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A polarimeter for spin transfer measurements of the [pi]d[right arrow]pp reaction Feltham, Andrew G. 1988

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A  P O L A R I M E T E R F O R SPIN T R A N S F E R T H E  M E A S U R E M E N T S  ird -» pp R E A C T I O N By  Andrew G . Feltham B.Sc(Hons) Carleton University  1986  A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T THE REQUIREMENTS FOR THE DEGREE MASTER OF  OF  SCIENCE  in THE  FACULTY  OF GRADUATE  STUDIES  D e p a r t m e n t of P h y s i c s  W e a c c e p t t h i s t h e s i s as c o n f o r m i n g to the required standard  T H E  UNIVERSITY OF BRITISH COLUMBIA  ©  September  1988  Andrew  Feltham  OF  OF  In  presenting  degree  this  at the  thesis  in  University of  partial  fulfilment  British Columbia,  freely available for reference and study. copying  of  department  this or  thesis by  for scholarly  his  publication of this thesis  or  her  the  I agree  requirements  for  may  representatives.  It  be is  Department The University of British Columbia Vancouver, Canada  advanced  that the Library shall make it  I further agree that permission  purposes  an  granted  for extensive  by the head  understood  that  for financial gain shall not be allowed without  permission.  DE-6 (2/88)  of  of  my  copying  or  my written  Abstract A p r o t o n p o l a r i m e t e r has been constructed at T R I U M F , w i t h design specifications i n t e n d e d t o m e a s u r e t h e p o l a r i z a t i o n o f p r o t o n s o v e r a n e n e r g y r a n g e o f 100 MeV t o 3 0 0 MeV.  It w a s b u i l t as t h e p r i n c i p l e d e t e c t o r i n a n e x p e r i m e n t t o d e t e r m i n e  t h r e e s p i n - t r a n s f e r p a r a m e t e r s o f t h e f u n d a m e n t a l ird —» pp r e a c t i o n . In this thesis, some theoretical a n d e x p e r i m e n t a l design aspects of t h e s p i n - t r a n s f e r m e a s u r e m e n t are discussed. T h e i n t e n t o f t h i s thesis is t o describe a n experiment  1  w h i c h m e a s u r e s t h e p o l a r i z a t i o n o f p r o t o n s e m i t t e d f r o m t h e nd —• pp  r e a c t i o n , u s i n g a n unpolarized t a r g e t . T h e sole p u r p o s e o f t h i s e x p e r i m e n t i s t o 2  demonstrate that o u r polarimeter a n d general apparatus are capable of identifying t h e nd —> pp e v e n t s f r o m a l a r g e b a c k g r o u n d p r e s e n c e , a n d t h a t t h e s y s t e m a t i c errors associated w i t h the p o l a r i z a t i o n extraction have been identified. T o this extent, t h e s y s t e m is r e a d y t o p r o d u c e t h e p r o t o n p o l a r i z a t i o n r e q u i r e d f o r t h e spin-transfer measurements.  This experiment is identical in all respects to the spin-transfer experiment, except that here, the target is unpolarized. The polarization of the protons is well know from the analyzing power, Ajvo, of the time reversed pp —• dir reaction. 1  2  ii  Table of Contents Abstract  ii  List of Tables  vi  List of Figures  vii  Acknowledgements  viii  I  II  Introduction  1  1.1  Theoretical Background  1  1.2  Goals of E 3 3 1  3  1.3  C h o i c e o f Trd —* pp C h a n n e l  6  1.4  Experimental Design Philosophy  8  1.5  Thesis Summary  10  Polarimeter Theory and Calibration  11  II. 1  Definition of A n a l y z i n g Power a n d Polarization for Spin | Particles  11  11.2  H o w are A n a l y z i n g Powers Measured?  14  11.3  ANO as a C a l i b r a t i o n S o u r c e o f P o l a r i z e d P r o t o n s  15  11.4  Polarization Estimators  18  II. 4.1  21  T h e Acceptance Test  III Description of Apparatus  24  III. 1 O u t l i n e o f t h e E x p e r i m e n t  24  III.2 D e s c r i p t i o n o f E x p e r i m e n t a l C o m p o n e n t s  27  III. 2.1 D e u t e r i u m T a r g e t  27  III.2.2 T h e C a r b o n A n a l y z e r  28 iii  111.2.3 T h e S c i n t i l l a t o r s  28  111.2.4 T h e M u l t i - W i r e D r i f t C h a m b e r s  29  111.2.5 T h e E 3 3 1 E v e n t T r i g g e r  33  111.2.6 T h e J - l l P r e p r o c e s s o r  33  111.2.7 O n l i n e D a t a A c q u i s i t i o n  36  IV Analysis of Data IV. 1 Geometry of the System  38  IV.2  O u t l i n e of the Software Philosophy  40  IV.3  C a l i b r a t i o n o f t h e REPDISK S o f t w a r e  41  IV.3.1  ONLINE  41  I V . 3.2  Offline parameter calibration  IV.4  V  38  wire chamber calibration  Polarization Software  46 47  IV.5 Diagnostics  48  Experimental Results  51  V. l  Performance of A p p a r a t u s  51  V. l . l  T h e Scintillators  51  V.l.2  T h e Wire Chambers  53  V.1.3  Software Definition of a " G o o d " Event  55  V.l.4  Kinematics Calculations  55  V.2  Polarization Results  59  VI Conclusions  65  Bibliography  67  A  Traceback Algebra  69  B  Spin Precession  72 iv  Extraction of Polarization Observables  List o f Tables I  T h e dimensions of the E331 event d e f i n i t i o n s c i n t i l l a t o r s  29  II  C a l c u l a t i o n of essential e x p e r i m e n t angles  40  III  A list of o u t p u t files u s e d for d i a g n o s t i c s  50  IV  A table of the polarizations obtained i n the polarimeter  60  V  A  VI  comparison predicted spin precessed polarizations w i t h those  measured  64  C o u p l i n g c o n s t a n t s as o b t a i n e d u s i n g F I N D A N G  78  vi  List of Figures 1  F e y n m a n d i a g r a m s r e p r e s e n t i n g e l a s t i c a n d i n e l a s t i c NN  2  S p i n - c o r r e l a t i o n o b s e r v a b l e s f o r t h e pp —> dix r e a c t i o n  3  S p i n - t r a n s f e r o b s e r v a b l e f i t s f o r t h e pp —> dw r e a c t i o n  4  Simple model of spin-orbit coupling  process' . .  2 4 5 12  5  A n a l y z i n g power geometry  14  6  A n a l y z i n g p o w e r p l o t s t a k e n f r o m t h e fits o f A p r i l l e - G i b o n i et a l . . .  16  7  A n g u l a r d e f i n i t i o n s o f Ayvo  17  a  n  (  l PN  8  A n g u l a r d e p e n d e n c e o f p p —> drr a n a l y z i n g p o w e r  9  D i a g r a m o f acceptance test. F i g u r e a) contains a cross-sectional v i e w  18  o f t h e a c c e p t a n c e test, b ) c o n t a i n s a n e n d - v i e w o f t h e a c c e p t a n c e t e s t . 2 3 10  T h e layout of the M - l l E x p e r i m e n t a l A r e a for E 3 3 1  25  11  Schematic layout of the polarimeter a r m  26  12  L a y o u t o f a Multi-wire  30  13  E l e c t r o n i c s d i a g r a m of the m a s t e r event trigger  14  T h e effect o f t h e J l l o n t h e d a t a w r i t t e n ( a c c e p t e d ) t o t a p e  35  15  T h e g e o m e t r y w h i c h defines t h e s c a t t e r i n g a n g l e s  39  Drift chamber p l a n e  34  16  T h e REPDISK flow c h a r t  42  17  A flow c h a r t o f t h e s u b r o u t i n e PION  43  18  Check calibrations  45  19  E x a m p l e o f t h e w i r e c h a m b e r combs  46  20  D e p e n d a n c e o f t h e o p e n i n g a n g l e (8  21  F l o w c h a r t o f t h e s o f t w a r e r o u t i n e POLAR  a  22  + 6^) o n 8  48  a  49  a n d TOF d i s t r i b u t i o n s f o r t h e s c i n t i l l a t o r s  52  ax  23  A typical resolution plot  24  T h e x , y a n d z p r o f i l e s a n d RDS o f t h e t r a c e b a c k t o t h e c a r b o n a n a l y z e r . 56  25  54  T h e x , y a n d z p r o f i l e s a n d RDS o f t h e t r a c e b a c k t o t h e d e u t e r a t e d target  26  57  T h e 1-dimensional profiles of the opening angle a n d coplanarity distributions  58  27  T w o dimensional kinematics distribution  61  28  P o l a r i z a t i o n d e p e n d a n c e o n 6c a n d r u n n u m b e r  62  29  P o l a r i z a t i o n d e p e n d a n c e o n 9A a n d  63  30  G e o m e t r y for target traceback  31  Polarization rotation due t o relativistic boost  70  vii  .  74  Acknowledgements  I w o u l d like to acknowledge the m a n y members of the E331 group, w i t h o u t w h o m an experiment of this nature w o u l d not be possible. In a d d i t i o n I w o u l d like to t h a n k Peter Trelle, Marcello P a v a n a n d Peter Weber for their valuable c o n t r i b u t i o n s t o the project a n d i n p a r t i c u l a r G a r t h Jones, w h o guided m e over m a n y h u r d l e s w i t h the project, especially i n the d y i n g days of t h i s thesis. F i n a l l y , I w o u l d like to acknowledge m y fellow physics groupies; M a r k , Chris, J o h n , R e e n a , G i o , J e a n , H o n a n d N i a l l , whose antics d u r i n g the last t w o years, 3  h a v e m a d e u n i v e r s i t y l i f e fun,  a t any h o u r o f t h e day. I r e a l l y h a v e t o t h a n k m y  r o o m m a t e H a r v , w h o h a s p u t u p w i t h a l o t as I p l o w e d t h r o u g h t h i s t h e s i s . T h a n k s everyone!  and of coarse, the list goes on.. vin  Chapter I Introduction 1.1  Theoretical Background  F o r m o r e t h a n t h i r t y years, physicists have s t u d i e d p i o n a b s o r p t i o n a n d p r o d u c t i o n r e a c t i o n s i n a n effort t o c o m p r e h e n d t h e m a n y c o n t r i b u t i n g p r o c e s s e s w i t h i n t h e nucleus. T h e i m p o r t a n c e of t h e pion's role t o t h e u n d e r s t a n d i n g of n u c l e i i s e x e m p l i f i e d b y t h e i t s s i g n i f i c a n t c o n t r i b u t i o n t o t h e n u c l e a r f o r c e s [1]. T h i s research has been accompanied b y extensive works o n nucleon-nucleon a n d p i o n - n u c l e o n (nN)  (NN)  scattering i n order t o provide theorists w i t h adequate  i n f o r m a t i o n o n w h i c h t o base a complete nuclear m o d e l . I n recent years, significant progress has been achieved i n p a r a m e t e r i z i n g e l a s t i c s c a t t e r i n g r e a c t i o n s u s i n g a One Boson Exchange ( O B E ) m o d e l elementary process, s h o w n i n figure  [2].  NN  This  l a , occurs v i a the transfer of a v i r t u a l boson  f r o m one nucleon t o another. U n f o r t u n a t e l y , a s i m i l a r s i m p l e m o d e l does n o t e x i s t f o r p i o n p r o d u c t i o n a n d absorption b y nuclei. A p i o n interacting w i t h a nucleon w i t h i n a nucleus, w i l l f r e q u e n t l y f o r m a n e x c i t e d i s o b a r s u c h as a N* o r A (see figures l b , l c ) ) . T h e c o u p l i n g o f these isobars w i t h t h e secondary nucleons o p e n m a n y c o m p l e x p a t h s b y w h i c h a r e a c t i o n c a n follow. In order t o f u l l y u n d e r s t a n d t h e nucleus, i t is i m p o r t a n t t o u n d e r s t a n d h o w t h e N* a n d A c o u p l e w i t h n u c l e o n s a n d p i o n s . T h i s complex nuclear environment can be greatly simplified through the s t u d y o f p i o n a b s o r p t i o n a n d p r o d u c t i o n b y few n u c l e o n s y s t e m s . T h e m o s t f u n d a m e n t a l o f t h e NNTT ^  NN  r e a c t i o n s i s ird ^ pp, b e c a u s e o f t h e t w o b o d y  1  A/N*  \ N  a)  A/N' v>  b)  Figure 1: a) F e y n m a n diagram for N — N elastic scattering; b), c) p r o d u c t i o n of A or TV* t h r o u g h p i o n absorption.  nature of its i n i t i a l a n d final states. T h i s p a t h also profits by i n v o l v i n g only charged particles i n b o t h entrance a n d exit channels, thus p e r m i t t i n g easy detection experimentally. A t intermediate energies, the pp —> dir reaction forms predominantly an intermediate state [3]. Information on how the ird ^ NA  NA  ^ pp reaction develops  is revealed i n terms of a p a r t i a l wave expansion of the process. Such an expansion i n terms of angular m o m e n t u m and spin components, provides a very useful interface between experiment a n d theory. T h e various contributions of the p a r t i a l waves provide theorists w i t h knowledge of the q u a n t u m numbers which characterize the i m p o r t a n t intermediate states [4]. Because of their angular m o m e n t u m basis, the p a r t i a l wave amplitudes are related to the spin dependent observables of the physical reaction [5]. Despite the fundamental importance of this reaction, it has only been i n the  2  past decade that experimentalists have been able t o measure such spin related parameters. T h e lack o f early progress was p r i m a r i l y d u e t o technological c o n s t r a i n t s , s u c h as l o w b e a m c u r r e n t s a n d d i f f i c u l t i e s i n p r o d u c i n g p o l a r i z e d beams a n d targets. Hence, o n l y i n recent years has there existed a sufficient d a t a base f r o m w h i c h one c o u l d begin t h e d e t e r m i n a t i o n of the various p a r t i a l wave a m p l i t u d e s [6] [7]. T o d a t e , t h e v a s t m a j o r i t y o f d a t a is b a s e d o n s p i n o b s e r v a b l e s  associated  w i t h t h e p r o t o n channels. These p r i m a r i l y consist of the s p i n correlation p a r a m e t e r s , ANN, ^-LL-I ASS, e t c . T h e s e o b s e r v a b l e s c h a r a c t e r i z e t h e s p i n - d e p e n d e n c e o f t h e pp —• dix c r o s s - s e c t i o n f o r v a r i o u s c o n f i g u r a t i o n s o f p r o t o n p o l a r i z a t i o n . A s d e m o n s t r a t e d i n f i g u r e 2, s u c c e s s f u l p a r t i a l w a v e a m p l i t u d e f i t s t o t h i s d a t a c a n b e r e a d i l y o b t a i n e d . U n f o r t u n a t e l y , t h e r e is n o t a u n i q u e s o l u t i o n , as t h e r e i s m o r e t h a n o n e set o f p a r t i a l w a v e a m p l i t u d e s w h i c h f i t t h e d a t a e q u a l l y well. These degenerate solutions lead to significant ambiguities i n t h e predictions of t h e d e u t e r o n s p i n - d e p e n d e n t o b s e r v a b l e s , as s h o w n i n figure 3. It is t h e r e f o r e clear that t h e e x p e r i m e n t a l d e t e r m i n a t i o n of the deuteron  spin-dependent  p a r a m e t e r s is n e c e s s a r y t o p r o v i d e i m p o r t a n t c o n s t r a i n t s o n t h e p o s s i b l e s o l u t i o n s of t h e p a r t i a l wave amplitudes.  1.2  Goals of E331  T h e only significant deuteron spin-dependent data, published t o date, is the m e a s u r e m e n t o f iT  u  a t S I N [8] . I n o r d e r t o fill t h i s v o i d , E 3 3 1 a t T R I U M F w a s 1  d e s i g n e d t o m e a s u r e t h e s p i n - t r a n s f e r o b s e r v a b l e s Kis,  KSS a n d KNN  of the  f u n d a m e n t a l ltd —* pp r e a c t i o n . T h e s e p a r a m e t e r s e v a l u a t e t h e e x c h a n g e o f p o l a r i z a t i o n f r o m a n i n i t i a l s t a t e p a r t i c l e t o o n e i n t h e final s t a t e . I n E 3 3 1 , t h e The vector analyzing power iTu, measures the right/left asymmetry of the reaction due to the deuteron being polarized normal to the reaction plane. This asymmetry is measured in the scattering plane, about the axis defined by the incident pion momentum. 1  4  • i i i i i i ii<  0.40 0.30  0.20 I I  •  0.10  /  \ 0.00  -0.10  1  -0.20  -0.30  -0.40  -0.30  I ' 45.  1 1 1  I 90. M  1 1  I 135.  180.  9" 0.30  K -o.io ss  i  1 1 1 1  45.  i  1 1 1 1  90.  i  1 11  135.  180.  