH OTHER EXPERIMENTS  d e s i r a b i l i t y of having r e a c t i o n models a v a i l a b l e with  which to compare e x p e r i m e n t a l data was  mentioned i n Chapter  I.  This  reaction  E2  is  especially  strength enough  true  in  (p,Y)  studies  where, even with p o l a r i z e d beams, t h e r e i s not information  components.  available  Moreover,  to  separate  the  E1  so i n c a s e s where i t i s d e s i r a b l e  c o l l e c t i v e E2 components ( f o r effective  charges)  be  made,  it  since  between these two model  in  example,  strength Then  will  a  to  have on the  experimental to which the  of  the  observations  Thus the  the  understand  from  of  purpose of  (p,Y)  reactions  direct  and/or  the  a  data  predictions  w i l l give information  t h e r e i n ) , Snover and  Ebisawa  (SN 75)  direct  capture  model  predict  collective  with  o t h e r s (PO  reaction  i s to  e x p e r i m e n t a l l y measured  work of Potokar and  semi-direct  components  better  assumptions about the E2 s t r e n g t h  the  are  experiment i t s e l f cannot d i s t i n g u i s h  presence  comparison  Following  the  components.  the  E2  i s necessary to have a model a v a i l a b l e  the p a r t i c u l a r case of  what e f f e c t s  and  t o l e a r n about  with which comparisons of g u a n t i t i e s e x t r a c t e d can  usually  proton r a d i a t i v e capture r e a c t i o n s  p a r t i c u l a r l y s e n s i t i v e to the presence o f d i r e c t E2 (HA 73b),  of  E2  guantities. the  actual  on the  extent  are  justified.  73 and  references  have r e c e n t l y extended  the  to  and  include  direct  78  collective  E2.  The  model  will  fee  described  comparisons  of the model p r e d i c t i o n s t o the present  briefly,  and  data  will  be made. .. This  chapter  also  contains  comparisons  r e s u l t s t o the i s o s c a l a r EWSR, and t o similar  experiments.  4.1  DSD  The  the  of t h e present  results  of  other,  Model  Lane and Lynn (LA 59) c o n s i d e r e d a d i r e c t c a p t u r e model t o explain  the  observation  of  Cohen  (CO 55)  that  the  cross  s e c t i o n s f o r (p,Y) r e a c t i o n s i n the energy range from 8 MeV 22  MeV  were approximately c o n s t a n t .  These c r o s s s e c t i o n s were  expected t o be f a l l i n g r a p i d l y as a f u n c t i o n of energy compound  nucleus  model  of  Bohr (BO 36) was  Lynn c o n s i d e r e d the case where energy  and  was  compound nucleus was mainly  the  captured d i r e c t l y formed.  extra-nuclear,  so  The that  incoming  were  still  the  too  the  valid.  Lane and  proton  radiated  process was c o n s i d e r e d to details  of  be  the nuclear  Although t h i s model gave  c r o s s s e c t i o n s t h a t were i n order of magnitude they  if  i n t o a bound s t a t e b e f o r e a  i n t e r i o r were r e l a t i v e l y unimportant.  experiment,  to  agreement  with  s m a l l by a f a c t o r o f about  four. Brown (BR 64) extended where  the  giant  (dipole)  scattered  this  model  to  include  the  case  incoming proton e x c i t e d the t a r g e t nucleus i n t o i t s resonance  state  i n t o a bound s t a t e .  and  the  proton  was  then  The e x c i t e d core of the nucleus  79  p l u s bound proton system then de-excited GDB M  state  by e m i t t i n g  semi-direct"  to  a gamma ray.  distinguish  process  considered  by  pictured  schematically  it  Lane  from  from  the  showed  i n a treatment only that  the  and  Saporetti  in  the  1  * C e (p, Y) 2  subsequent paper, of  above  MeV.  20  section and  a d i r e c t E2  investigated  More  direct  and  E2  cross  section  reaction to  to the c r o s s and  to 50  section  MeV.  Longo  found f u r t h e r  improvement  studies  (LO 69)  showed  PO  In a that  important f o r e n e r g i e s with  the  model  have  description  73)., DSD  amplitudes.  model t o  new  calculate,  in  s e c t i o n , the angular d i s t r i b u t i o n s of  the  (p,Y)  the  Because  include the  of  satisfactorily  measurable q u a n t i t i e s . The  process improved  reactions,  analyzing  they  power.  of  In order f o r a  model to be c o n s i d e r e d s u c c e s s f u l , i t should  describe  Brown»s,  f o r t h i s experiment.  (ZI 70,  Book  term between  Ebisawa have extended the  d e t a i l s being measured i n addition  from  approaches to h a n d l i n g the  collective  semi-direct  interference  amplitude was  of the c o l l e c t i v e e x c i t a t i o n Snover and  processes are  measured c r o s s  Saporetti  recent  different  the  p a r t s and  cross Longo  and  P r from 10 MeV  included  semi-direct  calculated  inclusion  1v 3  direct  Clement, Lane and  i n c l u s i o n of the s e m i - d i r e c t  (LO 68)  the d i r e c t and  two  slightly different  the agreement between the c a l c u l a t e d f o r the r e a c t i o n  step  e f f e c t s of a  amplitude i n nucleon capture r e a c t i o n s . 65),  The  one  IV-1.  Other workers have c o n s i d e r e d the  {CL  collective  Brown termed t h i s process  and Lynn.  in Figure  the  all  of  An o u t l i n e of the  d i f f e r e n t i a l cross s e c t i o n  these  be  able  experimentally  model f o l l o w s .  f o r a process undergoing  a  80  ENERGY  initial  COLLECTIVE  P+A A+ I  Fig.  final  IV-1 : Schematic r e p r e s e n t a t i o n o f d i r e c t and s e m i - d i r e c t processes. The i n i t i a l p + A s c a t t e r i n g system can proceed d i r e c t l y t o the f i n a l bound A + 1 s t a t e with t h e emission o f a gamma r a y , or i t can f i r s t excite the GDR of the core b e f o r e t h e proton i s captured i n t o a bound state. The core • bound proton system then d e - e x c i t e s by the e m i s s i o n o f a gamma r a y .  81  transition by  from an i n i t i a l  s t a t e i to a f i n a l s t a t e f i s given  (ME 65b)  IV-1  In t h i s e x p r e s s i o n , state  <f> and <t> are the i  f  initial  and  final  wave f u n c t i o n s , r e s p e c t i v e l y , v i s the i n c i d e n t p a r t i c l e p(E) i s the d e n s i t y  velocity, appropriate  electric  of f i n a l  multipole  states  operator.  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 reduces  to  and  £  i s the  Thus c a l c u l a t i o n o f calculating  matrix  elements of t h e form M.^  Now  the  Hamiltonian  f  = «fr | £!<!,.>  IV-2  f  f o r t h e i n t e r a c t i o n of the i n c i d e n t  nucleon with the t a r g e t nucleus can be w r i t t e n H = H where  H T(r)  and  V (r,t)  ?  as (BS 59b)  + T(r) + V ( r , t )  IV-3  i s the Hamiltonian f o r the A n u c l e a r  particles  i s the k i n e t i c energy of the i n c i d e n t p r o j e c t i l e i s the sum of t h e the  interaction  potentials  between  i n c i d e n t nucleon at l o c a t i o n r and each of the  target  p a r t i c l e s , whose  location  in  totality  is  r e p r e s e n t e d by 5. A  solution  to  the  Schroedinger  eguation  is  given by  $ ( r , 5 ) , where H*(?,t) = E*(r,t)  and  the wave f u n c t i o n  V satisfies  IV-4  82  H f ( r \ t ) = m(r,t)  IV-5  o  where H  o  The  = H + T (r). £ p o t e n t i a l V i s too  solution  to  potential V  IV-4,  complicated  so  the  ^ i s introduced-  where  opt  V = V and  opt  to  complex  permit  an  exact  optical  model  + 6V  IV-6  6V, the r e s i d u a l p a r t i c l e - h o l e i n t e r a c t i o n , i s t r e a t e d  perturbation. enabled  I t was c o n s i d e r a t i o n  Brown and B o s t e r l i  energy of the GDR Adding Hamiltcnian  the E  Q  enables  6V  that  (BR 59a) to c o r r e c t l y c a l c u l a t e the  i n the s h e l l optical  of the i n t e r a c t i o n  as a  model (see Chapter I ) .  