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Energy measurement of pion beams Suzuki, Takenori 1976

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ENERGY MEASUREMENT OF PION BEAMS  by  Takenori  Suzuki  M . S c , Tokyo U n i v e r s i t y , 1970  A THESIS SUBMITTED IN PARTIAL FULLFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics  We accept t h i s t h e s i s as conforming required  standard  The University of B r i t i s h Columbia October, 1976 © Takenori Suzuki, 1976  t o the  In  presenting  an  advanced  the I  Library  further  for  degree shall  agree  scholarly  by  his  of  this  thesis  in  at  University  the  make  that  thesis  may  be  It  University  of  British  1W5  October 6, 1976  of  British  available for  gain  Columbia  for  extensive by  the  understood  Physics  of  fulfilment of  granted  is  financial  2075 Wesbrook Place Vancouver, Canada  Date  freely  permission  purposes  for  partial  permission.  Department  V6T  it  representatives.  written  The  this  shall  reference  Head  be  requirements  Columbia,  copying  that  not  the  of  copying  agree  and  of my  I  this  that  study. thesis  Department or  for  or  publication  allowed without  my  Abstract Pion  energies  between  30 MeV a n d 60  determined by three methods: of  flight.  2 #  The  by range  methods  pion  energy  and 5 %  were  magnetic  time  with  one  of  500  cyclotron.  focussed  M9  (or  s t o p p e d ny^j ) c h a n n e l .  channel give  define  the  and copper were used curve  momentum o f  one m e t h o d o f Differential  to  the In  effect  the  time  discriminator different llator. energies.  energies methods.  the  which  from the  curves  of  f l i g h t method,  give; a time  particles  serious  effect  constant  give dependable reasonably  the  and  thus  were  differential  examined. l e a d i n g edge; t y p e walk,  i n the  due t o  plastic  the  scinti-  i n determining the  pion  fraction discriminator  results  well  this  f o r aluminum  t a k i n g the was  of  the  energy.  In  stopping plates  TRIUMF  a beam u s i n g  range.  However, the  agreed  The p i o n s  d i f f e r e n t i a l curves  of  a  three  beam p a r t i c l e s pion  time  and the  and the  energy-loss; of  was f o u n d t o  the  and i n t e g r a l range  was; f o u n d t o  This has  into  and  accuracy  bending magnets  determining the  taken  determine  The  an  flight,  MeV p r o t o n s  been  range  to  another.  were u s e d were produced by The p i o n s w e r e  field,  was m e a s u r e d  by the  consistant  MeV h a v e  with  and the those  of  measured the  other  -iiiEnergy Measurement o f Pion Beams table of contents: I  II  III  page  Introduction A  General  1  B  The Theory o f Energy-Loss  7  (1)  The Stopping Power o f Charged P a r t i c l e s  7  (2)  Range o f a Charged P a r t i c l e  9  E x p e r i m e n t a l Equipment A  Pion Production  12  B  M9 Pion Channel  14  C  Counters and the Degrader  19  D  Electronics  23  The B a s i c Experiment f o r t h e Measurement o f the  Range-Energy R e l a t i o n and Time o f F l i g h t  A  The Range Curve  B  The Height o f Peak i n t h e D i f f e r e n t i a l Range Curve  IV  26  30  C  The P r i n c i p l e  o f t h e T i m e - o f - F l i g h t Method  D  The Time Walk o f t h e D i s c r i m i n a t o r  35 40  R e s u l t s and D i s c u s s i o n A  Time o f F l i g h t  45  B  Range Measurement  49  C  The Energy S t r a g g l i n g and Inherent Energy Spread o f the Beam  55  D  Conclusion  References  -v-  l i s t of tables: 2.1  page  The s p e c i f i c a t i o n o f p l a s t i c  scintillators  and p h o t o - m u l t i p l i e r used f o r c o u n t e r s 2.2  Components and t h i c k n e s s e s o f absorbers used f o r the degrader  3.1  22  The c u r r e n t s o f magnets f o r the measurement on F-3  3.2  20  33  The e f f e c t o f the s t o p p i n g p l a t e t o the p i o n energy d e t e r m i n a t i o n and the comparison o f energy between the range and time o f f l i g h t method  3.3  The e n e r g i e s o f  34 pions and muons a r e measured  by the time o f f l i g h t method.  In o r d e r t o  check the s y s t e m a t i c e r r o r , two types o f d i s c r i m i n a t o r s a r e employed, LRS edge) and C F .  (leading  (constant f r a c t i o n ) , and two  k i n d s o f t a r g e t s , copper and b e r y l l i u m . 4.1  Magnet c u r r e n t s f o r the time o f f l i g h t and range measurements  4.2  4.3  43  46  The energy i n MeV o b t a i n e d by the time o f f l i g h t and range measurements  46  The d e t a i l o f range c a l c u l a t i o n  53  vi  l i s t of figures: 1.1  page  Energy-Loss curves f o r d i f f e r e n t charged p a r t i c l e s i n aluminum and copper  10  2.1  M9 pion beam channel  15  2.2  Pion k i n e t i c energy vs. angle i n a laboratory system  17  2.3  CH  18  2.4  Configuration of counters and degrader  18  2.5  Scheme f o r the measurements of the range curve  24  3.1  The d i f f e r e n t i a l and i n t e g r a l range curve f o r  2  target  30 MeV pion stopped by A l 3.2  27  The e f f e c t of the stopping plate between the t h i r d and fourth counter  3.3  31  C i r c u i t diagram of the time of f l i g h t measurement  3.4  36  Timing diagram of TAC and the display of p a r t i c l e s on the multi-channel analyzer  3.5  Time c a l i b r a t i o n spectrum  (periodic input, 10  nsec) 3.6  36  38  Time-walk due to three d i f f e r e n t input pulses of electrons, muons and pions with the same momentum.  Since the discriminator l e v e l of a  leading edge type i s fixed, the outputs s h i f t i n time spectrum.  41  -vii-  3.7  Constant f r a c t i o n o f h e i g h t p i c k - o f f  4.1  R.F.-referenced  4.2  time o f f l i g h t spectrum f o r  40,  50 and  60 MeV  The  range curves o f p i o n i n aluminum  e n e r g i e s , 30, 40, 4.3  41  The  of pion energies  50 and  e n e r g i e s , 30 and  47 (pion  60 MeV)  range curves o f p i o n i n copper  30,  50 (pion  50 MeV)  51  i  -viii-  acknowledgment I am roost g r a t e f u l t o P r o f e s s o r D.F. Measday f o r h i s v e r y p a t i e n t e x p l a n a t i o n s t o a n o v i c e i n the f i e l d , and f o r h i s a d v i c e on t h i s t h e s i s .  I would l i k e t o thank  members o f TINA group, D r s . M. H a s i n o f f , M. Salomon, J.M. P o u t i s s o u and o t h e r graduate students discussions.  f o r t h e i r h e l p and  The o p e r a t i o n o f TRIUMF by the c y c l o t r o n  crew i s g r e a t l y a p p r e c i a t e d .  I would a l s o l i k e t o thank my  w i f e , Yoshiko, f o r her e x c e l l e n t t y p i n g .  -1-  I  Introduction A  General  The e n e r g y - l o s s o f charged p a r t i c l e s i n matter has been s t u d i e d , d i s c u s s e d and reviewed i n many p u b l i c a t i o n s . T a b l e s and f i g u r e s o f range-energy r e l a t i o n s which are a v a i l a b l e i n these summaries are u s e f u l t o experimental p h y s i c i s t s i n d e s i g n i n g and e v a l u a t i n g t h e i r  experiment.  Though the a c c u r a c y i s somewhat worse than a magnetic spectrometer, the energy o f charged p a r t i c l e s can be def i n e d q u i t e a c c u r a t e l y by measuring t h e i r range i n matter. G e n e r a l l y , charged p a r t i c l e s l o s e t h e i r energy i n p a s s i n g through matter by e x c i t a t i o n and i o n i z a t i o n o f the atoms and m o l e c u l e s .  These p r o c e s s e s o f e n e r g y - l o s s are  s i m i l a r f o r a l l charged p a r t i c l e s , but t h e r e are some d i f f e r e n c e s between e l e c t r o n s and h e a v i e r charged p a r t i c l e s (muons, p i o n s , p r o t o n s , i o n s , e t c ) .  S i n c e an e l e c t r o n i s  very l i g h t , i t can l o s e energy by bremsstrahlung  radiation.  Although p i o n s , muons and e l e c t r o n s are c o n t a i n e d i n beam i n our experiment, we s h a l l p r i n c i p a l l y d i s c u s s the energyl o s s o f heavy charged p a r t i c l e s i n t h i s work. I t w i l l be shown l a t e r t h a t the s t o p p i n g power o f heavy charged p a r t i c l e s depends upon the charge and  kinetic  energy o f the i n c i d e n t p a r t i c l e s and t o a l e s s e r e x t e n t on the m a t e r i a l o f the s t o p p i n g medium.  I f we know the s t o p p i n g  power o f one k i n d o f charged p a r t i c l e , then we can apply the s c a l i n g r e l a t i o n o f the s t o p p i n g power t o any o t h e r k i n d o f heavy charged p a r t i c l e s .  In most a r t i c l e s , the  -2stopping Using  power and  the  scaling  powers and  range of protons r e l a t i o n , we  ranges o f other  can  are  discussed^"**  calculate  heavy p a r t i c l e s  the  stopping  from those  of  proton. The for  experimental  pions  results  p r o d u c e d by  discussed  of the  range-energy  t h e meson f a c i l i t y , TRIUMF, w i l l  i n t h i s work.  Although  the  -  s t u d i e s ^ o f p i o n r a n g e have b e e n r e p o r t e d . experiment concerns the  passing this  range o f p i o n  problem of the  energy-loss  t h r o u g h m a t t e r was  century.  Bohr  mechanics, then  considered  Bethe  u s i n g quantum m e c h a n i c s .  of  stopping  section  power and  experimental  The  present  by  copper.  particle  i n the e a r l y p a r t  t h e p r o b l e m by  successfully  A s h o r t summary o f  range i n matter w i l l  of  classical  be  the  given  theory in  B.  R e v i e w s o f r a n g e e n e r g y r e l a t i o n s have b e e n every  has  i n aluminum and  t r e a t e d the problem  by  few  of charged  discussed  be  range o f protons  been m e a s u r e d f o r v a r i o u s e n e r g y r a n g e s ^ * *  The  relation  few  years.  