Figure 3: Spin transfer observable fits for the pp —> dw reaction (T = 500 MeV). Note that this corresponds to T = 105 MeV of the wd —> pp reaction. p  v  5  degree o f s p i n transfer f r o m t h e p o l a r i z e d d e u t e r o n i n t h e i n i t i a l state, t o o n e o f the final state protons was measured. T o measure these parameters, three distinct p o l a r i z a t i o n configurations were c o n s i d e r e d . KT,S r e f e r s t o t h e c o n t r i b u t i o n o f l o n g i t u d i n a l l y p o l a r i z e d d e u t e r o n s t o t h e s i d e w a y s p o l a r i z a t i o n o f t h e f i n a l s t a t e p r o t o n . S i m i l a r l y , Kss,  {KNN) r e f e r s t o  the influence of sideways (normal) p o l a r i z a t i o n o f t h e deuteron, o n t h e sideways ( n o r m a l ) p o l a r i z a t i o n o f t h e p r o t o n . I n t h i s a p p l i c a t i o n , longitudinal  polarization  of a p a r t i c l e refers t o i t s c o m p o n e n t p a r a l l e l t o i t s m o m e n t u m i n t h e center o f m a s s s y s t e m a n d normal i s d e n n e d b y t h e n o r m a l t o t h e s c a t t e r i n g p l a n e . Sideways p o l a r i z a t i o n i s t h e c o m p o n e n t o r t h o g o n a l t o t h e normal a n d  longitudinal  c o m p o n e n t s i n c o m p l i a n c e w i t h t h e M a d i s o n C o n v e n t i o n [9]. Since i t has been c l a i m e d that a measurement of these observables w i l l r e s o l v e t h e e x i s t i n g a m b i g u i t i e s o f t h e p a r t i a l w a v e a m p l i t u d e s [10], t h e g o a l o f E 3 3 1 i s t o f i n a l l y p i n d o w n a l l t h e i m p o r t a n t a m p l i t u d e s o f t h e red ^ pp r e a c t i o n . T h i s d a t a w i l l also help t o clarify the c o n t r i b u t i o n s of t h e various i n t e r m e d i a t e s t a t e s o f t h e irNN  ^  NN  reaction, allowing theorists t o develop a more accurate  model of pion absorption a n d production b y nuclei.  1.3  Choice of nd —+ pp Channel  T h e r e are two channels available t o a n experimenter intending t o study t h e Tvd ^ pp r e a c t i o n . T i m e r e v e r s a l i n v a r i a n c e s u g g e s t s t h a t ird —> pp a n d pp —> dir are f u n d a m e n t a l l y t h e same reactions. In other words, one c a n o b t a i n t h e same b a s i c i n f o r m a t i o n f r o m a b s o r p t i o n as p r o d u c t i o n . T h e r e f o r e b o t h o p t i o n s m u s t b e weighed when designing a n experiment. T r a d i t i o n a l l y , m o s t e x p e r i m e n t s h a v e b e e n p e r f o r m e d i n t h e pp —> dn d i r e c t i o n , because f o r m a n y years t h e q u a l i t y o f p r o t o n b e a m s were b y f a r superior to p i o n beams. A s well, protons have always been m u c h easier t o p o l a r i z e t h a n 6  have deuterons, i n b o t h beams a n d targets. Hence the advantage of experience a n d convention favoured this channel. In fact, an experiment i n t e n d i n g to measure t h e s p i n t r a n s f e r p a r a m e t e r ^ 5 5 is c u r r e n t l y u n d e r w a y at T R I U M F [11]. T h i s e x p e r i m e n t is u t i l i z i n g a p o l a r i z e d p r o t o n b e a m a n d a d e u t e r o n p o l a r i m e t e r . A l t h o u g h this experiment benefits f r o m the m u c h higher statistics, resulting f r o m t h e s i g n i f i c a n t l y h i g h e r c u r r e n t p r o t o n b e a m , i t suffers f r o m t w o i m p o r t a n t disadvantages. F i r s t of all, the deuteron polarization contains b o t h vector a n d t e n s o r t e r m s . B o t h t e r m s w i l l c o n t r i b u t e t o t h e s p i n t r a n s f e r , yet t h e i r effects a r e 2  c o u p l e d as f a r as s c a t t e r i n g w i t h i n t h e p o l a r i m e t e r is c o n c e r n e d . T h i s l e a d s t o a significantly m o r e c o m p l i c a t e d analysis . Secondly, a n d m o r e significantly, there 3  exists l i t t l e or n o a n a l y z i n g power i n f o r m a t i o n for a d e u t e r o n p o l a r i m e t e r at the energies i n v o l v e d . C o n v e r s e l y , f o r t h e ltd —> pp r e a c t i o n , t h e r e e x i s t s w e l l e s t a b l i s h e d p r o t o n p o l a r i m e t e r techniques. A s well, successful technology to produce a n d measure p o l a r i z e d d e u t e r o n t a r g e t s h a s r e c e n t l y b e e n d e v e l o p e d at T R I U M F  [12].  D u e t o the difference i n the center of mass energies of the t w o reactions, the -nd —> pp d i r e c t i o n a l l o w s us t o i n v e s t i g a t e t h e r e a c t i o n o v e r a m u c h l a r g e r k i n e m a t i c range t h a n w o u l d be possible w i t h available T R I U M F energies i n the r e v e r s e sense. F o r e x a m p l e , e x p e r i m e n t s c a n b e c a r r i e d o u t o v e r a p i o n k i n e t i c e n e r g y r a n g e o f 105 t o 2 5 5 M e V . T h i s c o r r e s p o n d s t o a n e q u i v a l e n t p r o t o n e n e r g y range of 500 to 800 M e V . S u c h a range allows m e a s u r e m e n t of t h e s p i n transfer o b s e r v a b l e s o v e r t h e r e g i o n of t h e S - w a v e NA MeV  (T  p  « 565 M e V )  r e s o n a n c e , w h i c h p e a k s at T — v  140  [1].  T h e p r i n c i p l e d i s a d v a n t a g e s o f t h e ird —> pp c h a n n e l a r e l o w p i o n f l u x e s a n d  Tensor terms are a result of the fact that the deuteron has a spin of lh. The magnitude of the tensor term is well known for a polarized deuteron target, since it is a direct function of the magnitude of the vector polarization, a quantity which is measured using well established NMR techniques. 2  3  7  o n l y b a r e l y a d e q u a t e d e u t e r o n t a r g e t p o l a r i z a t i o n s ( < 4 0 % ) . A s w e l l t h e r e is a considerable presence of b a c k g r o u n d d u e t o quasi-free d e u t e r o n a b s o r p t i o n o f t h e pions b y other materials i n the region of the target. Fortunately,  background  r e m o v a l t e c h n i q u e s d o e x i s t , as w i l l b e d i s c u s s e d i n m o r e d e t a i l i n t h e n e x t s e c t i o n .  1.4  Experimental Design Philosophy  T h e factors w h i c h influenced the design of this experiment were, p r i m a r i l y t h e p r e s e n c e o f c o m p e t i n g b a c k g r o u n d r e a c t i o n s , t h e l a r g e m a g n e t i c field i n t h e r e g i o n of t h e target a n d t h e requirement of t r a j e c t o r y d e t e r m i n a t i o n w i t h i n t h e p o l a r i m e t e r for t h e e x t r a c t i o n of its observables. M u l t i - w i r e ionization chambers are widely used for charged particle position identification. T h e y are m o r e effective t h a n other types of detectors i n p r o v i d i n g a c c u r a t e p o s i t i o n i n f o r m a t i o n o v e r l a r g e d i s t a n c e s , as is r e q u i r e d b y t h e polarimeter. I n o r d e r t o s e p a r a t e t h e t r u e ird —* pp e v e n t s f r o m t h e b a c k g r o u n d , i t s t w o b o d y i n i t i a l a n d final s t a t e f e a t u r e w a s u s e d . A p i o n i n c i d e n t u p o n a d e u t e r o n a t r e s t w i l l p r o d u c e t w o p r o t o n s w i t h a d i s t i n c t k i n e m a t i c r e l a t i o n s h i p . If o n e i d e n t i f i e s t h e e n e r g y o r a n g l e o f o n e o f t h e final s t a t e p r o t o n s , t h e c o m p l i m e n t a r y p r o t o n must travel at a well defined angle a n d energy w i t h respect t o it. A l s o , conservation of m o m e n t u m requires a l l particles to move i n t h e same plane. O n t h e c o n t r a r y , i f t h e p i o n is a b s o r b e d b y a " q u a s i - f r e e " d e u t e r o n w i t h i n a larger nucleus, t h e amount of energy available to t h e protons w i l l be affected by the F e r m i m o t i o n of the " d e u t e r o n " w i t h i n the nucleus. T h i s w i l l lead to a significant broadening i n t h e energy a n d angle correlations between t h e protons. A n efficient d e t e c t i o n s y s t e m m u s t b e able t o a c c u r a t e l y i d e n t i f y p r o t o n p a i r s having the appropriate kinematic relationship. T h r e e standard techniques are available for this purpose. 8  Time of flight  (TOF)  a n d total energy absorption (TEA)  are two m e t h o d s for i d e n t i f y i n g the  t o t a l energy o f the t w o protons. T h e s u m o f t h e i r energies is a constant for a t r u e nd —• pp r e a c t i o n . T h e use o f TEA w a s r u l e d o u t , s i m p l y b e c a u s e a n y p r o t o n w h i c h was s c a t t e r e d b y t h e c a r b o n analyzer, w o u l d lose m u c h o f its energy before entering the a b s o r p t i o n detector, therefore degrading the system's a b i l i t y t o r e s o l v e energy. It a l s o w o u l d h a v e r e q u i r e d a n e x o r b i t a n t t h i c k n e s s o f s c i n t i l l a t o r t o s t o p t h e s e h i g h e n e r g y p r o t o n s . T h e TOF t e c h n i q u e w o u l d r e q u i r e t h e d e t e c t o r s to be placed at a large distance f r o m the target i n order t o o p t i m i z e the energy resolution. T h i s , however, w o u l d reduce the overall acceptance o f the p o l a r i m e t e r a n d thereby reduce statistics. Such systems, however, w o u l d r e t a i n their energy resolution despite the trajectory distortions introduced b y the target's magnetic field, since the proton's energy is not affected b y this field.  W i t h the t h i r d m e t h o d , one identifies the trajectories o f the t w o protons, u s i n g w i r e chambers, a n d checks the k i n e m a t i c a n g u l a r c o r r e l a t i o n s b e t w e e n t h e m . T h i s p r o c e d u r e a l s o e n s u r e s t h a t t h e p a r t i c l e s a r e c o - p l a n a r . S u c h a m e t h o d does suffer f r o m c o m p l i c a t i o n s r e s u l t i n g f r o m t h e d i s t o r t i o n o f t h e t r a j e c t o r i e s d u e t o the large m a g n e t i c field i n the region o f the target. However, since this field was w e l l k n o w n , i t i s p o s s i b l e t o c o r r e c t f o r i t s affect i n t h e s u b s e q u e n t  analysis.  T h e detector f a c i l i t y at T R I U M F was able t o p r o d u c e h i g h l y accurate, fast M u l t i - W i r e D r i f t C h a m b e r s a t a r e a s o n a b l e cost. S i n c e p o l a r i m e t e r s r e q u i r e p o s i t i o n i n f o r m a t i o n a n y w a y , i t w a s d e c i d e d t o use t h e t r a j e c t o r y t e c h n i q u e . A s a result, two a r m s c o n t a i n i n g w i r e chambers were constructed. T h i s choice allowed us t o p r o f i t f r o m t h e l a r g e r a c c e p t a n c e w h i c h c o u l d b e o b t a i n e d b y l o c a t i n g t h e p o l a r i m e t e r closer t o the target. I n a d d i t i o n , the p o s i t i o n a l i n f o r m a t i o n p r o v e d t o h a v e s i g n i f i c a n t v a l u e i n t h e a n a l y s i s , as s e v e r a l p a r a m e t e r s w e r e f o u n d t o h a v e a t r a j e c t o r y d e p e n d e n c e , (see c h a p t e r 5 )  9  1.5  Thesis  Summary  In the preceding sections, the need for f u r t h e r deuteron  spin-dependent  p a r a m e t e r s o f t h e ird —> pp r e a c t i o n , h a s b e e n d i s c u s s e d a n d a n e x p e r i m e n t ( E 3 3 1 ) , i n t e n d e d t o m e a s u r e t h r e e o f t h e m w a s d e s c r i b e d . It is t h e p u r p o s e o f t h i s thesis to describe the testing a n d c a l i b r a t i o n of the E331 system, b o t h w i t h respect to d e m o n s t r a t i n g irs a b i l i t y to reject b a c k g r o u n d reactions a n d to c o n f i r m s u c c e s s f u l o p e r a t i o n o f t h e p o l a r i m e t e r itself. In this r e g a r d , this thesis describes a n e x p e r i m e n t a i m e d at m e a s u r i n g the p o l a r i z a t i o n o f p r o t o n s p r o d u c e d f r o m t h e ird —> pp r e a c t i o n , a p r o c e s s  involving  a n u n p o l a r i z e d d e u t e r a t e d t a r g e t . S u c h a r e a c t i o n is u s e f u l b e c a u s e t h e 4  p o l a r i z a t i o n o f t h e o u t g o i n g p r o t o n is s t r i c t l y n o r m a l t o t h e s c a t t e r i n g p l a n e a n d e q u a l t o t h e w e l l - k n o w n a n a l y z i n g p o w e r , A^o, o f t h e t i m e r e v e r s e d pp —> dn c h a n n e l . H e n c e , o n e is p r o v i d e d w i t h a s o u r c e o f p r o t o n s w i t h k n o w n p o l a r i z a t i o n . T h i s c a l i b r a t e d s o u r c e c a n t h u s b e u s e d t o c h e c k f o r s y s t e m a t i c effects w h i c h influence the e x t r a c t i o n of the p o l a r i z a t i o n . C h a p t e r 2 of t h i s t h e s i s i n t r o d u c e s t h e d e f i n i t i o n s a n d t h e o r y r e q u i r e d f o r a n u n d e r s t a n d i n g o f t h e p o l a r i m e t e r . I n c h a p t e r 3, a d e s c r i p t i o n o f t h e e x p e r i m e n t a l a p p a r a t u s is p r e s e n t e d . T h i s e x p e r i m e n t a l s e t u p is i d e n t i c a l t o t h a t i n t e n d e d f o r the spin-transfer measurements.  C h a p t e r 4 contains a d e s c r i p t i o n of the software  analysis routines. Finally, chapter 5 summarizes the performance of the apparatus, presents the p o l a r i z a t i o n results a n d deals w i t h the systematics of the experiment.  4  T h e target configuration was identical to that used in the actual spin-transfer measurements. 10  Chapter II Polarimeter Theory and Calibration II. 1 Definition of Analyzing Power and Polarization for Spin ^ Particles I n t h i s s e c t i o n , t h e c o n c e p t s o f polarization  a n d analyzing power f o r s p i n |  p a r t i c l e s a r e d i s c u s s e d . A d e s c r i p t i o n o f p o l a r i z a t i o n i s best p r e c e d e d b y i t s definition. A n ensemble of particles whose spins are arranged i n a non-random f a s h i o n i s s a i d t o b e polarized. F o r s p i n | p a r t i c l e s , p o l a r i z a t i o n i s d e t e r m i n e d f r o m t h e difference between t h e number of particles aligned parallel t o a preferred a x i s (N+) a n d t h o s e d i r e c t e d a n t i - p a r a l l e l  (iV_). T h i s d i f f e r e n c e i s n o r m a l i z e d b y  t h e t o t a l n u m b e r o f p a r t i c l e s i n t h e e n s e m b l e a n d e x p r e s s e d as a p e r c e n t a g e .  p  =^rfi  xl00%  (1)  T h e i d e a o f analyzing power i s u s e f u l i n d e s c r i b i n g t h e s p i n d e p e n d e n c e o f n u c l e a r c r o s s - s e c t i o n s . It i s a w e l l - k n o w n p h e n o m e n o n  [13] t h a t a p o l a r i z e d b e a m  of particles i m p i n g i n g o n a nuclear target, w i l l produce a scattering distribution w h i c h is a z i m u t h a l l y asymmetric. Similarly, a n unpolarized b e a m o f particles s c a t t e r e d o f f n u c l e i w i l l e x p e r i e n c e v a r y i n g degrees o f p o l a r i z a t i o n , i n t h e f i n a l state, w h i c h is a f u n c t i o n o f the scattering angle a n d incident particle m o m e n t u m . B y u t i l i z i n g t h e s e effects, o n e h a s a m e t h o d f o r b o t h m e a s u r i n g a n d p r o d u c i n g beams of polarized particles. These phenomenon are a result of a non-central tensor force w i t h i n t h e nucleus, w h i c h preferentially deflects particles o f a c o m m o n s p i n i n a s i m i l a r d i r e c t i o n . S u c h a f o r c e c a n b e s i m p l y i l l u s t r a t e d as b e i n g a c o n s e q u e n c e o f t h e  11  o n (relative angular momentum) >  (particle momentum)  (common spin direction)  Figure 4: Given a group of spin | particles, with spins pointing put of the page and incident on a nuclear potential, the effect of the spin-orbit coupling will behave in opposite manners depending on which side of the potential the particle passes.  