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29 MeV elastic scattering differential cross section ratio of 12C/13C Gyles, William 1979

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29  M e V TT  ELASTIC  CROSS  SCATTERING  SECTION  RATIO  OF  DIFFERENTIAL 1  2  C /  1  3  C .  by WILLIAM  GYLES  B.Sc. U n i v e r s i t y of Manchester,  1976  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department  of Physics  We a c c e p t t h i s t h e s i s as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA August,  @  1979  W i l l i a m G y l e s , 1979  In p r e s e n t i n g t h i s  thesis in partial  an a d v a n c e d d e g r e e a t the L i b r a r y I further for  shall  the U n i v e r s i t y  make i t  agree that  freely  of  extensive  s c h o l a r l y p u r p o s e s may be g r a n t e d  this  written  thesis for  It  Department  of  f i n a n c i a l gain shall  fHYSICS  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5  the requirements I agree  r e f e r e n c e and copying of  this  that  not  copying or  for  that  study. thesis  by t h e Head o f my D e p a r t m e n t  i s understood  permission.  of  B r i t i s h Columbia,  available for  permission for  by h i s r e p r e s e n t a t i v e s . of  fulfilment  or  publication  be a l l o w e d w i t h o u t  my  i i  ABSTRACT  The l a r g e f l u x e s of p o s i t i v e and negative in  meson  factories  accumulation energy  over  the  pions  available  l a s t few years have l e a d t o an  o f p r e c i s e n-nuclear  data  and atomic number. L i t t l e  over  nuclear  wide  ranges  in  structure information  has been e x t r a c t e d from the data s i n c e a m i c r o s c o p i c  model  the  developed.  pion  interaction  However the e l a s t i c low  in  the nucleus  i s not f u l l y  scattering d i f f e r e n t i a l cross  for  s e c t i o n s for.  energy pions are p r e d i c t e d w e l l over a wide range of atomic  mass using a p o t e n t i a l i n which some parameters are e m p i r i c a l l y derived.,  since  the. p o t e n t i a l  i s semi-empirical  s t r u c t u r e i n f o r m a t i o n can not r e l i a b l y  be d e r i v e d d i r e c t l y from  the data. Measurement of d i f f e r e n c e s i n the between  neighbouring  nuclear  n u c l i d e s , however, should  the p o t e n t i a l produces the c o r r e c t  some n u c l e a r  variation  structure  be r e l i a b l e i f  of  differential  c r o s s s e c t i o n i n t h i s mass r e g i o n . In  the  section ratio is  measured  experiment  here the d i f f e r e n t i a l  f o r e l a s t i c s c a t t e r i n g of 29 MeV t r  -  on  1 3  using s c i n t i l l a t o r range t e l e s c o p e s . S o l i d  t a r g e t s of pressed measurement  reported  is  powder were  made  the  used.. s i n c e  only  a  cross C/  l 2  C  carbon  relative  e r r o r s i n the r a t i o are s t a t i s t i c a l  only. A l a r g e peak i n the  distribution  of  the  cross  section  iii ratio  i s produced  between the  1 2  C  and  by c h a n g e s i n t h e  of  the  of  pion  measurement o f t h e n e u t r o n  1 3  C  because  with  rms. r a d i u s  rms.  parameters  radius of  1 3  C  are  made.  The  minimum  ratio i s sensitive of  the  neutrons  d e p e n d e n c e on t h e shape o f t h e n e u t r o n potential  interference  » 3 C . The c r o s s s e c t i o n  neutron d i s t r i b u t i o n  interaction  s-p  of  1 3  in C  large the  to  s-wave  nucleus. A  and some t e s t s o f  distribution  and o p t i c a l  neutron  distribution  i s f o u n d t o be 2.365±.025 fm.  iv  Table of Contents  Chapter I .  Introduction  1  Chapter I I .  Apparatus and Experimental Technique  8  1. a) Apparatus  8  b) T a r g e t s  20  2. Experimental Procedure  -  21  Chapter I I I . A n a l y s i s and R e s u l t s  24  1. A n a l y s i s  24  2. R e s u l t s  •  Chapter IV.  Theoretical Interpretation  32  38  1. P o i n t Nucleus Impulse Approximation  38  2. O p t i c a l P o t e n t i a l C a l c u l a t i o n s  42  a) Core + Valence Neutron  47  Model  b) Neutron  D i s t r i b u t i o n rms. Radius  51  Chapter V.  Summary and D i s c u s s i o n  59  Bibliography  61  V  L i s t of Tables  Table I .  Eesults f o r Ar=r -rp n  of * C a  ( Taken from Varma and  3  8  Zamick >. ) 8  Table I I .  Dimensions of s c i n t i l l a t o r s .  Table I I I .  Inelastic scattering contribution  Table IV.  Results.  Table V.  to R. ......  -  Eesults  ( continued )  Elastic  scattering cross  of  16 33 34 35  section r a t i o s  36  »3C/i2c  Table VI.  O p t i c a l p o t e n t i a l parameters  44  Table V I I .  Eesults  53  of c a l c u l a t i o n s using d i f f e r e n t ......  neutron d i s t r i b u t i o n s with equal rms.  radius.  vi  L i s t of F i g u r e s  Figure  1.  Elastic scattering differential  6  cross section r a t i o s . Figure  2.  M8 beamline  Figure  3.  Time of f l i g h t spectrum of i n c i d e n t beam ....  10  Figure  4.  Eeam p r o f i l e s  12  Figure  5.  Experimental  Figure  6.  Range t e l e s c o p e s  14  Figure  7.  Range d i s t r i b u t i o n s of pions i n t e l e s c o p e s ..  17  Figure  8.  Experimental  c a l c u l a t e d .,  18  calculated  20  stopping Figure  9.  9  layout  and  12  patterns  Experimental  and  stopping p a t t e r n s at 0°. F i g u r e 10.  Electronic logic  22  Figure  11.  Time of f l i g h t spectrum from arm  Figure  12.  E vs. dE/dX s c a t t e r p l o t showing c u t s made ...  26  F i g u r e 13.  E vs. dE/dX s c a t t e r p l o t at 30°  28  F i g u r e 14.  Elastic  37  1  25  scattering cross section  r a t i o s of i 3 C / i 2 C F i g u r e 15.  Comparison of * C  and  3  1 2  C  differential  39  cross sections. Figure  16.  F i g u r e 17.  Impulse approximation c a l c u l a t i o n s of fi .....  41  C a l c u l a t i o n s of  45  1 2  C  c r o s s s e c t i o n s ..........  with parameter s e t s A, B and F i g u r e 18.  C.  O p t i c a l p o t e n t i a l c a l c u l a t i o n s of R.  ........  49  neutron + core model ..  50  Valence neutron + core model. Figure  19.  X z curves  f o r valence  vii  Figure  20.  C  1 3  neutron d i s t r i b u t i o n s with  rms. Figure  21.  t  plot  calculations Figure  22.  Results  23.  Besults  24.  X  2  using  optical potential  with  54  p a r a m e t e r s e t ft.  l 3  r p =2.306  ................  curves  with  * r =2.240 3  p  57 fm.  of o p t i c a l p o t e n t i a l  calulaticns  with  56  fm.  cf o p t i c a l p o t e n t i a l  calculations Figure  from  cf o p t i c a l potential  calculations Figure  52  radius  contour  z  equal  ..............  p a r a m e t e r s e t s A,  B and  C.  58  viii  ACKNOWLEDGEMENTS  I am pleased t o Research  Supervisor,  have  this  opportunity  Dr. Richard  and encouragement d u r i n g the course I would a l s o l i k e members  of  to  express  of  thanking  R. Johnson, f o r h i s guidance of t h i s work. my  thanks  to  the  pion  other  the PISCAT group, f o r t h e i r v a l u a b l e a s s i s t a n c e ,  and t o t h e Batho Memorial B i o m e d i c a l F a c i l i t y , f o r the their  my  channel.  use o f  1 CHAPTER I Introduction  Since t h e advent  o f the meson f a c t o r i e s  SIN)  a great d e a l of data on low energy  has  accumulated  developed cross  Pion-nucleus  LAMPF,  pion-nucleus  TRIUMF,  scattering  o p t i c a l p o t e n t i a l s have been  t o the extent t h a t they can  reasonably  predict  s e c t i o n s f o r e l a s t i c s c a t t e r i n g over a wide energy  and over for  .  (  a l a r g e spread i n atomic  this  effort  weight  The  pion-nucleus  to  produce  fits  goal  interaction  complex mechanism r e q u i r i n g account  range  impetus  has been the p o s s i b i l i t y of s t u d y i n g n u c l e a r  s t r u c t u r e with pions, but so f a r t h i s achieved.  . P a r t o f the  the  many  has  has  factors  hardly  proved  to  be  to be a  taken  nuclear  into  t o the data. T h i s , coupled with t h e  l a c k o f s t r u c t u r e i n the c r o s s s e c t i o n s , makes i t d i f f i c u l t separate  been  structure  to  e f f e c t s from the u n c e r t a i n t i e s i n  the i n t e r a c t i o n . In p a r t i c u l a r , i f o p t i c a l p o t e n t i a l parameters are f i t t e d structure is  to  experimental  information  pion-nucleus  the  nuclear  that can then be e x t r a c t e d from the f i t  limited. Ideally,  consideration  a  potential  of  the  would  multiple  be  pion-nucleon  scattering  derived  scattering  nucleus and a l l parameters f o r such a the  data,  strictly  of  potential  amplitudes  a pion i n the derived  1  and  short  range  that  are  two  c o r r e l a t i o n s between nucleonsi»2>  ( the E r i c s o n - E r i c s o n / L o r e n t z - L o r e n z e f f e c t ) . T h i s parameters  from  . I t i s necessary,  however , t o i n c l u d e such nuclear e f f e c t s as a b s o r p t i o n on nucleons )  from  usually  determined  by  introduces  pion-nucleus  2  experiments **' ' ' >. S i n c e 3  5  6  the  7  little  structure,  nuclear  structure details  these  details  experimental The  variations  can  cross  r e p o r t e d here  relative  measurement i t i s  effects  can  of  ir  -  on  1 2  also  normalisation  errors  long  instabilities  errors. cross of  1 3  and  each  predict  A r ~ 0.2  proton r^ in  be  neutron  fm.  * C  and  the  cross  making  out,  a  structure  potential  used  measurement  thus the  removing  problem  of  Systematic  comparison 1 3  C  of  shows t h e  between n e u t r o n  techniques.  to the  neutron  radii  i n Table  I is  Ar=r -rp n  in  * Ca 8  8 ). Hartree-foch c a l c u l a t i o n s only with the high  Accurate  Foch an  because  protons.  of  rms.  and  reliable  Indeed  1 6  >  energy  values  f o r t h e o r e t i c a l approximations  Hartree  distributions  F o r example,  values  , which a g r e e s  useful test  and  from  n e g l i g i b l e compared  ,a  at present  p i o n m i g h t seem t o be  with neutrons  nuclear  apparatus.  IV 2  By  systematic  are thus  of  reference  d e n s i t y dependent The  so  C.  from  a  that  MeV.  eliminating  Chapter  and <* p a r t i c l e r e s u l t s .  would  a t 29  the  a summary o f r e c e n t e x p e r i m e n t a l taken  C  other  in  sections  using d i f f e r e n t  ( table  1 3  Any  virtually  In  Discrepancies exist measured  ,  differential  I n a c c u r a c i e s i n the  cancel and  C  measurement e r r o r s i n t h e r a t i o  distribution  mimic  distributions  measures  extent.  will  differential  show  may  derived directly  suggested  isolated.  should c a n c e l to a l a r g e  statistical  parameters  density be  generally  section.  ratios  term  as  sections the  reliably  section  be  in  such  not  experiment  errors  cross  of used  calculations. obvious of  choice  for  i t s different  several  attempts  studying  interactions have  been  Eesults f o r A r = r - r ^ of * C a . Taken from Varma and Zamick. *  Table  n  8  8  Method  |  A r  (fm)  j T—  TI t o t a l cross sections  Bef. | 1  -+  GeV | GeV | to | MeV |  .19±.05 .21±.05  I I  .39±.10  1  79 MeV | | < 166 MeV 1.37 GeV |  .03±.08 .38±.12 .20±.06  j  1 1  1 3  .08±.05  1  1 5  1.05 1.0 p s c a t t e r i n g <! 10.8 16.3 oi e l a s t i c scattering  |  90 t o 240 MeV  | | L  i  I  9  0  1  1 1  1  1  |  12 1  1 1  4  1 .  i  4  made  to  exploit  interaction sections  the  isovector  nature  i n t h i s way. , B a t i o s  were  measured  1 7  * *  of  c+  ( o r fl+p ) i n t e r a c t i o n .  is insensitive  t o reasonable  the  and  at energies  1 8  e n e r g y , t h e c - p ( o r ti+n) i n t e r a c t i o n t h a n t h e n-n  of  pion-nucleon  fl-  total  cross  n e a r 1 GeV . . A t t h i s  i s about 3 times The r a t i o 6*/6~  v a r i a t o n s i n the  stronger  # however,  neutron  density  d i s t r i b u t i o n s . B e c a u s e o f t h i s i n s e n s i t i v i t y and b e c a u s e o f t h e dependence  on  calculations  distributions  extracted  of  from  Coulomb  these  effects, the density  measurements  are  not  reliable. When  measuring  the  differences  in  radii  between  neighbouring  i s o t o p e s many u n c e r t a i n t i e s i n t h e e x p e r i m e n t  theory  cancelled.  are  e x t r a c t neutron pion of  1 6  total 0  rms r a d i i  method  for 0 l 8  a n d *°Ca r e s p e c t i v e l y . a black  thus the r e s u l t s used.  scattering  and  l 9  4 8  C a by  At these  energies,  were c o n s i d e r e d  Jansen  et  al  2 0  >  insensitive  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 from  so s t r o n g  distribution  as  the  m o d e l must be used  to  relate  the neutron  those  the nucleus i s 3,3  resonance,  to the interaction  radii.  1 8  0  As  and * C a a t t h e 8  the  interaction  t h e p i o n samples o n l y t h e s u r f a c e o f t h e n u c l e u s ,  such  value.  their  h a v e compared T T * and T f - e l a s t i c  so t h e c r o s s s e c t i o n s a r e n o t s e n s i t i v e  rms.  comparing  d i s c because o f the strong  3,3 r e s o n a n c e a n d e x t r a c t e d n e u t r o n is  was u s e d by C o o p e r > t o  c r o s s s e c t i o n s i n t h e 100—200 MeV r a n g e w i t h  essentially  model  This  and  As  pointed  rms. r a d i i  of o p t i c a l parameters  t o l o w moments  rms. r a d i u s . A m a t t e r the  effective  of  the  distribution  radius  o u t by S t e r n h e i m and Y o o > , 2 l  to  the  however,  e x t r a c t e d depend upon t h e p a r t i c u l a r s e t used.  This  i s  caused  by  the  simple  5  diffractive largely  by  nature the  of  product of  r,  the  e f f e c t i v e radius  in  the  absorptivity  the  nucleus,  rms.  radii.  affect  the  necessary  the  potential can  absorptive,  and  3,3  giving the  of  fits  the  cross  Dytman  The  the  1 2  C  et and  is  radius  of  neutron  potential  also  for different r a d i i .  p o t e n t i a l should i f i t i s to  al. *C  a shift  small  2 2  >  also  A  predict  independantly f i x  spin  MeV  with  s,p  pipns  n+.  only a  be  dependent  generally  1 shows  free  would  be  k.k«  term,  where  and  .005  fm  s-wave i s o v e c t o r  scattering  length  . 13  fm  p-wave i s o s c a l e r s c a t t e r i n g  volume  .64  fm  3  p-wave i s o v e c t o r  volume  .43  fm  3  tr-N s c a t t e r i n g  It  produced  length  from the  low  n-nucleon  s-wave i s o s c a l e r s c a t t e r i n g  determined  that  minimum.  coefficients  scattering  cross  s m a l l peak a t  the  so  parametrised  X.x* ( c +c t v t ) x  lower  differential Fig.  effect  may 0  not  interference  a larger  of  nucleus i s not  are  the  considering  amplitude +bj  data  50  flat  i n the by  0  using  measured  with  3  scattering  a  section  quite  f(6) = b  are  and  Changes  different  r e s o n a n c e . Here t h e  amplitude, that  neglecting  for  part  section  anticipated,  scattering  fits  real  again  cross  the  distribution  w i t h n~.  to  occurs.  effective  a l l e v i a t e t h i s p r o b l e m by  r a t i o cf  be  absorption  nucleus,  parameters.  a n g l e s c a u s e d by may  k,  total  diffractive.  the  of  away from  section  leading  i s determined  wavenumber i n t h e  which t h e  requirement i s that  measured  energy  at  the  which  cause changes i n the  thus  value  scattering,  k,  Changes i n t h e  the  One  the  lengths  2 3  *.  the  Figure  1.  Elastic  Scattering  Differential  C r o s s S e c t i o n E a t i o s o f 50MeV n D a t a from  Dytman.  +  2 2 )  on  » C/» C. 3  2  7  Tc and T c 1 are the wavevectors o f the incoming and outgoing pion.  "t and X are t h e i s o s p i n v e c t o r s  t.T =+1  f o r n p or n~n and -1 f o r n-p or -n+n.  For  +  the  contribute  i s o s c a l e r nucleus,  the  1 2  C , only the i s o s c a l e r terms  . For H+n, the i s o s c a l e r  c a n c e l t o a l a r g e extent, have  of the nucleon and p i o n .  and  isovector  amplitudes  and the e x t r a neutron o f * C w i l l not 3  a l a r g e e f f e c t on the T I c r o s s s e c t i o n . With n~, however, +  isoscaler  proportions  and  of  s  isovector and  amplitudes  add.  The  relative  p wave a r e a l t e r e d from the completely  i s o s c a l e r case because o f the l a r g e s-wave i s o v e c t o r amplitude, t h e r e f o r e a l a r g e change i n the s-p i n t e r f e r e n c e minimum may be expected. Since where  there  the i s o v e c t o r  is  amplitudes  are  effective  only  a neutron o r proton excess , t h e if should  s e n s i t i v e t o the d i f f e r e n c e between proton and neutron  radii.  be  8  CHAPTEB I I Apparatus  1.  a)  experiment  beamline  , M8.  kinetic  Particle The  was  energy  a  performed  of the  fractions  beam  of about flux  last  the  should  quadrupole.  stable  over  beam was  also  monitored  plastic  scintillators,  production  from  M1  ( about  from  M2,  was  T2,  spectrum  taken  the  to  by  used  a  pair  in  also  flight  way  the  ) .  