The  L i v i n g s t o n e and  first  Bethel  0  extensive I n 1937,  summaries were  they  discussed  t h e o r y o f s t r a g g l i n g and  energy-loss  after  length of matter.  traversing a given  article  was  published  by  of charged  T a y l o r } ^ who  published  A  given  the  particles  short  summarized  summary  the  12 situation together  up  a l l o f the  high energies survey  on  t o 1951.  Then A l l i s o n  stopping  i n 1953.  power d a t a  I n 1954,  the energy-loss  and  Warshaw covering  gathered low  Uehling*^ published  o f heavy charged p a r t i c l e s .  and a He  -3reviewed  t h e a r g u m e n t s on w h i c h t h e e n e r g y - l o s s  are based  and  also experimental  the experimental  results.  data c o v e r i n g the energy  The  equations  survey  less  than  of 10  14 MeV  was  done by W h a l i n g  i n 1958.  some a d d i t i o n a l measurements on MeV  have been done.  including  Fano  Serre^  10 MeV  calculated  and  than  1 % i n the energy  this  t h e s i s we  and  700  literatures  t o 1963  and  the  reviewed  s t o p p i n g power  k a o n s , p i o n s and  s t o p p i n g powers a r e  agreement w i t h e x p e r i m e n t a l  v a l u e s o f ranges  to  f o r the wide energy  f o r protons,  ranges  mostly  the  t h e r a n g e and  (12 k i n d s )  t o 30 GeV  calculated  The  surveyed  summary,  in detail.  of various materials  The  a p r o t o n beam up  t h e a d d i t i o n a l r e s u l t s up  s t o p p i n g power t h e o r y  from  1 5  After Uehling's  muons.  i n good  v a l u e s t o an a c c u r a c y b e t t e r  range from refer  regions  10 MeV  t o 1 GeV.  In  to her r e p o r t f o r numerical  s t o p p i n g powers.  most r e c e n t summary a p p e a r e d  i n the  American  17 I n s t i t u t e Handbook.  In t h i s  article  r e c e n t r e f e r e n c e s , t h e commonly u s e d d a t a on The  the passage o f f a s t  t a b l e s o f s t o p p i n g powers and  calculations and  charged  using a correction  the continuous  a r e g i v e n t h e most formulae particles  ranges  in  matter. on  for inner shell electrons  s l o w i n g down a p p r o x i m a t i o n s .  problem o f energy-loss  principal  are based  a r e g i v e n f o r p r o t o n e n e r g i e s b e t w e e n 1 and The  and  i n matter  precisely  a s d i s c u s s e d i n summaries c i t e d  recently,  some m i s c e l l a n e o u s  is  1000  Values MeV.  understood  above.  But,  e f f e c t s have b e e n p o i n t e d  - 4 -  out  i n t h e r a n g e s and  Bethe's  formula.  According to the  18 measurements o f Heckman and  Lindstrom,  r a t e s o f p o s i t i v e p i o n s exceed  the energy-loss  that of negative pions  amounts o f 0 t o 60 MeV/cm i n t h e v e l o c i t y <£<  0.178  difference  (200 k e v < E < 2 . 3 MeV).  This  interval i s due  formula, which  0.051  t o the  i n t h e Coulomb i n t e r a c t i o n w i t h a t o m i c  Bethe's  by  electrons.  i s d e r i v e d by a quantum-  m e c h a n i c a l t h e o r y b a s e d on t h e f i r s t  Born  approximation,  g i v e s good agreement w i t h t h e e n e r g y - l o s s e x p e r i m e n t fast particles  but does not e x p l a i n  range o f o p p o s i t e charged p a r t i c l e s .  the d i f f e r e n c e  of  i n the  Such an e f f e c t i s  predicted  by t h e s e c o n d o r d e r B o r n a p p r o x i m a t i o n w h i c h 3 g i v e s a term p r o p o r t i o n a l t o Z o f the i n c i d e n t p a r t i c l e s . T h i s h i g h e r o r d e r c o r r e c t i o n o f t h e s t o p p i n g power, 3 19 p r o p o r t i o n a l t o Z, i s d i s c u s s e d by J a c k s o n and M c C a r t h y . 3 T h e y h a v e shown t h a t t h e Z t e r m a t low v e l o c i t y l e a d s t o the  range  difference  f o r p a r t i c l e s o f t h e same mass, t h e 20  same i n i t i a l also  energy,  indicated  velocity The  but o f o p p o s i t e charge.  t h e d e p a r t u r e from Bethe's  Inokuti  formula at  low  i n h i s summary. time o f f l i g h t  method f o r m e a s u r i n g  method i s a r e l a t i v e l y  simple  absolute velocity of p a r t i c l e s  u s e f u l o v e r a wide range  of p a r t i c l e  and  t y p e s and e n e r g i e s . 21  A r e v i e w o f t i m e i n t e r v a l measurements was who  d i s c u s s e d the time  second  t o micro-second.  intervals The  been r e p o r t e d f o r measuring  g i v e n by  i n the range o f  time o f f l i g h t c y c l o t r o n beam  Porat  pico-  method h a s 22-25 energy.  -5u s i n g the c y c l o t r o n o s c i l l a t o r deuteron  energy  as a time  w i t h momentum 1025  the time o f f l i g h t  between two  standard.  The  MeV/c was  measured  by  scintillation  counters  and  26 a l s o by  t h e r a n g e measurement.  For n e u t r a l p a r t i c l e s  (neutron), the usual techniques  a p p l i e d t o charged  such  and  as a m a g n e t i c s p e c t r o m e t e r  c a n n o t be has of  applied.  However, t h e t i m e  been a p p l i e d s u c c e s s f u l l y 27-29 neutrons.  from  o f p i o n by  the time  t h e meson f a c i l i t i e s  Those machines can  0  "  3  of f l i g h t  2  method energies  r e s u l t s o f measuring  of f l i g h t 3  relation  f o r measuring the  Recently the experimental energy  range-energy  particles  the  have b e e n r e p o r t e d  (SIN, LAMPF, TRIUMF).  accelerate high currents of proton  p r o d u c e t h e h i g h i n t e n s i t y o f p i o n beam.  to  From t h e meson  p r o d u c t i o n t a r g e t , t h e beam, w h i c h i n c l u d e s p i o n s , muons, e l e c t r o n s and reaction, through  the  same momentum b u t  method, we  the v e l o c i t i e s The of  Each p a r t i c l e  t o t h e d i f f e r e n t mass.  flight  pions  by  the  i s e x t r a c t e d to the experimental  the p i o n channel.  beam has due  o t h e r p a r t i c l e s produced  can  30 t o 60 MeV. a r e m e a s u r e d by the energy  The  obtained  velocity  the p a r t i c l e s  and  of  determine  area.  f o r the energy  e n e r g i e s o f p i o n s under the  from  time  show t h e r a n g e measurements  copper  the time o f  passing  contained i n the  the d i f f e r e n t  i n the experimental  i n aluminum and  area  T h e r e f o r e , u s i n g the  identify  present report w i l l  nuclear  flight  method.  the range w i t h  We  region  from  same c o n d i t i o n shall  compare  t h a t o f the time  of  -6-  flight. I n t h e measurement the  time d i f f e r e n c e  which t r a v e l  energy losses  discriminator.  and  we measure  between e l e c t r o n s  production  target  a n d mesons  to a  detector.  muons a n d p i o n s w i t h t h e same momentum h a v e  have a d i f f e r e n t  pick  i n arrival  from t h e p i o n  As e l e c t r o n s , different  o f t h e time o f f l i g h t ,  i n the p l a s t i c  t r i g g e r i n g time i n the leading We w i l l  discuss  o f f of discriminators the constant  scintillator,  fraction  the difference  between t h e l e a d i n g i n section  they  edge  type  i n time edge  type  3-D.  S i n c e o u r e x p e r i m e n t was done o n t h e M9 c h a n n e l , where t h e b e n d i n g magnets d e f i n e we s h a l l b e a b l e strength  to utilize  o f t h e magnet  the p a r t i c l e  the r e l a t i o n  currents  momentum,  between t h e  and t h e p i o n  energy.  -7B The  The T h e o r y  o f Energy-Loss  energy-loss o f charged  quite w e l l understood. and  particles  i n matter i s  I t i s m a i n l y due t o i o n i z a t i o n  e x c i t a t i o n o f t h e atoms i n m a t t e r .  The n u c l e a r  i n t e r a c t i o n w h i c h h a s a much s h o r t e r r a n g e magnetic has  i n t e r a c t i o n may become i m p o r t a n t when t h e p a r t i c l e  enough e n e r g y  find  t o come c l o s e t o t h e n u c l e u s .  the c l a s s i c a l  i n many e l e m e n t a r y  (1)  The  than the e l e c t r o -  calculation  o f the energy-loss i n matter  33-35 books on N u c l e a r P h y s i c s .  The S t o p p i n g Power o f C h a r g e d classical  We c a n  Particles o  t h e o r y was d e v e l o p e d b y B o h r .  t h a t an i o n w i t h c h a r g e  I t assumes  Ze and v e l o c i t y v p a s s e s w i t h i n a  d i s t a n c e b f r o m a n e l e c t r o n w h i c h i s assumed t o be f r e e a n d at r e s t .  While  the ion i s affected  by t h e e l e c t r o m a g n e t i c  i n t e r a c t i o n w i t h t h e s t o p p i n g medium, t h e a t o m i c a r e assumed t o move v e r y s l o w l y . is  Since the c o l l i s i o n  assumed t o b e s h o r t , t h e y a c q u i r e a n i m p u l s e  changing  their position.  approximation,  According t o t h i s  without  t h e momentum a c q u i r e d b y t h e e l e c t r o n must b e particle  c a n be c a l c u l a t e d b y a p p l y i n g G a u s s ' t h e o r e m t o a  c y l i n d e r w i t h t h e r a d i u s b. k i n e t i c energy  lost  The momentum g i v e s t h e c l a s s i c a l  by t h e i o n a t an impact  Then, t h e r a t e o f e n e r g y - l o s s p e r u n i t p a t h is  time  impulse  perpendicular t o the trajectory o f the incident and  electrons  g i v e n by i n t e g r a t i n g t h e c l a s s i c a l  f r o m a minimum t o a maximum.  parameter  b.  l e n g t h , -dE/dx,  kinetic  energy  over b  T h u s , t h e s t o p p i n g power o f  -8t h e a b s o r b i n g medium 9  Bethe  was o b t a i n e d  i n the c l a s s i c a l  10 '  developed  t h e t h e o r y d i s c u s s e d a b o v e by  c o n s i d e r i n g quantum t h e o r y , r e l a t i v i s t i c effects.  form.  