potential  a distorted  to  the  free  wave f u n c t i o n  ¥  o p t  particle (r,f)  to be  found from the s o l u t i o n of  H ° V ( r \ f ) = ET° (?,l) P  where H  o p t  = H  pt  IV-7  pt  +V  o  opt  I t i s shown i n Messiah  (ME 65b) t h a t t h e s o l u t i o n  to  IV-4  satisfies  * '^ ( l i i T ' ) * ° (?  =  l+  6V  P t  <^)  where e + 0 i n the l i m i t . Substitution  o f IV-8 i n t o IV-2 g i v e s  IV-8  83  M. . = < ^ | £ | T ° P > . + V ' •' l+f f l i-, E - H ±ie t  1  f  IV-9  1  1  A  where the |<|>^> are the i n t e r m e d i a t e Now two  the  electric  multipole  collective states. o p e r a t o r e can he s p l i t  into  parts S-*  where e  acts on t h e nucleon and e  N  addition, defined  + V  N  the  state  multipolarity  |<j) > have,  in  and  R  ,  Therefore,  , ODt  M = <w e U P > + i+f*f' Nri W  The  <  first  e  term  on  the  width  target.,  1  E - E  IV-11  r  for a  R  IV-9 f i n a l l y  ^ f- ' :^ '•V  +  in  acts  In  t h e present case, only one w e l l  |(|>> of energy transition.  T  iv-to  «!' l l-'? 6V 4  given  becomes  Pt>  R  IV-11  +.ir/2  represents  the d i r e c t  capture  p r o c e s s , and the second term the s e m i - d i r e c t one. Following (DO  67),  functions functions  the d i r e c t  Snover  and  capture  Ebisawa  calculations  expanded the i n i t i a l  i n terms of r a d i a l , angular momentum as  usual.  Since  the  t a r g e t i n i t s ground s t a t e and an electromagnetic  operator  of  initial  of  and  a  sum  of  Snover and Ebisawa carry out a f r a c t i o n a l  expansion  of  final  bound  s e l e c t s out only those p a r t s o f parentage  in  the i n i t i a l  angular momentum a l g e b r a .  spin  nucleon,  operators,  the  s t a t e wave  and the one-body parentage  s t a t e at the b e g i n n i n g . the  final  wave  state consists of a  incoming  consists  Donnelly  state  which  This have  s t a t e , and t h e r e f o r e  s i m p l i f i e s the  Upon  the  reduction  of  resultant  84  angular  momentum  algebra,  they  o b t a i n an e x p r e s s 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 of the d i r e c t capture w r i t t e n i n terms o f t h e r e a c t i o n matrix 3.6)  and  emitted  a  Legendre polynomial  gamma  proportional  ray. to  a  The  p a r t which t  expansion i n the angle  radial  is  elements ( E of s e c t i o n  reaction  direct  f o r the  matrix  of the  elements  are  element, fif., v i a  matrix  13  various  statistical  momentum c o u p l i n g  and  phase  space  c o e f f i c i e n t s , where / X ,(r) |  R  final  X^j( ) r  a  state  ^  n  i J ^  \j^ ^ r  wave  =  J  a  r  }  f  e  t l i e  r  functions,  a  d  =f  i  a  ( r )  I  respectively,  and f ^ (r)  normalized  by  factor e ^ ,  LJ  for  where  ±a  phase, and U {r) i s normalized  V  "  1  2  p a r t s o f the i n i t i a l and  l  d i r e c t e l e c t r i c multipole operator. phase  angular  i s given by  appropriate  a  and  U (r)  ,  where  factors  by  the  i s the  The ^ j ( r )  are  i s the Coulomb  spectroscopic  factor  the f i n a l s t a t e .  With  the  r a d i a l matrix  i n c l u s i o n o f the s e m i - d i r e c t  part o f IV-11, the  element i s modified to U (r) L J  A  where  /  a ^ i s the s t r e n g t h T  r  F  f  ( ) r  T  T*  R  2  D  with which the given  R  resonance  of  order <£, i s o s p i n T, i s e x c i t e d and  ^-T^ ^ ^ r  s  a  f°  r n 3  factor  which  d e s c r i b e s t h e manner i n  which the c o l l e c t i v e s t a t e i s e x c i t e d . Snover and Ebisawa  (SN 76)  choose  a  hydrodynamic  model  85  form  factor  for  ^(r)»  that  d e s c r i p t i o n o f the GDR i s taken a l l the protons  i n t h e nucleus  nucleus,  the  case,  with  -F-QC*)  collective  model  to be that o f an o s c i l l a t i o n of against a l l the neutrons i n the  nuclear  s u r f a c e remaining  rigid.  In t h i s  i s given by r V ^ r ) where V^rJ/4 i s t h e r e a l  symmetry  term i n the o p t i c a l p o t e n t i a l . a  i s , the  For proton  radiative  capture.  i s given by 3h|ze "11  &  i s the  IV-14  4M A<r >E_.  11  P  where  n  z VP 2  fraction  of the c l a s s i c a l d i p o l e sum r u l e  (equation 1-1) exhausted by the resonance o f energy E  and  l l  <r > i s t h e 2  mean  distribution The  extension  squared  radius  of  the  charge  direct  E2 i s  i n t h e nucleus.  of  the  model  to  include  s t r a i g h t f o r w a r d and i n v o l v e s , f o r example, using X - 2 f o r the direct  form f a c t o r  ( r ) . C o l l e c t i v e i s o s c a l a r E2 s t r e n g t h i s  introduced by using a form f a c t o r F^r^r) given by dV (r) F  =  - —  IV-15  r  20  dr  where V {r) i s the r e a l c e n t r a l nuclear p o t e n t i a l . Q  cv  f T  The s t r e n g t h  i s given by * 20 20 2M~E7 p 20 2 p  a  =  I  V  n  where 3  2Q  i s t h e f r a c t i o n of t h e EHSR (eguation 1-2)  '  1  6  exhausted  by the E2 resonance. The model.  guantities  E  ^ » T  a n <  * ^ r  a  r  e  n  o  t  T  They are adjusted t o f i t the  total  s p e c i f i e d by the cross  section  as  86  well as p o s s i b l e .  4.2 The O p t i c a l Model and C a l c u l a t i o n o f t h e Wave F u n c t i o n s The  radial  functions,  parts  of  the i n i t i a l s c a t t e r i n g s t a t e wave  x„.(r), are c a l c u l a t e d i n t h e o p t i c a l S.3  proposed by Fernbach, Serber and Taylor scattering  model,  first  (FE 49) t o d e s c r i b e t h e  o f i n c i d e n t p r o j e c t i l e s o f f a nucleus.  The o p t i c a l  p o t e n t i a l i s given by V  where V orbit  C N  and V  opt  = v + V + V , CN SO coul  TV-17 ' '  A  a r e t h e (complex) c e n t r a l  S Q  p o t e n t i a l s , r e s p e c t i v e l y , and V  c o u l  be that of a u n i f o r m l y charged sphere  of  nuclear  and  V  spin  i s normally taken t o radius  B , c  and i s  given by  —I -**) 3  c  v  couli  Z Z e 1  where  z  and  target n u c l e i , Normally potentials  Z  \  T  T  F O R R < R  '  2  —  for r > R  c  a r e t h e atomic numbers of the i n c i d e n t and  respectively. to  taken  i n c i d e n t nucleon  calculate from  the  elastic  (in this  case,  wave  functions,  optical  s c a t t e r i n g data i n v o l v i n g the a  proton)  and  the  target  87  nucleus  under  Unfortunately,  consideration  of  would  l2  no o p t i c a l p o t e n t i a l e x i s t s which  d e s c r i b e s both the d i f f e r e n t i a l power  ( C)  the  reaction  1 2  cross  section  C{p,p )i C  satisfactorily and  analyzing  i s t h a t both  of  these  guantities  vary r a p i d l y as a f u n c t i o n o f energy; thus the smoothly optical rapid  potential  varying  i s not a b l e t o match the f l u c t u a t i o n s .  v a r i a t i o n s are  structures  used.  i n the energy r e g i o n of  2  o  i n t e r e s t here. . The problem  be  caused  throughout  by  this  the  dominance  region,  as  of  The  resonance  shown i n the data of  Meyer et a l . (ME 76) . ftn attempt t o d e s c r i b e the e l a s t i c s c a t t e r i n g from carbon from E-> = 12 MeV  t o 20 HeV  was  of  protons  made by Nodvik, Duke  P  and  Melkanoff  (NO 62).  These  authors  assumed  the o p t i c a l  p o t e n t i a l to be of the f o l l o w i n g form;  R e ( V ) = ,- V f ( r )  IV-18  C N  Im(V ) = - W exp{-(r-R) /b } 2  IV-19  2  CN  / ^ v  \  V  S  df (r) + «  so " - ^ T c j - t e  0  ^  "  IV  f ( r ) = [l + exp{(r-r )/a}J o  where V, W and V  s  20  IV-21  _1  are the depths o f the v a r i o u s p o t e n t i a l w e l l s  and 1i/m c = i/2.0 fm. ir  Nodvik et a l . were a b l e sections  and  analyzing  to  powers  f i t the guite  differential  well  cross  individually  by  88  l e t t i n g s e v e r a l of the parameters i n eguations IV-18 vary,  but  they  could  not  fit  both  to  IV-21  guantities  well  simultaneously.  I n a d d i t i o n , the parameters f l u c t u a t e d  as  o f energy, thereby v i o l a t i n g the s p i r i t of  a  function  optical  model.  