of  The  large  c o i n c i d e n c e to stop  determined  the  was  needed  perform  3 t o 5 hours  primary  this  gas  directly  not  in  t h e muon time  pickup placed i n f r o n t  in  a  being t h a t i t  t h e t a r g e t , which  scintillators  s e p a r a t e d i n time  A pair  with  placed  required  the t a r g e t and  a capacitive  target,  time-of-flight clearly  The  time  after  7uA,  t h e i o n i z a t i o n chamber  c o n t a m i n a t i o n s by m e a s u r i n g  particles  typically  primarily  atmosphere,  three targets  Ap/p=4.2%.  .  —  Although  the  protons, a r i s i n g  counter.  electron  s *  s  was  The  electrons=84%.  measurement, t h e o n l y r e q u i r e m e n t  on t h e  first  run  2  measurements  eliminate  with  1% a c c u r a c y * * , t h e c a l i b r a t i o n  ratio be  the  MeV  muons=3% and  monitored  after  to  i s shown i n F i g . 2. 30  week's  1.4x10  was  chamber, open t o  calibrated  was  were p i o n s = 1 3 % ,  ionization the  on t h e TBIUMF b i o m e d i c a l p i o n  beam  d u r i n g the  pion f l u x  The  for  Technique.  A sketch of the beamline  proton current  giving  Experimental  Apparatus  The  pion  and  the and  of  the  of the  pion  proton  beam.  A  shows n - ' s ,  u's and  e's  ( F i g . 3 ).  o f w i r e chambers g i v i n g  mounted b e h i n d t h e s c i n t i l l a t o r s  M1  X and and  Y beam p r o f i l e s  M2.  They  were  we.re t r i g g e r e d  9  Figure 2 .  Channel  M 8 Beamline.  8 m take-off  length  Momentum  0  range  Momentum  acceptance  Momentum  resolution  flux  1.5* ir  at  momentum Be  180  MeV/c,  acceptance  full  10  cm  target  C o n t a m i n a t i o n a t 180 M e V / c i r " 10 cm c o o l e d B e t a r g e t Dose r a t e field  to  a  5  x  5  220  ±6.7%  x  5  cc  AP/P AP/P IT"  8  *  10  ! 1.3  x  10  J  +  30"  MeV/c  or  Polarity TT  -  angle  s  6  FWHM  FWHM  ir /sec/yA  proton  ir"/sec/yA  proton  +  ~2k%  electrons  ~11*  muons  ~0.2  rad/min/uA  proton  Figure  3.  Time o f F l i g h t of Incident  Beam.  Spectrum  11  by the c o i n c i d e n c e M1»M2 . The mounted  on r a i l s ,  wire chambers and  c o u l d be s l i d  scintillators,  along the beam d i r e c t i o n . Beam  p r o f i l e s were taken, with the corresponding beam tune, f o r both t a r g e t p o s i t i o n s . The beam p r o f i l e s at the t a r g e t p o s i t i o n s are shown i n F i g . 4 along with on-line  program.  a  Profiles  contour  plot  Fig. 5  is  by  an  were a l s o taken a t i n t e r v a l s along  the beam to estimate the.beam divergence be approximately  generated  , which was  found  to  ±1.4°. a  diagram  of  the  experimental  l a y o u t . The  d e t e c t o r arms, mounted on a t a b l e , p i v o t t o r o t a t e  around  t a r g e t c e n t r e though angles up to 90° f o r Arm  70° f o r Arm  2.  Because  of  1 and  space r e s t r i c t i o n s i n the experimental area, a  t a b l e which would allow r o t a t i o n o f the arms t o angles than  90°  could  not  angles the t a b l e was the new  be  used.  t o t a l energy formation  To make measurements a t these  turned around and the beam  pions  cannot  measurement  as  nuclear  disintegration  ) , cn s t o p p i n g , d e s t r o y s the energy  the e l a s t i c a l l y  used  s c a t t e r e d pions may  scintillator  range  of  the  solid  stops the pions and  star  be separated  energy,  and  by t h e i r range  ( see F i g . 6 ) d i f f e r e d only i n the elements t h a t  define  (  s i g n a t u r e o f the  stopped  f i r s t two counters i n each  angle and s e l e c t only p a r t i c l e s coming  from the d i r e c t i o n o f the t a r g e t . The next t h i c k counter  pions  on  t e l e s c o p e . The two range t e l e s c o p e s  the e l a s t i c a l l y s c a t t e r e d pions. The arm  refocussed  be r e c o g n i s e d s o l e l y by dE/dX and  pion . T i e pion range, however, i s a f u n c t i o n  a  greater  target position.  Negative  in  the  d e f i n e s the energy A E / A X .  In  Arm  almost 1,  stop i n the next counter, whose t h i c k n e s s i s designed  the to  Figure  H.  Beam  Profiles.  Contour plot of the beam at downstream target po s i t i o n  13  Figure  v;  s;;;;x  6.  Range  Telescopes.  V ////  C1  Arm1  rW/////l  /  A  C 1  Arm 2  C2  I  C 3  / / / //A  C2 C 3  15  stop  2 8 . 4 ± 3 MeV  pions  energy  loss  i n the  by  thin  counters  five  to  i n each  through  the  target  arm  incident  ) . I n Arm  pion  2 this  beam  difference  s e r v e s as a v e t o  stack.  The  dimensions effective  were a p p r o x i m a t e l y  i s replaced  being  equivalent  i n pion energy. for  particles  of the  solid  6 mb/sr  The  final  that  pass  scintillators  angles 1)  (Arm  1.6MeV  and  counter  , the t h i c k n e s s o f each  g i v e n i n T a b l e I I . The telescopes  MeV  1 . 9 MeV  approximately  counter  ( 30  of  and  are  the  two  8 mb/sr  (Arm  2) . The  energy  stopping incident  resolution  counters beam and  different  ,  i s limited  range  t h e energy  depths  in  the  target  superimposed  on  The  stopping pattern,  stopping  pattern  range The  energy  range  2 6  *.  has The  evaluated  from  accounting  f o r the  Ar/r an  0  for  intrinsic  See,  30  of  .  f o r e x a m p l e , E . D.  The  with  is  1.2 MeV  FWHM  E v a n s , The  The  factors  experimental  done  with  for the  .032. is  a  assumed f o r t h e  the  mean  range.  pions same  i n masses o f p i o n s t o be  from  convolution to  was  s h a p e was  values  protons  pions i s found of  the  t a b u l a t e d f o r protons over  difference  resolution  these  the  distribution  corresponding  those  MeV  been  of  geometries.  with  0  0  the  , a r e shown i n F i g . 7.  w i t h A r = . 042 r , where r„  straggling  of  scattering  from  Fig. 8  . A Gaussian  2 5  Ar/r  along  in  the range  program >  ratio  distribution  shown  on  by  some  stopping counters  is  combine t h e e f f e c t s Monte-Carlo  range  for  thickness resolution  spread caused  to the  resulting  the  straggling,  contributions  the  by  and  a may  be  range  by  protons  Using A E = A r obtained  Atomic Nucleus,  wide  dE/dr  for  pg.  the  664.  Table I I .  Dimensions o f S c i n t i l l a t o r s i n Telescopes. ..  i!  ARM 1  ARM 2  (Counter| Diameter |Thickness I|Counter I Diameter (cm) (cm) I (cm)  i 1  _ _ _ _ _ _ j__  1—  |  C1  |  I  C2  |  I  C3  i  s  |  VETO | 10.0X7.0 |  L  ++  l_  3.0  +-  C1  |  3.0  |  0.33  ||  3.0  t  0.33  ||  C2  |  |  5.0  |  1.97  i i  C3  I  |  5.0  I  1-27  || S1-S5 |  —1  .j  0,-640 || J _ _  |Thickness| I (cm) |  •\  -i  |  0.33  |  3.0  |  0.33  |  5.0  |  1.27  |  5.0X5.0 |  0.425 |  VETO | 10.0X7.0 |  0.640 |  x_  .j.  J  17  Figure  7.  Range D i s t r i b u t i o n  of Pions i n Telescope,  Range d i s t r i b u t i o n s i n c l u d i n g e f f e c t s of fa Range s t r a g g l i n g  Arm 1 stopping counter Arm 2 stopping counters  B  + 4.2% Ap/p  C  + target thickness  F i g u r e 8. Calculated  E x p e r i m e n t a l and Stopping Patterns.  Experimental  Arm 2 at 60° -Target at 45°  51  Calculated  S2  S3  S5  19  telescopes.  Fitting  in  of A r / r  the value  resolution  stopping  with  target  the  distribution  i n front range  formation  the  tail  intrinsic  9 compares t h e measured and  30MeV  of  t o an  pions  telescope.  caused  by  Ap/p=4.2%  with  The measured  charged  nuclear  t r a v e l l i n g beyond t h e p i o n  range,  l i m i t i n g the resolution,  Targets The  1 3  C  powders i n t o the  three  cm.  and thin  * C targets mylar frames.  frames  cm.  was  thick  than  being  0.85  as  of  frame  density The  * C 3  .327  possible.  target,of  The 5.17  0.019  cm. t h i c k  mass/cm?  of  the targets.  less  Over 9 9 % o f t h e  mass was a t t h e c i r c u m f e r e n c e , where t h e p i o n  target  density  The a r e a  2  while t h e  1 2  was l i m i t e d by t h e r e q u i r e m e n t  i n the pion  gm./cm.  precise  each  as  made,  cm. deep a n d t h e two windows were o f 0.00025  the  available.  plastic  similar  also  flux  was l e s s t h a n 5% o f t h e peak.  uncertainty MeV.  An empty frame was  carbon  m y l a r . The t h i c k n e s s o f t h e windows r e p r e s e n t e d  0.2%  target  were made by c o m p r e s s i n g  2  diameter c i r c u l a r r i m of  mylar,  ±2  patterns f o r  star  p a t t e r n s o f Arm 2 r e s u l t s  =.042 u s e d , c o r r e s p o n d i n g  has a l o n g  fragments from  b)  0  o f 1.6 MeV FWHM. F i g u r e  calculated  further  t o the stopping  energy  scattering  be  was t h e n r e s t r i c t e d by t h e amount  The t a r g e t  ( C)  densities  . The * C t a r g e t  12  3  C target  holders  on  was o f n a t u r a l  were g l u e d  relocation  were  that the  less  than  ( 7gm. ) o f  .330 gm./cm.  2  (  1 3  C ) and  was o f 9 9 . 