The B e t h e ' s e q u a t i o n  i s as  effects  and  shell  follows:  (1.1)  where t h e r e a r e n atoms w i t h a t o m i c volume o f t h e a b s o r b e r , the  i-th shell.  and  is  Z i n the  unit  i s the s h e l l correction  Since Bethe's equation  u s i n g t h e Born approximation can  number  of  i s o b t a i n e d by  f o r the c o l l i s i o n process, i t  be a p p l i e d when t h e v e l o c i t y  of the incident  particle  l a r g e compared t o t h e v e l o c i t y  o f the atomic  electrons.  Bethe's e q u a t i o n depends on t h e v e l o c i t y number  of incident  particles.  dE/dx(m^,Z)=dE/dx(m ,Z) 2  particles  of different  following relation  and  charge  In o r d e r f o r the e q u a t i o n  t o h o l d f o r two k i n d s o f  incident  mass b u t t h e same c h a r g e , t h e  i s needed;  (1.2)  where  and T  incident  particles.  2  are the k i n e t i c  power f o r p r o t o n s  From t h i s  energy  o f two k i n d s o f  result,  i f the stopping  i s known, t h e n  p a r t i c l e w i t h mass M  v  t h e s t o p p i n g power o f t h e  a n d one e l e c t r i c a l c h a r g e  i s obtained  -9by  s h i f t i n g t h e v a l u e s f o r p r o t o n by a f a c t o r  for pion).  Similarly,  i f particles  M /M (=0.150 x p with d i f f e r e n t Z are  c o n s i d e r e d , the e n e r g y - l o s s o f a p a r t i c l e w i t h charge  Z^ i s  2 shifted  by a f a c t o r  with charge proton,  Z .  2  from  the case  of a particle  F o r t h e c a s e o f an a l p h a p a r t i c l e  2  2 ) =4.  (Ztf/Z  (Z^/Z )  and a  I n t h e r e f e r e n c e 6, t h e e x p e r i m e n t a l  ir  s t o p p i n g powers f o r p r o t o n e n e r g i e s o f 0.05 t o 12 MeV a r e given.  Using  the rule  t o convert the proton  s t o p p i n g power  to  a p i o n s t o p p i n g power, t h e s t o p p i n g power f o r p i o n  of  0.05 t o 10 MeV  aluminum a n d If  absorber of  i s known f o r a c e r t a i n  u n d e r t h e same e n e r g y  the r e l a t i v e  t h e s t o p p i n g power  the r a t i o  s t o p p i n g power a n d i s u s e d t o  the thickness o f d i f f e r e n t  standard absorber. to calculate  scintillator  (2)  f o r any  (1.3)  i s called  used  at a  = Const.  calculate the  condition,  c a n be d e r i v e d f r o m  absorber  p a r t i c l e , then,  =  This  be  copper.  any a b s o r b e r  Q  i s shown i n F i g . 1.1 f o r t h e c a s e o f  t h e s t o p p i n g power o f a s t a n d a r d  given energy  energies  In t h i s r e p o r t , t h i s  the r e l a t i v e  Range o f a C h a r g e d  relation  thickness of p l a s t i c  t o aluminum and copper  Since the equation  materials relative to  absorbers.  Particle  f o r the energy-loss of  charged  will  Pions iip aluminum Protons inaluminibm CSI e °  £  2 10  i  O I LUI  x TO  -a r o 1^  1. 10"  3  10  2  1  Kinetic Fig.  1.1  1.0  10"  10  10'  energy , MeV  Energy-Loss curves f o r d i f f e r e n t  charged p a r t i c l e s  i n aluminum a n d c o p p e r .  -11-  particles  i s obtained  i n the  t o i n t e g r a t e from the at rest  to find  matter.  We  the  last  initial  total  section  (1), i t i s possible  i n c i d e n t energy t o the  energy  range o f t r a v e r s i n g p a r t i c l e s  in  have  (1.4)  where t h e t o the  continuous  slowing-down a p p r o x i m a t i o n  integration.  Since  the  stopping  power d e p e n d s  and  of p a r t i c l e s  known r a n g e f o r a c e r t a i n p a r t i c l e  by  a scaling rule.  of  stopping  the  theory  that the  power t o g e t has  the  last  o b t a i n the  t o use  the  equation  energy,  by  approximation  using the  This gives a l i t t l e  i n t e g r a t e d range.  the  integral  can  be  0 < E < Ej=l  MeV  In o r d e r  separated and  to avoid  i n t o two  E^<E<CE . q  since  enough e n e r g i e s  atomic e l e c t r o n s are a t r e s t , as  section.  range  r a n g e a t low  i n c i d e n t p a r t i c l e s have h i g h  the  i.e.,  the  been o b t a i n e d  assume t h a t t h e in  It i s difficult  can  on  the p a r t i c l e v e l o c i t y from the  c h a r g e , we  i s applied  discussed  uncertainty this  energy  Then,  to  to  uncertainty, regions,  for  the  low  energy 17  region, the For be  the  high  experimental  shown i n F i g . 1.1  energy r e g i o n , the  used s u c c e s s f u l l y .  mostly r e f e r  data  In the  i n t h i s work, t h e  stopping  reference  can  be  power e q u a t i o n 1 6 t o w h i c h we  experimental  results  of  36 Barkas  have been employed  f o r the  low  energy  region.  used. can will  -12II  E x p e r i m e n t a l Equipment A  Pion Production  The  e x t r a c t e d p r o t o n beam f r o m t h e TRIUMF c y c l o t r o n  be v a r i e d  i n energy c o n t i n u o u s l y  t h i s experiment,  t h e 500  MeV  t o 520  p r o t o n beam was  c y c l o t r o n and  t h e c u r r e n t was  5 nano-amps.  The  cyclotron  f r o m 150  ORTEC 437A) a s t h e s t o p s i g n a l  single  type of  In  Converter  f o r the time o f  because of  R.F.  (TAC,  flight  signal  peaks i n the time s p e c t r a  was  for a  particle.  t h i s e x p e r i m e n t , t h e b e r y l l i u m t a r g e t o f 10  t h i c k n e s s was  the  MHz  When t h e  measurement, e v e r y s e c o n d p u l s e o f t h e R.F, two  p r o v i d e d by  r a d i o f r e q u e n c y i s 23.04  f e d t o t h e Time t o A m p l i t u d e  erased t o produce  In  kept i n the range o f 1 t o  w h i c h c o r r e s p o n d s t o a 43.403 n s e c - p e r i o d . s i g n a l was  MeV.  can  cm  u s e d m a i n l y f o r t h e meson p r o d u c t i o n t a r g e t  of the high p r o d u c t i o n r a t e .  Since,  for a  target  low a t o m i c number, t h e p r o d u c t i o n r a t e o f e l e c t r o n s i s  s m a l l , t h e c o p p e r p r o d u c t i o n t a r g e t was the  peak p o s i t i o n o f e l e c t r o n s  i n a t i m e - d e l a y spectrum  with the case o f b e r y l l i u m t a r g e t . will  be g i v e n i n t h e s e c t i o n  e m p l o y e d t o compare  3-D.  The  result  A l s o a CH  2  and  discussion  target  was  used t o measure t h e e f f e c t o f t h e t h i c k n e s s o f a s t o p p i n g p l a t e between t h e t h i r d The  and  fourth counter  ( F i g . 2.4).  p i o n s , muons and e l e c t r o n s p r o d u c e d  at the  T2  o  t a r g e t p a s s t h r o u g h t h e M9  c h a n n e l w h i c h i s a t 135  d i r e c t i o n o f t h e p r o t o n beam We  can choose  to the  ( F i g . 2.1).  p o s i t i v e o r n e g a t i v e p i o n s by  changing  -13the p o l a r i t y o f t h e magnets. production  i s larger  than  the negative  positive  p i o n s were employed  negative  pions  r a n g e and  the  are captured reaction  Since the p o s i t i v e  i n our by  pion  a t t h e end  p r o d u c e s p a r t i c l e s and  these  p a r t i c l e s produce a background s i g n a l ,  curve  for a positive  for a negative  pion.  37 production,  experiment.  nuclei  pion  pion  Furthermore, of  their  photons. the  range  i s g e n e r a l l y c l e a n e r than  that  As  -14B The  M9 P i o n  Channel  l a y o u t o f M9 c h a n n e l i s shown i n F i g . 2.1.  f i x e d p a r t o f t h e M9 c h a n n e l c o n s i s t s o f f i v e  quadrupole  m a g n e t s , two b e n d i n g magnets a n d t h e c o l l i m a t i n g slit  i s s e t between t h e f i r s t  The  slit.  b e n d i n g magnet and t h e t h i r d  q u a d r u p o l e magnet s o t h a t we c a n a d j u s t t h e beam s i z e . has  It  30 cm x 30 cm beam s i z e when w i d e o p e n a n d i n o u r e x p e r i -  ment we u s e d of  The  vertical  2 cm w i d t h o f h o r i z o n t a l slit.  After the f i f t h  slit  and 10 cm w i d t h  q u a d r u p o l e magnet, t h e  meson beam e n t e r t h e e x p e r i m e n t a l a r e a t h r o u g h a vacuum window. for  I n t h e e x p e r i m e n t a l a r e a , t h e r e a r e two p l a c e s  setting counters.  fifth The  position  right  after the  q u a d r u p o l e magnet i s c a l l e d F - 2 , ( t h e s e c o n d  second p o s i t i o n  F-3,  The f i r s t  (the t h i r d  focus).  i s b e h i n d t h e C o r v a l l i s magnet a n d c a l l e d  focus).  When F-3 i s u s e d , t h e t r i p l e t  q u a d r u p o l e magnet i s i n s t a l l e d  i n t h e p l a c e o f F-2 t o f o c u s  the  experiment, both  beam down t o F - 3 .  In t h i s  positions  were used because o f t h e e x p e r i m e n t a l s c h e d u l e o f o t h e r groups. in  From t h e p o i n t o f v i e w o f t h e s e p a r a t i o n o f p a r t i c l e s  t h e time-delay spectrum,  from t h e t a r g e t , is  i s b e t t e r t h a n F-2.  distance  B u t t h e f l u x a t F-2  l a r g e r than t h a t a t F-3. The p i o n e n e r g y  and  F-3, which has l o n g  the p r o f i l e  magnets.  i s d e f i n e d b y t h e two b e n d i n g m a g n e t s  o f beam i s s h a p e d  by t h e e i g h t  quadrupole  The e f f e c t s o n beam e n e r g y o f v a r y i n g t h e c u r r e n t 38  for  t h e e a c h magnet h a v e b e e n c a l c u l a t e d .  Therefore, i n  o r d e r t o d e f i n e t h e momentum o f t h e p i o n s , we c a n s e t t h e  Shielding Experimental area  Radiation area  Bi -Bending  magnet Q -Quadrupole magnet W;-Window location  A Qc  Pion focus,F-2  I  I  Pa Corvallis  Proton beam Pion beam Pion focus,F-3 Fig.  2.1  M9 p i o n beam  channel  c u r r e n t o f e a c h magnet a c c o r d i n g t o t h e t a b l e in  4.1.  o r d e r t o g e t t h e optimum beam o f mesons, we  the  Then,  can a d j u s t  c u r r e n t s o f t h e s e c o n d b e n d i n g magnet and t h e e i g h t  q u a d r u p o l e magnets w h i l e h o l d i n g first  bending The  the  the current of the  magnet.  