potentials  Their  which  final  generally  results did  included  reasonably  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 , but d i d not do analyzing  set  Watson, Singh and differential  of  compromise  enough  for  the  well  for  the  so  o p t i c a l model parameters was  Segel  (W&  E^- = 10 MeV  69).  c r o s s s e c t i o n and  of p - s h e l i n u c l e i , i n c l u d i n g  same  to 50 as  MeV.  that  1 2  These  obtained  authors  fitted  replaced  C,  over the  energy  range  by  Nodvik  et  by a s u r f a c e d e r i v a t i v e form given  slightly  a^.  of  the  d i f f e r e n t from  the  was the  potential  by IV-22  d f ( r )  XT  diffuseness  from  a l . , except that  Im(V„ ) = 4 a W CN' I dr  The  the  T h e i r form of the o p t i c a l p o t e n t i a l  used  by  p o l a r i z a t i o n data f o r a v a r i e t y  Gaussian shape f o r the imaginary part of the c e n t r a l was  the  powers.  another  the  wildly  T  imaginary p a r t was diffusenesses  of  given the  a  value  real  and  s p i n - o r b i t p a r t s o f the p o t e n t i a l . The  f i t s obtained  s c a t t e r i n g from obtained  by  1 2  C  by Watson et a l . t o the data on  were not,  Nodvik  et  al.  in  general,  T h i s was  as  parameters  were  constrained  as  to be expected  s i n c e Watson e t a l . were f i t t i n g a wider range their  good  to be  of  elastic those (PE 70) ,  nuclei  smooth f u n c t i o n s  and of  energy, while those o f Nodvik et a l . were not. Thus, i t was  decided  to attempt to f i n d a  better  set  of  89  optical  model  parameters  satisfactorily analyzing  both  power  the  for  that  would  differential  the  12  C(p,p  ) C 1 2  describe  cross  more  section  reaction.  and  The data of  o  Meyer et a l . (ME 76) parameters fitted  were  were used i n the a n a l y s i s .  The  starting  taken to be those of Sene et a l . (SE 70),  p o l a r i z e d neutron s c a t t e r i n g data from  l 2  c at  E-* =  who  14.1  n MeV r e a s o n a b l y s u c c e s s f u l l y . was  The form o f the o p t i c a l  potential  taken to be the same as that used by Watson e t a l . , except  that a l l three allowed  to  optical  model  of  have  calculations.  the  independent  code A  shapes  radii  ABACUS-2  slight  f (r)  was  1  (eguation and  IV-21)  were  diffusenesses.  used  to  The  perform  m o d i f i c a t i o n to the code was  the  made so  that the shape parameters f o r the s p i n - o r b i t p o t e n t i a l c o u l d given values independent of those used f o r central  nuclear  potential..  parameters, i t was obtained  found  a  systematically fairly  good  shape  of  the  varying  the  f i t could  listed  in  The parameters which best  T a b l e IV-1.  f i t the  data  Also l i s t e d i n t h i s t a b l e are the  parameters o b t a i n e d by Watson e t a l . , t o be r e f e r r e d t o as and by B e c c h e t t i and Greenlees (BE 69b), t o be r e f e r r e d BG.  be  to 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 and a n a l y z i n g power  data s i m u l t a n e o u s l y . are  that  By  the  be  to  WSS, as  T h i s l a s t s e t was o b t a i n e d f o r a wide range of n u c l e i with  A>40,  E<50  MeV.  Some  comments on the new  s e t o f parameters  follow.  Written by E. H. Auerbach at Brookhaven National Laboratory and adapted f o r use a t the U n i v e r s i t y o f B r i t i s h Columbia by T. W. Donnelly and A. L. Fowler.  90  Table IV-1 Optical Model Parameters Parameter*  Potential-  ( R o ) .  ( R o )  SO  WSS  BG  1.15-.001E  1.17  1.13  1..15-.001E  1.32  1.40  1.15-.001E  1.01  .88  .57  .75  .55  .50  .51  .12  .57  .75  .24  *R  'SO  . 60-.28E+.4Z/A  1/3  (  /  .59E  (E<15 MeV)  54- .32E+.4Z/A / 1  5.5  SO  58.4  3  .22E-2 .7 (>0) Gaussian  W 6-.055E (E>15 MeV)  NEW  11.8-2 •5E  25 .42-. (>0) Surface derivative 6.2  7 .28-.  * r a d i i and diffusenesses are i n fm, potentials are i n MeV, E i s the laboratory energy. R refers to the c o e f f i c i e n t of i n r=R A^ . Terms proportional to (N-Z) have been suppressed since N=Z f o r C . 3  Q  0  1 2  91  The w e l l depths f o r the new reasonable  potential  dependence  because  term p r o p o r t i o n a l to ZA / -1  i n the WSS  and BG  3  so  it  E^> = 16.964 8eV. P  the  analyzing  of  was  and  left  is  expected  These a r e seen  an o v e r a l l trend was  constant  varied  resonances i s apparently radius  too  not too  parameters  at  the  rapidly  a Coulomb  value  cross  section  for  to  i t had  a few  at nor  hundred  effect  for  the imaginary  of  and s p i n - o r b i t  for  However, i t  NEW  f i t s to both t h e d i f f e r e n t i a l c r o s s s e c t i o n and  power.  an  strong.  found t h a t the extreme values obtained  best  have  difficult  p a r t s of a l l t h r e e p o t e n t i a l s are g u i t e d i f f e r e n t . was  The  explicitly  k i l o v o l t s e i t h e r above or below t h i s energy, so the  The  fairly  However, i n the present a n a l y s i s  Neither the d i f f e r e n t i a l power  to  i t s n o n - l o c a l i t y and  (PE 63) .  potentials.  V fluctuated considerably detect,  are  when compared to other o p t i c a l model a n a l y s e s .  depth of the r e a l c e n t r a l p o t e n t i a l energy  (NEW)  An i n c r e a s e of 10% i n ( R ) o  x  the  analyzing  o r i n g i t more i n t o l i n e  t o S 0  with the other analyses worsened the  gave  2  of the f i t (defined as  N  the sum  going  analyzing {R ) q  i  over both  power  the  differential  cross  data) by about a f a c t o r of 20.  of 10% worsened x  2  section  A decrease i n  by about a f a c t o r of 3.  While the value f o r the d i f f u s e n e s s of the r e a l well  a  R  markedly. gives  a  is  similar  to  the other r e s u l t s , a-^ and a  A c t u a l l y , the v a l u e of & shape to the imaginary  very s i m i l a r to t h a t  and  obtained  1  in  the  present  potential g 0  differ  analysis  p a r t of the p o t e n t i a l which i s by  Nodvik  et  al.  The  fits  92  deteriorated optimum  r a p i d l y when e i t h e r a  section  obtained sets. NEW,  and  the  analyzing  fits  power  to  especially  characteristic  at  the  section  at  and NEW  this  NEW  parameter  f i t much b e t t e r  angles,  and  o f the f i t s at a l l e n e r g i e s .  to choose between NDM  differential  a t E ^ = 16.964 MeV and  is clearly  forward  the  data  with the Nodvik et a l . (NDM), WSS  The a n a l y z i n g power data  cross  was v a r i e d from i t s  value.  Shown i n Figure IV-2 are cross  or a  for the f i t s  with  this  was  There i s not much  t o the  differential  or any other energy; WSS c o n s i s t e n t l y  underestimates t h i s g u a n t i t y at a l l the energies. One d i s a p p o i n t i n g aspect model  sets  is  of a l l  and  Shown i n  of r e a c t i o n c r o s s s e c t i o n data  S j ^ p a r t i a l waves, and the  these  partial The  3  do turn over s i m i l a r l y s 1  the  reaction  Figure  is a  (ME 77) f o r the d" / 3  cross  /  2  partial  cross to  sections  sections  the  data.  continue It  is  2  for  (PE 70),  and  in  t o know how  improving  to  differential  d e t e r i o r a t e badly.  cross  section  2  to  search f o r  weight  this  the f i t to the r e a c t i o n  and  Hence, i t was decided  reaction cross section i n the f i t t i n g  The  to r i s e at possible  t o t a l r e a c t i o n c r o s s s e c t i o n i n the x  c r o s s s e c t i o n , i t may happen t h a t the g u a l i t y of the  IV-3  r e a c t i o n c r o s s s e c t i o n s which  2  the best f i t , but i t i s d i f f i c u l t guantity  optical  to t h e data, but at a lower energy.  lower e n e r g i e s , c o n t r a r y include  these  waves c a l c u l a t e d with the v a r i o u s p o t e n t i a l s .  