7 % i s o t o p i c  purity  i s o t o p i c composition.  Thin  t o t h e base o f each t a r g e t  on t h e s c a t t e r i n g  table.  t o allow  20  F i g u r e 9. E x p e r i m e n t a l and C a l c u l a t e d S t o p p i n g P a t t e r n s a t 0°. (  Experimental V  Calculated  r • I I l  a  I S1  1 S2  I S3  I S4  I S5  VETO  21  2.  Experimental  Procedure  L a s e r r e f e r e n c e beams, which had p r e v i o u s l y by  the  Vertical  biomedical  group,  were  along  the  beam  aligned  used t o a l i g n our apparatus.  and h o r i z o n t a l planar beams were  intersection  been  aligned  with  their  d i r e c t i o n and c l o s e to the beam  c e n t r e . A 3" t h i c k lead block with a 3/8" hole d r i l l e d  through  was p l a c e d a t the t a r g e t p o s i t i o n with the hole along the l a s e r beam  intersection.  A  wire  chamber  p r o f i l e of the beam  just  behind the l e a d was taken, c l e a r l y showing the p o s i t i o n of  the  h o l e . The l e a d was removed and another beam  centre  could  be  r e l a t e d t o the l a s e r i n t e r s e c t i o n . The  t a b l e and t a r g e t  were  aluminum  with r e l o c a t i n g  blocks  to provide automatic additional  p r o f i l e taken so now the  aligned  using  alignment  the  laser  if  the  table  was  moved..  check could be made on the t a b l e alignment  differential  cross  and  s l o t s were f i x e d t o the f l o o r  beam d i r e c t i o n using the approximate 1/sin*(9/2) the  system  section  at  small  with the  dependence  angles.  An  This  of is  d i s c u s s e d i n Chapter I I I . A diagram o f the e l e c t r o n i c l o g i c Many  of  the  electrons  were  is  given  in  r e j e c t e d by the d i s c r i m i n a t o r s ,  which were adjusted t o accept a l l pions. In each arm was  defined  connected the  an  by the c o i n c i d e n c e C1*C2«C3. The C2 counters  event were  t o constant f r a c t i o n d i s c r i m i n a t o r s and the t i m i n g of  event  was  discriminator. ( the output start  F i g . 10.  the  defined The  the  coincidence  pulse of the TDC's,  by  which  output  C1«C2«C3«BF,  capacitive were  C1«C2«C3, and a l s o t o i n t e r r u p t  pulse  the  by  this  timed by the EF  pickup ),  stopped  of  the  computer.  was  used  event  to  pulse  Time-of-flight  /  CN CN  Cl  -CD-  C2  U •H  m o  ARM  C3  1  VETO  U •rH  •CD-  c o  1-1 +J  o  0) rH  W  o 01 M 3  S1  ARM.S2 2 S  S3  cn  •H  S5 VETO  -czt  23  and  analogue  signals  were  recorded  modules.,Bits were s e t i n a each  24  f o r each event i n CAMAC  b i t C212  pattern  unit  for  element of the two arms which t r i g g e r e d i n the event..The  data were t r a n s m i t t e d v i a a 2MHz s e r i a l CAMAC t r a n s m i s s i o n  link  to a NOV A 1200 computer  then  recorded  on  a  interrogated  magnetic  the  time—of-flight,  where tape.  buffer  pulse  they A  and  height  were  buffered  real-time on  and  FOETBAN program  command  or stopping  pattern  displayed information  while t h e experiment was i n progress. Each angle i n v o l v e d runs targets  and  stopping  d i s t r i b u t i o n depended on  angle  of  with  scatter,  the empty  with  each  frame.  of  Since  the t a r g e t  the t a r g e t angle  the two  carbon  the width of t h e angle  and the  was a d j u s t e d t o the best  compromise f o r the two arms. Each i n d i v i d u a l run was l i m i t e d t o l e s s than 4 hours ( mostly l e s s than 2 hours ) so t h a t any l o n g term d r i f t s i n d e t e c t o r e f f i c i e n c i e s , i o n chamber e f f i c i e n c y or pion f r a c t i o n would not be a p p r e c i a b l e i n the: time the  two  taken f o r  carbon t a r g e t s . Near the minimum o f the c r o s s s e c t i o n  (60°—70°) the t a r g e t s were were c o l l e c t e d .  interchanged  until  enough  events  24  CHAPTER I I I Analysis  1.  Analysis A modified  computer on  each event  way c u t s  and  could  parameters  ) , energy d e p o s i t e d  e l e c t r o n s produced  peak however c o n t a i n s from  pion  angles  AE/AX  they  many muons from particles  The s t o p p i n g  the  the signal  stopping  signals  Since  strengths these  assumed  deposited counters  production  ( E ),  spectrum  separated  in  from  from muons  t a r g e t . The p i o n  pion  decays  be s e p a r a t e d counter  from gains.  in  the  by c u t s  i s composed  these  were  The r e l a t i v e  beam  are  on t h e E and of 5 elements  added, gains  weighted were  found  o f t h e 96.3 MeV/c e l e c t r o n s f r o m  the  have a r a n g e o f 39 cm. i n t h e s c i n t i l l a t o r to  d e p o s i t t h e same e n e r g y  i n each  of the  elements.  Scatterplots time-of-flight requirement cuts  may  t o the r e l a t i v e  were  a  a t low  signals.  channel.  on  d e c a y s between t h e t a r g e t and d e t e c t o r s . A l s o  arm 2 and  from  energy  clearly  from  made  were s u c c e s s f u l . I n t h i s  A time-of—flight  i n the pion  protons  displayed  i n the stopping  signal.  were  muons  These  according  event  Tests  370  t h e c a r b o n t a r g e t and  (<40°)  detected.  the  i f the tests  1 ( F i g . 11 ) shows t h e p i o n s  and  by  of  be made on t i m e - o f - f l i g h t ,  or t h e absence o f a veto Arm  r o u t i n e on t h e OBC IBM  i n the analysis of the data.  or scatterplot  (AE/Ax  C3  KIOWA h i s t o g r a m m i n g  was used  histogram  in  and R e s u l t s  of E vs. AE/AX peak  were  C1»C2»C3«C4»VETO. on  AE/AX  of  produced,  particles with  Muons and p r o t o n s  in  the were  the  pion  coincidence separated  ( F i g . 12) . Note t h a t t h e p i o n s have a w e l l  • • X X 2 3 7 9 .  XX  2 3 1 3 .  XX XX  2 1 9 6 .  XX  21 ii. 2 C 1<*.  30  l  XX  xx  2 0 1 3 . 19  XX  5 2 .  1 0 4 1  » x x  .  X X X  11. 3 0 .  A X X *  17t;  .  x x x x  1 7 DP  .  H fD  A X X X  1647.  A X X X  15 3 6 .  x x x x  1525.  A X X X  1464. 1403  A X X X .  A X X X  1342.  A  12 81  .  1220  .  Hi M O 5  X X X  A X X X G A X X X F  1 1 5 9 .  x x x x x x  109b.  3>  n  X A X X X X  1037.  X A X X X X  976.  B  X A X X X X  915.  -» •  X A X X X X  R54 .  X A X X X X  7 3. e  X A X X X X  732 .  X X A X X X X X  electrons  43d .  X X A X X X X X  *27. 3fc6.  3P  305.  X X  244.  + X X  1fc3 .  X X X  ••>.  ill  11  2 2 2 4 C O  I  N H H> i Q  X X X A X X X X X +  unions  +  X X X X X X X + P 4  n:5444^5S<^566fc 6 6 7 7 7 7 7 6 3 33  FOGF  X X A X X X X X  X X X A X X X X X  L X X X A X X X X X X X X X X A X X X X X X 4  1 • X X X X +  122.  B! N  O  HI  W T3 rt) O r+ H C B  • X A X X X X  5 4 9 .  Olr.iF.  I  r+  X X A X X X X *  61 ,  H) HB (D  X A X X X X  67L .  L  c  pions  11  1  11353  1 5 Z 1 2 9 D A 5 4 2 5 7 5 5 & J P » X X X X A X X X X X X X r 6  2  23  i i i m i i i i i i i u m m n m i m i nm i l i u i i i i m i | S 9 < : 9 9 9 0 0 0 C 01 1 1 1 1 2 2 2 2 2 5 3 3 33 4 4 4 4 4 5 5 5 5 1 6 6 6 66 7 7 7 7 7 3 3 8 e S ? 9 ? ^  5 7 9 1 3 ^ 7 " 1 3 5 7 9 1 3 < 7 9 1 3 5 7 0 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3 5 79 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 * 3 5 7 9 1 3 5 7 9 1 3 5 *  Yc ^oooconorooooooooccoocoooorooccccccoocoooooooocooooooooojoooooojoocoocooooooootl  to  Ul  26  12- E v s . A E / A X S c a t t e r p l o t Showing C u t s Made.  Figure  -« rg tr»  J  IM  • O. f- — f «M .  rJ  . <_ — e» *oW  x  MA ^if\J -  L U  !  < !  (j.  (T w CT- CO O  • I  • i I I INJ  .  ^ r- L, j> >o o ;y tn o o cr m c CT> tvJ o o> — o o- o o CC ? O IC 13 C X! r»- O tr >o O a •« o ^> «J" o t) f* o CC — C cc i" o f* (?• o r— o N h O f- -G 0 r- m c r~ o — f fi o p» !*J o f- — o o o 43 0 o n o s  T — ~>  kioo Lo m o _ O I ,o o  in _ O r— O i in <o o tn x» ~• ir\ o tn •*» o in rsi o • in O O J c o s f x> o - o» r~ O «4- in O 4- —  • 4- o n Im J u  m a o o o im ir. o . -4- C \ O m rj o i m —* o • r*i c- O M U D <M CD O CM f- C r\j C rsi m f~ (si -r T , :M f\* O i fV -* c > rvv O •i l_T O  •0  H  -« • i i I-* .  CD  C «o o in o O i*. O (Nt O o c cr o CO c  N  1  r- o  -o o m c o m c (M CJ  L U u"\ c if. C> x !>n c m i. • *nio C if' c • l_' f~ "-^ ^ C ^ ^ ir. ^ c ^ j i.i rsj c_- r- tr. f N J ' f — J~. x to s r— r- r-jr— <c -o vO o ini*.