energy  d e f i n e d by t h e magnets was  pion produced  kinetic  fixed  energy  from t h e r e a c t i o n ,  checked  by  using  + 39 p + p — » TC +d. The p i o n  a s a f u n c t i o n o f a n g l e i s shown  i n F i g . 2.2  40 for  the proton energy  from  p i o n e n e r g y o f t h e M9 the  target  pions a r e produced  The  e n e r g y - l o s s o f 500 MeV x 1 x 0.5  it  has l o s t  a s shown  cm)  protons i n the CH  considering  2  that  target.  target  and a t t h e m i d d l e o f t h e t a r g e t  a h a l f o f 2.9 MeV.  assumed t h a t  assume  a t the middle o f t h e CHj  i s 2.9 MeV  so l a r g e a n d w i t h i n  initial  i n F i g . 2.3,  t h e mean e n e r g y o f t h e beam we  the  (1  The  c h a n n e l h a s t o be o b t a i n e d b y  thickness of the  To c a l c u l a t e  300 t o 600 MeV.  As t h e e n e r g y - l o s s i s n o t  t h e u n c e r t a i n t y o f beam e n e r g y ,  the pion  i s produced  i t is  by t h e r e a c t i o n o f 500  proton a t the center of the target. 41  Thus, t h e p i o n s  MeV  produced  0  have an energy  30.9 MeV  Then t h e p i o n energy in  the  target.  middle o f target target. 30 MeV  i n F i g . 2.2.  T h e r e f o r e the p i o n s produced  h a v e an e n e r g y  T h u s when we  adjust  p i o n beam p r o d u c e d  MeV.  a s shown  i s r e d u c e d b y t h e amount o f 1.5  magnet c u r r e n t s a c t u a l l y 29.4  a t 135  29.4 MeV  MeV  a t the  a t the surface of  t h e magnets t o o b t a i n  a  nominal  by t h e pp—»7L d r e a c t i o n , t h e +  correspond t o a p i o n energy o f  -17-  Lab Trt  MeV 320 P+P  11+4  240 T = 6do MeV p  160  80  0 0  40  80 Lab  Pig.  2.2  Pion k i n e t i c  120  9Degrees)  energy v s . angle i n a l a b o r a t o r y  system.  -18-  1 cm Proton beam 45° Pion beam, M9 Channel  0 5 cm  Fig.  2.3  CH  0  target  Mylar window  Si S  Degrader 2  M9 Channel  c  S3, b  4  Sj; Scintillators >Pion beam  m  H F-2  13.2U0.10 m  F-3  8.50 i 0,05  Fig.  2.4  Configuration  o f c o u n t e r s and  Stopping plate  degrader  -19C  Counters  and t h e D e g r a d e r  A f t e r emerging from which has t h e seven  t h e vacuum p i p e o f t h e M9  channel,  magnets i n s i d e o f t h e r a d i a t i o n  area,  t h e mesons come o u t i n t o t h e e x p e r i m e n t a l a r e a c a l l e d The  vacuum i s e n c l o s e d by m y l a r  window w i t h  As e x p l a i n e d i n t h e s e c t i o n B,  ness.  experimental  0.025 mm  thick-  there i s another  a r e a F-3 a f t e r t h e t r i p l e t  C o r v a l l i s magnet.  F-2.  quadrupole  and  In the e x p e r i m e n t a l a r e a , f o u r  scinti42  llation  counters with a p l a s t i c  were s e t t o m e a s u r e t h e r a n g e has  an anode p u l s e r i s e  four counters and  A RCA  2.1.  to exclude  and t h e r a t i o  NE102A H:C  8575 p h o t o m u l t i p l i e r was u s e d t i m e o f 2.5 n s .  i s shown i n E l g . 2.4  the thickness of p l a s t i c  table  (NE102A)  and t i m e o f f l i g h t .  t h e f a s t d e c a y t i m e o f 2.4 n s e c  atoms o f 1.104. has  scintillators  The l a y o u t o f t h e  and t h e c r o s s - s e c t i o n  scintillators  a r e shown i n  E a c h p l a s t i c was w r a p p e d i n b l a c k v i n y l light.  and  The t h i c k n e s s o f t h e b l a c k v i n y l  tape tape,  180 m i c r o n s ,  must a l s o be c o n s i d e r e d when c a l c u l a t i n g t h e  pion range.  A small size o f the p l a s t i c  (5 cm x 5 cm) was u s e d  especially  d e f i n e t h e s m a l l beam p r o f i l e  scintillator  f o r t h e second  i n front o f the degrader.  S i n c e t h e beam s p r e a d s o u t a f t e r p a s s i n g t h r o u g h the p l a s t i c  s c i n t i l l a t o r s of the third  were made q u i t e l a r g e .  counter t o  and f o u r t h  the absorber, counter  The f o u r t h c o u n t e r w h i c h g a v e  c o i n c i d e n c e s i g n a l h a d a c r o s s - s e c t i o n o f 20 cm x 20 The  cross-section of the f i r s t  l a r g e r than  t h a t o f t h e second  anticm.  counter  (10 cm x 10 cm)  was  counter  (5 cm x 5 cm).  The  -20-  NO  S  S  l 2  THICKNESS 0.64  C  m  SIZE 10x10  MATERIAL  PHOTO  C T n 2  PLASTIC; 0.16  NE  5x5  102A  RCA  8575  DENSITY? S  3  S  4  T a b l e 2.1  MUL  0.16  15x15  0.32  20x20  3 1.03 H/C;  The s p e c i f i c a t i o n  of plastic  p h o t o - m u l t i p l i e r used  g/cm  1.104  s c i n t i l l a t o r s and  f o r counters.  -21coincidence s i g n a l of the f i r s t  and  s e c o n d c o u n t e r was  t o d e f i n e t h e number o f i n c o m i n g p i o n s . first  The  used  area o f the  c o u n t e r i s l a r g e r t h a n t h a t o f t h e second c o u n t e r so  t h a t t h e beam i s d e f i n e d a l m o s t e n t i r e l y counter  by t h e  only.  The  d e g r a d e r , w h i c h was  counter, has  s e t between t h e s e c o n d and  s i x p l a t e s o f absorber which  i n d e p e n d e n t l y by a i r p i s t o n s . 2  c a n be  g/cm  f o r aluminum.  Each p l a t e o f a b s o r b e r has a  x 18 cm  f o r c o p p e r and  square.  The  were m e a s u r e d by a m i c r o m e t e r micron.  The  c o r n e r s and  f r o m 0 t o 13.9  2.2.  w h i c h was  range g/cm  2  cross-section  thicknesses o f the p l a t e  f o r t h e d e g r a d e r a r e shown i n t a b l e  third  moved  I t can v a r y the t o t a l  f r o m 0 t o 21.3  18 cm  second  used  These t h i c k n e s s e s accurate to  2.5  p l a t e s were m e a s u r e d a cm o r so i n a t t h e t h e a v e r a g e o f t h e s e numbers was  taken.  four  It  was  f o u n d t h a t a l l p l a t e s h a d a c o n s t a n t t h i c k n e s s w i t h i n 0.2 deviation  ( t a b l e 2.2).  available alloys All  and  The  p l a t e s were made o f c o m m e r c i a l l y  t h e c o m p o s i t i o n i s shown i n t a b l e  aluminum p l a t e s and  thin  %  c o p p e r p l a t e s had a  2.2.  well-finished  s u r f a c e and were p o l i s h e d w i t h f i n e  sand paper  surface before the experiment.  t h i c k copper p l a t e s  with  by a l o c a l m a c h i n e s h o p so  that  1.1  cm  t h i c k n e s s were g r o u n d  t h e t h i c k n e s s was  The  uniform to within  30  micron.  to clean  the  -22-  PLATE NO  COPPER  ALUMINUM  COPPER LEAD ZINC IRON  99.5  %  0.5  %  MAGNESIUM SILICON CHROMIUM COPPER ALUMINUM DENSITY  DENSITY  2.70 g/cm  0.273 ± 0.0012  2  0.728 ± 0.0013  0.443 ± 0.001  3  1.388 ± 0.0027  0.897 ± 0.001  4  2.890 ± 0.0034  1.770 ± 0.001  5  5.787 ± 0.005  3.537 ± 0.002  6  10.241 ± 0.011  6.992 ± 0.007  21.307 ± 0.013  13.857 ± 0.008  2.2  g/cm  3  0.218 ± 0.0005  Components a n d t h i c k n e s s e s o f a b s o r b e r s degrader.  % % % % % 3  3  1  TOTAL  Table  8.93 g / c m  1.0 0.6 0.25 0.25 97.9  used  g/cm  3  f o r the  -23D  Electronics  The  schematic diagram  of the e l e c t r o n i c s  measurements i s shown i n F i g . 2.5. the d i f f e r e n t i a l In  range  c u r v e and  integral  order t o take the d i f f e r e n t i a l  c o i n c i d e n c e i n the f i r s t , a n t i - c o i n c i d e n c e o f the  second  In t h i s  and  these p a r t i c l e s channel. Each The to  and  are expected  output  from  second  f o u r d e t e c t o r s i s f e d t o each  from the f i r s t ,  third  and  output  from the second  t h e l e a d i n g edge t y p e and  counter.  the  The number  Most o f  a r e e l e c t r o n s , p i o n s and  for  M9  muons.  discriminator.  fourth counter are fed (LRS q u a d - r d i s c r i m i n a t o r ) .  (ORTEC 463) The  i s used  difference  constant fraction  be d i s c u s s e d i n t h e f o l l o w i n g c h a p t e r .  only  between  discriminator The  output  from  q u a d - d i s c r i m i n a t o r i s a r e c t a n g u l a r p u l s e o f which  the width  i s v a r i a b l e w i t h t h e a m p l i t u d e o f 0.8  width of output 20 n s e c w i d e and  from the f i r s t  and  The  third  a width  wide r e c t a n g u l a r p u l s e i s used  from another  200  The  nsec  l e s s than  f o r the  output which i s the a n t i - c o i n c i d e n c e s i g n a l o v e r l a p completely the s i g n a l  volt.  discriminator i s  from the f o u r t h d i s c r i m i n a t o r  o u t p u t o f c o n s t a n t f r a c t i o n has  nsec.  the  counter.  t o h a v e emerged f r o m t h e  constant f r a c t i o n discriminator  t h e LRS  of  counter g i v e s the  t h e l e a d i n g edge t y p e d i s c r i m i n a t o r  will  taken.  counter with  both c o u n t e r s .  The  The  and  second  through  These p a r t i c l e s  signal  both  f o u r t h c o u n t e r were n o r m a l i z e d b y  coincidence of the f i r s t p a r t i c l e s which pass  c u r v e were  third  range  experiment,  curve, the counts  counts of c o i n c i d e n c e i n the f i r s t  of  f o r the  wide.  10  fourth  i n order to discriminator.  S;, Scintillators Lj,Light guides  Dj, Delay E.G.G.DP463  Pj, Photomultipliers Si  LRS DISCRI.  n  CF. DISCRI. ORTEC 463  L2 Degrader  LRS DISCRI.  L3  Counters * E.G.G. , AND  2.5  Scheme  ,  i  to  *  I  Stopping plate  Fig.  PRESET  f o r t h e measurements o f t h e r a n g e  curve  The  s i g n a l f o r the time o f f l i g h t i s the output o f the  c o i n c i d e n c e between the f i r s t and important  second counter.  t o d e f i n e the d i s t a n c e from the p i o n  t a r g e t t o measure the time o f f l i g h t .  production  T h e r e f o r e , i f the  c o i n c i d e n c e s i g n a l i s f e d t o the time-to-amplitude (TAC,  ORTEC 437A) we  counters  have to know which s i g n a l o f  i s a s t a r t o r stop s i g n a l .  