p o t e n t i a l s e t s give d ^  calculated  of  t h a t none of them reproduce s a t i s f a c t o r i l y the  p a r t i a l reaction cross sections. compendium  three  analyzing not  to  procedure.  the  fits  to  power  will  include  the  93  •6j Fig.  IV-2 :  m  (degrees)  C(p,p )* C differential cross section and a n a l y z i n g power at E J = 16.964 MeV. The data a r e from reference (ME 76). The c u r v e s a r e from o p t i c a l model f i t s using the p o t e n t i a l s shown.  l 2  2  D  F i g . IV-3 : Comparison of the o p t i c a l model a n a l y s e s with some experimental p a r t i a l r e a c t i o n c r o s s s e c t i o n s . The s o l i d curve i s a compendium of data (ME 77).  95  It  is  indeed  g u a n t i t i e s - the  not  too  surprising  differential  cross  that  the  section,  observed  the  analyzing  power and the p a r t i a l r e a c t i o n c r o s s s e c t i o n - are not a l l w e l l reproduced  by  this  simple  been shown i n the work (MI 71)  that  coupling  between  scattering E  x  - 4.43  E->- = 4  it  in  is  Mikoshiba,  important  scattering  to  in  Terasawa consider  the  These  to  8  authors  MeV  and the  elastic  the i n e l a s t i c channel t o the 2  Mey.  MeV  of  use of the o p t i c a l model.  investigated  +  It has Tanifuji  e f f e c t s of  channel  and  state in * C 2  the  region  at  from  with a coupled channel c a l c u l a t i o n  and  P  found that i t was p o s s i b l e t o f i t reasonably w e l l the excitation  functions  cross section order  to  and  angular  distributions  and a n a l y z i n g power at these  extend  this  low  observed  of both the  energies.  In  type of c a l c u l a t i o n t o the e n e r g i e s o f  i n t e r e s t i n t h e present work i t would be necessary t o  consider  the c o u p l i n g of other i n e l a s t i c channels i n a d d i t i o n to the a t 4.43 of  MeV  to the e l a s t i c channel.  Johnson  (JO 74)  who  extended  T h i s was shown i n the work  the c a l c u l a t i o n s o f Mikoshiba  e t a l . t o higher e n e r g i e s and found that break  down  seriously  at  one  about  the  model  E-> = 10  began  MeV.  A  to  major  P  computational e f f o r t would t h e r e f o r e be r e g u i r e d to extend approach  to higher e n e r g i e s and t h i s was  not attempted  in  this the  present work. All  three  parameter  c a l c u l a t e the i n i t i a l and was  the  final  sets  scattering  (NDM,  »SS and NEI)  state  wave  bound s t a t e wave f u n c t i o n , 0  found by s e t t i n g H = 0, and v a r y i n g V to  were used t o  f u n c t i o n s Xg..(r) L J  (r).  The  f i t the  latter binding  96  energy (E  (E )  of  B  = -1.94  a  HeV).  {Pjy ^  proton  2  The  real  i n the  parts of the  ground s t a t e  radial  wave  of  function  B  calculated s t a t e s 2_/  for  S  It to  can  ^3/2  a n f l 2  collective  the  part  t  b  e  the  f  s t a t e , the o  r  f  m  a c t  or  be dominant i n the  *r ')  leading  o  first  is  IV-4.  expected H  a  v  e  n  a  s  i s shown  contribution  D i r e c t Semi-Direct Capture  parameter  from  a  d i r e c t semi-direct  must be matched t o This  strength  cannot  including  E2  solid  wave  0*3/2  (the n u c l e a r r a d i u s  i s true  experiment  be accounted  strength of  and  dotted  dot-dash l i n e  the  capture,  of  t o t a l cross  dubious  sections  (NEW).  The  ,  an  cross  open  section  indistinguishable  near E  The  MeV.  from  reference  Figure  dotted  three p o t e n t i a l s are = 20.8  the  total  total  cross  if  value.  including  E2 capture o n l y are shown i n (NDM)  and  capture  the  E1  f o r c o r r e c t l y , then c a l c u l a t i o n s  would be  line  is  because most of the  s e c t i o n a r i s e s from e l e c t r i c d i p o l e  d i r e c t E1  the  shown i n F i g u r e  t o a p a r t i a l c a n c e l l a t i o n of i t s  section.  calculation  for  section.  c a l c u l a t i o n that cross  the  used  scattering  c a p t u r e ; i t i s because the  nuclear i n t e r i o r  C a l c u l a t i o n of the The  initial  rV^r)  E1 c a p t u r e , are  be seen from t h e s e p l o t s why  t o the c r o s s  4.3  a n < 3  of  a node i n s i d e the as  bound  line  Thus the  GDR  and  IV-5  as  a  (HSS)  and  a  calculated  i n the  the  region  with  the  of the  GDR  s o l i d curve i s the t o t a l c r o s s  section  x  taken width r  n  were  adjusted  (BE to  76b). match  The the  position  shape  of  E  and  the  GDR,  97  Fig.  : The real parts o f the r a d i a l wave functions c a l c u l a t e d with t h e NEW p o t e n t i a l . S c a t t e r i n g wave f u n c t i o n s <s > and d , ) l e a d i n g to E1 capture only a r e shown. The form f a c t o r r f ( r ) f o r volume c o u p l i n g t o the GDR i s a l s o shown. r indicates the nuclear radius. ° 2  3  2  98  40  1  1  [—j  1  1  T  1  GDR PLUS • NDM POTENTIAL DIRECT x4 CAPTURE ooooooWSS POTENTIAL ONLY -NEW POTENTIAL  30 «3-  PYGMY INCLUDED  :  w  s  s  POTENTIAL  2 O  to O UJ  20  CO CO  o ce o o  10  J  I  I  L 30  Et (MeV) Qb  Fig.  I V - 5 : F i t s to the » 2 C { p , ) i 3 t o t a l cross section. The solid curve i s a f i t (by eye) to the data o f Berghofer e t a l . (BE 76b). Y  0  N  99  and  3,,  was  adjusted  to  match  the t o t a l c r o s s s e c t i o n .  The  11  symmetry p o t e n t i a l V^O) estimate  given  by  was  Bohr  square r a d i u s <r > was results  and  on  l  *N  constant f o r p - s h e l l n u c l e i It  is  clear  that  to  be  Mottelson  taken  2  scattering  taken  to  be  MeV  fm  (<r >  from  2  is  2  calculation  ^3/2  (PH 75)) .  a l l three  potential  sets  partial  to  give  no  Presumably,  of the wave f u n c t i o n s has been ignored. shape somewhat resembles  absorption  cross  F a i l u r e to reproduce the l a t t e r failure  electron  the c o u p l i n g with the i n e l a s t i c channels i n the  t h a t the pygmy resonance the  an  approximately  i n d i c a t i o n of the presence o f the pygmy resonance. t h i s i s because  from  (BO 69c), and the mean  6.0  (ME 59)  100  reproduce  section  Note a l s o  the shape  (Figure  of  IV-3).  might have some e f f e c t  on  the  the pygmy i f the pygmy i s mainly a  d ^ 3  2  resonance. Some evidence s u p p o r t i n g the importance  of c o n s i d e r i n g the  c o u p l i n g of i n e l a s t i c channels i n c a l c u l a t i o n s  of  c r o s s s e c t i o n s i s found i n the work of Johnson a coupled channel c a l c u l a t i o n energies  up  to  E  p  = 9  f o r the  MeV.  l 2  G(p,  Y 0  the  )  1 3  N  reaction for  been  the energy  energy  the  His c a l c u l a t i o n of the  low 90°  which r e s u l t s mostly from E1 c a p t u r e , reproduces  the experimental already  0  T h i s i s below the lowest  energy t a i l of the pygmy resonance. curve,  Y  (JO 74), who d i d  measured i n the present work, but i t does extend i n t o  yield  (P, )  measurements  stated  where  the  very  well.  However,  it  has  that h i s model begins t o break down near present  measurements  begin,  so  more  complicated c o u p l i n g s would have t o be i n t r o d u c e d . T h e r e f o r e , i n order t o reproduce the presence of the pygmy  100  resonance,  i t was  amplitude  into  necessary to i n t r o d u c e a second c o l l e c t i v e the  present  calculation.  T h i s has sometimes  been necessary i n previous c a l c u l a t i o n s with the DSD example, Snover et a l . {SN GDR  in  The  i S  N  by  76)  including a  dashed l i n e i n F i g u r e IV-5  second The  model; f o r  p r o v i d e f o r fragmentation  small  second  E1  resonance  of the  amplitude.  shows the r e s u l t of i n c l u d i n g a  amplitude i n the present case u s i n g the  WSS  potential.  