spin-orbit coupling (L • S), between the nucleus and interacting particle. As is seen in figure 4, the sign of the relative angular momentum L = fx p and thus the sign of the spin-orbit potential will be determined by which side of the nucleus the incident particle passes. Hence for particles with 100% polarization, the potential will either be attractive or repulsive, tending to scatter the particles in the same general direction. Since L is always perpendicular to p, any projection of spin in the direction of motion of the particle will not contribute to the tensor coupling effect. For a more complete treatise on this topic, the reader is referred to the text of Goldberger and Watson [14]. Within the nucleus, this tensor coupling is superimposed on a large central force, so that a scattered particle will experience more than just the 12  s p i n - d e p e n d e n t effects. T h u s t h e analyzing power o f a r e a c t i o n i s a n u m b e r  which  relates t h e size o f t h e s p i n dependent c o u p l i n g t o a l l o t h e r forces o f t h e i n t e r a c t i o n . W h e n u s e d i n t h e c o n t e x t o f a p o l a r i m e t e r , t h e c o n c e p t o f analyzing power i s u s e d to describe the p o l a r i z a t i o n measurement efficiency o f a p a r t i c u l a r reaction. Analyzing  power f o r s p i n | p a r t i c l e s i s d e f i n e d m a t h e m a t i c a l l y a s t h e r a t i o o f  t h e s c a t t e r i n g a s y m m e t r y ( i l l u s t r a t e d i n f i g u r e 5 ) t o t h e i n c i d e n t p o l a r i z a t i o n P.  A  (  ,  E  $  )  =  « m  ( 2 )  T h i s q u a n t i t y i s a v e r a g e d o v e r a l l t h e c o l l i s i o n processes, c, t h e a s y m m e t r y , i s a n easily measurable quantity defined b y e —  N  L  andN  R  represent t h e  n u m b e r s c a t t e r e d t o t h e left a n d right o f t h e i n c i d e n t p r o t o n d i r e c t i o n i n t h e r e a c t i o n p l a n e . Left is i n t h e d i r e c t i o n o f t h e c r o s s - p r o d u c t o f t h e i n c i d e n t unscattered vector a n d the n o r m a l to the scattering plane. E q u a t i o n 2 follows directly f r o m the more general expression for t h e spin-dependent cross-section o f polarized spin | particles incident o n a n u n p o l a r i z e d target (or vice-versa).  da (6,<f>) _ do- {6,<t>) p  0  dn  -  dn dn do (6, <f>) [1 + PA(9)} dn 0  (3)  w h e r e n i s d e f i n e d i n e q u a t i o n 6. F r o m t h i s e x p r e s s i o n , o n e c a n o b t a i n t h e analyzing power, dat (6,4.)  MO) =  (4)  Here, t h e a n a l y z i n g power weights the c o n t r i b u t i o n o f the p o l a r i z e d cross-section w i t h respect t o the total cross-section.  13  Analyzer. Polarization G  Counters  Proton  F i g u r e 5: T h e analyzing  power i s m e a s u r e d b y c o m p a r i n g t h e n u m b e r o f p r o t o n s  w h i c h s c a t t e r left a n d r i g h t , f o r a g i v e n i n c i d e n t p o l a r i z a t i o n .  II.2  How are Analyzing Powers Measured?  In a n e x p e r i m e n t a l a p p l i c a t i o n , t h e effective nucleon-nucleus a n a l y z i n g p o w e r is somewhat m o r e complex t h a n t h e niave m o d e l discussed i n t h e previous section. T h e s c a t t e r i n g is a m i x t u r e o f several types o f elastic a n d i n e l a s t i c i n t e r a c t i o n s . E a c h process has i t s o w n a n a l y z i n g p o w e r d u e t o t h e different c o u p l i n g s involved. T h e t o t a l a n a l y z i n g p o w e r i s a w e i g h t e d a v e r a g e o f a l l o f t h e s e effects a n d i s p r i m a r i l y dependent o n incident b e a m energy scattering angle a n d t h e energy acceptance o f the detector. Because of the m a n y complex contributions, the most p r a c t i c a l means of d e t e r m i n i n g a n a n a l y z i n g power is t o m e a s u r e i t e x p e r i m e n t a l l y . T h i s is done b y s c a t t e r i n g a b e a m o f nucleons, w i t h k n o w n p o l a r i z a t i o n o n a desired target. G i v e n a m o n o e n e r g e t i c b e a m , t h e a n a l y z i n g power f o r a p a r t i c u l a r s c a t t e r i n g angle is p r o p o r t i o n a l t o t h e s c a t t e r i n g a s y m m e t r y e a t t h a t a n g l e , as e x p r e s s e d b y e q u a t i o n 2 a n d d e s c r i b e d i n figure 5.  14  A n appropriate target for the measurement of proton polarization is carbon because its principle component,  1 2  C , has zero spin and the nuclear scattering 1  cross-section is reasonably high. The p-C analyzing powers required for E331 have been previously measured over a large range of proton energies by several groups [15] [16]. Their data were fitted by a similar set of angle dependent polynomials with energy dependent coefficients. For our purposes, the fit of Aprille-Giboni et al. [15] was chosen to 2  represent the analyzing powers because their apparatus was quite similar to ours in design and energy acceptance. When designing an experiment, an important consideration is the optimal thickness of analyzer. A thicker target increases the scattering probability, and also acts as a filter to range out breakup type reactions which tend to have a lower analyzing power. O n the other hand there is a greater uncertainty in the analyzing power with a thicker target. This is a result of the increased uncertainty in the energy of the proton due to energy loss straggling and dispersion of the 3  scattering angle due, to the coulombic multiple scattering. In figure 6, plots of the average p — C analyzing power's functional dependance on energy and scattering angle are given. The data is taken from the fits of Aprille-Giboni, who have optimized the carbon thicknesses for different energies [15].  II.3  ANO as a Calibration Source of Polarized Protons  Analyzing power information is also useful when describing the production of polarized beams. In this case, the spin orbit coupling effect, discussed in A spin zero nucleus is free from the extra complexity of spin-spin coupling. It should be noted that there are some errors in this paper [17] which have been identified after communications with the authors. 0ur convention, adopted from Aprille-Giboni, defines the proton energy for which the analyzing power is defined, to be its average value, after the inherent energy loss, at the center of the analyzer. 1  2  3  15  -0.1  H 0  1  100  1  1  1  200 300 400 Proton Energy (MeV)  1  500  h  600  Figure 6: a) Demonstrates the analyzing power dependence on energy. At each energy, the analyzing power has been averaged over all polar angles (6c)- b) Analyzing power distribution for 7 cm thick carbon, where the average proton energy at the analyzer center is 200 MeV.  16  cm  P  p  pp -* dn  TT  P  P  6  cm  7Td  -> pp  F i g u r e 7: A n g u l a r d e f i n i t i o n s f o r ANO a n d PN o f t h e  —• dn a n d 7rd —• p p r e a c t i o n s .  s e c t i o n II. 1, is a p p l i e d i n t h e r e v e r s e sense. P a r i t y arguments, ensure that only n o r m a l l y p o l a r i z e d protons can be p r o d u c e d b y t h e ltd —• p p r e a c t i o n i f a n u n p o l a r i z e d t a r g e t is e m p l o y e d . T i m e r e v e r s a l i n v a r i a n c e i n d i c a t e s t h a t t h e i r p o l a r i z a t i o n , PN, is e q u a l t o t h e w e l l - k n o w n a n a l y z i n g p o w e r , ANO for p p —* dn [18]. I n t h e u s u a l a n g l e c o n v e n t i o n ANO  is  e x p r e s s e d w i t h r e s p e c t t o t h e o u t - g o i n g p i o n , 8„ [19] a n d PN w i t h r e s p e c t t h e f o r w a r d p r o t o n 8j.  T h i s c o n v e n t i o n is d e f i n e d i n figure 7, w i t h |#* | =  i n the  center of mass. ANO  can be represented by the ratio between the polarized a n d unpolarized  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 as d e s c r i b e d b e e q u a t i o n 4. T h e s e c r o s s s e c t i o n s m a y b e fitted  to associated Legendre a n d Legendre polynomials respectively. Using the  m o s t r e c e n t e n e r g y - d e p e n d e n t c o e f f i c i e n t s ( p r o v i d e d b y W a l d e n [20]) c o m p u t e r p r o g r a m y i e l d e d t h e 8* d e p e n d e n t ANO  a simple  d i s t r i b u t i o n at t h e p i o n energies  t y p i c a l l y s t u d i e d i n E331. T h e d i s t r i b u t i o n corresponding to the p i o n energy used 17  .0 0  200  150  50  Figure 8: Plot of Analyzing Power A^ for pp —• dn with T = sponding to TTT = 205 MeV in the time reversed reaction. 0  p  700 MeV corre-  in this thesis (T* = 205 MeV) is presented in figure 8.  II.4  Polarization Estimators  This section, describes how essential quantities are extracted from the scattering distributions. Rewriting equation 3, the scattering distribution of a polarized proton beam incident on a carbon target can be expressed as [15];  I(e,<f>) = I (6)[l + 0  n =  A(8)P -h]  (5)  B  x Pf P, XP> Pi  (6)  where n is normal to the p - C scattering plane, defined by equation 6, P  B  is  the polarization of the incident proton beam, A(9) is the p - C analyzing power, and J (0) is the unpolarized cross-section. The geometry of this system is defined o  18  i n figure 15. T h e C a r t e s i a n c o m p o n e n t s o f t h e p o l a r i z a t i o n a r e  PB  = Px + Py x  (7)  + PJ  y  P i s i g n o r e d s i n c e c o m p o n e n t s i n t h e d i r e c t i o n o f m o m e n t u m d o n o t affect z  t h e s c a t t e r i n g d i s t r i b u t i o n . 8 a n d <f> a r e t h e p o l a r a n d a z i m u t h a l a n g l e s o f t h e scattering. In terms of components,  1(8, <f>) = I (8)[l 0  + A(8)P  y  cos(<j>) + A(6)P sin(<£)]  x Ac(6, <f>)  X  F r o m e q u a t i o n 2, o n e c a n w r i t e e^(8) = A(8)P  y  a n d es(8) = A(8)P , X  (8)  w h e r e e# i s  t h e s c a t t e r i n g a s y m m e t r y p r o j e c t e d o n t o t h e p — C s c a t t e r i n g p l a n e a n d es i s t h e a s y m m e t r y projected onto the orthogonal plane defined b y the initial proton m o m e n t u m pi a n d t h e n o r m a l t o t h e r e a c t i o n p l a n e . B o t h a s y m m e t r i e s a r e m e a s u r e d a b o u t t h e a x i s d e f i n e d b y p~. T h e t e r m Ac(8, (f>) i s a d d e d t o r e f l e c t t h e acceptance of the system. T h i s factor accounts for the instrumental constraints o n the d e t e c t i o n o f t h e s c a t t e r i n g p a r t i c l e s , w h i c h , i f ignored, c o u l d p r o d u c e false a s y m m e t r i e s . F o r e x a m p l e , p a r t i c l e s w h i c h t r a v e l near t h e edge o f t h e detector a r e subjected t o bias since those w h i c h scatter towards the center o f t h e apparatus w i l l b e d e t e c t e d whereas those w h i c h deflect o u t w a r d s m a y n o t . T o simplify the following discussion, the derivations are performed for the s u b s e t o f a l l e v e n t s h a v i n g a c o m m o n s c a t t e r i n g a n g l e , 8k. E x p a n s i o n t o t h e c o m p l e t e d a t a set w i l l b e d e s c r i b e d  subsequently.  O n e c a n c o m p e n s a t e f o r t h e effect o f t h e a c c e p t a n c e f u n c t i o n i f c e r t a i n s y m m e t r y c o n d i t i o n s a r e m e t . F o r a g i v e n 8k, B e s s e t e t a l . [21] h a v e s h o w n t h a t i f t h e a c c e p t a n c e f u n c t i o n Ac(4>) i s s y m m e t r i c w i t h a p e r i o d o f 7 r 4  This is referred to in this thesis as azimuthal symmetry.  19  4  (Ac(<f>) = Ac(<f> + TT)) , the following relations are true 5  ( * f(<f>) cos(<f>)d<f> =Jo e j * f(<f>) cos\<f>)d<f> + e I * f(<t>) sm(<f>) cos(<t>)d4> Jo Jo F f(<f>) sm(<t>)d<f> =Joe F f{4>) sin(<£) cos(<£)<tyJo+ e F f(<f>) sm (<f>)d<j> ( 9 ) Jo r2ir  t7n  r7*  N  s  2  N  s  Where f(<f>) is the <f> dependent distribution of 1(6, <f>) from equation 8. To reflect the finite angular resolution of the experimental apparatus, the continuous variable <f> is subdivided in terms of discrete bins <f>i and the integrals are replaced by the following sums. — ^ c o s ( ^ , ) « j f f((j>)cos(<j>)d<t> o  — £ s i n ( < ^ ) « J f(<t>)s\n(4>)d<t> o  1 £ (<M * jf* fit) ( W cos2  cos2  f(<f>)sm\<f>)d4>  ^£sin (tf,) « 2  f  2w  1  — y^sin(^/)cos(^/) « / f(4>)sm(<t>)cos(4>)d<f> iv , JO Using an obvious matrix notation, equations 9 can be written as; / £,cos(<^) \  /  £,cos (<^)  £,sin(^,)cos(<^) W  2  V Eisin(^/) / ~ \ E;sin(^j)cos(^/)  from which one may solve e = F k  k  £/sin (<£,) 2  \  J \ e  s  /  nn)  B. k  For each 6 , the polarization may be calculated using k  The overall polarization, P , can be estimated by weighting each 6 contribution by k  its variance. Besset et al. [21] have demonstrated that the covariance matrix V(e) is given to a good approximation by the inverse of the Fk matrix. V (e) « Fl  1  k  5  This is ensured by the subroutine AC.TST  described in section II.4.1  20  (12)  and V(P ) k  =  (13)  A(e y k  So  (14) V / i t h Bp and Fp now denned as £/ A(8 )cos(<t>i)  (15)  k  E,A(^)sin(^,) D A ^ c o s k  2  ^ , )  Ei  £; A (^)sin(^,)cos(^)  Now we can write P = Fp B  P  and V ( P ) = F p  A (f?*)sin(<£,)cos(<^) ZiA (6 ) sin (4>,) 2  2  2  2  (16)  k  1  It should be noted that the acceptance function Ac(8,<f>) does not appear i n the calculation of the estimators. Hence, provided the Ac(4>) = Ac(4> + 7r) symmetry applies, no false asymmetries will be introduced by the geometry or efficiencies of the detection apparatus. In order to meet these requirements, one must be sure there are no regional inefficiencies i n the detectors a n d apply an "acceptance test" to insure the azimuthal symmetry of the apparatus.  II.4.1  The Acceptance Test  A s indicated i n the previous section, the acceptance test is required to ensure azimuthal symmetry of the detector. Because of the Cartesian nature of the polarimeter, such a test is performed with respect to the x a n d y components of the wire chambers. T h e geometry and layout of the following discussion are denned in detail in sections  III.l and IV. 1, respectively.  A s illustrated i n figure 9, the proton vector incident upon the carbon, is extrapolated to the last wire chamber (wc 6) of the polarimeter a r m . In order to 6  6  This chamber (WC 6) defines the solid angle of the second reaction (p - C).  21  determine whether an azimuthally symmetric scattering (<f> + 7r) would have been accepted by the system, one inverts the x and y components about the projection on wc6 of the extrapolated unscattered (straight through) vector. One now checks whether these new coordinates lie within the active region of this last chamber. In the situation where an event fails this test, it is rejected from the "good" event sample.  