It i s  converter two  In o r d e r t o make sure  t h a t the second d i s c r i m i n a t o r s i g n a l always t r i g g e r s  the  c o i n c i d e n c e s i g n a l , the second d i s c r i m i n a t o r i s a d j u s t e d t o g i v e the narrow p u l s e .  Furthermore, i t i s important  to  use  the c o n s t a n t f r a c t i o n d i s c r i m i n a t o r t o feed the s t a r t o r stop p u l s e t o the time-to-amplitude l e a d i n g edge type d i s c r i m i n a t o r .  converter instead of  The e f f e c t s of  the  these  d i s c r i m i n a t o r s i n the time o f f l i g h t w i l l be d i s c u s s e d i n the f o l l o w i n g c h a p t e r .  The R.F.  s i g n a l i s employed as a  stop s i g n a l , because the frequency  o f the R.F.  to use the s i g n a l as a s t a r t s i g n a l f o r the The output o f the TAC analyzer  TAC.  i s analyzed by the  (Northern S c i e n t i f i c Model, NS  i s too h i g h  900)  multi-channel and the time  d e l a y spectrum o f the a n a l y z e r i s typed out by a t e l e p r i n t e r . The count  o f the  (1,2) c o i n c i d e n c e i s used as the p r e s e t  count i n the d u a l c o u n t e r / t i m e r c o i n c i d e n c e i s counted. the f o l l o w i n g c h a p t e r .  (ORTEC 715)  and  (1,2,3,4)  T h i s w i l l be d i s c u s s e d i n d e t a i l i n When the count r a t e i s h i g h , 1 0  p r e s e t count i s employed but, when i t i s low, 4 10 i s employed.  a h a l f of  4  of  -26III  The B a s i c Experiment  f o r t h e Measurement o f t h e Range-  E n e r g y R e l a t i o n a n d Time o f F l i g h t T h e r e a r e s e v e r a l methods t o m e a s u r e t h e momentum and e n e r g y o f c h a r g e d p a r t i c l e s .  In o u r experiment, the  e n e r g y o f p i o n was d e f i n e d b y t h r e e methods, t h e r a n g e energy r e l a t i o n , current. problem be  t h e time o f f l i g h t  and t h e bending  magnet  I n t h i s c h a p t e r , t h e b a s i c i d e a and fundamental i n measuring  t h e range and t h e time o f f l i g h t  discussed.  A  The Range  Curve  When c h a r g e d p a r t i c l e s p a s s t h r o u g h m a t t e r , t h e y energy by i o n i z a t i o n .  I f we know t h e r a n g e  energy o f charged p a r t i c l e s  aluminum a n d c o p p e r .  In our e x p e r i -  i s d e t e r m i n e d by i t s range  In the section  2-D a b o v e , t h e method  o f measurement o f r a n g e h a s b e e n shown b r i e f l y . dences and  curve, respectively.  parentheses s i g n i f i e s  coincidence signal of the d i f f e r e n t i a l 3.1.  These  The number  curve  enclosed with  t h e c o u n t e r i n F i g . 2.5 a n d t h e b a r  on t h e h e a d o f t h e number  Fig.  The c o i n c i -  (1,2,3,4) a n d (1,2,3) g i v e t h e d i f f e r e n t i a l  integral  lose  i n matter, the  c a n be o b t a i n e d .  ment, t h e e n e r g y o f a p o s i t i v e p i o n in  will  4 designates that  from t h e f o u r t h c o u n t e r .  i t i s an a n t i The example  a n d i n t e g r a l r a n g e c u r v e i s shown i n  d a t a were t a k e n u s i n g a C l ^ p i o n p r o d u c t i o n  t a r g e t u n d e r t h e c o n d i t i o n t h a t t h e p i o n s were s t o p p e d i n 2 a 0.16 g/cm  thick  scintillator,  the horizontal s l i t  c l o s e d t o 2 cm a n d t h e a b s o r b e r u s e d was aluminum.  was  Fig.  3.1  The  differential  arid i n t e g r a l  range  c u r v e f o r 30.MeV p i o n s t o p p e d by A l .  With no absorber and p e r f e c t geometry, one might expect t h a t the counts o f (1,2,3,4) and (1,2,3) would equal 0 and 4 4 10, r e s p e c t i v e l y , f o r t h e case o f (1,2)=10.  But as shown  i n F i g . 3.1, (1,2,3,4)=80 and (1,2,3)=9400 w i t h o u t t h e absorber.  T h i s means t h a t 600 p a r t i c l e s o u t o f 10, which  f i r e t h e f i r s t and second c o u n t e r , go a s i d e t o miss t h e t h i r d counter.  As the absorber t h i c k n e s s i n c r e a s e s , t h e  d i f f e r e n t i a l curve shows c l e a r l y two peaks.  One o f them  i s the p i o n peak and t h e o t h e r the muon peak.  These peaks  r e s u l t from t h e s t o p p i n g o f p i o n s and muons i n t h e s c i n t i l l a t o r o f the t h i r d c o u n t e r .  So, as the matter which stops  the p a r t i c l e s i n c r e a s e s , these peaks g e t much h i g h e r .  The  p i o n peak i s h i g h e r than the muon peak due t o t h e h i g h e r percentage o f p i o n s i n t h e beam.  The p i o n s and muons have  d i f f e r e n t ranges because t h e two p a r t i c l e s have t h e same momentum but d i f f e r e n t e n e r g i e s (the muon mass i s 105.7 MeV/c  2  2 and t h e p i o n mass i s 139.6 MeV/c ) .  We a l s o  observe  an e l e c t r o n peak but the peak p o s i t i o n i s o f f s c a l e due t o the l a r g e  range.  The d i s t r i b u t i o n o f two peaks should be Gaussian.  When  a beam o f p a r t i c l e s l o s e s energy by i o n i z a t i o n , t h e p a r t i c l e s do not a l l stop a f t e r p a s s i n g through t h e same t h i c k n e s s o f material. dR o f R,  p  The p r o b a b i l i t y o f a p a r t i c l e s t o p p i n g w i t h i n P(R)dR o f a p a r t i c l e , i s g i v e n by  (  R  )  d  R  =  ^ e x  P  i - ^ - } d R  -29where Rg i s the mean range o b t a i n e d by i n t e g r a t i o n over the average  energy-loss  (Eq. 1.4) o r , e q u i v a l e n t l y , from t h e  p o s i t i o n o f peak on the d i f f e r e n t i a l range c u r v e . In P i g . 3.1, the i n t e g r a l range curve shows two s l o p e s around the peaks o f the d i f f e r e n t i a l c u r v e . second  The f i r s t and  s l o p e g i v e t h e range o f p i o n and muon, r e s p e c t i v e l y .  The mean range o f t h e p a r t i c l e s can be o b t a i n e d from t h e i n t e g r a l curve.  In the f i g u r e , we f i n d t h a t the p i o n s t e p  has i t s h a l f v a l u e a t  RS=R  Q * which i s t h e p o s i t i o n o f t h e  peak o f t h e d i f f e r e n t i a l c u r v e .  By drawing a tangent a t  the s t e e p e s t p o i n t o f the p i o n s l o p e and o b t a i n i n g t h e i n t e r s e c t i o n o f t h e tangent w i t h the R-axis, we f i n d the extrapolated r a n g e * ' 0  4 3  R  „ t . » which i s g i v e n by v  r  (3.2)  The d i f f e r e n c e  R  ~ n R  e x t r  *  s  d e f  i n e d as the s t r a g g l i n g p a r a -  meter S. Since i t i s c l e a r from P i g . 3.1 t h a t both t h e i n t e g r a l and d i f f e r e n t i a l curve g i v e t h e same range, we w i l l use t h e d i f f e r e n t i a l curve t o d e f i n e the range o f p a r t i c l e i n matter by u s i n g Gaussian curve  fitting.  -30-  B  The Height o f Peak i n the D i f f e r e n t i a l Range Curve  In the l a s t s e c t i o n i t was  noted t h a t the p a r t i c l e s  which a r e counted i n the d i f f e r e n t i a l range be counted  in scintillator  l l a t o r #4.  (1,2,3,4) must  #3 y e t not be counted i n s c i n t i -  These pions must t h e r e f o r e stop some way  scintillator  #4  (or i n the wrapping m a t e r i a l ) .  into  In our 2  experiment  a thin plastic s c i n t i l l a t o r  (0.16 g/cm  ) was  used f o r the t h i r d counter t o reduce the a b s o r p t i o n o f the pion.  I t was  expected t h a t the width o f peak i n the  curve might be narrower  by u s i n g the t h i n s c i n t i l l a t o r f o r  the t h i r d counter without any absorber between the and f o u r t h c o u n t e r .  range  third  The width o f the peak i s caused  mostly  by the i n i t i a l energy spread o f beam and the s t r a g g l i n g s i n the degrader and p l a s t i c s c i n t i l l a t o r s c o n t r i b u t e a little  t o the w i d t h .  I f t h e r e i s an absorber behind the  t h i r d c o u n t e r , the energy  s t r a g g l i n g i s i n c r e a s e d and  width o f peak seems t o be wider.  the  In o r d e r t o see the  e f f e c t o f the t h i c k n e s s such an' absorber behind the t h i r d counter, the range curves were taken f o r t h r e e c a s e s ; no absorber,  (2) w i t h 0.8  (3) w i t h 1.5 mm 30 MeV  mm  (1)  t h i c k n e s s aluminum absorber,  t h i c k n e s s aluminum a b s o r b e r .  For t h i s  p i o n s were employed and the CHj t a r g e t was  test,  used a t  the T 2 - t a r g e t . The range curves are shown i n F i g . 3.2. was  performed  magnet.  This  experiment  on the F-3 p o s i t i o n behind the C o r v a l l i s  The experimental arrangement i s shown i n F i g . 2.4  and the c u r r e n t s o f bending magnets and guadrupole  magnets  -31-  0  1 Absorber  Fig.  3.2  The e f f e c t and  fourth  •  2  thickness ( cm,Al)  o f t h e s t o p p i n g p l a t e between t h e t h i r d counter.  are given i n table Fig.  3.2  shows t h a t  3.1.  The  results  a r e shown i n t a b l e  t h e peak w i t h A l a b s o r b e r i s h i g h e r t h a n  t h e c a s e w i t h o u t t h e a b s o r b e r and peak p o s i t i o n s  shift  according t o the t h i c k n e s s of the absorber behind the counter.  The  total  f o r the a d d i t i o n a l  ranges o f the t h r e e cases  view,  i t does  third  c o u n t e r i s added  3.2  and  A l s o t h e energy o b t a i n e d from  i s c o n s i s t a n t w i t h the energy  of the time o f f l i g h t .  third  (correcting  absorber) are given i n t a b l e  agree v e r y w e l l each o t h e r . the range  3.2.  f r o m t h e measurement  T h e r e f o r e , from t h e energy p o i n t  not matter whether the a b s o r b e r b e h i n d or not.  But  from the p o i n t o f  o f measurement, a s s e e n f r o m F i g . 