r e s u l t s of the c a l c u l a t i o n f o r a l l three p o t e n t i a l s  agreed  to w i t h i n ±2055 f o r the t o t a l c r o s s s e c t i o n and a l s o f o r most of the other q u a n t i t i e s c a l c u l a t e d results  using  WSS  by  the  model,  so  only  the  (the most general o f the three p o t e n t i a l s )  w i l l be r e f e r r e d t o i n f u t u r e .  The  only e x c e p t i o n was t h a t  HEW  tended t o g i v e E2 d i r e c t c a p t u r e c r o s s s e c t i o n s t h a t were about U0% l a r g e r than NDM used  t o reproduce  or WSS  above E+ = 16 HeV.  parameters  the t o t a l c r o s s s e c t i o n shown i n F i g u r e IV-5  are l i s t e d i n T a b l e  IV-2.  Table GDR  The  IV-2  Parameters Used t o Reproduce the T o t a l Cross S e c t i o n Using t h e WSS P o t e n t i a l E  i l  P  l l  20.5  MeV  4.0  MeV  0.6  Pygmy  13.8  MeV  6.0  MeV  1.8  very l a r g e  resonance  the GDR.  r  GDR  The pygmy  l l  value  for g  required  to  reproduce  i n d i c a t e s t h a t the pygmy i s not a fragment of  I t i s not the r e s u l t of i s o s p i n s p l i t t i n g of t h e  f o r example.  the  N e v e r t h e l e s s , r e f e r e n c e t o the f i g u r e s  shown  GDR, in  101  Chapter  III  i n d i c a t e s that t h e model i s reasonably  i n reproducing most of the experimental q u a n t i t i e s . lines  in  above.  these  successful The  solid  f i g u r e s represent the c a l c u l a t i o n r e f e r r e d  In cases where a dotted l i n e i s shown, an i s o s c a l a r  resonance i s assumed to l i e at E  •= 25 MeV  {see  to E2  below).  X  In follow  Figure the  III-7,  trend  coefficients  of  fairly  i s possibly a l i t t l e the  calculation  it  can  the  measured  well. low,  The  and  angular  in  Figure  The  structure  III-8.  calculation  in  The  a ;  this  2  d-wave and  w e l l reproduced f o r the s o l u t i o n calculated  phase  difference  value of the measured one. it  stands  to  produce  is  such  phase  to  likely  s-wave amplitudes  represents a f a i r l y  a  fails  very  however,  3  than  phase d i f f e r e n c e  values  a  and  as are the  good average model  as  v a r i a t i o n ; here again,  with allowance f o r  coupling  to  i n e l a s t i c channels would be very i n t e r e s t i n g . Neither  difference  the is  differences  magnitude of the  given  correctly  difference  is  a little  reproduced beyond the U  the  low.  structure  p/f r a t i o nor the p,f phase  by  are not too severe.  where i t i s a l s o noted that  and  I  also  Nothing i s present i n the  c a l c u l a t i n g the wave f u n c t i o n s the  distribution  magnitude of the c a l c u l a t e d  r e l a t e d to i t s f a i l u r e t o reproduce the s,d seen  calculations  b^ i s perhaps more constant  indicates.  reproduce the broad  be seen t h a t the  the  model  although  T h i s i s seen i n F i g u r e magnitude  The  of  the  p,d  the  III-9, phase  f , d phase d i f f e r e n c e i s w e l l  that i s observed between 10  MeV  MeV. Finally,  capture  alone  i t can  be seen i n Figure  satisfactorily  111-12  that  accounts f o r the  direct  E2  experimentally  102  measured  cross  sections  if  the  "low"  consistent  set  of  s c l u t i o n s are the c o r r e c t ones. The  dotted l i n e s i n the f i g u r e s are the r e s u l t of assuming  t h a t an i s o s c a l a r E2 resonance e x i s t s at the expected energy of E^ = 25 Me? taken  to  {see Chapter I ) .  be the same as that of the GDR  E2 resonance was 0.5). HeV  The {BO  that  In the c a l c u l a t i o n , the width  the  presence  Over  cross of  r 2  o  =  4  M e ¥  )  assumed t o exhaust 505? of the sum  c e n t r a l nuclear  6 9c).  (  p o t e n t i a l was  most  section  such  # and  was the  {B Q  rule  2  taken t o be V (0) Q  =  -50  of the r e g i o n . Figure I I I - 1 2 shows is  relatively  insensitive  a resonance, but at E->- =  16 HeV  and  to  the  17  HeV,  P  the c a l c u l a t i o n i s i n disagreement with the data. , I t i s in  Figure  I I I - 7 t h a t the assumption of an E2 resonance b r i n g s  the c a l c u l a t e d a^ i n t o b e t t e r agreement with c a l c u l a t e d a^ becomes l a r g e r than the present Overall, collective inclusion  shown  E2 of  there  appears  strength only  into  direct  d e s c r i p t i o n of the angular  to  be  the  no  the data, but measurements.,  need  to i n c o r p o r a t e  calculation,  since  E2 amplitudes provides a distribution  the  coefficients  the  reasonable and  the  e x t r a c t e d E2 c r o s s s e c t i o n s .  4.4  Sum  Rules  Another  t e s t f o r the presence of c o l l e c t i v e s t r e n g t h i n a  r e a c t i o n i s t h a t an a p p r e c i a b l e f r a c t i o n o f the a p p r o p r i a t e r u l e should necessary  be  exhausted  {BE  76a).  Of  course  t h a t t h i s s t r e n g t h be concentrated  in a  it  is  sum also  sufficiently  =  103  narrow energy range t h a t i t w i l l appear as a does  not  seem  Nevertheless, can o f t e n  to  be  resonance,  which  t h e case f o r t h e present measurements.  the d i r e c t capture c o n t r i b u t i o n to the sum  be s i g n i f i c a n t . ,  After converting  rules  the measured ( p , ^ )  E2 c r o s s s e c t i o n s t o the i n v e r s e  ( ,P ) E2 c r o s s s e c t i o n s  the  {Appendix  detailed  balance  theorem  Y  o  A)  eguation 1-2 from E+ = 10.0 MeV to 17.0 MeV 17.6  the i n t e g r a l of  {E  p  using  =11.1  MeV  to  X  MeV) was found t o be 0.44 ± 0.17 yb/MeV, c o r r e s p o n d i n g t o  10.3  ± 4.0% o f t h e EHSB {Appendix A ) .  ±20%  o v e r a l l normalization  These e r r o r s i n c l u d e the  e r r o r i n the  determination  t o t a l c r o s s s e c t i o n by Berghofer e t a L , {BE 76b).  of the  The f r a c t i o n  of the EwSB exhausted by t h e c a l c u l a t e d d i r e c t c a p t u r e E2 c r o s s section  (HSS p o t e n t i a l ) i s 6.8%.  exhausted  in  the  1 2  C{p,Y )i3N  Thus t h e f r a c t i o n of the ESSB  reaction i s consistent  Q  c a l c u l a t e d f r a c t i o n assuming the E2 part proceeding s o l e l y by d i r e c t  4.5 Comparison The  E2  strength  Detailed  complete  (SN 76, AD 77). cross  reaction  is  with Other Work  of s t u d i e s of since  the  capture.  that  measurements i s very t y p i c a l reactions.  of  with the  the  1 4  i s extracted of  that  from  found  the  in  present  other  {p,^)  comparisons can be made with the r e s u l t s C ( p , Y )»s$j  analyses In t h e  a n  0  1 4  of  c[  these  is  N (  p,Y )i*o 0  reactions  C<p,Y )*reaction,  sections are t y p i c a l l y  Q  1.0 yb from E  are the  reactions, available total  E2  = 19.5 MeV t o 27.0  MeV, compared t o E2 c r o s s s e c t i o n s o f the order of 0.2 yb  from  104  E  = 11.1  x  MeV  to  17.6  MeV  because o f the energy-sguared  However,  f a c t o r i n the denominator  of the  EWSR,  both  limit;  (10.3 ± 4.0)% o f the sum r u l e l i m i t i s depleted  present  reactions  i n t h e present study.  case  i*C(p,Y )i5 0  exhaust s i m i l a r amounts o f the sura r u l e  compared  to  channel  o  is  region  for a  GQR  of 1  of  et  Y 0  )  1 S  the N ( p , l s  i n t h e new  a l . (HA 74a)  difference  N  Y 0  r e a c t i o n , Adelberger  )  1 6  the EWSR  Hanna  et  T-matrix elements  i n the that  a l . (AD 77)  C o n s i d e r a b l y l e s s E2  measurement,  a l . excluded  although x  A  strength  was  there was s t i l l an  = 20.6 HeV and 24.8 MeV. I t of  the o r i g i n a l  and Bj from t h e f i t s  1  and so the E2 c r o s s  subject  et  this  ° r e a c t i o n from E+ = 8 MeV t o 18 MeV  should be p o i n t e d out that i n t h e a n a l y s i s  presumably  found  approximately 30% o f the  the l a r g e  excess over d i r e c t capture near E  data,  In a study  E2 s t r e n g t h found i n t h i s r e a c t i o n from  = 19.6 MeV t o 29 MeV).  