* m ir 4 *r 3* CT- C 1  —  i -, * O o O ir. c I A c i^io in ^ if\ c ir> o • • • , t~ ir. c- o f-''-*\ rv n r- if. m o «n c> fnrnf^r-ir\Iirsjr>JfNJ—«—««j--if-inf\) •  27  27  defined AE/AX nuclear  but  a l a r g e spread  disintegration  mucns from  pion  stopping  made i s f o u n d empty  by  ( F i g . 13).  the A E / A X  examining  frame  nuclear The  run  energy  At a n g l e s  are rejected  picns producing  counter  target  stopping.  decay i n f l i g h t  E to include only the  on  i n E, b e c a u s e o f  by  from  l e s s than  45°  making a c u t  on  disintegrations  position  of  the c u t t o  vs. E s c a t t e r p l o t  which c o n t a i n s mostly  of  muons a t  in be the  these  an g l e s . In empty  a l l cases  the  same c u t s were  t a r g e t frame r u n s .  detecting three  elastically  targets,  appreciably  efficiencies  s c a t t e r e d pions  assuming  during the  Differential data  The  made  for  1 2  of the  was  the  f o r the  three  c r o s s s e c t i o n s may  * C  and  3  detectors for same  p h o t o m u l t i p l i e r gains  time  C,  for  d i d not  the  change  runs.  be  calculated  from  the  using  doydft  = (H / I C -N F  F  B  /IC  8  ) c o s VS)  SJ (0) / (Fft j DE^T e  )  f  where N  and  F  target IC F  F  N  out  and is  are  g  number o f %  i s the  number o f p i o n s  are the  corresponding  conversion  target  f a c t o r from  pions i n c i d e n t angle  fraction  without  on  the  i o n chamber the  target i n  and  solid  of  readings,  i o n chamber  reading  to  the  target,  between t h e t a r g e t n o r m a l and  Slejtis t h e e f f e c t i v e D i s the  detected during  runs,  ICj a  the  the  beam,  angle,  pions  reaching  the  detector  from  the  decaying,  En  i s the  efficiency  S  i s the  fraction  o f t h e d e t e c t o r arm  of the  for pions,  range d i s t r i b u t i o n  of the  elastically  PIONS  VETO,  HMSCA  1 1 1 1 i 11 11 12<!ii22?.'Z2i33}3333 iih','-^'. <,<,5555555 5 5 5 6 6 6 6 6 6 6 j 6 6 7 7 7 7 7 7 7 7 7 7 8 C o ? 3 OS 8 6 9 9 9 9 9 9 9 * 3 9 1 2 3 « 5 6 7 H ' " ~ 1 2 3 ' » 5 6 7 B < : f : i 2 3<f5t 7 8 ^ 0 1 2 3 4 5 6 78 9 0 1 2 3 4 5 6 7 H 9 0 1 2 345671) 701 2 3 4 56 / 8 9 0 1 2 3 * 5 6 7 8 9 0 1 , : 3 4 5 6 7 3 5 0 1 2 3 ^ 6 7 8 9 OO3C00C W O O J O C O O J O C J C O C )OOOCOOOOOCCCOOCOOOOOtOOO3OO3OO0OO 3 0 C 0 0 0 0 0 0 J O O O O O S C P O O r O O O w C O O O O O O O O O O W n O i  :  f. I N 0 0 0 0 3 0 C C O 0 00 0 3 0 0 C O O C O 0 0 0 0 O O O C O O C C C C O O J O 0 0 0 0 0 1CO0O0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O C C O 0 0 0 0 O J O O O O O O O O O O C ' 0 0 0 0 0  co  29  scattered  pions covered by the s t o p p i n g  J (0) i s the J a c o b i a n dJl^j,/d5l T i s the number of n u c l e i / c m In  calculating  the  ( J /  i2 j= =[ <  B  Superscripts with  1 2  C,  1 3  C  J(8) all  1 2  i 3  i  N  (  cross  section  ratio  , leaving  3/[i N /i IC 2  -%/ICj, ]* T/i3T 2  2  F  F  s i g n i f y q u a n t i t i e s a t t r i b u t a b l e to  1 3  the run  respectively.  the same f o r C  and * C t o b e t t e r than 0.02% at  1 2  angles . U n c e r t a i n t i e s  therefore  i n the t a r g e t .  2  / i s ^ - t y / i c ,  :  target is  t0 m  differential  however most f a c t o r s c a n c e l  i3  element  3  »Jleff  i n f a c t o r s such as F  A  &E/E={[8 N 13  F  M i n E)/6  1 3  N  F  J  2  + [ S  +1 8N 3 i(lnH)/dN  *  E  TT  8  L  2  N  from  M i n E)/d* N J 2  F  2  F  323 1 ^  * N -i N L IC * IC Y N,F _ - N V * N ~ 3  2  F  3  <  do not a f f e c t the r e s u l t s .  S t a t i s t i c a l e r r o r s were c a l c u l a t e d  i IC,  N  1 3NF 3 1 NF 1 3  icF  12N  N ICj B  12IC  F  F  1 Z  IC  6  IC3  \13X C  F  F  2  13  8  F  2  f  1 N - N 2  F  f  1 3  F  IC ]ii2ICp B  where ox s i g n i f i e s the s t a t i s t i c a l e r r o r i n x. Where angle  the  calculated.  more  than  statistically In  set  of runs had been taken at one  weighted  mean  of  the  ratios  was  a l l such cases t h e d i s t r i b u t i o n of v a l u e s f o r  the r a t i o was c o n s i s t a n t The  one  with s t a t i s t i c a l  angle of the beam r e l a t i v e  to  errors. the  0°  line  o f the  30  scattering  table  the  arm  the  same cross  and target  section  scattering, may b e  was e v a l u a t e d  from  a t 30° l e f t  a t 30° v a r i e s  then  the  three  pairs  of  runs  with  a n d 30° r i g h t . A s s u m i n g  as 1/sin*(0/2)  ratios of left  from  and r i g h t  Eutherford  cross  sections  written  (5 (9-C+X) /o*(9-C-X) = s i n * ( ( 9 - C - X ) /2) / s i n * ( (0-C+X) /2)  C= t h e c o r r e c t i o n  t o t h e mean s c a t t e r i n g a n g l e  angle  ) .  X=  ( see below  the  angle  scattering Writing as  between  the  beam  f o rf i n i t e  direction  and  0°  solid  on  the  table.  9-C  as 9  and l e f t ,  C  <5 i 6% r e s p e c t i v e l y  L  /6* 3 / = 1  cross  sections  one g e t s  L  lO  right differential  s i n [ (0 - X ) / 2 ] / s i n [  4  (0 +X)/2}  C  C  =[ s i n ( 9 /2) - c o s ( 0 /2) X/2 J/[ s i n ( 0 /2) + c o s ( 0 /2) X/2 ] C  C  C  C  X=[ 2 s i n ( 9 /2) / c o s ( 9 /2) J£ 1- (6 /6 ) C  6L/6K  i s evaluated  ( 0 . 4 3 , 0 . 61,0. 37) giving  C  where  of the three  An  scattering  additional angle over  differential here.  The  cross mean  from  the  1 / 4  M  1 + (6 / d * ) *' J 4  L  remains  simply  angle  right  the  of  X  runs,  standard  X i s added t o a l l t h e from i n  at small  varies  values  30° l e f t ,  and subtracted  the detector  scattering  i s  The a n g l e  problem  section  three  error  values.  a n g l e s f o r arms on t h e l e f t right.  %  i n t h e same way a s E. T h r e e  are obtained  X= 0.5°±.1°  deviation  L  defining  angles,  rapidly should  those  and be  on  the  t h e mean  since  the  non-linearly  evaluated  by  31  averaging finite  the d i s t r i b u t i o n  beam  spot  size  d e t e c t o r s , weighting angle,  or  to  a  this by  region  angle and  solid  each a n g l e  beam  the  second  f o r a point source.  spot.  by t h e by  by  1/sin*(9/2).  evaluated  The  that Since,  rapidly  in  approximately  integration  detector  the  was  then  which d e f i n e s t h e s o l i d  The r e s u l t s f o r 30° and 4 0 ° a r e 2 9 . 6 °  39.8.  the  inherent  width  inelastically counters. MeV  the  scattered  of * C  pions  arms  to these  small  excited  cross  contributions small  combined  to  to  and  cancel  in  kn e s t i m a t e R  was  0 ( i n e l a s t i c ) /<$ ( e l a s t i c ) Table I I I .  in  the  and  MeV  s t a t e of * C  from  t h e 4.4  were  evaluated  previously to  simulate  3  mentioned  stopping  s t a t e s were c a l c u l a t e d by E. R o s t  section  effect,  contribution  distribution  . The c r o s s s e c t i o n s f o r t h e e l a s t i c s c a t t e r i n g  efficiencies  elastic  t o be d e t e c t e d  and t h e 3.68  2  range  f o r detecting the pions  t h e Monte C a r l o program two  of the  due t o r a n g e s t r a g g l i n g i t i s p o s s i b l e f o r  The e f f i c i e n c i e s  state  using  very  subtended  o f c r o s s s e c t i o n s does n o t v a r y  a point  ever  produced  the cross section at  approximation  Because of t h e broadening  and  with  angle  , t h e c o r r e c t i o n was o n l y  considering  performed  and  good  however, t h e r a t i o  of s c a t t e r i n g angles  with a  a small r a t i o  tendency the r a t i o of the  for  >.  The  of i n e l a s i c to the  inelastic  a l l contrive to give  size  of  made f o r l a r g e a n g l e s i s greatest.  2 7  the  a  inelastic  where t h e r a t i o  The r e s u l t s a r e  shown  in  32  2.  Results 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 r a t i o  of  runs  and  the  weighted  ratios  were  very  close  c o m b i n e d by e v a l u a t i n g t h e w e i g h t e d  weighting the angles s i m i l a r l y  t o f i n d t h e mean a n g l e  and  in  F i g u r e 14 ) . The  results  set  means f o r e a c h a n g l e a r e g i v e n i n  T a b l e i v . I n c a s e s where two a n g l e s were the  r e s u l t s f o r each  analysis discussed i n the following  Table V  were  sections.  used  ( < 1° ) mean a n d ( Table V i n the  33  Table  III. Inelastic Scattering Contribution to B.  1  I I  12c Cross Sections (mb/sr)  I  |Angle | I  I  I  |Elastic|4.4MeV r-  130  | I  | I i  140 150  8.4 ~[o.49  | I  |  I  |  I  9.1 9.6  I  i 3 c Cross Sections (mb/sr)  | I I  I I  Elastic!!.68MevJ 11.6  | E* 3 II B / I f  I  I  I  34|.998|  E*2  I  |  | 0.40  | .61  |  I  I  I  | .59  | .34|1.00|  |0.62  12.5  | 0.49  |0.73  13.1  | 0.57  I  I  I  | .62  |  I  I  I  I  I  I  (  I  36 | 1.00 |  E and E are the r a t i o s of the e f f i c i e n c i e s f o r e l a s t i c / i n e l a s t i c s c a t t e r i n g o f * c and C. 1  2  1  3  2  E' = the measured scattering.  