3.1,  of  the view  the stopping r a t e with  the absorber i s g r e a t e r than the s t o p p i n g r a t e without the absorber. is  I t i s also clear  from t h e t a b l e  n o t a g r e a t change i n t h e f u l l - w i d t h  w i t h t h e aluminum a b s o r b e r .  As  3.2  half-maximum  t o improve  the count r a t e ;  c h o s e n was  0.8  mm  even  t h e measurements f o r t h i s  t h e s i s were made w i t h a s m a l l p r o t o n c u r r e n t 5 n A ) , a s t o p p i n g p l a t e was  that there  (between  1 and  used i n our range  experiment  as a compromise t h e  thickness  o f aluminum  plate.  -33-  MAGNET  CURRENT*  MAGNET  (Amp.)  (Amp.) B  l  245 .5  B  2  247.0 168.1  Q  4  3  Table 3.1  *  128.7  168 .9 178.6  Q  6  205.3 167.4  163.7 Q  CURRENT*  Q  8  119.0  The c u r r e n t s o f magnets f o r the measurement on F-3.  The numbers i n the table a c t u a l l y show the voltage which i s an output of the amplifier, 0-50 mV->0-5 V. I t amplifies the voltage across the shunt, 50 mV-500 A,+0.25%, which i s i n s t a l l e d i n series with the magnet. Thus the reading of the amplifier gives d i r e c t l y the amount of magnetic current.  THICKNESS OF ^ ^ ^ A B SORBE R  PEAK POSITION  000  0.8  mm  mm  1.5  mm  3.59+0.24  3.38*0.22  3.29+.0.23  3.62±0.24  3.51±0.22  3.51*0.23  2 . (g/cm ) RANGE OF ALUMINUM INCLUDING THE THICKNESS OF STOPPING PLATE . . 2.  j  ., ' (g/cm ) HALF WIDTH OF P  E  A  1.30  1.20  1.32  1.19  1.29  1.29  (g/cm )  K  2  CORRECTION FROM P L A S T I C AND VINYL COVER,IN ALUMINUM *™GE > (g/cm ) 2  F: ••:  . •• •',  TOTAL RANGE IN ALUMINUM RANGE • '; 2 •?>•"!:• (g/cm ) ENERGY OBTAINED FROM ALUMINUM RANGE i (MeV)  4.81±0.25  4.80±0.23  4.80±0.24  29.3±0.9  29.3±0.8  29.3±0.8  ENERGY OBTAINED FROM THE TIME OF FLIGHT METHOD " (MeV)  30.02±0.87  l  Table  3.2  The e f f e c t o f t h e s t o p p i n g determination  p l a t e t o t h e pion energy  a n d t h e c o m p a r i s o n o f e n e r g y between  t h e r a n g e a n d t i m e o f f l i g h t method.  C  The P r i n c i p l e o f the T i m e - o f - F l i g h t Method  The M9 meson beam channel  i s shown i n F i g . 2.1.  The  500 MeV proton s t r i k e s the T2 t a r g e t and produce p i o n s (as w e l l as h e a v i e r i o n s such as deuterons, Some o f the charged  alphas,  etc.).  p i o n s decay i n t o muons; the n e u t r a l  p i o n s a l l decay i n t o two gamma rays which immediately  create  e l e c t r o n - p o s i t r o n s p a i r s i n the surrounding m a t e r i a l . A l l these p a r t i c l e s , beam.  emitted a t 135°,  form i n t o the M9  As a l l these events o c c u r v e r y q u i c k l y , the time  s t r u c t u r e o f the M9 beam i s always r e l a t e d t o the time s t r u c t u r e o f the proton beam and thus t o the R.F. o f the accelerator.  In our experiment, s i g n a l s o f the R.F. and  the s c i n t i l l a t i o n counter were employed t o measure the velocity of particles. The e l e c t r o n i c s f o r the time o f f l i g h t i s shown i n F i g . 3.3.  We used the time-to-amplitude  437A) and m u l t i - c h a n n e l a n a l y z e r  converter  (TAC, ORTEC  (N.S. 900) t o measure the  time o f f l i g h t o f p a r t i c l e s from the T2 p r o d u c t i o n t a r g e t to the s c i n t i l l a t i o n counter i n the experimental a r e a . t h i s type o f t i m e - i n t e r v a l measurement, the output o f the TAC i s l i n e a r l y p r o p o r t i o n a l t o the time t  2  In  voltage  difference  - t ^ = 4 t , where the a r r i v a l o f the s t a r t p u l s e and the  stop p u l s e i s a t t ^ and t  2  , respectively.  The t i m i n g diagram  ( F i g . 3.4) shows the s t a r t and s t o p , the time counted MCA.  difference  by the TAC and the p l a c e o f each p a r t i c l e on the  S i n c e the h i g h frequency  s i g n a l o f the R.F. can not  be accepted as the s t a r t p u l s e , the R.F. s i g n a l i s f e d t o  (Cf.  Fjg. 2.5) START E.G.G. AND  Li  ' PION  D  3.3  PROTON  437 TAC  DIS'CRI STOP  MECL3I PRESCALER 86.8 ns  43.4 ns R.F.  Fig.  2  i ORTEC  MCA NS900 PRINTER  C i r c u i t diagram o f the time o f f l i g h t measurement.  T2  O  k—  43.4 ns TIME  START (COUNTER)  86 £ ns  L_  STOP (R.F.) TAC  TIME  A H  ©2  r  OUT PUT VOLTAGE OF TAC DISPLAY OF MCA  V=0  •43.4ns 7fr  x CD  (—4  LU X  Fig.  3.4  CHANNEL NUMBER Timing diagram o f TAC and the d i s p l a y o f p a r t i c l e s on the m u l t i - c h a n n e l a n a l y z e r .  -37the s t o p and the s i g n a l o f t h e s c i n t i l l a t i o n counter t o the s t a r t as shown i n F i g . 3.4.  Consequently, the o r d e r o f  p a r t i c l e s on the MCA i s r e v e r s e d so t h a t the e l e c t r o n peak appears a f t e r the p i o n and muon peaks. The R.F. has a 43.4 nsec r e p e t i t i o n time but i n our experiment a s c a l e o f two was added t o the R.F. p u l s e s , thus p r o v i d i n g a stop p u l s e every 86.8 nsec.  Therefore,  t h e r e a r e two peaks f o r each p a r t i c l e i n the time-delay spectrum.  S i n c e these two peaks have a time d i f f e r e n c e o f  43.4 ns, t h i s g i v e s n a t u r a l c a l i b r a t i o n f o r the MCA. a l s o c a l i b r a t e d the MCA u s i n g the time c a l i b r a t o r 462).  We  (ORTEC  The c a l i b r a t i o n r e s u l t s are shown i n F i g . 3.5.  The p e r i o d i c i n p u t o f 10 ns s i g n a l i s f e d t o TAC.  (The  time c a l i b r a t o r produces a p a i r o f p u l s e s s e p a r a t e d by 10, 20, 40 and 80 ns; t h e s p e c i f i c a t i o n s o f t h e instrument i n d i c a t e t h a t the e r r o r i s +10 psec f o r the 10 nsec and 0.005 % f o r the o t h e r i n t e r v a l s . )  interval  As shown i n F i g .  3.5, a l l peaks a r e almost e q u a l l y separated and each channel corresponds t o 178 psec.  S i n c e the MCA shows good  linearity,  the u n c e r t a i n t y o f the MCA i s n e g l i g i b l e i n our experiment. In o r d e r t o c a l c u l a t e the v e l o c i t y o f p a r t i c l e s , we have t o know the d i s t a n c e from the T2 t o the counter which g i v e s the s t a r t s i g n a l t o the TAC and, a l s o , the time d i f f e r e n c e i n a r r i v a l a t t h e counter between the e l e c t r o n and another particle.  Assuming t h a t the e l e c t r o n s t r a v e l w i t h t h e v e l o -  c i t y o f l i g h t , t h e time o f f l i g h t f o r e l e c t r o n from T2 t o the counter i s known.  Then, adding the time d i f f e r e n c e between  15  xicr —  A  2  ^  A  2  — *  A  2  —$K  A,  A 1,2 =10 10  si nsec  Ai  =56.2 channels  A2  =56.3 channels  in *->  c  Z3  O  o  0  100  Fig.  3.5  -i  .  L  200  Time c a l i b r a t i o n  _!_L  300 Channel number  spectrum  (periodic  input,  400 10 n s e c )  500  -39the  particle  particle the  and  electron,  i s calculated.  particle.  the time o f f l i g h t  This  f o r the  time g i v e s the v e l o c i t y  of  -40D  The  Time Walk o f t h e D i s c r i m i n a t o r  In the time  of f l i g h t  time walk a r i s e s  measurement, t h e p r o b l e m  from the d i s c r i m i n a t o r .  serious d i f f e r e n c e i n the p a r t i c l e the is  suitable discriminator. caused  I f we the  by  the d i f f e r e n t  The  e n e r g y , we  and muons i n t h e p l a s t i c  MeV/c, t h e p u l s e muons and  1,  2 and  shown i n F i g . 3.6. flight,  has  Since these  to avoid the  d i s c r i m i n a t o r was  used  the  d i s c r i m i n a t o r b e c a u s e i t has  triggering  are  leading  time  as  and  time  overestimated.  The  fraction  constant  a l e a d i n g edge t i m i n g  same t r i g g e r i n g  independent o f the pulse amplitude  principle  to electrons,  fixed discriminator  experiment.  the  energy-  100  employ t h e  time walk, a c o n s t a n t  i n our  type  d i f f e r e n c e s g i v e a wrong  f r a c t i o n d i s c r i m i n a t o r i s b e t t e r than  45  correspond  Suppose we  the energies of p a r t i c l e s  In o r d e r  The  o f l e a d i n g edge  3 i n F i g . 3.6  the d i f f e r e n t  matter.  energy-loss  C o n s i d e r i n g the  type d i s c r i m i n a t o r with  each p a r t i c l e  in  same momentum a r o u n d  pions, r e s p e c t i v e l y .  edge t r i g g e r level,  outputs  the  choose  electrons with  scintillator.  shown i n F i g . 3.6.  l o s s o f each p a r t i c l e with  have t o  of p a r t i c l e s  same momentum, e l e c t r o n s h a v e a s m a l l e r  d i s c r i m i n a t o r are  gives  time walk o f d i s c r i m i n a t o r  energy-loss  e x a m p l e s o f d e t e c t o r p u l s e and  of  Since t h i s  h a v e a beam i n c l u d i n g p i o n s , muons and  than pions  of  44  rise  time,  time.  The  46 '  is illustrated  pulse i s delayed  (a) and  i n F i g . 3.7.  and  detector  a f r a c t i o n o f the undelayed  (b) i s s u b t r a c t e d f r o m i t ( c ) . cancels the delayed  The  The  attenuated  i n v e r t e d pulse at the  pulse  pulse exactly fraction  Out put of photomultiplier  Time  Fig.  3.6  T i m e - w a l k due t o t h r e e d i f f e r e n t  input pulses of  e l e c t r o n s , muons a n d p i o n s w i t h t h e same momentum. Since the discriminator is  fixed,the outputs  level  shift  o f a l e a d i n g edge t y p e  i n time  spectrum.  a, Inverted and delayed anode pulse b, Attenuated anode pulse  c, Resulting zero- crossing pulse d, Current gate e, Zero-crossing trigger  Fig.  3.7  Constant  f r a c t i o n of height  pick-off.  -42phase p o i n t on the delayed  pulse.  The time o f f l i g h t experiment was done a t t h e F-3 f o c u s s i n g p o s i t i o n w i t h 50 MeV pions i n o r d e r t o see the e f f e c t o f the time walk o f a d i s c r i m i n a t o r . fraction discriminator  (ORTEC 463) and the LRS l e a d i n g edge  type d i s c r i m i n a t o r were employed. i n t a b l e 3.4.  