found  Hanna  which exhausted  Because  found i n t h e * C ( p ,  x  of  i n the  f o r the c a l c u l a t e d E2 d i r e c t c a p t u r e through  concentration  (E  exists  = 20.2 MeV and 26.8 MeV (about 7% o f  x  (SN 7 6 ) ) .  remeasured  reaction,  o  exhausted  possibly l 6  N  EWSR between E  situation  o f t h e p h o t o d i s i n t e g r a t i o n of 0 .  of the i s ( p , Y ) i 6 0 evidence  i n t h e case  .  N  A somewhat d i f f e r e n t (Y,P )  (6.8 ± 1.4)%  i n the  sections  obtained  to the were  t o the l i m i t a t i o n s d i s c u s s e d at the end o f  s e c t i o n 3.8. A l l t h r e e of these similarities., trends;  (p, ) Y  0  reactions  show  a  great  many  The a^ and b^ c o e f f i c i e n t s show roughly the same  a^. and  b^  increasing excitation  both  become  energy,  slowly  more  f o r example, and a  p o s i t i v e with Q  becomes  more  105  negative.  As  a  result  of  the  s i m i l a r i t i e s i n the angular  d i s t r i b u t i o n s , the behaviour of the T-matrix the  same.  Ml  elements  three r e a c t i o n s are a f f l i c t e d  of secondary  solutions  statistical  sense.  some In  of  which  is  with the  are  much  presence  acceptable  f a c t , i n the case of the  in  *C (p, Y )  1  Q  r e a c t i o n , some of the second s o l u t i o n s were " p r e f e r r e d " i n sense I.  that  excluded  sections  ISJJ  the  they had lower c h i - s g u a r e s than those f o r s o l u t i o n  A l l t h r e e r e a c t i o n s show t h e same behaviour when  are  a  and  B  1  from the a n a l y s i s ; namely, the e x t r a c t e d E2 c r o s s  are  systematically  coefficients  that  are  higher  reproduced  and  the  from  the  A-^  and  B^  analysis  which  excluded them are c o n s i s t e n t with the measured v a l u e s only when the e r r o r s account. to  be  of  the  reproduced  Finally, correct,  no  then  a l l three  collective  c o l l e c t i v e E1  excitation  E2  evidence l l  B(p,Y )i2c 0  The also  For  example,  (p,^ ) 0  Noe  et  functions  are  E2 capture and  collective  E2  measurements i n other a l . (NO 76)  find  by  model, that t h e r e i s no  strength  above  the  GDH  i n the  reaction.  amount of E2 s t r e n g t h seen i n  similar  are taken  £ 2  capture) i n t e r f e r i n g with d i r e c t  t o c a l c u l a t i o n s with the DSD for  into  capture.  nuclei.  comparison  taken,  by c o n s i d e r i n g o n l y d i r e c t  T h i s r e s u l t i s i n accord with light  are  i f the s o l u t i o n I values f o r a  s a t i s f a c t o r i l y reproduced <with  coefficients  to  the  strength  Most o f the i n f o r m a t i o n about  then  be  present  study  is  seen i n many ( a , Y ) r e a c t i o n s .  these r e a c t i o n s comes from  s t u d i e s on n u c l e i with ground s t a t e J p a r i t y s t a t e s can  the  formed  (a,Y ) Q  = 0 , s i n c e only n a t u r a l in  the  compound  system,  106  thereby E2  example,  C{a,Y )  measurements o f t h e angular  Snover  i&0  0  E  an unambiguous d e t e r m i n a t i o n o f the E1 and  s t r e n g t h s through  For 1Z  permitting  and  e t a l . (SN 74)  found  distributions.  studied  the  reaction  17% of the EWSR was exhausted  between  = 1 2 MeV and 28 MeV. , S i m i l a r r e s u l t s a r e obtained f o r other x  ( a ,  s t u d i e s on s p i n 0 n u c l e i ranging up to A = 60;  )  Y 0  about 1%/MeV o f t h e EWSR i s exhausted  namely,  (HA 7 4 c ) .  Thus, t h e present measurement o f t h e E2 s t r e n g t h i n  1  in  agreement  with  similar  other l i g h t n u c l e i .  capture  0.  As  I t i s a l s o i n agreement with a v a r i e t y  mentioned  i n the i n t r o d u c t i o n ,  s c a t t e r i n g s t u d i e s on these n u c l e i have strength  t o be  course, that < »P )  t h e present  experiment 1 3  0  shown  nuclei the the  1  2  of  c and  inelastic guadrupole  very much spread out. I t should be noted, o f  channel i n t h e decay o f N .  Y  Ni s  r e a c t i o n measurements i n  i n e l a s t i c s c a t t e r i n g measurements on the near-by l 6  3  investigated  o n l y the  Many other p o s s i b l e decay  branches e x i s t , f o r example, proton decays t o e x c i t e d s t a t e s i n i 2 C and neutron, alpha and levels  i n other  deuteron,  neighbouring  channels are not open  until  etc.  decays  to  various  n u c l e i , although many o f these  higher  excitation  energies are  reached. Moreover, t h e r e o f t e n appears other  charged  t o be l i t t l e  p a r t i c l e channels., For example, Weller and Blue  (WE 73) i n v e s t i g a t e d r a d i a t i v e deuteron E  = 19.5  MeV  to  x  22.3 MeV. '  c r o s s s e c t i o n was measured  s t r e n g t h i n the  capture  They found  by  3  from  ll  that the t o t a l  (Y,d )  •  13% o f  i n the i * C ( p , Y ) * N 3  0  the t o t a l reaction  o  ( ,P ) Y  o  cross  section  (FI 6 3 ) . I f E2 deuteron  decay e x h i b i t s t h e same c h a r a c t e r i s t i c s as E2 proton and  alpha  107  { a_, = 1% a _ , ) ,  decay  0  then  t h e amount of E2 s t r e n g t h i n the  deuteron channel must be very s m a l l indeed. mass J 1  B  13  s t u d i e d with  (DE 74).  near  (d,^ ) r e a c t i o n s i n c l u d e * 5 N (DE 76) and 0  Some evidence f o r Ml o r E2 r a d i a t i o n o c c u r r i n g i n  the r e g i o n o f the GDR non-zero  Other n u c l e i  of  coefficients,  these  reactions  i s evident  from  but no g u a n t i t a t i v e e s t i m a t e s were  made. S i m i l a r l y , s m a l l amounts o f E2 or Ml s t r e n g t h a r e seen i n 3  He  capture  reactions  on  1 2  C  and  l  *0  o b s e r v a t i o n o f non-zero a^ c o e f f i c i e n t s . (EB 77) have r e c e n t l y EWSR i n the The in  l 3  Snover  and  found a c o n t r i b u t i o n o f about  (3He,Y ) 0  present  (SH 74)  »60  r e a c t i o n from E  x  through Ebisawa  0.3% of the  = 24 MeV t o 38 MeV.  r e s u l t s , then, i n d i c a t e t h a t t h e E2 s t r e n g t h  N i s very spread out. No other g u a n t i t a t i v e measurements have been made  i n the  mass 13 n u c l e i , b u t Shin et a l . (SH 71) observed s m a l l non-zero a  1  from  and  a  3  coefficients  inelastic  energies  electron  i n photoproton angular d i s t r i b u t i o n s scattering  r a n g i n g from 21 t o 32 MeV.  on  l 3  C  at  Shin e t a l . a l s o  much l a r g e r a^ c o e f f i c i e n t s i n a s i m i l a r study on authors  C.  These 2  the t a i l  regions  Inelastic  alpha  more  clearly  of the GDR o f t h i s nucleus than w i l l be  seen i n the t a i l r e g i o n s o f t h e more spread s c a t t e r i n g on  1 2  than 20% o f t h e EWSR i s exhausted MeV  l 2  observed  point out that t h e GDR i s more concentrated i n * C and  thus t h e e f f e c t s o f E2 i n t e r f e r e n c e w i l l be seen in  excitation  C  out GDR  (KN 76) has shown that  between E  =15  i n * C , and s o i t would seem t h a t t h e present 2  measurement has accounted  of  MeV 1 2  1 3  C.  less  and 30  C (p,^)  1 3  N  f o r a c o n s i d e r a b l e p o r t i o n o f t h e E2  108  strength  expected  in  mass  13,  present c r o s s s e c t i o n s are a c t u a l l y  especially lower  noting  limits.  t h a t the  109  Chapter V  SUMMARY AND CONCLUSIONS  The extend  main purposes our  of  knowledge  the present  o f the nucleus  measurements 1 3  were  N by i n v e s t i g a t i n g t h e  nature of the E2 s t r e n g t h i n t h e r e g i o n below t h e GDR provide  a  Ebisawa. variety  simple  and t o  t e s t o f the DSD capture model o f Snover and  These aims have been of  to  reasons  to  be  largely  achieved,  but f o r a  given below, success has not been  complete., I t would appear t h a t capture gamma ray r e a c t i o n s by  polarized  protons  do  not permit  determination o f  the capture  amplitudes  believed.  