ratio  with  no  1 3  correction  for  inelastic  Table IV.  r  Angle| co. m| h  (B)  Results.  Weighted Mean —I  +  0. 953±. 093 0.971±, 065  29. 5  r +1.018±.10 130. 5 1.052±. 065 0.888±. 041  139-5  0.965±.054 +  1  +— 0.935±.035  161.2 1.64 1.81 1.34 1.53  166- 2  ±. 22 ±. 35 ±. 16 ±. 13  1.59 ±. 32 1.59 ±. 22  170- 3 |71. 3  1.83 ±. 14 1.70 ±. 13 1.79 ±. 14  180. 3  r  + —  -Hi  1.77±. 11  1.46 ±. 10  85.3 [90.4 I  [91.4  1.59 ±.18  1.63 ±. 12 +-  T  1. 5 1 U . 0 8 9 +—  H  1.47 ±. 11 1.427±. 084  1.442±.066  1.46 ±. 11 1.37 ±. 13  1.421±.084  35  Table IV.  [  T  | Angle | jc.o.ml  l  *G/  1 2  (R)  6  (continued)  T  I Weighted  j  Mean  100.5| 1.58 ±.13 | | 111.31  1.56 ±.13  | 1.575±.094  1.217±.073 |  115.2I 1.222±.0eV~[ 131. i t 1.2811.069 1 141. 11 1. 098±.064 | 149.91  1.134±.090 J  j 150-91 1.26 ±.13 | | | 1.153±.064 | 1.175±.057  I  I  I  36  T a b l e V. E l a s t i c Ratios  Angle Cm  O. m.  Scattering  Cross  of 13C/12C.  6/ 6  13  lz  (K)  4 29.8  i  0.976±.047  39.8  0.9351.035  45.0  0.93U.041  50. 1  0.985±.066  60.6  1.2521.075  66.2  1.5111.089  71.0  1.61 ±.10  80.3  1.77  90.7  1.434±.052  101.3  1.575±.094  110.3  1.217±.073  115.2  1.222±.061  130.5  1.28H.069  140.5  1.098±.064  150.6  1.1631.048  ±.079  Section  Figure 14. E l a s t i c Scattering Cross Section E a t i o s of C/ C. 1 3  1 2  1-8"  1-7-  1-6-  V 5 -  1-4  •  1-3-  1-2-  1-1  •  1 0 -  0-9-  2 0  4 0  6 0  8 0  100  ©cm.  120  140  160  38  CHAPTER IV Theoretical  It  can  be  more s e n s i t i v e than  is  v+  seen  7 0 ° - 8 0 ° i s due  scattering  to  optical gives the  1 3  C  a  change  differential  calculation  line  of  ratio  of  scattering the e x t r a  the l 2  with  ( see  indicated.  2 8  *  I t can  been  values  Impulse  Approximation  calculation  cross  was  by  made  the  an  of  R  l 2  C  is  optical  in  the  Table V  t h e minimum  slightly  and  of  raised  s e c t i o n s was  amplitude  for  1 3  Model to  C.  2  potential  f o r 30  MeV  parameters  far  this  amplitudes.  nucleons f o r  0  - 1  how  1 2  The  together the C  T h i s gives f o r the  2  0  see  e v a l u a t e d by a d d i n g  R=[ 13b +b, +(13c + C i ) k c o s (9) 3 / t 12b  Optical  s-p  Multiplying  pion-nucleon  amplitudes of the twelve  where k=.48 f m  C  elastic  „- on  of  seen t h a t  shifted  the  The  MeV  2 ).  be  in  3  results  section  1 3  peak c e n t r e d a t  * C.  o f 29  in  2  i s predicted  n~n  large  and  the  by t h e  neutron  structure  C  the n~ i s  » C.  simple  structure  F i g . 14 t h a t  valence  cross s e c t i o n  together  Point Nucleus A  in  between  c r o s s s e c t i o n has  above t h a t  1«  of the  potential calculation the  from  MeV. ( F i g . 1 ) . The  minimum  shown i n F i g . 15 potential  immediately  to the e f f e c t s a t 50  interference  Interpretation  and  adding  ratio  +12c k c o s (9) 3 2 2  0  0  pions b  0  ,b  a  ,c  0  and  C j were  used  rather  39  F i g u r e 15. Comparison of C and D i f f e r e n t i a l Cross Sections. 1 3  i  •  20  i  40  i  1  60  i  1  i  80  ®c.m.  1  1  1  1  100  * C 2  1  1  i  1  120  1  1  i  1—  140  40  than  free  nN  scattering increase  amplitudes  in in  McManus and  the  nucleus.  b .  The  0  Carr >,  t e r m s were added t o b  The  sizes  0  bo=  of  give The  1  and  c -  .020  the  fm.;  difference  were  taken  (Table VI,  The  0  a better approximation  main  parameters  (SMC), S e t  5  Im  to  imaginary  Im  imaginary  Set  c = terms  were  v a l u e f o r B a t 8 0 ° , but  sizes  b  c  imaginary SMC  Set  and  0  Im  were  0  taken  from  p a r t s t o t h e s and  p wave a n n i h i l a t i o n  result  as t h e d a s h e d l i n e  is  shown  p o s i t i o n and  large  At low  angles  is  agreement i s n o t  good, a l t h o u g h  ( dotted l i n e reduced  only  terms.  were  to  the  give  relative  ratio  of  the  strengths  of  in Fig. 1 6 .  The  expect  relative  those of a f i n i t e nucleus  may  size  amounts  B=1  i s predicted. is  the a b s o r p t i o n the  that  the  by  the  s-p  1 3  a point  of s and  proportionaly  t h e same amount i n b o t h  k to  , produces  the  solid  dip i s C  nucleus,  angle.  one  would  p waves t o d i f f e r  the  from  i n the  scattering,  p-wave a m p l i t u d e  n u c e i , by  curve;  is  interference  r e d u c i n g t h e p-wave Reducing  also  imaginary  nucleus. Also multiple scattering  have the e f f e c t o f  fm-i  the  i s also raised  uses  the Lorenz-Lorentz e f f e c t .  .32  a n g l e s , however,  deep as i t i s moved t o a s m a l l e r  model e s s e n t i a l l y  the  at  t h e c r o s s s e c t i o n minimum of  terms a r e such  less  the  terms s e t t o zero  increasing  i n angle but  imaginary  becomes  Since t h i s  ) . By  trend  t h e d i p below  with imaginary  s i n c e now  shifted  The  minimum  by  roughly c o r r e c t .  same c a l c u l a t i o n  greatly not  Imaginary  the  maximum i s p r e d i c t e d a t t h e c o r r e c t  shown  B).  1.  The  The  Strieker,  chosen  the c o r r e c t  Im  an  from  terms  approximately of  in  fm?  .047  0  is  to  simply  This curve  by  changing  represents  41  Figure  16-  Impulse  Approximation  Calculations  of  R.  42  the  best  f i t to the  essential features curve  although  data  when  only  quantitatively  f a c t o r i s shown i n F i g . good  and  near the c r o s s shifted in  section  with  k  reduced  by  the smaller due  to  the  minimum  amount and i n t h e o p p o s i t e The  i n the cross  the  same  size of the r a t i o  imaginary  section  terms  being  direction  then  which i s c a n c e l l e d  than  produce  partly  a  by t h e  i n number o f n u c l e o n s .  Such r e a s o n a b l e q u a l i t a t i v e a g r e e m e n t u s i n g model  A l l the  1. A g a i n q u a l i t a t i v e l y t h e agreement i s  minimum i s  t h e 30 MeV fl- c a s e *  increase  varied.  t h e a g r e e m e n t i s n o t good. The  i t i s seen t h a t  by a s m a l l e r  decrease  is  o f t h e d i s t r i b u t i o n o f E c a n be seen i n t h i s  same c a l c u l a t i o n f o r 50 MeV it*  quite  k  suggests  incorporating  that  calculations  the c o r r e c t  using  k i n e m a t i c s and  so  simple  an o p t i c a l finite  a  potential  size  effects  s h o u l d be r e l i a b l e !  2«  Optical The  and  Potential cptical  Carr > 5  Calculations  p o t e n t i a l used  was t h a t  of S t r i e k e r ,  McManus  and was o f t h e form  2w0 ,t=-4n{b(r)+pj, B f* 01  (r)+ <c (  0  +  (C  0  0  (P -1)/2p £  +4n[V.L(r)c(r)V  e  Pj  -1)/2 V  2  f(r))  § 7 2j)2 (r)}  + C /p,V -f (r)\7 2  0  P l  3 +2wV (r) c  where b(r)=  P l  c(r)=1/  (b f(r)-£ b 8/>(r) ) 0;  P l  TT  1  (c f (r)-t^ 0  C l  if(r))  ty(r)=ft<r)-fl,(r) ETT=- 1 f o r rr- and +1 f o r -a* L (r) = (1+4nA(A-1) c ( r ) / 3 A ) —  1  i s the Ericson-Ericson  factor  43  V = Coulomb  potential  c  w=w/(1+E/A) i s t h e p i o n r e d u c e d e n e r g y , where total  energy  and  M=nuclear  The  factors  in  the pion-nucleus centre  w  i s the  pion  o f mass f r a m e , E=w/M  mass/A.  = (1 + E)/(1+E/A)  P l  and p,, =(1+E/2)/(1 + E/A) a r e r e d u c e d mass The  terms,  factors. Vp  involving  t h e nH  transformation  from  system *.  potential  2 9  The  This  and V  z  2  ^  are  2  account  for  o f mass s y s t e m t o t h e rtk  centre  i s described  fully  wave e g u a t i o n u s e d t o c a l c u l a t e  cross sections  to  i n SMC.  the e l a s t i c  scattering  was o f t h e form ( V k + 2 w U t ) J 0 ' ( r ) =0 2+  2  op  where k i s t h e wave v e c t o r o f t h e p i o n - n u c l e u s c e n t r e  of  system.  was used  to  A program  evaluate  differential The VI) B  0  the  d e v e l o p e d by A. Thomas and M. K r e l l partial  cross  t a k e n from  and C , a d j u s t e d 0  potential  to f i t  The  parameters used  the  1 2  C  differential  real  Gaussian j>(*)=J> l 0  where f = n u c l e o n 0  hence  the  ( s e t A, T a b l e parameters,  cross  and i m a g i n a r y p a r t s o f B  kept e q u a l i n accordance with p i o n i c A modified  and  i n F i g . 17 a r e t h e combined  2 8  runs.  shifts  SMC s e t 1, w i t h t h e a b s o r p t i o n  ( F i g . 17 ) . The d a t a > several  phase  s e c t i o n s f r o m t h e wave e q u a t i o n .  