The c o n s t a n t  The r e s u l t s a r e shown  I t i s apparent t h a t the energy o f LRS  d i s c r i m i n a t o r g i v e s h i g h e r energy than t h a t o f the C F . discriminator. there  From the time-delay spectrum o f the MCA,  i s 1.0 nsec time d i f f e r e n c e between LRS and C F .  d i s c r i m i n a t o r i n the time o f f l i g h t o f the p i o n T2 t o the c o u n t e r . about 0.8 nsec.  increases  In the case o f muon, t h i s d i f f e r e n c e i s  Since the e n e r g y - l o s s o f 30 MeV p i o n  plastic scintillator MeV p i o n  from the  (0.76 MeV)  i n the  i s l a r g e r than t h a t o f 50  (0.56 MeV), the time d i f f e r e n c e f o r 30 MeV  pion  t o 2.0 nsec.  In o r d e r t o  d e f i n e t h e meson v e l o c i t y by the time o f  f l i g h t method, we have t o know the peak p o s i t i o n o f the pions and e l e c t r o n s on the time-delay spectrum. Fig.  As seen i n  4.1, the h e i g h t o f the e l e c t r o n peak comes down as  the energy o f p i o n i n c r e a s e s . of e l e c t r o n i s increased  S i n c e the contamination  by u s i n g the t a r g e t w i t h a l a r g e  atomic number, we compare the energy d i f f e r e n c e between the b e r y l l i u m t a r g e t and copper t a r g e t . i n t a b l e 3.1.  Though the height  The r e s u l t s a r e shown  o f the e l e c t r o n peak i n case  o f copper i s almost twice as h i g h as t h a t i n case o f b e r y l l i u m , we do not d e t e c t any d i f f e r e n c e i n the r e l a t i v e  COPPER TA RGET LRS K.E.  (MeV)  MOMENTUM  (MeV/c)  BETA  54.67 ± 2.3 135.3  + 3.3  BERYLLIUM TARGET CF.  51.51 ± 1.91 130.7  + 2.8  LRS  CF.  55.22 ± 2.31 136.1  ± 3.3  50.93 ± 1.94 129.8  ± 2.9  0.695 + 0.008  0.682 + 0.008  0.697 ± 0.009  0.680 ± 0.008  TIME*  (nsec)  57.13 ± 0.54  58.18 ± 0.45  56.97 ± 0.53  58.39 ± 0.45  K.E.  (MeV)  63.91 ± 3.8  59.91 ± 3.1  62.79 ± 3.6  59.33 ± 3.0  MOMENTUM  (MeV/c)  BETA * TIME  (nsec)  132.8 ±4.8  127.6  0.782 ± 0.01  0.769 ± 0.01  50.80 ± 0.58  ± 4.0  51.61 + 0.50  131.4  ± 4.6  126.9 ± 3 . 9  0.778 ± 0.01  0.767 ± 0.01  51.02 ± 0.56  51.74 ± 0.48  * Time o f f l i g h t between the p i o n p r o d u c t i o n t a r g e t (T2) and counter. Table 3.3  The e n e r g i e s o f p i o n s and muons a r e measured by the time o f f l i g h t method. In o r d e r t o check the s y s t e m a t i c e r r o r , two types o f d i s c r i m i n a t o r s a r e employed, LRS ( l e a d i n g edge) and C F . (constant f r a c t i o n ) , and two k i n d s of  t a r g e t s , copper and b e r y l l i u m .  -44positions  o f the e l e c t r o n  no d i f f e r e n c e targets.  and p i o n p e a k s .  i n the energy d e t e r m i n a t i o n  Thus,  there  between t h e  i  -45IV  R e s u l t s and The  made on and 30 for  F-3.  We  f o r the case with  Time o f  see  The  two  two  30,  and  The  40,  two  e l e c t r o n i c s and  have been d i s c u s s e d  p i o n p e a k s on  the  f o r the  i n the  s p e c t r u m and  60  p i o n peaks corresponds  MeV  t o the  case. R.F.  are  last  a l s o muon  The  I n t h e e n e r g y c a l c u l a t i o n , we  period of  43.40  MeV,  t h e beam w i t h  because o f  97 MeV/c, i n w h i c h p i o n s  t h e beam c o n s i s t s m a i n l y Though we  can  energy.  i t s large to  F o r example, e l e c t r o n s  h a v e a v e l o c i t y o f 0.99999  electrons.  o b t a i n the pion  T h i s assumption g i v e s n e g l i g i b l e e f f e c t  energy c a l c u l a t i o n o f p i o n .  target  assume t h a t t h e e l e c t r o n  i s the v e l o c i t y of l i g h t ,  momentum.'  in  can  and  separation  counter,  we  of  chapter.  to  (8.50+0.05 m),  shown  principle  Knowing t h e d i s t a n c e f r o m t h e p i o n p r o d u c t i o n  velocity  currents  4-1.  nsec. the  50  energies,  magnet  spectra of four energies  experimental  e l e c t r o n peaks except of  o f aluminum a b s o r b e r  shown i n t a b l e  r a n g e were  energies,  the copper absorbers.  flight  time of f l i g h t can  and  Flight  time o f  F i g . 4.1.  time o f f l i g h t  chose f o u r d i f f e r e n t  each energy are  The  the  We  50 MeV  A  in  measurements o f t h e  60 MeV and  Discussion  (=@).  of pions with expect  contained  have a K.E.  F i g . 4.1 a few  the  of  30  shows t h a t  muons  and  muons f r o m t h e d e c a y  of  p i o n s , most o f t h e muons i n t h e muon peak come f r o m  the  production  correspond-  ing  target.  t o muons w i t h  B e c a u s e t h e peak has  a velocity  t h e momentum d e t e r m i n e d by  the  channel.  -46-  ENERGY,  MeV  MOMENTUM, MeV/c MAGNET, l  B  2 l  Q  4.1  39.5  50.5  59,0  95.4  112.3  129.2  141.2  247.4  291.2  335.0  366.0  252.6  297.0  342.0  373.0  192.0  226.0  260.0  284.0  159.5  187.8  216.0  235.0  129.2  152.0  175.0  191.0  169.1  199.0  229.0  250.0  178.7  210.0  242.0  265,0  Amp B  Q  Table  29.4  2  Q  3  Q  4  Q  5  Magnet c u r r e n t s f o r t h e t i m e o f f l i g h t  and range  measurements. TIME OF FLIGHT  RANGE OF A l RANGE OF C u  PION  30.9 ± 0.9  29.8 ± 0.8  30.1 ± 1.2  MUON  37.4 ± 1.6  36.1 ± 1.3  36.4 + 1.1  39.5  41.5 ± 1.5  39.6 + 1.1  50.5  52.4 ± 2.2  MAGNET ENERGY  29.4  49.7 ± 1.1*  50.3  i 1.8  50.1 ± 1.4** 58.6 ± 1.0  59.0  60.6 ± 2.9 59.0 ± 1.4  * ** Table  F i t t e d with the Gaussian d i s t r i b u t i o n . F i t t e d w i t h t h e extreme v a l u e d i s t r i b u t i o n . 4.2  The e n e r g y and  i n MeV o b t a i n e d b y t h e t i m e  range measurements.  of flight  0  200  100  -47-  300  500  400  xic/*  7C  30 MeV Pion  0 .x10  3  8  n  7L •  . 40 MeV  «  x10 3 ..e 2  *  0  *  •  «  t  '*  *-> x10 c  H  o  50 MeV  o  x10  0 2  x10  H  60 MeV n.  1\  .  x10  7t  1 0  Fig.  100  4.1  200 300 Channel number  R.F.-referenced and  time o f f l i g h t  60 MeV o f p i o n  energies.  400  spectrum  500  f o r 30,40,50  The  d e c a y muons w o u l d a p p e a r b e t w e e n t h e muon and e l e c t r o n  peak b u t t h e e x p e r i m e n t a l for  such The  evidence  s p e c t r a do n o t show c l e a r  muons. p i o n peak h a s a t i m e  spread.  In our time  of  flight  m e a s u r e m e n t s , t h e R.F. s i g n a l o f t h e a c c e l e r a t o r i s u s e d as t h e s t o p p u l s e a g a i n s t t h e s t a r t counter. has  The p r o t o n beam a t t h e p i o n p r o d u c t i o n t a r g e t  a bunch o f about 4 n s e c .  c o n t r i b u t o r t o t h e time The  peak w i d t h  equals  p u l s e o f t h e second  spread  a s m e a s u r e d i n o u r method.  a t h a l f maximum i s 3.6 n s e c  t h e p r o t o n beam t i m e  resolution  This i s the p r i n c i p a l  spread.  and n e a r l y  A l s o t h e energy  (-|r-) a n d momentum r e s o l u t i o n  (-p—) o f t h e 30  MeV p i o n a r e 8.2 % a n d 4.5 % a t FWHM, r e s p e c t i v e l y . In flight  order to calculate  t h e peak p o s i t i o n  spectrum, the data p o i n t s a r e f i t t e d  i n t h e time o f with a  Gaussian  47 distribution.  The v a l u e o f t h e m e a s u r e d p o s i t i o n  as X + ry where # = = and  i s the standard  n i s t h e number o f d a t a p o i n t s .  p r o g r a m i s w r i t t e n b y u s i n g t h e LQF one  o f the l e a s t squares 48  library  program.  fitting  The r e s u l t s  i s written  e r r o r o f t h e mean The c u r v e  fitting  subroutine which i s  p r o g r a m i n t h e UBC  a r e shown i n t a b l e  computer  4.2.  e n e r g i e s have b e e n d e f i n e d w i t h a d e v i a t i o n o f 3 t o 5  The  -49B  Range Measurement  The range curves f o r A l and Cu a r e shown i n F i g . 4.2 and 4.3,  respectively.  The e n e r g i e s o b t a i n e d from  ranges  The experimental data are  fitted  are shown i n t a b l e 4.2.  with a Gaussian d i s t r i b u t i o n as d i s c u s s e d i n the section.  last  The curves o f the Gaussian d i s t r i b u t i o n and  data are i l l u s t r a t e d i n F i g . 4.2  and 4.3  the  by a s o l i d l i n e  and  dots, respectively. In o r d e r t o c a l c u l a t e the energy o f p i o n , we have t o know the t h i c k n e s s and content of the absorber i n d e t a i l . The t h i c k n e s s e s o f the metal o f the degrader  are g i v e n by  the peak p o s i t i o n o f the f i t t e d curve w i t h the  Gaussian  distribution.  Elements c o n t a i n e d i n the absorber are shown  i n t a b l e 2.2.  S i n c e , i n both the aluminum and  absorber, 99.5  % o f the c o n s t i t u e n t s can be c o n s i d e r e d as  copper  elements w i t h s i m i l a r atomic numbers, the e f f e c t s from  the  d i f f e r e n t elements can be n e g l e c t e d i n the range c a l c u l a t i o n . We  have t o count the t h i c k n e s s of p l a s t i c  scintillators  and a l s o the windows which keep a vacuum i n the M9  channel.  The t h i c k n e s s e s o f p l a s t i c s c i n t i l l a t o r a r e shown i n t a b l e 2.1.  Each p l a s t i c i s covered by b l a c k v i n y l tape, w i t h a  t h i c k n e s s o f 180 microns,  i n o r d e r t o keep out l i g h t .  vacuum window i s made o f mylar o f 0.25 density of p l a s t i c s c i n t i l l a t o r  mm  (NE 102A)  thickness. i s 1.032  g/cm  The The 3 and  i t s t o t a l t h i c k n e s s i n c l u d i n g the f i r s t , second and t h i r d 2 counter i s 0.983 + 0.1 g/cm. Assuming the d e n s i t y of the 3 b l a c k v i n y l tape and mylar as 1.0 g/cm,,we get 0.037 g/cm 2  ' '  1 1' • , 111  1 1  j 1-3.37  xicr  6.47  - I — "- • • *  >  11  1  • 30 MeV A 40 MeV o 50 MeV x 60 MeV  1  •A  I  —  Gaussian distri DUtion  —  Extreme value distribution  .12  ic-10.27 \\ \\  \ r  1  5  \  13.64 184  I 7 I  \v  or '  7 /  0 '  v  *  0  /  A  O  \  O  0  • . '  *  A  0  0  A  0  0  i  i  i  i  i  •  5  I  i  A  A  A  i  1 0 . 