22  b)  stroight through scottered event  symmetrically accepted event  Figure 9: Diagram of acceptance test. Figure a) contains a cross-sectional view of the acceptance test, b) contains an end-view of the acceptance test.  23  Chapter III Description of Apparatus III.l  Outline of the Experiment  T h e experiment was performed i n the M - l l experimental area at T R I U M F (see f i g u r e 1 0 ) . T h e M - l l b e a m - l i n e i s c a p a b l e o f p r o d u c i n g l a r g e p i o n f l u x e s o v e r a g r e a t r a n g e o f energies. A b e a m p r o f i l e , a t t h e t a r g e t , o f 1.5 c m ( F W H M ) i n t h e h o r i z o n t a l a n d 1 c m ( F W H M ) i n t h e v e r t i c a l [22], w i t h a c o r r e s p o n d i n g d i v e r g e n c e o f Ad = ± 0 . 6 7 ° ( h o r z . ) a n d A</> = ± 3 . 2 ° ( v e r t . ) [22] a r e t y p i c a l l y obtained b y M - l l . Such specifications c o m m i t t e d a l l b u t the tails of the b e a m to h i t t h e s m a l l t a r g e t a r e a (see s e c t i o n III.2.1). T h e deuterated b u t a n o l target was m o u n t e d i n the same cryogenic system u s e d f o r t h e p o l a r i z e d t a r g e t ( i n o r d e r t o test t h e b a c k g r o u n d r e j e c t i o n ) a n d w a s l o c a t e d at t h e focus o f t h e p i o n b e a m . T h e protons p r o d u c e d i n t h e r e a c t i o n were i d e n t i f i e d w i t h p l a s t i c s c i n t i l l a t o r s a n d Multi Wire Drift Chambers  (MWDC's)  configured i n t w o arms. T h e forward a r m , ( a r m A ) defined t h e solid angle of t h e detector . T h e relative position of the two arms was governed b y t h e appropriate 1  2-body kinematic relationship between the protons. T h e p o l a r i m e t e r w a s c o n t a i n e d i n a r m A . It c o n s i s t e d o f a set o f t h r e e M W D C ' s o n e i t h e r s i d e o f a c a r b o n a n a l y z e r , as s h o w n i n figure 1 1 . T h e t h r e e chambers d o w n s t r e a m f r o m t h e c a r b o n were significantly larger t h a n t h e f o r w a r d ones, i n order t o t r a c k large angle scatterings f r o m t h e analyzer. S c i n t i l l a t o r s were  Wire chamber 3 defined the solid angle (about 40 mSter.) since it was fully illuminated by the outgoing protons. 1  24  F i g u r e 10: T h e l a y o u t o f t h e M - l l E x p e r i m e n t a l A r e a f o r E 3 3 1 .  p l a c e d a t detection points a t b o t h t h e e n t r a n c e a n d e x i t o f t h e l a r g e c h a m b e r set i n order t o identify particles w h i c h h a d passed f r o m the analyzer through the latter half o f the polarimeter. A r m B w a s u s e d t o i d e n t i f y t h e t r a j e c t o r y o f t h e b a c k w a r d p r o t o n . It contained three small M W D C ' s for positional information. A g a i n , scintillators w e r e s i t u a t e d a t detection points o n b o t h e n d s o f t h i s a r m t o q u i c k l y s i g n a l t h e passage of a charged particle. Several p e r i p h e r a l detectors were present i n t h e area, w h o s e p r i n c i p l e f u n c t i o n w a s t o p r o v i d e b e a m m o n i t o r i n g a n d d i a g n o s t i c s . A s m a l l finger scintillator ( S i ) placed i n the primary beam, provided a measure of the incident p i o n b e a m rate. D o w n s t r e a m f r o m the s c i n t i l l a t o r S i , was a fast w i r e c h a m b e r [23,24]. It p r o v i d e d a h o r i z o n t a l p r o f i l e o f t h e b e a m i n c i d e n t o n t h e target, a n d a l l o w e d one t o check for shifts o r b r o a d e n i n g o f t h e p i o n b e a m . D i s p l a c e m e n t o f t h e b e a m was a l s o m o n i t o r e d b y a d o w n s t r e a m h o d o s c o p e . 25  This  63-cfrJ  \J/  ZU  63 cm  21 cm 21 cm  54" to hgt.  111 cm to tgt.  — >  <  small chamber set  large chamber set  carbon analyzer Positions refer to middle of chambers  Figure 11: Schematic layout of the polarimeter arm.  26  was a four p a d d l e device w h i c h w a s centrally located i n the b e a m , such that a n y shift w o u l d create a relative increase of the count rate i n o n e of the panels. F i n a l l y , t w o sets o f c o u n t e r s w e r e u s e d t o m o n i t o r f l u c t u a t i o n s i n t h e p i o n b e a m ' s i n t e n s i t y . A m u o n c o u n t e r (pi • fa)  w  a  s  placed beside t h e M - l l b e a m pipe  to count decay m u o n s f r o m the p i o n beam. T h e m o n i t o r telescope (MT1  • MT2  • M T 3 ) w a s p l a c e d a w a y f r o m t h e b e a m , a i m e d a t t h e t a r g e t . It  counted a l l charged particles r a d i a t i n g f r o m t h e target. B o t h o f these techniques p r o d u c e d count rates w h i c h were p r o p o r t i o n a l t o the incident p i o n b e a m rate. A l l s i g n a l s p r o d u c e d b y t h e d e t e c t o r s w e r e sent v i a f a s t , l o w a t t e n u a t i o n c o a x i a l c a b l e s t o t h e M - l l c o u n t i n g r o o m w h e r e t h e event l o g i c w a s p r o c e s s e d , a n d EVENT i n f o r m a t i o n was stored o n magnetic tape. T h e d a t a a c q u i s i t i o n w a s p e r f o r m e d w i t h a D i g i t a l E q u i p m e n t PDP-11/34  whose p r i m a r y function was to  w r i t e t h e event d a t a onto m a g n e t i c tape, b u t w h i c h also a n a l y z e d a f r a c t i o n of the events a n d p r e p a r e d histograms f r o m these f o r o n l i n e diagnostics. i n f o r m a t i o n w a s p a s s e d t o t h e c o m p u t e r v i a Analog to Digital  Instrumental  Converters  (ADC)  w h i c h m e a s u r e d t h e electrical charge of the signal pulses f r o m each event a n d Time to Digital  Converters  (TDC), w h i c h m e a s u r e d t i m e i n t e r v a l s b e t w e e n  pulses. T h e s e u n i t s were s i t u a t e d i n a CAMAC crate, w h i c h w a s i n t e r f a c e d t o t h e PDP-11  t h r o u g h a C A E Starburst J - l l p r e p r o c e s s o r .  III.2  Description of Experimental Components  III.2.1  Deuterium Target  T h e target used i n this experiment consisted of 1 m m diameter beads of d e u t e r a t e d b u t a n o l (C4.D10O), 5 % D 2 O , d o p e d w i t h 6 x 1 0 d e u t e r a t e d EHBA  - Cr  v  1 9  molecules/mL of  complex. T h e beads were h e l d i n a teflon ( F E P )  c o n t a i n e r w h o s e d i m e n s i o n s w e r e 18 x 2 2 x 6 mm? (2.4 c m ) d u r i n g t h e J a n u a r y 3  27  1987 r u n a n d 2 0 x 2 0 x 10 m m (4.0 c m ) f o r t h e M a y / J u n e 1 9 8 7 r u n [25]. 3  3  T h e t a r g e t w a s s u r r o u n d e d b y l i q u i d He a n d He w h i c h w a s c o o l e d b y a 3  4  d i l u t i o n r e f r i g e r a t o r t o a f e w h u n d r e d m i l l i k e l v i n . T w o sets o f s u p e r c o n d u c t i n g h e l m h o l t z c o i l s p r o d u c e d a 2.5 T m a g n e t i c f i e l d t o d e f i n e t h e a x i s o f p o l a r i z a t i o n for E 3 3 1 , the spin-transfer experiment. T h e field was m a i n t a i n e d d u r i n g the c o a r s e o f t h i s e x p e r i m e n t i n o r d e r t o s i m u l a t e , as c l o s e l y as p o s s i b l e , t h e conditions of E 3 3 1 .  111.2.2  The Carbon Analyzer  T h e c a r b o n analyzer consisted of layered graphite of dimensions 7 x 30 x 30 c m a n d d e n s i t y 1.7 g/cm . 3  3  T h e layers of v a r y i n g thicknesses provide t h e freedom t o  change the overall analyzer depth, although this feature was not employed i n o u r experiment. T h e analyzer was laterally positioned o n the a r m A cart, such that its c e n t e r w a s c o l l i n e a r w i t h t h e c e n t e r s o f t h e w i r e c h a m b e r s o n b o t h sides.  Along  the central axis of the polarimeter, the carbon was located equidistant between the t h i r d a n d fourth chambers.  111.2.3  The Scintillators  T h e c o m m o n p l a s t i c scintillators used i n t h e e x p e r i m e n t , p r o v i d e d fast logic s i g n a l s f o r t h e m a s t e r t r i g g e r , as w e l l as e n e r g y d e p o s i t i o n (dE/dx) flight  a n d time of  (TOF) i n f o r m a t i o n needed for particle identification. T h e large scintillators of a r m A h a d p h o t o m u l t i p l i e r s p l a c e d at b o t h their  u p p e r a n d lower ends. T h i s increased their light collection efficiency, a n d r e d u c e d position-dependent t i m i n g errors b y averaging b o t h t o p a n d b o t t o m times w i t h a m e a n t i m e r . T h e scintillators of a r m B were m u c h s m a l l e r a n d hence o n l y r e q u i r e d single ended light collection. T h e dimensions of t h e scintillators are listed i n t a b l e I. 28  T a b l e I: T h e s c i n t i l l a t o r d i m e n s i o n s  Detector A r m A  Position  Width x Height x thickness  18 x x 18 x 18 x  forward backward  B  36  forward backward  34 70 35 35  x x x x  cm  3  0.4 0.7 0.4 0.4  P a i r s of scintillators were p l a c e d at each d e t e c t i o n p o i n t . B y r e d u c i n g their i n d i v i d u a l sizes, o n e c o u l d i n c r e a s e d e t e c t o r e f f i c i e n c y . A r e d u n d a n t s c i n t i l l a t o r w a s p o s i t i o n e d c e n t r a l l y a f t e r t h e l a s t s c i n t i l l a t o r p a i r o f a r m A . It  (S5A)  p r o v i d e d a u s e f u l c h e c k o f t h e r e l a t i v e efficiencies o f t h e p o l a r i m e t e r a r m scintillator pairs, since a difference i n efficiency at this p o i n t c o u l d i n t r o d u c e artificial scattering asymmetries.  III.2.4 Multi-Wire  T h e Multi-Wire Drift  Chambers  Drift Chambers ( M W D C ' s ) were u s e d t o p r o v i d e p o s i t i o n a l  i n f o r m a t i o n for charged particles passing through the apparatus. These detectors use i n f o r m a t i o n f r o m b o t h t h e anodes a n d cathodes t o d e t e r m i n e t h e coordinates of a track. Discrete nearest w i r e positions are o b t a i n e d d i r e c t l y f r o m t h e anodes w h i c h are situated 8 m m apart. In addition, t h e electrons p r o d u c e d b y t h e i o n i z i n g particles have a highly position sensitive drift time. T h e c o m b i n a t i o n of the t w o pieces o f i n f o r m a t i o n p r o v i d e a p o s i t i o n a c c u r a c y o f t h e o r d e r o f 500 m i c r o n s [26]. F u r t h e r d e t a i l s o f t h e i r o p e r a t i n g c h a r a c t e r i s t i c s w i l l b e c o n t a i n e d i n t h e t h e s i s o f P a v a n [26]. E l e v e n chambers were c o n s t r u c t e d b y the T R I U M F detector f a c i l i t y f o r t h e E 3 3 1 g r o u p . D u r i n g t h e e x p e r i m e n t s i x s m a l l chambers o f a c t i v e a r e a 30 x 30 c m , 2  a n d three large chambers o f a c t i v e a r e a 60 x 60 c m were e m p l o y e d . A single spare 2  c h a m b e r o f e a c h size w a s a v a i l a b l e i n case o f b r e a k d o w n . E a c h c h a m b e r 29  consisted  Cathode Bus  X  EVEN  RIGHT  Anode Delay Line  F i g u r e 12:  T h e layout of a single p l a n e of a M W D C . T h e cathodes are r e a d out  o n one side t h r o u g h the  ODD/EVEN b u s , a n d t h e a n o d e s f r o m t h e RIGHT a n d LEFT  sides o f t h e a n o d e d e l a y l i n e .  of two independent planes of wires, oriented orthogonally, to p r o v i d e b o t h x a n d y i n f o r m a t i o n . A d i a g r a m o f a t y p i c a l p l a n e is p r o v i d e d i n f i g u r e  12.  A plane contained alternating anode a n d cathode wires, whose centroids w e r e s p a c e d 4 mm  apart. T h e anode wires were a l l c o n n e c t e d at a c o m m o n e n d to  a delay line, w h i c h was r e a d out at b o t h ends. T h e c a t h o d e wires were a l t e r n a t e l y attached to  ODD a n d EVEN busses.  A signal induced on an anode w i r e  2  w i l l s p l i t a n d t r a v e l t o w a r d s b o t h e n d s of  t h e d e l a y l i n e . T h e t i m e s o f a r r i v a l o f t h e s e s i g n a l s a t t h e r i g h t a n d left  TDC  channels d e p e n d o n a n u m b e r of p a r a m e t e r s . 3  tR = td + nd  a  +  tk  R  The reader is referred to Sauli [27] for more information on the detection mechanism of gas chambers. The START being provided from the master trigger. 2  3  30  t  L  Where t  d  = t + (N -n)d d  + t  a  (17)  kL  is the c o m m o n drift time i n the chambers, n is the anode wire number,  N is the t o t a l n u m b e r o f wires, d is the a m o u n t o f delay per w i r e p r o v i d e d b y the a  a n o d e d e l a y l i n e , a n d t (t ) kR  kl  is the respective delay constant associated w i t h each  line a n d associated electronics. T h e wire p o s i t i o n can b e o b t a i n e d b y c a l c u l a t i n g the difference between i # and  tR-t  L  = U + nd + t a  -t  kR  d  -(N  - n)d a  t  = 2d n + (t +t ) a  + Nd s/ constant  kR  kL  kh  (18)  a  T h i s q u a n t i t y i s a l i n e a r f u n c t i o n o f t h e w i r e n u m b e r n. T h e drift t i m e is calculated f r o m the s u m of  tfi + t  L  = t + nd + t d  a  kR  a n d ti-  + t + (N - n)d + t d  a  = 2t + (t +t ) d  N  kR  kL  v  kL  (19)  + Nd  a  constant=k [ anoC  '  e  T h i s q u a n t i t y (twice the drift t i m e plus a constant) is independent o f the w i r e n u m b e r , n. U n f o r t u n a t e l y , there is a n inherent a m b i g u i t y associated w i t h the drift t i m e i n f o r m a t i o n , as i t i s n o t k n o w n o n w h i c h s i d e o f t h e a n o d e w i r e t h e e v e n t h a d passed. T o resolve this ambiguity, the a m p l i t u d e s o f the cathode analog signals are measured. It h a s been shown  ODD a n d EVEN  [28] t h a t t h e s i g n a l i n d u c e d o n  t h e c a t h o d e w h i c h i s closest t o t h e e v e n t w i l l b e 10 — 15 % l a r g e r t h a n t h a t i n d u c e d o n the next nearest c a t h o d e ( o n the other side o f the anode). Hence the c o r r e c t s i d e i s d e t e r m i n e d b y c a l c u l a t i n g t h e d i f f e r e n c e i n a m p l i t u d e o f t h e ODD and  EVEN s i g n a l s , (ODD  - g•  (ODD  + g • EVEN) 31  EVEN) K  '  W h e r e g is t h e r e l a t i v e g a i n m a t c h i n g c o n s t a n t f o r t h e O D D a n d E V E N c h a n n e l s . T h e d i f f e r e n c e is n o r m a l i z e d b y t h e s u m t o r e m o v e e n e r g y d e p e n d e n c e . F i n a l l y , t h e checksum is a p a r a m e t e r w h i c h c a n b e u s e d t o i d e n t i f y e v e n t s w i t h b a d c h a m b e r i n f o r m a t i o n o r m u l t i p l e h i t s . E s s e n t i a l l y , o n e is c o m p a r i n g t h e drift t i m e measurements o b t a i n e d f r o m the cathodes a n d anodes. T h e cathode d r i f t t i m e is m e a s u r e d b y t h e t i m e o f a r r i v a l o f t h e O D D s i g n a l . E l e m e n t s w h i c h c o n t r i b u t e t o t h e ODD t i m e are,  to = tj. + xd  c  w h e r e td is a g a i n t h e d r i f t t i m e , d  c  +  (21)  t  ko  is t h e d e l a y p e r u n i t d i s t a n c e o f t h e c a t h o d e  b u s , x is t h e p o s i t i o n a l o n g t h e a n o d e b u s , a n d tk  Q  is t h e d e l a y a s s o c i a t e d w i t h t h e  O D D s i g n a l c a b l e s a n d a s s o c i a t e d e l e c t r o n i c s . T h e xd  c o m b i n a t i o n is a  c  n o n - n e g l i g i b l e d e l a y - l i n e effect i n t r o d u c e d b y t h e c a t h o d e b u s c o n s t r u c t i o n . U s i n g the k n o w n position i n f o r m a t i o n f r o m equation  18, o n e c a n c o r r e c t f o r t h i s effect.  