s e t of o p t i c a l  was  wave  mass  0  section results of  and C  0  were  atom d a t a * * .  form, 1t"(r/a)  2  3 e x p l - (r/a) 3  density at the centre,  2  was used  f o r the proton  Table VI.  |  SET A  Optical  |  Potental  SET B  |  Parameters.  SET C  1  l~ IUnits|  j -.040+i.002 | -.040+i.002  | -.040+.002  1f  I  ho  |  bi  | -. 11 - i . 0 0 1  Bo  I  Co  I  • 75 +i.007  |  .75 +i.007 |  C1  i  .62 +i.004  |  .62  Co  |  -.79 + i . ? 9  A  |  1 .0  --17 +i.17  | -.11  -i.001  | -.11  -i.001  1 f m,  I  | -.13  +i.13  | -.17  +i.17  1 fm.*  1  .75  +i.007  |fm.3  |  .62  +i.004  |fm.3  |  | -.75 1  +i.004 | +i.75 LO  | -.79 + i . 7 9 |  0.6  1 f m.  6  1  Figure  17. C a l c u l a t i o n s of C Cross Section w i t h P a r a m e t e r S e t s A, B and C. 1 2  46  and  n e u t r o n d i s t r i b u t i o n s of  the  shell  1 2  C . The p a r a m e t e r  , <x , was s e t t o  model v a l u e , oc = ( Z , N - 2 ) / 3  for  neutron  , proton d i s t r i b u t i o n s .  The  rms. r a d i u s  i s  then  g i v e n by <r 2 >i/2 = I f  The  proton  the  2.5-2/Z,N  n  rms. r a d i i ,  r  i]»/2 a  , were o b t a i n e d by s u b t r a c t i n g o u t  p  p r o t o n s i z e from t h e charge r a d i i ,  <r,2>V2  g i v e n by e l e c t r o n  scattering, using <r > = <rf>2  The equal  n e u t r o n and p r o t o n d i s t r i b u t i o n s f o r C  were  l2  a s i s r e a s o n a b l e from  Hartree-Foch c a l c u l a t i o n s 12  . 82  r =2.46±.025 c  fm.,  1 6  their  *.  The  gives  closed  s h e l l n a t u r e and from  charge  proton  assumed  radius  and  3 0  *  of  neutron  C,  l2  radii  of  i 2 r . = 2 . 3 2 6 ± . 0 2 6 fm. There rms.  i s  charge  radii  some  radius  discrepency of  was measured t o  S t a n f o r d . The momentum was  .7 t o 1.7 f m  the of  - 1  1 3  C.  be  in  the  The r a t i o  .96±.01  by  t r a n s f e r range  of  literature 1 3  C/  c  1 2  3 2  of  experiment They  .968±.015.  In  h i g h e r energy  Heisenberg e t a l .  attribute  with the.low  3  3  *  purity  later  to  electron  found a r a t i o  measured  transfer  results  range  give  a  scattering  o f .9907±.0004  the discrepency with the results  b e i n g due t o t h e d i f f i c u l t y  This  a  two  a l . at  i n the experiment  r a t i o t c be .975± . 0 2 , c o v e r i n g t h e momentum  ratio  et  3  . ¥ang e t a l . a t S a s k a t o o n *  .2 t o 1.1 f m - * . They c o m b i n e d t h e  the  rms. c h a r g e  Crannell ** covered  on  o f C r a n e l l as  i n determining the t a r g e t t h i c k n e s s  liquified  methane t a r g e t u s e d  by  Cranell.  i s suggested by t h e d i f f e r e n c e i n c r o s s s e c t i o n r a t i o s of  47  13C/12C measured by transfer.  t h e two  Yang u s e d  a gas  experiments  a n a l y s i s , but checks  proton  neutron  different  with  added t o  a  the  Sensitivity  the  p-wave  to  7  the  case  so t h a t  the  1 3  C  dependence  The  on  the  The  used t o d e s c r i b e t h e  the d e n s i t y  harmonic  size  was  oscillator of  of the neutron  and  a  at  f i t , this  a  point  wave f u n c t i o n  was  left  out  a spherical  was  on t h e X  2  of  of  X  2  a was  core.  was  the  used  neutron  varied.  p-wave n e u t r o n  The  was  number.  evaluations minima. at  The this  be g r e a t ,  wavefunction  over  of a p a r t i c l e  left in  by where  n  radial  averaged  w e l l s t r e n g t h , V , was  jar (r)=H [ (r) Yf (6) t h e number o f i n t e r n a l  any  Model  o f the  nI|n  2  with  t o converge  minima would n o t  harmonic w e l l i s g i v e n  momentum guantum  C  investigated.  of t h e X  tended  harmonic o s c i l l a t o r  t o be  neutron  distribution  determination  however,  + Valence Neutron  as a p a r a m e t e r  6  core  of  wavefunction  100° must be i n d i s a g r e e m e n t  calculations,  a n g l e s . The  distribution  1 3  made.  so t h e i n f l u e n c e  Core  were  i t would n o t e f f e c t t h e p o s i t i o n s  different  all  momentum  h a s been u s e d i n  c  distribution  neutrons  r a d i u s of  reasonable  a)  r = 2 . 4 4 fm.,  a modified Gaussian  Since the point  point  1 3  a r e made o f  . Firstly  Gaussian  the second  rms.  ,  approaches  distribution  neutron  for  same  radius.  Two  In  the  methane t a r g e t , e n r i c h e d t o 84%.  h i g h momentum t r a n s f e r r e s u l t this  at  nodes  and  1  is  the  n  is  angular  48  B-ni ( r ) = / -Jm  3/22 -n+2 (2n+21+1)! n! ((21+1) !!) 2  (vr2)l/2  1  L1+»  (  2  V r  z) -vrV2 e  and n  L  L  i  ,  R  (21 + 1)11 >3 (2l+2k+1)!»  l+i/2(/3)=XI(-1) 2 ( n \ k=0  •v=mW/-n  ,  ft  \kj  where  m=  the  reduced  mass  and  U) = t h e  n=0,  1=1.  oscillator  frequency. For is  the  given  p-wave n e u t r o n o f  1 3  C,  6 neutrons  of  1 2  C  t h e rms.  [ (2x3/2 assuming  a  d e p t h , v»  the  effective than  Using rms.  angles.  of R  were with  is  f o r the e x t r a  the neutron and  for  (equal to  1 2  data  1  c  sharper  best  f i t well  not  parabolic.  be  of  good,  1 3  C  1 3  positions  will  strength, and  be  V.  best f i t s  (Fig.  18)  perhaps  same  ,  core case  of core)  neutron  The core  i n the  and  the  f o r small  although  the  ( F i g . 19).  Also  curve f o r a neutron core  seen t h a t  C  (reduced  The  except  are the  f o r the reduced 2  of  that  distributions  r =2.306 fm  identical,  is  the  expected  neutron  radius).  radius  of  as t h e c o r e r a d i u s i n c r e a s e s t h e  strength increases. The  3  2.326 fm.  i s unpaired.  core  for  essentially the  I t can  be  proton  the  repeated  shown i n F i g . 19 i s t h e X 2.360 fm.  radius of  i t might  2  f o r the best f i t s  Z  rms.  V i  were made f o r v a r y i n g w e l l  were  The ~X  minimum  depth  an  fm~ .  neutron  o f 2.326 fm.  agreement  i s .40  2.306 fm.  calculations  cases  c  since  equal  calculations  two  1 2  well  this  radius=  radius  for  radius i s then  +4x5/2)/6VJ  harmonic w e l l . F o r  well  smaller  radius  5/2"V  l  the  rms.  by <nl|r2|nl>=(2n+l+3/2)V~ =  For  The  The TC  2  of  the  curves minima  are are  generally therefore  Figure  18.  Optical Potential Calculations V a l e n c e Neutron + Core Model.  o f E.  F i g u r e 19. X Curves f o r V a l e n c e Neutron + Core Model. z  51  defined  as  midway between t h e two  than the  lowest  neutron  distributions  2.370±.020  v a l u e . E v a l u a t i n g t h e rms.  fox  fm.  Respectively.  The  This suggests  that  neutron  of  were  d i s t r i b u t i o n s of the  distributions  i s t o the  were  same  formed  by  almost  radius of  radius=2.36  combining  well  s t r e n g t h with m o d i f i e d  distributions  of d i f f e r i n g  rms.  radii.  measured  3  Gaussian  by  Heisenberg  distribution,  proton  b u t oc d i f f e r s  by  about  6%  v a r i a t i o n s i n the d i s t r i b u t i o n  20)  are  ±10%. the  here  The  resulting  total  suggests which form from may  X  varies  2  that  is  less  significantly  of the neutron  permit  by  f o r R are than  of  for  1.5  very  1  3  the r ^  shell (Fig.  similar  and  ( Table VII ) . T h i s  measurement  Combination  pion s c a t t e r i n g  measurement  from  of  l  3  r  n  d e p e n d e n t on u n c e r t a i n t i e s i n t h e  distribution.  energy  distribution  e q u i v a l e n t t o v a r y i n g c< by  i t i s possible to obtain a  not  higher  approximately distributions  Gaussian  i s r e p r e s e n t e d as a m o d i f i e d  model v a l u e . The used  fm.  p-wave n e u t r o n  of d i f f e r e n t  * C  the  different  wavefunctions  of  fm.  constant.  with  rms.  The  and  structure.  made  total  total  2.360  rms.  detailed  therefore  f o r the  and  r a d i u s remains  not t o t h e  R  2.326  greater  2.365±.019  R  the s e n s i t i v i t y and  radii,  3  (core)=2.306,  is 1  2  * r =2.354±.014,  b e s t f i t rms.  Calculations  The  gives r  distribution  neutron  p i o n t s where X  more  and  with  with  detailed  information  pionic  structure  atom  data  in  the  distribution. b)  Neutron  Distribution  Calculations distribution 1  3  r  p  and  l  3  r  n  of  rms. R  Radius  were  made, u s i n g a m o d i f i e d  f o r b o t h t h e n e u t r o n s and . A X  2  contour p l o t  ( F i g 21)  protons, was  Gaussian  varying  produced  from  both the  52  Figure  <N  *~  O  ~"  20. * C Neutron D i s t r i b u t i o n s with E q u a l rms. B a d i u s 3  CO  o  (£_wj  CD  o  suojjneu)  <t  ^  o  o (/  Table V I I . R e s u l t s of C a l c u l a t i o n s Osing D i f f e r e n t Neutron D i s t r i b u t i o n s with Equal rms. Radius.  |Line |Fig 20  B  r  I  n  (core) |  (fm.)  j (fm7 ) 2  2.280  .32  20.77  2.306  .353|  19.81  2.340  .4 1  19.40  Gaussian distribution I  2.380  | I  .50  19.30 19.93  F i g u r e 21. "X C o n t o u r P l o t f r o m O p t i c a l P o t e n t i a l C a l c u l a t i o n s u s i n g P a r a m e t e r S e t A. 2  55  results  (Figs 22  changes  and  from at  * r = p  XT-  is  1 3  changes  of  1  3  r  from  is  n  at  * r  at the X  3  n  * rp  =  3  2.240 fm.  3  correspondingly  value of  value of  2.370±.022 fm.  2.337±.025 fm.  measurement  23). The  The  .064  therefore  2.306 fm.  to  3  3  n  .103  fm.  s l i g h t l y dependent On  the the  Z  sensitive  to  the  p  The  r ^ assumed. The HL contour shows, however, t h a t  more  to  i r -i r  guantity  fm.  minimum  2  neutron d i s t r i b u t i o n than  the  proton d i s t r i b u t i o n . Some t e s t s were made of the of  1  was  r  3  on  the  f i x e d at  (  B  and A,  n  and  C ).  Set  0  o p t i c a l p o t e n t i a l parameters. The  2.306 fm'. Two  C,  C i s SMC  Fig.  fm.  and  17.  Parameter  1 with the  f i t s t o the  resulting TC  The  set  the  ( F i g . 24,  1  C 2  r  r  n  n  c  used was l 2  i r - i r p = .44  fm.  . This demonstrates the  2  n  i s r e l a t i v e to the  r  2  size  1 2  C  assumed.  investigated  = rp=2.300  fm.  12  t l  = 2.344 fm--.  i rp=2.305 fm,. t h e r e f o r e  3  shown  3  3  3  0  respectively.  with  l  B  parameter,  s e c t i o n are  used wiith * rj,=2. 280. fm. i s at  used  give 1 r =2.374±. 015  ( F i g . 24)  2  and  p  radius  1 (no v a r i a t i o n of  cross  f o r s e t s B and  curve D)  proton  Ericson-Ericson C  1 2  calculations  A was  Set  =2.326 fm.  1 2  r  B i s SMC  dependence on the value of  performing  minimum  The  2.387±.016 fm.  The by  Set  measurement  a d d i t i o n a l parameter s e t s were  Table VI ). Set  changed to 0.6.  in  dependence of the  The The  resulting two  X  2  cases,  both give values of measurement of  1 3  r-  n  F i g u r e 22. R e s u l t s of O p t i c a l P o t e n t i a l C a l c u l a t i o n s with *3r„=2.306  fm.  57  F i g u r e 23. E e s u l t s of O p t i c a l P o t e n t i a l C a l c u l a t i o n s with r =2.240 1  3  D  fm.  58  F i g u r e 24. "X c u r v e s o f O p t i c a l C a l c u l a t i o n s w i t h P a r a m e t e r S e t s k, 2  Potential  B and  C.  59  CHAPTER V Summary and D i s c u s s i o n  The e l a s t i c s c a t t e r i n g 29 MeV n~ on " c / i a c Since  the  d i f f e r e n t i a l cross section  were measured at angles from 30° to  measurement  was  a  r e l a t i v e one the e r r o r s  r a t i o are purely s t a t i s t i c a l . A about 80° shows that the  valence  i s quite  r a t i o s of  large  peak  in  the  the n- i s more s e n s i t i v e . t o  neutron than t r a t 50 MeV, +  150°. on the  ratio  the e f f e c t s o f  where the d i s t r i b u t i o n  flat.  The peak i n the d i s t r i b u t i o n i s due to structure  of the s-p i n t e r f e r e n c e  which i s due mainly t o the l a r g e Chapter  IV,  section  1,  a  change  in  minimum between * C and 2  s-wave  the  n~n  n-nucleus  the data are f i t q u i t e  well  the 1 3  interaction  was shown  with t h i s simple model i f one  parameter , the r e l a t i v e amount of s and p wave s c a t t e r i n g , allowed t o vary. T h i s suggests that potential  calculation  any  >errors  i n using a d e t a i l e d or  information the  such  interpretation  as  density  i f the  d i s t r i b u t i o n s may  calculation  behaviour f o r v a r i a t i o n s i n the n u c l e a r A  6  neutron  the  produce  structure  be i s o l a t e d i n  produces  the  correct  structure..  + harmonic o s c i l l a t o r valence neutron model  was used f o r the neutron d i s t r i b u t i o n of the  in  they w i l l  same e f f e c t i n both n u c l e i . In t h i s way, n u c l e a r  is  optical  uncertainties  p o t e n t i a l w i l l c a n c e l t o a l a r g e e x t e n t since the  C,  i n t e r a c t i o n . , In  r e p r e s e n t e d by a sum of AN s c a t t e r i n g amplitudes. I t was that  at  optical potential calculations  1 3  C.  I t was found  that  were i n s e n s i t i v e to the s i z e  of the core but depended l a r g e l y on the neutron rms. r a d i u s .  A  60  measurement o f tests  were  the  made  parameters. For i3C  rms.  fm.  1 3  C  neutron  for  the  radius  using  three of  s e t s of  neutron value this  to  has  neutron  fm.  by  d e v i a t i o n s o f the  standard  those  with The  should  the  where t h e  rms.  2.365±.025  neutron  reliability be  tested.  The  fm.  radius  to  this  remaining  method o f  the  experiment apart of the  neutron  where  previous  be  radius  i s then  f r o m Coulomb  effects. be  measurements  fm.  The out  to  be  covered  including  made on and  neutron  differences  measured  situation  radius should  found  different  range  measuring  from e l e c t r o n s c a t t e r i n g . In t h i s  and  as  Leaving  measurements  already used  r^  and  the  isotones  be  of  1 3  rp,  measured  neighbouring exist  1 2  model.  proton may  rp-  2.387  is  of  2.387±.016  others.  e r r o r s guoted are  + core of  values  than the  z  1 3  parameters  2.354  values  and in  some  potential  used,  uncertainty The  and  optical  2.374±.015  potential  span from  made  parameters  the  a much l a r g e r %  measurement t h e  the  are n e g l i g i b l e .  optical  distributions 2.387  on  2.370±.022,  Heisenberg  different  r a d i u s was  dependence  were f o u n d . E r r o r s due  measured by  rms.  between  where a c c u r a t e case  u  analogous to the Secondly an  Hartree  a  isotope Foch  radii  values would  +  present  measurement such  as  Ca  4 8  calculations  exist. By  combining  scattering only  with  and  piohic  i s sampled  , i t may  information  on  the  information atom be  data  from  , where t h e  possible to  neutron density  higher  obtain  distribution.  energy  surface more  pion  density detailed  61  BIBLIOGRAPHY  M. E r i c s o n and T.E.O. E r i c s o n , Ann. Phys. 36, 323 (1966) J . Eisenberg, J . Hufner and E . J . Moniz, Phys. L e t t . .47B, 381 (1973) M. K r e l l and T.E.O. E r i c s o n ,  Nucl. Phys. JBJL1, 521  (1969)  J . Hufner, P h y s i c s Reports 2JG, 1 (1975) i  K. S t r i e k e r , H. McManus and J . Carr, Phys. Rev. £19, 929 (1979),  W. Gibbs, B.F. Gibson and G.J. Stephenson, Phys. Rev. L e t t . 39, 1317 (1977) N.J. Digiacomo, Phys. L e t t . 66B,  421  (1977)  G.K.  Varma and L. Zamick, Nucl. Phys. A306, 343  G.K.  Varma, B u l l . Am.  G.K.  Varma and L. Zamick, Phys. Rev. C1.6, 308  Phys. Soc. 22, 1009  (1978)  (1977) (1977)  J.C. Lombardi e t a l . , Nucl. Phys. A188, 103 (1972) G.W. Greenless et a l . , Phys. Rev. 20, 1063 (1970); Phys. Rev. C3, 1560 (1971) G.M.  Lerner e t a l . , Phys. Rev. £ 1 2 , 778  (1975)  B. T a t i s c h e f f et a l . , Phys. Rev. C5, 234 I . B r i s s a u d et a l . , Nucl. Phys. A191, G.D.  (1972)  145,  Alkazov et a l . , Nucl. Phys. A280, 365  M.J. Jackobson e t a l . , Phys. Rev. L e t t . 38, J.W.  Negele, Phys. Rev. CJ, 1260  A. Abashian, R. Cool and J.W. Phys. Rev. J04, 855 (1956)  (1977) 1201  (1970)  Cronin,  B. W.  A l l e r d i c e et a l . , Nucl. Phys. A209, 1 (1973)  M.D.  Cooper, A.I.P. Conf. Proc. 33, 237  J . Jansen e t a l . , Phys. L e t t . 77B, 359  (1976) (1978)  (1977)  62  2 1 . ) M.M.  S t e r n h e i m and Kwang-Bock Y o o ,  Phys. Eev. L e t t . 41,1781,(1978) 2 2 . ) S.A. Dytman e t a l . , P h y s . E e v . .C.l 8, 2316 2 3 . ) G. Eowe, M. S a l o m o n and R.H. P h y s . E e v . J 8 C , 584 24. ) K. S h o r t y  (1978)  Landau,  (1978)  private  communication.  25. ) T. M a r k s , p r i v a t e  communication.  26. ) J . F . J a n n i , A i r F o r c e Weapons  Laboratory,  T e c h n i c a l E e p o r t N° AWFL-TE-65-150 27. ) E. E o s t , p r i v a t e  communication.  28. ) R.E. J o h n s o n e t a l . ,  t o be p u b l i s h e d .  29. ) M. T h i e s s , P h y s . L e t t . 6 3 B , 43  (1976)  3 0 . ) I . S i c k and J . S . M c C a r t h y , N u c l . 3 1 . ) H. C r a n e l l e t a l . , 3 2 . ) C.S. Yang e t a l . ,  Nucl. Nucl.  P h y s . A150,631  P h y s . A103, 677 Phys.  A162. 71  (1967)  (1971)  3 3 . ) J . H e i s e n b e r g , J . S . M c C a r t h y and I . S i c k , N u c l . P h y s . A 1 5 7 , 435 (1979)  (1970)  

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