had  been  This  **C(p,Y )i5N 0  and  l s  already  1 2  C(p,Y )i3N 0  as  shown  unambiguous  was p r e v i o u s l y  t o be t r u e f o r t h e  Q  and above the GDR.  reaction  reported  here  The r e s u l t s extend  u n c e r t a i n t i e s to t h e r e g i o n below t h e GDR as w e l l . is  twofold;  a  N ( p , Y ) * * o r e a c t i o n s , where t h e measurements  were i n the r e g i o n through the  as  induced  from  these  The problem  two s o l u t i o n s e x i s t at s e v e r a l e n e r g i e s which a r e  both s t a t i s t i c a l l y  acceptable,  and  there  might  be  seme  Ml  r a d i a t i o n u n d e r l y i n g the s t r u c t u r e i n t h i s r e g i o n . So  f a r as  the f i r s t  problem  i s concerned,  s e v e r a l reasons t o p r e f e r t h e s o l u t i o n I r e s u l t s . s o l u t i o n s g i v e an E2 c r o s s s e c t i o n which v a r i e s smoothly  with  energy  and t h e observed  there are  First, more  or  these less  smooth v a r i a t i o n of the  110  angular d i s t r i b u t i o n c o e f f i c i e n t s would  arise  naturally  from  t h i s , but not from a cross s e c t i o n t h a t was f l u c t u a t i n g widely. In  a d d i t i o n , the second s o l u t i o n s are not always a c c e p t a b l e i n  a statistical  sense, and those which  physical solutions.  cannot  be the  F i n a l l y , the second s o l u t i o n cannot always  be  found,  the  f a u l t of the i n i t i a l  which  a r e not  f o r example at E+ = 12.8 MeV, although t h i s may be  comes  and  goes  search parameters. might  be  Thus  believed  to  a  solution  be  just  a  mathematical s o l u t i o n which has no p h y s i c a l s i g n i f i c a n c e . An attempt to l e a r n about strength  was  made  by  the  possible  performing  an  presence  analysis  c o e f f i c i e n t s most s e n s i t i v e to M1 r a d i a t i o n , excluded.  In  the E2 c r o s s  this  section  much l a r g e r e r r o r s .  analysis, systematically  of  M1  i n which the and  B^,  were  higher values o f  were found, although these values a l s o had i n the  errors  i s c l e a r enough - the £2 amplitudes were being e x t r a c t e d  from a  data  The reason f o r the increase  s e t that was l e s s s e n s i t i v e t o t h e i r presence.  However,  i t i s not c l e a r why t h e amplitudes were always l a r g e r . drawback t o t h i s a n a l y s i s was that t h e v a l u a b l e  x  2  significance  t e s t was l o s t s i n c e t h e r e were zero degrees of freedom fit.  A major  i n the  In any event, t h e s o l u t i o n s determined from using  the c o e f f i c i e n t s l a y a t s t a t i s t i c a l l y again these are t h e p r e f e r r e d Accepting  satisfactory  t h e low, c o n s i s t e n t  measurements made i n other l i g h t capture  into  s i m i l a r i t i e s and exhaust  * C, 2  levels,  so  solutions. s e t of E2 c r o s s s e c t i o n s as  being c o r r e c t , t h e r e s u l t s o f t h i s experiment agree  radiative  a l l of  nuclei. * N,  roughly  3  l 5  N  with  many  In p a r t i c u l a r , proton and  equivalent  **0  bear  amounts  many of the  111  EWSR.  The  reactions  latter  result  involving light  has been observed i n a v a r i e t y o f  nuclei.  Thus the present measurement  has  f u r t h e r confirmed the s y s t e m a t i c s of E2  for  low k n u c l e i . The  c a l c u l a t i o n s with t h e DSD  deficiency  that  model  photodisintegration  suffered  from  the  the pygmy resonance d i d n o t appear n a t u r a l l y  from t h e c a l c u l a t i o n .  T h i s was  since  showed  Johnson's  work  actually  an  expected  result  the importance o f the c o u p l i n g  between the ground s t a t e and t h e f i r s t e x c i t e d s t a t e o f C 1Z  reproducing  the pygmy*s  that s e v e r a l  attempts to d e s c r i b e  calculations  have  However, when introduced  t h e presence  into  capture.  a^ and that  the  pygmy  I t might be noted with  n o t been wholly s u c c e s s f u l of  the present  reasonably s u c c e s s f u l E2  low energy t a i l .  t h e pygmy  calculation,  (Kl 74, Jk 71).  was  artificially  t h e DSD model was to  In p a r t i c u l a r , t h e energy dependences o f the odd assuming  E2 d i r e c t c a p t u r e r a d i a t i o n was i n t e r f e r i n g with the  were  The various also  phases a s s o c i a t e d  E2  amplitudes  The  E2 d i r e c t capture c r o s s s e c t i o n s c a l c u l a t e d  with the  given reasonably w e l l by the model. with t h e model  with the measured c r o s s s e c t i o n s , assuming t h e s o l u t i o n  I values were t h e c o r r e c t In summary, no strong located  model  i n r e p r o d u c i n g the parameters r e l a t e d  dominant E1 r a d i a t i o n .  agreed  shell  c o e f f i c i e n t s were s a t i s f a c t o r i l y c a l c u l a t e d  only  in  i n the region  experimental comparisons  measurements  ones. evidence f o r the e x i s t e n c e of  t h e GDR  of  the E2  GQR  i s found e i t h e r i n the cross  section  o f t h e r e s u l t s with the DSD model.  section contributes  of a  an amount t o t h e EWSR (10.3  or i n  The E2 c r o s s  ± 4.0%) that i s  112  s i m i l a r t o the s t r e n g t h seen i n a v a r i e t y of <p, ) and Y  0  r e a c t i o n s i n the r e g i o n below the expected It higher  would  be  energies  strength.  (a,  Y D  )  GQB.  of i n t e r e s t to extend these measurements t o to  search  Extension  for  evidence  of the measurements  of  collective  E2  both above and below  the r e g i o n s t u d i e d i n the present work using a f i n e r g r i d would be of i n t e r e s t t o e s t a b l i s h  whether the "resonances" and " d i p s "  observed i n some o f t h e e x t r a c t e d q u a n t i t i e s are r e a l or merely statistical. channel  calculation  t o understand detailed  I t would a l s o be of i n t e r e s t t o extend a  coupled  through the pygmy resonance r e g i o n to t r y  i t s s t r u c t u r e i n a more s a t i s f a c t o r y  experimental  information  now  exists  comparisons o f such a c a l c u l a t i o n c o u l d be made.  way.  Much  with  which  113  Bibliography  AD 73  E. G. A d e l b e r g e r , M. D. Cooper and H. F. Swanson, U n i v e r s i t y o f Washington Annual Report (1973) p.11. „  AD 77  E. G. Adelberger, J . E. 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Now  reduced  use i s made o f t h e w e l l known sum r u l e c o n c e r n i n g transition  f i n a l states f. £  where <r  2L-2  p r o b a b i l i t y and t h e energy  summed over a l l  T h i s i s g i v e n by (NA 65) B(EL,E)] - L(2L+D 4TT  2  -h Z 2L-2 <r 2MA 2  2  > i s t h e {2L-2)th moment o f t h e charge  i n the ground  the  A-a  distribution  s t a t e of the nucleus, M i s the nucleon mass and A  i s the atomic number o f t h e nucleus.  122  With L = 2 , equation A - 4 becomes  Thus, u s i n g eguation  the EWSR given by  A-3,  A-6  reduces to  Upon s e t t i n g Gell-Mann-Telegdi  a'(E2)  Tr e  e /hc =  1 / 1 3 7 and  2  2  result  putting  R  ( i . e . the  fm  putting  Z  =  A / 2 , the  TT A <r > " 137 12 "M^  E  sum  f o l l o w s from = 1.2  2  2  d  oft-quoted  <r >  2  i s obtained as  o(E2)  The  Z  2  2  J  rule  <r >  =  2  g  (  E  |  3/5  charge  A  dE  )  R , 2  =  0. 2 2 Z A " V 2  where  ~  8  yb/MeV  3  R = R AV  3  Q  and  distribution  i s assumed t o be  nucleus  the  o  uniform  throughout  the n u c l e u s ) .  To o b t a i n the EWSR f o r the given  in  eguation A - 7 i s used.  charge d i s t r i b u t i o n value (ME  f o r **N.  59),  for N l 3  From 2  N,  expression  The mean sguare r a d i u s of the  was assumed to be the same  the  t h i s i s 6 . 0 fm .  1 3  work  Thus from  of  as  Meyer-Berkhout  equation  et a l . the  A-7,  the  EWSR  l i m i t i s 4 . 