1 5 Absorber thickness (gm/ m ,AD  20  2  C  . 4.2  The r a n g e c u r v e s o f p i o n i n aluminum( p i o n e n e r g i e s , 3 0 , 4 0 , 5 0  and 60  MeV)  x  ^  3.84—nl  A / \a )MeV /*  \  I*  \  \  11.74  iL  •  50 Me  \l  /A A /  • • A  n peak  A  0  0  » •1»  5  A •  i  i  i  •  i  i  i  A i  i  10 15 Absorber thickness (g/ m , Cu) 2  C  Fig.  4.3  The r a n g e  curves of pion  i n c o p p e r ( p i o n e n e r g i e s , 3 0 and 50 MeV),  A  •  i  20  f o r these a b s o r b e r s .  The e r r o r caused from the assumption 2 o f the d e n s i t y as 1.0 g/cm might be n e g l i g i b l e i n comparison w i t h the t o t a l t h i c k n e s s due t o the s m a l l number.  Thus, t h e 2  t h i c k n e s s o f absorber b e s i d e s the metal i s 1.02+0.1 g/cm. The r a t i o o f the number H versus C i s 1.104 plastic scintillator.  i n the  The s t o p p i n g powers o f CH^^  are  c a l c u l a t e d by u s i n g the numbers o f s t o p p i n g power i n the r e f e r e n c e 16 and show i n t a b l e 4.3. 2  Assuming t h a t the  t h i c k n e s s o f 1.02 g/cm o b t a i n e d above i s made o f CE^ ^, we can c a l c u l a t e the e q u i v a l e n t t h i c k n e s s o f CHj ^ f o r the aluminum and copper range curves  (table  4.3).  Since the M9 channel has a vacuum o f l e s s than 0.05 Hg, we can n e g l e c t the slowing down o f charged by a i r i n s i d e the channel.  mm  particles  A f t e r emerging from the vacuum  p i p e , the charged p a r t i c l e s s t i l l have t o go through a i r o f 1 m o r so, which has a d e n s i t y o f 1.29 mg/cm.  This contributes  t o the slowing down o f charged p a r t i c l e s . The e f f e c t 2 2 corresponds t o 0.15 g/cm i n aluminum range and 0.17 g/cm i n copper range. F o r 50 and 60 MeV range curves o f aluminum, the data 49 are a l s o f i t t e d w i t h the extreme v a l u e d i s t r i b u t i o n , because 2 "y^ - t e s t f o r the curve f i t t i n g w i t h the Gaussian  distribution  i s worse than t h a t o f the extreme v a l u e d i s t r i b u t i o n . p r o b a b i l i t y d e n s i t y f u n c t i o n i s g i v e n by  The  P(*)^pH^-ex (-^)} „.x, P  PION  STOPPING  ENERGY  POWER OF C H  U  RANGE  RANGE OBTAINED FROM RANGE CURVES  1  Al  Cu  -  -Al  1.1 2"  MeV  Mev-cm / g  30  EQUIVALENT OF C K  g/cm  g/cm  .Cu  / 2 g/cm  2  TOTAL RANGE**  g/cm  Al 2  Cu  / 2 g/cm  g/cm  2  1.29*0.10 1.49±0.10  3.40*0.23 3.84*0.37 4.95*0.23 5.86*0.37  1.28+0.10  6.47±0.37  4.443  40  3.723  8.01*0.37  10.12+0.41 50  3.230  60  * **  2.945  L.26±0.10  1.44±0.10  11.64*0.41  11•74+0•Iz 10.27*0.55  11.79*0.55  13.64*0.43  15.15*0.43  L3.84±0.6G  15.35*0.60  13.71+0.75  1.25+0.10  Range curves a r e f i t t e d w i t h t h e Extreme v a l u e The t h i c k n e s s o f s t o p p i n g p l a t e behind  T a b l e 4.3  •  The d e t a i l o f range  calculation.  distribution.  the t h i r d counter a r e c o n s i d e r e d .  The  curves  4.2.  fitted  I t i s clear  good f i t t i n g a little two  w i t h two d i s t r i b u t i o n s  a r e shown i n P i g .  t h a t t h e extreme v a l u e d i s t r i b u t i o n  a r o u n d t h e peak and t h e peak p o s i t i o n  from  the Gaussian  distributions  case.  gives  shifts  The e n e r g i e s d e f i n e d b y  a r e shown i n t a b l e  4.2. 2  The installed  aluminum p l a t e w i t h  0.220 g/cm  t o s t o p p i o n s between t h e t h i r d  A h a l f o f the thickness of the p l a t e to  the range.  copper  0.728 g/cm  The t o t a l  ranges  t h e summation o f r a n g e s air. range,  i s considered to contribute  thickness behind  shown i n t a b l e  i n aluminum  In o r d e r t o o b t a i n t h e energy the range-energy  the r e s u l t s  and f o u r t h c o u n t e r .  F o r t h e c o p p e r r a n g e c u r v e s , we e m p l o y e d t h e 2  plate with  counter.  t h i c k n e s s was  the t h i r d  4.3 a r e g i v e n b y  ( o r c o p p e r ) , CH^ ^ a n d o f the pions  from t h e  t a b l e o f r e f e r e n c e 16 i s u s e d , a n d  a r e shown i n t a b l e  4.2.  Thus, t h e range  g i v e t h e p i o n e n e r g i e s w i t h i n a n e r r o r o f °± 3 %.  curves  -55C  The Energy S t r a g g l i n g  and Inherent Energy Spread  of  the Beam The 3-A.  s t r a g g l i n g parameter S i s d i s c u s s e d i n the  From F i g . 3.1,  section  the v a l u e o f S can be o b t a i n e d and i s  2  about 0.59  g/cm  i n aluminum range.  It is difficult  to  get the exact v a l u e o f the s t r a g g l i n g parameter because the beam c o n t a i n s p i o n s , muons and e l e c t r o n s , for  the p i o n s can not be d e f i n e d e a s i l y .  The  S^J^-at, where  parameter i s a l s o g i v e n by  -  o<  the  R-axis  straggling  i s g i v e n by  Eq. 3.1 which i s employed t o f i t the d i f f e r e n t i a l  curve. 2  From the c a l c u l a t i o n o f curve f i t t i n g , 0C* =0.55 g/cm,  then  2  S=0.69 g/cm.  T h i s i s c l o s e t o the number o b t a i n e d  x i m a t e l y i n F i g . 3.1. o f S may  appro-  The d i f f e r e n c e between the two  values  come from the u n c e r t a i n t y i n choosing the R - a x i s .  The  s t r a g g l i n g o f the range o f charged p a r t i c l e s due  to  the f l u c t u a t i o n s o f the i o n i z a t i o n p r o c e s s i s c a l c u l a t e d t h e o r e t i c a l l y by  sternheimer. /  For the case o f 30  MeV  p i o n s , the s t r a g g l i n g parameter i n aluminum i s equal t o  0.19  2  g/cm.  The d i f f e r e n c e between the s t r a g g l i n g o f range o b t a i n e d  from the experiment  and the t h e o r e t i c a l s t r a g g l i n g parameter  mentioned above must be mainly a t t r i b u t e d t o the spread o f the p i o n beam. pions,Y  /  ctuation  Thus, the energy  2  ^ (R - RQ) } t i s equal t o 0.53  energy  spread o f the  g/cm.  2  The f l u 34 i n range and i n e n e r g y - l o s s i s r e l a t e d . For a 2  s m a l l t h i c k n e s s dR and a s m a l l e n e r g y - l o s s dE,  (~) (4 2  R ) 2  d E  where  E ) 2  = ^(E-E ) > 2  d R  Q  and  (AE  ( 4 R  2  )  )^R d  E  SB  =  2 <^(R-RQ)  > .  A c c o r d i n g t o t h i s r e l a t i o n , the energy  spread  -56o f p i o n beam,  -/ < ( E - E ) > 2  Q  yields  1.9 MeV, i d A E / E = 6 . 3 %  ( A P / P = 3 . 2 %) f o r t h e 30 MeV o f t h e p i o n beam. are s m a l l e r than spectrum.  These  values  t h e v a l u e s o b t a i n e d by t h e time o f f l i g h t  -57D  Conclusion  The  various systematic effects  the accuracy of  the  energy d e t e r m i n a t i o n .  the  l e a d i n g e d g e t y p e and c o n s t a n t f r a c t i o n  in  the l a s t  chapter.  We  limit  d i s c u s s e d t h e time walk o f  T h e n , we  could  discriminator  reduce  the time  by u s i n g t h e c o n s t a n t f r a c t i o n d i s c r i m i n a t o r . of  flight  measurement, t h e t i m e w a l k o f t h e  is  t h e most s e r i o u s  systematic error  walk  In the  time  discriminator  i n the energy  determi-  51 nation.  The  specification  of the constant f r a c i o n  discri-  m i n a t o r c l a i m s t h a t t h e walk i s l e s s  t h a n 0.51  nsec.  t h e r e i s s m a l l t i m e w a l k s u c h a s 0.1  nsec, the  energy  determined (0.6) MeV  by t h e t i m e o f f l i g h t f o r t h e 30  (60) MeV  will  pion.  be r e d u c e d by In the energy  by t h e r a n g e method, t h e r e i s t h e u n c e r t a i n t y energy  tabled  It  We  can expect t h a t  has b e e n shown t h a t  methods t o d e t e r m i n e magnetic  there are three  and  The  %.  calculated  energy.  the  The  particles  time  of  r a n g e methods a c t u a l l y work a s a c a l i b r a t i o n  t h i s work, we  have shown t h a t  channel.  three independent  p i o n e n e r g i e s a r e c o n s i s t a n t w i t h one  w i t h i n the s t a t i s t i c a l  can improve  In  another  e r r o r and p i o n e n e r g i e s c a n now  5 % by t h e t i m e o f f l i g h t  be  and  method.  the accuracy of the time o f  of  methods f o r  m e a s u r e d w i t h c o n f i d e n c e t o an a c c u r a c y 2 % by r a n g e  We  range-  independent  p i o n e n e r g y d e f i n e d by t h e magnets o f t h e M9  measuring  calculation  i n the  p i o n channel choose  w i t h t h e same momentum b u t d i f f e r e n t flight  0.2  i t i s b e t t e r t h a n 0.5  the p i o n energy.  c u r r e n t s o f t h e M9  If  flight  -58measurement by u s i n g a s t o p s i g n a l of  from  a counter instead  t h e R.F. s i g n a l o r by u s i n g a c h o p p e d beam w i t h a  bunch.  In t h e range-energy  sharp  d e t e r m i n a t i o n one m i g h t be  a b l e t o improve t h e geometry o f t h e f o u r c o u n t e r s  somewhat.  -59-  references:  1  W.H.  B a r k a s a n d S. v o n F r i e s e n , Nuovo C i m e n t o 1 9 , 4 1  (1961)  2  U.P. 461  Z r e l o v and G.D.  Stoletov, Soviet Physics  R. M a t h e r and E . S e g r e , P h y s . Rev.  4  C . J , B a k k e r and E . S e g r e , P h y s . Rev.  5  H.  Bichsel,  1788  R.F.  M o z l e y and W.A.  84, 191  (1951)  81, 489  (1951)  A r o n , P h y s . Rev. 1 0 5 ,  (1957)  6  E.A.  Uehling,  7  T.D.  Lagerlund,  M.  W.C.  Lam,  I n s t r . and M e t h . 1 2 8 , 5 2 5  Nuclear Science  Nucl.  Blecher,  Mag.  2 5 , 10  N. B o h r , P h i l .  9  H. B e t h e , Ann. P h y s . 7 ^ 5 , 3 2 5 M.S.  L i v i n g s t o n e a n d H.A.  261  (1937)  S e r i e s 29  (1960)  K. Gotow, D. J e n k i n s  8  B e t h e , Rev. Modern P h y s . 9_,  12  S.K.  Allison  779  (1953)  13  E.A.  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