T h e v a l u e r e m a i n i n g is s i m p l y t h e d r i f t t i m e p l u s a c o n s t a n t . A  "good"  checksum  w i l l be characterized by a constant value. E r r o n e o u s i n f o r m a t i o n i n c l u d i n g multiple hits w i l l lead to incompatible drift time measurements, such that e q u a t i o n 22 w i l l d e v i a t e f r o m t h i s c o n s t a n t .  checksum  — £R + ti — C(to — 2 t j - j - k de ano  — (tp, — Ctd  ti)d ) c  C'k th,ode ca  (22)  w h e r e C is t h e r e l a t i v e c a l i b r a t i o n o f t h e c a t h o d e d r i f t s c a l e t o t h e a n o d e s c a l e , a n d k hode cat  is t h e c o n s t a n t t i m e d e l a y s a s s o c i a t e d w i t h t h e c a t h o d e c h a n n e l . 32  111.2.5  The E331 Event Trigger  T h e m a s t e r event trigger's p r i m a r y p u r p o s e is t o i d e n t i f y those events f o r w h i c h a particle has passed through b o t h detector arms, i n coincidence. T h i s provides a n e f f e c t i v e f i r s t - o r d e r b a c k g r o u n d r e m o v a l . It t h u s d e f i n e s t h e c o m m o n START f o r these events, w i t h respect t o w h i c h a l l t i m i n g is c o m p a r e d . I n o r d e r t o d e f i n e a n EVENT, c o i n c i d e n c e s m u s t h a v e b e e n o b s e r v e d b y o n e o f the scintillators at all four detection points o n the detector carts.  START  = (SIA  v  + S2A) • (S3A + SAA + S5A) • (SIB  <v  v  + S2B)  A  • (S3B  ^ B  +  SAB)  .  (23) where t h e a r m A t i m i n g signals are generated b y m e a n timers connected t o t h e top a n d b o t t o m photomultipliers of the respective scintillators. T h e o v e r a l l t i m i n g w a s d e f i n e d b y t h e (SIA  + 5 2 A ) signal since t h e protons  t r a v e r s i n g a r m A w e r e o f h i g h e r e n e r g y a n d t h e r e f o r e w e r e less a f f e c t e d b y t h e t i m e v a r i a t i o n s d u e t o e n e r g y loss. T h e r e l a t i v e t i m i n g o f t h e p u l s e s a n d t h e l o g i c d e f i n i t i o n s a r e g i v e n i n figure 13.  111.2.6  The J - l l Preprocessor  A very important component of our data acquisition system was the Creative E l e c t r o n i c s S y s t e m s ( C E S ) Starburst F a s t P r o c e s s o r . T h i s d e v i c e w a s a s m a l l c o m p u t e r w h i c h s a t as a m o d u l e i n t h e C A M A C c r a t e . T h e J - l l C P U o f t h e s t a r b u r s t h a d a n a r c h i t e c t u r e v e r y s i m i l a r t o t h a t o f a D i g i t a l P D P - 1 1 . It w a s c a p a b l e o f u s i n g t h e C A M A C b u s as a n e x t e n s i o n o f i t s m e m o r y a n d t h e r e b y h a d e a s y access t o d a t a c h a n n e l s o f t h e o t h e r m o d u l e s i n t h e c r a t e . T h i s f e a t u r e a l l o w e d i t t o p e r f o r m r a p i d calculations o n t h e d a t a , a n d m a k e fast decisions f o r each event. Since t h e p — C analyzing power, resulting f r o m the nuclear interaction, 33  SIB S2B  S3B S4B  S1A S2A  S3A S4A S5A  Figure 13: Master EVENT trigger logic. A EVENT and in particular (S1A + S2A) is delayed so that it defines the START time.  vanishes for small polar angle scattering (see figure 6), whereas the electromagnetic cross-section is strongly peaked at 0°, only events which scatter more than 5 or 6 degrees are of any value. This reduces the useful data to merely a few percent of all the good ird  pp events. To reduce the number of useless  events being written to tape, the J - l l was used to calculate the x and y scattering angle projections, and select only those which had surpassed a predetermined minimum deflection. For every master event trigger, the J - l l would calculate changes in the x and y slopes from the position information available from wire chambers 1,3,4,6, using equation 24, where l — 420 mm, which is the distance c  between the outer chambers of each set. In this expression, the small-angle approximation, 6 « tan 8 was used. — (wc6x — wcAx — {wclx —  6  X  8  y  wc3x))/l  = (wc6y — wcAy — [wcly — wc3y))/l  34  c  c  (24)  J—11 Filter  OFF  QJN  o-l.  -200200  ::BS:  1)0  (T  Figure 14: The x and y projected scattering angles for accepted events with and without the J - l l filter.  35  It w a s t h e n r e q u i r e d t h a t t h e x o r y s l o p e c h a n g e d b y a t l e a s t 6° i n o r d e r t o accept the event. F o r such a " g o o d event," the J - l l w o u l d i n d i c a t e to the P D P - 1 1 t h a t t h e e v e n t w a s t o b e r e a d . T h e effect o f t h e J - l l f i l t e r o n t h e a c c e p t e d d a t a is d e m o n s t r a t e d i n f i g u r e 14.  III.2.7  T h e STAR  Online Data  Acquisition  system o f G . S m i t h [29] w a s c h o s e n as t h e d a t a a c q u i s i t i o n r o u t i n e . It  is a P D P - 1 1 s o f t w a r e p a c k a g e w h i c h d o e s s e v e r a l t a s k s . T h e s e i n c l u d e r e a d i n g t h e modules i n the C A M A C crate, p e r f o r m i n g calculations, p r o d u c i n g selected histograms a n d w r i t i n g the raw d a t a to magnetic tape. T h i s system benefits f r o m its simplicity a n d  flexibility.  It is e a s i l y a d a p t a b l e t o v a r i o u s e x p e r i m e n t a l  configurations. A m i n i m u m of " o n l i n e " calculations were performed o n the data, d u r i n g the a c q u i s i t i o n , as i t w a s i m p o r t a n t t o r e d u c e c o m p u t e r d e a d t i m e . T h e  philosophy  a d o p t e d was to ensure t h a t the basic components of the a p p a r a t u s were f u n c t i o n i n g p r o p e r l y , a n d save h i g h e r l e v e l c a l c u l a t i o n s f o r t h e o f f l i n e a n a l y s i s . T h e essential c o m p u t a t i o n s p e r f o r m e d online allowed one to m o n i t o r the progress o f t h e r u n a n d i d e n t i f y s y s t e m a t i c effects a n d f a i l i n g s as t h e y o c c u r r e d .  Important  calculations included;  • w i r e p o s i t i o n ( c o m b s ) , a n d c h e c k s u m s , t o i n d i c a t e t h e p e r f o r m a n c e q u a l i t y of the chambers.  • t h e J - l l x a n d y s c a t t e r i n g angles, t o ensure t h a t the cuts i n the J - l l h a d n o t d r i f t e d s i g n i f i c a n t l y f r o m t h e c e n t e r (see figure 14 a) ).  • T h e i n - b e a m fast w i r e c h a m b e r p o s i t i o n i n f o r m a t i o n , to check t h a t t h e p i o n b e a m h a d not shifted f r o m the target. 36  • the ^ distributions of the scintillators, to verify that their gains were set properly. • the ODD, EVEN and ODD-EVEN distributions to identify possible problems with the wire chamber cathodes and operating voltages. A similar version of STAR has been adapted for the VAX for the purpose of offline analysis. This is discussed in the next chapter.  37  Chapter IV Analysis of Data IV. 1 Geometry of the System T h e basic g e o m e t r y used t o describe t h e reactions i n t h e offline analysis follows t h e c o n v e n t i o n o f s e v e r a l a u t h o r s [30,31,32,21]. It is I l l u s t r a t e d i n f i g u r e 15. T w o angles are used t o describe t h e relationship between t h e scattered a n d incident p r o t o n vectors. T h e s e angles are defined w i t h respect t o a reference s y s t e m , i n w h i c h t h e i n c i d e n t p a r t i c l e m o m e n t u m defines t h e z-axis, a n d t h e o r t h o g o n a l y - a x i s is  fixed.  S i n c e t h i s e x p e r i m e n t as w e l l as E 3 3 1 w e r e d u a l s c a t t e r i n g e x p e r i m e n t s , i t w a s n e c e s s a r y t o d e f i n e t w o s u c h g e o m e t r i e s . I n t h e i n i t i a l nd —> pp r e a c t i o n , t h e y-axis (y) was chosen t o be vertical i n t h e l a b frame, w h i c h was a central value f o r the n o r m a l t o t h e scattering plane. T h i s axis was well defined a n d orthogonal t o the incident p i o n beam. F o r the p — C reaction, the y-axis was chosen to b e n o r m a l t o t h e s c a t t e r i n g p l a n e o f t h e first r e a c t i o n ( e q u a t i o n 6) (nd), a n d w a s c o n s i s t e n t w i t h t h e M a d i s o n C o n v e n t i o n [9]. T h e p o l a r a n g l e QQ r e p r e s e n t s t h e a n g l e b e t w e e n t h e z a x i s a n d t h e m o m e n t u m v e c t o r o f t h e o u t g o i n g p a r t i c l e . T h e a z i m u t h a l a n g l e <f>c d e n o t e s t h e angle between the previously defined y-axis a n d the n o r m a l to t h e scattering plane  (n). T o a v o i d using a basis that whose d e f i n i t i o n w o u l d d e p e n d o n the k i n e m a t i c details of e a c h event considered, t h e c o m p o n e n t s of a l l vectors i n t h e offline a n a l y s i s w e r e e x p r e s s e d i n t e r m s o f a c o o r d i n a t e s y s t e m , fixed t o t h e l a b o r a t o r y . 38  Figure 15: The geometry which defines the scattering angles  The z-axis was defined by the central axis of the wire chambers of arm A , and the 1  y component was vertical in the lab frame. The x component was defined by the standard right hand coordinate system. This system was chosen for its simplicity, since the trajectory vectors could thereby be calculated directly from the chamber information. The system was free from the complexities and errors introduced by extra transformations required by a trajectory dependent coordinate frame. In the analysis, the incoming pion beam direction, and the arm B trajectories were all transformed into the arm A basis. This allowed for straightforward calculation of the kinematic and traceback quantities. Table II illustrates the scattering angle calculations in terms of the vectors defined in the offline coordinate system. The p^s (k = 7r,p,,py) represent the momentum vectors (trajectories) of the particles, y is the vertical axis in the lab frame, and hj, is the normal of the ltd —> pp reaction plane. These formulae define J  T h i s is described in the chamber calibration section (IV.3.1).  39  T a b l e II: C a l c u l a t i o n o f e s s e n t i a l e x p e r i m e n t  o  -(^) - es  c d  Oc  <f>d  s i g n o f <f>  angles.  <f>c  cos(^c)-  s i g n o f <f>c  d  H d  X  (i^xft,;,)  (lfi..-xft,,|)  •PP.  the f u n d a m e n t a l variables u t i l i z e d i n the software.  IV.2  O u t l i n e of the Software P h i l o s o p h y  T h e software i n h e r i t e d b y the E331 g r o u p was a n ensemble of subroutines w r i t t e n by several authors. T h e u l t i m a t e goal was to produce a coherent analysis  package  designed specifically for the needs of E 3 3 1 , yet l e a v i n g it flexible e n o u g h t h a t the routines m a y be easily u t i l i z e d b y other experiments u t i l i z i n g the polarimeter. O t h e r r e q u i r e m e n t s were: u s e r f r i e n d l y , c o m p u t e r e f f i c i e n c y , a n d p r o v i s i o n o f a wide range of diagnostics available for trouble shooting. In order to s i m p l i f y the analysis, the process was d i v i d e d into two separate programs. T h e first  (REPDISK) r e a d a l l t h e e v e n t s f r o m t a p e a n d , b y a p p l y i n g a  series o f c a l c u l a t i o n s a n d c u t s , r e d u c e d t h e d a t a t o a s u b s e t o f " g o o d "  events,  w h e r e a " g o o d " event is defined i n section V . l . 3 . T h e s e events were a n a l y z e d b y the second program  (POLAR), w h i c h c a l c u l a t e d t h e p r o t o n p o l a r i z a t i o n f r o m t h e  s c a t t e r i n g a n g l e i n f o r m a t i o n , repdisk i s d e s c r i b e d i n t h e r e m a i n d e r o f t h i s s e c t i o n and the next, and  POLAR i s d e s c r i b e d i n s e c t i o n I V . 4 .  REPDISK w a s a v e r y g e n e r a l r o u t i n e w h i c h c o u l d b e a p p l i c a b l e t o m a n y e x p e r i m e n t a l s i t u a t i o n s . It s e r v e d t o i n i t i a l i z e r u n p a r a m e t e r s , d e f i n e h i s t o g r a m s , test files, t a k e c a r e o f b o o k - k e e p i n g a n d o u t p u t v a r i o u s o u t p u t d i a g n o s t i c files (see section IV.5). It o r c h e s t r a t e d the r e a d i n g of the event buffers f r o m t a p e o r disk 40  a n d t r e a t e d each according t o buffer t y p e . Different subroutines were called for 2  s c a l e r , h e a d e r , t r u e e v e n t , e n d o f f i l e etc., b u f f e r s . F o r e a c h r e a l e v e n t b u f f e r , REPDISK w o u l d c a l l t h e s u b r o u t i n e PION, w h i c h o r g a n i z e d a l l t h e c a l c u l a t i o n s p e r f o r m e d o n t h e d a t a . T h e f i n a l selected d a t a were then binned according to the histograms denned during the initialization. T h e flow o f REPDISK i s o u t l i n e d i n f i g u r e 16. T h e l o g i c a n d flow o f PION is i l l u s t r a t e d i n f i g u r e 17. F o r those wishing t o use t h e polarimeter i n other e x p e r i m e n t a l configurations, t h e p o l a r i m e t e r routines are m o s t l y self-sufficient a n d independent of o t h e r a p p a r a t u s . N e c e s s a r y c h a n g e s t o t h e s o f t w a r e f o r a d i f f e r e n t 3  e x p e r i m e n t a l s i t u a t i o n w i l l g e n e r a l l y b e m a d e i n t h e s u b r o u t i n e PION, s i n c e e a c h e x p e r i m e n t w i l l i n general require some different analysis routines. D u e t o t h e m o d u l a r n a t u r e o f PION, i t is s i m p l y n e c e s s a r y t o c a l l o r r e m o v e d i f f e r e n t calculation subroutines where required.  IV.3  C a l i b r a t i o n o f t h e REPDISK  IV.3.1  ONLINE  Software  wire chamber calibration  In this section, a s u m m a r y of the online wire chamber c a l i b r a t i o n procedures are p r e s e n t e d . F o r e x p e d i e n c y , o n l y c o a r s e w i r e r e s o l u t i o n ( ± 4 mm)  was required b y  the online routines. T h e c a l i b r a t i o n used for the chambers d u r i n g offline analysis, a l t h o u g h s i m i l a r w a s m u c h m o r e precise, offering a r e s o l u t i o n of a n order of m a g n i t u d e b e t t e r t h a n t h e online system. T h e offline technique w i l l be discussed i n t h e t h e s i s o f P a v a n [26]. T h e r e a r e several steps i n v o l v e d i n t h e c a l i b r a t i o n software. T h e order i n A buffer is a data structure in which the events were stored on magnetic tape. True events were placed sequentially in a buffer, until full. For other types of information (scaler info., header, etc) the remainder of the buffer was left empty. Each buffer type was assigned an identification code. These routines are available from the author upon request. 2  3  41  Initialize Repdisk Read Define  Read Scalers  in  ONLIN.SET Drift table histos/dot plots  R e a d E v e n t Buffer (from  disk)  7Tv Scaler Parity  Buffery  Error  3= 67  End  Normal of Run  \/  Save s c a l e r info  Proton? Scint.  \/ Save test  Cuts , Yes  \/  results  c a l l PION  \/ Save h i s t o info  \/ Bin Histos  \/  Write good event parameters to AMAT###.DAT Pass  info  to POLAR  \/ ->(END  B i n Dot P l o t s  Figure 16: The REPDISK flow chart  42  No  c a l l BOXSTUFE-appties 2 - d cuts to data  c a l l C B L K 2 : calculates combs, checksums and ODD-EVENs  \/  No  c a l l TSTBLK(2): applies cuts to checksums  Fes  \/  c a l l AC_TST: checks scattering symetry in detector  c a l l WCPOS: calculates drift position and adds to wire position  \/ c a l l CODE_CHAM: suff. good WC combinations  \/  c a l l TSTBLK(3):checfcs if event came from target  Yes  c a l l TRAJECTORY: fits slopes and intercepts from WC positions  c a l l CARBONTRBK: traces back to origin of scat, in carbon  No  t-H  Q  CU W  No  cc; o  Yes  d  \/ c a l l XYZTRBK trace back to deuteron tgt.  \/ c a l l TRAJ FIT: refits trajectories to include target as origin  u 0  c a l l C_ANG calculates scattering angles in analyzer  +-> CU  u  c  CD >  No  Tl CQ  c a l l TSTBLK(5): checks if event scat'd more than req'd minimum Yes  \/ c a l l C O P L : calculates and scat. kinematics angles of 1st react.  c a l l TSTBLK(4):cheefcs  if good traceback to carbon,  (^RETIJRNJ^'  c a l l AMAT : writes good evnt param. to disk  F i g u r e 17: A flow chart of the subroutine PION.  43  w h i c h they are performed is i m p o r t a n t , since many of the calibrations depend on earlier results. Initially, one must calibrate the i n d i v i d u a l chamber planes, a n d i n the end, their relative positions are adjusted i n software so that the coordinates of every trajectory are defined consistently i n a set of three chambers. T h e first i t e m for calibration is the checksum, since it defines a l l the good chamber events for the later calculations. T h e checksums require a two step c a l i b r a t i o n process. T h e constant C f r o m equation 22 relates the T D C scales of the ODD cathode t o the anode drift time measurements. C was obtained by p l o t t i n g ODD cathode T D C vs anode sumtime (equation 19) (see figure 18 a) ). In order to compensate for the delay line effect of the cathode bus, a position dependent correction, d , (see equation 22) was included. T h i s constant was c  obtained b y p l o t t i n g the checksum vs uncalibrated difftime (t^ — tn). T h i s procedure is demonstrated i n figure 18 b). To o b t a i n the position scale factor, it was necessary to produce a plot of the wire "combs" (figure 19) for each plane. T h e physical distance between each wire was k n o w n , so the relation between the d a t a bins a n d the distance is given as  scale factor  =  (data bin of right wire — data bin of left (n  wire)  — 1) • 8mm  where n is the number of wire peaks, a n d 8 mm is the physical distance between the centers of the anode wires. T h e next step was to o b t a i n a relative position c a l i b r a t i o n of the chambers. A straight line f r o m the target origin, through the m i d d l e of chamber three also passes t h r o u g h the centers of a l l other chambers of a r m A . T h i s designated the central z-axis. In software, a cut was placed a r o u n d the central two wires of chamber three . T h e required offset for each chamber was then obtained f r o m the 4  Chamber three was chosen because it was the only chamber which was fully illuminated.  4  44  -1500 -1000 -500 0 500 1000 1500 Chamber Position  Figure 18: Calculation of relative TDC and the position dependent calibration constants for the checksum.  45  1000  u i -150  i -100  i -50  Difftime  r 0  (mm)  r  50  (wire  i  100  i 150  comb)  Figure 19: Example of the wire chamber combs.  event distribution centroid of histograms of trajectories which had passed through the window on chamber three. This offset would compensate for any physical misalignment of the chambers, by defining an effective center in software. This condition was also used to constrain the centroids of arm B. In this case, the central axis of arm B is defined in terms of the angular correlation between the nd —»• pp events which defined the arm A z-axis, and those traversing arm B.  IV.3.2  Offline parameter calibration  For the offline analysis, in addition to the wire chamber information, two physical parameters were necessary to complete the set of software calibrations. The first was an estimate of the distance from the target to the middle of the closest wire chamber on each arm. This distance was difficult to measure physically because of the large vessel walls surrounding the target. The procedure used was the standard traceback routine, described in appendix A. The origin (target position) 46  was defined as the centroid of the distance of closest approach to where all trajectories passed closest to the central axis of the arm. T h i s calculation was also done for a r m B before its components were transformed into the a r m A frame. T h e second parameter involved the correction necessary for the nd —> pp opening angle dependence o n the scattering angle of the forward proton. T h i s dependance was a physical effect resulting from the kinematics of the reaction. T h e correction was obtained by plotting 8 + 6b ys 6 , as i n figure 20. A s a result, a  a  the opening angle was redefined as 6 + 8 = 8 + 8 + 6 • corr . Such a correction a  0  a  0  a  corresponded well with the actual kinematic dependence, which is easily calculated. T h e coplanarity was defined as the angle between the backward going proton a n d the normal to the scattering plane of the first reaction (defined b y p^ x pj).  It  was found that its width d i d not significantly depend o n any kinematic quantities and hence d i d not require any calibrations.  IV.4  Polarization Software  T h e p r o g r a m POLAR was used to analyze the subset of "good" events defined by the REPLAY program. POLAR read from a file A M A T # # # . D A T (produced by REPDISK) which contained, for each event;  • the scattering angles of the deuteron reaction, 8^, <f>d  • the scattering angles of the carbon reaction, 6 , <j) C  c  • the kinematic parameters, 6 + 6b a n d coplanarity a  In POLAR the carbon scattering angles were binned i n multi-dimensional arrays with respect to the variables contained i n A M A T # # # . D A T , with the intention of enabling investigation of the dependence of the polarization o n these 47  200 216 232 248 264 280 296 312 328 344 360 6 (tenths of degrees)  Figure 20: Dependance of the opening angle (8 + 8^) on 8 . This slope is obtained simply by hand. a  parameters. For example, for each bin of the 2-dimensional  8 , <f> C  c  <f>  a  d  distribution, there corresponds a  array. From arrays, the polarization could thus be calculated  as a function of <j>^. The polarizations, were calculated using techniques described in section II.4. By creating different subsets in POLAR, of  8c  and  <j>c  related arrays, the  polarization is evaluated as a function of the available parameters. The flow diagram for POLAR is illustrated in figure 21.  IV.5  Diagnostics  In order to easily identify systematic effects, both online and offline, a series of diagnostics were produced for each run. These are given in table III.  48  Enter run  parameters;  - f i r s t a n d last r u n n u m b e r s —minimum  scattering  angle i n c a r b o n  —2—D p o s i t i o n of peak i n 0„+0 —radius  b  vs Copl. dist.  of f o r e g r o u n d / b a c k g r o u n d  separation  i  Zero all arrays  V Read i n Event  >  1 get  the  Distinguish the foreground from the background  next event  Bin  V  a r r a y s a c c o r d i n g to;  f ( r u n no.,S ,0 ,status) C  C  S t a t u s is f l a g  f(0 ,e ,0 ,status) d  c  c  for  f(0 ,0 ,0 ,status) d  e  background  c  etc. no more events  \U  Calculate  p o l a r i z a t i o n as a  of t h e p a r a m e t e r s b i n n e d  to  function above  call SUMS(k,e ,0 ,status) o  o  read  8 and 0 C  O  all polarizatior. calculations  k is a f i x e d p a r a m e t e r w h i c h r e p r e s e n t s the p a r t i c u l a r b i n f o r w h i c h t h e p o l a r i z a t i o n is b e i n g e v a l u a t e d  Figure 21: Flow chart of the software routine  49  are  required for  POLAR.  Table III: A list of output files used for diagnostics  header  files  test files  scalar  RUN###.REP  TST###.REP  files  SCL###.REP  histogram files R ^ ^ ^ . H S T  logs a series of run queues from the offline analysis, errors due to opening a file etc. are written in here. contains a list of all the tests applied in the offline analysis, they are accompanied by the number of times each event was true, and its relative efficiency normalized to a given test. contains a list of all the final hardware scalar totals for each run. contains the histogram information used by the STAR system histogram routine. The following histograms are stored for each run: 6 + 9b, coplanarity, Be, carbon traceback, carbon R D S distribution, resolution plots for six chamber sets, the x, y J - l l angles, and the x, y, z profiles of the target traceback and its RDS distribution. a  scatter plots  RUN###.DOT  contains the dot plot information used by DOT. The following dot plots are stored for each run: kinematics and target traceback scatter plots.  scalar  DIA###.DAT  contains the incremental scalar values as a function of time. The following scalar information is stored for each run: counting rates of the beam, monitor telescope, and muon counters, the up/down and left/right hodoscopes and the clock counter. The scalers were read every 5 minutes.  files  50  Chapter V Experimental Results V.l V.l.l  Performance of Apparatus The Scintillators  T h e scintillators' performance was evaluated by how well the protons of the true ird —> pp e v e n t s c o u l d b e s e p a r a t e d f r o m o t h e r r e a c t i o n s . A s w e l l as t h e q u a s i - ird —• pp r e a c t i o n s , s i g n i f i c a n t b a c k g r o u n d w a s c o n t r i b u t e d b y q u a s i - f r e e TTN scattering where the resulting protons h a d a wide range of generally lower e n e r g i e s . F i g u r e 22 a ) s h o w s d i s t r i b u t i o n s o f e n e r g y d e p o s i t e d i n t y p i c a l s c i n t i l l a t o r s o f a r m s A a n d B . T h e l a r g e r p e a k is d u e t o p r o t o n s . T o i t s left is t h e residual u p p e r e n d of the m i n i m u m i o n i z i n g p i o n peak. T h e b u l k of this peak has been r e m o v e d b y t h e h a r d w a r e d i s c r i m i n a t o r t h r e s h o l d levels. T h e events t o t h e r i g h t o f t h e p r o t o n p e a k s a r e i n g e n e r a l t h e q u a s i - f r e e TTN p r o t o n s . T h e  ^  d i s t r i b u t i o n s i n t h e scintillators p r o v i d e d a first level b a c k g r o u n d separation. T h e mono-energetic protons produced a well defined peak w h i c h p e r m i t t e d a simple, yet effective one d i m e n s i o n a l c u t . T h e T O F d i s t r i b u t i o n s w e r e n o t as u s e f u l f o r b a c k g r o u n d r e m o v a l as t h e  ^  i n f o r m a t i o n . T h i s w a s a r e s u l t o f t h e s h o r t flight p a t h s b e t w e e n t h e t a r g e t a n d scintillators. However, since t h e b a c k g r o u n d separation w a s n o t c r i t i c a l l y d e p e n d e n t o n t h e TOF i n f o r m a t i o n , t h e s e r e s u l t s w e r e q u i t e a c c e p t a b l e . O n l y loose c u t s w e r e a p p l i e d t o t h i s d i s t r i b u t i o n , a n e x a m p l e o f w h i c h is s h o w n i n figure  22 b ) , b e c a u s e o f t h e p o o r r e s o l u t i o n . T O F i n f o r m a t i o n w a s o n l y a v a i l a b l e  51  Figure 22: Examples of the ^ and T O F distributions of the scintillators. Note the events in the zeroeth channel correspond to those which passed through the partner scintillator.  52  f r o m a r m B, because a r m A was u s e d to define the S T A R T .  V.1.2  The Wire Chambers  In the analysis, the w i r e chambers were s t r i c t l y used for v e c t o r c o n s t r u c t i o n . H e n c e it was i m p o r t a n t to k n o w h o w well the slope a n d i n t e r c e p t of a t r a j e c t o r y c o u l d be determined. T h e self-consistency of the slope c a l c u l a t i o n was e s t i m a t e d u s i n g e q u a t i o n 26, t h e r e s o l u t i o n c a l c u l a t i o n . T h i s e x p r e s s i o n e n a b l e d c o m p a r i s o n o f t h e expected position on a chamber (calculated on the basis of the p o s i t i o n i n the adjacent chambers) w i t h the actual position i n that chamber. T h e resolution plot, s h o w n i n f i g u r e 23 w a s p r o d u c e d u s i n g e v e n t s f o r w h i c h a l l c h a m b e r s  had  successful c h e c k s u m s a n d w i t h t h e c e n t r a l c h a m b e r h a v i n g t h e e x t r a c o n d i t i o n of good ODD/EVEN information  WC3 /y X  resolution  =  —  -  WCl /y x  — — wc2 / x  y  (26)  T h i s d i s t r i b u t i o n was a sensitive i n d i c a t o r of m i s c a l i b r a t i o n s of the software. If t h e r e l a t i v e offset o r g a i n o f o n e o f t h e c h a m b e r s h a d c h a n g e d , a s h i f t o r broadening i n the resolution peak would result, For a well calibrated system, the w i d t h s o f s u c h r e s o l u t i o n p e a k w e r e t y p i c a l l y 700 microns v a l u e o f less t h a n 3 0 0  F W H M , with a mean  microns.  T h e t r a c e b a c k s t o t h e s c a t t e r i n g o r i g i n at t h e c a r b o n ( d e u t e r i u m ) w e r e a r e f l e c t i o n o f t h e c o n s i s t e n c y b e t w e e n t h e sets ( a r m s ) o f c h a m b e r s . S i n c e , i n general, two a r b i t r a r y vectors do not intersect, the p o s i t i o n of m i n i m u m distance between t h e m defined the " o r i g i n " of the traceback. T h e m a g n i t u d e of this distance i n d i c a t e d how well the vectors agreed w i t h each other. T h i s distance  For such events the middle chamber had a well defined position because drift information was definitely included (since O D D / E V E N had good information). 1  53  2500  Front of arm A, y-planes  20001500 c  D O  o 1000  500-  -60  -40 -20 0 20 40 Resolution Plot (tenths of millimeters)  60  F i g u r e 23: A t y p i c a l r e s o l u t i o n p l o t .  d e n o t e d b y Root Difference  Squared  (RDS)  was calculated i n the s t a n d a r d  t r a c e b a c k r o u t i n e w h i c h is d e s c r i b e d i n a p p e n d i x A . O f the t w o target tracebacks, the deuteron's R D S was significantly broader, since it was e x t r a p o l a t e d over a m u c h greater distance, a n d w a s d i s t o r t e d b y t h e m a g n e t i c f i e l d s u r r o u n d i n g t h e t a r g e t . A c u t o n t h i s d i s t a n c e o f 5 0 mm  for the  d e u t e r o n a n d 10 m m f o r t h e c a r b o n w a s f o u n d t o b e a p p r o p r i a t e v a l u e s a t w h i c h t o r e j e c t p o o r d a t a , as seen b y f i g u r e s 24 a n d 25. B o t h t a r g e t t r a c e b a c k s p r o d u c e d g o o d d i s t r i b u t i o n s (see figures 24 a n d 25) about the regions where the targets were expected. Hence a cut o n the d a t a whose t r a c e b a c k i n d i c a t e d t h a t t h e events h a d o c c u r r e d o u t s i d e t h e target r e g i o n w o u l d e l i m i n a t e those events w i t h p o o r t r a c e b a c k i n f o r m a t i o n , a n d those w h i c h h a d t r u l y been scattered b y a n external target. T h e large angle between a r m s A a n d B (about 165°), r e s u l t e d i n t h e p o o r r e s o l u t i o n of t h e t a r g e t ' s z - c o m p o n e n t .  