2 9 yb/MeV. To  evaluate  /  dE f o r the e x p e r i m e n t a l  measurements,  i t i s f i r s t necessary to convert the capture (p, o) sections to t h e i n v e r s e ( *P ) photodisintegration Y  Y  Q  cross cross  123  sections.  T h i s i s accomplished using t h e p r i n c i p l e o f d e t a i l e d  balance,  which s t a t e s that  {DE 67) (21 +1)(21 +1) p  a(Y,P ) o  = o(p,Y )  2  (  2  I  +  1  ? -  JZ  T  and I  A-9  * l  A  where I , I  2  a r e the s p i n s of the t a r g e t  A  proton  and  the  nucleus,  residual  the  nucleus,  r e s p e c t i v e l y , i n a (p,Y) experiment and  Pj and p  a r e the c e n t r e  2  incident  of  particles  mass  momenta  of the  i n the {*,p) and (p,Y)  processes, r e s p e c t i v e l y . Substitution 1 2  C(p,' )  1 3  0  N  of  the q u a n t i t i e s  r e a c t i o n i n t o equation  appropriate  to  the  A-9 y i e l d s  793.3 E o(Y,P ) g-T" — a(p,Y ) yb/MeV ° Y ° L a b  A-10  2  o  After  Q  t h e E2 p h o t o d i s i n t e g r a t i o n c r o s s s e c t i o n s had been  determined i n t h i s manner, the EWSR was evaluated up  the energy  these  segments  measurements  region was  were  studied  bounded  by  by  breaking  i n t o e i g h t segments.  Each o f  energies  where  experimental  made. . The EWSR was then c a l c u l a t e d i n each  segment and t h e r e s u l t s added together  to yield  17.6 MeV  /  dE = 0.44 ± 0.17 yb/MeV  A-1 1  11.1 MeV  F u r t h e r d i s c u s s i o n of t h i s r e s u l t i s given i n Chapter IV.  124  Appendix B  T-MATBIX  ELEMENTS  In t h i s appendix, the e x p r e s s i o n s connecting  the angular  d i s t r i b u t i o n c o e f f i c i e n t s t o the r e a c t i o n amplitudes will  be developed.  and phases  T r a n s i t i o n s i n v o l v i n g M1 r a d i a t i o n w i l l be  considered i n a d d i t i o n to those i n v o l v i n g E1 and E2 r a d i a t i o n . I t has a l r e a d y been s t a t e d i n Chapter case to  of a 0  +  t a r g e t , s ^ and  E1 c a p t u r e , and  These  states  will  respectively., p y and 1  2  P3/2  a n <  a  r  t  i l a  f  w  a  v  e  n c o n ,  momentum  above  reduced  and p a r i t y  matrix  eguations  polarized  are l i s t e d  been  between  elements and the u n p o l a r i z e d angular can be  written  down  between the reduced  matrix  immediately.  elements and  angular d i s t r i b u t i o n c o e f f i c i e n t s .  The r e s u l t s  i n T a b l e B-1.  omitted  for clarity.  f o r t h e A^ c o e f f i c i e n t s , and a  e x p r e s s i o n s f o r the B^ c o e f f i c i e n t s .  phase  angles  I t i s to be understood  where a term i n t and t' o c c u r s , there i s a cos expressions  f,  Making use o f  connection  In t h i s t a b l e , the c o s i n e s and s i n e s of the have  and  I I I - 5 and I I I - 9 , i t i s a t r i v i a l extension t o  o b t a i n the connection the  capture.  conservation,  waves l e a d t o M1 capture.  distribution coefficients Using  t o E2  as s, d, p  the t a b l e s of Carr and B a g l i n (CA 12), the the  the  n <  lead  s  for  i - ? p a r t i a l waves l e a d  be abbreviated  By angular P  ^5/2  i  I I I that  in  (<|> -<t> ,) t  t  sin(4> -<f' i) t  t  that the  i n the  125 fable  B-1  R e l a t i o n s between the Angular D i s t r i b u t i o n C o e f f i c i e n t s Reduced T-matrix Elements * A  = 3s* * 3d* + 5p2 * 5f* + 3 p ?  0  r  r  + 3p* 3/2  / 0  l/2  A^ = 9.487sp - 1.342pd + 9.859df - 7 . 3 4 8 s p A  - 5.196p  3/2  d  = 2.5p2 - 1.5d« + 2.857fz • 2.243sd - 1.75pf - 3 . 0 p 2  2  r  y  + 1.5p2 3/2 A  and t h e  3  - 9.487p  P  l /  2  f - 11,619p p 3/2 *3/2  = 7.746sf • 8.05pd - 4.382df  A^ •= -2.857f2 + 13.997pf B  = 4.744sp + 2.682pd - 4.930df - 3.674sp  B  B  , • 10.392p , d 3/2 3/2 r  1  2  = -2. 122sd + 1.458pf + 23.718p f 3/2  3  = -2.582sf - 2.683pd + .365df  o/  B  = -3.499pf  * Note t h a t u n s u b s c r i p t e d  To are  simplify  p*s r e f e r to  E  2  c  a  P  these e x p r e s s i o n s , the f o l l o w i n g  t  u  r  e  »  replacements  made. s  s/-JT  d  d//T  P  P//5"  f  f//5"  l/2  P  3/2  P  P  P  ->  1 / 2  //3"  3 / 2  /^r  B-1  126  These s u b s t i t u t i o n s l e a d to the s e t of eguations l i s t e d i n Table  B-2.  The c o s i n e s and s i n e s have again been omitted f o r  Table Relations  A  = s  2  between the angular D i s t r i b u t i o n C o e f f i c i e n t s and the Reaction amplitudes *  * p  o  B-2  • d  2  + f  2  2  + p  r  r  2  * p  l/2  2  *3/2  A^ = 2.450sp - .347pd + 2. 546df - 2 . 4 4 9 s p A^ = .5p  2  - .5d  2  • .571f  - 3.000p A  3  + 1.414sd - .350pf - p  p - 2.450p  3 / 2  2 / 2  3/2  d  + -5p*  /2  f  = 2.000sf + 2.079pd - 1.131df  A^ = - 5 . 7 1 f B  3/2  2  - 1.732p  i/2  2  * 2.799pf  = 1. 225sp + .692pd - 1.273df - 1 . 2 2 5 s p  B  2  = -.737sd + ,291pf + 6 . 1 2 4 p  B  3  = -.667sf - .693pd • ,094df  3 / 2  3/2  >  2.683p  3/2  d  f  B^ = -,7 00pf * Note t h a t u n s u b s c r i p t e d p's r e f e r to P / 3  clarity. matrix of Table  E 2  ^ capture.  I n t h i s t a b l e , the t«s and t'»s are now the elements B-1  r e f e r r e d t o i n Chapter I I I .  are  the  actual  reduced  reaction  The t * s and t ' * s  matrix  elements  equations B-1 g i v e the connections between the two,  and  127  Table  III-2  has  been  dropping terms i n v o l v i n g p ^ terms i n v o l v i n g M1  radiation.  obtained from Table B-2 simply by and p  p a r t i a l waves; that  is,  128  Appendix C  POLARIZED PROTON BEAM ASYMMETRIES  In  this  appendix,  the  asymmetries  measurable  developed.  The  p o l a r i z e d proton beam w i l l be used  i n the  right  (R) , with s o l i d  polarizations  experiment  will  angles ^  with  two  a  detectors  be r e f e r r e d t o as l e f t  (L)  and  and fi , r e s p e c t i v e l y , the beam  L  R  as up (+) and down {+), the y i e l d s as Y, and the  amount o f beam d e l i v e r e d  t o the t a r g e t as Q.  Then  there  are  four q u a n t i t i e s measured, which are (HA 65)  \i  Y  Y  Y  V  - V  R+ = V  L  +  1  left  detector  P  L i  P  A  R+  " L P  +  Q  +  In these e x p r e s s i o n s , Y the  +  W --  = \  R* = R  1  V  L +  (  1  +  P  R  )  C  A )  A  C  )  C  +  A  )  C  +  "  "  2  ~  "  1  3  4  r e f e r s t o the counts recorded i n  when t h e i n c i d e n t beam i s p o l a r i z e d up, e t c .  I t has been assumed t h a t the d e t e c t o r s  are not s e n s i t i v e t o the  polarization  power of  and thus A, t h e a n a l y z i n g  the  reaction,  i s constant throughout. The  following  further  assumptions  are  now  made.  number of protons i n c i d e n t on the t a r g e t i s assumed t o  The  be the  129  sane  whether  thus Q beam  = Q does  viewed  from  e i t h e r the l e f t  f o r each s p i n s t a t e . not s h i f t  or r i g h t  detector;  I t w i l l be assumed t h a t  the  when the s p i n s t a t e i s a l t e r e d , thus ft. = T  fi f o r +  each d e t e c t o r .  magnitude states.  of  the  Finally, i t will  polarization  the  assumed  that  the  same i n the two  spin  Hith t h e s e assumptions, equations C-1 t o C-4  Rf " W  Y  The  is  be  polarized  beam  Y  L4-  Y  R+  W  =  =  Q  A  "  1  become  P A )  C  ~  l  1  +  asymmetries  6  "  PA)  (  "  C  P A )  C  follow  by  taking  "  7  8  various  combinations of these e q u a t i o n s . The a n a l y z i n g  power f o l l o w s from  p - l A + _((! + PA) Y  2  which l e a d s to 1 A  =  i/I -  1  ?jTTT  c-10  The charge r a t i o asymmetry f o l l o w s Y  Y  Q  2  • *Lt R t _ i?t ;  Y  L+  Y  0" 2  R+  from  C-11  130  2±  the  /  Rt  Li  4-  J L-)- R4-  solid  angle  Q  Similarly,  =  Y  considering  Y  which  gives  Rt R+  \ "R  C-12  Y  ^R  Y  Y  Lt LI  J *R+  Y  R+  asymmetry  arises  from  

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