This  w i d t h w a s g r e a t l y i m p r o v e d b y c o n s t r a i n i n g t h e o r i g i n t o l i e close t o t h e axis of 54  the pion beam. In s u m m a r y , the tests a p p l i e d to the t r a j e c t o r i e s e n s u r e d t h a t the tracebacks w e r e k n o w n t o a r e s o l u t i o n o f a b o u t 2 cm.  W i t h these cuts a b o u t | of the d a t a  was r e m o v e d at the d e u t e r o n traceback a n d a b o u t h a l f of t h e d a t a tested i n the c a r b o n traceback was rejected.  V.1.3 A  Software Definition of a " G o o d "  Event  " g o o d " event h a d to satisfy the f o l l o w i n g c r i t e r i a i n o r d e r to be r e t a i n e d for  polarization analysis i n  POLAR.  1. p r o v i d e c o i n c i d e n c e e v e n t i n b o t h a r m s , i n d i c a t i n g a 2 - b o d y final s t a t e . 2. l i e w i t h i n t h e ^  a n d TOF  cuts imposed by the scintillators, thus being  i d e n t i f i e d as p r o t o n s i n e a c h a r m . 3. h a v e a t l e a s t 2 o f 3 w i r e c h a m b e r s p r o v i d e g o o d e v e n t s f o r b o t h x a n d y p l a n e s i n e a c h o f t h e 3 sets. 4. t r a j e c t o r i e s m u s t e n a b l e g o o d q u a l i t y t r a c e b a c k s t o t h e s c a t t e r i n g o r i g i n at the d e u t e r a t e d target a n d c a r b o n analyzer. 5. m u s t p a s s t h e a c c e p t a n c e test d e s c r i b e d i n s e c t i o n II.4.1. 6. m u s t s c a t t e r i n t h e c a r b o n b y 9 ° , o r m o r e . 2  A b o u t 5 % o f a l l e v e n t s w r i t t e n t o t a p e q u a l i f i e d as " g o o d "  V.1.4  Kinematics  events.  Calculations  T h e q u a s i - f r e e nd —» pp b a c k g r o u n d w a s t h e m o s t d i f f i c u l t t o d i s t i n g u i s h f r o m t h e f o r e g r o u n d s i n c e t h e final s t a t e p a r t i c l e s w e r e t h e s a m e . H o w e v e r , figure 26 s h o w s t h a t t h e t r u e nd —> pp s i g n a l w a s w e l l s e p a r a t e d f r o m t h e m u c h  flatter  background.  A 9° cut was applied in the offline analysis in order to not be affected by the cuts introduced by the J - l l in the acquisition (as seen in figure 24). 2  55  n  1  r  -200-160-120-80 -40 0 40 80 120 160 200 Carbon Traceback; x-component (mm)  -200-160-120-80 -40 0 40 80 120 160 200 Carbon Traceback; z-component (mm)  i  1  r  200-160-120-80 -40 0 40 80 120 160 200 Carbon Traceback; y-component (mm)  5  10 15 20 25 30 35 40 45 50 Carbon Traceback; RDS (mm)  ure 24: The x, y and z profiles and RDS of the trace back to the carbon analyzer.  56  5000  8000  4000  6000  ,3000 54000 J  2000H 2000  1000  -100-80 - 6 0 - 4 0 - 2 0 0  20 40 60 80 100  Target Traceback; x-component (mm)  -100-80 - 6 0 - 4 0 - 2 0  0  20 40 60 80 100  Target Traceback; y-component (mm)  Figure 25: The x, y and z profiles and RDS of the trace back to the deuterated target.  57  16001400 1200 -  1425 1475 1525 1575 1625 1675 1725 1775 1825 c? + 8 (tenths of degrees) o  b  1000  Coplanarity (tenths of degrees)  Figure 26: The 1-dimensional profiles of the opening angle and coplanarity distributions.  58  Since t h e opening angle of the t w o final state protons w a s a f u n c t i o n of p i o n energy, i t s r e s o l u t i o n was l i m i t e d b y t h e u n c e r t a i n t y o f the p i o n b e a m m o m e n t u m . T h e c o p l a n a r i t y , t o f i r s t o r d e r , w a s i n d e p e n d e n t o f energy, s i n c e a l l ird —> pp r e a c t i o n s m u s t b e c o p l a n a r , r e g a r d l e s s o f t h e a v a i l a b l e energy. It w a s t h e uncertainty of trajectories which broadened this function. T h e p i o n beam, whose divergence was not monitored, contributed most significantly t o the w i d t h of the c o p l a n a r i t y d i s t r i b u t i o n . I t s d i v e r g e n c e i n t h e y - d i r e c t i o n a t t h e t a r g e t [22] i s a b o u t 5° as c o m p a r e d w i t h o n l y 1° i n t h e x - d i r e c t i o n . S e p a r a t i o n o f t h e b a c k g r o u n d f r o m t h e f o r e g r o u n d is o b t a i n e d b y p l a c i n g a t w o d i m e n s i o n a l c u t a r o u n d t h e k i n e m a t i c s p e a k , as seen i n f i g u r e 2 7 . T h e a m o u n t o f b a c k g r o u n d r e m a i n i n g u n d e r t h e m a i n p e a k i s a b o u t 10 % . S i n c e t h e r e m a i n i n g 3  b a c k g r o u n d has satisfied t h e p r o t o n ^  a n d TOF c u t s o f t h e s c i n t i l l a t o r s o f b o t h  a r m s , a n d a r e b r o a d l y accepted b y t h e d u a l a r m coincidence, these events are a t t r i b u t e d t o t h e q u a s i - ltd —• pp r e a c t i o n . F o r t h e p u r p o s e s o f t h i s t h e s i s , n o f u r t h e r e f f o r t w a s m a d e t o r e m o v e t h e b a c k g r o u n d . T h e effect o f a 10 % b a c k g r o u n d o n t h e r e a l vd —> pp p o l a r i z a t i o n s i g n a l w a s d i s r e g a r d e d a t t h i s p o i n t , b u t f o r E 3 3 1 , i t is a n t i c i p a t e d t h a t c o r r e c t i o n s w i l l b e a p p l i e d t o e l i m i n a t e t h e background's influence.  V.2  Polarization Results  T h e p o l a r i z a t i o n results presented f o r this thesis were m e a s u r e d w i t h t h e deuterons i n a n u n p o l a r i z e d state. T h e k i n e t i c energy of the incident pions w a s 205 MeV  a n d t h e p r o t o n s w e r e d e t e c t e d a t a c e n t r a l a n g l e o f 27° ( l a b ) w i t h a n  acceptance of ± 4°. T h e p o l a r i z a t i o n of the foreground a n d b a c k g r o u n d are presented i n t a b l e I V . T h e b a c k g r o u n d p o l a r i z a t i o n is o b t a i n e d b y m e a s u r i n g t h e In an identical experimental set-up, we measured the kinematics distribution for an undeuterated butanol target. Although the measurement was only at the nearby pion energy of 205 MeV, the shape was similar to the tails of the background seen in this experiment. 3  59  T a b l e I V : P r o t o n p o l a r i z a t i o n s as m e a s u r e d b y t h e p o l a r i m e t e r .  Measured Polarizations  Predicted (%)  Polarizations (from  (%)  A ) N 0  Foreground; N o r m a l component Sideways  component  36.5  39  ± 2 4  10.8  0  ± 2 5  Background; N o r m a l component Sideways  component  25.2 n  0  -  ± 3 6  ±3.5  p o l a r i z a t i o n o f t h e e v e n t s o u t s i d e o f t h e c u t s h o w n i n figure 27. T h e e r r o r s q u o t e d a r e p u r e l y s t a t i s t i c a l , a n d a r e c a l c u l a t e d u s i n g t h e f o r m u l a e o f s e c t i o n II.4.  A s is  seen i n t a b l e I V , t h e n o r m a l p o l a r i z a t i o n is c o n s i s t e n t w i t h t h a t p r e d i c t e d b y t h e ANQ fits ( a l t h o u g h s l i g h t l y l o w ) . M o r e s i g n i f i c a n t is t h e c o n s i s t e n t n o n - z e r o sideways p o l a r i z a t i o n for b o t h the foreground a n d b a c k g r o u n d events. A s  was  d i s c u s s e d i n s e c t i o n II.3, p r o t o n s p r o d u c e d f r o m u n p o l a r i z e d d e u t e r i u m (nd —> pp), should contain only a normal polarization component. S e a r c h e s f o r s y s t e m a t i c effects, p r e s e n t e d i n figures 28 a n d 29, v e r i f i e d t h a t t h e p o l a r i m e t e r w a s p e r f o r m i n g as e x p e c t e d . T h e i n d e p e n d e n c e o f p o l a r i z a t i o n as a f u n c t i o n o f c a r b o n s c a t t e r i n g a n g l e (9c),  a n d r u n n u m b e r is w e l l c o n f i r m e d .  F i g u r e 29 i l l u s t r a t e s t h e d e p e n d e n c e o f t h e p o l a r i z a t i o n o n 9  d  varies r a p i d l y w i t h 9 , d  a n d (f>d. S i n c e  ANO  s u c h a d e p e n d e n c e o f PN, is n o t u n e x p e c t e d . H o w e v e r ,  because of the large error bars w i t h this data, no further analysis was a t t e m p t e d . T h e s y s t e m a t i c effect w h i c h c o u l d e x p l a i n t h e s i g n i f i c a n t s i d e w a y s p o l a r i z a t i o n observed by the p o l a r i m e t e r , was the precession of the p r o t o n ' s s p i n d u e t o t h e l a r g e 2.5 T m a g n e t i c field i n t h e r e g i o n o f t h e t a r g e t . W i t h s o m e r o u g h c a l c u l a t i o n s , i t w a s q u i c k l y s h o w n t h a t t h i s effect w a s s i g n i f i c a n t . A m o d e l was developed to p r e d i c t the n a t u r e of the s p i n precession. 60  computer  105 100-  95O  c  90-  D CL  0  u  8580-  75 150  155  160  165  170  175  F i g u r e 27: 2 - d i m e n s i o n a l d i s t r i b u t i o n o f t h e k i n e m a t i c s . T h e c i r c l e r e p r e s e n t s t h e designated cut w h i c h distinguished the foreground f r o m the background.  R e s u l t s f r o m t h i s c o m p u t e r m o d e l , w h i c h is d e s c r i b e d i n a p p e n d i x C , s h o w e d t h a t there was a significant dependance of the s p i n precession o n the p a r t i c l e ' s trajectory. T h i s was due to the v a r y i n g a m o u n t a n d r e l a t i v e d i r e c t i o n of the field e x p e r i e n c e d b y the s p i n c o m p o n e n t s over different trajectories. A l s o the i n i t i a l p o l a r i z a t i o n c o m p o n e n t s were e x p e c t e d to be a f u n c t i o n of s c a t t e r i n g angle 9  a  (the  f o r w a r d s c a t t e r i n g angle of the p r o t o n ) . T h e r e f o r e it was not a s i m p l e m a t t e r to d i r e c t l y o b t a i n t h e final p o l a r i z a t i o n f o r a l l t h e e v e n t s i n t h e p o l a r i m e t e r ' s acceptance. T h e proceedure developed to account for this was the following. T h e total p r e c e s s i o n w a s c a l c u l a t e d f o r a set o f e v e n l y s p a c e d t r a j e c t o r i e s , w h i c h s p a n n e d the acceptance of the detector. T h e s e were weighted by the r e l a t i v e n u m b e r of e v e n t s w h i c h f o l l o w e d e a c h p a t h f r o m t h e e x p e r i m e n t a l d a t a set. T h e w e i g h t e d s u m t h e n r e p r e s e n t e d t h e average p o l a r i z a t i o n m e a s u r e d b y t h e p o l a r i m e t e r .  61  100 -  100  75"  75  50-  50"  25  25  o-25" -50 -75  44-  0 -25P = 0.365  -50 -75  Foreground Normal Polarization P 8  6  t 10  I 12  I 14  I 16  I 18  20  H  P_= 0.108 Foreground Sideways Polarization  10  12  14  16  18  20  e  100"  I  I  I  I  I I  75" 50"  i ii  25"  [  0 -25-50  -25  t0 . 0 2 4  P = 0.365  -50  N  -75 -100  ti  -75  Foreground Normal Polarization  i i i  11  0  1  ±0 . 0 2 5  P = 0.108 s Foreground Sideways Polarization  -100 295 296297 298 299300 301302303304305 Run Number  —i—i—i—i—i—i—i—T  295 296 297 298 299 300 301302 303304305 Run Number  Figure 28: Polarization dependance on 9c and run number.  62  I  100  I  I  I  I  I  I  I I  100  I  I  I  I  I  I  I  I I  75"  75 'o  50"  50  i  25  i  25' f  -  0  rizatic  c -25  ± 0.024  Foreground Normal Polarization  oton  CL  -75  -  i  f 1  -25-  o o -50-  -  P = 0.365  -50  o-  1  t  ± 0.025 P = 0.108 s  -75"  Foreground Sideways Polarization  -100  I 1  CL  -100 0  I I I I 1 I I I I 1 2 3 4 5 6 7 8 9 10  I I I I 2 3 4 5  I  I I I 7 8 9 10  I  I  e  100  i  i  i  i  i  i  i  i i  100  75"  75  50  50  25  25  o-  0  -25"  -25  -50 -75"  P = 0.365  I  L_  -50  N  -75  Foreground Normal Polarization  -i—i—i—i—i—i—r - i o o i5—-r4 - 3 - 2 - 1 0 1 2 3 4  _i  -100 5  ure 29: Polarization dependance on  63  Foreground Sideways Polarization i i i i 5 -4 -3 -2 -1 0  I 1  i 2  i  r  3  4  5  and <j>d. Note the units are arbitrary.  comparison of polarizations of those predicted b y the program FINDANG and those measured b y the polarimeter.  Table V : A  Measured  Predicted  Polarizations (%)  Polarizations (%)  Normal polarization  36.5  Sideways polarization  10.8 "  ± 2 4  ± 2  5  36.5 12.6  I n o u r case w h e r e t h e p r o t o n i n i t i a l l y h a d a p u r e n o r m a l p o l a r i z a t i o n , i t i s quite straight forward t o calculate t h e s p i n precession o f the p r o t o n t o t h e analyzer. A l t h o u g h the polarimeter only measured the t w o components orthogonal to the particle's direction of motion, one could unambiguously  compare  the polarization measured experimentally w i t h that predicted b y t h e precession c a l c u l a t i o n (see a p p e n d i x C ) . F i n a l l y , table V compares t h e predicted p o l a r i z a t i o n components w i t h those m e a s u r e d b y t h e p o l a r i m e t e r . It i s s e e n t h a t t h e s p i n p r e c e s s i o n  successfully  a c c o u n t s f o r t h e p o l a r i z a t i o n seen b y t h e p o l a r i m e t e r t o w e l l w i t h i n e x p e r i m e n t a l error.  64  Chapter VI Conclusions T h i s thesis has demonstrated the successful operation of the polarimeter detector apparatus, designed to measure spin-transfer observables of the ird —* pp rection. It has been demonstrated that the scintillators are effective i n removing p i o n events f r o m the event sample. A standard has been defined w i t h respect to which the wire chamber trajectory information can be evaluated. T h e resolution plot determines the self-consistency of a set of wire chambers a n d acts as a check of the software position calibration. T h e Root Difference Squared (RDS) is a check of consistence amongst the sets of wire chambers. T h e quantity is also a check of the traceback to the target origin, which allows one to reject events whose trajectories are not likely to have a c o m m o n origin. W i t h respect to background separation, it has been demonstrated that our system c a n eliminate a l l b u t about 10 % of the background which lies under the true event kinematics peak. P r o t o n s p i n precession i n the target magnetic field has been identified as the principle systematic effect i n our polarimeter system. T h o u g h rigorous numerical calculations w i t h a computer, we have been able to reproduce the results observed by the polarimeter, after p r o v i d i n g a n i n i t i a l p r o t o n p o l a r i z a t i o n equal to A^ of 0  the identical time reversed reaction. T h i s successful comparison indicates that the A p r i l l e - G i b o n i analyzing powers used i n the p o l a r i z a t i o n extraction are appropriate for our system, i n p a r t i c u l a r for T = 200 MeV (the t y p i c a l p r o t o n p  energy obtained i n this experiment). In this regard, i t w o u l d be useful to repeat 65  t h i s e x p e r i m e n t f o r a l l e n e r g i e s at w h i c h t h e s p i n - t r a n s f e r m e a s u r e m e n t w i l l b e p e r f o r m e d , i n o r d e r to v e r i f y t h e usefulness of these a n a l y z i n g powers over t h e entire range. F i n a l l y , it has now been established t h a t a v a l i d d a t a analysis package exists for use i n f u r t h e r p o l a r i m e t e r experiments.  66  Bibliography [1] D. A s h e r y a n d J . P . S c h i f f e r . Ann. Rev. Nucl. Part. Sci., 36:207, 1 9 8 6 . [2] R. M a c h e i d t . T h e m e s o n t h e o r y o f n u c l e a r f o r c e s a n d n u c l e a r m a t t e r . 1 9 8 5 . Lectures presented at t h e W o r k s h o p o n R e l a t i v i s t i c D y n a m i c s a n d Quark-Nuclear  Physics.  [3] D . V . B u g g . Ann. Rev. Nucl. Part. Sci., 35:295-320, 1 9 8 5 . [4] M . B e t z et a l . T h e o r i e s o f p i o n p r o d u c t i o n i n n u c l e o n - n u c l e o n c o l l i s i o n s . Invited paper presented at the Workshop o n P i o n P r o d u c t i o n a n d A b s o r p t i o n in Nuclei. [5] B . B l a n k l e i d e r a n d L R . A f n a n . Phys. Rev. C, 31:1380, 1 9 8 5 . [6] D . V . B u g g . Nucl. Phys., A416:227, 1984. [7] D . V . B u g g . Nucl. Phys., A437:534, 1 9 8 5 . [8] G . R . S m i t h et a l . Phys. Rev. C , 30:980, 1984. [9] Madison  convention,  in Polarization  Phenomenon  in Nuclear  Reactions,  U n i v e r s i t y of W i s c o n s i n Press, M a d i s o n W i s c o n s i n , 1971. [10] G . J o n e s , p r i v a t e c o m m u n i c a t i o n . [11] D. H u t c h e o n . p r i v a t e c o m m u n i c a t i o n . [12] G . R . S m i t h et a l . Nucl. Instr. Meth., A 2 5 4 : 2 6 3 - 2 6 9 , 1 9 8 7 . [13] M . A . P r e s t o n a n d R . K . B h a d u r i . Structure  of the Nucleus.  Addison-Wesley,  R e a d i n g Massachusetts, 1975. [14] M . L . G o l d b e r g e r a n d K . M . W a t s o n .  Collision  theory. J o h n W i l e y a n d S o n s ,  N e w Y o r k N . Y . , 1964. [15] E . A p r i l e - G i b o n i et a l . Nucl. Instr. Meth., 215:147-157, 1 9 8 3 . [16] M . W . M c N a u g h t o n et a l . Nucl. Inst. Meth., A 2 4 1 : 4 3 5 - 4 4 0 , 1 9 8 5 . [17] P. W e b e r , p r i v a t e c o m m u n i c a t i o n . [18] F . S p e r i s e n , W . G r u e b l e r , a n d V . K o n i g . Nucl. Instr. Meth., 1983. 67  204:491-503,  [19] G . C a n t a l e et a l . Helv. Phys. Act, 60:398-455, 1987. [20] P. W a l d e n . p r i v a t e c o m m u n i c a t i o n . [21] D . B e s s e t , Q . H . D o , B . F a v i e r , L . G . G r e e n i a u s , R. H e s s , D . W . W e r r e n , a n d C . H . W e d d i g e n . Nucl. Instr. Meth., 166:373-389, 1 9 7 9 . [22] T r i u m f u s e r ' s h a n d b o o k .  Second E d i t i o n , unpublished.  [23] R. H e n d e r s o n et a l . IEEE  Trans. Nucl. Sci, NS-34:528, 1 9 8 7 .  [24] R . H e n d e r s o n et a l . IEEE  Trans. Nucl. Sci., NS-35-.477, 1 9 8 8 .  [25] G . D . W a i t a n d D . C . H e a l e y . Summary of Deuteron Target Polarization Analysis for Experiment SSI. T e c h n i c a l R e p o r t , T R I U M F , 1987. u n p u b l i s h e d . [26] M . P a v a n . p r i v a t e c o m m u n i c a t i o n . [27] F . S a u l i . Principles Chambers.  of Operation on Multiwire Proportional  and Drift  T e c h n i c a l R e p o r t , C E R N , 1979. Y e l l o w r e p o r t 77-09.  [28] A . H . W a l e n t a . Nucl. Instr. Meth., 151:461-472, 1978. [29] G . S m i t h . STAR unpublished.  system online manual. T e c h n i c a l R e p o r t , T R I U M F , 1987.  [30] D . B e s s e t , B . F a v i e r , L . G . G r e e n i a u s , R. H e s s , C . L e c h a n o i n e , D . R a p i n , a n d D. W . W e r r e n . Nucl. Instr. Meth., 166:515, 1 9 7 8 . [31] R . D . R a n s o m et a l . Nucl. Instr. Meth., 201:309-313, 1 9 8 2 . [32] G . W a t e r s e t a l . Nucl. Inst. Meth., 153:401-408, 1978. [33] M . S e v i o r . p r i v a t e c o m m u n i c a t i o n .  68  Appendix A Traceback Algebra T h e traceback routines were used b y t h e software t o decide where t h e scattering origin was situated. In order t o determine the scattering origin, the positions o n t h e i n i t i a l a n d final s c a t t e r i n g t r a j e c t o r i e s c o r r e s p o n d i n g t o t h e c l o s e s t  approach  w a s first d e t e r m i n e d . T h e " o r i g i n " w a s t h e n d e f i n e d as t h e m i d p o i n t o f t h e l i n e j o i n i n g t h e s e t w o p o i n t s . F o r t h i s p u r p o s e , t h e g e o m e t r y d e f i n e d i n figure 3 0 w a s used. R  A  andR  B  are vectors w h i c h e x t e n d towards a k n o w n p o i n t i n space f r o m —*  — *  the coordinate origin . A a n d B are vectors defined b y the wire chamber position information with e  a n d e g as t h e i r u n i t d i r e c t i o n a l v e c t o r s . T h e v e c t o r s t  A  t  A  and  are defined b y the following relations.  B  t f  where k  A  andK  a  A  andt  f l  = R  A  = R  - ke  (27)  - ke  (28)  A  B  B  A  B  e variable parameters.  r  B  Since t  A  indicate points o n vectors A a n d B respectively, t h e a i m is  B  t o m i n i m i z e t h e difference o f these vectors. T h e m a g n i t u d e o f t h i s difference is e x p r e s s e d a s \t  \t -t \  2  A  B  =  —t \, 2  A  B  where  \R -RB\ +k +kl+2k (R -R )-e -2k (R 2  A  A  B  A  B  B  A  A  (29) T h i s expression is a m i n i m u m w h e n b o t h p a r t i a l derivatives w i t h respect t o t h e 1  For example, the coordinate from one of the wire chambers. 69  apparatus origin t o be m i n i m i z e d  F i g u r e 30: G e o m e t r y f o r t a r g e t t r a c e b a c k .  parameters k  A  and k  B  are e q u a l t o zero. d\t -t \* A  d\t -t \  B  A  dk,  2  B  dk B  u n d e r these constraints;  k  A  k  B  = ke B  = Ke A  •e  A  B  + (R  B  - (R  B  —*  •e  A  - R) • e  A  A  —*  - R) • e  A  B  B  A f t e r d e c o u p l i n g these equations, one o b t a i n s the f o l l o w i n g expressions;  k  B  =  ( i _ (g* . g )2) B  [{{^  ~  A  '  ^  A  ' ^  ~  i ^  A  ~  '  and, k  A  — k (e B  A  •e) • e B  B  T h e s e p a r a m e t e r s w e r e s u b s t i t u t e d b a c k i n t o e q u a t i o n s 27, 28 t o o b t a i n t h e d e s i r e d p o i n t s o n t h e v e c t o r s . T h e o r i g i n is t h e n (t  A  0(x,y,z) 70  + t) B  B y evaluating the m i n i m u m distance of \t  A  — ts\,  one can evaluate the  quality of the traceback. If this distance is large, it is an i n d i c a t i o n that the trajectory i n f o r m a t i o n was poor, or the two vectors d i d not share a c o m m o n i n reality. \t  A  — 2B| is denoted as the R o o t Difference Squared ( R D S ) .  71  Appendix B Spin Precession Unfortunately, i t was not possible to correct for the spin precession based solely on i n f o r m a t i o n recorded b y the detectors. B o t h arms were situated well outside the effective range of the magnetic field, hence a l l the trajectories appeared devoid of deflection. T h e approach henceforth taken, was t o calculate the effect of precession o n a k n o w n polarization (from A ) a n d compare its final components y0  outside the effective magnetic field w i t h those measured by the polarimeter. T h e obscure geometry of the system d i d not allow for a simple analytic solution, so it was necessary t o perform the calculations numerically. T h e p r o g r a m F I N D A N G [33] h a d previously been w r i t t e n t o calculate energy losses, kinematic angles, a n d trajectories of charged particles passing i n a n d out of the magnetic field of the polarized deuterium target. FINDANG  allows  one t o determine the magnetic field a n d particle velocity components at any point along a particle's p a t h . B y choosing a n appropriate step size, the s p i n precession could be calculated at each interval, a n d integrated over the entire p a t h length. E q u a t i o n 34 is the B a r g m a n n , M i c h e l , Telegdi (BMT) equation w h i c h describes the time rate of change of the spin of a particle i n its rest frame w i t h respect t o the electric field, magnetic field a n d velocity components expressed i n the laboratory frame. ds  R  dt  e  _  •s mc  x  'f-^)*-(5-0^(W  +  electric field  term  (34) Since no electric fields were present i n the lab system, the last t e r m was ignored.  72  T h i s equation was t h basis for the spin precession calculation. A t this p o i n t i t is useful t o o u t l i n e the frames o f reference used b y FINDANG  t o describe the p o l a r i z a t i o n , a n d e x p l a i n the subtle differences between  the s p i n a n d p o l a r i z a t i o n t r a n s f o r m a t i o n s l i n k i n g these frames. A l t h o u g h s p i n is a n invariant quantity, the m a g n i t u d e o f its c o m p o n e n t s are f r a m e d e p e n d e n t . T h i s i s d e s c r i b e d b y e q u a t i o n s 37, 38. H e n c e t h e s p i n c o m p o n e n t p a r a l l e l t o t h e d i r e c t i o n o f m o t i o n w o u l d appear t o i n c r e a s e f o r a n o b s e r v e r m o v i n g a t r e l a t i v i s t i c speeds c o m p a r e d w i t h t h a t seen b y a n o b s e r v e r i n t h e p a r t i c l e ' s rest f r a m e . T h i s c a n a l s o l e a d t o a n apparent r o t a t i o n o f t h e s p i n w h e n i t is observed f r o m different frames. V e c t o r p o l a r i z a t i o n o n the other h a n d is a s t a t i s t i c a l q u a n t i t y described b y e q u a t i o n 1. I t s d i r e c t i o n a l c o m p o n e n t s d e s c r i b e t h e m e a n d i r e c t i o n o f a l l t h e s p i n s i n a b e a m o f particles. W h e n this m e a n is b o o s t e d into a n o t h e r f r a m e , i t w i l l also r o t a t e as d e s c r i b e d above, b u t i t is not correct t o t h i n k t h a t i t s m a g n i t u d e w i l l also change. T h e n u m b e r s aligned p a r a l l e l a n d a n t i - p a r a l l e l w i t h t h e m e a n d i r e c t i o n w i l l not change. T h r e e frames o f reference were used b y the c o m p u t e r m o d e l . T h e f i n a l p o l a r i z a t i o n s w e r e e x p r e s s e d i n t h e l a b o r a t o r y f r a m e as d e s c r i b e d i n s e c t i o n I V . 1, since this was the f r a m e relevant t o the polarimeter. T h e s p i n precession calculations (equation 34) required the spin components t o be defined i n the p a r t i c l e ' s rest f r a m e , whereas the i n i t i a l p o l a r i z a t i o n was d e s c r i b e d i n the center o f m a s s f r a m e as i s t h e c o n v e n t i o n f o r s p i n t r a n s f e r o b s e r v a b l e s [5]. A s i s seen i n f i g u r e 3 1 , o n e c a n r e l a t e t h e c e n t e r o f m a s s p o l a r i z a t i o n t o t h a t seen i n t h e l a b f r a m e b y a s i m p l e W i g n e r r o t a t i o n about the n o r m a l t o the s c a t t e r i n g p l a n e b y a n a n g l e u.  cos u = cos 9* cos 91 + 7CM s i n 9* s i n 9L 73  (35)  F i g u r e 31: R o t a t i o n d u e t o r e l a t i v i s t i c b o o s t f r o m c e n t e r o f m a s s t o l a b f r a m e , f o r the general r e a c t i o n c(a,c)d.  TYl  s i n a; = — ( s i n # * cos 6L — *)CM hi*  c  o  s  9*  s  m  (36)  ®L)  W h e r e 9*, a n d 6^ a r e t h e a n g l e s b e t w e e n t h e o u t g o i n g p r o t o n a n d t h e i n c i d e n t p i o n i n t h e c e n t e r o f m a s s a n d l a b f r a m e s r e s p e c t i v e l y , a n d to = 9  cm  — 9T,.  This  r o t a t i o n leaves a n y n o r m a l c o m p o n e n t u n a f f e c t e d . T o b o o s t b e t w e e n t h e r e s t a n d l a b o r a t o r y s y s t e m s , e q u a t i o n s 37, 38 w e r e used.  SR = S ~  T~T fe •  L  2 s  L  =  ?R + -  1  — (s  7 + 1  v  R  •  P) P  (37)  0) 0 '  (38)  W h e r e 7, a n d ft d e s c r i b e t h e m o t i o n o f t h e r e s t f r a m e ( p r o t o n ) i n t h e l a b s y s t e m . T h i s b o o s t is t r a j e c t o r y d e p e n d e n t . In summary, the c o m p u t e r p r o g r a m rotates the p o l a r i z a t i o n to the lab frame f r o m a p r e d e t e r m i n e d i n i t i a l v a l u e i n t h e c e n t e r o f m a s s f r a m e . T h i s v a l u e is 74  boosted to the rest frame as a spin whose magnetic precession is integrated over the entire p a t h length t h r o u g h the magnetic field. T h e final s p i n is boosted back to the lab where it can again be renormalized to a p o l a r i z a t i o n , w i t h conserved magnitude.  75  Appendix C Extraction of Polarization Observables It is known that the effect of the computer program F I N D A N G on the polarizations is a rotation of one form or another. Spin precession is simply a rotation of the spin components which are orthogonal to the magnetic field, whereas boosts from the center of mass to the lab frames are associated with rotations about the normal to the scattering plane. For this reason, the polarization observed at the analyzer (P,) can be expressed as a linear combination of the polarization components (Pj) characterizing the proton when its produced by the reaction within the target. INPN P's = fsP  N  + gNPs  +  hP  + gsPs +  hP  N  s  (39)  L  (40)  L  where / , g, h are the coupling constants which express how the initial normal, sideways and longitudinal components relate respectively to the final components, indicated by the subscripts. Equation 41 describes the dependence of the proton polarization on the polarization of the deuteron, where the P 's are the various components of k  deuteron vector polarization, and the K^s are the spin-transfer parameters. The tensor terms are assumed to be small, having a sign which is independent of the spin of the vector terms. P <x Ps- K s s  S  + Pi • LS K  + tensor  PN OC ANO + PN ' KNN + tensor  76  terms terms  (41)  PL oc Pg • KSL + PL ' KLL + tensor  terms  A s d e m o n s t r a t e d b y e q u a t i o n s 41 t h e s i z e a n d s i g n o f t h e p r o t o n ' s  sideways  a n d l o n g i t u d i n a l p o l a r i z a t i o n is a linear f u n c t i o n of the sideways a n d l o n g i t u d i n a l vector p o l a r i z a t i o n of the deuteron (together w i t h tensor terms). T h e n o r m a l p o l a r i z a t i o n o f t h e d e u t e r o n , o n t h e o t h e r h a n d , is i n d e p e n d e n t o f t h e d e u t e r o n p o l a r i z a t i o n i n t h e s c a t t e r i n g plane. U s i n g these features, o n e c a n c o m p a r e results for r u n s where t h e deuteron p o l a r i z a t i o n i n t h e scattering p l a n e is of opposite sign, a n d simplify the above equations.  PN  +)  -{PN\ PN  -PN'  =  JNPN  +  Rg Ps  +  =  INPN  -  g Ps  -  N  N  =  {\ + R)g P N  s  +  Rh P N  L  (42)  hP) N  L  (l +  R)h P N  L  and  P£ set K  N  - P*  +)  P^  =  fsP  +  RgsPs  +  -(Ps~\  =  ISPN  -  gsPs  -  -Ps^ - P^  N  =  (i + R)g Ps s  and K  = P*  +)  s  - P^  ( _ )  +  Rh P s  L  (43)  hP) s  (i +  L  R)h P s  L  .  N o w w e h a v e t w o e q u a t i o n s a n d t w o u n k n o w n s , w h e r e R is t h e r a t i o o f t h e magnitudes of the polarization corresponding to the positively a n d negatively p o l a r i z e d d e u t e r i u m targets.  R = l^tli  m  \P*(+)\  {  ]  N o w s o l v i n g f o r P$ a n d PT, o n e o b t a i n s , {Ks P S  l  - ^K ) N  (1 + * )  ~ (Ks - ^ K ) N  ^  i  In order t o o b t a i n t h e c o u p l i n g constants, one c a n use t h e spin precession p r o g r a m discussed i n a p p e n d i x B. B y setting t h e i n i t i a l center o f m a s s p o l a r i z a t i o n t o p u r e n o r m a l , sideways o r l o n g i t u d i n a l , one c a n measure h o w m u c h  77  Table VI: Coupling constants as obtained using F I N D A N G  IN  p' LJL  fs  PN  P LM. 1  gN h  N  gs  ? P' P  LM.  hs  Pf,  p' PN P'  Is. p  ?  P' LS.  Pr.  of it will transfer to the sideways or longitudinal components after traversing the magnetic field. The coupling constants are defined with respect to the initial and final components in table VI. In the case of a purely normal initial polarization, equations 39, 40 yield P/y directly; Ps = ^JN  (47)  PN = TJs  (48)  or  This technique is simple and unambiguous since no components of P 5 or Pi can contribute.  78  

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